Appendix B: Teacher Lesson Plans & Station Guide

B.1 Running the 45-Minute Workshop & Foundational Fluency Stations

The StackCalc SC-6 station curriculum aligns with the Grades 3–6 Common Core State Standards for Mathematics (CCSSM, 2010) and follows the Math Workshop Rotation Model, where every mathematical concept moves from physical blocks (Dienes, 1960; Bruner, 1966) to visual bar and array models (Singapore MOE, 2014) to the four-shelf number stack. Grounded in research on elementary mathematics instruction (Ma, 1999), relational understanding of equality (Falkner et al., 1999), and structured problem-solving heuristics (Pólya, 1945), each 45-minute lesson unfolds in four phases:

Time Window	Phase	Activity & Student Experience
00:00 – 05:00	Whole-Class Warm-Up	Share the real-world story challenge and pose the main question.
05:00 – 20:00	Station Investigation	Hands-on physical block build at studio stations.
20:00 – 35:00	Stack Verification & Practice	SC-6 four-shelf stack verification and keystroke logging.
35:00 – 45:00	Discourse & Exit Ticket	Mathematical reflection, quick check, and notebook log.

B.1.1 Classroom Hand-Signals for the Four Shelves (\(X, Y, Z, T\))

When introducing the four-shelf stack (\(X, Y, Z, T\)) at the classroom carpet or whiteboard, pair each shelf with a simple hand-height cue so students connect the physical blocks on their desk with the shelves on the screen:

Stack Shelf	Classroom Hand-Signal Cue	How It Connects to the Desk Tray & Screen
Ground Shelf \(X\)	Hands Flat on the Desk (The Workbench)	Every number you type starts right here on Ground Shelf \(X\), matching the front/bottom row of your block tray.
Waiting Shelf \(Y\)	Hands Raised to Chest Height	Tapping [ENTER] lifts your first number up to Waiting Shelf \(Y\), matching the back/top row of your block tray so both numbers sit in plain sight before an operation combines them.
Upper Shelf \(Z\)	Hands Raised to Shoulder Height	Whenever you start building a second group in a multi-step problem, your finished first group rides automatically up to Shelf \(Z\) to wait safely out of the way.
Top Shelf \(T\)	Hands Raised Overhead (The Constant Dispenser)	Pressing [g] [REP] fills all four shelves up to Top Shelf \(T\), which feeds a fresh copy of your step size downward every time the stack drops during skip-counting.

B.1.2 Setting Up the Five-Table Studio Rotation & 60-Second Tray Cleanup

In a typical classroom of 20 to 30 students, the 45-minute workshop runs smoothly when desks are clustered into five station tables of 4 to 6 students (two or three partner pairs per table). Each table receives one modular studio tray (or a pair of student kits) alongside two or three SC-6 calculators so every pair shares one physical block tray and one calculator. On the first problem at a station, Partner A (The Architect) builds the physical model with the 3D-printed blocks while Partner B (The Stack Navigator) keys the matching sequence onto the SC-6 stack and calls out what appears on Shelf \(Y\) and Ground Shelf \(X\); on the next problem, partners swap roles so every student builds with both their hands and the keypad.

Fast finishers sometimes want to jump straight to the calculator buttons without touching the blocks. Placing a Quest Scoreboard lid at the center of the table solves this naturally through the Two-Token Quest Scoreboard Rule: partners earn their first +10 token only when their physical block model is built on the desk, and they stack their second +10 token (+20 total) when the SC-6 screen confirms the exact same result.

Because every StackCalc studio tray is engineered with recessed sockets for its exact part count—10 pockets per row in the Ten-Frame, 12 color columns in the Counting & Array case, and stepped wells in the \(84\text{ mm}\) Fraction and Prime Factor cases—you never have to count loose plastic pieces by hand at the end of class. Give a 60-Second Gridfinity Socket Count-In chime at minute 34:00: if every socket in the tray is filled and the lid slides shut flush, zero pieces are left on the floor!

