📖 StackCalc Master Answer Key & Keystroke Verification Portal
The Paper-First Principle: Calculators should never replace reasoning. Students first build the problem with physical 3D blocks or write their mathematical setup, solve on paper by hand, and verify on their distraction-free StackCalc instrument. This portal provides complete worked solutions and canonical keystrokes across all volumes in The StackCalc Library.
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📗 Book 2A: Early Elementary Math Lab (Grades K–2 • SC-6)
Covers Units 01 through 10 plus 4 Cumulative Benchmark & Capstone Mastery Exams: Step Counting, Number Bonds, Make-10 Trades, Doubling, Place Value, Subtraction, Regrouping Carry, Two-Digit Operations, Equal Groups & Arrays, and Measurement/Money (840 active problems).
Unit 01: Counting, Cardinality & The One-More Step Counter
Grades: Pre-K – Kindergarten | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (Digit Blocks 0–9 & Gold [+1] Step Tokens)
Key Standards: CCSS.MATH.K.CC.A.1, CCSS.MATH.K.CC.B.4, CCSS.MATH.K.OA.A.1
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Place the [4] Digit Block in the Ones column of your Place Value Tray and add 1 Gold [+1] step token. What total quantity do you have now?
- Verified Answer:
5 ([5] Digit Block; 4 + 1 = 5).
- Keystrokes:
4 ENTER 1 +
- Warm-Up B (Extension Lab): Row 1 holds [6] and Row 0 holds [2] on your Place Value Tray. Which row has more, and how many more does it have?
- Verified Answer:
Row 1 has 4 more (6 - 2 = 4).
- Keystrokes:
6 ENTER 2 -
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Place [3] in the top row and [2] in the bottom row of your Place Value Tray. Combine both groups—what is the total? |
5 |
3 ENTER 2 + |
| 2 |
You have 5 ones on the tray and take away 2. What Digit Block shows how many ones remain? |
3 |
5 ENTER 2 - |
| 3 |
You have 3 blocks on the tray and take away all 3. How many blocks are left, and which Digit Block shows an empty tray? |
0 (Zero) |
3 ENTER 3 - |
| 4 |
Start with [1] in the Ones socket and add one more [+1] token three times in a row. What count do you reach? |
4 |
1 ENTER 1 + 1 + 1 + |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
4
3 ENTER 1 + |
4
5 ENTER 1 - |
6 cars
5 ENTER 1 + |
6
5 ENTER 1 + |
2
6 ENTER 4 - |
6
2 ENTER 5 + 1 - |
| #2 |
7
6 ENTER 1 + |
8
9 ENTER 1 - |
5 eggs
3 ENTER 1 + 1 + |
8
5 ENTER 3 + |
3
8 ENTER 5 - |
7
8 ENTER 3 - 2 + |
| #3 |
9
8 ENTER 1 + |
5
7 ENTER 1 - 1 - |
7 frogs
8 ENTER 1 - |
9
5 ENTER 4 + |
3
9 ENTER 6 - |
6
5 ENTER 4 + 3 - |
| #4 |
6
4 ENTER 1 + 1 + |
8
10 ENTER 1 - 1 - |
7 claps
4 ENTER 3 + |
10
5 ENTER 5 + |
3
10 ENTER 7 - |
8
10 ENTER 6 - 4 + |
| #5 |
7
5 ENTER 1 + 1 + |
7
6 ENTER 1 + |
9 dots
6 ENTER 3 + |
5
2 ENTER 1 + 1 + 1 + |
2
7 ENTER 5 - |
10
1 ENTER 2 + 3 + 4 + |
| #6 |
1
0 ENTER 1 + |
6
7 ENTER 1 - |
Floor 6
2 ENTER 4 + |
7
3 ENTER 4 + |
5
0 ENTER 5 + |
4
10 ENTER 1 - 2 - 3 - |
| #7 |
9
7 ENTER 2 + |
3
6 ENTER 3 - |
8 blocks
10 ENTER 1 - 1 - |
4
6 ENTER 1 - 1 - |
0
8 ENTER 8 - |
6
9 ENTER 3 - |
| #8 |
9
6 ENTER 1 + 1 + 1 + |
4
10 ENTER 6 - |
6 shells
5 ENTER 2 + 1 - |
5
8 ENTER 3 - |
9
9 ENTER 0 + |
7
10 ENTER 3 - |
| #9 |
10
9 ENTER 1 + |
7
6 ENTER 2 + 1 - |
4 cm
2 ENTER 2 + |
6
10 ENTER 4 - |
8
3 ENTER 3 + 2 + |
9
2 ENTER 4 + 3 + |
| #10 |
8
5 ENTER 1 + 1 + 1 + |
6
5 ENTER 3 + 2 - |
5
4 ENTER 1 + |
6
4 ENTER 3 + 1 - |
4
9 ENTER 2 - 3 - |
7
6 ENTER 3 + 2 - |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Step up 1 more on the Ten-Frame: 6 + 1 = |
10 pts |
7 |
6 ENTER 1 + |
| 2 |
Core Fluency |
Step up 2 times on the number track: 8 + 1 + 1 = |
10 pts |
10 |
8 ENTER 1 + 1 + |
| 3 |
Core Fluency |
Remove 1 block from the tray: 9 - 1 = |
10 pts |
8 |
9 ENTER 1 - |
| 4 |
Core Fluency |
Step back 2 times on the tray: 7 - 1 - 1 = |
10 pts |
5 |
7 ENTER 1 - 1 - |
| 5 |
Core Fluency |
Find the missing [+1] steps: 5 + [ ? ] = 8 |
10 pts |
3 |
8 ENTER 5 - |
| 6 |
Core Fluency |
Follow the 2-step tray path: 4 + 3 - 1 = |
10 pts |
6 |
4 ENTER 3 + 1 - |
| 7 |
Application |
Emma has 6 crayons and gets 2 more. How many crayons in all? |
10 pts |
8 crayons |
6 ENTER 2 + |
| 8 |
Application |
A shelf holds 9 books. 2 books are checked out. How many books remain? |
10 pts |
7 books |
9 ENTER 2 - |
| 9 |
Application |
An ant walks 5 steps forward, 2 more forward, and 1 step back. Total steps from start? |
10 pts |
6 steps |
5 ENTER 2 + 1 - |
| 10 |
Error Analysis |
Error Analysis: Ben says 1 more than 8 is 7 because he counted backward. Give the right answer. |
10 pts |
9 |
1 ENTER 8 + |
5. STEM Lab Case Study: The Garden Stepping Stone Trail
Scenario: A landscape gardener designs a walking path with stone markers spaced 1 meter apart. Students calculate steps, distances, and total stones needed.
- Case Question 1: Start at Stone 4 and lay 3 more stones. What stone number are you on?
- Expected Result: 7 | Keystrokes:
4 ENTER 1 + 1 + 1 +
- Case Question 2: A trail has 8 stones. A bird lands on stone 5. How many stones remain to the end?
- Expected Result: 3 | Keystrokes:
8 ENTER 5 -
- Case Question 3: Two parallel paths have 4 stones and 5 stones. How many stones total?
- Expected Result: 9 | Keystrokes:
4 ENTER 5 +
- Case Question 4: Error Analysis: Leo says stepping forward 2 stones from 6 gives stone 9. Find the true stone.
- Expected Result: Stone 8 (6 + 2 = 8, not 9). | Keystrokes:
6 ENTER 2 +
Unit 02: Part-Part-Whole & Number Bonds on the Place Value Tray
Grades: Kindergarten – Grade 1 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (Number Bond Stage & Digit Blocks 0–9)
Key Standards: CCSS.MATH.K.OA.A.3, CCSS.MATH.1.OA.B.3, CCSS.MATH.1.OA.C.6
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Place [5] in Row 1 (Part 1) and [4] in Row 0 (Part 2) of your Place Value Tray. Combine the two parts to find the Whole.
- Verified Answer:
9 ([9] Digit Block; 5 + 4 = 9).
- Keystrokes:
5 ENTER 4 +
- Warm-Up B (Extension Lab): A number bond has a Whole of 8 and a known Part of 5. Find the missing Part 2.
- Verified Answer:
Missing Part is 3 (8 - 5 = 3).
- Keystrokes:
8 ENTER 5 -
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
To make a Whole of 6, Part 1 is 2. What is the missing Part 2? |
4 |
6 ENTER 2 - |
| 2 |
In a number bond, Part 1 is 7 and Part 2 is 2. What is the Whole when you combine both parts? |
9 |
7 ENTER 2 + |
| 3 |
Three parts join together: first combine 2 and 3, then add 4 to that sum. What is the Whole? |
9 |
2 ENTER 3 + 4 + |
| 4 |
Ten birds sit on a fence. Seven fly to a tree. How many stay on the fence? |
3 |
1 0 ENTER 7 - |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
5
3 ENTER 2 + |
4
7 ENTER 3 - |
9 apples
5 ENTER 4 + |
9
10 ENTER 1 - |
2
3 ENTER 4 + 5 - |
2
10 ENTER 8 - |
| #2 |
8
5 ENTER 3 + |
4
9 ENTER 5 - |
3 players
10 ENTER 7 - |
8
10 ENTER 2 - |
4
6 ENTER 2 + 4 - |
5
9 ENTER 4 - |
| #3 |
10
6 ENTER 4 + |
4
10 ENTER 6 - |
5 stickers
8 ENTER 3 - |
7
10 ENTER 3 - |
3
5 ENTER 5 + 7 - |
3
9 ENTER 6 - |
| #4 |
9
2 ENTER 7 + |
6
8 ENTER 2 - |
4 slots
9 ENTER 5 - |
6
10 ENTER 4 - |
3
2 ENTER 7 + 6 - |
3
10 ENTER 2 - 5 - |
| #5 |
8
4 ENTER 4 + |
6
10 ENTER 4 - |
9 pennies
6 ENTER 3 + |
5
10 ENTER 5 - |
4
8 ENTER 1 - 3 - |
2
9 ENTER 4 - 3 - |
| #6 |
10
7 ENTER 3 + |
3
8 ENTER 5 - |
6 puppies
10 ENTER 4 - |
8
2 ENTER 3 + 3 + |
4
9 ENTER 3 - 2 - |
3
10 ENTER 3 - 4 - |
| #7 |
9
1 ENTER 8 + |
3
9 ENTER 6 - |
6 blocks
8 ENTER 2 - |
10
4 ENTER 1 + 5 + |
5
10 ENTER 4 - 1 - |
1
10 ENTER 5 ENTER 4 + - |
| #8 |
9
5 ENTER 4 + |
7
10 ENTER 3 - |
3 birds
10 ENTER 7 - |
9
3 ENTER 4 + 2 + |
9
5 ENTER 4 + |
9
3 ENTER 6 + |
| #9 |
9
3 ENTER 6 + |
4
4 ENTER 5 + 5 - |
6
3 ENTER 5 + 2 - |
3
9 ENTER 2 - 4 - |
10
7 ENTER 3 + |
3
10 ENTER 3 - 4 - |
| #10 |
10
8 ENTER 2 + |
7
3 ENTER 7 + 3 - |
5
8 ENTER 3 - |
2
10 ENTER 3 - 5 - |
5
9 ENTER 4 - |
4
10 ENTER 4 - 2 - |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Combine the parts: 6 + 3 = |
10 pts |
9 |
6 ENTER 3 + |
| 2 |
Core Fluency |
Combine the parts: 4 + 6 = |
10 pts |
10 |
4 ENTER 6 + |
| 3 |
Core Fluency |
Subtract to find the part: 8 - 3 = |
10 pts |
5 |
8 ENTER 3 - |
| 4 |
Core Fluency |
Subtract to find the part: 10 - 8 = |
10 pts |
2 |
10 ENTER 8 - |
| 5 |
Core Fluency |
Find the missing addend: 5 + [ ? ] = 8 |
10 pts |
3 |
8 ENTER 5 - |
| 6 |
Core Fluency |
Find the missing addend: [ ? ] + 7 = 10 |
10 pts |
3 |
10 ENTER 7 - |
| 7 |
Application |
There are 9 bees in a hive. 4 fly out. How many bees are still inside? |
10 pts |
5 bees |
9 ENTER 4 - |
| 8 |
Application |
Lucas has 3 blue marbles and 7 yellow marbles. How many marbles altogether? |
10 pts |
10 marbles |
3 ENTER 7 + |
| 9 |
Application |
A carton holds 10 eggs. 6 eggs are left. How many eggs were used? |
10 pts |
4 eggs |
10 ENTER 6 - |
| 10 |
Error Analysis |
Error Analysis: Zoe solved 6 + [ ? ] = 9 by adding 6 + 9 = 15. Explain & give the right number. |
10 pts |
3 |
9 ENTER 6 - |
5. STEM Lab Case Study: The Space Station Cargo Balancing Tray
Scenario: Astronauts balance payloads on the supply shuttle. Each cargo tray holds exactly 10 kg across two compartments (Part 1 and Part 2).
- Case Question 1: Compartment A holds 6 kg. How many kg fit into Compartment B to make 10 kg?
- Expected Result: 4 kg | Keystrokes:
10 ENTER 6 -
- Case Question 2: Compartment A holds 3 kg. What is the missing cargo weight for a 10 kg tray?
- Expected Result: 7 kg | Keystrokes:
10 ENTER 3 -
- Case Question 3: Two small containers weigh 4 kg and 5 kg. Can they both fit in a 10 kg tray?
- Expected Result: Yes, 9 kg total (1 kg space left). | Keystrokes:
4 ENTER 5 +
- Case Question 4: Error Analysis: Sam says 10 − 8 = 3. Use the number bond to find the correct missing part.
- Expected Result: 2 kg (8 + 2 = 10, not 3). | Keystrokes:
10 ENTER 8 -
Unit 03: Complements to 10 & Make-10 Regrouping with Carry Tokens
Grades: Grade 1 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (Ten-Frame Trays & Gold [+1] Carry Tokens)
Key Standards: CCSS.MATH.1.OA.C.6, CCSS.MATH.1.NBT.B.2, CCSS.MATH.2.OA.B.2
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): The Ones socket holds [6]. How many more ones are needed to complete a full 10 and trade for 1 Ten?
- Verified Answer:
4 ones (10 - 6 = 4).
- Keystrokes:
1 0 ENTER 6 -
- Warm-Up B (Extension Lab): Use the Make-Ten strategy on your Place Value Tray to add 9 + 6. What is the sum?
- Verified Answer:
15 (9 + 1 = 10, then 10 + 5 = 15).
- Keystrokes:
9 ENTER 1 + 5 +
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
You have 4 ones on your tray. How many more ones do you need to make a full 10? |
6 |
1 0 ENTER 4 - |
| 2 |
Solve 7 + 5 by bridging through 10 on your Place Value Tray. |
12 |
7 ENTER 3 + 2 + |
| 3 |
The Ones socket holds [9]. How many more ones are needed to complete 10 and regroup into 1 Ten? |
1 |
1 0 ENTER 9 - |
| 4 |
Solve 8 + 6 by bridging through 10 on your Place Value Tray. |
14 |
8 ENTER 2 + 4 + |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
1
10 ENTER 9 - |
16
10 ENTER 6 + |
15 strawberries
9 ENTER 6 + |
14
9 ENTER 5 + |
7
16 ENTER 9 - |
23
16 ENTER 7 + |
| #2 |
3
10 ENTER 7 - |
19
10 ENTER 9 + |
15 passengers
8 ENTER 7 + |
17
9 ENTER 8 + |
7
14 ENTER 7 - |
25
17 ENTER 8 + |
| #3 |
13
9 ENTER 4 + |
7
17 ENTER 10 - |
12 stars
7 ENTER 5 + |
18
9 ENTER 9 + |
9
17 ENTER 8 - |
34
28 ENTER 6 + |
| #4 |
16
9 ENTER 7 + |
18
6 ENTER 4 + 8 + |
18 muffins
10 ENTER 8 + |
11
8 ENTER 3 + |
7
13 ENTER 6 - |
34
25 ENTER 9 + |
| #5 |
12
8 ENTER 4 + |
15
7 ENTER 5 + 3 + |
9 ducks
14 ENTER 5 - |
13
7 ENTER 6 + |
6
20 ENTER 14 - |
17
22 ENTER 5 - |
| #6 |
14
8 ENTER 6 + |
8
12 ENTER 4 - |
8 beads
16 ENTER 8 - |
8
11 ENTER 3 - |
3
20 ENTER 17 - |
16
24 ENTER 8 - |
| #7 |
15
8 ENTER 7 + |
8
14 ENTER 6 - |
14 pinecones
7 ENTER 4 + 3 + |
7
13 ENTER 6 - |
24
18 ENTER 6 + |
25
31 ENTER 6 - |
| #8 |
12
7 ENTER 5 + |
8
15 ENTER 7 - |
7 ones
15 ENTER 8 - |
6
15 ENTER 9 - |
24
19 ENTER 5 + |
19
6 ENTER 9 + 4 + |
| #9 |
11
6 ENTER 5 + |
5
13 ENTER 8 - |
13
10 ENTER 3 + |
9
17 ENTER 8 - |
16
8 ENTER 6 + 2 + |
20
7 ENTER 8 + 3 + 2 + |
| #10 |
15
7 ENTER 8 + |
7
16 ENTER 9 - |
13
8 ENTER 5 + |
9
18 ENTER 9 - |
17
9 ENTER 7 + 1 + |
19
18 ENTER 7 + 6 - |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Make 10 with a [+1] Carry Token: 8 + 5 = |
10 pts |
13 |
8 ENTER 5 + |
| 2 |
Core Fluency |
Make 10 on the Ten-Frame: 9 + 6 = |
10 pts |
15 |
9 ENTER 6 + |
| 3 |
Core Fluency |
Split to fill the Ten-Frame: 7 + 6 = |
10 pts |
13 |
7 ENTER 6 + |
| 4 |
Core Fluency |
Compose 1 Ten and 8 Ones: 10 + 8 = |
10 pts |
18 |
10 ENTER 8 + |
| 5 |
Core Fluency |
Bridge back through 10: 14 - 5 = |
10 pts |
9 |
14 ENTER 5 - |
| 6 |
Core Fluency |
Unbundle 1 Ten to subtract: 15 - 8 = |
10 pts |
7 |
15 ENTER 8 - |
| 7 |
Application |
A shelf holds 9 science books and 8 history books. How many books in all? |
10 pts |
17 books |
9 ENTER 8 + |
| 8 |
Application |
Liam has 13 apple slices and eats 6 of them. How many slices are left? |
10 pts |
7 slices |
13 ENTER 6 - |
| 9 |
Application |
Find the sum by pairing the 10-bond first: 8 + 7 + 2 = |
10 pts |
17 |
8 ENTER 7 + 2 + |
| 10 |
Error Analysis |
Error Analysis: Mia says the '1' in 16 means 1 One. What does the '1' actually represent? |
10 pts |
1 Ten (10) |
16 ENTER 6 - |
5. STEM Lab Case Study: The Farm Fresh Egg Carton Packaging
Scenario: A poultry farm packages organic eggs into cartons of 10. When a carton fills, workers close the lid and pack a new carton.
- Case Question 1: A basket has 8 brown eggs and 5 white eggs. How many full 10-cartons and loose eggs?
- Expected Result: 1 full carton of 10 and 3 loose eggs (13 eggs). | Keystrokes:
8 ENTER 5 +
- Case Question 2: A carton has 7 eggs. How many more eggs are needed to fill the 10-frame carton?
- Expected Result: 3 eggs | Keystrokes:
10 ENTER 7 -
- Case Question 3: Two nests have 9 eggs and 6 eggs. What is the total egg harvest?
- Expected Result: 15 eggs (1 ten and 5 ones). | Keystrokes:
9 ENTER 6 +
- Case Question 4: Error Analysis: A student writes 8 + 6 = 15. Count the dots to find the true sum.
- Expected Result: 14 eggs (8 + 2 = 10, 10 + 4 = 14). | Keystrokes:
8 ENTER 6 +
Unit 04: Doubles, Near-Doubles & Addition Facts within 20
Grades: Grade 1 – 2 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (Twin-Row Doubling Mat & Digit Blocks)
Key Standards: CCSS.MATH.1.OA.C.6, CCSS.MATH.2.OA.B.2
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Place [7] in both rows of your Place Value Tray to build a twin-row double. What is the sum of two 7s?
- Verified Answer:
14 ([1] Ten and [4] Ones on the tray; 7 + 7 = 14).
- Keystrokes:
7 ENTER +
- Warm-Up B (Extension Lab): Use the known double 6 + 6 to solve the near-double 6 + 7.
- Verified Answer:
13 (6 + 6 = 12, then 12 + 1 = 13).
- Keystrokes:
6 ENTER + 1 +
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Two matching rows on the Place Value Tray each hold 8. What is double 8? |
16 |
8 ENTER + |
| 2 |
Use the double 8 + 8 to solve the near-double 8 + 9. |
17 |
8 ENTER + 1 + |
| 3 |
A carton has two equal rows of 9 eggs. How many eggs are in the carton? |
18 |
9 ENTER + |
| 4 |
Use the double 5 + 5 to solve the near-double 5 + 6. |
11 |
5 ENTER + 1 + |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
8
4 ENTER 4 + |
8
3 ENTER 5 + |
8 legs
4 ENTER 4 + |
20
10 ENTER 10 + |
14
5 ENTER 5 + 4 + |
12
3 ENTER + ENTER + |
| #2 |
12
6 ENTER 6 + |
10
4 ENTER 6 + |
18 cupcakes
9 ENTER 9 + |
22
11 ENTER 11 + |
15
6 ENTER 6 + 3 + |
16
4 ENTER + ENTER + |
| #3 |
16
8 ENTER 8 + |
12
5 ENTER 7 + |
17 points
8 ENTER 9 + |
26
13 ENTER 13 + |
19
7 ENTER 7 + 5 + |
20
5 ENTER + ENTER + |
| #4 |
18
9 ENTER 9 + |
14
6 ENTER 8 + |
14 colors
7 ENTER 7 + |
28
14 ENTER 14 + |
20
8 ENTER 8 + 4 + |
10
1 ENTER 2 + 3 + 4 + |
| #5 |
24
12 ENTER 12 + |
16
7 ENTER 9 + |
11 bicycles
5 ENTER 6 + |
30
15 ENTER 15 + |
13
9 ENTER 9 + 5 - |
15
1 ENTER 2 + 3 + 4 + 5 + |
| #6 |
11
5 ENTER 6 + |
5
10 ENTER 5 - |
13 shells
6 ENTER 7 + |
6
12 ENTER 2 ÷ |
7
6 ENTER 6 + 5 - |
20
2 ENTER 4 + 6 + 8 + |
| #7 |
13
6 ENTER 7 + |
6
12 ENTER 6 - |
14 birds
6 ENTER 8 + |
8
16 ENTER 2 ÷ |
7
8 ENTER 8 + 9 - |
6
8 ENTER + 5 ENTER + - |
| #8 |
17
8 ENTER 9 + |
7
14 ENTER 7 - |
16 studs
8 ENTER 8 + |
10
20 ENTER 2 ÷ |
5
7 ENTER 8 + 10 - |
6
9 ENTER + 6 ENTER + - |
| #9 |
13
7 ENTER 6 + |
8
16 ENTER 8 - |
7 pairs
14 ENTER 7 - |
21
10 ENTER 11 + |
8
9 ENTER 9 + 10 - |
19
6 ENTER 7 + 6 + |
| #10 |
14
6 ENTER 8 + |
9
18 ENTER 9 - |
17
8 ENTER 9 + |
25
12 ENTER 13 + |
18
4 ENTER 4 + 5 ENTER 5 + + |
19
8 ENTER + 3 + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Clone on the Twin-Row Tray (Double): 8 + 8 = |
10 pts |
16 |
8 ENTER 8 + |
| 2 |
Core Fluency |
Use Double + 1 (Near-Double): 7 + 8 = |
10 pts |
15 |
7 ENTER 8 + |
| 3 |
Core Fluency |
Use Double + 1 (Near-Double): 8 + 9 = |
10 pts |
17 |
8 ENTER 9 + |
| 4 |
Core Fluency |
Slide 1 block to make a Hidden Double: 6 + 8 = |
10 pts |
14 |
6 ENTER 8 + |
| 5 |
Core Fluency |
Spot the double in the 3-addend chain: 6 + 6 + 5 = |
10 pts |
17 |
6 ENTER 6 + 5 + |
| 6 |
Core Fluency |
Halve the Twin-Row Double: 18 - 9 = |
10 pts |
9 |
18 ENTER 9 - |
| 7 |
Application |
A paint box has 2 matching rows of 7 watercolors. How many watercolors are in the box? |
10 pts |
14 watercolors |
7 ENTER 7 + |
| 8 |
Application |
Leo read 7 pages on Monday and 8 pages on Tuesday. How many pages in all? |
10 pts |
15 pages |
7 ENTER 8 + |
| 9 |
Application |
Young Gauss Pairing: Add the stair-step numbers 1 + 2 + 3 + 4 = |
10 pts |
10 |
1 ENTER 2 + 3 + 4 + |
| 10 |
Error Analysis |
Error Analysis: Sam says 7 + 9 = 15. Slide 1 block from 9 to 7 to show the true Hidden Double! |
10 pts |
16 (8 + 8 = 16) |
7 ENTER 9 + |
5. STEM Lab Case Study: The School Bus Morning Seating Chart
Scenario: A school bus has two identical columns of seats. Children sit in pairs: one on the window side, one on the aisle side.
- Case Question 1: Row 1 through 7 each have 2 students (7 on left, 7 on right). How many students total?
- Expected Result: 14 students (7 + 7 = 14). | Keystrokes:
7 ENTER +
- Case Question 2: 8 students sit on the left and 9 on the right. Use near-doubles to find total passengers.
- Expected Result: 17 students ((8 + 8) + 1 = 17). | Keystrokes:
8 ENTER + 1 +
- Case Question 3: 6 boys and 6 girls sit in the front section. How many students in the front section?
- Expected Result: 12 students | Keystrokes:
6 ENTER +
- Case Question 4: Error Analysis: Mia doubles 9 and gets 17. Use the stack to find double 9.
- Expected Result: 18 (9 [ENTER] [+] = 18). | Keystrokes:
9 ENTER +
Unit 05: Two-Digit Place Value: Tens & Ones on the Place Value Tray
Grades: Grade 1 – 2 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (6×2 Whole Number Tray & Digit Blocks 0–9)
Key Standards: CCSS.MATH.1.NBT.B.2, CCSS.MATH.2.NBT.A.1
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Build 47 on your Place Value Tray. Add 30. What is the sum?
- Verified Answer:
77 (7 tens and 7 ones).
- Keystrokes:
47 ENTER 30 +
- Warm-Up B (Extension Lab): Start with 82 on your Place Value Tray. Subtract 40. What number remains?
- Verified Answer:
42 (4 tens and 2 ones).
- Keystrokes:
82 ENTER 40 -
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Build 25 on your Place Value Tray. Add 30: 25 + 30 = |
55 |
25 ENTER 30 + |
| 2 |
Build 58 on your Place Value Tray. Subtract 20: 58 − 20 = |
38 |
58 ENTER 20 - |
| 3 |
Build 41 on your Place Value Tray. Add 40: 41 + 40 = |
81 |
41 ENTER 40 + |
| 4 |
Build 93 on your Place Value Tray. Subtract 50: 93 − 50 = |
43 |
93 ENTER 50 - |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
24
20 ENTER 4 + |
55
45 ENTER 10 + |
58 marbles
50 ENTER 8 + |
50
53 ENTER 3 - |
39
29 ENTER 10 + |
58
12 ENTER 40 + 6 + |
| #2 |
37
30 ENTER 7 + |
52
32 ENTER 20 + |
54 books
34 ENTER 20 + |
80
89 ENTER 9 - |
51
61 ENTER 10 - |
88
31 ENTER 50 + 7 + |
| #3 |
49
40 ENTER 9 + |
56
26 ENTER 30 + |
45 stickers
75 ENTER 30 - |
7
67 ENTER 60 - |
30
64 ENTER 34 - |
55
89 ENTER 30 - 4 - |
| #4 |
50
50 ENTER 0 + |
91
51 ENTER 40 + |
80 rolls
60 ENTER 20 + |
4
94 ENTER 90 - |
60
87 ENTER 27 - |
29
76 ENTER 50 - 3 + |
| #5 |
63
60 ENTER 3 + |
57
67 ENTER 10 - |
8 crayons
48 ENTER 40 - |
70
7 ENTER 10 × |
6
58 ENTER 52 - |
40
100 ENTER 60 - |
| #6 |
76
70 ENTER 6 + |
38
58 ENTER 20 - |
79 beads
29 ENTER 50 + |
40
4 ENTER 10 × |
65
15 ENTER 20 + 30 + |
70
100 ENTER 30 - |
| #7 |
82
80 ENTER 2 + |
54
84 ENTER 30 - |
44 students
64 ENTER 20 - |
8
86 ENTER 10 PV |
42
92 ENTER 30 - 20 - |
74
70 ENTER 4 + |
| #8 |
95
90 ENTER 5 + |
53
93 ENTER 40 - |
7 bags
70 ENTER 10 ÷ |
6
86 ENTER 1 PV |
66
46 ENTER 30 + 10 - |
45
83 ENTER 38 - |
| #9 |
70
40 ENTER 30 + |
40
65 ENTER 25 - |
39¢
30 ENTER 9 + |
59
30 ENTER 20 + 4 ENTER 5 + + |
63
83 ENTER 40 - 20 + |
95
20 ENTER 40 + 35 + |
| #10 |
90
50 ENTER 40 + |
50
86 ENTER 36 - |
62
60 ENTER 2 + |
88
50 ENTER 30 + 6 ENTER 2 + + |
6 Tens
78 ENTER 18 - 10 ÷ |
78
40 ENTER 30 + 8 + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Compose from Tens and Ones on the tray: 50 + 7 = |
10 pts |
57 |
50 ENTER 7 + |
| 2 |
Core Fluency |
Add 2 Tens on the Place Value Tray: 38 + 20 = |
10 pts |
58 |
38 ENTER 20 + |
| 3 |
Core Fluency |
Add 4 Tens on the Place Value Tray: 46 + 40 = |
10 pts |
86 |
46 ENTER 40 + |
| 4 |
Core Fluency |
Remove 3 Tens from the tray: 74 - 30 = |
10 pts |
44 |
74 ENTER 30 - |
| 5 |
Core Fluency |
Find the missing Tens value: 82 = [ ? ] + 2 |
10 pts |
80 |
82 ENTER 2 - |
| 6 |
Core Fluency |
Follow the Tens-column jump: 25 + 50 - 20 = |
10 pts |
55 |
25 ENTER 50 + 20 - |
| 7 |
Application |
A crate holds 6 packs of 10 juice boxes and 4 single boxes. Total juice boxes? |
10 pts |
64 boxes |
60 ENTER 4 + |
| 8 |
Application |
Elena has 85¢. She spends 3 dimes. How many cents does she have left? |
10 pts |
55¢ |
85 ENTER 30 - |
| 9 |
Application |
On a 100-chart, start at 33, move down 4 rows (+40) and right 5 (+5). Where do you land? |
10 pts |
78 |
33 ENTER 40 + 5 + |
| 10 |
Error Analysis |
Error Analysis: Eli says the '5' in 58 means 5 Ones. What value does the '5' actually have? |
10 pts |
50 (5 Tens) |
5 ENTER 10 × |
5. STEM Lab Case Study: The Classroom Library Book Cart
Scenario: Student librarians organize books onto a rolling cart. Each large bin holds exactly 10 books (1 Ten), and the top rack holds single display books (Ones).
- Case Question 1: Cart A has 4 full bins of 10 books and 7 display books. How many books are on Cart A?
- Expected Result: 47 books | Keystrokes:
40 ENTER 7 +
- Case Question 2: 2 full bins of 10 books are added to Cart A. How many books are on the cart now?
- Expected Result: 67 books | Keystrokes:
47 ENTER 20 +
- Case Question 3: Cart B starts with 86 books. Classrooms check out 3 full bins of 10 books. How many books remain?
- Expected Result: 56 books | Keystrokes:
86 ENTER 30 -
- Case Question 4: Error Analysis: Mia counts 5 bins of 10 and 3 single books and writes 35. Find the true total!
- Expected Result: 53 books (50 + 3 = 53, not 35). | Keystrokes:
50 ENTER 3 +
Unit 06: Subtraction as Take-Away & Comparison on the Tray
Grades: Grade 1 – 2 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (2-Row Tray, Digit Blocks & Gold [+1] Tokens)
Key Standards: CCSS.MATH.1.OA.B.4, CCSS.MATH.2.OA.B.2
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Place [1][6] in Row 1 of your Place Value Tray and [9] in Row 0. What is 16 − 9?
- Verified Answer:
7.
- Keystrokes:
16 ENTER 9 -
- Warm-Up B (Extension Lab): Row 1 holds [1][7] and Row 0 holds [1][2] on your Place Value Tray. Find the difference.
- Verified Answer:
5.
- Keystrokes:
17 ENTER 12 -
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Place [1][4] in Row 1 and subtract [6] in Row 0: 14 − 6 = |
8 |
14 ENTER 6 - |
| 2 |
Compare [1][9] in Row 1 and [1][3] in Row 0: 19 − 13 = |
6 |
19 ENTER 13 - |
| 3 |
Place [1][7] in Row 1 and subtract [8] in Row 0: 17 − 8 = |
9 |
17 ENTER 8 - |
| 4 |
Find the difference between 20 and 14 on the tray: 20 − 14 = |
6 |
20 ENTER 14 - |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
7
12 ENTER 5 - |
6
13 ENTER 7 - |
8 birds
15 ENTER 7 - |
7
11 ENTER 1 - 3 - |
10
17 ENTER 4 - 3 - |
6
14 ENTER 5 - 3 - |
| #2 |
7
13 ENTER 6 - |
8
17 ENTER 9 - |
8 stickers
17 ENTER 9 - |
5
12 ENTER 2 - 5 - |
8
19 ENTER 5 - 6 - |
6
16 ENTER 8 - 2 - |
| #3 |
6
14 ENTER 8 - |
9
15 ENTER 6 - |
8 jumps
20 ENTER 12 - |
6
13 ENTER 3 - 4 - |
7
20 ENTER 8 - 5 - |
3
17 ENTER 9 - 5 - |
| #4 |
8
15 ENTER 7 - |
8
19 ENTER 11 - |
9 cookies
18 ENTER 9 - |
5
14 ENTER 4 - 5 - |
6
16 ENTER 6 - 4 - |
3
20 ENTER 7 - 10 - |
| #5 |
6
15 ENTER 9 - |
12
20 ENTER 8 - |
5 pages
16 ENTER 11 - |
7
16 ENTER 6 - 3 - |
9
8 ENTER 7 + 6 - |
4
12 ENTER 15 ENTER 7 - - |
| #6 |
9
16 ENTER 7 - |
9
14 ENTER 5 - |
8 crayons
14 ENTER 6 - |
6
15 ENTER 9 - |
8
11 ENTER 6 + 9 - |
7
16 ENTER 18 ENTER 9 - - |
| #7 |
8
16 ENTER 8 - |
6
17 ENTER 11 - |
6 points
19 ENTER 13 - |
9
17 ENTER 8 - |
12
15 ENTER 8 - 5 + |
17
9 ENTER 8 + |
| #8 |
9
17 ENTER 8 - |
7
19 ENTER 12 - |
7 shells
15 ENTER 8 - |
6
13 ENTER 7 - |
13
18 ENTER 9 - 4 + |
7 apples
20 ENTER 6 - 7 - |
| #9 |
9
18 ENTER 9 - |
14
8 ENTER 6 + |
6 seats
20 ENTER 14 - |
14
20 ENTER 6 - |
14
14 ENTER 7 - 7 + |
6
8 ENTER + 5 ENTER + - |
| #10 |
13
20 ENTER 7 - |
16
7 ENTER 9 + |
5
13 ENTER 8 - |
9
20 ENTER 11 - |
6
9 ENTER 8 + 6 ENTER 5 + - |
7
19 ENTER 7 - 5 - |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Compare Row 1 and Row 0 on the tray: 14 - 6 = |
10 pts |
8 |
14 ENTER 6 - |
| 2 |
Core Fluency |
Bridge back through 10 to subtract: 16 - 9 = |
10 pts |
7 |
16 ENTER 9 - |
| 3 |
Core Fluency |
Find the distance on the number line: 17 - 8 = |
10 pts |
9 |
17 ENTER 8 - |
| 4 |
Core Fluency |
Take away from 2 full Ten-Frames: 20 - 13 = |
10 pts |
7 |
20 ENTER 13 - |
| 5 |
Core Fluency |
Find the missing subtrahend: 15 - [ ? ] = 6 |
10 pts |
9 |
15 ENTER 6 - |
| 6 |
Core Fluency |
Follow the 2-step take-away path: 18 - 7 - 4 = |
10 pts |
7 |
18 ENTER 7 - 4 - |
| 7 |
Application |
Lucas has 16 trading cards and gives 7 to his friend. How many cards does he have left? |
10 pts |
9 cards |
16 ENTER 7 - |
| 8 |
Application |
A sunflower is 19 inches tall and a daisy is 11 inches tall. How much taller is the sunflower? |
10 pts |
8 inches |
19 ENTER 11 - |
| 9 |
Application |
15 ducks swim in a pond. 8 fly away, then 4 new ducks land. How many ducks now? |
10 pts |
11 ducks |
15 ENTER 8 - 4 + |
| 10 |
Error Analysis |
Error Analysis: Why must 17 be entered BEFORE 9 when computing 17 - 9 on the RPN stack? |
10 pts |
Y holds 17 (17 - 9 = 8) |
17 ENTER 9 - |
5. STEM Lab Case Study: The Frog Jump Number Line Olympics
Scenario: Biologists track tree frogs hopping along a 20-meter measuring tape beside a pond. Students use subtraction to find jump distances and remaining track lengths.
- Case Question 1: Frog A starts at meter 6 and lands at meter 15. How long was its jump?
- Expected Result: 9 meters | Keystrokes:
15 ENTER 6 -
- Case Question 2: Frog B is at meter 13. How many meters left to reach the finish line at meter 20?
- Expected Result: 7 meters | Keystrokes:
20 ENTER 13 -
- Case Question 3: Frog C jumps from 0 to 18, then hops backward 7 meters. What marker is it on now?
- Expected Result: Meter 11 | Keystrokes:
18 ENTER 7 -
- Case Question 4: Error Analysis: Leo subtracts 16 − 9 and gets 8. Use the number line to find the true distance.
- Expected Result: 7 meters (9 + 7 = 16, not 8). | Keystrokes:
16 ENTER 9 -
Unit 07: Make-10 Regrouping with Gold [+1] Carry Tokens
Grades: Grade 1 – 2 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (2-Row Tray & Gold [+1] Carry Tokens)
Key Standards: CCSS.MATH.1.NBT.C.4, CCSS.MATH.2.NBT.B.5
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Build 37 ([3][7]) on your Place Value Tray and add [6] in Ones. Snap a Gold [+1] Carry Token onto Tens. What is 37 + 6?
- Verified Answer:
43.
- Keystrokes:
37 ENTER 6 +
- Warm-Up B (Extension Lab): Add 58 + 25 on your Place Value Tray using a Gold [+1] Carry Token. What is the sum?
- Verified Answer:
83.