Foundational fluency stations (F.1–F.11) build quick ten-frame pattern recognition, spotting groups of 5 and 10 at a glance, tower comparisons, place-value bundling, and skip-counting fluency as warm-up and practice stations for Grades 3–6 classrooms using physical Ten-Frame trays, Place Value Trays, and Counting Tiles alongside [▦] Ten-Frame Mode.

Station F.1: Spotting 5-and-Some-More in Ten-Frame Mode

Place a Ten-Frame Studio tray and ten two-color flip counters (yellow and red) between each pair of students to launch your first workshop rotation (CCSS 3.NBT.A.2; TEKS 3.2.A). To kick off the five-minute warm-up, have partners fill the top row of their tray with 5 red counters and place 3 yellow counters in the bottom row, then ask how they can tell the total is 8—with 2 empty pockets left to reach 10—without counting from one.

Invite students to hold up their SC-6 calculator, tap the Ten-Frame key [▦], and compare the glass display to the plastic tray on their desk. They will immediately notice the same two rows of five cells separated by the center 5-anchor divider. Have them build the warm-up tray step by step on both the desk and the screen, dropping one counter into the top row each time they tap 1 [+] until all five top pockets are filled. Once they see that a full top row always represents an instant group of 5 on Shelf \(X\), ask them to add the 3 yellow counters into the bottom row (3 [+]) so their eyes read \(5 + 3 = 8\) and spot the 2 empty cells remaining to make 10.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
[▦]	0	0	0	0 (10F)	Open Ten-Frame Mode (0 dots filled)
1 [+]	0	0	0	1 (10F)	Add 1st counter into top row (1 dot)
1 [+]	0	0	0	2 (10F)	Add 2nd counter (2 dots)
1 [+]	0	0	0	3 (10F)	Add 3rd counter (3 dots)
1 [+]	0	0	0	4 (10F)	Add 4th counter (4 dots)
1 [+]	0	0	0	5 (10F)	Complete the full top row of 5 dots

During partner practice, challenge teams to build 7 on the tray (5 on top, 2 on bottom) and match it on screen ([▦] 7), fill all 10 dots to confirm that two complete rows of 5 make 10, and then clear the frame ([C] [▦]) to enter 3 dots and name the 7 empty boxes needed to reach 10.

As you circulate among tables, watch for two common habits: if a student taps calculator keys faster than they place counters and overshoots the count, coach them to pair each physical counter with a single key press ("touch the block, tap the key"). If a student scatters counters randomly across both rows, guide them to fill the top row of 5 from left to right first so the 5-and-some-more pattern jumps out at a glance.

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Station F.2: Building & Splitting Benchmark Number Bonds

What happens when you shake 5 two-color counters inside a cup and spill them onto the desk—does the total ever change when the colors flip? In this number-bond investigation (CCSS 3.OA.C.7; TEKS 3.4.A), partners use a Ten-Frame Studio tray, a small cup, and 5 (or 10) two-color counters to discover how decomposing a whole into partner pairs (\(4+1, 3+2, 2+3, 1+4\)) always preserves the total. Begin by having one partner shake 5 counters in the cup, spill them onto the desk, and sort the yellow and red counters into the tray.

When the first spill lands on 3 yellow and 2 red counters, walk students through loading those two parts onto the SC-6 elevator: typing 3 [ENTER] lifts the 3 yellow counters up to Shelf \(Y\), and typing 2 places the 2 red counters on Shelf \(X\). With both parts visible side by side on the screen, pressing [+] combines them into 5 on the bottom shelf. Have the second partner shake and spill the same 5 counters again—perhaps landing on 4 yellow and 1 red—and key in 4 [ENTER] 1 [+]. Discuss why every spill of the cup lands on the exact same sum of 5: no counters ever entered or left the cup, so only the yellow-and-red split changed.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
3	0	0	0	3	Enter 3 yellow counters onto Shelf \(X\)
[ENTER]	0	0	3	3	Copy 3 up onto Shelf \(Y\)
2	0	0	3	2	Enter 2 red counters onto Shelf \(X\)
[+]	0	0	0	5	Combine both parts: \(3 + 2 = \mathbf{5}\)

Once the first two spills are verified, have partners log all remaining splits of 5 (1 [ENTER] 4 [+], 2 [ENTER] 3 [+], and 5 [ENTER] 0 [+]) before repeating the cup-spill investigation with 10 counters.

Keep an eye on the screen during the first few entries: if 32 appears instead of 5, the student typed 3 and 2 without pressing [ENTER] between them; remind them to press [ENTER] so the 3 rides up to Shelf \(Y\) before 2 steps onto Shelf \(X\). And if a student wonders why \(3+2\) and \(2+3\) give the same answer, have them rotate their Ten-Frame tray \(180^\circ\) on the desk to see that the total count never changes.

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Station F.3: Finding the Difference Between Two Towers

Many young learners think subtraction only means "taking things away"—which makes comparison questions like "How much taller is the blue tower than the gold tower?" surprisingly tricky. This station reframes subtraction as the physical height gap between two side-by-side towers (CCSS 3.NBT.A.2; TEKS 3.2.D). Give each pair the Counting & Array Studio with its \(6 \times 2\) docking lid, 10 Cobalt Blue tiles, 10 Amber Gold tiles, and a straightedge ruler. Start by having students dock a tower of 7 blue tiles in the left socket and a tower of 4 gold tiles in the adjacent right socket, laying the ruler across the top of the shorter tower to see the step gap.

Ask partners how many tiles sit where the two towers are neck-and-neck (4 tiles) and how many extra blue tiles rise above the ruler (3 tiles). To measure that gap on the SC-6 stack, have them load the taller tower onto Shelf \(Y\) first (7 [ENTER]), enter the shorter tower on Shelf \(X\) (4), and press [-]. The 3 on the bottom shelf measures the exact 3-tile step between the towers. Then have students snap 3 more gold tiles onto the shorter tower and run a finger across both tops: standing flush at 7 tiles physically proves that \(4 + 3 = 7\) and \(7 - 4 = 3\) are inverse partners.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
7	0	0	0	7	Enter the taller 7-tile tower onto Shelf \(X\)
[ENTER]	0	0	7	7	Copy 7 up onto Shelf \(Y\)
4	0	0	7	4	Enter the shorter 4-tile tower onto Shelf \(X\)
[-]	0	0	0	3	Subtract to find the height gap: \(7 - 4 = \mathbf{3}\)

Invite partners to build and compare tower pairs of 9 and 5 (9 [ENTER] 5 [-] \(\to 4\)), 8 and 8 (8 [ENTER] 8 [-] \(\to 0\), level tops), and 6 and 2 (6 [ENTER] 2 [-] \(\to 4\)).

Whenever a student enters the shorter tower first (4 [ENTER] 7 [-])—in [f] [LINE] Number Line mode or with [NUM] Whole Number Mode toggled off—and sees -3, explain that -3 tells us the gold tower is 3 tiles shorter than the blue tower, and show how tapping [x⇄y] before [-] puts the taller tower on Shelf \(Y\) first. Also make sure both towers are seated firmly in the lid sockets so the tile steps align cleanly.

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Station F.4: Bridging Through 10 with the Ten-Frame

Instead of counting on their fingers one by one to add \(8 + 5\), fluent mathematicians build a bridge through 10 (\(8 + 2 + 3 = 13\)). Set two Ten-Frame Studio trays side by side with 20 two-color counters at each partner station (CCSS 3.NBT.A.2; TEKS 3.4.A), and launch the activity by having students place 8 yellow counters into Tray 1 while holding 5 red counters in their hand. Ask: "How many of the red counters in your hand do we need just to finish filling Tray 1?"