- Keystrokes:
58 ENTER 25 +
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Build 27 + 8 on your Place Value Tray with a [+1] token: 27 + 8 = |
35 |
27 ENTER 8 + |
| 2 |
Build 46 + 17 on your Place Value Tray: 46 + 17 = |
63 |
46 ENTER 17 + |
| 3 |
Build 39 + 24 on your Place Value Tray: 39 + 24 = |
63 |
39 ENTER 24 + |
| 4 |
Build 55 + 28 on your Place Value Tray: 55 + 28 = |
83 |
55 ENTER 28 + |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
33
27 ENTER 6 + |
43
26 ENTER 17 + |
44 shells
28 ENTER 16 + |
35
28 ENTER 2 + 5 + |
8
34 ENTER 26 - |
28
100 ENTER 72 - |
| #2 |
43
38 ENTER 5 + |
63
35 ENTER 28 + |
52 students
25 ENTER 27 + |
57
49 ENTER 1 + 7 + |
7
55 ENTER 48 - |
44
100 ENTER 56 - |
| #3 |
53
45 ENTER 8 + |
63
47 ENTER 16 + |
84¢
46 ENTER 38 + |
73
67 ENTER 3 + 3 + |
15
80 ENTER 65 - |
62
100 ENTER 38 - |
| #4 |
63
59 ENTER 4 + |
82
58 ENTER 24 + |
63 tulips
39 ENTER 24 + |
62
38 ENTER 2 + 22 + |
25
62 ENTER 37 - |
19
100 ENTER 81 - |
| #5 |
73
66 ENTER 7 + |
73
39 ENTER 34 + |
82 pages
37 ENTER 45 + |
84
59 ENTER 1 + 24 + |
55
17 ENTER 25 + 13 + |
52
26 ENTER + |
| #6 |
83
74 ENTER 9 + |
75
46 ENTER 29 + |
75 passengers
48 ENTER 27 + |
45
36 ENTER 10 + 1 - |
59
26 ENTER 19 + 14 + |
74
37 ENTER + |
| #7 |
94
88 ENTER 6 + |
83
65 ENTER 18 + |
83 beads
54 ENTER 29 + |
66
57 ENTER 10 + 1 - |
75
38 ENTER 22 + 15 + |
96
48 ENTER + |
| #8 |
40
36 ENTER 4 + |
93
57 ENTER 36 + |
85 trees
68 ENTER 17 + |
64
45 ENTER 20 + 1 - |
87
29 ENTER 31 + 27 + |
8
64 ENTER 26 - 30 - |
| #9 |
60
53 ENTER 7 + |
92
44 ENTER 48 + |
90 blocks
45 ENTER 45 + |
87
68 ENTER 20 + 1 - |
77
18 ENTER 24 + 35 + |
70
15 ENTER 25 + 18 + 12 + |
| #10 |
80
72 ENTER 8 + |
100
63 ENTER 37 + |
63
38 ENTER 25 + |
83
54 ENTER 30 + 1 - |
81
27 ENTER 27 + 27 + |
90
47 ENTER 28 + 15 + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Regroup with a Gold [+1] Carry Token: 28 + 7 = |
10 pts |
35 |
28 ENTER 7 + |
| 2 |
Core Fluency |
Make the next ten on the tray: 46 + 8 = |
10 pts |
54 |
46 ENTER 8 + |
| 3 |
Core Fluency |
Add two-digit numbers with a Carry Token: 37 + 26 = |
10 pts |
63 |
37 ENTER 26 + |
| 4 |
Core Fluency |
Combine Tens and Ones with regrouping: 58 + 35 = |
10 pts |
93 |
58 ENTER 35 + |
| 5 |
Core Fluency |
Find the missing partner to 100: 67 + [ ? ] = 100 |
10 pts |
33 |
100 ENTER 67 - |
| 6 |
Core Fluency |
Pair 10-partners first on the stack: 24 + 19 + 16 = |
10 pts |
59 |
24 ENTER 19 + 16 + |
| 7 |
Application |
A library shelf has 38 fiction books and 27 science books. How many books in all? |
10 pts |
65 books |
38 ENTER 27 + |
| 8 |
Application |
Maya has 45¢ and earns 39¢ more. How many cents does she have altogether? |
10 pts |
84¢ |
45 ENTER 39 + |
| 9 |
Application |
Double 36 on the stack. What is the total? |
10 pts |
72 |
36 ENTER + |
| 10 |
Error Analysis |
Error Analysis: Zoe added 48 + 25 and got 63 because she forgot to carry 1 Ten. Give the true sum! |
10 pts |
73 |
48 ENTER 25 + |
5. STEM Lab Case Study: The Coin Bank Regrouping Vault
Scenario: An automated coin sorting machine collects pennies in the Ones chute. Whenever 10 pennies accumulate, it trades them for 1 Dime (10¢) in the Tens chute.
- Case Question 1: The vault holds 37¢ (3 dimes, 7 pennies) and 8 more pennies drop in. What is the new total?
- Expected Result: 45¢ (4 dimes, 5 pennies). | Keystrokes:
37 ENTER 8 +
- Case Question 2: Deposit A is 46¢ and Deposit B is 27¢. Combine them and state total cents:
- Expected Result: 73¢ | Keystrokes:
46 ENTER 27 +
- Case Question 3: Three coin rolls hold 25¢, 38¢, and 17¢. What is the total value in the vault?
- Expected Result: 80¢ | Keystrokes:
25 ENTER 38 + 17 +
- Case Question 4: Error Analysis: A clerk adds 58¢ + 26¢ and writes 74¢. Find the true sum with the carry token!
- Expected Result: 84¢ (8 + 6 = 14¢, so 50 + 20 + 14 = 84¢). | Keystrokes:
58 ENTER 26 +
Unit 08: Two-Digit Addition & Subtraction (0–99)
Grades: Grade 2 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (2-Row Tray, Digit Blocks & [+1] Tokens)
Key Standards: CCSS.MATH.2.NBT.B.5, CCSS.MATH.2.NBT.B.7
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Solve 45 + 39 on your Place Value Tray and paper first, then verify on your SC-6.
- Verified Answer:
84.
- Keystrokes:
45 ENTER 39 +
- Warm-Up B (Extension Lab): Place [6][4] in Row 1 and [2][8] in Row 0 of your Place Value Tray. Unbundle 1 Ten to solve 64 − 28.
- Verified Answer:
36.
- Keystrokes:
64 ENTER 28 -
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Solve 36 + 48 on paper & Place Value Tray, then verify: 36 + 48 = |
84 |
36 ENTER 48 + |
| 2 |
Solve 71 − 35 on paper with unbundling: 71 − 35 = |
36 |
71 ENTER 35 - |
| 3 |
Solve 54 + 29 on paper & Place Value Tray, then verify: 54 + 29 = |
83 |
54 ENTER 29 + |
| 4 |
Solve 83 − 47 on paper with unbundling: 83 − 47 = |
36 |
83 ENTER 47 - |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
59
34 ENTER 25 + |
34
50 ENTER 16 - |
27 seats
85 ENTER 58 - |
35
54 ENTER 19 - |
48
36 ENTER 27 + 15 - |
36
62 ENTER 26 - |
| #2 |
88
52 ENTER 36 + |
37
70 ENTER 33 - |
83 stickers
44 ENTER 39 + |
57
76 ENTER 19 - |
45
58 ENTER 29 - 16 + |
45
83 ENTER 38 - |
| #3 |
44
68 ENTER 24 - |
33
80 ENTER 47 - |
56 feet
92 ENTER 36 - |
36
65 ENTER 29 - |
61
74 ENTER 38 - 25 + |
45
94 ENTER 49 - |
| #4 |
52
87 ENTER 35 - |
36
90 ENTER 54 - |
17 points
64 ENTER 47 - |
53
82 ENTER 29 - |
55
49 ENTER 33 + 27 - |
121
47 ENTER 74 + |
| #5 |
85
47 ENTER 38 + |
62
100 ENTER 38 - |
27 bagels
75 ENTER 48 - |
54
93 ENTER 39 - |
42
85 ENTER 25 - 18 - |
23
38 ENTER 25 + 40 - |
| #6 |
85
56 ENTER 29 + |
26
74 ENTER 48 - |
$83
38 ENTER 45 + |
23
52 ENTER 29 - |
36
96 ENTER 38 - 22 - |
26
82 ENTER 36 - 20 - |
| #7 |
33
52 ENTER 19 - |
35
91 ENTER 56 - |
37 pages
90 ENTER 53 - |
27
65 ENTER 38 - |
27
100 ENTER 45 - 28 - |
17
100 ENTER 54 ENTER 29 + - |
| #8 |
36
63 ENTER 27 - |
37
65 ENTER 28 - |
81 acorns
52 ENTER 29 + |
34
81 ENTER 47 - |
30
42 ENTER 35 + 28 ENTER 19 + - |
47
75 ENTER 28 - |
| #9 |
36
71 ENTER 35 - |
83
49 ENTER 34 + |
56 meters
81 ENTER 25 - |
47
73 ENTER 20 - 6 - |
27
64 ENTER 18 + 90 ENTER 35 - - |
85
46 ENTER 39 + |
| #10 |
36
82 ENTER 46 - |
84
47 ENTER 37 + |
34
72 ENTER 38 - |
47
85 ENTER 30 - 8 - |
47 riders
52 ENTER 19 + 24 - |
76
84 ENTER 37 - 29 + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Add with a Gold [+1] Carry Token: 46 + 38 = |
10 pts |
84 |
46 ENTER 38 + |
| 2 |
Core Fluency |
Add two-digit numbers on the tray: 57 + 29 = |
10 pts |
86 |
57 ENTER 29 + |
| 3 |
Core Fluency |
Unbundle 1 Ten to 10 Ones to subtract: 64 - 28 = |
10 pts |
36 |
64 ENTER 28 - |
| 4 |
Core Fluency |
Unbundle 1 Ten on the Place Value Tray: 81 - 45 = |
10 pts |
36 |
81 ENTER 45 - |
| 5 |
Core Fluency |
Subtract across a zero in the Ones place: 90 - 37 = |
10 pts |
53 |
90 ENTER 37 - |
| 6 |
Core Fluency |
Follow the 2-step stack chain: 52 + 29 - 34 = |
10 pts |
47 |
52 ENTER 29 + 34 - |
| 7 |
Application |
A store has 74 balloons and sells 38 of them. How many balloons are left? |
10 pts |
36 balloons |
74 ENTER 38 - |
| 8 |
Application |
Leo has 48¢. How many more cents does he need to buy a toy that costs 85¢? |
10 pts |
37¢ |
85 ENTER 48 - |
| 9 |
Application |
A school garden harvested 35 tomatoes Monday, 28 Tuesday, and gave 19 away. How many left? |
10 pts |
44 tomatoes |
35 ENTER 28 + 19 - |
| 10 |
Error Analysis |
Error Analysis: Mia solved 73 - 28 and got 55 because she did 8 - 3 = 5. Find the true answer! |
10 pts |
45 |
73 ENTER 28 - |
5. STEM Lab Case Study: The Weather Station Thermometer Log
Scenario: Junior meteorologists record daily high and low temperatures (in °F) and monthly rainfall totals at the school rooftop weather station.
- Case Question 1: Monday's high was 74°F and the overnight low was 48°F. What was the temperature swing?
- Expected Result: 26°F | Keystrokes:
74 ENTER 48 -
- Case Question 2: Morning temperature was 56°F and rose 27°F by afternoon. What was the afternoon high?
- Expected Result: 83°F | Keystrokes:
56 ENTER 27 +
- Case Question 3: Rainfall was 38 mm in April and 45 mm in May. How much short of 100 mm is the 2-month total?
- Expected Result: 17 mm | Keystrokes:
100 ENTER 38 ENTER 45 + -
- Case Question 4: Error Analysis: A student computes 82°F − 47°F = 45°F by doing 7 − 2 = 5. Correct the reading!
- Expected Result: 35°F (unbundle 1 ten: 12 − 7 = 5, 70 − 40 = 30 -> 35°F). | Keystrokes:
82 ENTER 47 -
Unit 09: Equal Groups, Repeated Addition & Arrays
Grades: Grade 2 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (2-Row Tray & Equal-Group Digit Blocks)
Key Standards: CCSS.MATH.2.OA.C.3, CCSS.MATH.2.OA.C.4
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Place four [5] Digit Blocks on your Place Value Tray (4 groups of 5). What is 5 + 5 + 5 + 5?
- Verified Answer:
20 (4 groups of 5).
- Keystrokes:
5 ENTER 5 + 5 + 5 +
- Warm-Up B (Extension Lab): Place [9] in Row 1 and [9] in Row 0 of your Place Value Tray ([9] [ENTER] [+]). What even number is formed?
- Verified Answer:
18 (Even).
- Keystrokes:
9 ENTER +
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Place three [5] blocks on the tray (3 groups of 5): 5 + 5 + 5 = |
15 |
5 ENTER 5 + 5 + |
| 2 |
Place four [3] blocks on the tray (4 groups of 3): 3 + 3 + 3 + 3 = |
12 |
3 ENTER 3 + 3 + 3 + |
| 3 |
Double [8] on the tray using the instant shortcut: 8 [ENTER] + = |
16 |
8 ENTER + |
| 4 |
Place [9] in Row 1 and [9] in Row 0 of the tray: 9 + 9 = |
18 |
9 ENTER + |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
12
6 ENTER 6 + |
16
2 ENTER 8 × |
12 panes
4 ENTER 3 × |
14
7 ENTER + |
12
6 ENTER 2 × |
4
2 x² |
| #2 |
18
9 ENTER 9 + |
16
8 ENTER 2 × |
12 wheels
6 ENTER 2 × |
22
11 ENTER + |
18
9 ENTER 2 × |
9
3 x² |
| #3 |
12
4 ENTER 4 + 4 + |
24
3 ENTER 8 × |
15 wheels
5 ENTER 3 × |
28
14 ENTER + |
30
6 ENTER 5 × |
16
4 x² |
| #4 |
18
6 ENTER 6 + 6 + |
28
4 ENTER 7 × |
16 tires
4 ENTER 4 × |
32
16 ENTER + |
40
8 ENTER 5 × |
25
5 x² |
| #5 |
12
3 ENTER 3 + 3 + 3 + |
30
5 ENTER 6 × |
24 eggs
2 ENTER 12 × |
7
14 ENTER 2 ÷ |
70
7 ENTER 10 × |
36
6 x² |
| #6 |
20
5 ENTER 5 + 5 + 5 + |
30
6 ENTER 5 × |
27 books
3 ENTER 9 × |
10
20 ENTER 2 ÷ |
19
3 ENTER 5 × 4 + |
24
6 ENTER + ENTER + |
| #7 |
15
5 ENTER 3 × |
4
12 ENTER 3 ÷ |
25 plants
5 ENTER 5 × |
12
24 ENTER 2 ÷ |
22
4 ENTER 4 × 6 + |
32
8 ENTER + ENTER + |
| #8 |
25
5 ENTER 5 × |
6
18 ENTER 3 ÷ |
35 fingers
7 ENTER 5 × |
6 R 1
13 ENTER 2 ÷R |
17
5 ENTER 4 × 3 - |
9
5 x² 4 x² - |
| #9 |
21
7 ENTER 7 + 7 + |
4
20 ENTER 5 ÷ |
8 stickers
16 ENTER 2 ÷ |
8 R 1
17 ENTER 2 ÷R |
20
2 ENTER 4 × 3 ENTER 4 × + |
11
6 x² 5 x² - |
| #10 |
24
6 ENTER 4 × |
6
24 ENTER 4 ÷ |
18
3 ENTER 6 × |
14
8 ENTER 6 + |
30
3 ENTER 6 × 2 ENTER 6 × + |
39
4 ENTER 6 × 3 ENTER 5 × + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Evaluate 3 equal groups of 5: 5 + 5 + 5 = |
10 pts |
15 |
5 ENTER 5 + 5 + |
| 2 |
Core Fluency |
Evaluate 4 rows of 4 tiles: 4 × 4 = |
10 pts |
16 |
4 ENTER 4 × |
| 3 |
Core Fluency |
Evaluate 5 rows of 6 tiles: 5 × 6 = |
10 pts |
30 |
5 ENTER 6 × |
| 4 |
Core Fluency |
Double 13 on the stack: 13 + 13 = |
10 pts |
26 |
13 ENTER + |
| 5 |
Core Fluency |
Split 18 into 2 equal rows: 18 ÷ 2 = |
10 pts |
9 |
18 ENTER 2 ÷ |
| 6 |
Core Fluency |
Array plus loose blocks: (3 × 4) + 7 = |
10 pts |
19 |
3 ENTER 4 × 7 + |
| 7 |
Application |
A baking sheet has 4 rows of 5 cookies. How many cookies are on the sheet? |
10 pts |
20 cookies |
4 ENTER 5 × |
| 8 |
Application |
6 boxes each hold 3 tennis balls. How many tennis balls are there in all? |
10 pts |
18 balls |
6 ENTER 3 × |
| 9 |
Application |
Divide 19 blocks into 2 equal rows. How many in each row and how many left over? |
10 pts |
9 R 1 |
19 ENTER 2 ÷R |
| 10 |
Error Analysis |
Error Analysis: Why is 17 an odd number when you build a 2-row array? |
10 pts |
1 block left over (8 R 1) |
17 ENTER 2 ÷R |
Scenario: Second-grade botanists lay out rectangular raised beds in the school garden, spacing vegetable seedlings in equal rows and columns.
- Case Question 1: Bed A has 4 rows of 5 carrot plants. How many carrot plants are in Bed A?
- Expected Result: 20 plants | Keystrokes:
4 ENTER 5 ×
- Case Question 2: Bed B has 3 rows of 6 tomato plants. How many tomato plants are in Bed B?
- Expected Result: 18 plants | Keystrokes:
3 ENTER 6 ×
- Case Question 3: Combine Bed A and Bed B. How many vegetable plants are in both beds?
- Expected Result: 38 plants | Keystrokes:
20 ENTER 18 +
- Case Question 4: Error Analysis: A student plants 4 rows of 6 sunflowers and writes 4 + 6 = 10. Find the true count!
- Expected Result: 24 sunflowers (6 + 6 + 6 + 6 = 24). | Keystrokes:
4 ENTER 6 ×
Unit 10: Measurement, Length & Coin Money Pipelines
Grades: Grade 2 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (2-Row Tray, Digit Blocks & [+1] Carry Tokens)
Key Standards: CCSS.MATH.2.MD.A.1, CCSS.MATH.2.MD.B.5, CCSS.MATH.2.MD.C.8
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): A ribbon is 28 cm long in Row 1. You cut off 12 cm in Row 0. How much ribbon remains?
- Verified Answer:
16 cm.
- Keystrokes:
28 ENTER 12 -
- Warm-Up B (Extension Lab): You buy a pencil for 35¢ and an eraser for 25¢. How much change do you receive from $1.00?
- Verified Answer:
40¢.
- Keystrokes:
100 ENTER 35 ENTER 25 + -
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
A board is 42 inches long. You cut 15 inches off. How much remains? |
27 inches |
42 ENTER 15 - |
| 2 |
Find the total value in cents of 2 quarters and 3 dimes: |
80¢ |
50 ENTER 30 + |
| 3 |
You spend 72¢ at the store. What is your change from $1.00? |
28¢ |
100 ENTER 72 - |
| 4 |
Combine lengths: side A is 18 cm and side B is 24 cm. Find the total length: |
42 cm |
18 ENTER 24 + |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
33 cm
18 ENTER 15 + |
50¢
25 ENTER 25 + |
9 cm
11 ENTER 2 - |
32 cm
10 ENTER 6 + 2 × |
45¢
100 ENTER 25 ENTER 30 + - |
17 cm
23 ENTER 6 - |
| #2 |
42 in
24 ENTER 18 + |
75¢
25 ENTER 3 × |
64 cm
38 ENTER 26 + |
40 cm
14 ENTER 6 + 2 × |
20¢
100 ENTER 45 ENTER 35 + - |
28 in
42 ENTER 14 - |
| #3 |
62 ft
35 ENTER 27 + |
100¢
4 ENTER 25 × |
45 in
80 ENTER 35 - |
50 in
15 ENTER 10 + 2 × |
26¢
100 ENTER 28 ENTER 46 + - |
4 dimes
90 ENTER 50 - 10 ÷ |
| #4 |
84 cm
46 ENTER 38 + |
45¢
25 ENTER 10 + 10 + |
5¢ left
50 ENTER 30 + 75 - |
60 cm
18 ENTER 12 + 2 × |
27¢
100 ENTER 39 ENTER 34 + - |
4 nickels
45 ENTER 25 - 5 ÷ |
| #5 |
32 cm
50 ENTER 18 - |
65¢
50 ENTER 10 + 5 + |
75¢
45 ENTER 30 + |
36 cm
4 ENTER 9 × |
36 in
3 ENTER 12 × |
6 cm
30 ENTER 2 ÷ 9 - |
| #6 |
35 in
64 ENTER 29 - |
49¢
30 ENTER 15 + 4 + |
32¢
100 ENTER 68 - |
60 cm
4 ENTER 15 × |
20 in
12 ENTER 8 + |
7 cm
40 ENTER 2 ÷ 13 - |
| #7 |
27 in
72 ENTER 45 - |
82¢
50 ENTER 30 + 2 + |
24 cm
43 ENTER 19 - |
45 cm
12 ENTER 15 + 18 + |
18 ft
6 ENTER 3 × |
40 cm
90 ENTER 22 - 28 - |
| #8 |
37 cm
85 ENTER 48 - |
60¢
100 ENTER 40 - |
62¢
40 ENTER 15 + 7 + |
54 in
20 ENTER 20 + 14 + |
85 min
60 ENTER 25 + |
9 cm
45 ENTER 5 ÷ |
| #9 |
60 cm
15 ENTER 20 + 25 + |
35¢
100 ENTER 65 - |
50 cm
100 ENTER 25 - 25 - |
10 cm
40 ENTER 4 ÷ |
35 min
50 ENTER 15 - |
99¢
75 ENTER 20 + 4 + |
| #10 |
64 cm
100 ENTER 36 - |
22¢
100 ENTER 78 - |
35¢
15 ENTER 20 + |
15 cm
50 ENTER 16 - 19 - |
37¢
100 ENTER 63 - |
40
100 ENTER 42 - 18 - |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Combine the two rod lengths: 26 cm + 38 cm = |
10 pts |
64 cm |
26 ENTER 38 + |
| 2 |
Core Fluency |
Find the remaining board length: 75 in - 29 in = |
10 pts |
46 in |
75 ENTER 29 - |
| 3 |
Core Fluency |
Find the total value in cents of 2 Quarters, 2 Dimes, and 1 Nickel: |
10 pts |
75¢ |
50 ENTER 20 + 5 + |
| 4 |
Core Fluency |
Find the change from $1.00 after spending 57¢: |
10 pts |
43¢ |
100 ENTER 57 - |
| 5 |
Core Fluency |
Find the perimeter of a rectangle 11 cm long by 7 cm wide: |
10 pts |
36 cm |
11 ENTER 7 + 2 × |
| 6 |
Core Fluency |
Compute the two-item change from 100¢: 100 - (36 + 24) = |
10 pts |
40 |
100 ENTER 36 ENTER 24 + - |
| 7 |
Application |
A key starts at the 4 cm mark on a ruler and ends at the 11 cm mark. How long is the key? |
10 pts |
7 cm |
11 ENTER 4 - |
| 8 |
Application |
Maya buys a 45¢ sticker and a 28¢ pencil. How much change does she get from $1.00? |
10 pts |
27¢ |
100 ENTER 45 ENTER 28 + - |
| 9 |
Application |
A square garden has sides that are each 12 feet long. What is the perimeter of the garden? |
10 pts |
48 feet |
4 ENTER 12 × |
| 10 |
Error Analysis |
Error Analysis: Leo found the perimeter of a 10 cm by 5 cm rectangle by adding 10 + 5 = 15. Fix it! |
10 pts |
30 cm (all 4 sides) |
10 ENTER 5 + 2 × |
5. STEM Lab Case Study: The Toy Maker's Measuring & Coin Shop
Scenario: Young craftspeople cut balsa wood strips to build model gliders and sell their finished models at the school STEM fair.
- Case Question 1: A glider needs a 28 cm main wing and an 18 cm tail wing. Total wood strip length needed?
- Expected Result: 46 cm | Keystrokes:
28 ENTER 18 +
- Case Question 2: You cut the 46 cm needed from an 80 cm balsa strip. How many cm of wood are left over?
- Expected Result: 34 cm | Keystrokes:
80 ENTER 46 -
- Case Question 3: A customer buys a glider for 45¢ and a launcher for 30¢ with $1.00. Find their change:
- Expected Result: 25¢ (1 Quarter) | Keystrokes:
100 ENTER 45 ENTER 30 + -
- Case Question 4: Error Analysis: A student measures a wing from the 3 cm mark to 15 cm and says 15 cm. Correct it!
- Expected Result: 12 cm (15 − 3 = 12 cm). | Keystrokes:
15 ENTER 3 -
Appendix C: Cumulative Benchmark & Capstone Mastery Exams (Pages 117–120)
Cumulative Exam 1: Units 01–05 Benchmark (p. 117 • 100 Points)
Covers Step Counting, Number Bonds to 10, Make-10 Regrouping, Twin-Row Doubling, and Tens & Ones on the Place Value Studio Tray.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Step Count by 1s: Start at 6 on your Ten-Frame and add 1 block three times. Where do you land? |
10 pts |
9 |
6 ENTER 1 + 1 + 1 + |
| 2 |
Core Fluency |
Number Bond to 10: A full Ten-Frame holds 10 blocks. If 4 blocks are blue, how many are yellow? |
10 pts |
6 |
10 ENTER 4 - |
| 3 |
Core Fluency |
Make-10 Bridge: Place 8 blocks in Frame 1 and 5 in Frame 2. Trade blocks to fill 10 and find 8 + 5: |
10 pts |
13 |
8 ENTER 2 + 3 + |
| 4 |
Core Fluency |
Nine-Plus Bridge: Use a Make-10 trade on your Place Value Tray to solve 9 + 7 = ? |
10 pts |
16 |
9 ENTER 7 + |
| 5 |
Core Fluency |
Twin-Row Doubling: Place 8 blocks on Row Y and 8 blocks on Row X of your tray. What is the double of 8? |
10 pts |
16 |
8 ENTER + |
| 6 |
Core Fluency |
Near-Double Plus One: Use your double of 7 plus 1 more block to find 7 + 8 = ? |
10 pts |
15 |
7 ENTER + 1 + |
| 7 |
Application |
Place Value Tens & Ones: Build 3 Tens and 6 Ones on the Place Value Tray. What 2-digit number is formed? |
10 pts |
36 |
30 ENTER 6 + |
| 8 |
Application |
Three-Addend Make-10: Snap together the 10-pair first to solve 7 + 3 + 8 = ? |
10 pts |
18 |
7 ENTER 3 + 8 + |
| 9 |
Application |
Story Application: Mia stacks 6 red blocks and 6 gold blocks, then adds 4 more. How many blocks in all? |
10 pts |
16 |
6 ENTER 6 + 4 + |
| 10 |
Error Analysis |
Error Spotter: Leo typed 40 + 5 and got 405 instead of placing [4] in Tens and [5] in Ones. What is 40 + 5? |
10 pts |
45 |
40 ENTER 5 + |
Cumulative Exam 2: Units 06–10 Benchmark (p. 118 • 100 Points)
Covers Fact-Family Subtraction, Gold [+1] Carry Tokens, Two-Digit Regrouping, Equal-Groups Block Arrays, and Measurement/Money.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Fact Family Take-Away: Unbundle 1 Ten on the Place Value Tray to solve 14 − 6 = ? |
10 pts |
8 |
14 ENTER 6 - |
| 2 |
Core Fluency |
Gold [+1] Carry Token: Add 27 + 16 on the Place Value Tray by trading 10 Ones for a [+1] Ten token: |
10 pts |
43 |
27 ENTER 16 + |
| 3 |
Core Fluency |
Two-Digit Regrouping Sum: Combine 38 + 25 using your Place Value Digit Blocks and [+1] Carry Token: |
10 pts |
63 |
38 ENTER 25 + |
| 4 |
Core Fluency |
Two-Digit Unbundling Subtraction: Trade 1 Ten for 10 Ones on the tray to solve 52 − 18 = ? |
10 pts |
34 |
52 ENTER 18 - |
| 5 |
Core Fluency |
Two-Step Tray Chain: Compute (24 + 19) − 13 on the 4-level stack without parentheses: |
10 pts |
30 |
24 ENTER 19 + 13 - |
| 6 |
Core Fluency |
Equal Groups Array: Build 4 rows of 5 blocks on your desk grid: 4 × 5 = ? |
10 pts |
20 |
4 ENTER 5 × |
| 7 |
Application |
Skip-Count Array: Build 3 rows of 6 blocks and verify on the SC-6 stack: 3 × 6 = ? |
10 pts |
18 |
3 ENTER 6 × |
| 8 |
Application |
Coin Bank Total: Find the total value in cents of 2 quarters, 3 dimes, and 1 nickel: |
10 pts |
85 |
50 ENTER 30 + 5 + |
| 9 |
Application |
Length Comparison: A blue block tower is 45 cm tall and a red tower is 28 cm tall. How much taller is the blue tower? |
10 pts |
17 |
45 ENTER 28 - |
| 10 |
Error Analysis |
Error Spotter: Sam added 29 + 14 and forgot the Gold [+1] Carry Token, getting 33. What is the true sum 29 + 14? |
10 pts |
43 |
29 ENTER 14 + |
Cumulative Exam 3: Multi-Step Synthesis (p. 119 • 100 Points)
Synthesizes Make-10 Regrouping, Place Value Studio Carry/Unbundling, Block Arrays, and Multi-Step Story Problems across Units 01–10.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Make-10 & Subtract: Add 8 + 7 on your Ten-Frame, then take away 5 blocks: (8 + 7) − 5 = ? |
10 pts |
10 |
8 ENTER 7 + 5 - |
| 2 |
Core Fluency |
Double & Add Ten: Double 9 on your Twin-Row Tray, then add 1 Ten on the Place Value Tray: |
10 pts |
28 |
9 ENTER + 10 + |
| 3 |
Core Fluency |
Place Value Expansion: Combine 50 + 20 + 8 on the Whole Number Tray: |
10 pts |
78 |
50 ENTER 20 + 8 + |
| 4 |
Core Fluency |
Regrouping & Unbundling Chain: Solve (36 + 17) − 24 step-by-step on paper and verify on the SC-6: |
10 pts |
29 |
36 ENTER 17 + 24 - |
| 5 |
Core Fluency |
Array Plus Extra Blocks: Build 5 rows of 4 blocks, then add 6 more blocks. How many blocks in all? |
10 pts |
26 |
5 ENTER 4 × 6 + |
| 6 |
Core Fluency |
Two Arrays Combined: Combine a 3 × 4 block array and a 2 × 5 block array. How many blocks in all? |
10 pts |
22 |
3 ENTER 4 × 2 ENTER 5 × + |
| 7 |
Application |
Making Change from $1.00: You buy a 45¢ pencil and a 25¢ eraser. How much change do you receive from $1.00? |
10 pts |
30 |
100 ENTER 45 ENTER 25 + - |
| 8 |
Application |
Three-Number Regrouping Sum: Add 18 + 22 + 35 on your Place Value Tray: |
10 pts |
75 |
18 ENTER 22 + 35 + |
| 9 |
Application |
Garden Perimeter Walk: A rectangular garden bed is 12 feet long and 8 feet wide. What is its perimeter in feet? |
10 pts |
40 |
12 ENTER 8 + 12 + 8 + |
| 10 |
Error Analysis |
Error Spotter: Ava solved 60 − 27 without unbundling a Ten and wrote 47. What is the true difference 60 − 27? |
10 pts |
33 |
60 ENTER 27 - |
Cumulative Exam 4: Final Capstone Mastery (p. 120 • 100 Points)
Final K–2 mastery exam integrating Place Value Studio Trays, Gold [+1] Carry Tokens, Block Arrays, and SC-6 Stack Verification.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Place Value Tray Challenge: Build 4 Tens and 8 Ones, then add 3 Tens and 7 Ones on the Place Value Tray: |
10 pts |
85 |
48 ENTER 37 + |
| 2 |
Core Fluency |
Zero-Placeholder Vault: Unbundle 1 Ten on the Place Value Tray to solve 80 − 36 = ? |
10 pts |
44 |
80 ENTER 36 - |
| 3 |
Core Fluency |
Double-Double Chain: Double 6, then double that result again on the stack: |
10 pts |
24 |
6 ENTER + ENTER + |
| 4 |
Core Fluency |
Block Tower Difference: Tower A has 6 rows of 5 blocks and Tower B has 18 blocks. How many more blocks are in Tower A? |
10 pts |
12 |
6 ENTER 5 × 18 - |
| 5 |
Core Fluency |
Classroom Supply Kit: 4 tables each have 6 markers, plus 8 spare markers in the caddy. How many markers in all? |
10 pts |
32 |
4 ENTER 6 × 8 + |
| 6 |
Core Fluency |
Two-Step Bookstore Purchase: You have 90¢. You spend 35¢ and then earn 20¢ back. How many cents do you have now? |
10 pts |
75 |
90 ENTER 35 - 20 + |
| 7 |
Application |
Three-Stack Addition: Combine 26 + 34 + 19 using Gold [+1] Carry Tokens on the Place Value Tray: |
10 pts |
79 |
26 ENTER 34 + 19 + |
| 8 |
Application |
Symmetric Array Difference: Subtract a 3 × 3 square array from a 5 × 5 square array: |
10 pts |
16 |
5 ENTER 5 × 3 ENTER 3 × - |
| 9 |
Application |
Maker Lab Ribbon Cut: Start with 72 inches of ribbon and cut off two 18-inch pieces. How much ribbon remains? |
10 pts |
36 |
72 ENTER 18 - 18 - |
| 10 |
Error Analysis |
Capstone Error Analysis: Noah evaluated (15 + 9) − 8 and got 24 because he forgot to subtract 8. Find (15 + 9) − 8: |
10 pts |
16 |
15 ENTER 9 + 8 - |
📘 Book 2B: Upper Elementary Math Lab (Grades 3–5 • SC-6)
Covers Units 01 through 10 plus 4 Cumulative Benchmark & Capstone Mastery Exams: Multiplication Arrays, Division & Remainders, Unit Fractions, Multi-Digit Operations, Long Division & Factors, Fraction Equivalence, Prime Factorization/GCF/LCM, 4-Register Stack PEMDAS, Mixed Numbers, and Fraction Multiplication & Division (840 active problems).
Unit 01: Multiplication Concepts, Commutativity & Area Models
Grades: Grade 3 | Instrument: StackCalc SC-6 | Visual / Block Model: Rectangular Area Grid Diagram & Place Value Studio
Key Standards: CCSS.MATH.3.OA.A.1, CCSS.MATH.3.OA.B.5
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): A garden bed is 7 feet long by 9 feet wide. What is its total square footage?
- Verified Answer:
63 sq ft.
- Keystrokes:
7 ENTER 9 ×
- Warm-Up B (Extension Lab): Solve 6 × 40 using place value tens on paper and verify on the stack.
- Verified Answer:
240 (24 tens).
- Keystrokes:
6 ENTER 40 ×
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
A solar array has 7 rows of 8 panels. How many panels in all? |
56 |
7 ENTER 8 × |
| 2 |
Multiply a single digit by a multiple of 10: 5 × 40 = |
200 |
5 ENTER 40 × |
| 3 |
A storage rack holds 9 shelves with 6 bins each. How many bins in all? |
54 |
9 ENTER 6 × |
| 4 |
Solve 8 × 30 using place value scaling: 8 × 30 = |
240 |
8 ENTER 30 × |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
28
4 ENTER 7 × |
28
4 ENTER 7 × |
56 trees
7 ENTER 8 × |
120
3 ENTER 40 × |
42
6 ENTER 5 × 6 ENTER 2 × + |
70
5 ENTER 6 × 4 ENTER 7 × + 3 ENTER 4 × + |
| #2 |
36
6 ENTER 6 × |
45
5 ENTER 9 × |
63 squares
9 ENTER 7 × |
150
5 ENTER 30 × |
72
8 ENTER 5 × 8 ENTER 4 × + |
240
3 ENTER 4 × 20 × |
| #3 |
40
5 ENTER 8 × |
48
6 ENTER 8 × |
48 muffins
8 ENTER 6 × |
300
6 ENTER 50 × |
54
5 ENTER 6 × 4 ENTER 6 × + |
240
5 ENTER 8 × 6 × |
| #4 |
42
7 ENTER 6 × |
7
42 ENTER 6 ÷ |
54 spaces
6 ENTER 9 × |
280
4 ENTER 70 × |
47
4 ENTER 8 × 3 ENTER 5 × + |
9
6 ENTER 6 × 4 ÷ |
| #5 |
32
8 ENTER 4 × |
8
64 ENTER 8 ÷ |
64 musicians
8 ENTER 8 × |
240
8 ENTER 30 × |
78
6 ENTER 9 × 4 ENTER 6 × + |
4
8 ENTER 6 × 12 ÷ |
| #6 |
54
9 ENTER 6 × |
7
63 ENTER 9 ÷ |
81 sq m
9 ENTER 9 × |
350
7 ENTER 50 × |
52
9 ENTER 8 × 5 ENTER 4 × - |
60
9 ENTER 8 × 3 ENTER 4 × - |
| #7 |
49
7 ENTER 7 × |
5
35 ENTER 7 ÷ |
42 books
7 ENTER 6 × |
480
6 ENTER 80 × |
48
8 ENTER 8 × 4 ENTER 4 × - |
460
7 ENTER 40 × 6 ENTER 30 × + |
| #8 |
56
8 ENTER 7 × |
9
72 ENTER 8 ÷ |
45 boxes
5 ENTER 9 × |
360
9 ENTER 40 × |
220
4 ENTER 30 × 5 ENTER 20 × + |
150
9 ENTER 70 × 8 ENTER 60 × - |
| #9 |
72
9 ENTER 8 × |
9
54 ENTER 6 ÷ |
50 sq ft
6 ENTER 5 × 4 ENTER 5 × + |
560
8 ENTER 70 × |
390
6 ENTER 40 × 3 ENTER 50 × + |
72
7 ENTER 2 + 5 ENTER 3 + × |
| #10 |
81
9 ENTER 9 × |
1
8 ENTER 8 × 7 ENTER 9 × - |
48
6 ENTER 8 × |
540
9 ENTER 60 × |
160
8 ENTER 50 × 6 ENTER 40 × - |
470
8 ENTER 40 × 5 ENTER 30 × + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Evaluate area product: 7 × 8 = |
10 pts |
56 |
7 ENTER 8 × |
| 2 |
Core Fluency |
Evaluate area product: 9 × 6 = |
10 pts |
54 |
9 ENTER 6 × |
| 3 |
Core Fluency |
Multiply by a multiple of 10: 6 × 40 = |
10 pts |
240 |
6 ENTER 40 × |
| 4 |
Core Fluency |
Multiply by a multiple of 10: 8 × 70 = |
10 pts |
560 |
8 ENTER 70 × |
| 5 |
Core Fluency |
Find the missing factor: 9 × [ ? ] = 72 |
10 pts |
8 |
72 ENTER 9 ÷ |
| 6 |
Core Fluency |
Evaluate distributive split: (6 × 5) + (6 × 4) = |
10 pts |
54 |
6 ENTER 5 × 6 ENTER 4 × + |
| 7 |
Application |
A solar roof has 8 rows of 9 panels. How many panels in all? |
10 pts |
72 panels |
8 ENTER 9 × |
| 8 |
Application |
Five buses each carry 50 passengers. How many passengers total? |
10 pts |
250 passengers |
5 ENTER 50 × |
| 9 |
Application |
Find the composite area of a 7 m by 8 m room and a 4 m by 6 m porch: |
10 pts |
80 |
7 ENTER 8 × 4 ENTER 6 × + |
| 10 |
Error Analysis |
Error Analysis: Sam says 7 × 30 = 21. Calculate the true product: |
10 pts |
210 |
7 ENTER 30 × |
5. STEM Lab Case Study: Warehouse Pallet Storage System
Scenario: Logistics managers calculate warehouse floor space in square meters using area models.