Students quickly see that Tray 1 has 2 empty pockets waiting. Have them drop 2 red counters into those pockets to complete a full 10 in Tray 1, then place the remaining 3 red counters into Tray 2 so their eyes read \(10 + 3 = 13\) immediately. Next, have them key 8 [ENTER] 5 [+] on the SC-6 to see 13 on Shelf \(X\), and then press [▦]: the LCD displays one completely filled ten-frame of 10 dots plus 3 dots in the second frame, mirroring the two plastic trays on the table.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
8	0	0	0	8	Enter 8 counters in Tray 1
[ENTER]	0	0	8	8	Copy 8 up onto Shelf \(Y\)
5	0	0	8	5	Enter 5 counters to add
[+]	0	0	0	13	Bridge through 10: \(8 + 2 + 3 = \mathbf{13}\)
[▦]	0	0	0	13(10F)	View as 1 full ten-frame plus 3 dots

From there, partners can practice the same make-ten bridge with \(9 + 6\) (9 [ENTER] 6 [+] [▦] \(\to 10 + 5 = 15\)), \(7 + 5\) (7 [ENTER] 5 [+] [▦] \(\to 10 + 2 = 12\)), and \(8 + 7\) (8 [ENTER] 7 [+] [▦] \(\to 10 + 5 = 15\)).

Whenever a student reverts to counting up one by one on their fingers, point to the empty pockets in Tray 1 and ask: "How many empty seats are left to finish 10? Fill Tray 1 completely before any counters spill over into Tray 2."

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Station F.5: The Turn-Around Rule of Addition (\(3 + 7 = 7 + 3\))

Snap 3 Cobalt Blue tiles onto 7 Amber Gold tiles to build a 10-tile tower, then flip the whole tower upside down in your hands—did the height of the tower change at all? Pairing Four-Color Counting Tiles with the SC-6 swap key [x⇄y] gives students a tangible proof of the Turn-Around Rule of addition (CCSS 3.NBT.A.2; TEKS 3.4.A).

To mirror that physical flip on the calculator, have students enter 3 [ENTER] 7 so Shelf \(Y\) holds 3 and Shelf \(X\) holds 7. When they tap [x⇄y], the two shelves trade places right on the screen—putting 7 on Shelf \(Y\) and 3 on the bottom shelf—and pressing [+] yields 10. Tapping [f] [LASTX] 3 [+] confirms that both orders land on 10 because swapping the order of two addends never changes the combined height of the tower.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
3	0	0	0	3	Enter 3 blue tiles onto Shelf \(X\)
[ENTER]	0	0	3	3	Copy 3 up onto Shelf \(Y\)
7	0	0	3	7	Enter 7 gold tiles onto Shelf \(X\)
[x⇄y]	0	0	7	3	Swap Shelves \(X\) and \(Y\) (\(7\) on \(Y\), \(3\) on \(X\))
[+]	0	0	0	10	Add both shelves: \(7 + 3 = \mathbf{10}\)

Encourage partners to test \(4 + 8\) vs. \(8 + 4\) (both 12), \(2 + 9\) vs. \(9 + 2\) (both 11), and \(6 + 5\) vs. \(5 + 6\) (both 11) using [x⇄y].

Before closing the station, pose a thought-provoking extension question: does subtraction also obey the Turn-Around Rule (\(7 - 3\) vs. \(3 - 7\))? Having students switch to [f] [LINE] Number Line mode (or toggle [NUM] off) and key 3 [ENTER] 7 [-] on the SC-6 so -4 appears on screen sparks a memorable conversation about why order matters in subtraction even though addition is turn-around friendly.

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Station F.6: Number Bonds & Part-Whole Missing Numbers

An overturned cup hiding a secret pile of tiles turns missing-addend equations like \(? + 6 = 14\) into a classroom detective game (CCSS 3.OA.D.8; TEKS 3.5.A). Give each pair 14 counting tiles and an opaque cup. During the opening mystery, one partner counts out all 14 tiles onto the desk, hides a secret cluster under the cup while the other partner closes their eyes, and leaves 6 tiles visible outside the cup.