- Case Question 1: A pallet bay holds 6 rows of 8 pallets. How many pallets fit in the bay?
- Expected Result: 48 | Keystrokes:
6 ENTER 8 ×
- Case Question 2: If 4 bays each have 30 boxes, how many boxes total?
- Expected Result: 120 | Keystrokes:
4 ENTER 30 ×
- Case Question 3: Rotate a 7 by 9 pallet layout by 90 degrees. Does the area change?
- Expected Result: No, 63 pallets either way. | Keystrokes:
7 ENTER 9 ×
- Case Question 4: Error Analysis: A clerk calculates 6 × 80 = 48. Find the true product.
- Expected Result: 480 (6 × 8 tens = 48 tens = 480). | Keystrokes:
6 ENTER 80 ×
Unit 02: Division as Fair-Sharing, Grouping & Remainders
Grades: Grade 3 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio & Prime Factor Studio
Key Standards: CCSS.MATH.3.OA.A.2, CCSS.MATH.3.OA.B.6
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): 36 apples are packed into bags of 4. How many bags can be filled?
- Verified Answer:
9 bags.
- Keystrokes:
36 ENTER 4 ÷
- Warm-Up B (Extension Lab): A baker puts 29 cookies into boxes of 6. How many full boxes and how many cookies left over?
- Verified Answer:
4 full boxes, 5 leftover cookies (29 = 6 × 4 + 5).
- Keystrokes:
29 ENTER 6 ÷R
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Divide 35 candies equally into 5 treat bags. How many per bag? |
7 |
35 ENTER 5 ÷ |
| 2 |
How many groups of 8 are in 48? |
6 |
48 ENTER 8 ÷ |
| 3 |
Share 27 marbles among 4 friends. Find the quotient and remainder: |
6 R 3 |
27 ENTER 4 ÷R |
| 4 |
Divide 56 books equally across 7 classroom shelves. How many per shelf? |
8 |
56 ENTER 7 ÷ |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
7
28 ENTER 4 ÷ |
8
56 ENTER 7 ÷ |
8 seedlings
40 ENTER 5 ÷ |
6 R 1
19 ENTER 3 ÷R |
25
6 ENTER 4 × 1 + |
8
18 ENTER 14 + 4 ÷ |
| #2 |
6
36 ENTER 6 ÷ |
9
54 ENTER 6 ÷ |
7 boxes
56 ENTER 8 ÷ |
5 R 3
23 ENTER 4 ÷R |
38
7 ENTER 5 × 3 + |
8
27 ENTER 29 + 7 ÷ |
| #3 |
9
45 ENTER 5 ÷ |
4
32 ENTER 8 ÷ |
5 pieces
45 ENTER 9 ÷ |
5 R 4
29 ENTER 5 ÷R |
53
8 ENTER 6 × 5 + |
7
50 ENTER 8 - 6 ÷ |
| #4 |
6
42 ENTER 7 ÷ |
7
49 ENTER 7 ÷ |
9 students
63 ENTER 7 ÷ |
5 R 4
34 ENTER 6 ÷R |
67
9 ENTER 7 × 4 + |
8
80 ENTER 8 - 9 ÷ |
| #5 |
6
48 ENTER 8 ÷ |
8
72 ENTER 9 ÷ |
8 panels
72 ENTER 9 ÷ |
6 R 3
45 ENTER 7 ÷R |
71
8 ENTER 8 × 7 + |
9
6 ENTER 6 × 4 ÷ |
| #6 |
6
54 ENTER 9 ÷ |
40
5 ENTER 8 × |
8 liters
48 ENTER 6 ÷ |
6 R 2
50 ENTER 8 ÷R |
5 R 1
31 ENTER 6 ÷R |
6
4 ENTER 12 × 8 ÷ |
| #7 |
8
64 ENTER 8 ÷ |
63
7 ENTER 9 × |
30 chairs
180 ENTER 6 ÷ |
6 R 6
60 ENTER 9 ÷R |
11 carts
44 ENTER 4 ÷ |
13
56 ENTER 7 ÷ 45 ENTER 9 ÷ + |
| #8 |
9
63 ENTER 7 ÷ |
72
9 ENTER 8 × |
50 packs
350 ENTER 7 ÷ |
8 R 5
53 ENTER 6 ÷R |
2 markers
50 ENTER 6 ENTER 8 × - |
4
72 ENTER 8 ÷ 30 ENTER 6 ÷ - |
| #9 |
9
72 ENTER 8 ÷ |
30
120 ENTER 4 ÷ |
4 boxes
36 ENTER 4 - 8 ÷ |
8 R 3
67 ENTER 8 ÷R |
8 beads
62 ENTER 6 ENTER 9 × - |
61
8 ENTER 7 × 5 + |
| #10 |
9
81 ENTER 9 ÷ |
40
240 ENTER 6 ÷ |
7
35 ENTER 5 ÷ |
8 R 4
76 ENTER 9 ÷R |
5 boats
4 ENTER 1 + |
23
64 ENTER 8 ÷ 3 ENTER 5 × + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Evaluate exact quotient: 54 ÷ 6 = |
10 pts |
9 |
54 ENTER 6 ÷ |
| 2 |
Core Fluency |
Evaluate exact quotient: 63 ÷ 9 = |
10 pts |
7 |
63 ENTER 9 ÷ |
| 3 |
Core Fluency |
Find the missing divisor: 48 ÷ [ ? ] = 6 |
10 pts |
8 |
48 ENTER 6 ÷ |
| 4 |
Core Fluency |
Find quotient and remainder: 29 ÷ 4 = |
10 pts |
7 R 1 |
29 ENTER 4 ÷R |
| 5 |
Core Fluency |
Find quotient and remainder: 47 ÷ 6 = |
10 pts |
7 R 5 |
47 ENTER 6 ÷R |
| 6 |
Core Fluency |
Verify division using (Q × D) + R: (8 × 7) + 4 = |
10 pts |
60 |
8 ENTER 7 × 4 + |
| 7 |
Application |
72 books are placed equally onto 8 library shelves. How many books per shelf? |
10 pts |
9 books |
72 ENTER 8 ÷ |
| 8 |
Application |
38 tennis balls are packed in cans of 3. Find full cans and leftover balls: |
10 pts |
12 R 2 |
38 ENTER 3 ÷R |
| 9 |
Application |
Evaluate two-step pipeline: (42 + 14) ÷ 8 = |
10 pts |
7 |
42 ENTER 14 + 8 ÷ |
| 10 |
Error Analysis |
Error Analysis: Ben says 37 ÷ 5 = 6 R 7 because 5×6+7=37. Find the true quotient and remainder: |
10 pts |
7 R 2 |
37 ENTER 5 ÷R |
5. STEM Lab Case Study: Wildlife Rescue Meal Distribution
Scenario: Marine biologists divide daily fish rations and vitamin packs among orphaned seal pups and sea otters.
- Case Question 1: Share 36 kg of herring equally among 4 seal pups. How many kg per pup?
- Expected Result: 9 kg each | Keystrokes:
36 ENTER 4 ÷
- Case Question 2: Divide 45 vitamin tablets into packs of 6. Find quotient and remainder.
- Expected Result: 7 R 3 (6×7+3=45). | Keystrokes:
45 ENTER 6 ÷R
- Case Question 3: If 54 fish are shared among 9 otters, how many fish does each otter receive?
- Expected Result: 6 fish | Keystrokes:
54 ENTER 9 ÷
- Case Question 4: Error Analysis: Sam enters 6 [ENTER] 30 [÷] and gets 0.2. What should he do?
- Expected Result: Enter dividend first: 30 [ENTER] 6 [÷] = 5. | Keystrokes:
30 ENTER 6 ÷
Unit 03: Unit Fractions, Denominator Strips & Comparator Heights
Grades: Grade 3 – 4 | Instrument: StackCalc SC-6 | Visual / Block Model: Fraction Studio (84 mm Proportional Towers)
Key Standards: CCSS.MATH.3.NF.A.1, CCSS.MATH.3.NF.A.2
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Place two 1/4 blocks into the comparator. What fraction height do they match?
- Verified Answer:
2/4 = 1/2 (42.0 mm).
- Keystrokes:
. 1 . 4 ENTER . 1 . 4 +
- Warm-Up B (Extension Lab): Compare 1/4 and 1/8. How many 1/8 blocks equal 1/4?
- Verified Answer:
2 eighths equal 1/4 (21.0 mm).
- Keystrokes:
. 1 . 8 ENTER . 1 . 8 +
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Combine two 1/3 fraction blocks: 1/3 + 1/3 = |
2/3 |
. 1 . 3 ENTER . 1 . 3 + |
| 2 |
Subtract 1/4 from 1/2: 1/2 − 1/4 = |
1/4 |
. 1 . 2 ENTER . 1 . 4 - |
| 3 |
Find the height of three 1/6 blocks: 1/6 + 1/6 + 1/6 = |
1/2 |
. 1 . 6 ENTER . 1 . 6 + . 1 . 6 + |
| 4 |
Subtract 1/8 from 1/4: 1/4 − 1/8 = |
1/8 |
. 1 . 4 ENTER . 1 . 8 - |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
1/2
.1.4 ENTER .1.4 + |
1/4
.1.2 ENTER .1.4 - |
42 mm
84 ENTER 2 ÷ |
1/2
1 ENTER .1.2 - |
1/2
.1.6 ENTER .1.6 + .1.6 + |
4
.1.2 ENTER .1.8 ÷ |
| #2 |
3/4
.1.4 ENTER .2.4 + |
1/6
.1.3 ENTER .1.6 - |
21 mm
84 ENTER 4 ÷ |
3/4
1 ENTER .1.4 - |
3/8
.1.8 ENTER .1.8 + .1.8 + |
4
.2.3 ENTER .1.6 ÷ |
| #3 |
5/6
.1.6 ENTER .4.6 + |
1/8
.1.4 ENTER .1.8 - |
14 mm
84 ENTER 6 ÷ |
1/4
1 ENTER .3.4 - |
1/2
.1.8 ENTER .1.8 + .2.8 + |
9
.3.4 ENTER .1.12 ÷ |
| #4 |
1/3
.1.6 ENTER .1.6 + |
1/12
.1.6 ENTER .1.12 - |
7 mm
84 ENTER 12 ÷ |
5/6
1 ENTER .1.6 - |
1/4
.1.12 ENTER .1.12 + .1.12 + |
5/8
.7.8 ENTER .1.4 - |
| #5 |
1/4
.1.8 ENTER .1.8 + |
1/6
.1.2 ENTER .1.3 - |
10.5 mm
84 ENTER 8 ÷ |
1/6
1 ENTER .5.6 - |
1/3
.1.12 ENTER .1.12 + .2.12 + |
1/2
.5.6 ENTER .1.3 - |
| #6 |
5/8
.3.8 ENTER .2.8 + |
3/8
.1.2 ENTER .1.8 - |
56 mm
28 ENTER 2 × |
5/8
1 ENTER .3.8 - |
1
.1.2 ENTER .1.4 + .1.4 + |
2/3
.11.12 ENTER .1.4 - |
| #7 |
1/2
.3.4 ENTER .1.4 - |
1/4
.1.3 ENTER .1.12 - |
70 mm
14 ENTER 5 × |
1/8
1 ENTER .7.8 - |
2/3
.1.3 ENTER .1.6 + .1.6 + |
5/8
.7.8 ENTER .1.4 - |
| #8 |
2/3
.5.6 ENTER .1.6 - |
1/6
.1.4 ENTER .1.12 - |
3/4
.1.4 ENTER .2.4 + |
7/12
1 ENTER .5.12 - |
1/2
.1.4 ENTER .1.8 + .1.8 + |
1/3
.5.6 ENTER .1.2 - |
| #9 |
1/2
.7.8 ENTER .3.8 - |
5/12
.1.2 ENTER .1.12 - |
1/2
.7.8 ENTER .3.8 - |
1
.1.4 ENTER .1.4 + .1.4 + .1.4 + |
5/8
.3.4 ENTER .1.8 + .1.4 - |
14 mm
42 ENTER 28 - |
| #10 |
1/2
.11.12 ENTER .5.12 - |
1/2
.2.3 ENTER .1.6 - |
1/8
.1.4 ENTER .1.8 - |
1/6
1 ENTER .2.6 - .3.6 - |
2/3
.5.6 ENTER .1.3 - .1.6 + |
1
.1.2 ENTER .1.6 + .1.3 + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Add unit fractions and simplify: 1/4 + 1/4 = |
10 pts |
1/2 |
.1.4 ENTER .1.4 + |
| 2 |
Core Fluency |
Subtract like fractions and simplify: 5/6 − 1/6 = |
10 pts |
2/3 |
.5.6 ENTER .1.6 - |
| 3 |
Core Fluency |
Find the comparator difference: 1/3 − 1/4 = |
10 pts |
1/12 |
.1.3 ENTER .1.4 - |
| 4 |
Core Fluency |
Find the comparator difference: 1/2 − 1/6 = |
10 pts |
1/3 |
.1.2 ENTER .1.6 - |
| 5 |
Core Fluency |
Find the complement to 1 Whole: 1 − 3/8 = |
10 pts |
5/8 |
1 ENTER .3.8 - |
| 6 |
Core Fluency |
Evaluate chained sum: 1/6 + 1/6 + 1/6 = |
10 pts |
1/2 |
.1.6 ENTER .1.6 + .1.6 + |
| 7 |
Application |
If 1 Whole is 84 mm tall, how tall is a 1/3 block in mm? |
10 pts |
28 mm |
84 ENTER 3 ÷ |
| 8 |
Application |
How many 1/8 blocks are needed to match 3/4? |
10 pts |
6 |
.3.4 ENTER .1.8 ÷ |
| 9 |
Application |
A carpenter glues a 3/8 in strip to a 1/8 in strip. Total thickness? |
10 pts |
1/2 in |
.3.8 ENTER .1.8 + |
| 10 |
Error Analysis |
Error Analysis: Noah added 1/4 + 1/4 = 2/8. Simplify 1/4 + 1/4 correctly: |
10 pts |
1/2 |
.1.4 ENTER .1.4 + |
5. STEM Lab Case Study: Architectural Glass Window Panes
Scenario: An architect proportions structural glass windows using unit fraction ratios.
- Case Question 1: Clerestory vs. Transom Pane: An upper clerestory pane is 1/3 of the frame height and a transom pane is 1/4. Compare your 84 mm towers to find the fraction difference:
- Expected Result: 7 mm = 1/12 | Keystrokes:
. 1 . 3 ENTER . 1 . 4 -
- Case Question 2: Modular Louver Stack: Each glass louver vent is 1/6 of a meter tall. How many 1/6 louver blocks stack up in the comparator well to fill a 1/2-meter window opening?
- Expected Result: 3 sixths (14×3 = 42 mm = 1/2). | Keystrokes:
. 1 . 2 ENTER . 1 . 6 ÷
- Case Question 3: Compound Mullion Frame: The architect stacks a 1/4-meter tinted pane above a 1/8-meter acoustic sill. What total fraction of a meter is the combined assembly?
- Expected Result: 3/8 (31.5 mm). | Keystrokes:
. 1 . 4 ENTER . 1 . 8 +
- Case Question 4: Error Analysis: A junior drafter claims a 1/8-meter glass strip is taller than a 1/4-meter strip because 8 > 4. Use the 84 mm Fraction Comparator to explain why this is false:
- Expected Result: Larger denominator means smaller pieces: 1/4 (21 mm) > 1/8 (10.5 mm). | Keystrokes:
. 1 . 4 ENTER . 1 . 8 -
Unit 04: Multi-Digit Multiplication & Partial Products
Grades: Grade 4 | Instrument: StackCalc SC-6 | Visual / Block Model: Partial-Product Area Grid Diagram & Place Value Studio
Key Standards: CCSS.MATH.4.NBT.B.5
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Use partial products to find 38 × 7 on paper, then check on the stack.
- Verified Answer:
266 ((30×7) + (8×7) = 210 + 56 = 266).
- Keystrokes:
38 ENTER 7 ×
- Warm-Up B (Extension Lab): Multiply 43 × 26 using an area grid model on paper, then check on the stack.
- Verified Answer:
1118 ((40×20)+(40×6)+(3×20)+(3×6) = 800+240+60+18 = 1118).
- Keystrokes:
43 ENTER 26 ×
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Multiply using partial products: 52 × 8 = |
416 |
52 ENTER 8 × |
| 2 |
Multiply two 2-digit numbers: 28 × 14 = |
392 |
28 ENTER 14 × |
| 3 |
Multiply using area decomposition: 64 × 7 = |
448 |
64 ENTER 7 × |
| 4 |
Multiply two 2-digit numbers: 45 × 32 = |
1440 |
45 ENTER 32 × |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
92
23 ENTER 4 × |
130
20 ENTER 5 × 6 ENTER 5 × + |
372
124 ENTER 3 × |
288
24 ENTER 12 × |
325
200 ENTER 60 + 50 + 15 + |
420
16 ENTER 12 + 10 ENTER 5 + × |
| #2 |
170
34 ENTER 5 × |
301
40 ENTER 7 × 3 ENTER 7 × + |
870
145 ENTER 6 × |
392
28 ENTER 14 × |
748
600 ENTER 60 + 80 + 8 + |
600
40 ENTER 15 - 20 ENTER 4 + × |
| #3 |
138
46 ENTER 3 × |
348
50 ENTER 6 × 8 ENTER 6 × + |
872
218 ENTER 4 × |
560
35 ENTER 16 × |
690
400 ENTER 200 + 60 + 30 + |
720
35 ENTER 24 × 12 ENTER 10 × - |
| #4 |
162
27 ENTER 6 × |
268
60 ENTER 4 × 7 ENTER 4 × + |
1530
306 ENTER 5 × |
966
42 ENTER 23 × |
70
27 ENTER 16 × 200 - 120 - 42 - |
1000
18 ENTER 25 × 22 ENTER 25 × + |
| #5 |
416
52 ENTER 8 × |
592
70 ENTER 8 × 4 ENTER 8 × + |
1080 panels
8 ENTER 135 × |
1440
45 ENTER 32 × |
384
24 ENTER 16 × |
900
12 ENTER 15 × 5 × |
| #6 |
448
64 ENTER 7 × |
765
80 ENTER 9 × 5 ENTER 9 × + |
1840 miles
460 ENTER 4 × |
1350
54 ENTER 25 × |
980
28 ENTER 35 × |
1600
25 ENTER 16 × 4 × |
| #7 |
450
75 ENTER 6 × |
558
90 ENTER 6 × 3 ENTER 6 × + |
1715 lb
7 ENTER 245 × |
1116
62 ENTER 18 × |
864
36 ENTER 24 × |
15
360 ENTER 24 ÷ |
| #8 |
432
48 ENTER 9 × |
174
30 ENTER 6 × 1 ENTER 6 × - |
1440 pencils
9 ENTER 160 × |
888
37 ENTER 24 × |
720
25 ENTER 18 × 15 ENTER 18 × + |
25
450 ENTER 18 ÷ |
| #9 |
581
83 ENTER 7 × |
392
50 ENTER 8 × 1 ENTER 8 × - |
670
100 ENTER 5 × 30 ENTER 5 × + 4 ENTER 5 × + |
1904
56 ENTER 34 × |
440
45 ENTER 24 × 32 ENTER 20 × - |
625
25 ENTER 25 × |
| #10 |
768
96 ENTER 8 × |
693
100 ENTER 7 × 1 ENTER 7 × - |
832
208 ENTER 4 × |
2100
75 ENTER 28 × |
170
32 ENTER 15 × 310 - |
924
34 ENTER 18 × 26 ENTER 12 × + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Multiply 2-digit by 1-digit: 47 × 6 = |
10 pts |
282 |
47 ENTER 6 × |
| 2 |
Core Fluency |
Multiply 2-digit by 1-digit: 68 × 7 = |
10 pts |
476 |
68 ENTER 7 × |
| 3 |
Core Fluency |
Multiply 3-digit by 1-digit: 135 × 4 = |
10 pts |
540 |
135 ENTER 4 × |
| 4 |
Core Fluency |
Evaluate partial products: (40 × 8) + (6 × 8) = |
10 pts |
368 |
40 ENTER 8 × 6 ENTER 8 × + |
| 5 |
Core Fluency |
Multiply 2-digit by 2-digit: 32 × 15 = |
10 pts |
480 |
32 ENTER 15 × |
| 6 |
Core Fluency |
Multiply 2-digit by 2-digit: 45 × 24 = |
10 pts |
1080 |
45 ENTER 24 × |
| 7 |
Application |
A battery array has 26 modules with 15 cells each. Total cells? |
10 pts |
390 cells |
26 ENTER 15 × |
| 8 |
Application |
Find the area of a 38 m by 22 m field minus a 10 m by 8 m shed: |
10 pts |
756 |
38 ENTER 22 × 10 ENTER 8 × - |
| 9 |
Application |
Find the missing factor: 16 × [ ? ] = 400 |
10 pts |
25 |
400 ENTER 16 ÷ |
| 10 |
Error Analysis |
Error Analysis: Mia wrote 24 × 13 = (20×10) + (4×3) = 212. Find the true product 24 × 13: |
10 pts |
312 |
24 ENTER 13 × |
5. STEM Lab Case Study: Solar Battery Storage Bank
Scenario: Electrical engineers calculate total energy cells across multi-tier battery racks.
- Case Question 1: A residential battery rack holds 24 rows of 6 lithium-iron cells. Decompose 24 into 20 + 4 on your Place Value Tray to find total cells:
- Expected Result: 144 cells | Keystrokes:
24 ENTER 6 ×
- Case Question 2: A commercial microgrid bank has 32 modules with 15 supercapacitors each. Use a 2×2 area model to find the total capacitors:
- Expected Result: 480 capacitors | Keystrokes:
32 ENTER 15 ×
- Case Question 3: Backup Rack Expansion: 7 inverter trays each hold 43 battery cells. Split 43 into 40 + 3 partial products to find the total cells:
- Expected Result: 301 (280 + 21 = 301). | Keystrokes:
43 ENTER 7 ×
- Case Question 4: Error Analysis: To size a 25-row by 14-column solar array, an apprentice computed (20×10) + (5×4) = 220, forgetting the two cross-rectangles. Find the true cell count:
- Expected Result: True product is 350 ((20×14) + (5×14) = 280 + 70 = 350). | Keystrokes:
25 ENTER 14 ×
Unit 05: Long Division, Quotients & Remainders
Grades: Grade 4 | Instrument: StackCalc SC-6 | Visual / Block Model: Place Value Studio (6×2 Whole Number Tray)
Key Standards: CCSS.MATH.4.NBT.B.6
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Divide 95 ÷ 6 on paper. State quotient and remainder, then verify on stack.
- Verified Answer:
15 R 5 (6 × 15 + 5 = 95).
- Keystrokes:
95 ENTER 6 ÷R
- Warm-Up B (Extension Lab): If 134 ÷ 5 = 26 R 4, verify by evaluating 26 × 5 + 4 on the stack.
- Verified Answer:
134 (Verified).
- Keystrokes:
26 ENTER 5 × 4 +
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Divide 76 ÷ 3 on paper: state quotient and remainder = |
25 R 1 |
76 ENTER 3 ÷R |
| 2 |
Verify division on stack: 25 × 3 + 1 = |
76 |
25 ENTER 3 × 1 + |
| 3 |
Divide 98 ÷ 4 on paper: state quotient and remainder = |
24 R 2 |
98 ENTER 4 ÷R |
| 4 |
Divide 125 ÷ 6 on paper: state quotient and remainder = |
20 R 5 |
125 ENTER 6 ÷R |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
34
68 ENTER 2 ÷ |
25 R 1
76 ENTER 3 ÷R |
76
25 ENTER 3 × 1 + |
102
408 ENTER 4 ÷ |
28 crates
168 ENTER 6 ÷ |
88
84 ENTER 92 + 88 + 3 ÷ |
| #2 |
24
96 ENTER 4 ÷ |
21 R 3
87 ENTER 4 ÷R |
94
18 ENTER 5 × 4 + |
205
615 ENTER 3 ÷ |
30 chairs
270 ENTER 9 ÷ |
50
42 ENTER 48 + 54 + 56 + 4 ÷ |
| #3 |
17
85 ENTER 5 ÷ |
18 R 4
94 ENTER 5 ÷R |
95
15 ENTER 6 × 5 + |
204
816 ENTER 4 ÷ |
26 R 4
134 ENTER 5 ÷R |
12
480 ENTER 8 ÷ 5 ÷ |
| #4 |
13
91 ENTER 7 ÷ |
15 R 5
95 ENTER 6 ÷R |
98
24 ENTER 4 × 2 + |
107
535 ENTER 5 ÷ |
26 R 7
215 ENTER 8 ÷R |
12
648 ENTER 9 ÷ 6 ÷ |
| #5 |
18
108 ENTER 6 ÷ |
24 R 2
98 ENTER 4 ÷R |
115
16 ENTER 7 × 3 + |
107
642 ENTER 6 ÷ |
27 boxes
324 ENTER 12 ÷ |
40
24 ENTER 15 × 9 ÷ |
| #6 |
18
144 ENTER 8 ÷ |
16 R 3
115 ENTER 7 ÷R |
125
20 ENTER 6 × 5 + |
104
728 ENTER 7 ÷ |
30 drums
450 ENTER 15 ÷ |
63
36 ENTER 14 × 8 ÷ |
| #7 |
18
162 ENTER 9 ÷ |
20 R 5
125 ENTER 6 ÷R |
146
24 ENTER 6 × 2 + |
120
480 ENTER 4 ÷ |
33 tiles
528 ENTER 16 ÷ |
10
288 ENTER 8 ÷ 156 ENTER 6 ÷ - |
| #8 |
45
225 ENTER 5 ÷ |
24 R 2
146 ENTER 6 ÷R |
173
21 ENTER 8 × 5 + |
150
750 ENTER 5 ÷ |
12 boxes
100 ENTER 4 - 8 ÷ |
70
315 ENTER 7 ÷ 225 ENTER 9 ÷ + |
| #9 |
48
336 ENTER 7 ÷ |
21 R 5
173 ENTER 8 ÷R |
230
32 ENTER 7 × 6 + |
26
120 ENTER 6 ÷ 36 ENTER 6 ÷ + |
32 trucks
140 ENTER 116 + 8 ÷ |
313
34 ENTER 9 × 7 + |
| #10 |
54
432 ENTER 8 ÷ |
24 R 3
219 ENTER 9 ÷R |
6
150 ENTER 18 ENTER 8 × - |
36
210 ENTER 7 ÷ 42 ENTER 7 ÷ + |
90
104 ENTER 14 - |
0
816 ENTER 4 ÷ 204 - |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Evaluate exact quotient: 156 ÷ 6 = |
10 pts |
26 |
156 ENTER 6 ÷ |
| 2 |
Core Fluency |
Evaluate exact quotient with zero in tens: 816 ÷ 4 = |
10 pts |
204 |
816 ENTER 4 ÷ |
| 3 |
Core Fluency |
Evaluate 2-digit divisor quotient: 360 ÷ 15 = |
10 pts |
24 |
360 ENTER 15 ÷ |
| 4 |
Core Fluency |
Find quotient and remainder: 87 ÷ 4 = |
10 pts |
21 R 3 |
87 ENTER 4 ÷R |
| 5 |
Core Fluency |
Find quotient and remainder: 146 ÷ 6 = |
10 pts |
24 R 2 |
146 ENTER 6 ÷R |
| 6 |
Core Fluency |
Verify division on stack: (26 × 5) + 4 = |
10 pts |
134 |
26 ENTER 5 × 4 + |
| 7 |
Application |
252 books are packed into 7 equal cartons. How many books per carton? |
10 pts |
36 books |
252 ENTER 7 ÷ |
| 8 |
Application |
95 containers are stacked 6 high. Find full stacks and leftover containers: |
10 pts |
15 R 5 |
95 ENTER 6 ÷R |
| 9 |
Application |
Find the average of 3 container loads: 115, 125, and 135: |
10 pts |
125 |
115 ENTER 125 + 135 + 3 ÷ |
| 10 |
Error Analysis |
Error Analysis: A student says 98 ÷ 4 = 23 R 6. Find the true quotient and remainder: |
10 pts |
24 R 2 |
98 ENTER 4 ÷R |
5. STEM Lab Case Study: Cargo Ship Container Loading
Scenario: Port logistics engineers pack intermodal shipping containers into uniform vessel bays and verify remainders.
- Case Question 1: 87 intermodal containers are loaded into vessel bays of 4 containers each. Find the full bays and remaining containers with [÷R]:
- Expected Result: 21 R 3 | Keystrokes:
87 ENTER 4 ÷R
- Case Question 2: Manifest Audit: Rebuild the total container count on the stack from 21 full bays of 4 plus 3 deck containers (21 × 4 + 3):
- Expected Result: 87 containers | Keystrokes:
21 ENTER 4 × 3 +
- Case Question 3: Bulk Cement Hold: 146 export pallets are crane-lifted in slings of 6 pallets per lift. Find full lifts and leftover pallets:
- Expected Result: 24 R 2 | Keystrokes:
146 ENTER 6 ÷R
- Case Question 4: Error Analysis: Distributing 816 tons of ballast equally across 4 hull tanks (816 ÷ 4), a clerk wrote 24 tons per tank. Find the true load:
- Expected Result: 204 (tens place has 0 tens). | Keystrokes:
816 ENTER 4 ÷
Unit 06: Fraction Equivalence & Common Denominators
Grades: Grade 4 | Instrument: StackCalc SC-6 | Visual / Block Model: Fraction Studio (Comparator Well & Proportional Towers)
Key Standards: CCSS.MATH.4.NF.A.1, CCSS.MATH.4.NF.A.2
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Test 3/6 and 4/8 in the comparator well. What single simplified fraction do both equal?
- Verified Answer:
1/2 (42.0 mm).
- Keystrokes:
. 3 . 6
- Warm-Up B (Extension Lab): Find the common denominator and sum of 1/2 + 1/3 on paper, then check on the stack.
- Verified Answer:
5/6 (3/6 + 2/6 = 5/6).
- Keystrokes:
. 1 . 2 ENTER . 1 . 3 +
- Hands-On Block Focus (Fraction Multiplication & Division):
- Multiplication: Scaling one tower by another (e.g., 2/3 of 3/8 = 1/4 matching the 21 mm block).
- Division (Split & Rebuild 1 Full Tower): To divide
1/2 ÷ 3/4, split the 1/2 block (42 mm) into 3 equal 1/6 blocks (÷ 3 -> 14 mm is 1/4 of the mystery tower), then stack 4 of those 1/6 blocks (× 4 -> 56 mm = 2/3) to prove why ÷(3/4) = ×(4/3)!
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Find equivalent fraction: 4/8 simplifies to = |
1/2 |
. 4 . 8 |
| 2 |
Add unlike fractions: 1/3 + 1/6 = |
1/2 |
. 1 . 3 ENTER . 1 . 6 + |
| 3 |
Add unlike fractions: 1/4 + 3/8 = |
5/8 |
. 1 . 4 ENTER . 3 . 8 + |
| 4 |
Find common denominator and sum: 2/5 + 1/10 = |
1/2 |
. 2 . 5 ENTER . 1 . 1 0 + |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
1/2
.2.4 ENTER |
3/4
.1.2 ENTER .1.4 + |
5/6
.1.2 ENTER .1.3 + |
6
.1.2 ENTER 12 × |
3/4 cup
.1.2 ENTER .1.4 + |
1
.1.2 ENTER .1.3 + .1.6 + |
| #2 |
1/2
.3.6 ENTER |
1/2
.1.3 ENTER .1.6 + |
7/12
.1.3 ENTER .1.4 + |
4
.1.3 ENTER 12 × |
5/6 mile
.1.3 ENTER .1.2 + |
7/8
.1.2 ENTER .1.4 + .1.8 + |
| #3 |
1/2
.4.8 ENTER |
3/8
.1.4 ENTER .1.8 + |
11/12
.2.3 ENTER .1.4 + |
8
.2.3 ENTER 12 × |
5/8 inch
.3.8 ENTER .1.4 + |
3/4
.1.3 ENTER .1.4 + .1.6 + |
| #4 |
1/3
.2.6 ENTER |
5/8
.1.4 ENTER .3.8 + |
1 1/12
.1.3 ENTER .3.4 + |
3
.1.4 ENTER 12 × |
1/3 liter
.5.6 ENTER .1.2 - |
1/2
.1.4 ENTER .1.6 + .1.12 + |
| #5 |
1/3
.4.12 ENTER |
7/8
.1.2 ENTER .3.8 + |
9/10
.1.2 ENTER .2.5 + |
9
.3.4 ENTER 12 × |
5/8 inch
.7.8 ENTER .1.4 - |
3/4
.2.3 ENTER .1.4 + .1.6 - |
| #6 |
3/4
.6.8 ENTER |
5/6
.2.3 ENTER .1.6 + |
1/6
.1.2 ENTER .1.3 - |
10
.5.6 ENTER 12 × |
5/6 mile
.2.3 ENTER .1.6 + |
3/8
.3.4 ENTER .1.8 + .1.2 - |
| #7 |
3/4
.9.12 ENTER |
3/4
.1.3 ENTER .5.12 + |
1/12
.1.3 ENTER .1.4 - |
6
.3.4 ENTER 8 × |
7/12
.1.3 ENTER .1.4 + |
5/12
.3.4 ENTER .1.3 - |
| #8 |
2/3
.4.6 ENTER |
1/2
.2.5 ENTER .1.10 + |
5/12
.3.4 ENTER .1.3 - |
6
.3.5 ENTER 10 × |
5/12
1 ENTER .7.12 - |
7/12
.5.6 ENTER .1.4 - |
| #9 |
2/3
.8.12 ENTER |
3/8
.3.4 ENTER .3.8 - |
5/12
.2.3 ENTER .1.4 - |
49 mm
84 ENTER .7.12 × |
1/24 m
.2.3 ENTER .5.8 - |
5/12
1 ENTER .1.3 ENTER .1.4 + - |
| #10 |
5/6
.10.12 ENTER |
1/2
.5.6 ENTER .1.3 - |
3/10
.4.5 ENTER .1.2 - |
70 mm
84 ENTER .5.6 × |
7/12
.1.3 ENTER .1.4 + |
1
.2.3 ENTER .1.4 + .1.12 + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Simplify to lowest terms: 6/8 = |
10 pts |
3/4 |
.6.8 ENTER |
| 2 |
Core Fluency |
Simplify to lowest terms: 8/12 = |
10 pts |
2/3 |
.8.12 ENTER |
| 3 |
Core Fluency |
Add related fractions: 1/3 + 1/6 = |
10 pts |
1/2 |
.1.3 ENTER .1.6 + |
| 4 |
Core Fluency |
Add related fractions: 1/4 + 3/8 = |
10 pts |
5/8 |
.1.4 ENTER .3.8 + |
| 5 |
Core Fluency |
Add unlike fractions: 1/2 + 1/3 = |
10 pts |
5/6 |
.1.2 ENTER .1.3 + |
| 6 |
Core Fluency |
Add unlike fractions: 1/3 + 1/4 = |
10 pts |
7/12 |
.1.3 ENTER .1.4 + |
| 7 |
Application |
Subtract unlike fractions: 3/4 − 1/3 = |
10 pts |
5/12 |
.3.4 ENTER .1.3 - |
| 8 |
Application |
How many twelfths are equivalent to 2/3? |
10 pts |
8 |
.2.3 ENTER 12 × |
| 9 |
Application |
A stained-glass border combines 1/2 m, 1/4 m, and 1/8 m strips. Total length? |
10 pts |
7/8 m |
.1.2 ENTER .1.4 + .1.8 + |
| 10 |
Error Analysis |
Error Analysis: A student adds 1/2 + 1/3 = 2/5. Find the true sum: |
10 pts |
5/6 |
.1.2 ENTER .1.3 + |
5. STEM Lab Case Study: Stained Glass Window Assembly
Scenario: Architectural glass artisans cut proportional strips and match lead-came joints using common denominators.
- Case Question 1: Cathedral Medallion Check: Two 1/4-meter ruby glass panels are soldered side-by-side. Compare them in the 84 mm well to simplify 2/4:
- Expected Result: 1/2 (42.0 mm). | Keystrokes:
.2.4 ENTER
- Case Question 2: Lead-Came Border Joint: An artisan joins a 1/3-meter cobalt strip and a 1/4-meter amber strip. Trade for twelfths in the comparator well to find total length:
- Expected Result: 7/12 (4/12 + 3/12 = 7/12). | Keystrokes:
.1.3 ENTER .1.4 +
- Case Question 3: Rose Window Arch: Combine a 1/2-meter central panel with a 1/6-meter border arch and simplify to lowest terms:
- Expected Result: 2/3 (3/6 + 1/6 = 4/6 = 2/3). | Keystrokes:
.1.2 ENTER .1.6 +
- Case Question 4: Error Analysis: An apprentice glazier added 1/2 + 1/3 = 2/5 m of glass and cut the frame too short! Find the true combined length:
- Expected Result: 5/6 (3/6 + 2/6 = 5/6, not 2/5). | Keystrokes:
.1.2 ENTER .1.3 +
Unit 07: Prime Factorization, GCF/LCM & Factor Trees
Grades: Grade 4 – 5 | Instrument: StackCalc SC-6 | Visual / Block Model: Prime Factor Studio (Logarithmic Prime Blocks 2, 3, 5, 7)
Key Standards: CCSS.MATH.4.OA.B.4, CCSS.MATH.6.NS.B.4
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Find the prime factorization of 18 and verify the product on your stack.
- Verified Answer:
18 (2 × 3 × 3 = 18).
- Keystrokes:
2 ENTER 3 × 3 ×
- Warm-Up B (Extension Lab): Given GCF(8, 12) = 4, find LCM(8, 12) on the stack.
- Verified Answer:
24 ((8 × 12) ÷ 4 = 24).