Ask students how they can figure out how many tiles are hiding under the cup without peeking. Because the whole collection is 14 tiles and the visible part is 6 tiles, subtracting the visible part from the whole reveals the hidden part. Have students key the starting whole first (14 [ENTER]), enter the visible part second (6), and press [-] to uncover 8 on Shelf \(X\)—then lift the cup to count the 8 hidden tiles (\(8 + 6 = 14\)).

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
14	0	0	0	14	Enter the starting whole of 14 tiles
[ENTER]	0	0	14	14	Copy 14 up onto Shelf \(Y\)
6	0	0	14	6	Enter the 6 visible tiles onto Shelf \(X\)
[-]	0	0	0	8	Subtract to uncover the 8 hidden tiles

Partners can then trade roles hiding tiles for \(? + 5 = 12\) (12 [ENTER] 5 [-] \(\to 7\)), \(? + 9 = 16\) (16 [ENTER] 9 [-] \(\to 7\)), and \(? + 8 = 15\) (15 [ENTER] 8 [-] \(\to 7\)).

Watch out for a common reflex: a student sees the + sign in \(? + 6 = 14\) and immediately adds \(14 + 6 = 20\). Point to the overturned cup and ask: "Can the hidden pile under the cup have 20 tiles if we only started with 14 tiles on the whole table?" That physical check steers them right back to subtracting the visible part from the whole.

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Station F.7: Bundling 10 Ones into 1 Ten

Count out 13 loose \(1\text{U}\) unit blocks (\(7.0\text{ mm}\) tall each) and stack 10 of them into a single column right beside the solid \(70.0\text{ mm}\) 10-Rod from the Ten-Frame Studio (CCSS 3.NBT.A.1; TEKS 3.2.A). Using the Place Value Studio (Whole Numbers Tray, digit miniblocks, and +1 regrouping unit cubes), this station shows students why 10 loose ones trade cleanly for 1 ten.

When students run a finger across the top of their 10-cube tower and the solid 10-Rod, they feel that both stand at the exact same \(70.0\text{ mm}\) height. Have them trade those 10 loose ones for a 1 digit miniblock in the TENS socket and dock a 3 miniblock in the ONES socket of the Place Value Tray, then key 10 [ENTER] 3 [+] on the SC-6. Comparing the tray to the 13 on Shelf \(X\) makes it clear that the 1 in 13 means 1 full bundle of ten (\(10\)), while the 3 means 3 loose ones.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
10	0	0	0	10	Enter 1 bundle of ten (\(10\))
[ENTER]	0	0	10	10	Copy 10 up onto Shelf \(Y\)
3	0	0	10	3	Enter 3 loose ones onto Shelf \(X\)
[+]	0	0	0	13	Combine 1 ten and 3 ones into 13

Once 13 is docked on the tray, ask partners to build and verify 2 tens and 5 ones (20 [ENTER] 5 [+] \(\to 25\)), 3 tens and 8 ones (30 [ENTER] 8 [+] \(\to 38\)), and 4 tens and 0 ones (40 [ENTER] 0 [+] \(\to 40\)), pressing [PV] to see the calculator decompose each number back into its place-value parts.

A classic place-value misconception is writing 103 for thirteen by placing 10 and 3 side by side. If you see this, have the student slide the 3 miniblock directly over the 0 placeholder in the ONES socket of the Place Value Tray so they see why 0 was only holding the ones seat empty until the 3 arrived.

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Station F.8: Skip-Counting Ladders with Constant Repeat

How does tapping equal towers of Counting Tiles across a desk turn into multiplication readiness? At this station (CCSS 3.OA.A.1; TEKS 3.4.E), partners snap together six towers of 5 tiles each, line them up in a row across the table, and touch each tower as they skip-count aloud (5, 10, 15, 20, 25, 30) before matching that cadence with the SC-6 Constant Repeat key ([g] [REP]).

Demonstrate how to load Top Shelf \(T\) with [g] [REP] so the calculator skip-counts right alongside their fingers. Typing 5 and pressing [g] [REP] copies 5 into all four shelves (\(T, Z, Y,\) and \(X\)). Have students touch the second 5-tile tower and tap [+] once to see 10 on Shelf \(X\), then tap [+] four more times as they touch the third, fourth, fifth, and sixth towers (15, 20, 25, 30). Because Top Shelf \(T\) keeps feeding a fresh 5 down onto Shelf \(Y\) after every addition, each tap of [+] adds one more tower of 5.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
5	0	0	0	5	Enter step size of 5 onto Shelf \(X\)
[g] [REP]	5	5	5	5	Fill all four shelves (\(T, Z, Y, X\)) with 5
[+]	5	5	5	10	Two 5-towers \(= 10\); Shelf \(T\) refills \(5\) from the top
[+]	5	5	5	15	Three 5-towers \(= 15\)
[+]	5	5	5	20	Four 5-towers \(= \mathbf{20}\)

After mastering fives, let partners build and step through skip-counting ladders by twos up to 16 (2 [g] [REP] [+]...), by tens up to 70 (10 [g] [REP] [+]...), and by threes up to 21 (3 [g] [REP] [+]...).

Sometimes a student taps [ENTER] repeatedly after [g] [REP] and wonders why the display stays stuck on 5; remind them that [ENTER] only copies Shelf \(X\) upward to Waiting Shelf \(Y\), whereas [+] combines Shelf \(Y\) and the bottom shelf to step up the ladder.

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Station F.9: Two-Digit Addition with Regrouping

Why do we "carry a 1" when adding \(38 + 27\), and what is that carried 1 really worth? On the Place Value Studio Whole Numbers Tray (1000 | 100 | 10 | 1), digit miniblocks (0–9) and stackable +1 regrouping unit blocks make two-digit regrouping concrete (CCSS 3.NBT.A.2; TEKS 3.4.A). Begin the activity by having students dock 3 tens and 8 ones in the top row of the tray (\(38\), matching Shelf \(Y\)) and 2 tens and 7 ones in the bottom row (\(27\), matching Shelf \(X\)).

When students combine the ones column (\(8 + 7 = 15\) ones), ask why all 15 unit blocks cannot remain in the ONES socket. Because every group of 10 ones forms a new ten, students stack 10 of the +1 unit blocks into a \(70.0\text{ mm}\) tower, trade that tower for +1 in the TENS column, and leave 5 ones in the ONES socket. Counting the TENS column (\(3 + 2 + 1\text{ traded ten} = 6\text{ tens}\)) and the ONES column (\(5\text{ ones}\)) yields 65, which students verify on the SC-6 by keying 38 [ENTER] 27 [+].

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
38	0	0	0	38	Enter first addend (\(38\): 3 tens, 8 ones)
[ENTER]	0	0	38	38	Copy 38 up onto Shelf \(Y\)
27	0	0	38	27	Enter second addend (\(27\): 2 tens, 7 ones)
[+]	0	0	0	65	Regroup 15 ones into 1 ten and 5 ones: 65

For independent practice, teams model and check \(46 + 28\) (46 [ENTER] 28 [+] \(\to 74\)), \(57 + 35\) (57 [ENTER] 35 [+] \(\to 92\)), and \(19 + 64\) (19 [ENTER] 64 [+] \(\to 83\)).

This physical trade cures the classic paper-and-pencil 515 error (where a student adds \(3+2=5\) and \(8+7=15\) and writes 515): whenever a student forgets to carry the new ten, have them physically stack 10 of the 15 unit cubes and move that tower into the TENS column before reading the result.

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Station F.10: Two-Digit Subtraction with Unbundling

Dock 5 in the TENS socket and 2 unit blocks in the ONES socket of the Place Value Studio Whole Numbers Tray to represent \(52\), then challenge partners to lift 6 ones out of the ONES socket (\(52 - 26\), CCSS 3.NBT.A.2; TEKS 3.4.A).