- Keystrokes:
8 ENTER 12 × 4 ÷
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Multiply prime factorization of 20: 2 × 2 × 5 = |
20 |
2 ENTER 2 × 5 × |
| 2 |
Multiply prime factorization of 30: 2 × 3 × 5 = |
30 |
2 ENTER 3 × 5 × |
| 3 |
Find the Greatest Common Factor (GCF) of 12 and 18: |
6 |
2 ENTER 3 × |
| 4 |
Find the Least Common Multiple (LCM) of 6 and 8 given GCF(6, 8) = 2: |
24 |
6 ENTER 8 × 2 ÷ |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
32
2 ENTER 5 yˣ |
30
2 ENTER 3 × 5 × |
12 bouquets
24 ENTER 36 GCF |
8
64 √x |
96
8 ENTER 12 × |
3
36 ENTER 12 ÷ |
| #2 |
81
3 ENTER 4 yˣ |
84
2 x² 3 × 7 × |
48 teeth
12 ENTER 16 LCM |
12
144 √x |
150
10 ENTER 15 × |
5
225 ENTER 45 ÷ |
| #3 |
2³ × 3
3 ENTER 2 |
40
2 x³ 5 × |
24 (3 & 4 packs)
8 ENTER 6 LCM |
3
27 ENTER 3 ˣ√𝑦 |
12
6 ENTER 36 × 18 ÷ |
2
100 ENTER 50 ÷ |
| #4 |
2² × 3 × 5
3 ENTER 2 |
45
3 x² 5 × |
14 teeth
42 ENTER 14 GCF |
10
6 x² 8 x² + √x |
12
4 ENTER 60 × 20 ÷ |
12
36 ENTER 48 GCF |
| #5 |
29 (Prime)
29 ENTER 1 ÷ |
50
2 ENTER 5 x² × |
51 years
17 ENTER 3 LCM |
8
10 x² 6 x² - √x |
28
2 ENTER 3 + 5 + 7 + 11 + |
80
16 ENTER 20 LCM |
| #6 |
8
16 ENTER 24 GCF |
72
2 x³ 3 x² × |
221 years
13 ENTER 17 × |
6
3 ENTER 2 × |
24
11 ENTER 13 + |
10
5 ENTER 30 × 15 ÷ |
| #7 |
15
30 ENTER 45 GCF |
7
42 ENTER 6 ÷ |
42
2 ENTER 3 × 7 × |
9
3 ENTER 3 × |
36
17 ENTER 19 + |
60
2 ENTER 5 × 6 × |
| #8 |
24
6 ENTER 8 LCM |
7
28 ENTER 4 ÷ |
8 factors
24 ENTER 3 ÷ |
10
2 ENTER 3 + 5 + |
47
50 ENTER 3 - |
36
1 x³ 2 x³ + 3 x³ + |
| #9 |
60
12 ENTER 15 LCM |
5
75 ENTER 15 ÷ |
18 counts
6 ENTER 9 LCM |
6
12 ENTER 18 GCF 30 GCF |
17
30 ENTER 13 - |
60
6 ENTER 10 × |
| #10 |
36
12 ENTER 18 LCM |
15
135 ENTER 9 ÷ |
64
4 ENTER 4 × 4 × |
24
4 ENTER 6 LCM 8 LCM |
8
2 x³ |
15
9 x² 12 x² + √x |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Write the prime factorization of 45: |
10 pts |
3² × 5 |
5 ENTER 3 |
| 2 |
Core Fluency |
Find GCF(24, 36) using shared prime factors: |
10 pts |
12 |
24 ENTER 36 GCF |
| 3 |
Core Fluency |
Find LCM(8, 10) using union of prime factors: |
10 pts |
40 |
8 ENTER 10 LCM |
| 4 |
Core Fluency |
Stack the Prime Factor blocks: 2² × 3 × 5 = |
10 pts |
60 |
2 x² 3 × 5 × |
| 5 |
Core Fluency |
How many positive factors does 18 = 2¹ × 3² have? |
10 pts |
6 |
2 ENTER 3 × |
| 6 |
Core Fluency |
Given GCF(a, 12) = 4 and LCM(a, 12) = 24, find a: |
10 pts |
8 |
4 ENTER 24 × 12 ÷ |
| 7 |
Application |
Bus A arrives every 10 min and Bus B every 15 min. Both arrive at 9:00 AM. Minutes until next overlap (LCM)? |
10 pts |
30 min (9:30 AM) |
10 ENTER 15 LCM |
| 8 |
Application |
32 sixth-graders and 40 seventh-graders form equal-sized trivia teams with no mixed grades. Max team size? |
10 pts |
8 students |
32 ENTER 40 GCF |
| 9 |
Application |
17-Year Prime Cicadas emerge every 17 years; a predator cycles every 2 years. Years until both overlap? |
10 pts |
34 years |
17 ENTER 2 LCM |
| 10 |
Error Analysis |
Error Analysis: Sam claims 51 is prime. Divide 51 ÷ 3 on your stack to find its hidden prime factor! |
10 pts |
17 (51 = 3 × 17) |
51 ENTER 3 ÷ |
5. STEM Lab Case Study: Gear Train Planetary Mechanism
Scenario: Mechanical engineers design clockwork gear trains using prime tooth counts (2, 3, 5, 7) so wear distributes evenly across every rotation cycle.
- Case Question 1: Pinion Tooth Decomposition: A 12-tooth brass pinion gear is machined for a clockwork train. Stack your Prime Factor blocks to build 12:
- Expected Result: 2 × 2 × 3 = 12 | Keystrokes:
2 ENTER 2 × 3 ×
- Case Question 2: Shared Gear Pitch (GCF): An 18-tooth drive gear meshes with a 24-tooth idler gear. Find the largest shared tooth grouping GCF(18, 24):
- Expected Result: 6 (2 × 3 = 6). | Keystrokes:
18 ENTER 24 GCF
- Case Question 3: Full Mesh Alignment Cycle (LCM): A 12-tooth gear drives an 18-tooth gear. After how many tooth-meshes do the exact same two marked teeth meet again?
- Expected Result: 36 | Keystrokes:
12 ENTER 18 LCM
- Case Question 4: Error Analysis: An apprentice claims a 9-tooth cog has a prime number of teeth so it cannot share factors. Divide 9 ÷ 3 to prove 9 is composite:
- Expected Result: 3 (9 = 3 × 3, composite). | Keystrokes:
9 ENTER 3 ÷
Unit 08: The 4-Register Stack: Four Operations & Automatic Lift
Grades: Grade 4 – 5 | Instrument: StackCalc SC-6 | Visual / Block Model: Expression Tree Diagram & 4-Register Stack
Key Standards: CCSS.MATH.5.OA.A.1, CCSS.MATH.5.OA.A.2
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): A baker bakes 15 cookies, sells 6 of them, and packs the remaining cookies equally into 3 bags. How many cookies go in each bag?
- Verified Answer:
3 cookies per bag ((15 - 6) ÷ 3 = 3).
- Keystrokes:
1 5 ENTER 6 - 3 ÷
- Warm-Up B (Extension Lab): Evaluate (24 ÷ 6) + 15 by solving the division group first and then adding 15 to that result.
- Verified Answer:
19 (24 ÷ 6 = 4, then 4 + 15 = 19).
- Keystrokes:
2 4 ENTER 6 ÷ 1 5 +
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Solve (14 - 5) × 3 by finding the difference inside the parentheses first, then multiplying by 3. |
27 |
1 4 ENTER 5 - 3 × |
| 2 |
Solve 30 ÷ (2 + 3) by adding the divisor group (2 + 3) first, then dividing 30 by that sum. |
6 |
2 ENTER 3 + 3 0 x⇄y ÷ |
| 3 |
Evaluate (9 + 7) ÷ (8 - 4) by finding the numerator sum and denominator difference first, then dividing. |
4 |
9 ENTER 7 + 8 ENTER 4 - ÷ |
| 4 |
A farmer collects 20 eggs, sets aside 2 cracked eggs, and divides the rest equally among 6 cartons. How many per carton? |
3 eggs |
2 0 ENTER 2 - 6 ÷ |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
32
12 ENTER 5 ENTER 4 × + |
6
15 ENTER 9 + 4 ÷ |
58 books
65 ENTER 18 + 25 - |
16
2 ENTER 15 ENTER 4 ENTER 3 + - × |
120 cm³
4 ENTER 5 × 6 × |
20
50 ENTER 2 ENTER 3 x² 6 + × - |
| #2 |
60
18 ENTER 6 ENTER 7 × + |
45
24 ENTER 9 - 3 × |
80 people
3 ENTER 24 × 8 + |
20
4 ENTER 20 ENTER 3 ENTER 5 × - × |
96 cm²
6 ENTER 4 x² × |
24
4 ENTER 18 ENTER 10 ENTER 4 - 2 × - × |
| #3 |
16
40 ENTER 3 ENTER 8 × - |
27
45 ENTER 5 ÷ 18 + |
$38
15 ENTER 4 × 22 - |
4
36 ENTER 12 - 2 ENTER 4 + ÷ |
150 cm²
6 ENTER 5 x² × |
2
5 x² 3 x² + 4 x² 1 + ÷ |
| #4 |
18
60 ENTER 7 ENTER 6 × - |
34
10 ENTER 6 ENTER 4 × + |
9 boxes
84 ENTER 12 - 8 ÷ |
10
7 ENTER 8 × 6 - 2 ENTER 3 + ÷ |
26
2 ENTER 3 × 2 ENTER 4 × + 3 ENTER 4 × + |
4
6 x² 2 x² - 2 x³ ÷ |
| #5 |
24
15 ENTER 36 ENTER 4 ÷ + |
15
50 ENTER 7 ENTER 5 × - |
$51
3 ENTER 9 × 4 ENTER 6 × + |
4
40 ENTER 24 + 5 x² 9 - ÷ |
52
2 ENTER 3 × 2 ENTER 4 × + 3 ENTER 4 × + 2 × |
6
7 ENTER 8 + 90 x⇄y ÷ |
| #6 |
42
50 ENTER 48 ENTER 6 ÷ - |
21
12 ENTER 36 ENTER 4 ÷ + |
46 m
100 ENTER 3 ENTER 18 × - |
6
14 ENTER 16 + 2 x³ 3 - ÷ |
94
3 ENTER 4 × 3 ENTER 5 × + 4 ENTER 5 × + 2 × |
46
6 ENTER 9 × 100 x⇄y - |
| #7 |
8
14 ENTER 10 + 3 ÷ |
39
4 ENTER 6 × 3 ENTER 5 × + |
7 teams, R 1
29 ENTER 4 ÷R |
30
4 ENTER 5 × 6 ENTER 3 × + 8 - |
91 cm³
4 x³ 3 x³ + |
100
12 ENTER 4 × 13 ENTER 4 × + |
| #8 |
80
35 ENTER 15 - 4 × |
32
8 ENTER 7 × 6 ENTER 4 × - |
36 m
45 ENTER 18 + 27 - |
12
20 ENTER 4 ÷ 7 + |
14 faces
3 ENTER 6 × 4 - |
90
25 ENTER 6 × 15 ENTER 4 × - |
| #9 |
46
10 ENTER 3 x² 4 × + |
60
7 ENTER 5 + 11 ENTER 6 - × |
150 seats
5 ENTER 12 × 6 ENTER 15 × + |
30
6 ENTER 4 + 5 ENTER 2 - × |
20
12 x² 8 x² - 12 ENTER 8 - ÷ |
88
82 ENTER 86 + 90 + 94 + 4 ÷ |
| #10 |
9
5 x² 2 ENTER 8 × - |
6
28 ENTER 14 + 13 ENTER 6 - ÷ |
26
6 ENTER 4 ENTER 5 × + |
32
50 ENTER 3 ENTER 6 × - |
68
2 ENTER 3 × 4 ENTER 5 × + 6 ENTER 7 × + |
5
45 ENTER 15 + 18 ENTER 6 - ÷ |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Multiply before adding: 14 + 5 × 6 = |
10 pts |
44 |
14 ENTER 5 ENTER 6 × + |
| 2 |
Core Fluency |
Divide & power order: 24 - 16 ÷ 4 + 2³ = |
10 pts |
28 |
24 ENTER 16 ENTER 4 ÷ - 2 x³ + |
| 3 |
Core Fluency |
Inside-out parentheses: (16 + 8) ÷ 3 = |
10 pts |
8 |
16 ENTER 8 + 3 ÷ |
| 4 |
Core Fluency |
Two-branch expression tree: (9 - 4) × (6 + 2) = |
10 pts |
40 |
9 ENTER 4 - 6 ENTER 2 + × |
| 5 |
Core Fluency |
Two-branch expression tree: (12 + 8) × (15 - 9) = |
10 pts |
120 |
12 ENTER 8 + 15 ENTER 9 - × |
| 6 |
Core Fluency |
Horizontal fraction bar: (5² + 11) ÷ (4 + 2) = |
10 pts |
6 |
5 x² 11 + 4 ENTER 2 + ÷ |
| 7 |
Application |
A crate holds 6 rows of 8 apples, plus 12 pears. How many pieces of fruit in all? |
10 pts |
60 fruit |
6 ENTER 8 × 12 + |
| 8 |
Application |
Noah has $50. He buys 4 notebooks at $7 each. How much change does he receive? |
10 pts |
$22 |
50 ENTER 4 ENTER 7 × - |
| 9 |
Application |
A maker lab orders 4 kits at $18 each and 3 sensors at $9 each. Total cost? |
10 pts |
$99 |
4 ENTER 18 × 3 ENTER 9 × + |
| 10 |
Error Analysis |
Error Analysis: Why does (9 + 3) ÷ (6 - 2) need NO parentheses on a 4-register RPN stack? |
10 pts |
Y holds 12 while X computes 4 (3) |
9 ENTER 3 + 6 ENTER 2 - ÷ |
5. STEM Lab Case Study: Space Elevator Cargo Transfer
Scenario: Astrodynamicists balance orbital payload pods on a 4-Register Stack Stage without parentheses.
- Case Question 1: Pod A carries 15 tons and transfers 6 tons to the station; the remaining cargo is split across 3 shuttles: (15 − 6) ÷ 3 =
- Expected Result: 3 | Keystrokes:
15 ENTER 6 - 3 ÷
- Case Question 2: Two thruster banks fire together: (8 + 2) pods each burning (9 − 4) kg of fuel. Evaluate (8 + 2) × (9 − 4):
- Expected Result: 50 | Keystrokes:
8 ENTER 2 + 9 ENTER 4 - ×
- Case Question 3: From a 100-ton counterweight reserve, subtract 4 racks of 5 tons plus a 10-ton tether clamp: 100 − (4 × 5 + 10) =
- Expected Result: 70 | Keystrokes:
4 ENTER 5 × 10 + 100 x⇄y -
- Case Question 4: Error Analysis: Student executes left-to-right on 10 + 2 × 5 and gets 60.
- Expected Result: Multiplication comes first: 2 × 5 = 10; 10 + 10 = 20. | Keystrokes:
2 ENTER 5 × 10 +
Unit 09: Adding & Subtracting Fractions & Mixed Numbers
Grades: Grade 5 | Instrument: StackCalc SC-6 | Visual / Block Model: Fraction Studio (84 mm Proportional Towers)
Key Standards: CCSS.MATH.5.NF.A.1, CCSS.MATH.5.NF.A.2
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Add 1 3/4 + 2 1/8 on paper and verify on your calculator.
- Verified Answer:
3 7/8 (1 6/8 + 2 1/8 = 3 7/8).
- Keystrokes:
1 . 3 . 4 ENTER 2 . 1 . 8 +
- Warm-Up B (Extension Lab): Solve 4 1/3 − 2 1/2 on paper with regrouping and check on the stack.
- Verified Answer:
1 5/6 (3 8/6 − 2 3/6 = 1 5/6).
- Keystrokes:
4 . 1 . 3 ENTER 2 . 1 . 2 -
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Add mixed numbers: 1 1/2 + 2 1/4 = |
3 3/4 |
1 . 1 . 2 ENTER 2 . 1 . 4 + |
| 2 |
Subtract mixed numbers: 3 1/2 − 1 1/4 = |
2 1/4 |
3 . 1 . 2 ENTER 1 . 1 . 4 - |
| 3 |
Add mixed numbers with unlike denominators: 2 1/3 + 1 1/6 = |
3 1/2 |
2 . 1 . 3 ENTER 1 . 1 . 6 + |
| 4 |
Subtract with regrouping: 4 1/4 − 1 3/4 = |
2 1/2 |
4 . 1 . 4 ENTER 1 . 3 . 4 - |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
3 2/3
1.1.3 ENTER 2.1.3 + |
3 5/6
2.1.3 ENTER 1.1.2 + |
2 1/2
3.3.4 ENTER 1.1.4 - |
1 3/4
3 ENTER 1.1.4 - |
3 3/4 cups
2.1.4 ENTER 1.1.2 + |
5
1.1.2 ENTER 2.1.4 + 1.1.4 + |
| #2 |
3 1/2
1.1.4 ENTER 2.1.4 + |
3 7/12
1.1.3 ENTER 2.1.4 + |
2 2/3
4.5.6 ENTER 2.1.6 - |
2 1/3
5 ENTER 2.2.3 - |
5 5/6 miles
3.1.3 ENTER 2.1.2 + |
5
2.1.3 ENTER 1.1.2 + 1.1.6 + |
| #3 |
3 2/3
2.1.6 ENTER 1.3.6 + |
4 11/12
3.2.3 ENTER 1.1.4 + |
2 1/4
3.1.2 ENTER 1.1.4 - |
1 1/2
3.1.4 ENTER 1.3.4 - |
3 1/4 ft
6 ENTER 2.3.4 - |
2 5/8
3.1.2 ENTER 1.1.4 + 2.1.8 - |
| #4 |
3 3/4
1.1.2 ENTER 2.1.4 + |
4
1.1.2 ENTER 2.1.2 + |
3 1/2
4.5.6 ENTER 1.1.3 - |
2 2/3
4.1.3 ENTER 1.2.3 - |
6 5/8 lb
4.1.4 ENTER 2.3.8 + |
2 11/12
2.1.2 ENTER 1.3.4 + 1.1.3 - |
| #5 |
3 1/2
2.1.3 ENTER 1.1.6 + |
4 1/4
1.3.4 ENTER 2.1.2 + |
3 5/8
5.7.8 ENTER 2.1.4 - |
1 3/4
3.1.4 ENTER 1.1.2 - |
2 5/6 liters
4.1.3 ENTER 1.1.2 - |
6 1/12
5.1.4 ENTER 1.1.2 - 2.1.3 + |
| #6 |
4 5/8
3.1.4 ENTER 1.3.8 + |
4 1/6
2.2.3 ENTER 1.1.2 + |
2 1/6
3.1.2 ENTER 1.1.3 - |
1 5/6
4.1.3 ENTER 2.1.2 - |
4 3/8 in
1.3.4 ENTER 2.5.8 + |
2 1/2
3.1.2 ENTER 1.1.4 + 1.1.2 ENTER .3.4 + - |
| #7 |
3 7/8
1.3.4 ENTER 2.1.8 + |
4 5/12
2.2.3 ENTER 1.3.4 + |
3 5/12
4.2.3 ENTER 1.1.4 - |
2 2/3
5.1.6 ENTER 2.1.2 - |
11 1/2 ft
3.1.2 ENTER 3.1.2 + 2.1.4 + 2.1.4 + |
2 5/6
5.1.6 ENTER 2.1.3 - |
| #8 |
5 2/3
2.1.2 ENTER 3.1.6 + |
5 1/6
3.5.6 ENTER 1.1.3 + |
3 5/12
5.3.4 ENTER 2.1.3 - |
2 7/12
4.1.4 ENTER 1.2.3 - |
1 7/12 ft
5.1.4 ENTER 3.2.3 - |
2 7/12
4.1.3 ENTER 1.3.4 - |
| #9 |
5 2/3
4.1.4 ENTER 1.5.12 + |
4 3/8
2.5.8 ENTER 1.3.4 + |
3 1/3
6.5.6 ENTER 3.1.2 - |
3 5/8
6.1.8 ENTER 2.1.2 - |
4 1/6 yd
8 ENTER 2.1.3 - 1.1.2 - |
3 3/4
6.1.4 ENTER 2.1.2 - |
| #10 |
3 1/2
2.3.10 ENTER 1.1.5 + |
7 7/12
4.3.4 ENTER 2.5.6 + |
3 3/10
4.4.5 ENTER 1.1.2 - |
2 7/12
5.1.3 ENTER 2.3.4 - |
1 3/4
3.1.4 ENTER 1.1.2 - |
5
4.1.3 ENTER 1.1.2 - 2.1.6 + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Add mixed numbers: 1 1/2 + 2 1/4 = |
10 pts |
3 3/4 |
1.1.2 ENTER 2.1.4 + |
| 2 |
Core Fluency |
Add unlike mixed numbers: 2 1/3 + 1 1/2 = |
10 pts |
3 5/6 |
2.1.3 ENTER 1.1.2 + |
| 3 |
Core Fluency |
Add with fraction carry: 1 3/4 + 2 1/2 = |
10 pts |
4 1/4 |
1.3.4 ENTER 2.1.2 + |
| 4 |
Core Fluency |
Subtract mixed numbers: 4 5/6 − 1 1/3 = |
10 pts |
3 1/2 |
4.5.6 ENTER 1.1.3 - |
| 5 |
Core Fluency |
Subtract from a whole number: 4 − 1 3/8 = |
10 pts |
2 5/8 |
4 ENTER 1.3.8 - |
| 6 |
Core Fluency |
Subtract with regrouping: 3 1/4 − 1 1/2 = |
10 pts |
1 3/4 |
3.1.4 ENTER 1.1.2 - |
| 7 |
Application |
Subtract with regrouping: 4 1/3 − 2 1/2 = |
10 pts |
1 5/6 |
4.1.3 ENTER 2.1.2 - |
| 8 |
Application |
A baker combines 2 2/3 cups oats and 1 3/4 cups flour. Total dry ingredients? |
10 pts |
4 5/12 cups |
2.2.3 ENTER 1.3.4 + |
| 9 |
Application |
Evaluate multi-step pipeline: 3 1/2 + 1 1/3 − 2 1/6 = |
10 pts |
2 2/3 |
3.1.2 ENTER 1.1.3 + 2.1.6 - |
| 10 |
Error Analysis |
Error Analysis: Mia wrote 4 1/4 − 1 3/4 = 3 2/4. Find the true difference: |
10 pts |
2 1/2 |
4.1.4 ENTER 1.3.4 - |
5. STEM Lab Case Study: The Chef's Artisan Bread Recipe
Scenario: A master baker combines sourdough flours across multiple dough troughs.
- Case Question 1: Multi-Grain Trough Blend: The baker blends 2 1/3 cups of dark rye flour with 1 1/2 cups of whole wheat flour. Find the combined flour volume:
- Expected Result: 3 5/6 cups | Keystrokes:
2 . 1 . 3 ENTER 1 . 1 . 2 +
- Case Question 2: Levain Starter Portioning: A proofing tub holds 3 1/4 cups of sourdough starter. After scooping out 1 1/2 cups for baguettes, how much starter remains?
- Expected Result: 1 3/4 cups | Keystrokes:
3 . 1 . 4 ENTER 1 . 1 . 2 -
- Case Question 3: Brioche Hydration Mix: Combine 1 3/8 cups of warm buttermilk and 2 1/4 cups of whole milk in the mixer bowl. What is the total liquid volume?
- Expected Result: 3 5/8 cups | Keystrokes:
1 . 3 . 8 ENTER 2 . 1 . 4 +
- Case Question 4: Error Analysis: Subtracting 1 1/2 cups of flour from 3 1/4 cups, an apprentice subtracted 1/2 − 1/4 backward and wrote 2 1/4 cups. Trade 1 Whole for 4/4 to fix it:
- Expected Result: 1 3/4 (must unbundle 1 whole to make 2 5/4 − 1 2/4 = 1 3/4). | Keystrokes:
3 . 1 . 4 ENTER 1 . 1 . 2 -
Unit 10: Fraction Studio: Proportions, Multiplication & Tower Division
Grades: Grade 5 | Instrument: StackCalc SC-6 | Visual / Block Model: Fraction Studio (84 mm Proportional Towers)
Key Standards: CCSS.MATH.5.NF.B.4, CCSS.MATH.5.NF.B.7
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Find 2/3 of 3/4 by multiplying (2/3) × (3/4) and simplifying to lowest terms.
- Verified Answer:
1/2 (6/12 = 1/2, 42.0 mm).
- Keystrokes:
.2.3 ENTER .3.4 ×
- Warm-Up B (Extension Lab): Use Split & Rebuild to evaluate (3/4) ÷ (3/8) on paper and check on your SC-6.
- Verified Answer:
2 ((3/4) × (8/3) = 2 wholes).
- Keystrokes:
.3.4 ENTER .3.8 ÷
- Hands-On Block Focus (Fraction Multiplication & Division):
- Multiplication: Scaling one tower by another (e.g., 2/3 of 3/8 = 1/4 matching the 21 mm block).
- Division (Split & Rebuild 1 Full Tower): To divide
1/2 ÷ 3/4, split the 1/2 block (42 mm) into 3 equal 1/6 blocks (÷ 3 -> 14 mm is 1/4 of the mystery tower), then stack 4 of those 1/6 blocks (× 4 -> 56 mm = 2/3) to prove why ÷(3/4) = ×(4/3)!
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Multiply fraction scaling: 1/2 × 3/4 = |
3/8 |
.1.2 ENTER .3.4 × |
| 2 |
Multiply fractions and simplify: 2/3 × 3/8 = |
1/4 |
.2.3 ENTER .3.8 × |
| 3 |
Divide by a whole number: 1/2 ÷ 3 = |
1/6 |
.1.2 ENTER 3 ÷ |
| 4 |
Divide by a fraction (Split & Rebuild): 1/2 ÷ 3/4 = |
2/3 |
.1.2 ENTER .3.4 ÷ |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
3/5
3 ENTER .1.5 × |
3/8
.1.2 ENTER .3.4 × |
24 cm
.1.2 ENTER 48 × |
16
2 ENTER 3 ÷ 24 × |
7
.1.3 ENTER .1.4 + 12 × |
63 mm
.3.4 ENTER 84 × |
| #2 |
5/8
5 ENTER .1.8 × |
2/5
.2.3 ENTER .3.5 × |
27 cm
.3.4 ENTER 36 × |
27
3 ENTER 4 ÷ 36 × |
8
.1.2 ENTER .1.6 + 12 × |
70 mm
.5.6 ENTER 84 × |
| #3 |
2 2/3
4 ENTER .2.3 × |
1/2
.3.4 ENTER .2.3 × |
1/4
.1.2 ENTER .1.2 × |
28
4 ENTER 5 ÷ 35 × |
5
.3.4 ENTER .1.3 - 12 × |
1/6
1 ENTER .1.2 - .1.3 - |
| #4 |
4 1/2
6 ENTER .3.4 × |
1/2
.4.5 ENTER .5.8 × |
8 glasses
6 ENTER .3.4 ÷ |
25
5 ENTER 6 ÷ 30 × |
1/4
.1.2 ENTER 2 × 3 ÷ 3 × 4 ÷ |
1/8
1 ENTER .1.2 - .1.4 - .1.8 - |
| #5 |
12
.2.3 ENTER 18 × |
1/3
.3.4 ENTER .4.9 × |
9 students
.3.8 ENTER 24 × |
25%
1 ENTER 4 ÷ 100 × |
2/5
.2.3 ENTER 3 × 4 ÷ 4 × 5 ÷ |
1
.1.2 ENTER .1.3 + .1.6 + |
| #6 |
21
.3.4 ENTER 28 × |
12
3 ENTER .1.4 ÷ |
6 loaves
4.1.2 ENTER .3.4 ÷ |
60%
3 ENTER 5 ÷ 100 × |
10
2.1.2 ENTER 4 × |
4
1.3.4 ENTER 2.1.4 + |
| #7 |
35
.5.6 ENTER 42 × |
12
4 ENTER .1.3 ÷ |
7.5 m
1.25 ENTER 6 × |
70%
7 ENTER 10 ÷ 100 × |
3
4.1.2 ENTER 1.1.2 ÷ |
1/2
.3.4 ENTER .2.3 × |
| #8 |
27
.3.5 ENTER 45 × |
1/6
.1.3 ENTER 2 ÷ |
3/4 (63mm > 56mm)
.1.4 ENTER 3 × |
20
25 ENTER 100 ÷ 80 × |
1
1 ENTER .1.2 ENTER .1.2 + ÷ |
3
3.3.4 ENTER 1.1.4 ÷ |
| #9 |
28
.7.8 ENTER 32 × |
2
.3.4 ENTER .3.8 ÷ |
[.] [5] [.] [8] (5/8)
.5.8 ENTER |
15
10 ENTER 100 ÷ 150 × |
2
1 ENTER .1.3 ENTER .1.6 + ÷ |
15 oz
4.5 ENTER 3 ÷ 10 × |
| #10 |
16
.4.9 ENTER 36 × |
2
.5.6 ENTER .5.12 ÷ |
2
.1.2 ENTER .1.4 ÷ |
60 mph
180 ENTER 3 ÷ |
1.25 (1 1/4)
.3.4 ENTER 0.5 + |
2
.5.6 ENTER .1.3 - .1.4 ÷ |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Find the fractional part of a bar model: 3/4 of 24 = |
10 pts |
18 |
.3.4 ENTER 24 × |
| 2 |
Core Fluency |
Multiply fraction by fraction: 2/3 × 3/4 = |
10 pts |
1/2 |
.2.3 ENTER .3.4 × |
| 3 |
Core Fluency |
Divide a whole number by a unit fraction: 5 ÷ 1/3 = |
10 pts |
15 |
5 ENTER .1.3 ÷ |
| 4 |
Core Fluency |
Multiply fractions and simplify: 3/5 × 10/9 = |
10 pts |
2/3 |
.3.5 ENTER 10 × 9 ÷ |
| 5 |
Core Fluency |
Tower Division in the 84 mm Well: 3/4 ÷ 3/8 = |
10 pts |
2 |
.3.4 ENTER .3.8 ÷ |
| 6 |
Core Fluency |
Convert 5/8 to an exact decimal: |
10 pts |
0.625 |
5 ENTER 8 ÷ |
| 7 |
Application |
A ribbon is 4 1/2 yards long. How many 3/4-yard pieces can be cut from it? |
10 pts |
6 pieces |
4.1.2 ENTER .3.4 ÷ |
| 8 |
Application |
Pythagoras's Perfect Fifth: Stop a 30 cm harp string at 2/3 of its length. Vibrating length? |
10 pts |
20 cm |
.2.3 ENTER 30 × |
| 9 |
Application |
In the 84 mm Fraction Studio, each 1/12 block is 7 mm tall. How tall is 5/12? |
10 pts |
35 mm |
7 ENTER 5 × |
| 10 |
Error Analysis |
Error Analysis: Why is 1/2 ÷ 3/4 equal to 2/3 instead of 3/8? |
10 pts |
Dividing by 3/4 scales by 4/3 (2/3) |
.1.2 ENTER .3.4 ÷ |
5. STEM Lab Case Study: Suspension Bridge Tower Cable Geometry
Scenario: Civil engineers calculate tower tension ratios using fraction multiplication and division.
- Case Question 1: Secondary Suspender Scaling: A main bridge cable carries 3/4 of a kilonewton unit load, and a secondary suspender carries 1/2 of that load (1/2 × 3/4). Find the suspender load:
- Expected Result: 3/8 meter (31.5 mm). | Keystrokes:
. 1 . 2 ENTER . 3 . 4 ×
- Case Question 2: Three-Strand Load Sharing: A 1/2-meter anchor rod is split equally among 3 steel tie-down brackets (1/2 ÷ 3). What fraction of a meter does each bracket hold?
- Expected Result: 1/6 (14.0 mm). | Keystrokes:
. 1 . 2 ENTER 3 ÷
- Case Question 3: Full Tower Reconstruction: A 1/2-meter scale test cable represents 3/4 of the design span. Use Split & Rebuild (1/2 ÷ 3/4) to find the full 1-tower design span:
- Expected Result: 2/3 (56.0 mm tower). | Keystrokes:
. 1 . 2 ENTER . 3 . 4 ÷
- Case Question 4: Error Analysis: A structural intern computed 1/2 ÷ 3/4 = 3/8 by multiplying instead of dividing. Use the 84 mm Split & Rebuild model to explain why 1/2 ÷ 3/4 = 2/3:
- Expected Result: Dividing by 3/4 scales up by 4/3: 1/2 × 4/3 = 2/3 (56 mm). | Keystrokes:
. 1 . 2 ENTER . 3 . 4 ÷
Appendix C: Cumulative Benchmark & Capstone Mastery Exams (Pages 117–120)
Cumulative Exam 1: Units 01–05 Benchmark (p. 117 • 100 Points)
Covers Area-Model Multiplication, Division, 84 mm Unit Fraction Towers, Multi-Digit Place Value Trays, and Long Division.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Area Model Multiplication: Decompose 16 × 7 into Tens and Ones partial products on your grid and find 16 × 7: |
10 pts |
112 |
16 ENTER 7 × |
| 2 |
Core Fluency |
Equal-Sharing Division: Divide 96 blocks equally into 8 columns on the comparator tray. How many per column? |
10 pts |
12 |
96 ENTER 8 ÷ |
| 3 |
Core Fluency |
Fraction Tower Height: One whole tower is 84 mm tall. How tall is a 3/4 tower in millimeters? |
10 pts |
63 |
84 ENTER 3 × 4 ÷ |
| 4 |
Core Fluency |
Two-Dot Unit Fraction Sum: Stack three 1/8 blocks and two 1/8 blocks in the Fraction Studio comparator well: |
10 pts |
5/8 |
. 3 . 8 ENTER . 2 . 8 + |
| 5 |
Core Fluency |
Multi-Digit Place Value Addition: Combine 468 + 375 using Gold [+1] Carry Tokens across Tens and Hundreds: |
10 pts |
843 |
468 ENTER 375 + |
| 6 |
Core Fluency |
Zero-Borrowing Vault: Unbundle across the zero Tens column on the Place Value Tray to solve 704 − 258 = ? |
10 pts |
446 |
704 ENTER 258 - |
| 7 |
Application |
Two-Digit Area Product: Multiply 24 × 15 using a 4-quadrant area model and verify on the stack: |
10 pts |
360 |
24 ENTER 15 × |
| 8 |
Application |
Long Division Quotient: Divide 378 ÷ 14 on paper using partial quotients and check on the SC-6: |
10 pts |
27 |
378 ENTER 14 ÷ |
| 9 |
Application |
Multi-Step Crate Packing: 6 crates hold 24 books each, plus 18 extra books on the cart. How many books in all? |
10 pts |
162 |
6 ENTER 24 × 18 + |
| 10 |
Error Analysis |
Error Spotter: Maya solved 500 − 163 and wrote 463 without unbundling Hundreds. What is 500 − 163? |
10 pts |
337 |
500 ENTER 163 - |
Cumulative Exam 2: Units 06–10 Benchmark (p. 118 • 100 Points)
Covers Fraction Equivalence Towers, Logarithmic Prime Factor Blocks, GCF/LCM, 4-Register PEMDAS, and Fraction Multiplication & Division.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Common Denominator Tower Duel: Add 1/3 + 1/4 using 1/12 towers in the Fraction Studio comparator: |
10 pts |
7/12 |
. 1 . 3 ENTER . 1 . 4 + |
| 2 |
Core Fluency |
Unlike Fraction Difference: Compare a 3/4 tower and a 1/6 tower in the comparator well: 3/4 − 1/6 = ? |
10 pts |
7/12 |
. 3 . 4 ENTER . 1 . 6 - |
| 3 |
Core Fluency |
Prime Factor Tower Product: Stack two [2] blocks, one [3] block, and one [5] block in the Prime Factor Studio. What composite number is formed? |
10 pts |
60 |
2 ENTER 2 × 3 × 5 × |
| 4 |
Core Fluency |
Prime Factor GCF Overlap: Find the Greatest Common Factor of 24 and 36 from their shared prime blocks: |
10 pts |
12 |
2 ENTER 2 × 3 × |
| 5 |
Core Fluency |
Prime Factor LCM Union: Find the Least Common Multiple of 12 and 18 using their prime factor towers: |
10 pts |
36 |
2 ENTER 2 × 3 × 3 × |
| 6 |
Core Fluency |
4-Register Two-Branch PEMDAS: Evaluate (18 + 6) ÷ (9 − 5) from the inside out without parentheses: |
10 pts |
6 |
18 ENTER 6 + 9 ENTER 5 - ÷ |
| 7 |
Application |
Mixed Number Addition: Combine 1 1/2 and 2 1/4 using Fraction Studio towers and two-dot entry: |
10 pts |
15/4 |
1.1.2 ENTER 2.1.4 + |
| 8 |
Application |
Fraction Tower Scaling (Multiplication): Find 2/3 of a 3/4 tower: (2/3) × (3/4) = ? |
10 pts |
1/2 |
. 2 . 3 ENTER . 3 . 4 × |
| 9 |
Application |
Split & Rebuild Fraction Division: Divide a 1/2 tower by 3/4 using Split & Rebuild: (1/2) ÷ (3/4) = ? |
10 pts |
2/3 |
. 1 . 2 ENTER . 3 . 4 ÷ |
| 10 |
Error Analysis |
Error Spotter: Ethan evaluated 8 + 4 × 5 left-to-right as 60 instead of multiplying 4 × 5 first. Find 8 + (4 × 5): |
10 pts |
28 |
4 ENTER 5 × 8 + |
Cumulative Exam 3: Multi-Step Synthesis (p. 119 • 100 Points)
Synthesizes Fraction Tower Operations, Prime Factor Trees, Place Value Decimal Shifts, Volume, and Nested Order of Operations.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Two-Branch Product: Evaluate (14 + 11) × (19 − 11) on the 4-level stack: |
10 pts |
200 |
14 ENTER 11 + 19 ENTER 11 - × |
| 2 |
Core Fluency |
Four-Fraction Flush Tower: Add 1/2 + 1/4 + 1/6 + 1/12 on the Fraction Studio lid: |
10 pts |
1 |
. 1 . 2 ENTER . 1 . 4 + . 1 . 6 + . 1 . 12 + |
| 3 |
Core Fluency |
Fraction of a Whole Quantity: Find 3/8 of 64 miles: |
10 pts |
24 |
64 ENTER 8 ÷ 3 × |
| 4 |
Core Fluency |
Fraction Division by a Unit Fraction: How many 1/8 blocks fit inside a 3/4 tower? |
10 pts |
6 |
. 3 . 4 ENTER . 1 . 8 ÷ |
| 5 |
Core Fluency |
Decimal Place Value Shift: Multiply 3.45 × 100 by sliding digits two columns left on the Place Value Tray: |
10 pts |
345 |
3.45 ENTER 100 × |
| 6 |
Core Fluency |
Rectangular Prism Volume: A 3D block storage bin measures 8 cm × 5 cm × 6 cm. Find its volume V = l × w × h: |
10 pts |
240 |
8 ENTER 5 × 6 × |
| 7 |
Application |
Nested Order of Operations: Evaluate 120 ÷ (6 + 3 × 2) from the inside out: |
10 pts |
10 |
120 ENTER 3 ENTER 2 × 6 + ÷ |
| 8 |
Application |
Prime Tower Quotient: Remove the prime factor blocks of 12 from the prime factor tower of 84. What factor block remains? |
10 pts |
7 |
84 ENTER 12 ÷ |
| 9 |
Application |
Combined Recipe Scaling: A baker triples a 2/3-cup flour scoop and adds 1/2 cup. Find the total flour in cups: |
10 pts |
5/2 |
3 ENTER . 2 . 3 × . 1 . 2 + |
| 10 |
Error Analysis |
Error Spotter: To solve (2/3) ÷ (4/5), a student multiplied (2/3) × (4/5) = 8/15 instead of splitting ÷4 and rebuilding ×5. Find (2/3) ÷ (4/5): |
10 pts |
5/6 |
. 2 . 3 ENTER . 4 . 5 ÷ |
Cumulative Exam 4: Final Capstone Mastery (p. 120 • 100 Points)
Final Grades 3–5 mastery assessment uniting the Place Value Studio, Fraction Studio, and Prime Factor Studio.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Place Value & Decimal Sum: Combine 14.75 + 8.50 − 3.25 on the docked Whole & Decimal Place Value Trays: |
10 pts |
20 |
14.75 ENTER 8.5 + 3.25 - |
| 2 |
Core Fluency |
Three-Tower Flush Sum: Stack 1/2 + 1/3 + 1/6 in the Fraction Studio comparator well: |
10 pts |
1 |
. 1 . 2 ENTER . 1 . 3 + . 1 . 6 + |
| 3 |
Core Fluency |
Split & Rebuild Mastery: Divide a 3/4 tower by 3/8 using Split & Rebuild: (3/4) ÷ (3/8) = ? |
10 pts |
2 |
. 3 . 4 ENTER . 3 . 8 ÷ |
| 4 |
Core Fluency |
Fundamental Theorem Check: Compute 2³ × 3 × 5 by chaining prime factors on the stack: |
10 pts |
120 |
2 ENTER 2 × 2 × 3 × 5 × |
| 5 |
Core Fluency |
GCF × LCM Symmetry Law: Verify that GCF(12, 18) × LCM(12, 18) equals 12 × 18: |
10 pts |
216 |
12 ENTER 18 × |
| 6 |
Core Fluency |
Composite Surface Area: Find the total area of two 9 × 4 walls and two 6 × 4 walls: |
10 pts |
120 |
2 ENTER 9 × 4 × 2 ENTER 6 × 4 × + |
| 7 |
Application |
Average of Four Lab Trials: Find the mean of {18, 22, 26, 30} by summing and dividing by 4: |
10 pts |
24 |
18 ENTER 22 + 26 + 30 + 4 ÷ |
| 8 |
Application |
Multi-Step Fraction Scaling: Evaluate (3/4 + 1/2) × (4/5) on the SC-6 stack: |
10 pts |
1 |
. 3 . 4 ENTER . 1 . 2 + . 4 . 5 × |
| 9 |
Application |
Engineering Beam Cut: From a 10-foot beam, saw off three pieces of length 2 1/2 feet. How much beam remains? |
10 pts |
5/2 |
10 ENTER 3 ENTER . 5 . 2 × - |
| 10 |
Error Analysis |
Capstone Error Analysis: A student computed (36 − 12) ÷ (4 + 2) as 36 − (12 ÷ 4) + 2 = 35. Find the true value of (36 − 12) ÷ (4 + 2): |
10 pts |
4 |
36 ENTER 12 - 4 ENTER 2 + ÷ |
📙 Book 3: Middle School Math Lab (Grades 6–8 • SC-6)
Full 45-Section Two-Method Student Workbook (900 problems + 10 Unit Mastery Quizzes).