Students immediately spot the bottleneck: there are only 2 loose ones sitting in the ONES column! Ask how they can get more loose ones on the tray without changing the starting total of 52. By lifting 1 ten out of the TENS column (leaving 4 tens) and trading it for ten +1 unit blocks in the ONES column, the tray now holds 4 tens and 12 ones (\(40 + 12 = 52\)). From those 12 ones and 4 tens, students easily lift away 6 ones and 2 tens to leave 2 tens and 6 ones (26), then verify on the stack with 52 [ENTER] 26 [-].

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
52	0	0	0	52	Enter starting total (\(52\): 5 tens, 2 ones)
[ENTER]	0	0	52	52	Copy 52 up onto Shelf \(Y\)
26	0	0	52	26	Enter amount to subtract (\(26\))
[-]	0	0	0	26	Unbundle 1 ten (\(4\text{ tens} + 12\text{ ones} - 26\)): 26

Students can then practice unbundling on the tray and verifying on the SC-6 with \(63 - 27\) (63 [ENTER] 27 [-] \(\to 36\)), \(71 - 38\) (71 [ENTER] 38 [-] \(\to 33\)), and \(80 - 45\) (80 [ENTER] 45 [-] \(\to 35\)).

Watch for the upside-down subtraction habit where a student flips the ones digits on paper (\(6 - 2 = 4\)) to get 34 instead of 26. Pointing to the physical tray—where only 2 loose ones sit in the ONES column until a ten is unpacked into 10 ones—makes the reason for unbundling unmistakable.

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Station F.11: Even and Odd Numbers via Equal Pairing

Give each pair of students 15 counters and ask them to line up 11 counters in two equal rows of partners to open the investigation (CCSS 3.OA.D.9; TEKS 3.4.I). Linking physical partner-pairing to remainder division ([g] [÷R]) turns even-and-odd rules into something students can see and touch.

Students see right away that 11 counters form 5 complete pairs with 1 leftover counter standing alone at the end of the line. Explain that this lone leftover counter is the physical signature of an odd number, and show how the SC-6 Quotient & Remainder key ([g] [÷R]) tests any number for partners: typing 11 [ENTER] 2 [g] [÷R] places the 5 complete pairs on Shelf \(X\) and stores the leftover remainder on Shelf \(Y\). Tapping [x⇄y] brings that remainder of 1 to the front, proving that 11 is odd (whereas any even number leaves a remainder of 0 when divided by 2).

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
11	0	0	0	11	Enter 11 counters to test
[ENTER]	0	0	11	11	Copy 11 up onto Shelf \(Y\)
2	0	0	11	2	Divide into pairs of 2
[g] [÷R]	0	0	1 (Rem)	5 (Quot)	5 full pairs in \(X\); remainder 1 in \(Y\)
[x⇄y]	0	0	5 (Quot)	1 (Rem)	Swap to view the 1 leftover counter (ODD)

Have students test 14 (14 [ENTER] 2 [g] [÷R] [x⇄y] \(\to\) Remainder 0, even), 17 (17 [ENTER] 2 [g] [÷R] [x⇄y] \(\to\) Remainder 1, odd), and 20 (20 [ENTER] 2 [g] [÷R] [x⇄y] \(\to\) Remainder 0, even), and then test a three-digit mystery number like 348.

When a student guesses that 16 is odd because its tens digit 1 is odd, have them pair up a full 10-frame first: because every 10 is already 5 complete pairs of 2, only the ones digit decides whether a counter is left without a partner.

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B.2 Grade 3 Lesson Plans (Ages 8–9)

Grade 3 stations move from counting into 2D rectangular arrays ([f] [RECT]), fair-share division with remainders ([g] [÷R]), unit fraction towers, and polygon perimeters ([Σ+]).

Station 3.1: Arrays, Equal Groups & Rectangular Area

When students snap 24 tiles from their Four-Color Counting & Array Studio onto a \(4 \times 4\) Gridfinity coordinate baseplate in 6 columns of 4 rows, equal-group multiplication and rectangular floor area become the exact same picture (CCSS 3.OA.A.1, 3.MD.C.7; TEKS 3.4.D, 3.6.C).