Complete section-by-section answers, calculator keystroke traces, and 100-point quiz solutions for Book 3 are published in the dedicated Middle School Portal:
👉 Middle School Math Lab Complete Solutions Directory
📕 Book 4: High School & STEM Lab (Grades 9–12 & AP • SC-32)
Covers Units 01 through 10 plus 4 Cumulative Benchmark & Capstone Mastery Exams: Linear Functions & Slope, Factoring Quadratics, Quadratic Formula & [POLY], Higher-Degree Polynomials & Horner's Method, Compound Growth & Exponents, Logarithms & Base e, Right-Triangle Trig, 2D Polar Vectors, Combinatorics & Statistics, and Binary/Hex Digital Logic (840 active problems).
Unit 01: Linear Equations, Rate of Change & Slope Pipelines
Grades: Grade 9 – 10 (Algebra 1) | Instrument: StackCalc SC-32 | Visual / Block Model: Coordinate Slope Triangle Diagram & 4-Level Stack
Key Standards: CCSS.MATH.HSA.CED.A.2, CCSS.MATH.HSF.IF.B.6
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Find the slope passing through (1, 4) and (5, 16) on paper, then check on SC-32.
- Verified Answer:
3 (12 ÷ 4 = 3).
- Keystrokes:
16 ENTER 4 - 5 ENTER 1 - ÷
- Warm-Up B (Extension Lab): Given line with m = 3 through (2, 4), find y when x = 8.
- Verified Answer:
22 (3(8−2) + 4 = 18 + 4 = 22).
- Keystrokes:
8 ENTER 2 - 3 × 4 +
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Find slope between (3, 5) and (7, 17): m = |
3 |
17 ENTER 5 - 7 ENTER 3 - ÷ |
| 2 |
Find slope between (-2, 8) and (4, -4): m = |
-2 |
-4 ENTER 8 - 4 ENTER -2 - ÷ |
| 3 |
Line has m = 4 through (1, 3). Find y when x = 6: y = |
23 |
6 ENTER 1 - 4 × 3 + |
| 4 |
Find the x-intercept of 3x + 6y = 24: x = |
8 |
24 ENTER 3 ÷ |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
2
10 ENTER 2 - 5 ENTER 1 - ÷ |
13
4 ENTER 5 × 7 - |
7%
350 ENTER 5000 ÷ 100 × |
10
40 ENTER 4 ÷ |
-0.5
-1 ENTER 2 ÷ |
6
15 ENTER 3 - 4 ENTER 2 - ÷ |
| #2 |
3
17 ENTER 5 - 6 ENTER 2 - ÷ |
8
-3 ENTER 4 × 20 + |
0.0833
15 ENTER 180 ÷ |
8
40 ENTER 5 ÷ |
0.2
-1 ENTER -5 ÷ |
6
19 ENTER -5 - 7 ENTER 3 - ÷ |
| #3 |
4
24 ENTER 4 - 8 ENTER 3 - ÷ |
26
2.5 ENTER 8 × 6 + |
60 mph
210 ENTER 30 - 4 ENTER 1 - ÷ |
8
48 ENTER 6 ÷ |
-2.5
-1 ENTER 0.4 ÷ |
4
32 ENTER 12 - 3 ENTER -2 - ÷ |
| #4 |
-2
5 ENTER 15 - 6 ENTER 1 - ÷ |
17
9 ENTER 3 - 2 × 5 + |
-30 L/min
320 ENTER 500 - 6 ÷ |
-6
48 ENTER -8 ÷ |
8
2 ENTER 14 + 2 ÷ |
24
3 ENTER 4 × 12 + |
| #5 |
3
13 ENTER -2 - 2 ENTER -3 - ÷ |
23
6 ENTER 1 - 4 × 3 + |
-30 ft/s
60 ENTER 420 - 12 ÷ |
9
36 ENTER 9 - 3 ÷ |
7
5 ENTER 9 + 2 ÷ |
100
1200 ENTER 20 ENTER 8 - ÷ |
| #6 |
-2
-3 ENTER 9 - 2 ENTER -4 - ÷ |
4
7 ENTER 4 - -2 × 10 + |
$245
65 ENTER 45 ENTER 4 × + |
11
43 ENTER 12 + 5 ÷ |
2
-6 ENTER 10 + 2 ÷ |
120
2400 ENTER 35 ENTER 15 - ÷ |
| #7 |
5
15 ENTER -5 - 4 ENTER 0 - ÷ |
17
3 ENTER 2 + 5 × 8 - |
-6°C
20 ENTER 6.5 ENTER 4 × - |
-5
-20 ENTER 15 - 7 ÷ |
-2
3 ENTER -7 + 2 ÷ |
30
60 ENTER 10 - 10 ÷ 4 × 10 + |
| #8 |
0.5
11 ENTER 7 - 10 ENTER 2 - ÷ |
-3
4 ENTER 1 + -3 × 12 + |
$240
120 ENTER 0.08 ENTER 1500 × + |
4
2 ENTER 18 - -4 ÷ |
16
6 ENTER 2 - 3 × 4 + |
24
0.5 ENTER 8 × 6 × |
| #9 |
0
6 ENTER 6 - 3 ENTER -5 - ÷ |
16
10 ENTER 2 - 1.5 × 4 + |
2%/min
95 ENTER 15 - 40 ÷ |
6
20 ENTER 4 + 6 ENTER 2 - ÷ |
2
5 ENTER 1 - -2 × 10 + |
8
6 x² 2 x² - 6 ENTER 2 - ÷ |
| #10 |
-3
-15 ENTER -3 - 8 ENTER 4 - ÷ |
4
19 ENTER 3 ENTER 5 × - |
3
14 ENTER 2 - 6 ENTER 2 - ÷ |
7
40 ENTER 5 - 9 ENTER 4 - ÷ |
8
25 ENTER 5 - 4 ÷ 3 + |
4
25 ENTER 5 - 8 ENTER 3 - ÷ |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Find slope between (2, 5) and (6, 21): m = |
10 pts |
4 |
21 ENTER 5 - 6 ENTER 2 - ÷ |
| 2 |
Core Fluency |
Find slope between (-3, 10) and (5, -6): m = |
10 pts |
-2 |
-6 ENTER 10 - 5 ENTER -3 - ÷ |
| 3 |
Core Fluency |
Evaluate point-slope y = 3(x − 4) + 7 at x = 10: y = |
10 pts |
25 |
10 ENTER 4 - 3 × 7 + |
| 4 |
Core Fluency |
Find x-intercept of 5x + 4y = 60 (set y = 0): x = |
10 pts |
12 |
60 ENTER 5 ÷ |
| 5 |
Core Fluency |
Solve 6x − 14 = 34 on the stack: x = |
10 pts |
8 |
34 ENTER 14 + 6 ÷ |
| 6 |
Core Fluency |
Find the perpendicular slope to m = 4: m_perp = |
10 pts |
-0.25 |
-1 ENTER 4 ÷ |
| 7 |
Application |
A road rises 180 m over a 3,000 m horizontal run. Find percent grade: |
10 pts |
6% |
180 ENTER 3000 ÷ 100 × |
| 8 |
Application |
Find intersection x of y = 5x + 4 and y = 2x + 19: x = |
10 pts |
5 |
19 ENTER 4 - 5 ENTER 2 - ÷ |
| 9 |
Application |
Given slope m = 4 through (2, 3) and (7, y2), find y2 = |
10 pts |
23 |
7 ENTER 2 - 4 × 3 + |
| 10 |
Error Analysis |
Error Analysis: A student found slope between (1, 3) and (5, 15) as (5−1)÷(15−3) = 0.33. Find true slope: |
10 pts |
3 |
15 ENTER 3 - 5 ENTER 1 - ÷ |
5. STEM Lab Case Study: Civil Highway Grade & Mountain Pass Surveying
Scenario: Companion to Vol. II, Ch. 2 & Ch. 12: Civil engineers survey elevation changes Δy (Register Y) over horizontal runs Δx (Register X) across mountain passes.
- Case Question 1: West Ascent Ramp: A survey team measures milepost (2, 3) to summit (6, 11). Find the highway slope m = Δy ÷ Δx:
- Expected Result: 2 | Keystrokes:
11 ENTER 3 - 6 ENTER 2 - ÷
- Case Question 2: East Descent Grade: From ridge (-4, 10) down to valley (2, -8), compute the descent slope m = Δy ÷ Δx:
- Expected Result: -3 | Keystrokes:
-8 ENTER 10 - 2 ENTER -4 - ÷
- Case Question 3: Viaduct Pier Elevation: Using point-slope profile y = 3(x − 2) + 5, find the pier height y at station x = 7:
- Expected Result: 20 | Keystrokes:
7 ENTER 2 - 3 × 5 +
- Case Question 4: Error Analysis: Student calculates Δx ÷ Δy = 4 ÷ 8 = 0.5. Correct the mistake.
- Expected Result: Slope is Rise / Run = Δy ÷ Δx = 8 ÷ 4 = 2. | Keystrokes:
8 ENTER 4 ÷
Unit 02: Algebraic Factoring, Area Models & Quadratic Trinomials
Grades: Grade 9 – 10 (Algebra 1) | Instrument: StackCalc SC-32 | Visual / Block Model: Prime Factor Studio (Factor Pairs) & 2×2 Algebraic Area Grid
Key Standards: CCSS.MATH.HSA.SSE.B.3, CCSS.MATH.HSA.REI.B.4
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Factor x² + 7x + 12 by decomposing 12 with Prime Factor blocks to find the factor pair whose sum is 7, and find both roots of x² + 7x + 12 = 0.
- Verified Answer:
3 × 4 = 12 and 3 + 4 = 7; factored form (x + 3)(x + 4), roots x = -3 and x = -4.
- Keystrokes:
POLY 1 B 7 C 1 2 D XEQ
- Warm-Up B (Extension Lab): Factor x² - 5x + 6 into two binomials by finding two negative factors of +6 whose sum is -5, and state both roots.
- Verified Answer:
(-2) × (-3) = +6 and (-2) + (-3) = -5; factored form (x - 2)(x - 3), roots x = 2 and x = 3.
- Keystrokes:
POLY 1 B 5 +/- C 6 D XEQ
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Factor x² + 6x + 8 by decomposing 8 into a factor pair that sums to 6, and find the roots. |
(x + 2)(x + 4); roots -2, -4 |
POLY 1 B 6 C 8 D XEQ |
| 2 |
Factor x² - 4 (difference of two squares, where the linear coefficient b = 0). |
(x - 2)(x + 2); roots 2, -2 |
POLY 1 B 0 C 4 +/- D XEQ |
| 3 |
Factor x² + 5x + 4 into two binomials and state the solutions to x² + 5x + 4 = 0. |
(x + 1)(x + 4); roots -1, -4 |
POLY 1 B 5 C 4 D XEQ |
| 4 |
Factor x² - 5x + 6 using signed factor pairs and state both roots. |
(x - 2)(x - 3); roots 2, 3 |
POLY 1 B 5 +/- C 6 D XEQ |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
18
3 ENTER 6 × |
x² + 7x + 10
2 ENTER 5 + |
156 (12 × 13)
10 x² 5 ENTER 10 × + 6 + |
2x² + 9x + 4
2 ENTER 4 × 1 + |
9
6 ENTER 2 ÷ x² |
600
53 ENTER 47 - 53 ENTER 47 + × |
| #2 |
11
4 ENTER 7 + |
x² + x - 12
3 ENTER 4 × |
x² + 8x + 15
3 ENTER 5 + |
3x² + 7x - 6
3 ENTER 3 × 2 - |
36
12 ENTER 2 ÷ x² |
3000
65 ENTER 35 - 65 ENTER 35 + × |
| #3 |
35
3 ENTER 2 + 3 ENTER 4 + × |
x² - 36
6 ENTER 6 × |
t = 6 sec
POLY -1 B 6 C 0 D XEQ |
x² + 10x + 25
5 x² |
3
39 ENTER 25 + √x 5 - |
5
15 ENTER 3 ÷ |
| #4 |
30
3 x² 5 ENTER 3 × + 6 + |
(x + 2)(x + 4) (roots -2, -4)
POLY 1 B 6 C 8 D XEQ |
4 and 7
POLY 1 B -11 C 28 D XEQ |
4x² - 12x + 9
-3 x² |
2
16 ENTER 9 + √x 3 - |
-5
-20 ENTER 4 ÷ |
| #5 |
12
5 x² 4 ENTER 5 × - 7 + |
(x + 4)(x + 5) (roots -4, -5)
POLY 1 B 9 C 20 D XEQ |
13 and 7
20 ENTER 6 + 2 ÷ |
(2x + 1)(x + 2) (roots -0.5, -2)
POLY 2 B 5 C 2 D XEQ |
(4, 7)
8 ENTER 2 ÷ |
2
8 ENTER 4 ÷ |
| #6 |
14 (5x² + 9x)
5 ENTER 9 + |
(x - 2)(x - 5) (roots 2, 5)
POLY 1 B -7 C 10 D XEQ |
9 (25 - 16)
-5 x² 4 ENTER 2 × 2 × - |
3(x - 2)(x + 2)
POLY 3 B 0 C -12 D XEQ |
4 (at x = 5)
5 x² 10 ENTER 5 × - 29 + |
19
5 x² 2 ENTER 3 × - |
| #7 |
6x² + 15x
3 ENTER 2 × |
(x - 5)(x + 3) (roots 5, -3)
POLY 1 B -2 C -15 D XEQ |
x = 3, y = 7
10 ENTER 1 - 3 ÷ |
x + 4
16 ENTER 4 ÷ |
10 (at x = 3)
3 x² +/- 6 ENTER 3 × + 1 + |
37
4 ENTER 3 yˣ 3 ENTER 3 yˣ - |
| #8 |
6x(2x + 3)
12 ENTER 18 GCD |
x = 2, 4
POLY 1 B -6 C 8 D XEQ |
x = 4 (y = 3)
8 ENTER 2 ÷ |
x + 5
10 ENTER 2 ÷ |
0 (1 repeated root)
-6 x² 4 ENTER 1 × 9 × - |
(x - 2)(x² + 2x + 4)
8 ENTER 3 ˣ√𝑦 |
| #9 |
75 (x² - 25)
10 x² 25 - |
x = 2, -4
POLY 1 B 2 C -8 D XEQ |
-5 ([+/-] negates X)
5 +/- |
20
15 ENTER 5 + |
-16 (2 complex roots)
2 x² 4 ENTER 1 × 5 × - |
x = 2 or x = 3
POLY 1 B -5 C 6 D XEQ |
| #10 |
156
12 ENTER 13 × |
Sum = 9, Prod = 14
POLY 1 B -9 C 14 D XEQ 2 ENTER 7 + |
x = -2, -5
POLY 1 B 7 C 10 D XEQ |
x = 3
POLY 2 B -7 C 3 D XEQ |
c = 16
8 ENTER 2 ÷ x² |
b = -5 (negate X)
5 +/- |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Evaluate the quadratic area 2x² + 3x - 4 when x = 3: |
10 pts |
23 |
2 ENTER 3 x² × 3 ENTER 3 × + 4 - |
| 2 |
Core Fluency |
Area Grid Expansion: Expand (x + 3)(x + 4) and evaluate at x = 2: |
10 pts |
30 |
2 ENTER 3 + 2 ENTER 4 + × |
| 3 |
Core Fluency |
Al-Khwarizmi's Square: Find c = (b/2)² to complete the square for x² + 14x + c: |
10 pts |
49 |
14 ENTER 2 ÷ x² |
| 4 |
Core Fluency |
Factor completely: x² + 8x + 15 = |
10 pts |
(x + 3)(x + 5) (roots -3, -5) |
POLY 1 B 8 C 15 D XEQ |
| 5 |
Core Fluency |
Find both roots of x² - 8x + 12 = 0: |
10 pts |
x = 2, 6 |
POLY 1 B -8 C 12 D XEQ |
| 6 |
Core Fluency |
Find both roots of x² - 25 = 0: |
10 pts |
x = 5, -5 |
25 √x |
| 7 |
Application |
A rectangle has sides (x + 1) and (x + 5) and area 21. Find the positive value of x: |
10 pts |
x = 2 |
POLY 1 B 6 C -16 D XEQ |
| 8 |
Application |
Find two numbers whose sum is 13 and product is 40: |
10 pts |
5 and 8 |
POLY 1 B -13 C 40 D XEQ |
| 9 |
Application |
What is the sum of the roots of x² - 11x + 24 = 0? |
10 pts |
11 |
POLY 1 B -11 C 24 D XEQ 3 ENTER 8 + |
| 10 |
Error Analysis |
Error Analysis: Why does (x + 4)² equal x² + 8x + 16 instead of x² + 16? |
10 pts |
Two 4x rectangles (+8x) |
2 ENTER 4 × |
5. STEM Lab Case Study: Renaissance Garden Quadrangle Proportions
Scenario: Companion to Vol. II, Ch. 6 & Ch. 7: Renaissance architects proportioned walled courtyards using 2×2 algebraic area grids (x + p)(x + q) and Prime Factor blocks.
- Case Question 1: Main Cloister Lawn: Factor the courtyard area x² + 5x + 6 on the 2×2 area grid and verify the corner constant product:
- Expected Result: (x + 2)(x + 3) (2×3 = 6) | Keystrokes:
2 ENTER 3 ×
- Case Question 2: Fountain Terrace Cutout: Factor the difference-of-squares plaza x² − 16 and verify the corner square area:
- Expected Result: (x − 4)(x + 4) (4² = 16) | Keystrokes:
4 x²
- Case Question 3: Pavilion Stone Count: Evaluate the expanded courtyard area (x + 3)(x + 4) when base module x = 5 m:
- Expected Result: 72 (8 × 9 = 72). | Keystrokes:
5 ENTER 3 + 5 ENTER 4 + ×
- Case Question 4: Error Analysis: Student claims (x + 3)² = x² + 9 at x = 2 (getting 13). Find true (2 + 3)²:
- Expected Result: 25 (x² + 6x + 9 = 4 + 12 + 9 = 25). | Keystrokes:
2 ENTER 3 + x²
Grades: Grade 9 – 11 (Algebra 1 & 2) | Instrument: StackCalc SC-32 | Visual / Block Model: Parabola Root Diagram & SC-32 [POLY] Solver
Key Standards: CCSS.MATH.HSA.REI.B.4.B, CCSS.MATH.HSN.CN.C.7
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Evaluate the discriminant for 2x² − 7x + 3 = 0 on paper and verify on the stack.
- Verified Answer:
25 (49 − 24 = 25; 2 real rational roots).
- Keystrokes:
-7 x² 4 ENTER 2 × 3 × -
- Warm-Up B (Extension Lab): Use [POLY] on the SC-32 to solve 2x² − 5x − 3 = 0.
- Verified Answer:
x1 = 3.0, x2 = -0.5.
- Keystrokes:
POLY 2 B -5 C -3 D XEQ
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Evaluate discriminant for x² + 6x + 9 = 0: Δ = |
0 |
6 x² 4 ENTER 1 × 9 × - |
| 2 |
Solve roots with [POLY]: x² − 7x + 12 = 0 -> |
x = 4, 3 |
POLY 1 B -7 C 12 D XEQ |
| 3 |
Solve roots with [POLY]: x² + 2x − 15 = 0 -> |
x = 3, -5 |
POLY 1 B 2 C -15 D XEQ |
| 4 |
Evaluate discriminant for x² + 4x + 13 = 0: Δ = |
-36 |
4 x² 4 ENTER 1 × 13 × - |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
9
7 x² 4 ENTER 1 × 10 × - |
x = 3, 2
POLY 1 B -5 C 6 D XEQ |
x = 3, 0.5
POLY 2 B -7 C 3 D XEQ |
4
8 ENTER 2 ÷ |
5
POLY -16 B 64 C 80 D XEQ |
7
14 ENTER 2 ÷ |
| #2 |
0
-6 x² 4 ENTER 1 × 9 × - |
x = 5, 3
POLY 1 B -8 C 15 D XEQ |
x = -0.5, -2
POLY 2 B 5 C 2 D XEQ |
-3
-12 ENTER 4 ÷ |
2 s
-64 ENTER -32 ÷ |
4.5
9 ENTER 2 ÷ |
| #3 |
49
3 x² 4 ENTER 1 × -10 × - |
x = 3, -5
POLY 1 B 2 C -15 D XEQ |
x = 2.5, -1
POLY 2 B -3 C -5 D XEQ |
3
-18 ENTER -6 ÷ |
144 ft
2 x² -16 × 64 ENTER 2 × + 80 + |
-5
-20 ENTER 4 ÷ |
| #4 |
9
5 x² 4 ENTER 2 × 2 × - |
x = 5, -2
POLY 1 B -3 C -10 D XEQ |
x = 1.5, -0.5
POLY 4 B -4 C -3 D XEQ |
2.5
20 ENTER 8 ÷ |
3 s
POLY -16 B 0 C 144 D XEQ |
-2.75
-11 ENTER 4 ÷ |
| #5 |
64
-10 x² 4 ENTER 3 × 3 × - |
x = -3, -4
POLY 1 B 7 C 12 D XEQ |
x = 3, -0.4
POLY 5 B -13 C -6 D XEQ |
5
25 √x |
5 s
POLY -5 B 20 C 25 D XEQ |
3
4 ENTER -7 + +/- |
| #6 |
0
-12 x² 4 ENTER 4 × 9 × - |
x = 5, 4
POLY 1 B -9 C 20 D XEQ |
x = 4, 0.5
POLY 2 B -9 C 4 D XEQ |
8
64 √x |
45 m
2 x² -5 × 20 ENTER 2 × + 25 + |
-28
4 ENTER -7 × |
| #7 |
-36
4 x² 4 ENTER 1 × 13 × - |
x = 3, -7
POLY 1 B 4 C -21 D XEQ |
x = 2.5, -2.5
POLY 4 B 0 C -25 D XEQ |
3
7 ENTER 5 + 4 ÷ |
5
POLY 1 B 4 C -45 D XEQ |
5
4 x² 8 ENTER 4 × - 21 + |
| #8 |
49
-3 x² 4 ENTER 2 × -5 × - |
x = 4, -4
POLY 1 B 0 C -16 D XEQ |
x = 1, 0.6
POLY 5 B -8 C 3 D XEQ |
0.5
7 ENTER 5 - 4 ÷ |
5
POLY 2 B 14 C -120 D XEQ |
8
3 x² -2 × 12 ENTER 3 × + 10 - |
| #9 |
-16
2 x² 4 ENTER 5 × 1 × - |
5
POLY 1 B -10 C 25 D XEQ |
x = 0.75, 0.5
POLY 8 B -10 C 3 D XEQ |
4.2361
4 ENTER 20 √x + 2 ÷ |
26.06 mph
POLY 0.05 B 1 C -60 D XEQ |
12
4 ENTER 36 × √x |
| #10 |
361
-11 x² 4 ENTER 6 × -10 × - |
x = 8, 3
POLY 1 B -11 C 24 D XEQ |
x = 1.5, 0.2
POLY 10 B -17 C 3 D XEQ |
1.618
1 ENTER 5 √x + 2 ÷ |
16
-6 x² 4 ENTER 1 × 5 × - |
5
POLY 2 B -10 C 12 D XEQ 2 ENTER 3 + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Evaluate discriminant Δ = b² − 4ac for 2x² − 7x + 3 = 0: |
10 pts |
25 |
-7 x² 4 ENTER 2 × 3 × - |
| 2 |
Core Fluency |
Evaluate discriminant Δ = b² − 4ac for x² + 4x + 13 = 0: |
10 pts |
-36 |
4 x² 4 ENTER 1 × 13 × - |
| 3 |
Core Fluency |
Solve roots with [POLY]: x² − 7x + 12 = 0 -> |
10 pts |
x = 4, 3 |
POLY 1 B -7 C 12 D XEQ |
| 4 |
Core Fluency |
Solve roots with [POLY]: x² + 2x − 15 = 0 -> |
10 pts |
x = 3, -5 |
POLY 1 B 2 C -15 D XEQ |
| 5 |
Core Fluency |
Solve roots with [POLY]: 2x² − 5x − 3 = 0 -> |
10 pts |
x = 3, -0.5 |
POLY 2 B -5 C -3 D XEQ |
| 6 |
Core Fluency |
Find the axis of symmetry x = -b ÷ (2a) for y = 3x² − 18x + 7: |
10 pts |
3 |
18 ENTER 6 ÷ |
| 7 |
Application |
A projectile follows h(t) = -16t² + 48t + 64. Find positive landing time t: |
10 pts |
4 s |
POLY -16 B 48 C 64 D XEQ |
| 8 |
Application |
Find Vieta's product of roots (c ÷ a) for 5x² − 17x + 30 = 0: |
10 pts |
6 |
30 ENTER 5 ÷ |
| 9 |
Application |
Find vertex minimum value of f(x) = x² − 6x + 13 at x = 3: |
10 pts |
4 |
3 x² 6 ENTER 3 × - 13 + |
| 10 |
Error Analysis |
Error Analysis: A student got Δ = -11 for x² − 3x − 5 by subtracting 4(1)(-5) as +20. Find true Δ: |
10 pts |
29 |
-3 x² 4 ENTER 1 × -5 × - |
5. STEM Lab Case Study: Trajectory Ballistics Flight Arc Analysis
Scenario: Companion to Vol. II, Ch. 7: Aerospace guidance engineers analyze parabolic flight equations on the 4-level stack using [POLY] and [SOLVE] to find real crossings and impact times.
- Case Question 1: Radar Target Clearance: Evaluate discriminant Δ = b² − 4ac for flight clearance curve 2x² − 7x + 3 = 0:
- Expected Result: 25 (2 real roots) | Keystrokes:
-7 x² 4 ENTER 2 × 3 × -
- Case Question 2: Wind-Tunnel Calibration Arc: Extract the reference roots of x² + 5x + 6 = 0 using [POLY]:
- Expected Result: x1 = -2, x2 = -3 | Keystrokes:
POLY 1 B 5 C 6 D XEQ
- Case Question 3: Downrange Impact Solution: Extract the roots of trajectory profile 2x² − 5x − 3 = 0 using [POLY]:
- Expected Result: x1 = 3.0, x2 = -0.5 | Keystrokes:
POLY 2 B -5 C -3 D XEQ
- Case Question 4: Error Analysis: Student says Δ < 0 for x² + 4 = 0 means no roots. Evaluate Δ:
- Expected Result: -16 (2 complex conjugate roots ±2i). | Keystrokes:
0 x² 4 ENTER 1 × 4 × -
Unit 04: Polynomial Operations, Horner's Rule & Synthetic Division
Grades: Grade 10 – 11 (Algebra 2) | Instrument: StackCalc SC-32 | Visual / Block Model: Nested Polynomial Pipeline Diagram & SC-32 Stack
Key Standards: CCSS.MATH.HSA.APR.A.1, CCSS.MATH.HSA.APR.B.2
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Evaluate P(x) = 3x³ − 2x² + 4x − 8 at x = 2 using Horner's rule on the stack.
- Verified Answer:
16 (((3(2)−2)(2)+4)(2)−8 = (4(2)+4)(2)−8 = 12(2)−8 = 16).
- Keystrokes:
3 ENTER 2 × -2 + 2 × 4 + 2 × -8 +
- Warm-Up B (Extension Lab): Test if (x − 3) is a factor of P(x) = x³ − 4x² + x + 6 using synthetic division on stack.
- Verified Answer:
Remainder = 0; (x − 3) is a factor.
- Keystrokes:
1 ENTER 3 × -4 + 3 × 1 + 3 × 6 +
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Evaluate P(x) = x² − 6x + 8 at x = 4: P(4) = |
0 |
1 ENTER 4 × -6 + 4 × 8 + |
| 2 |
Evaluate P(x) = 2x³ + 3x² − 5x + 1 at x = 2: P(2) = |
19 |
2 ENTER 2 × 3 + 2 × -5 + 2 × 1 + |
| 3 |
Find remainder of (x³ + 2x² − 7x − 2) ÷ (x − 2): R = |
0 |
1 ENTER 2 × 2 + 2 × -7 + 2 × -2 + |
| 4 |
Evaluate P(x) = 4x² − 12x + 9 at x = 1.5: P(1.5) = |
0 |
4 ENTER 1.5 × -12 + 1.5 × 9 + |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
6
1 ENTER 2 × 3 + 2 × -4 + |
3
1 ENTER 3 × -7 + 3 × 15 + |
0
1 ENTER 2 × 0 + 2 × -7 + 2 × 6 + |
9
1 ENTER 2 × -2 + 2 × 3 + 2 × -4 + 2 × 5 + |
80 cm³
1 ENTER 10 × 8 × |
4
12 ENTER 4 + 8 - 2 ÷ |
| #2 |
10
2 ENTER 3 × -5 + 3 × 7 + |
15
2 ENTER 2 × 5 + 2 × -3 + |
0
1 ENTER -3 × 0 + -3 × -7 + -3 × 6 + |
21
1 ENTER 2 × 1 + 2 × -2 + 2 × 3 + 2 × -1 + |
96 cm³
2 ENTER 8 × 6 × |
7
10 ENTER 2 ENTER 9 × 15 - - |
| #3 |
54
3 ENTER 4 × 4 + 4 × -10 + |
3
1 ENTER 2 × -2 + 2 × 4 + 2 × -5 + |
14
1 ENTER 3 × 0 + 3 × -5 + 3 × 2 + |
4
2 ENTER 2 × -3 + 2 × 1 + 2 × -5 + 2 × 2 + |
72 cm³
3 ENTER 6 × 4 × |
x = 3, 2
POLY 1 B -5 C 6 D XEQ |
| #4 |
11
1 ENTER 2 × 2 + 2 × -3 + 2 × 1 + |
2
1 ENTER 1 × 3 + 1 × -4 + 1 × 2 + |
2
2 ENTER 2 × 0 + 2 × -9 + 2 × 4 + |
0
2 x² x² 5 ENTER 2 x² × - 4 + |
21
1 ENTER 2 × 2 + 2 × 2 + 2 × 1 + |
x = 2, -1
POLY 1 B -1 C -2 D XEQ |
| #5 |
19
2 ENTER 2 × 3 + 2 × -5 + 2 × 1 + |
19
2 ENTER 3 × -5 + 3 × 1 + 3 × 7 + |
15
1 ENTER 2 × 4 + 2 × 0 + 2 × -9 + |
40
3 x² x² 5 ENTER 3 x² × - 4 + |
52
1 ENTER 3 × 2 + 3 × 2 + 3 × 1 + |
22
2 ENTER 2 × 1 + x² 3 - |
| #6 |
7
1 ENTER 3 × -4 + 3 × 6 + 3 × -2 + |
-4
1 ENTER -1 × 4 + -1 × 1 + -1 × -6 + |
0
1 ENTER 3 × 0 + 3 × 0 + 3 × -27 + |
0
2 x² x² 16 - |
2
1 ENTER 2 × -6 + 2 × 9 + 2 × |
3
2 x² 3 - 2 × 1 + |
| #7 |
2
2 ENTER -1 × -3 + -1 × 1 + -1 × 8 + |
-16
1 ENTER -2 × -3 + -2 × 2 + -2 × 8 + |
0
1 ENTER -2 × 0 + -2 × 0 + -2 × 8 + |
80
3 x² x² 1 - |
0
1 ENTER 3 × -6 + 3 × 9 + 3 × |
21
4 ENTER 3 yˣ 1 - 4 ENTER 1 - ÷ |
| #8 |
0
4 ENTER 1.5 × -12 + 1.5 × 9 + |
4
3 ENTER 2 × -4 + 2 × -5 + 2 × 6 + |
0
1 ENTER 3 × -4 + 3 × 1 + 3 × 6 + |
16
2 x² x² 2 ENTER 2 x² × + 8 - |
625
-2 ENTER 5 × 15 + 5 × 100 + 5 × |
3
3 ENTER 2 - 5 + 7 - 4 + |
| #9 |
0
1 ENTER -2 × 5 + -2 × 2 + -2 × -8 + |
6
1 ENTER 4 × -6 + 4 × 11 + 4 × -6 + |
0
1 ENTER -1 × -4 + -1 × 1 + -1 × 6 + |
15
3 ENTER 3 yˣ 4 ENTER 3 × - |
262
2 ENTER 10 × 5 + 10 × 12 + |
-27
-3 ENTER 3 yˣ |
| #10 |
32
3 ENTER 3 × -7 + 3 × 5 + 3 × -1 + |
0
2 ENTER 1 × 3 + 1 × -8 + 1 × 3 + |
-3
1 ENTER 2 × -5 + |
-15
-3 ENTER 3 yˣ 4 ENTER -3 × - |
0
1 ENTER 2 × 0 + 2 × -7 + 2 × 6 + |
0
1 ENTER 3 × -4 + 3 × 1 + 3 × 6 + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Evaluate P(x) = 2x² − 4x + 5 at x = 3 via Horner's rule: |
10 pts |
11 |
2 ENTER 3 × -4 + 3 × 5 + |
| 2 |
Core Fluency |
Evaluate P(x) = 3x³ − 2x² + 4x − 8 at x = 2 via Horner's rule: |
10 pts |
16 |
3 ENTER 2 × -2 + 2 × 4 + 2 × -8 + |
| 3 |
Core Fluency |
Find the remainder of (x³ + 2x² − 5x + 4) ÷ (x − 2): |
10 pts |
10 |
1 ENTER 2 × 2 + 2 × -5 + 2 × 4 + |
| 4 |
Core Fluency |
Test if (x − 2) is a factor of x³ + 2x² − 7x − 2: P(2) = |
10 pts |
0 |
1 ENTER 2 × 2 + 2 × -7 + 2 × -2 + |
| 5 |
Core Fluency |
Find the remainder of (x³ − 7x + 6) ÷ (x − 3): |
10 pts |
12 |
1 ENTER 3 × 0 + 3 × -7 + 3 × 6 + |
| 6 |
Core Fluency |
Evaluate quartic P(x) = x⁴ − 5x² + 4 at x = 3: |
10 pts |
40 |
3 x² x² 5 ENTER 3 x² × - 4 + |
| 7 |
Application |
Find box volume V(x) = x(14 − 2x)(10 − 2x) at cut x = 2 cm: |
10 pts |
120 cm³ |
2 ENTER 10 × 6 × |
| 8 |
Application |
Given x = 2 is a root of x³ − 6x² + 11x − 6 = 0, solve depressed x² − 4x + 3 = 0 with [POLY]: |
10 pts |
x = 3, 1 |
POLY 1 B -4 C 3 D XEQ |
| 9 |
Application |
Find the sum of all coefficients of P(x) = 4x³ − 3x² + 6x − 2: |
10 pts |
5 |
4 ENTER 3 - 6 + 2 - |
| 10 |
Error Analysis |
Error Analysis: Dividing by (x − 2), a student evaluated P(-2) instead of P(2) on x² − 5x + 6. Find true P(2): |
10 pts |
0 |
1 ENTER 2 × -5 + 2 × 6 + |
5. STEM Lab Case Study: Signal Processing Filter Transfer Functions
Scenario: Companion to Vol. II, Ch. 7, Ch. 8 & Ch. 14: Audio engineers evaluate high-order filter transfer polynomials on the stack using Horner's nested pipeline.
- Case Question 1: Stage-1 Gain Response: Evaluate filter polynomial P(x) = 2x² − 4x + 5 at frequency ratio x = 3 via Horner's rule:
- Expected Result: 11 | Keystrokes:
2 ENTER 3 × -4 + 3 × 5 +
- Case Question 2: Notch Filter Null Check: Evaluate cubic transfer function P(x) = x³ − 4x² + x + 6 at x = 2 via Horner's rule:
- Expected Result: 0 ((x − 2) is a factor) | Keystrokes:
1 ENTER 2 × -4 + 2 × 1 + 2 × 6 +
- Case Question 3: Remainder Theorem Telemetry: Evaluate filter polynomial P(x) = 2x² − 3x + 5 at x = 3 to find the remainder when dividing P(x) by (x − 3):
- Expected Result: Remainder = 14 (Remainder Theorem) | Keystrokes:
2 ENTER 3 × -3 + 3 × 5 +
- Case Question 4: Error Analysis: Student omits the 0x² coefficient in P(x) = x³ − 7x + 6 at x = 2. Evaluate P(2) with the 0x² term included:
- Expected Result: 0 (Missing degree terms must have coefficient 0 in synthetic division: 1, 0, -7, 6) | Keystrokes:
1 ENTER 2 × 0 + 2 × -7 + 2 × 6 +
Unit 05: Exponential Growth, Decay & Continuous Compounding
Grades: Grade 10 – 12 (Algebra 2 & Precalculus) | Instrument: StackCalc SC-32 | Visual / Block Model: Prime Factor Studio & SC-32 Scientific Stack ([yˣ], [eˣ])
Key Standards: CCSS.MATH.HSF.LE.A.1, CCSS.MATH.HSF.LE.A.4
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): A bacteria culture starts with P = 500 cells and triples every hour. Calculate the population after t = 4 hours.
- Verified Answer:
3^4 = 81; 500 × 81 = 40,500 bacteria.
- Keystrokes:
3 ENTER 4 yˣ 5 0 0 ×
- Warm-Up B (Extension Lab): A bacterial colony of 500 cells grows continuously at rate r = 0.12 per hour. Find population at t = 8 hrs.
- Verified Answer:
1,305.85 cells (500 × e^0.96 ≈ 1,305.85).
- Keystrokes:
0.12 ENTER 8 × eˣ 500 ×
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Invest $2,000 at 6% compounded annually for 5 years: A = |
$2,676.45 |
2000 ENTER 1.06 ENTER 5 yˣ × |
| 2 |
Continuous compounding: $5,000 at 4% for 8 years: A = |
$6,885.64 |
0.04 ENTER 8 × eˣ 5000 × |
| 3 |
Depreciation: Car drops 15% per year from $30,000 after 4 years: A = |
$15,660.19 |
30000 ENTER 0.85 ENTER 4 yˣ × |
| 4 |
Half-life: 100g substance with half-life 6 days after 18 days: A = |
12.5g |
100 ENTER 0.5 ENTER 3 yˣ × |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
1024
2 ENTER 10 yˣ |
$1,343.92
1000 ENTER 1.03 ENTER 10 yˣ × |
2.7183
1 eˣ |
10 g
80 ENTER 0.5 ENTER 3 yˣ × |
$6,768.39
10000 ENTER 1.05 ENTER 8 yˣ ÷ |
$1,531.02
1000 ENTER 1.05 ENTER 4 yˣ × 1.08 ENTER 3 yˣ × |
| #2 |
729
3 ENTER 6 yˣ |
$2,737.14
2000 ENTER 1.04 ENTER 8 yˣ × |
7.3891
2 eˣ |
12.5 g
100 ENTER 0.5 ENTER 3 yˣ × |
$3,524.80
5000 ENTER 1.06 ENTER 6 yˣ ÷ |
$3,256.31
2000 ENTER 1.06 ENTER 5 yˣ × 1.04 ENTER 5 yˣ × |
| #3 |
1.6289
1.05 ENTER 10 yˣ |
$1,346.86
1000 ENTER 1.015 ENTER 20 yˣ × |
1.6487
0.5 eˣ |
15 g
240 ENTER 0.5 ENTER 4 yˣ × |
$12,130.61
20000 ENTER 0.5 eˣ ÷ |
$312.71
0.6 eˣ 10000 × 10000 ENTER 1.06 ENTER 10 yˣ × - |
| #4 |
$1,628.89
1000 ENTER 1.05 ENTER 10 yˣ × |
$5,072.97
4000 ENTER 1.02 ENTER 12 yˣ × |
$1,648.72
0.05 ENTER 10 × eˣ 1000 × |
88.39 mg
500 ENTER 0.5 ENTER 2.5 yˣ × |
160000
2500 ENTER 2 ENTER 6 yˣ × |
450.85
-2 eˣ 9 × 1 + 1000 x⇄y ÷ |
| #5 |
$2,676.45
2000 ENTER 1.06 ENTER 5 yˣ × |
$5,862.89
5000 ENTER 1.01 ENTER 16 yˣ × |
$6,885.64
0.04 ENTER 8 × eˣ 5000 × |
$15,660.19
30000 ENTER 0.85 ENTER 4 yˣ × |
50.625 Gbps
10 ENTER 1.5 ENTER 4 yˣ × |
100
0 eˣ 9 × 1 + 1000 x⇄y ÷ |
| #6 |
$6,842.85
5000 ENTER 1.04 ENTER 8 yˣ × |
$1,348.85
1000 ENTER 1.005 ENTER 60 yˣ × |
$2,699.72
0.06 ENTER 5 × eˣ 2000 × |
$6,553.60
20000 ENTER 0.8 ENTER 5 yˣ × |
62.46°C
-0.1 ENTER 5 × eˣ 70 × 20 + |
6.3212 V
1 ENTER -1 eˣ - 10 × |
| #7 |
$6,756.57
3000 ENTER 1.07 ENTER 12 yˣ × |
$3,809.20
3000 ENTER 1.01 ENTER 24 yˣ × |
$20,137.53
0.07 ENTER 10 × eˣ 10000 × |
55.59 kPa
-0.12 ENTER 5 × eˣ 101.3 × |
$744.09
1000 ENTER 1.03 ENTER 10 yˣ ÷ |
1.5431
1 eˣ -1 eˣ + 2 ÷ |
| #8 |
$21,589.25
10000 ENTER 1.08 ENTER 10 yˣ × |
0.1268
1.01 ENTER 12 yˣ 1 - |
1305.85
0.12 ENTER 8 × eˣ 500 × |
1.624 V
-0.5 ENTER 4 × eˣ 12 × |
$1,360.38
4000 ENTER 1.05 ENTER 6 yˣ × 4000 - |
1.1752
1 eˣ -1 eˣ - 2 ÷ |
| #9 |
25600
200 ENTER 2 ENTER 7 yˣ × |
0.0824
1.02 ENTER 4 yˣ 1 - |
6273.25
0.03 ENTER 15 × eˣ 4000 × |
89.25 mg
-0.25 ENTER 6 × eˣ 400 × |
0.0052
0.1 eˣ 1.1 - |
1
1 eˣ -1 eˣ + 2 ÷ x² 1 eˣ -1 eˣ - 2 ÷ x² - |
| #10 |
$103,680
50000 ENTER 1.2 ENTER 4 yˣ × |
$10.62
1348.85 ENTER 1338.23 - |
$6,795.70
1 eˣ 2500 × |
43.05%
100 ENTER 0.9 ENTER 8 yˣ × |
110.25
100 ENTER 1.05 x² × |
1359.14
0.05 ENTER 20 × eˣ 500 × |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Invest $1,000 at 5% compounded annually for 10 years: A = |
10 pts |
$1,628.89 |
1000 ENTER 1.05 ENTER 10 yˣ × |
| 2 |
Core Fluency |
Invest $2,000 at 6% compounded annually for 5 years: A = |
10 pts |
$2,676.45 |
2000 ENTER 1.06 ENTER 5 yˣ × |
| 3 |
Core Fluency |
Quarterly compounding: $1,000 at 8% annual rate for 3 years: |
10 pts |
$1,268.24 |
1000 ENTER 1.02 ENTER 12 yˣ × |
| 4 |
Core Fluency |
Continuous compounding: $1,000 at 5% for 10 years: A = |
10 pts |
$1,648.72 |
0.05 ENTER 10 × eˣ 1000 × |
| 5 |
Core Fluency |
Continuous compounding: $5,000 at 4% for 8 years: A = |
10 pts |
$6,885.64 |
0.04 ENTER 8 × eˣ 5000 × |
| 6 |
Core Fluency |
Depreciation: A $30,000 car loses 15% per year for 4 years: A = |
10 pts |
$15,660.19 |
30000 ENTER 0.85 ENTER 4 yˣ × |
| 7 |
Application |
Half-life decay: 160 g of an isotope after 4 half-lives: A = |
10 pts |
10 g |
160 ENTER 0.5 ENTER 4 yˣ × |
| 8 |
Application |
Find Present Value of $10,000 due in 5 years at 6% annual: |
10 pts |
$7,472.58 |
10000 ENTER 1.06 ENTER 5 yˣ ÷ |
| 9 |
Application |
Evaluate hyperbolic sine sinh(1) = (e¹ − e⁻¹) ÷ 2: |
10 pts |
1.1752 |
1 eˣ -1 eˣ - 2 ÷ |
| 10 |
Error Analysis |
Error Analysis: A student multiplied 1000 × 1.05 first, then raised 1050² instead of 1000 × (1.05)². True value: |
10 pts |
$1,102.50 |
1000 ENTER 1.05 x² × |
5. STEM Lab Case Study: Global Fiber Optic Bandwidth Expansion
Scenario: Companion to Vol. II, Ch. 3 & Ch. 11: Network architects and financial engineers model compound traffic growth, TVM cash flows, and optical attenuation using exponential functions.
- Case Question 1: Discrete Backbone Expansion: A 1,000 Gbps node grows 5% annually for 10 years. Find the capacity A = 1000(1.05)¹⁰:
- Expected Result: $1,628.89 | Keystrokes:
1000 ENTER 1.05 ENTER 10 yˣ ×
- Case Question 2: Continuous Traffic Surge: Model 1,000 units growing continuously at 5% for 10 years (A = Peʳᵗ):
- Expected Result: $1,648.72 | Keystrokes:
0.05 ENTER 10 × eˣ 1000 ×
- Case Question 3: Optical Laser Attenuation: An 80 mW signal halves every 50 km. Find the power after 3 half-distances:
- Expected Result: 10 mW | Keystrokes:
80 ENTER 0.5 ENTER 3 yˣ ×
- Case Question 4: Error Analysis: Student enters rate 5% as 5 instead of 0.05. Convert 5% to its decimal rate on the stack:
- Expected Result: 0.05 (5% = 5/100 = 0.05; using 5 multiplies by 6 instead of 1.05!) | Keystrokes:
5 ENTER 100 ÷
Unit 06: Natural Logarithms, Inverse Exponentials & Doubling Time [LN]
Grades: Grade 11 – 12 (Algebra 2 & Precalc) | Instrument: StackCalc SC-32 | Visual / Block Model: Prime Factor Studio (Logarithmic Height Towers) & SC-32 [LN]
Key Standards: CCSS.MATH.HSF.BF.B.5, CCSS.MATH.HSF.LE.A.4
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Find the exact doubling time for a population growing at 4% continuously.
- Verified Answer:
17.33 years (ln(2) ÷ 0.04 ≈ 17.329).
- Keystrokes:
2 LN 0.04 ÷
- Warm-Up B (Extension Lab): Evaluate log_2(64) on the stack using natural logarithms.
- Verified Answer:
6 (2⁶ = 64).
- Keystrokes:
64 LN 2 LN ÷
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Find the doubling time at 8% continuous interest: t = |
8.66 years |
2 LN 0.08 ÷ |
| 2 |
Find the tripling time at 5% continuous growth: t = |
21.97 years |
3 LN 0.05 ÷ |
| 3 |
Evaluate log_3(81) using change of base: |
4 |
81 LN 3 LN ÷ |
| 4 |
Solve e^(0.03 t) = 5 for t: t = |
53.65 |
5 LN 0.03 ÷ |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
7
128 LN 2 LN ÷ |
17.33 years
2 LN 0.04 ÷ |
pH = 4
0.0001 LOG +/- |
x = 3
32 LN 2 LN ÷ 2 - |
2718.28 (1000e)
0.05 ENTER 20 × eˣ 1000 × |
8 × 10⁷
4 ENTER 3 10ˣ × 2 ENTER 4 10ˣ × × |
| #2 |
4
81 LN 3 LN ÷ |
8.66 years
2 LN 0.08 ÷ |
100 times
7 ENTER 5 - 10ˣ |
x = 1.5
27 LN 9 LN ÷ |
13.86 years
2 LN 0.05 ÷ |
3 × 10⁶
9 ENTER 10 ENTER 8 yˣ × 3 ENTER 10 x² × ÷ |
| #3 |
0.6931
2 LN |
21.97 years
3 LN 0.05 ÷ |
20 dB
100 LOG 10 × |
x = 5/3
32 LN 8 LN ÷ |
9.90 years
2 LN 0.07 ÷ |
2.5 × 10⁷
5 ENTER 3 10ˣ × x² |
| #4 |
2.3026
10 LN |
27.73 years
4 LN 0.05 ÷ |
$3,200
400 ENTER 2 ENTER 3 yˣ × |
x = 81
3 ENTER 4 yˣ |
35°C
0.5 ENTER 3 yˣ 80 × 25 + |
-9
-9 10ˣ LOG |
| #5 |
2.3026
2 LN 5 LN + |
3 half-lives
800 ENTER 100 ÷ LN 2 LN ÷ |
3 half-lives
64 ENTER 8 ÷ LN 2 LN ÷ |
x = 13
5 x² 1 + 2 ÷ |
2 hours
45 ENTER 25 - 80 ÷ LN 0.5 LN ÷ |
1,000 times
6 ENTER 3 - 10ˣ |
| #6 |
4
10000 LOG |
12 years
72 ENTER 6 ÷ |
28 mm
16 LN 2 LN ÷ 7 × |
2
400 LOG 4 LOG - |
100
1000 ENTER 1 ENTER 9 + ÷ |
1,000 times
30 ENTER 10 ÷ 10ˣ |
| #7 |
6
64 LN 2 LN ÷ |
8 years
72 ENTER 9 ÷ |
100
0 eˣ 100 × |
6
4 LN 2 LN ÷ 3 × |
2.20
9 LN |
8
32 ENTER 5 ˣ√𝑦 ENTER 3 yˣ |
| #8 |
5
5 eˣ LN |
t = 4
16 LN 2 LN ÷ |
x = 4
100000 LOG 1 - |
3
8 LN 2 LN ÷ |
0.0625 (1/16)
0.5 ENTER 4 yˣ |
16
64 ENTER 3 ˣ√𝑦 x² |
| #9 |
2
25 LOG 4 LOG + |
x = 1,000
3 10ˣ |
5 (2 + 3)
2 ENTER 3 + |
8
2 LN 3 × eˣ |
5 doublings
32 LN 2 LN ÷ |
27
81 ENTER 4 ˣ√𝑦 ENTER 3 yˣ |
| #10 |
x = 4
27 LN 3 LN ÷ 1 + |
3
64 LN 4 LN ÷ |
log₁₀(10) = 1
2 LOG 5 LOG + |
36
6 LOG 2 × 10ˣ |
0.0693
2 LN 10 ÷ |
0.2 (0.04 × 5)
0.04 ENTER 5 × |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Prime Log Tower: How many [2] blocks stack to reach 64 (find log₂(64))? |
10 pts |
6 |
64 LN 2 LN ÷ |
| 2 |
Core Fluency |
Change of Base on SC-32: Evaluate log₃(81) = ln(81) ÷ ln(3): |
10 pts |
4 |
81 LN 3 LN ÷ |
| 3 |
Core Fluency |
Evaluate base-10 logarithm: log₁₀(1,000) = |
10 pts |
3 |
1000 LOG |
| 4 |
Core Fluency |
Evaluate natural logarithm: ln(e⁴) = |
10 pts |
4 |
4 eˣ LN |
| 5 |
Core Fluency |
Napier Product Law: log₂(8) + log₂(4) = |
10 pts |
5 |
8 LN 2 LN ÷ 4 LN 2 LN ÷ + |
| 6 |
Core Fluency |
Solve for x: 5^x = 125 |
10 pts |
x = 3 |
125 LN 5 LN ÷ |
| 7 |
Application |
Logarithmic Doubling Time: A colony of 50 bacteria doubles every hour. How many hours to reach 400? |
10 pts |
3 hours |
400 ENTER 50 ÷ LN 2 LN ÷ |
| 8 |
Application |
160 mg of medicine halves every 4 hours. How much remains after 8 hours (2 half-lives)? |
10 pts |
40 mg |
160 ENTER 0.5 x² × |
| 9 |
Application |
What is the pH = -log₁₀[H⁺] of a solution with [H⁺] = 10⁻⁷ M? |
10 pts |
pH = 7 |
-7 10ˣ LOG +/- |
| 10 |
Error Analysis |
Error Analysis: Verify log(a·b) = log(a) + log(b) by evaluating ln(2) + ln(3) on the stack: |
10 pts |
1.7918 (ln(6) = ln(2)+ln(3)) |
2 LN 3 LN + |
5. STEM Lab Case Study: Carbon-14 Archeological Dating Laboratory
Scenario: Companion to Vol. II, Ch. 1 & Ch. 3: Geochemists calibrate spectrometer ratios and Willard Libby Carbon-14 decay constants using natural and base-10 logarithms.
- Case Question 1: Isotope Benchmark Rate: Find the continuous doubling/halving time t = ln(2) ÷ 0.07 for a 7% rate:
- Expected Result: 9.90 years | Keystrokes:
2 LN 0.07 ÷
- Case Question 2: Decade Ratio Calibration: Evaluate the natural logarithm of a 10× isotope ratio, ln(10):
- Expected Result: 2.3026 | Keystrokes:
10 LN
- Case Question 3: Spectrometer Base Conversion: Evaluate log₁₀(1000) using the natural log change-of-base pipeline ln(1000) ÷ ln(10):
- Expected Result: 3.0 | Keystrokes:
1000 LN 10 LN ÷
- Case Question 4: Error Analysis: Student tries to calculate ln(-4). Explain the error.
- Expected Result: Logarithms of negative numbers are undefined in real numbers (eˣ > 0 always). | Keystrokes:
4 LN
Grades: Grade 10 – 12 (Geometry & Precalculus) | Instrument: StackCalc SC-32 | Visual / Block Model: Right-Triangle & Unit-Circle Diagram & SC-32 Trig Keys
Key Standards: CCSS.MATH.HSG.SRT.C.6, CCSS.MATH.HSG.SRT.C.8
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): A 24 m guy wire makes a 30° angle with the ground. Find the tower height.
- Verified Answer:
12.0 m (24 × 0.5 = 12.0 m).
- Keystrokes:
24 ENTER 30 SIN ×
- Warm-Up B (Extension Lab): A flagpole 12 m tall casts a 9 m shadow. Find the sun's angle of elevation.
- Verified Answer:
53.13° (atan(12/9) ≈ 53.13°).
- Keystrokes:
12 ENTER 9 ÷ ATAN
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Find opposite leg: hyp = 20, θ = 30°: opp = |
10.0 |
20 ENTER 30 SIN × |
| 2 |
Find adjacent leg: hyp = 50, θ = 45°: adj = |
35.36 |
50 ENTER 45 COS × |
| 3 |
Find angle θ where opp = 5, adj = 12: θ = |
22.62° |
5 ENTER 12 ÷ ATAN |
| 4 |
Evaluate tan(60°): tan(60°) = |
1.7321 |
60 TAN |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
0.5
30 SIN |
5
10 ENTER 30 SIN × |
16
8 ENTER 30 SIN ÷ |
30°
0.5 ASIN |
124.97 m
200 ENTER 32 TAN × |
1
37 SIN x² 37 COS x² + |
| #2 |
0.5
60 COS |
12
24 ENTER 30 SIN × |
30
15 ENTER 60 COS ÷ |
60°
0.5 ACOS |
224.11 m
500 ENTER 24 TAN × 1.5 + |
0
2 ENTER 30 SIN × 30 COS × 60 SIN - |
| #3 |
1
45 TAN |
20
40 ENTER 60 COS × |
14.14
10 ENTER 45 SIN ÷ |
45°
1 ATAN |
9.66 m
10 ENTER 75 SIN × |
0.5
30 COS x² 30 SIN x² - |
| #4 |
1
90 SIN |
35.36
50 ENTER 45 COS × |
30.52
25 ENTER 35 COS ÷ |
36.87°
3 ENTER 4 ÷ ATAN |
2.59 m
10 ENTER 75 COS × |
16.97
12 ENTER 45 SIN × 30 SIN ÷ |
| #5 |
1
0 COS |
33.81
80 ENTER 25 SIN × |
28
18 ENTER 40 SIN ÷ |
53.13°
4 ENTER 3 ÷ ATAN |
28543 m
3000 ENTER 6 TAN ÷ |
24.49
20 ENTER 60 SIN × 45 SIN ÷ |
| #6 |
0.7071
45 SIN |
72.5
80 ENTER 25 COS × |
13
5 x² 12 x² + √x |
22.62°
5 ENTER 12 ÷ ATAN |
11.54°
1.2 ENTER 6 ÷ ASIN |
7
5 x² 8 x² + 2 ENTER 5 × 8 × 60 COS × - √x |
| #7 |
0.866
30 COS |
20
20 ENTER 45 TAN × |
17
8 x² 15 x² + √x |
28.07°
8 ENTER 17 ÷ ASIN |
30
0.5 ENTER 10 × 12 × 30 SIN × |
8.72
6 x² 10 x² + 2 ENTER 6 × 10 × 60 COS × - √x |
| #8 |
1.7321
60 TAN |
17.32
30 ENTER 30 TAN × |
25
7 x² 24 x² + √x |
73.74°
7 ENTER 25 ÷ ACOS |
51.96
0.5 ENTER 8 × 15 × 60 SIN × |
5
3 x² 4 x² + 2 ENTER 3 × 4 × 90 COS × - √x |
| #9 |
0.6
3 ENTER 5 ÷ |
32.49
100 ENTER 18 TAN × |
12
20 x² 16 x² - √x |
53.13°
12 ENTER 9 ÷ ATAN |
120.21 V
170 ENTER 45 SIN × |
3.1416
π 180 × 180 ÷ |
| #10 |
0.9231
12 ENTER 13 ÷ |
25.98
15 ENTER 30 TAN ÷ |
24
26 x² 10 x² - √x |
5.71°
0.1 ATAN |
53.13°
12 ENTER 9 ÷ ATAN |
1
30 SIN 60 COS + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Evaluate special angle ratio: sin(30°) = |
10 pts |
0.5 |
30 SIN |
| 2 |
Core Fluency |
Evaluate special angle ratio: tan(60°) = |
10 pts |
1.7321 |
60 TAN |
| 3 |
Core Fluency |
Find opposite leg: hyp = 20, θ = 30°: opp = |
10 pts |
10 |
20 ENTER 30 SIN × |
| 4 |
Core Fluency |
Find adjacent leg: hyp = 50, θ = 45°: adj = |
10 pts |
35.36 |
50 ENTER 45 COS × |
| 5 |
Core Fluency |
Find hypotenuse: opp = 15, θ = 30°: hyp = |
10 pts |
30 |
15 ENTER 30 SIN ÷ |
| 6 |
Core Fluency |
Find angle of elevation θ where opp = 3, adj = 4: |
10 pts |
36.87° |
3 ENTER 4 ÷ ATAN |
| 7 |
Application |
Find angle θ where opp = 5, adj = 12: |
10 pts |
22.62° |
5 ENTER 12 ÷ ATAN |
| 8 |
Application |
A surveyor 100 m from a building measures θ = 40°. Find the building height: |
10 pts |
83.91 m |
100 ENTER 40 TAN × |
| 9 |
Application |
Law of Cosines: find side c for a = 5, b = 8, and included angle C = 60°: |
10 pts |
7 |
5 x² 8 x² + 2 ENTER 5 × 8 × 60 COS × - √x |
| 10 |
Error Analysis |
Error Analysis: Verify sin²(25°) + cos²(25°) on the stack (must equal 1, not sin 50°): |
10 pts |
1 |
25 SIN x² 25 COS x² + |
5. STEM Lab Case Study: Geodetic Mountain Peak Triangulation
Scenario: Companion to Vol. II, Ch. 4 & Ch. 6: Topographical surveyors triangulate summit elevations using Eratosthenes sightlines and right-triangle trigonometry.
- Case Question 1: Laser Sightline Rise: A 10 m calibration mast is sighted at θ = 30°. Find the vertical rise opp = 10·sin(30°):
- Expected Result: 5.0 m | Keystrokes:
10 ENTER 30 SIN ×
- Case Question 2: Ridge Slope Angle: A ridge rises 3 m over a 4 m horizontal run. Find the angle of elevation θ = atan(3 ÷ 4):
- Expected Result: 36.87° | Keystrokes:
3 ENTER 4 ÷ ATAN
- Case Question 3: Direct Line-of-Sight Distance: Survey markers stand 5 km East and 12 km North. Find the hypotenuse distance:
- Expected Result: 13.0 | Keystrokes:
5 x² 12 x² + sqrt
- Case Question 4: Error Analysis: Calculator in Radian mode gives sin(30) = -0.988.
- Expected Result: Switch mode to Degrees (DEG annunciator) so 30° yields sin(30°) = 0.5. | Keystrokes:
30 SIN
Unit 08: 2D Vectors, Polar Conversions & Resultant Forces
Grades: Grade 11 – 12 (Precalc & Physics) | Instrument: StackCalc SC-32 | Visual / Block Model: 2D Vector Right-Triangle Diagram & SC-32 [▸θ,r]
Key Standards: CCSS.MATH.HSN.VM.A.1, CCSS.MATH.HSN.VM.A.3
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): A surveyor walks 60 m East (x = 60) and 80 m North (y = 80). Find straight-line distance r and angle θ.
- Verified Answer:
r = 100 m, θ = 53.13°.
- Keystrokes:
80 ENTER 60 ▸θ,r
- Warm-Up B (Extension Lab): Add vectors A = <6, 8> and B = <2, 7>. Find resultant magnitude and angle.
- Verified Answer:
r = 17.0, θ = 61.93° (<8, 15> -> 8²+15²=289, √289=17).
- Keystrokes:
8 ENTER 7 + 6 ENTER 2 + ▸θ,r
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Convert <5, 12> to polar: r = |
13.0 |
12 ENTER 5 ▸θ,r |
| 2 |
Find heading angle for <5, 12>: θ = |
67.38° |
12 ENTER 5 ▸θ,r x⇄y |
| 3 |
Convert <8, 6> to polar: r = |
10.0 |
6 ENTER 8 ▸θ,r |
| 4 |
Add vectors <4, 3> and <8, 6>. Find magnitude r = |
15.0 |
3 ENTER 6 + 4 ENTER 8 + ▸θ,r |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
10
8 ENTER 6 ▸θ,r |
5
10 ENTER 30 SIN × |
8 m
10 x² 6 x² - √x |
100 kt
80 ENTER 60 ▸θ,r |
5
2 ENTER 2 + 1 ENTER 2 + ▸θ,r |
10
8 ENTER 6 ▸θ,r |
| #2 |
13
12 ENTER 5 ▸θ,r |
6
12 ENTER 60 COS × |
5 ft
4 ENTER 3 ▸θ,r |
250 kt
70 ENTER 240 ▸θ,r |
13
12 ENTER 5 ▸θ,r |
25
24 ENTER 7 ▸θ,r |
| #3 |
17
15 ENTER 8 ▸θ,r |
14.14
20 ENTER 45 COS × |
10 miles
6 ENTER 8 ▸θ,r |
5 m/s
-3 ENTER 4 ▸θ,r |
17
9 ENTER -6 - 10 ENTER 2 - ▸θ,r |
60°
360 ENTER 6 ÷ |
| #4 |
12
15 x² 9 x² - √x |
43.30
50 ENTER 30 COS × |
15 m
9 ENTER 12 ▸θ,r |
13 Ω
5 ENTER 12 ▸θ,r |
15
4 ENTER 3 ▸θ,r 3 × |
4√3 (≈ 6.93)
3 √x 4 ÷ 4 x² × |
| #5 |
7
25 x² 24 x² - √x |
10
20 ENTER 30 SIN × |
25 N
50 ENTER 30 SIN × |
17 Ω
-8 ENTER 15 ▸θ,r |
14
4 ENTER 5 × 3 ENTER -2 × + |
6
6 ENTER 3 × 2 × 1 × √x |
| #6 |
15
12 ENTER -9 ▸θ,r |
7
14 ENTER 60 COS × |
0 (Perpendicular!)
3 ENTER 4 × 4 ENTER -3 × + |
10 N
8 ENTER 6 ▸θ,r |
k = -4
6 ENTER 2 × +/- 3 ÷ |
5
8 ENTER 6 ▸θ,r 2 ÷ |
| #7 |
5
3 ENTER -1 - 5 ENTER 2 - ▸θ,r |
45°
5 ENTER 5 ▸θ,r x⇄y |
13 m
3 x² 4 x² + 12 x² + √x |
17 N
8 ENTER 15 ▸θ,r |
3/5 (0.6)
3 ENTER 4 ENTER 3 ▸θ,r ÷ |
36.87°
3 ENTER 4 ▸θ,r x⇄y |
| #8 |
(6, 6)
8 ENTER 4 + 2 ÷ 2 ENTER 10 + 2 ÷ |
13
12 ENTER -5 ▸θ,r |
13
12 ENTER 5 ▸θ,r |
10 N
-8 ENTER 6 ▸θ,r |
58 J
10 ENTER 5 × 4 ENTER 2 × + |
53.13°
4 ENTER 3 ▸θ,r x⇄y |
| #9 |
m = 2
11 ENTER 3 - 5 ENTER 1 - ÷ |
15
-12 ENTER 9 ▸θ,r |
4.2 m
4.2 ENTER 45 TAN × |
12
8 ENTER 0.4 ÷ 0.6 × |
12
3 ENTER 9 + |
12π rad/s
6 ENTER 2 × |
| #10 |
<7, 5>
-2 ENTER 7 + 4 ENTER 3 + |
c = 7
8 x² 5 x² + 2 ENTER 8 × 5 × 60 COS × - √x |
√(3²+4²) = 5
4 ENTER 3 ▸θ,r |
60
10 ENTER 12 × 30 SIN × |
7
2 x² 3 x² + 6 x² + √x |
5.0 (y in Y, x in X)
4 ENTER 3 ▸θ,r |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Find the hypotenuse for legs a = 9 and b = 12: |
10 pts |
15 |
12 ENTER 9 ▸θ,r |
| 2 |
Core Fluency |
Find the distance between (1, 1) and (4, 5): |
10 pts |
5 |
5 ENTER 1 - 4 ENTER 1 - ▸θ,r |
| 3 |
Core Fluency |
Resultant magnitude of <2, 5> + <4, 3> using [▸θ,r]: |
10 pts |
10 |
8 ENTER 6 ▸θ,r |
| 4 |
Core Fluency |
If r = 16 and θ = 30°, find y = r·sin(30°): |
10 pts |
8 |
16 ENTER 30 SIN × |
| 5 |
Core Fluency |
Find the angle θ in degrees when Δx = 10 and Δy = 10: |
10 pts |
45° |
10 ENTER 10 ▸θ,r x⇄y |
| 6 |
Core Fluency |
Evaluate the dot product of <2, 5> and <3, 4>: |
10 pts |
26 |
2 ENTER 3 × 5 ENTER 4 × + |
| 7 |
Application |
A hiker walks 5 km South and 12 km East. What is the straight-line distance back to camp? |
10 pts |
13 km |
12 ENTER 5 ▸θ,r |
| 8 |
Application |
A 17-ft beam props up a wall 15 ft high. How far is the base of the beam from the wall? |
10 pts |
8 ft |
17 x² 15 x² - √x |
| 9 |
Application |
Find the maximum value of 3 sin(x) + 4 cos(x): |
10 pts |
5 |
4 ENTER 3 ▸θ,r |
| 10 |
Error Analysis |
Error Analysis: Why is the hypotenuse of a 3-4-5 right triangle shorter than walking 3 + 4 = 7? |
10 pts |
Triangle Inequality (5 < 7) |
-4 ENTER 3 ▸θ,r |
5. STEM Lab Case Study: Commercial Aircraft Crosswind Navigation
Scenario: Companion to Vol. II, Ch. 4 & Ch. 5: Flight navigators and electrical engineers compute wind triangles and Steinmetz AC phasors using 2D vector resolution and [▸θ,r].
- Case Question 1: Drone Test Flight: An aircraft drifts 3 km East (X) and 4 km North (Y). Convert <3, 4> to polar [▸θ,r]:
- Expected Result: r = 5.0, θ = 53.13° | Keystrokes:
4 ENTER 3 ▸θ,r
- Case Question 2: Jet Crosswind Leg: A jet travels 5 km East (X) and 12 km North (Y). Convert <5, 12> to polar [▸θ,r]:
- Expected Result: r = 13.0, θ = 67.38° | Keystrokes:
12 ENTER 5 ▸θ,r
- Case Question 3: Combined Thrust & Wind: Add force vectors <10, 15> and <20, 25> and find the resultant magnitude r:
- Expected Result: 50.0 N (θ = 53.13°) | Keystrokes:
40 ENTER 30 ▸θ,r
- Case Question 4: Error Analysis: Student adds magnitudes 3 + 4 = 7 instead of vector sum 5.
- Expected Result: Vector magnitude uses Pythagorean theorem: √(3² + 4²) = 5, not 7. | Keystrokes:
4 ENTER 3 ▸θ,r
Unit 09: Combinatorics, Probability Ratios & Series Accumulation
Grades: Grade 11 – 12 & AP (Precalc & AP Stats) | Instrument: StackCalc SC-32 | Visual / Block Model: Decision Slot Diagram, Statistical Plot & SC-32 [PROB] ([Pn,r]/[Cn,r])/[Σ+]
Key Standards: CCSS.MATH.HSS.CP.B.9
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): A lottery draws 4 winning numbers from a pool of 10 (order does not matter). How many winning combinations are possible?
- Verified Answer:
(10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 5040 / 24 = 210 combinations.
- Keystrokes:
1 0 x! 4 x! ÷ 6 x! ÷
- Warm-Up B (Extension Lab): Given the four data points {12, 15, 18, 21}, find the sample size n, the total sum of the values, and the sample mean x̄.
- Verified Answer:
Count n = 4, Sum = 66, Mean x̄ = 66 / 4 = 16.5.
- Keystrokes:
12 Σ+ 15 Σ+ 18 Σ+ 21 Σ+ x̄,ȳ
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Evaluate 6! (the number of ways to arrange 6 distinct books on a shelf). |
6 × 5 × 4 × 3 × 2 × 1 = 720 |
6 x! |
| 2 |
How many ways can 5 runners finish in 1st, 2nd, and 3rd place? |
5 × 4 × 3 = 60 |
5 x! 2 x! ÷ |
| 3 |
How many 2-person partnerships can be formed from 7 students? |
(7 × 6) / 2 = 21 |
7 x! 2 x! ÷ 5 x! ÷ |
| 4 |
Find the mean x̄ of the two data values 5 and 15. |
(5 + 15) / 2 = 10 |
5 Σ+ 1 5 Σ+ x̄,ȳ |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
120
5 x! |
1/2 (5/10)
5 ENTER 3 ENTER 5 + 2 + ÷ |
1,000 codes
3 10ˣ |
$8.00
20 ENTER 0.25 × 4 ENTER 0.75 × + |
1.5
90 ENTER 75 - 10 ÷ |
20
6 ENTER 3 PROB Cn,r |
| #2 |
56
8 x! 6 x! ÷ |
0.65
1 ENTER 0.35 - |
15 handshakes
6 ENTER 2 PROB Cn,r |
3.5
1 ENTER 6 + 2 ÷ |
-2
62 ENTER 70 - 4 ÷ |
24
4 ENTER 2 PROB Cn,r 2 x² × |
| #3 |
20
5 ENTER 2 PROB Pn,r |
1/8 (0.125)
0.5 ENTER 3 yˣ |
21 pizzas
7 ENTER 2 PROB Cn,r |
7
3.5 ENTER 2 × |
90
2.5 ENTER 4 × 80 + |
30
1 x² 2 x² + 3 x² + 4 x² + |
| #4 |
120
6 ENTER 3 PROB Pn,r |
6 ways
4 ENTER 2 PROB Cn,r |
12 arrangements
4 x! 2 x! ÷ |
3/10 (0.3)
3 ENTER 2 PROB Cn,r 5 ENTER 2 PROB Cn,r ÷ |
36
6 x² |
91
6 ENTER 7 × 13 × 6 ÷ |
| #5 |
10
5 ENTER 2 PROB Cn,r |
3/8 (0.375)
4 ENTER 2 PROB Cn,r 2 ENTER 4 yˣ ÷ |
0.81 (81%)
0.9 ENTER 0.9 × |
18 ways
4 ENTER 2 PROB Cn,r 3 ENTER 2 PROB Cn,r × |
18
8 ENTER 10 + |
100
4 ENTER 5 × 2 ÷ x² |
| #6 |
21
7 ENTER 2 PROB Cn,r |
18
CLEAR CLSTAT 12 Σ+ 15 Σ+ 18 Σ+ 21 Σ+ 24 Σ+ x̄,ȳ |
1/2 (4/8)
4 ENTER 8 ÷ |
24 ways
5 ENTER 1 - x! |
4 (unchanged)
4 ENTER 0 + |
2
1 ENTER 1 ENTER .1.2 - ÷ |
| #7 |
36
9 ENTER 7 PROB Cn,r |
8
6 ENTER 10 + 2 ÷ |
100
5 ENTER 88 × 4 ENTER 85 × - |
9 diagonals
6 ENTER 2 PROB Cn,r 6 - |
8
4 ENTER 2 × |
9
6 ENTER 1 ENTER .1.3 - ÷ |
| #8 |
56
8 ENTER 3 PROB Cn,r |
8 (12 - 4)
12 ENTER 4 - |
100
10 x² |
20 diagonals
8 ENTER 2 PROB Cn,r 8 - |
87
0.3 ENTER 80 × 0.7 ENTER 90 × + |
4 I's
11 ENTER 1 - 4 - 2 - |
| #9 |
16
2 ENTER 4 yˣ |
5
15 ENTER 5 - 2 ÷ |
32 patterns
2 ENTER 5 yˣ |
0.81
0.9 x² |
30
2.5 ENTER 8 × 10 + |
48
4 x! 2 x! × |
| #10 |
3
3 x! 2 x! ÷ |
210
20 ENTER 21 × 2 ÷ |
Order doesn't matter (10)
5 ENTER 3 PROB Cn,r |
15/16
1 ENTER 1 ENTER 2 ENTER 4 yˣ ÷ - |
3
33 ENTER 30 - |
5 (count n)
CLEAR CLSTAT 2 Σ+ 4 Σ+ 6 Σ+ 8 Σ+ 10 Σ+ |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Evaluate: 4! = |
10 pts |
24 |
4 x! |
| 2 |
Core Fluency |
Evaluate: 6P2 = |
10 pts |
30 |
6 ENTER 2 PROB Pn,r |
| 3 |
Core Fluency |
Evaluate: 6C3 = |
10 pts |
20 |
6 ENTER 3 PROB Cn,r |
| 4 |
Core Fluency |
Find the sum of Row 5 of Pascal's Triangle: |
10 pts |
32 |
2 ENTER 5 yˣ |
| 5 |
Core Fluency |
Find the Mean of {14, 18, 22, 26}: |
10 pts |
20 |
CLEAR CLSTAT 14 Σ+ 18 Σ+ 22 Σ+ 26 Σ+ x̄,ȳ |
| 6 |
Core Fluency |
Find the Sample SD of {6, 12, 18}: |
10 pts |
6 |
12 ENTER 6 - |
| 7 |
Application |
How many 2-person lab pairs can be chosen from 9 students? |
10 pts |
36 pairs |
9 ENTER 2 PROB Cn,r |
| 8 |
Application |
A fair coin is flipped 3 times. What is P(at least 1 Tail)? |
10 pts |
7/8 (0.875) |
1 ENTER 0.5 ENTER 3 yˣ - |
| 9 |
Application |
Evaluate the arithmetic series: 1 + 2 + 3 + ... + 15 = |
10 pts |
120 |
15 ENTER 16 × 2 ÷ |
| 10 |
Error Analysis |
Error Analysis: Why is 10C8 equal to 10C2 = 45 without multiplying 8 factors? |
10 pts |
Choosing 8 leaves 2 out (45) |
10 ENTER 2 PROB Cn,r |
5. STEM Lab Case Study: Cyber Security Password Entropy Cryptography
Scenario: Companion to Vol. II, Ch. 9 & Ch. 10: Security engineers and statisticians calculate key entropy, combinatorial search spaces, and small-sample distributions using [PROB] ([Pn,r], [Cn,r]) and [Σ+].
- Case Question 1: Ordered PIN Sequence: How many 3-digit ordered PINs can be formed from 10 distinct digits without repetition, P(10, 3)?
- Expected Result: 720 | Keystrokes:
10 ENTER 3 PROB Pn,r
- Case Question 2: Unordered Backup Keys: How many unordered 3-key subsets can be chosen from 10 master keys, C(10, 3)?
- Expected Result: 120 | Keystrokes:
10 ENTER 3 PROB Cn,r
- Case Question 3: Cryptographic Token Pool: How many 5-token subsets can be drawn from a 52-token hardware pool, C(52, 5)?
- Expected Result: 2,598,960 | Keystrokes:
52 ENTER 5 PROB Cn,r
- Case Question 4: Error Analysis: Student uses P(6, 2) = 30 for a 2-person committee from 6 people. Find true C(6, 2):
- Expected Result: 15 committees (C(6, 2) = 15). | Keystrokes:
6 ENTER 2 PROB Cn,r
Unit 10: Digital Logic, Base Conversions & Computer Architecture
Grades: Grade 9 – 12 & AP (AP CSP & CS) | Instrument: StackCalc SC-32 | Visual / Block Model: Place Value Studio (Binary [0]/[1] Bit Sockets) & SC-32 [BASE]
Key Standards: CCSS.MATH.HSN.RN.A.1, CSTA 3A-DA-07
1. Concept Labs & Warm-Up Solutions
- Warm-Up A (Core Lab): Express decimal 255 as a sum of powers of 2 to write its 8-bit binary and 2-digit hexadecimal representations.
- Verified Answer:
Binary: 11111111_2 (eight 1s); Hexadecimal: FF_16 (15 × 16 + 15 = 255).
- Keystrokes:
2 5 5 BASE
- Warm-Up B (Extension Lab): Compare 12 and 3 column by column in 4-bit binary to find the result of 12 OR 3 in both binary and decimal.
- Verified Answer:
1100_2 OR 0011_2 = 1111_2 = 15_10.
- Keystrokes:
1 2 ENTER 3 +
2. Hands-On Lab Investigations (Investigations 1–4)
| # |
Investigation Prompt |
Expected Solution |
Calculator Verification Keys |
| 1 |
Decompose decimal 42 into powers of 2 to write its 6-bit binary representation. |
32 + 8 + 2 -> 101010_2 |
4 2 BASE |
| 2 |
Convert hexadecimal 2A_16 to decimal using place value. |
32 + 10 = 42 |
2 ENTER 1 6 × 1 0 + BASE |
| 3 |
Evaluate the bitwise AND of 15 and 7 in binary and decimal. |
0111_2 = 7 |
1 5 ENTER 8 ÷R x⇄y |
| 4 |
Evaluate the bitwise XOR of 12 and 10 in binary and decimal. |
0110_2 = 6 |
4 ENTER 2 + |
3. Leveled Practice Worksheets (Practice A through F)
| Problem # |
Practice A (Core) |
Practice B (Multi-Step) |
Practice C (Real-World) |
Practice D (Mental Math) |
Practice E (STEM Apps) |
Practice F (Mastery) |
| #1 |
5
4 ENTER 1 + |
2 (0010₂)
6 ENTER 4 - |
256 (0 to 255)
2 ENTER 8 yˣ |
0111₂ (7)
5 ENTER 2 + |
-1
255 ENTER 256 - |
10 bits
1024 LN 2 LN ÷ |
| #2 |
11
8 ENTER 2 + 1 + |
7 (0111₂)
6 ENTER 1 + |
1,023
2 ENTER 10 yˣ 1 - |
1111₂ (15)
12 ENTER 3 + |
-2
254 ENTER 256 - |
65,536 (64 KB)
2 ENTER 16 yˣ |
| #3 |
15 (F₁₆)
8 ENTER 4 + 2 + 1 + |
5 (0101₂)
4 ENTER 1 + |
255
15 ENTER 16 × 15 + |
10000₂ (16)
9 ENTER 7 + |
FC₁₆
256 ENTER 4 - |
7 ((NOT A) OR (NOT B))
15 ENTER 8 - |
| #4 |
1001₂
8 ENTER 1 + |
4 (0100₂)
12 ENTER 8 - |
128
8 ENTER 16 × |
1001₂ (9)
14 ENTER 5 - |
-128 (80₁₆)
2 ENTER 7 yˣ +/- |
0
25 ENTER 0 × |
| #5 |
1110₂ (E₁₆)
8 ENTER 4 + 2 + |
13 (1101₂)
12 ENTER 1 + |
192
192 ENTER 16 ÷R 16 × |
F₁₆ (15)
10 ENTER 5 + |
+127 (7F₁₆)
2 ENTER 7 yˣ 1 - |
25 (x OR 0 = x)
25 ENTER 0 + |
| #6 |
26
1 ENTER 16 × 10 + |
6 (0110₂)
15 ENTER 9 - |
255 (FF₁₆)
170 ENTER 85 + |
10₁₆ (16)
12 ENTER 4 + |
12 (C₁₆)
201 ENTER 16 ÷R |
12 (1100₂)
8 ENTER 4 + |
| #7 |
47
2 ENTER 16 × 15 + |
0
42 ENTER 42 - |
0
85 ENTER 85 - |
20₁₆ (32)
31 ENTER 1 + |
13 (1101₂)
9 ENTER 4 + |
10 (original M!)
8 ENTER 2 + |
| #8 |
255
15 ENTER 16 × 15 + |
40
5 ENTER 2 ENTER 3 yˣ × |
FFF₁₆
2 ENTER 12 yˣ 1 - |
0F₁₆ (15)
255 ENTER 240 - |
14 (1110₂)
15 ENTER 1 - |
5 ones (Odd parity)
1 ENTER 1 + 1 + 1 + 1 + |
| #9 |
C7₁₆
12 ENTER 16 × 7 + |
12
48 ENTER 2 x² ÷ |
1000₂ (8)
7 ENTER 1 + |
111101₂
7 ENTER 8 × 5 + |
55
255 ENTER 200 - |
70
7 ENTER 8 × 7 ENTER 2 × + |
| #10 |
21
2 ENTER 8 × 5 + |
FF₁₆ (255)
256 ENTER 1 - |
0
8 ENTER 8 - |
56₈
5 ENTER 10 × 6 + |
0
16 ENTER 16 ÷R x⇄y |
2A (42₁₀)
2 ENTER 16 × 10 + |
4. 100-Point Unit Mastery Check
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Convert binary 1101₂ to decimal: |
10 pts |
13 |
8 ENTER 4 + 1 + |
| 2 |
Core Fluency |
Convert hexadecimal 3C₁₆ to decimal: |
10 pts |
60 |
3 ENTER 16 × 12 + |
| 3 |
Core Fluency |
Convert binary 11110101₂ to hexadecimal: |
10 pts |
F5₁₆ |
15 ENTER 16 × 5 + |
| 4 |
Core Fluency |
Evaluate: 10 AND 6 = |
10 pts |
2 |
2 ENTER 1 × |
| 5 |
Core Fluency |
Evaluate: 10 OR 6 = |
10 pts |
14 |
8 ENTER 4 + 2 + |
| 6 |
Core Fluency |
Evaluate: 10 XOR 6 = |
10 pts |
12 |
8 ENTER 4 + |
| 7 |
Application |
Evaluate left shift: 9 << 2 = |
10 pts |
36 |
9 ENTER 2 x² × |
| 8 |
Application |
What is the maximum unsigned decimal value stored in 4 binary bits? |
10 pts |
15 |
2 ENTER 4 yˣ 1 - |
| 9 |
Application |
Extract the low nibble: What is 0x4D AND 0x0F in decimal? |
10 pts |
13 |
77 ENTER 16 ÷R x⇄y |
| 10 |
Error Analysis |
Error Analysis: Why does grouping binary into 4-bit nibbles convert directly to Hex? |
10 pts |
2⁴ = 16 |
2 ENTER 4 yˣ |
5. STEM Lab Case Study: Microcontroller Firmware Register Masking
Scenario: Companion to Vol. II, Ch. 1, Ch. 10 & Ch. 13: Firmware engineers configure GPIO pins on the RP2350 microcontroller using hexadecimal masks and Shannon bitwise AND, OR, and XOR.
- Case Question 1: Full Port Enable: Convert decimal 255 to hexadecimal and binary for an 8-pin GPIO port:
- Expected Result: 0xFF = 255 (BIN: 11111111) | Keystrokes:
255 ENTER BASE
- Case Question 2: Alternating Pin Merge: Combine odd-pin mask 0xAA and even-pin mask 0x55 using bitwise OR:
- Expected Result: 0xFF = 255 (11111111) | Keystrokes:
170 ENTER 85 +
- Case Question 3: Lower-Nybble Status Mask: Isolate the lower 4 status bits of 0xFF using bitwise AND with 0x0F:
- Expected Result: 0x0F = 15 (00001111) | Keystrokes:
255 ENTER 16 ÷R x⇄y
- Case Question 4: Error Analysis: Student adds 0x0F + 1 as 0x10 and claims it is 10 decimal. Find 15 + 1 in decimal:
- Expected Result: 16 (0x10 = 16 decimal). | Keystrokes:
15 ENTER 1 +
Appendix C: Cumulative Benchmark & Capstone Mastery Exams (Pages 117–120)
Cumulative Exam 1: Units 01–05 Benchmark (p. 117 • 100 Points)
Covers Linear Slope & Distance, Quadratic Discriminants, [POLY] Roots, Horner's Nested Polynomial Evaluation, and Exponential Growth/Decay.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Coordinate Slope on the Stack: Find the slope m through (2, 5) and (6, 21): |
10 pts |
4 |
21 ENTER 5 - 6 ENTER 2 - ÷ |
| 2 |
Core Fluency |
2D Euclidean Distance: Find the Euclidean distance d between (1, 2) and (7, 10): |
10 pts |
10 |
7 ENTER 1 - x² 10 ENTER 2 - x² + √x |
| 3 |
Core Fluency |
Quadratic Discriminant: Evaluate Δ = b² − 4ac for 2x² − 7x + 3 = 0: |
10 pts |
25 |
-7 x² 4 ENTER 2 × 3 × - |
| 4 |
Core Fluency |
Quadratic Roots via [POLY]: Find the larger root of x² − 7x + 12 = 0: |
10 pts |
4 |
POLY 1 B -7 C 12 D XEQ |
| 5 |
Core Fluency |
Horner's Nested Polynomial: Evaluate P(x) = 2x³ − 3x² + 4x − 5 at x = 3 using Horner's nested method: |
10 pts |
34 |
2 ENTER 3 × 3 - 3 × 4 + 3 × 5 - |
| 6 |
Core Fluency |
Difference of Cubes Evaluation: Compute 5³ − 2³ on the SC-32 stack: |
10 pts |
117 |
5 ENTER 3 yˣ 2 ENTER 3 yˣ - |
| 7 |
Application |
Compound Growth: Find the balance A of $1,000 after 2 years of 5% annual compounding: |
10 pts |
1102.5 |
1 ENTER 0.05 + 2 yˣ 1000 × |
| 8 |
Application |
Exponential Half-Life Decay: Starting with 160 g, how many grams remain after 4 half-lives? |
10 pts |
10 |
0.5 ENTER 4 yˣ 160 × |
| 9 |
Application |
Parallel Resistor Network: Find the equivalent resistance R of a 6 Ω resistor and a 12 Ω resistor in parallel: |
10 pts |
4 |
6 x⁻¹ 12 x⁻¹ + x⁻¹ |
| 10 |
Error Analysis |
Error Spotter: A student squared −6 by typing 6 [x²] [+/-] and got −36. What is the true value of (−6)²? |
10 pts |
36 |
-6 x² |
Cumulative Exam 2: Units 06–10 Benchmark (p. 118 • 100 Points)
Covers Logarithms & Base e, Right-Triangle Trigonometry, 2D Polar Vectors [▸θ,r], Combinatorics [PROB] ([Cn,r]/[Pn,r]), Statistics [Σ+], and Digital Logic.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Logarithmic Prime Tower Identity: Evaluate log₁₀(1000) + log₁₀(100) using [LOG] on the SC-32: |
10 pts |
5 |
1000 LOG 100 LOG + |
| 2 |
Core Fluency |
Continuous Doubling Time: Find the continuous doubling time t (to 4 decimal places) for 5% continuous growth: |
10 pts |
13.8629 |
2 LN 0.05 ÷ |
| 3 |
Core Fluency |
Natural Log & Base-e Inverse: Evaluate e^(ln(25)) + 10 on the SC-32: |
10 pts |
35 |
25 LN eˣ 10 + |
| 4 |
Core Fluency |
Pythagorean Trig Identity: Evaluate sin²(30°) + cos²(30°) on the stack: |
10 pts |
1 |
30 SIN x² 30 COS x² + |
| 5 |
Core Fluency |
Right-Triangle Ramp Height: A 20 m ramp is inclined at 30°. Find the vertical rise h: |
10 pts |
10 |
20 ENTER 30 SIN × |
| 6 |
Core Fluency |
2D Vector Magnitude via [▸θ,r]: Convert rectangular vector (x = 5, y = 12) to polar magnitude r: |
10 pts |
13 |
12 ENTER 5 ▸θ,r |
| 7 |
Application |
Combinatorics Committee Count: How many 3-person teams can be chosen from 8 engineers, C(8, 3)? |
10 pts |
56 |
8 ENTER 3 PROB Cn,r |
| 8 |
Application |
Permutation Podium Orderings: In how many ways can 1st, 2nd, and 3rd place be awarded among 6 finalists, P(6, 3)? |
10 pts |
120 |
6 ENTER 3 PROB Pn,r |
| 9 |
Application |
Sample Mean via [Σ+]: Accumulate {14, 18, 22, 26} with [Σ+] and compute x̄ with [x̄,ȳ]: |
10 pts |
20 |
14 Σ+ 18 Σ+ 22 Σ+ 26 Σ+ x̄,ȳ |
| 10 |
Error Analysis |
Bitwise Masking (AP CSP): Apply bitwise AND between 0xF0 and 0x3C (report answer in decimal): |
10 pts |
48 |
60 ENTER 16 ÷R 16 × |
Cumulative Exam 3: Multi-Step Synthesis (p. 119 • 100 Points)
Synthesizes Difference Quotients, Law of Cosines, 3D Vectors, pH & Decibel Scales, Binomial Counts, and Bitwise Register Operations.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Difference Quotient (Average Rate of Change): Evaluate (f(5) − f(2)) ÷ (5 − 2) for f(x) = x²: |
10 pts |
7 |
5 x² 2 x² - 5 ENTER 2 - ÷ |
| 2 |
Core Fluency |
Quadratic Vertex Output: Evaluate y = 3x² − 12x + 17 at its axis of symmetry x = 2: |
10 pts |
5 |
3 ENTER 2 x² × 12 ENTER 2 × - 17 + |
| 3 |
Core Fluency |
Law of Cosines Side c: For a triangle with sides a = 5, b = 8, and included angle C = 60°, find side c: |
10 pts |
7 |
5 x² 8 x² + 2 ENTER 5 × 8 × 60 COS × - √x |
| 4 |
Core Fluency |
3D Space Diagonal: Find the space diagonal d of a 3 × 4 × 12 rectangular prism: |
10 pts |
13 |
3 x² 4 x² + 12 x² + √x |
| 5 |
Core Fluency |
pH Hydrogen Ion Scale: Calculate the pH of a solution with hydrogen ion concentration [H⁺] = 0.0001 M: |
10 pts |
4 |
0.0001 LOG +/- |
| 6 |
Core Fluency |
Decibel Gain Ratio: Find the decibel gain dB = 10 · log₁₀(P₂ / P₁) for a 1000× power ratio: |
10 pts |
30 |
1000 LOG 10 × |
| 7 |
Application |
Binomial Probability Count: Multiply C(6, 2) × 2² to count weighted 2-success paths: |
10 pts |
60 |
6 ENTER 2 PROB Cn,r 2 x² × |
| 8 |
Application |
Factorial Ratio: Evaluate 7! ÷ 5! on the SC-32 stack: |
10 pts |
42 |
7 x! 5 x! ÷ |
| 9 |
Application |
Bitwise OR Register Merge: Combine upper nybble 0xA0 and lower nybble 0x05 using bitwise OR (report in decimal): |
10 pts |
165 |
160 ENTER 5 + |
| 10 |
Error Analysis |
Bitwise XOR Toggle: Toggle bits of 0xFF with mask 0x0F using bitwise XOR (report in decimal): |
10 pts |
240 |
255 ENTER 15 - |
Cumulative Exam 4: Final Capstone Mastery (p. 120 • 100 Points)
Final Grades 9–12 & AP mastery assessment integrating Physics Kinematics, Harmonic & RMS Means, Golden Ratio Radicals, and Stack Architecture.
| # |
Category |
Assessment Prompt |
Points |
Expected Solution |
Verification Keys |
| 1 |
Core Fluency |
Root-Mean-Square (RMS) Speed: Compute the RMS speed of 6 m/s and 8 m/s on the 4-level stack: |
10 pts |
7.0711 |
6 x² 8 x² + 2 ÷ √x |
| 2 |
Core Fluency |
Kinetic Energy Formula: Find the kinetic energy KE (in Joules) for mass m = 4 kg and velocity v = 15 m/s: |
10 pts |
450 |
0.5 ENTER 4 × 15 x² × |
| 3 |
Core Fluency |
Projectile Range at 45°: Compute R = (v² · sin(90°)) ÷ g for v = 28 m/s and g = 9.8 m/s²: |
10 pts |
80 |
28 x² 90 SIN × 9.8 ÷ |
| 4 |
Core Fluency |
Continuous Compounding Return: Evaluate A = 500 · e^(0.05 × 20) (to 2 decimal places) on the SC-32: |
10 pts |
1359.14 |
0.05 ENTER 20 × eˣ 500 × |
| 5 |
Core Fluency |
Harmonic Mean of Two Rates: Find the average round-trip speed for 30 mph outbound and 60 mph return: |
10 pts |
40 |
2 ENTER 30 x⁻¹ 60 x⁻¹ + ÷ |
| 6 |
Core Fluency |
Golden Ratio Quadratic Root: Compute (1 + √5) ÷ 2 to 4 decimal places on the SC-32: |
10 pts |
1.618 |
1 ENTER 5 √x + 2 ÷ |
| 7 |
Application |
Polar-to-Rectangular X Component: For polar magnitude r = 10 and angle θ = 60°, find the horizontal component x: |
10 pts |
5 |
10 ENTER 60 COS × |
| 8 |
Application |
Acoustics Inverse-Square Ratio: If initial sound intensity is 120 units and drops by a factor of 2 per doubling, find the intensity after 3 doublings: |
10 pts |
15 |
120 ENTER 2 ENTER 3 yˣ ÷ |
| 9 |
Application |
Byte Address Offset: Compute base address 256 + (4 × 16) for a 32-bit register array: |
10 pts |
320 |
256 ENTER 4 ENTER 16 × + |
| 10 |
Error Analysis |
Capstone Error Analysis: To evaluate (20 + 16) ÷ (11 − 2), a student keyed 20 [ENTER] 16 [+] 11 [-] 2 [÷] and got 12.5. Find the true value: |
10 pts |
4 |
20 ENTER 16 + 11 ENTER 2 - ÷ |
⚡ Book 5: Stack Speed & Fluency (Full-Year 180-Test Edition)
Complete solutions for all 15 Speed Lessons, 90 Warm-Up Drills, 180 Timed Tests (3,600 problems), Appendix B Sprint Drills, and Champion Pipelines.
Section 1: Section 1: Step Counting, Benchmarks & Sequences
Grades: Foundational to Advanced | Focus: Unit steps, skip-counting, 10/100 benchmarks, and arithmetic series.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
76 + 4 + 19 = |
Complete 80 first, then + 19 |
99 |
| W2 |
150 - 25 - 25 = |
Combine 25 + 25 = 50, subtract from 150 |
100 |
| W3 |
1 + 3 + 5 + 7 = |
4 consecutive odd numbers -> 4² |
16 |
| W4 |
10 + 15 + 20 + 25 + 30 = |
5 terms × middle term (20) |
100 |
| W5 |
1 + 2 + 3 + ... + 8 = |
Gauss rule: 8 × 9 ÷ 2 |
36 |
| W6 |
2 + 4 + 6 + 8 + 10 = |
n(n + 1) for 5 even terms: 5 × 6 |
30 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
4 |
9 |
15 |
6 |
25 |
60 |
36 |
69 |
96 |
15 |
36 |
64 |
| #2 |
6 |
9 |
16 |
8 |
50 |
18 |
48 |
60 |
80 |
30 |
49 |
81 |
| #3 |
9 |
15 |
7 |
10 |
21 |
32 |
43 |
99 |
126 |
25 |
64 |
100 |
| #4 |
5 |
4 |
10 |
15 |
28 |
16 |
51 |
80 |
100 |
30 |
81 |
121 |
| #5 |
9 |
4 |
11 |
30 |
18 |
50 |
70 |
129 |
156 |
40 |
100 |
144 |
| #6 |
6 |
11 |
17 |
16 |
45 |
100 |
55 |
100 |
120 |
75 |
121 |
169 |
| #7 |
9 |
13 |
5 |
14 |
110 |
36 |
125 |
124 |
151 |
28 |
115 |
130 |
| #8 |
7 |
13 |
6 |
25 |
12 |
20 |
80 |
120 |
140 |
52 |
125 |
140 |
| #9 |
4 |
5 |
11 |
15 |
24 |
14 |
135 |
154 |
181 |
90 |
135 |
150 |
| #10 |
7 |
8 |
14 |
35 |
12 |
35 |
100 |
140 |
160 |
56 |
145 |
160 |
| #11 |
6 |
8 |
15 |
15 |
40 |
90 |
100 |
184 |
211 |
150 |
155 |
170 |
| #12 |
12 |
15 |
7 |
20 |
80 |
27 |
90 |
160 |
180 |
36 |
165 |
180 |
| #13 |
13 |
17 |
9 |
30 |
30 |
44 |
245 |
214 |
206 |
60 |
175 |
190 |
| #14 |
14 |
3 |
10 |
30 |
40 |
22 |
165 |
180 |
200 |
50 |
45 |
55 |
| #15 |
14 |
9 |
15 |
50 |
10 |
30 |
200 |
209 |
236 |
250 |
66 |
78 |
| #16 |
13 |
12 |
18 |
30 |
25 |
60 |
255 |
200 |
220 |
90 |
91 |
105 |
| #17 |
12 |
12 |
5 |
22 |
70 |
24 |
300 |
239 |
266 |
70 |
120 |
136 |
| #18 |
10 |
5 |
11 |
60 |
21 |
32 |
300 |
220 |
240 |
350 |
153 |
171 |
| #19 |
16 |
7 |
13 |
55 |
36 |
20 |
480 |
269 |
296 |
55 |
190 |
210 |
| #20 |
16 |
7 |
14 |
100 |
18 |
50 |
300 |
240 |
260 |
110 |
231 |
253 |
3. Optimal Champion Pipelines
- 1C #11 (68+12+20):
Pair 68 + 12 = 80 first, then + 20 = 100
- 1D #3 (1+3+5+7+9):
5 odd terms -> 5² = 25
- 1D #16 (14+16+18+20+22):
5 × middle term (18) = 90
- 1D #19 (1+2+...+10):
10 × 11 ÷ 2 = 55 ([10] [ENTER] [11] [×] [2] [÷])
Section 2: Section 2: Number Bonds & Missing Addends
Grades: Foundational to Advanced | Focus: Part-part-whole composition, decomposition, and missing-part equations.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
9 + [ ? ] = 17 |
17 - 9 (or 1 + 7) |
8 |
| W2 |
36 + [ ? ] = 60 |
Count up: +4 to 40, +20 to 60 |
24 |
| W3 |
82 - 35 = |
82 - 30 = 52, then 52 - 5 |
47 |
| W4 |
64 + 29 + 36 = |
Pair 64 + 36 = 100 first, then + 29 |
129 |
| W5 |
200 - 68 - 32 = |
Combine 68 + 32 = 100, then 200 - 100 |
100 |
| W6 |
85 + 45 = 100 + [ ? ] |
85 + 45 = 130, minus 100 |
30 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
7 |
10 |
16 |
14 |
9 |
14 |
39 |
21 |
28 |
190 |
117 |
119 |
| #2 |
7 |
10 |
6 |
13 |
12 |
16 |
59 |
26 |
33 |
155 |
120 |
122 |
| #3 |
8 |
13 |
11 |
15 |
13 |
7 |
64 |
59 |
70 |
65 |
123 |
125 |
| #4 |
8 |
9 |
12 |
15 |
4 |
9 |
63 |
36 |
43 |
82 |
126 |
128 |
| #5 |
9 |
8 |
14 |
17 |
9 |
13 |
33 |
41 |
48 |
175 |
129 |
131 |
| #6 |
9 |
8 |
11 |
7 |
8 |
13 |
36 |
83 |
58 |
135 |
132 |
134 |
| #7 |
9 |
18 |
9 |
6 |
10 |
4 |
37 |
15 |
22 |
129 |
135 |
137 |
| #8 |
4 |
7 |
10 |
8 |
6 |
10 |
35 |
20 |
27 |
70 |
138 |
140 |
| #9 |
3 |
13 |
12 |
7 |
14 |
8 |
22 |
71 |
82 |
100 |
141 |
143 |
| #10 |
5 |
6 |
9 |
9 |
5 |
10 |
26 |
30 |
37 |
30 |
144 |
146 |
| #11 |
3 |
16 |
14 |
8 |
15 |
7 |
33 |
35 |
42 |
40 |
36 |
42 |
| #12 |
11 |
5 |
8 |
9 |
9 |
14 |
43 |
95 |
106 |
35 |
39 |
45 |
| #13 |
11 |
11 |
17 |
9 |
11 |
5 |
34 |
45 |
16 |
185 |
42 |
48 |
| #14 |
11 |
11 |
7 |
9 |
12 |
16 |
56 |
14 |
21 |
320 |
45 |
51 |
| #15 |
12 |
14 |
12 |
11 |
4 |
9 |
83 |
83 |
94 |
200 |
48 |
54 |
| #16 |
12 |
10 |
6 |
7 |
6 |
11 |
95 |
24 |
31 |
275 |
51 |
57 |
| #17 |
12 |
9 |
15 |
14 |
9 |
13 |
37 |
29 |
36 |
265 |
54 |
60 |
| #18 |
6 |
9 |
12 |
7 |
10 |
4 |
52 |
107 |
118 |
180 |
57 |
63 |
| #19 |
5 |
12 |
10 |
15 |
12 |
6 |
45 |
39 |
46 |
275 |
60 |
66 |
| #20 |
4 |
8 |
11 |
9 |
6 |
10 |
46 |
44 |
15 |
400 |
63 |
69 |
3. Optimal Champion Pipelines
- 2C #17 (63 + ? = 100):
9-from-9, 10-from-10 rule -> 37
- 2D #6 (48 + 35 + 52):
Group (48 + 52) + 35 = 100 + 35 = 135
- 2D #9 (200 - 68 - 32):
200 - (68 + 32) = 200 - 100 = 100
- 2D #11 (76 + 54 = 90 + ?):
[76] [ENTER] [54] [+] [90] [-] -> 40
Section 3: Section 3: Complements & Make-10 Bridging
Grades: Foundational to Advanced | Focus: Complements to 10, 100, and 1,000, bridging across tens, and compensation.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
100 - 64 = |
9-from-9 (9-6=3), 10-from-10 (10-4=6) |
36 |
| W2 |
57 + 19 = |
Add 20 (77), subtract 1 |
76 |
| W3 |
1000 - 384 = |
9-3=6, 9-8=1, 10-4=6 |
616 |
| W4 |
10 - 3.65 = |
9 - 3 = 6 and 100 - 65 = .35 |
6.35 |
| W5 |
299 + 156 = |
300 + 155 |
455 |
| W6 |
99 + 98 + 97 = |
300 - (1 + 2 + 3) |
294 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
3 |
7 |
5 |
17 |
25 |
33 |
70 |
84 |
80 |
600 |
842 |
799 |
| #2 |
6 |
6 |
4 |
15 |
44 |
55 |
55 |
75 |
71 |
350 |
755 |
712 |
| #3 |
2 |
5 |
3 |
13 |
39 |
47 |
38 |
66 |
62 |
725 |
668 |
625 |
| #4 |
7 |
4 |
11 |
14 |
67 |
78 |
72 |
57 |
53 |
616 |
581 |
538 |
| #5 |
1 |
3 |
10 |
13 |
53 |
77 |
26 |
48 |
44 |
258 |
494 |
451 |
| #6 |
4 |
2 |
9 |
16 |
81 |
92 |
81 |
39 |
35 |
481 |
407 |
364 |
| #7 |
8 |
10 |
8 |
8 |
23 |
31 |
17 |
30 |
26 |
137 |
320 |
277 |
| #8 |
5 |
9 |
7 |
23 |
44 |
55 |
43 |
21 |
17 |
903 |
233 |
190 |
| #9 |
14 |
15 |
16 |
26 |
37 |
45 |
45 |
88 |
91 |
6.6 |
872 |
985 |
| #10 |
17 |
17 |
18 |
23 |
67 |
78 |
67 |
72 |
75 |
3.25 |
988 |
1101 |
| #11 |
19 |
12 |
13 |
34 |
67 |
75 |
66 |
86 |
89 |
57.5 |
606 |
719 |
| #12 |
13 |
14 |
15 |
37 |
81 |
92 |
84 |
100 |
103 |
146 |
722 |
835 |
| #13 |
15 |
16 |
17 |
42 |
21 |
29 |
67 |
84 |
87 |
163 |
840 |
953 |
| #14 |
14 |
18 |
12 |
24 |
44 |
55 |
93 |
98 |
101 |
283 |
956 |
1069 |
| #15 |
12 |
17 |
19 |
32 |
35 |
43 |
64 |
112 |
115 |
455 |
28.5 |
38.25 |
| #16 |
13 |
16 |
18 |
36 |
67 |
78 |
63 |
86 |
89 |
735 |
38 |
47.75 |
| #17 |
15 |
18 |
14 |
23 |
65 |
73 |
66 |
78 |
81 |
297 |
47.5 |
57.25 |
| #18 |
12 |
14 |
16 |
55 |
81 |
92 |
125 |
70 |
73 |
291 |
57 |
66.75 |
| #19 |
14 |
13 |
15 |
67 |
19 |
27 |
139 |
92 |
95 |
600 |
66.5 |
76.25 |
| #20 |
16 |
15 |
17 |
47 |
44 |
55 |
136 |
84 |
87 |
3355 |
76 |
85.75 |
3. Optimal Champion Pipelines
- 3C #6 (100 - 19):
9-1=8, 10-9=1 -> 81
- 3D #5 (1000 - 742):
Left-to-right 9-9-10 rule -> 258
- 3D #16 (498 + 237):
500 + 235 = 735
- 3D #18 (98 + 97 + 96):
300 - (2 + 3 + 4) = 291
Section 4: Section 4: Doubles, Equal Groups & Squares
Grades: Foundational to Advanced | Focus: Doubles, near-doubles, equal groups, halving, and mental squaring.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
2 × 46 = |
Double 40 (80) + double 6 (12) |
92 |
| W2 |
4 × 18 = |
Double 18 -> 36, double 36 -> 72 |
72 |
| W3 |
15² = |
Ends-in-5 rule: 1 × 2 = 2, append 25 |
225 |
| W4 |
25² = |
Ends-in-5 rule: 2 × 3 = 6, append 25 |
625 |
| W5 |
19 × 21 = |
20² - 1 = 400 - 1 |
399 |
| W6 |
15² - 5² = |
(15 + 5)(15 - 5) = 20 × 10 |
200 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
6 |
10 |
14 |
26 |
30 |
36 |
54 |
54 |
62 |
121 |
145 |
171 |
| #2 |
8 |
13 |
17 |
28 |
34 |
40 |
76 |
68 |
76 |
144 |
197 |
227 |
| #3 |
10 |
15 |
20 |
30 |
38 |
44 |
92 |
82 |
90 |
169 |
257 |
291 |
| #4 |
12 |
16 |
20 |
32 |
42 |
48 |
130 |
96 |
104 |
196 |
325 |
363 |
| #5 |
14 |
19 |
23 |
36 |
46 |
52 |
156 |
110 |
118 |
225 |
401 |
443 |
| #6 |
16 |
21 |
26 |
50 |
50 |
56 |
60 |
124 |
132 |
256 |
485 |
531 |
| #7 |
18 |
22 |
26 |
70 |
54 |
60 |
72 |
138 |
146 |
400 |
577 |
627 |
| #8 |
20 |
25 |
29 |
90 |
31 |
37 |
92 |
136 |
160 |
625 |
625 |
1225 |
| #9 |
9 |
27 |
32 |
31 |
43 |
50 |
100 |
88 |
100 |
900 |
1225 |
2025 |
| #10 |
11 |
28 |
32 |
49 |
57 |
65 |
140 |
216 |
240 |
1225 |
2025 |
3025 |
| #11 |
13 |
31 |
35 |
12 |
28 |
14 |
96 |
128 |
140 |
99 |
3025 |
4225 |
| #12 |
15 |
33 |
38 |
20 |
17 |
22 |
120 |
296 |
320 |
143 |
4225 |
5625 |
| #13 |
17 |
34 |
38 |
18 |
26 |
32 |
200 |
168 |
180 |
224 |
5625 |
7225 |
| #14 |
19 |
37 |
41 |
30 |
37 |
44 |
37 |
376 |
400 |
399 |
192 |
252 |
| #15 |
22 |
39 |
44 |
32 |
15 |
17 |
48 |
20 |
23 |
624 |
280 |
352 |
| #16 |
24 |
40 |
44 |
42 |
18 |
20 |
65 |
26 |
29 |
899 |
399 |
483 |
| #17 |
21 |
43 |
47 |
12 |
21 |
23 |
21 |
32 |
35 |
450 |
525 |
621 |
| #18 |
23 |
45 |
50 |
18 |
24 |
26 |
30 |
38 |
41 |
244 |
667 |
775 |
| #19 |
12 |
46 |
50 |
25 |
27 |
29 |
78 |
44 |
47 |
200 |
840 |
960 |
| #20 |
18 |
49 |
53 |
24 |
30 |
32 |
50 |
50 |
53 |
400 |
1020 |
1152 |
3. Optimal Champion Pipelines
- 4C #10 (4 × 35):
Double 35 -> 70, double 70 -> 140
- 4D #10 (35²):
3 × 4 = 12, append 25 -> 1225
- 4D #15 (24 × 26):
25² - 1 = 625 - 1 = 624
- 4D #20 (25² - 15²):
(25 + 15)(25 - 15) = 40 × 10 = 400
Section C1: Cumulative Review C1: Number Sense (Sections 1–4)
Grades: Foundational to Advanced | Focus: Integrated review across step counting, number bonds, complements, and doubles.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
1000 - 465 = |
Left-to-right 9-9-10: 5, 3, 5 |
535 |
| W2 |
398 + 165 = |
400 + 163 |
563 |
| W3 |
29 × 31 = |
30² - 1 = 900 - 1 |
899 |
| W4 |
420 ÷ 4 = |
Halve twice: 420 -> 210 -> 105 |
105 |
| W5 |
1 + 3 + 5 + 7 + 9 + 11 + 13 = |
7 odd terms -> 7² |
49 |
| W6 |
45 + 55 + 65 + 75 = |
Pair (45+75) + (55+65) = 120 + 120 |
240 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
8 |
33 |
52 |
27 |
44 |
70 |
63 |
60 |
84 |
21 |
44 |
48 |
| #2 |
10 |
40 |
49 |
20 |
17 |
20 |
32 |
67 |
77 |
42 |
5 |
6 |
| #3 |
9 |
18 |
21 |
9 |
78 |
112 |
84 |
84 |
112 |
535 |
78 |
84 |
| #4 |
5 |
34 |
57 |
8 |
21 |
24 |
36 |
127 |
143 |
218 |
7 |
8 |
| #5 |
2 |
87 |
102 |
16 |
45 |
72 |
76 |
180 |
220 |
446 |
120 |
128 |
| #6 |
7 |
19 |
22 |
32 |
25 |
28 |
55 |
82 |
90 |
563 |
9 |
10 |
| #7 |
16 |
90 |
40 |
32 |
115 |
156 |
96 |
170 |
209 |
196 |
170 |
180 |
| #8 |
14 |
77 |
88 |
48 |
29 |
32 |
76 |
114 |
128 |
625 |
5 |
6 |
| #9 |
16 |
25 |
28 |
35 |
175 |
84 |
120 |
147 |
184 |
399 |
95 |
100 |
| #10 |
13 |
96 |
130 |
21 |
33 |
36 |
55 |
146 |
162 |
899 |
7 |
8 |
| #11 |
13 |
59 |
62 |
24 |
108 |
150 |
23 |
140 |
176 |
5.65 |
147 |
154 |
| #12 |
15 |
26 |
29 |
40 |
37 |
40 |
56 |
135 |
147 |
36.2 |
9 |
10 |
| #13 |
7 |
75 |
108 |
16 |
210 |
266 |
65 |
264 |
312 |
390 |
207 |
216 |
| #14 |
8 |
130 |
143 |
23 |
41 |
44 |
100 |
165 |
179 |
260 |
5 |
6 |
| #15 |
15 |
32 |
35 |
36 |
111 |
160 |
144 |
112 |
150 |
240 |
100 |
104 |
| #16 |
18 |
52 |
84 |
57 |
45 |
48 |
225 |
106 |
116 |
280 |
7 |
8 |
| #17 |
15 |
99 |
108 |
35 |
195 |
252 |
90 |
216 |
261 |
105 |
162 |
168 |
| #18 |
9 |
33 |
36 |
65 |
49 |
52 |
40 |
148 |
160 |
475 |
9 |
10 |
| #19 |
20 |
160 |
68 |
36 |
329 |
150 |
165 |
248 |
297 |
650 |
232 |
240 |
| #20 |
15 |
191 |
206 |
55 |
53 |
56 |
67 |
246 |
264 |
49 |
5 |
6 |
3. Optimal Champion Pipelines
- C1C #16 (15²):
1 × 2 = 2, append 25 -> 225
- C1D #10 (29 × 31):
30² - 1 = 899
- C1D #14 (18² - 8²):
(18 + 8)(18 - 8) = 26 × 10 = 260
- C1D #20 (1+3+...+13):
7 odd numbers -> 7² = 49
Section 5: Section 5: Multi-Step Operations & Grouping
Grades: Grades 3–8+ | Focus: Multiplication, division, inside-out parentheses, and two-branch expressions.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
(18 - 6) × 3 = |
[18] [ENTER] [6] [-] [3] [×] (no extra ENTER!) |
36 |
| W2 |
45 ÷ (12 - 3) = |
Inner 12 - 3 = 9 first, then 45 ÷ 9 |
5 |
| W3 |
(15 × 4) + (18 ÷ 2) = |
60 + 9 (or two-branch stack) |
69 |
| W4 |
((20 - 8) × 4) ÷ 6 = |
12 × 4 = 48, then ÷ 6 |
8 |
| W5 |
(17 × 6) - (13 × 6) = |
Factor out 6: (17 - 13) × 6 |
24 |
| W6 |
(45 × 12) ÷ 18 = |
Cancel 9: (5 × 12) ÷ 2 |
30 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
24 |
33 |
52 |
36 |
44 |
70 |
69 |
60 |
84 |
110 |
44 |
48 |
| #2 |
35 |
40 |
49 |
5 |
17 |
20 |
28 |
67 |
77 |
108 |
5 |
6 |
| #3 |
48 |
18 |
21 |
5 |
78 |
112 |
4 |
84 |
112 |
8 |
78 |
84 |
| #4 |
63 |
34 |
57 |
5 |
21 |
24 |
10 |
127 |
143 |
10 |
7 |
8 |
| #5 |
8 |
87 |
102 |
36 |
45 |
72 |
72 |
180 |
220 |
100 |
120 |
128 |
| #6 |
9 |
19 |
22 |
8 |
25 |
28 |
72 |
82 |
90 |
10 |
9 |
10 |
| #7 |
9 |
90 |
40 |
40 |
115 |
156 |
90 |
170 |
209 |
100 |
170 |
180 |
| #8 |
7 |
77 |
88 |
90 |
29 |
32 |
28 |
114 |
128 |
96 |
5 |
6 |
| #9 |
18 |
25 |
28 |
80 |
175 |
84 |
8 |
147 |
184 |
600 |
95 |
100 |
| #10 |
5 |
96 |
130 |
10 |
33 |
36 |
56 |
146 |
162 |
5 |
7 |
8 |
| #11 |
18 |
59 |
62 |
8 |
108 |
150 |
24 |
140 |
176 |
15 |
147 |
154 |
| #12 |
8 |
26 |
29 |
40 |
37 |
40 |
32 |
135 |
147 |
32 |
9 |
10 |
| #13 |
27 |
75 |
108 |
30 |
210 |
266 |
17 |
264 |
312 |
31 |
207 |
216 |
| #14 |
22 |
130 |
143 |
25 |
41 |
44 |
120 |
165 |
179 |
24 |
5 |
6 |
| #15 |
8 |
32 |
35 |
35 |
111 |
160 |
6 |
112 |
150 |
90 |
100 |
104 |
| #16 |
10 |
52 |
84 |
10 |
45 |
48 |
60 |
106 |
116 |
30 |
7 |
8 |
| #17 |
20 |
99 |
108 |
40 |
195 |
252 |
60 |
216 |
261 |
120 |
162 |
168 |
| #18 |
20 |
33 |
36 |
48 |
49 |
52 |
5 |
148 |
160 |
16 |
9 |
10 |
| #19 |
30 |
160 |
68 |
5 |
329 |
150 |
10 |
248 |
297 |
10 |
232 |
240 |
| #20 |
5 |
191 |
206 |
72 |
53 |
56 |
120 |
246 |
264 |
10 |
5 |
6 |
3. Optimal Champion Pipelines
- 5C #5 ((8+4)×(9-3)):
[8] [ENTER] [4] [+] [9] [ENTER] [3] [-] [×] -> 72
- 5C #12 (100-(16×5-12)):
Inner 80 - 12 = 68 first, 100 - 68 = 32
- 5D #15 ((28×9)-(18×9)):
Distributive shortcut: (28 - 18) × 9 = 90
- 5D #16 ((45×12)÷18):
Cancel common factors: 45 × (12/18) = 45 × 2/3 = 30
Section 6: Section 6: Fractions, Decimals & Reciprocals
Grades: Grades 4–9+ | Focus: Fraction equivalence, unlike denominators, mixed numbers, decimals, and reciprocals.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
3/4 ÷ 1/2 (decimal) = |
3/4 is 1/2 of tower -> double to 6/4 = 1.5 |
1.5 |
| W2 |
3/4 + (1/2 × 1/4) (decimal) = |
0.75 + 0.125 |
0.875 |
| W3 |
1 ÷ (1/3 + 1/6) = |
1/3 + 1/6 = 1/2, reciprocal is 2 |
2 |
| W4 |
(1/3 + 1/4) × 24 = |
Distribute 24: 8 + 6 |
14 |
| W5 |
2/3 × 3/4 × 4/5 = |
Telescoping cancel 3 and 4 -> 2/5 |
2/5 |
| W6 |
36 × (7/9 - 1/4) = |
Distribute 36: 28 - 9 |
19 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
3/4 |
3/5 |
2/3 |
1/2 |
5/6 |
11/10 |
3.75 |
10 |
21 |
1 |
3/2 |
2 |
| #2 |
2/5 |
5/6 |
3/4 |
1/2 |
3/10 |
8/15 |
2.25 |
5 |
6.5 |
2 |
8/3 |
10/3 |
| #3 |
5/8 |
7/8 |
4/5 |
5/8 |
5/12 |
15/4 |
2.25 |
23 |
56 |
2 |
5/2 |
3 |
| #4 |
3/10 |
9/10 |
5/6 |
1/2 |
9/20 |
17/12 |
1.75 |
8 |
9.5 |
2.4 |
4 |
14/3 |
| #5 |
3/4 |
11/12 |
1/2 |
5/6 |
1/4 |
1/15 |
6 |
15 |
7 |
1 |
7/2 |
4 |
| #6 |
1/4 |
3/4 |
1 |
1/6 |
6/5 |
16/5 |
15 |
4.75 |
6.25 |
10 |
2 |
8/3 |
| #7 |
1 |
1 |
1 |
7/12 |
29/40 |
5/6 |
10 |
46 |
84 |
14 |
2 |
5/2 |
| #8 |
3/4 |
5/6 |
3/4 |
5/12 |
1/6 |
1/6 |
10 |
7.125 |
8.625 |
1 |
10/3 |
4 |
| #9 |
5/8 |
1 |
3/10 |
3/8 |
4/3 |
5/6 |
6 |
5 |
14 |
17 |
44 |
55 |
| #10 |
0.5 |
1/2 |
1 |
1/2 |
11/10 |
22/15 |
6 |
6.25 |
7.75 |
18 |
14 |
21 |
| #11 |
0.25 |
1/6 |
1/2 |
1/2 |
4/15 |
1/10 |
10 |
69 |
28 |
2/5 |
24 |
32 |
| #12 |
0.75 |
3/4 |
1 |
12 |
15/4 |
16/21 |
0.875 |
7.125 |
8.625 |
1/5 |
52 |
65 |
| #13 |
0.2 |
3 |
4 |
4/3 |
29/24 |
31/30 |
0.125 |
10 |
21 |
2 |
10 |
15 |
| #14 |
0.6 |
8 |
10 |
1.25 |
1/3 |
1/10 |
0.375 |
6 |
7.5 |
2 |
30 |
40 |
| #15 |
2/5 |
15 |
42 |
2.5 |
12/5 |
3/2 |
0.625 |
23 |
56 |
16 |
44 |
55 |
| #16 |
1/2 |
48 |
42 |
2 |
5/6 |
17/12 |
0.875 |
6.25 |
7.75 |
5/3 |
1/2 |
4/7 |
| #17 |
8 |
49 |
8 |
3 |
1/4 |
3/10 |
3 |
15 |
7 |
7/4 |
4/7 |
5/8 |
| #18 |
5 |
16 |
27 |
1.5 |
5/6 |
5/6 |
3 |
5.75 |
7.25 |
2.5 |
5/8 |
2/3 |
| #19 |
9 |
9 |
30 |
1 |
17/15 |
9/20 |
1.5 |
46 |
84 |
11 |
2/3 |
7/10 |
| #20 |
10 |
40 |
22 |
12 |
1/10 |
1/12 |
10 |
8.75 |
10.25 |
19 |
7/10 |
8/11 |
3. Optimal Champion Pipelines
- 6C #12 (3/4 + 1/2×1/4):
0.75 + 0.125 = 0.875 ([.1.2] [ENTER] [.1.4] [×] [.3.4] [+])
- 6D #4 (1 ÷ (1/4 + 1/6)):
Reciprocal pipeline: [4] [1/x] [6] [1/x] [+] [1/x] -> 2.4
- 6D #12 (1/2×2/3×3/4×4/5):
Telescoping cancellation leaves 1/5
- 6D #19 (24 × (5/6 - 3/8)):
Distribute 24: 20 - 9 = 11
Section 7: Section 7: Primes, GCF/LCM & Nested Expressions
Grades: Grades 5–10+ | Focus: Prime factorization, GCF, LCM, powers, and multi-branch expression trees.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
GCF(12, 18) × LCM(12, 18) = |
Use a × b = 12 × 18 |
216 |
| W2 |
Number of positive factors of 24 = |
24 = 2³ × 3¹ -> (3+1)(1+1) |
8 |
| W3 |
(15² - 5²) ÷ (15 + 5) = |
Simplifies to 15 - 5 |
10 |
| W4 |
(30 - (4 × 5)) × (12 ÷ (2 + 4)) = |
(30 - 20) × (12 ÷ 6) = 10 × 2 |
20 |
| W5 |
(2⁵ - 2³) ÷ (3² - 3) = |
(32 - 8) ÷ (9 - 3) = 24 ÷ 6 |
4 |
| W6 |
(12² + 5²) ÷ 13 = |
169 ÷ 13 (since 5-12-13 -> 13² ÷ 13) |
13 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
12 |
6 |
8 |
12 |
20 |
26 |
50 |
80 |
108 |
216 |
182 |
256 |
| #2 |
18 |
60 |
75 |
15 |
40 |
55 |
30 |
6 |
7 |
300 |
255 |
342 |
| #3 |
20 |
17 |
66 |
16 |
40 |
48 |
20 |
112 |
144 |
36 |
340 |
440 |
| #4 |
24 |
6 |
7 |
14 |
56 |
75 |
12 |
8 |
9 |
24 |
437 |
550 |
| #5 |
30 |
84 |
96 |
24 |
68 |
36 |
40 |
144 |
180 |
36 |
546 |
672 |
| #6 |
36 |
10 |
29 |
36 |
112 |
135 |
5 |
5 |
6 |
120 |
667 |
806 |
| #7 |
8 |
18 |
20 |
60 |
35 |
42 |
70 |
208 |
252 |
6 |
800 |
952 |
| #8 |
16 |
9 |
12 |
70 |
144 |
171 |
8 |
7 |
8 |
8 |
36 |
39 |
| #9 |
27 |
5 |
10 |
20 |
57 |
66 |
36 |
120 |
154 |
9 |
39 |
42 |
| #10 |
64 |
5 |
6 |
10 |
40 |
55 |
30 |
9 |
10 |
10 |
42 |
45 |
| #11 |
125 |
36 |
42 |
7 |
52 |
40 |
50 |
264 |
312 |
28 |
45 |
48 |
| #12 |
2 |
26 |
127 |
4 |
56 |
75 |
5 |
6 |
7 |
10 |
48 |
51 |
| #13 |
17 |
16 |
18 |
6 |
50 |
58 |
40 |
242 |
288 |
10 |
45 |
48 |
| #14 |
4 |
135 |
150 |
10 |
112 |
135 |
24 |
8 |
9 |
40 |
48 |
51 |
| #15 |
5 |
17 |
66 |
72 |
74 |
84 |
4 |
104 |
140 |
7 |
8 |
15 |
| #16 |
6 |
4 |
5 |
60 |
144 |
171 |
10 |
5 |
6 |
4 |
15 |
24 |
| #17 |
12 |
60 |
72 |
200 |
71 |
54 |
60 |
270 |
320 |
5 |
24 |
14 |
| #18 |
12 |
10 |
29 |
25 |
40 |
55 |
8 |
7 |
8 |
10 |
14 |
24 |
| #19 |
24 |
14 |
16 |
16 |
65 |
74 |
5 |
252 |
300 |
60 |
24 |
36 |
| #20 |
10 |
24 |
27 |
36 |
56 |
75 |
125 |
9 |
10 |
13 |
36 |
20 |
3. Optimal Champion Pipelines
- 7D #2 (GCF×LCM of 15,20):
Identity a × b = 15 × 20 = 300
- 7D #9 (Factors of 36):
36 = 2² × 3² -> (2+1)(2+1) = 9 factors
- 7D #14 ((25²-15²)÷(25-15)):
(a² - b²)/(a - b) = a + b = 25 + 15 = 40
- 7D #20 ((12²+5²)÷13):
Pythagorean 5² + 12² = 13², so 13² ÷ 13 = 13
Section 8: Section 8: Algebra, Quadratics & Coordinate Geometry
Grades: Grades 6–10+ | Focus: Linear equations, slope, Pythagorean distance, and quadratic roots.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
Sum of roots of x² - 9x + 20 = 0 : |
Vieta's rule: -b / a = 9 / 1 |
9 |
| W2 |
Product of roots of x² + 11x + 24 = 0 : |
Vieta's rule: c / a = 24 / 1 |
24 |
| W3 |
Vertex x-coord of y = x² - 8x + 19 : |
Axis of symmetry x = -b / (2a) = 8 / 2 |
4 |
| W4 |
Solve for x: x + y = 12, x - y = 6 : |
Average (12 + 6) ÷ 2 |
9 |
| W5 |
Hypotenuse for legs 7 and 24 = |
Pythagorean triple (7, 24, 25) |
25 |
| W6 |
3D diagonal √(2² + 3² + 6²) = |
√(4 + 9 + 36) = √49 |
7 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
17 |
4 |
5 |
3 |
3 |
4 |
20 |
3 |
4 |
9 |
10 |
13 |
| #2 |
13 |
15 |
25 |
2 |
4 |
5 |
25 |
3 |
4 |
20 |
15 |
17 |
| #3 |
18 |
6 |
7 |
3 |
5 |
6 |
4 |
7 |
8 |
-11 |
14 |
17 |
| #4 |
12 |
35 |
49 |
3 |
6 |
2 |
17 |
5 |
6 |
24 |
21 |
23 |
| #5 |
14 |
8 |
9 |
-2 |
2 |
3 |
-2 |
7 |
8 |
5 |
18 |
21 |
| #6 |
12 |
63 |
81 |
-0.5 |
3 |
4 |
-3 |
7 |
8 |
3.5 |
27 |
29 |
| #7 |
23 |
10 |
11 |
5 |
4 |
5 |
-3 |
5 |
6 |
16 |
22 |
25 |
| #8 |
9 |
33 |
49 |
7 |
5 |
6 |
-3 |
3 |
4 |
0 |
33 |
35 |
| #9 |
35 |
4 |
5 |
5 |
6 |
2 |
3 |
5 |
6 |
4 |
7 |
8 |
| #10 |
6 |
26 |
38 |
10 |
2 |
3 |
2 |
5 |
6 |
3 |
9 |
10 |
| #11 |
6 |
6 |
7 |
13 |
10 |
15 |
3 |
9 |
10 |
7 |
11 |
12 |
| #12 |
7 |
50 |
66 |
17 |
39 |
13 |
5 |
7 |
8 |
3 |
13 |
14 |
| #13 |
5 |
8 |
9 |
15 |
17 |
34 |
3 |
3 |
4 |
5 |
15 |
16 |
| #14 |
5 |
28 |
42 |
20 |
50 |
75 |
4 |
3 |
4 |
7 |
17 |
18 |
| #15 |
12 |
10 |
11 |
8 |
123 |
41 |
5 |
7 |
8 |
5 |
3 |
4 |
| #16 |
7 |
56 |
74 |
12 |
5 |
10 |
-5 |
5 |
6 |
25 |
5 |
6 |
| #17 |
-8 |
4 |
5 |
5 |
26 |
39 |
4 |
7 |
8 |
41 |
7 |
8 |
| #18 |
24 |
28 |
42 |
5 |
51 |
17 |
-0.5 |
7 |
8 |
13 |
9 |
10 |
| #19 |
-4 |
6 |
7 |
10 |
25 |
50 |
3 |
5 |
6 |
7 |
11 |
12 |
| #20 |
25 |
23 |
35 |
17 |
82 |
123 |
5 |
3 |
4 |
9 |
13 |
14 |
3. Optimal Champion Pipelines
- 8C #5 (Roots of x²+5x+6=0):
[↱][POLY] 1 [B] 5 [C] 6 [D] [XEQ] -> -3, -2
- 8D #5 (Sum of roots 2x²-10x+7):
Vieta's -b/a = 10 / 2 = 5
- 8D #10 (Min of x²-8x+19):
Complete square: (x - 4)² + 3 -> min is 3
- 8D #18 (√(3²+4²+12²)):
Chained (3,4,5) + (5,12,13) -> 13
Grades: Grades 5–10+ | Focus: Comprehensive review across fractions, expressions, number theory, and algebra.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
3/4 + (1/2 × 1/4) (decimal) = |
0.75 + 0.125 |
0.875 |
| W2 |
(2/3 + 3/4) × 12 = |
Distribute 12: 8 + 9 |
17 |
| W3 |
Distance from (3, 4) to (9, 12) = |
dx = 6, dy = 8 -> (6, 8, 10) triple |
10 |
| W4 |
Sum of roots of 2x² - 14x + 9 = 0 : |
Vieta's -b / a = 14 / 2 |
7 |
| W5 |
Minimum value of y = x² - 6x + 14 : |
(x - 3)² + 5 -> min is 5 |
5 |
| W6 |
2/3 × 3/4 × 4/5 × 5/6 = |
Telescoping cancel -> 2/6 = 1/3 |
1/3 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
27 |
6 |
8 |
1.75 |
20 |
26 |
0.875 |
80 |
108 |
55 |
182 |
256 |
| #2 |
64 |
60 |
75 |
3.75 |
40 |
55 |
2.4 |
6 |
7 |
36 |
255 |
342 |
| #3 |
36 |
17 |
66 |
1.25 |
40 |
48 |
14 |
112 |
144 |
1225 |
340 |
440 |
| #4 |
5 |
6 |
7 |
1.5 |
56 |
75 |
17 |
8 |
9 |
400 |
437 |
550 |
| #5 |
0.75 |
84 |
96 |
15 |
68 |
36 |
12 |
144 |
180 |
11 |
546 |
672 |
| #6 |
0.25 |
10 |
29 |
10 |
112 |
135 |
40 |
5 |
6 |
5/3 |
667 |
806 |
| #7 |
12 |
18 |
20 |
32 |
35 |
42 |
70 |
208 |
252 |
300 |
800 |
952 |
| #8 |
18 |
9 |
12 |
50 |
144 |
171 |
25 |
7 |
8 |
28 |
36 |
39 |
| #9 |
24 |
5 |
10 |
12 |
57 |
66 |
3 |
120 |
154 |
7 |
39 |
42 |
| #10 |
6 |
5 |
6 |
24 |
40 |
55 |
-0.5 |
9 |
10 |
13 |
42 |
45 |
| #11 |
12 |
36 |
42 |
3 |
52 |
40 |
-5 |
264 |
312 |
17 |
45 |
48 |
| #12 |
20 |
26 |
127 |
20 |
56 |
75 |
7 |
6 |
7 |
13 |
48 |
51 |
| #13 |
10 |
16 |
18 |
10 |
50 |
58 |
12 |
242 |
288 |
9 |
45 |
48 |
| #14 |
18 |
135 |
150 |
3 |
112 |
135 |
7 |
8 |
9 |
5 |
48 |
51 |
| #15 |
14 |
17 |
66 |
4 |
74 |
84 |
12 |
104 |
140 |
7 |
8 |
15 |
| #16 |
7 |
4 |
5 |
-2 |
144 |
171 |
15 |
5 |
6 |
25 |
15 |
24 |
| #17 |
14 |
60 |
72 |
2 |
71 |
54 |
8 |
270 |
320 |
30 |
24 |
14 |
| #18 |
5 |
10 |
29 |
28 |
40 |
55 |
24 |
7 |
8 |
616 |
14 |
24 |
| #19 |
10 |
14 |
16 |
225 |
65 |
74 |
399 |
252 |
300 |
294 |
24 |
36 |
| #20 |
4 |
24 |
27 |
625 |
56 |
75 |
624 |
9 |
10 |
1/3 |
36 |
20 |
3. Optimal Champion Pipelines
- C2C #2 (1 ÷ (1/4 + 1/6)):
[4] [1/x] [6] [1/x] [+] [1/x] -> 2.4
- C2D #5 (24 × (5/6 - 3/8)):
Distribute 24: 20 - 9 = 11
- C2D #11 (Dist (-3,5) to (5,-10)):
dx = 8, dy = 15 -> (8,15,17) triple = 17
- C2D #14 (Min of x²-6x+14):
Vertex at x = 3 -> 9 - 18 + 14 = 5
Section 9: Section 9: Trigonometry, Vectors & Polar Coordinates
Grades: Grades 8–12+ | Focus: Special angles, right-triangle projections, inverse trig, and vector magnitudes.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
Vector <5, 12> -> Magnitude r = |
Pythagorean triple or [12][ENTER][5][▸θ,r] |
13 |
| W2 |
5 sin²(20°) + 5 cos²(20°) = |
Factor 5(sin² + cos²) = 5(1) |
5 |
| W3 |
Area of Δ: a = 8, b = 10, C = 30° -> |
1/2 × a × b × sin(30°) = 0.5 × 80 × 0.5 |
20 |
| W4 |
2 sin(15°) cos(15°) = |
Double-angle identity: sin(30°) |
0.5 |
| W5 |
max value of 3 sin(x) + 4 cos(x) = |
Hypotenuse √(3² + 4²) |
5 |
| W6 |
Law of Cosines: a=8, b=5, C=60° -> c = |
c² = 64 + 25 - 2(8)(5)(0.5) = 49 -> c = 7 |
7 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
0 |
5 |
8 |
5 |
8 |
10 |
0.25 |
5 |
7 |
12.0208 |
10 |
15 |
| #2 |
1 |
7 |
10 |
6 |
10 |
12 |
0.75 |
7 |
9 |
8.6603 |
39 |
13 |
| #3 |
0.5 |
67 |
65 |
10 |
12 |
14 |
0.5 |
9 |
11 |
7.0711 |
17 |
34 |
| #4 |
0.5 |
11 |
14 |
8 |
14 |
16 |
0.5 |
11 |
13 |
5.7735 |
50 |
75 |
| #5 |
1 |
13 |
16 |
9 |
16 |
18 |
3 |
13 |
15 |
53.13 |
123 |
41 |
| #6 |
0 |
58 |
56 |
14 |
18 |
20 |
1 |
15 |
17 |
36.87 |
5 |
10 |
| #7 |
0 |
17 |
20 |
10 |
20 |
22 |
5 |
17 |
19 |
45 |
26 |
39 |
| #8 |
1 |
19 |
22 |
10 |
22 |
24 |
3 |
19 |
21 |
-90 |
51 |
17 |
| #9 |
1 |
49 |
47 |
0 |
24 |
26 |
4 |
62 |
72 |
19 |
25 |
50 |
| #10 |
2 |
23 |
26 |
8 |
26 |
28 |
0.25 |
81 |
93 |
7 |
82 |
123 |
| #11 |
1 |
25 |
28 |
7 |
10 |
15 |
5 |
87 |
96 |
20 |
120 |
90 |
| #12 |
5 |
40 |
38 |
9 |
39 |
52 |
10 |
110 |
121 |
0.5 |
90 |
72 |
| #13 |
30 |
29 |
32 |
5 |
68 |
17 |
13 |
75 |
84 |
0.8660 |
72 |
60 |
| #14 |
60 |
31 |
34 |
10 |
25 |
50 |
23 |
98 |
109 |
2 |
60 |
64 |
| #15 |
45 |
31 |
29 |
13 |
82 |
123 |
0 |
9 |
12 |
1 |
73 |
45 |
| #16 |
55 |
35 |
38 |
10 |
15 |
20 |
20 |
16 |
20 |
5 |
45 |
40 |
| #17 |
28 |
37 |
40 |
15 |
52 |
13 |
18 |
25 |
30 |
13 |
40 |
36 |
| #18 |
π |
22 |
20 |
17 |
17 |
34 |
10 |
36 |
42 |
17 |
36 |
144 |
| #19 |
π/2 |
41 |
44 |
20 |
50 |
75 |
4 |
49 |
56 |
180 |
157 |
30 |
| #20 |
π/3 |
43 |
46 |
25 |
123 |
164 |
2 |
64 |
72 |
90 |
30 |
196 |
3. Optimal Champion Pipelines
- 9D #1 (12 cos 45° + 5 sin 45°):
Factor (12 + 5) × sin(45°): [17] [ENTER] [45] [SIN] [×] -> 12.0208
- 9D #5 (Angle of <3, 4>):
[4] [ENTER] [3] [↰] [▸θ,r] [x⇄y] -> 53.13°
- 9D #10 (Law of Cosines a=8,b=5,C=60°):
cos(60°)=0.5 cancels the 2 in 2ab cos C: c² = a²+b²-ab = 64+25-40 = 49 -> 7
- 9D #17 (max of 5 sin x + 12 cos x):
Amplitude √(5² + 12²) = 13
Section 10: Section 10: Exponents, Logarithms & Growth Models
Grades: Grades 8–12+ | Focus: Powers, fractional exponents, base-10 and natural logarithms, and exponential growth.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
log₁₀(4) + log₁₀(25) = |
log₁₀(4 × 25) = log₁₀(100) |
2 |
| W2 |
Half-life (3 periods): 100 × (0.5)³ = |
100 ÷ 8 = 12.5 |
12.5 |
| W3 |
27^(2/3) = |
Cube root first (3), then square (3²) |
9 |
| W4 |
16^(3/4) = |
4th root first (2), then cube (2³) |
8 |
| W5 |
Change of base: ln(125) ÷ ln(5) = |
log₅(125) = 3 since 5³ = 125 |
3 |
| W6 |
Solve: 4^x = 32 (decimal) = |
Base 2: 2^(2x) = 2⁵ -> 2x = 5 |
2.5 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
1 |
9 |
18 |
0 |
2 |
3 |
2 |
60 |
80 |
9.9021 |
3 |
4 |
| #2 |
1 |
17 |
34 |
1 |
3 |
4 |
3 |
40 |
50 |
13.8629 |
4 |
2 |
| #3 |
16 |
33 |
66 |
2 |
4 |
5 |
2 |
25 |
30 |
610.7014 |
2 |
3 |
| #4 |
32 |
65 |
130 |
3 |
5 |
6 |
3 |
120 |
140 |
2473.08 |
3 |
4 |
| #5 |
64 |
129 |
258 |
4 |
6 |
7 |
3 |
70 |
80 |
281.79 |
4 |
2 |
| #6 |
256 |
257 |
514 |
-1 |
7 |
8 |
2 |
40 |
45 |
1331 |
2 |
3 |
| #7 |
1024 |
513 |
1026 |
-2 |
8 |
9 |
6 |
180 |
200 |
3 |
3 |
4 |
| #8 |
27 |
1025 |
2050 |
0 |
9 |
10 |
4 |
100 |
110 |
4 |
4 |
2 |
| #9 |
81 |
9 |
18 |
1 |
3 |
4 |
3 |
55 |
60 |
3 |
2 |
3 |
| #10 |
125 |
17 |
34 |
3 |
4 |
5 |
1000 |
240 |
260 |
5 |
3 |
4 |
| #11 |
625 |
3 |
4 |
-2 |
5 |
6 |
32 |
400 |
500 |
3 |
9 |
16 |
| #12 |
1000 |
4 |
5 |
5 |
6 |
7 |
50 |
1000 |
1200 |
2 |
64 |
125 |
| #13 |
10000 |
5 |
6 |
9 |
7 |
8 |
25 |
2400 |
2800 |
1.5 |
25 |
36 |
| #14 |
0.5 |
6 |
7 |
7 |
8 |
9 |
12.5 |
700 |
800 |
1.5 |
27 |
64 |
| #15 |
0.25 |
3 |
4 |
3 |
6 |
9 |
10 |
1600 |
1800 |
9 |
16 |
25 |
| #16 |
0.2 |
4 |
5 |
5 |
8 |
11 |
400 |
3600 |
4000 |
8 |
125 |
216 |
| #17 |
0.01 |
5 |
6 |
6 |
10 |
13 |
180 |
1000 |
1100 |
8 |
9 |
16 |
| #18 |
4 |
6 |
7 |
3 |
12 |
15 |
1100 |
2200 |
2400 |
16 |
64 |
125 |
| #19 |
3 |
3 |
4 |
3 |
14 |
17 |
1210 |
4800 |
5200 |
2.5 |
25 |
36 |
| #20 |
4 |
4 |
5 |
1 |
16 |
19 |
1102.5 |
1300 |
1400 |
1.5 |
27 |
64 |
3. Optimal Champion Pipelines
- 10D #1 (ln(2) ÷ 0.07):
[2] [LN] [.07] [÷] -> 9.9021
- 10D #3 (500 × e^(0.04×5)):
[.04] [ENTER] [5] [×] [eˣ] [500] [×] -> 610.7014
- 10D #5 (250 × (1+0.06/12)²⁴):
[.06] [ENTER] [12] [÷] [1] [+] [24] [yˣ] [250] [×] -> 281.79
- 10D #18 (64^(2/3)):
(∛64)² = 4² = 16
Section 11: Section 11: Combinatorics, Probability & Statistics
Grades: Grades 7–12+ | Focus: Factorials, permutations, combinations, probability ratios, and dataset statistics.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
7C2 = |
(7 × 6) ÷ 2 = 7 × 3 |
21 |
| W2 |
8C3 = |
(8 × 7 × 6) ÷ 6 = 8 × 7 |
56 |
| W3 |
Arrangements of BOOK (4! ÷ 2!) = |
24 ÷ 2 |
12 |
| W4 |
Row 5 sum: 5C0 + ... + 5C5 = |
Pascal's Row Sum = 2⁵ |
32 |
| W5 |
5 coins: P(exactly 2 Heads) = |
5C2 ÷ 2⁵ = 10 ÷ 32 |
0.3125 |
| W6 |
Sample SD of {2, 5, 8} = |
3 evenly spaced numbers -> step d = 3 |
3 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
1 |
8 |
11 |
5 |
30 |
42 |
0.25 |
21 |
36 |
16 |
16 |
32 |
| #2 |
1 |
10 |
13 |
6 |
35 |
56 |
0.125 |
28 |
45 |
32 |
32 |
64 |
| #3 |
2 |
12 |
15 |
20 |
28 |
36 |
0.875 |
36 |
55 |
64 |
64 |
128 |
| #4 |
6 |
14 |
17 |
30 |
72 |
90 |
36 |
45 |
66 |
0.375 |
128 |
256 |
| #5 |
24 |
16 |
19 |
120 |
45 |
55 |
6 |
55 |
78 |
0.25 |
256 |
512 |
| #6 |
120 |
18 |
21 |
42 |
165 |
220 |
1/6 |
66 |
91 |
10 |
512 |
1024 |
| #7 |
720 |
20 |
23 |
6 |
132 |
156 |
1/36 |
78 |
105 |
0.3125 |
1024 |
2048 |
| #8 |
15 |
22 |
25 |
10 |
286 |
364 |
1/36 |
91 |
120 |
0.3 |
4 |
5 |
| #9 |
6 |
24 |
27 |
10 |
15 |
21 |
6 |
105 |
136 |
0.4 |
5 |
6 |
| #10 |
20 |
26 |
29 |
15 |
42 |
56 |
24 |
120 |
153 |
18 |
6 |
7 |
| #11 |
8 |
14 |
16 |
20 |
28 |
36 |
120 |
52 |
64 |
24 |
7 |
8 |
| #12 |
9 |
17 |
19 |
21 |
84 |
120 |
3 |
75 |
90 |
120 |
8 |
9 |
| #13 |
8 |
20 |
22 |
35 |
90 |
110 |
12 |
102 |
120 |
8/3 |
9 |
10 |
| #14 |
18 |
23 |
25 |
28 |
165 |
220 |
30 |
133 |
154 |
3 |
10 |
11 |
| #15 |
5 |
26 |
28 |
56 |
66 |
78 |
15 |
168 |
192 |
4 |
3 |
4 |
| #16 |
0.5 |
29 |
31 |
36 |
156 |
182 |
28 |
92 |
104 |
2 |
3 |
4 |
| #17 |
1/6 |
32 |
34 |
45 |
15 |
21 |
9 |
125 |
140 |
-1.5 |
3 |
4 |
| #18 |
0.5 |
35 |
37 |
120 |
35 |
56 |
60 |
162 |
180 |
15 |
3 |
4 |
| #19 |
1/3 |
38 |
40 |
66 |
56 |
72 |
16 |
203 |
224 |
3.5 |
3 |
4 |
| #20 |
12 |
41 |
43 |
100 |
84 |
120 |
0.0667 |
248 |
272 |
7 |
3 |
4 |
3. Optimal Champion Pipelines
- 11B #15 (8C3):
(8 × 7 × 6) ÷ 6 = 56
- 11C #14 (Arrangements of LEVEL):
5! ÷ (2! × 2!) = 120 ÷ 4 = 30
- 11D #3 (Row 6 sum 6C0+...+6C6):
2⁶ = 64
- 11D #15 (Sample SD of {10,14,18}):
3 evenly spaced values with step d = 4 -> SD = 4
Section 12: Section 12: Base Conversions, Binary & Bitwise Logic
Grades: Grades 7–12+ | Focus: Binary, octal, and hexadecimal place value, bit shifting, and boolean gates.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
Hex 2F₁₆ in decimal = |
2 × 16 + 15 = 32 + 15 |
47 |
| W2 |
12 (1100₂) XOR 10 (1010₂) = |
Differing bits: 0110₂ |
6 |
| W3 |
Left shift: 7 << 3 = |
7 × 2³ = 7 × 8 |
56 |
| W4 |
Binary 10101100₂ in hexadecimal = |
Nibbles 1010 (A) and 1100 (C) |
AC |
| W5 |
Decimal 4095 (2¹² - 1) in Hex = |
Three 4-bit 1111 nibbles -> FFF |
FFF |
| W6 |
8-bit Two's Complement of -1 (Hex) = |
256 - 1 = 255 = FF₁₆ |
FF |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
1 |
6 |
11 |
16 |
24 |
31 |
1 |
0 |
1 |
15 |
37 |
44 |
| #2 |
2 |
0101 |
1000 |
20 |
23 |
2A |
0 |
15 |
15 |
18 |
4A |
57 |
| #3 |
3 |
12 |
17 |
31 |
46 |
53 |
1 |
2 |
4 |
10 |
5D |
6A |
| #4 |
4 |
0111 |
1010 |
32 |
39 |
40 |
0 |
22 |
28 |
10 |
70 |
7D |
| #5 |
5 |
18 |
23 |
64 |
68 |
75 |
1 |
12 |
13 |
30 |
83 |
90 |
| #6 |
6 |
1001 |
1100 |
255 |
4F |
56 |
8 |
15 |
19 |
15 |
96 |
A3 |
| #7 |
7 |
24 |
29 |
10 |
90 |
97 |
14 |
10 |
20 |
63 |
A9 |
B6 |
| #8 |
8 |
1011 |
1110 |
12 |
65 |
6C |
6 |
60 |
72 |
40 |
BC |
C9 |
| #9 |
9 |
30 |
5 |
15 |
112 |
119 |
2 |
0 |
1 |
AC |
CF |
DC |
| #10 |
10 |
1101 |
0001 |
16 |
7B |
82 |
7 |
27 |
31 |
F0 |
E2 |
EF |
| #11 |
12 |
6 |
11 |
26 |
134 |
141 |
5 |
30 |
24 |
10110100 |
FC |
FB |
| #12 |
15 |
1111 |
0011 |
32 |
91 |
98 |
9 |
152 |
176 |
FFF |
FA |
F9 |
| #13 |
0101 |
12 |
17 |
47 |
156 |
163 |
6 |
4 |
5 |
255 |
F8 |
F7 |
| #14 |
0110 |
0010 |
0101 |
255 |
A7 |
AE |
15 |
23 |
31 |
1023 |
F6 |
F5 |
| #15 |
1001 |
18 |
23 |
10 |
178 |
185 |
192 |
30 |
16 |
10 |
F4 |
F3 |
| #16 |
1010 |
0100 |
0111 |
1F |
BD |
C4 |
10 |
46 |
52 |
55 |
F2 |
F1 |
| #17 |
1100 |
24 |
29 |
40 |
200 |
207 |
12 |
8 |
9 |
FF |
F0 |
EF |
| #18 |
1101 |
0110 |
1001 |
FF |
D3 |
DA |
56 |
31 |
31 |
FE |
EE |
ED |
| #19 |
1110 |
30 |
5 |
10 |
222 |
229 |
6 |
30 |
24 |
15 |
EC |
EB |
| #20 |
1111 |
1000 |
1011 |
100 |
E9 |
F0 |
8 |
108 |
120 |
255 |
EA |
E9 |
3. Optimal Champion Pipelines
- 12C #15 (192 AND 240):
192 = 128 + 64, both inside top mask 240 -> 192
- 12D #9 (10101100₂ in Hex):
1010₂ = A, 1100₂ = C -> AC
- 12D #12 (4095 in Hex):
2¹² - 1 = 12 ones = three F nibbles -> FFF
- 12D #20 (170 XOR 85):
Complementary bit masks 10101010₂ ^ 01010101₂ = 11111111₂ = 255
Section C3: Cumulative Review C3: All-Level Championship (Sections 1–12)
Grades: Grades 7–12+ | Focus: Grand championship speed tests spanning arithmetic, algebra, trig, logs, stats, and CS.
1. Warm-Up Drills (W1–W6)
| Drill |
Prompt |
Shortcut Cue |
Verified Answer |
| W1 |
12 cos(45°) + 5 sin(45°) (4 dec) = |
(12 + 5) × sin(45°) = 17 × 0.7071 |
12.0208 |
| W2 |
Monthly: 250 × (1 + 0.06 ÷ 12)²⁴ = |
[.06][ENTER][12][÷][1][+][24][yˣ][250][×] |
281.79 |
| W3 |
1 + 2 + 3 + ... + 20 = |
Gauss rule: 20 × 21 ÷ 2 |
210 |
| W4 |
GCF(24, 36) × LCM(24, 36) = |
24 × 36 = 30² - 6² = 900 - 36 |
864 |
| W5 |
max value of 5 sin(x) + 12 cos(x) = |
Amplitude √(5² + 12²) |
13 |
| W6 |
5 coins: P(exactly 2 Heads) = |
5C2 ÷ 32 = 10 ÷ 32 |
0.3125 |
2. Progressive Practice Tests (Tests A1–A3, B1–B3, C1–C3, D1–D3)
| # |
Test A1 |
Test A2 |
Test A3 |
Test B1 |
Test B2 |
Test B3 |
Test C1 |
Test C2 |
Test C3 |
Test D1 |
Test D2 |
Test D3 |
| #1 |
36 |
6 |
11 |
29 |
24 |
31 |
3.0667 |
0 |
1 |
210 |
37 |
44 |
| #2 |
1.25 |
0101 |
1000 |
7 |
23 |
2A |
2.4 |
15 |
15 |
485 |
4A |
57 |
| #3 |
6 |
12 |
17 |
9 |
46 |
53 |
17 |
2 |
4 |
400 |
5D |
6A |
| #4 |
24 |
0111 |
1010 |
-2 |
39 |
40 |
25 |
22 |
28 |
11 |
70 |
7D |
| #5 |
7 |
18 |
23 |
5 |
68 |
75 |
11 |
12 |
13 |
864 |
83 |
90 |
| #6 |
13 |
1001 |
1100 |
3 |
4F |
56 |
12.0208 |
15 |
19 |
9 |
96 |
A3 |
| #7 |
1 |
24 |
29 |
17 |
90 |
97 |
1 |
10 |
20 |
13 |
A9 |
B6 |
| #8 |
8 |
1011 |
1110 |
10 |
65 |
6C |
7 |
60 |
72 |
3 |
BC |
C9 |
| #9 |
32 |
30 |
5 |
1024 |
112 |
119 |
9.9021 |
0 |
1 |
6 |
CF |
DC |
| #10 |
3 |
1101 |
0001 |
6 |
7B |
82 |
610.7014 |
27 |
31 |
7 |
E2 |
EF |
| #11 |
4 |
6 |
11 |
1102.5 |
134 |
141 |
2473.08 |
30 |
24 |
13 |
FC |
FB |
| #12 |
120 |
1111 |
0011 |
12.5 |
91 |
98 |
281.79 |
152 |
176 |
3 |
FA |
F9 |
| #13 |
10 |
12 |
17 |
120 |
156 |
163 |
120 |
4 |
5 |
16 |
F8 |
F7 |
| #14 |
10 |
0010 |
0101 |
35 |
A7 |
AE |
30 |
23 |
31 |
2.5 |
F6 |
F5 |
| #15 |
10 |
18 |
23 |
56 |
178 |
185 |
1/6 |
30 |
16 |
64 |
F4 |
F3 |
| #16 |
1111 |
0100 |
0111 |
FF |
BD |
C4 |
FFF |
46 |
52 |
0.3125 |
F2 |
F1 |
| #17 |
16 |
24 |
29 |
100 |
200 |
207 |
47 |
8 |
9 |
4 |
F0 |
EF |
| #18 |
8 |
0110 |
1001 |
14 |
D3 |
DA |
72 |
31 |
31 |
AC |
EE |
ED |
| #19 |
225 |
30 |
5 |
6 |
222 |
229 |
624 |
30 |
24 |
FF |
EC |
EB |
| #20 |
625 |
1000 |
1011 |
192 |
E9 |
F0 |
1225 |
108 |
120 |
255 |
EA |
E9 |
3. Optimal Champion Pipelines
- C3C #1 ((3 1/3 ÷ 1 1/4) + 2/5):
[3.1.3] [ENTER] [1.1.4] [÷] [.2.5] [+] -> 3.0667
- C3C #6 (12 cos 45° + 5 sin 45°):
17 × sin(45°) -> 12.0208
- C3D #5 (GCF(24,36)×LCM(24,36)):
24 × 36 = (30 - 6)(30 + 6) = 900 - 36 = 864
- C3D #10 (Law of Cosines a=8,b=5,C=60°):
c² = 64 + 25 - 40 = 49 -> c = 7