Ask students to count how many columns of 4 form their rectangle and how many total square tiles cover its floor (24). Introduce area as the number of unit squares covering a flat region, and have students enter the rectangle's length and width on the SC-6 (6 [ENTER] 4) before tapping [f] [RECT] to see 24 on Shelf \(X\). Next, have them rotate the baseplate \(90^\circ\) on the desk so it stands 4 columns wide by 6 rows tall and run 4 [ENTER] 6 [f] [RECT]: because turning the baseplate never adds or removes a single tile, \(6 \times 4\) and \(4 \times 6\) cover the exact same area of 24 square units.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
6	0	0	0	6	Enter 6 columns along the length
[ENTER]	0	0	6	6	Copy 6 up onto Shelf \(Y\)
4	0	0	6	4	Enter 4 rows along the width
[f] [RECT]	0	0	0	24	Compute rectangle area: \(6 \times 4 = \mathbf{24}\) tiles

To build fluency with [f] [RECT], have partners calculate and model a \(7 \times 5\) rug (7 [ENTER] 5 [f] [RECT] \(\to 35\)), an \(8 \times 3\) garden bed (8 [ENTER] 3 [f] [RECT] \(\to 24\)), and—combining two 24-tile partner kits—a \(9 \times 4\) patio (9 [ENTER] 4 [f] [RECT] \(\to 36\)).

Once partners finish the three practice builds, launch the "Capture the Rectangle" four-player table game using the shared 144-tile classroom case (Player 1 in Archival White, Player 2 in Cobalt Blue, Player 3 in Radiant Gold, and Player 4 in Slate Gray) and two dice (or [🎲]). On each turn, a player rolls two factors (such as 5 and 4), computes the area on the SC-6 (5 [ENTER] 4 [f] [RECT] \(\to 20\)), and snaps a \(5 \times 4\) rectangle of their color onto the shared board so that no two rectangles of the same color touch along a side. If a student confuses perimeter with area during gameplay, have them trace the outer edge with one fingertip ("that's the perimeter fence") and press their open palm flat across the tiles ("that's the area floor").

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Station 3.2: Division as Fair Sharing vs. Grouping

Deal 23 Four-Color Counting Tiles (from a single 24-tile student kit) one by one into 4 Gridfinity sorting trays like playing cards until you cannot complete another fair round (CCSS 3.OA.A.2; TEKS 3.4.H).

When the dealing stops, students see 5 tiles inside each of the 4 trays and 3 leftover tiles sitting on the desk because 3 is not enough to give all 4 trays a sixth tile. Have them verify their physical deal on the SC-6 by entering the starting total first (23 [ENTER]), the 4 trays second (4), and pressing [g] [÷R]. Shelf \(X\) displays the whole-number quotient 5 (tiles per tray), and tapping [x⇄y] brings Shelf \(Y\) to the front to reveal the remainder 3 (leftover tiles).

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
23	0	0	0	23	Enter 23 total tiles onto Shelf \(X\)
[ENTER]	0	0	23	23	Copy 23 up onto Shelf \(Y\)
4	0	0	23	4	Enter 4 equal sorting trays
[g] [÷R]	0	0	3 (Rem)	5 (Quot)	5 tiles per tray in \(X\); 3 leftover in \(Y\)
[x⇄y]	0	0	5 (Quot)	3 (Rem)	Swap to bring the 3 leftover tiles to \(X\)

Ask partners to solve three real-world sharing puzzles with [g] [÷R]: 19 markers shared among 3 pencil cases (19 [ENTER] 3 [g] [÷R] \(\to\) Quotient 6, Remainder 1), 31 apples packed into bags of 6 (31 [ENTER] 6 [g] [÷R] \(\to\) Quotient 5, Remainder 1), and 35 tiles shared among teams of 4 (35 [ENTER] 4 [g] [÷R] \(\to\) Quotient 8, Remainder 3).

Then let partners play "Remainder Race to Bank" with a shared pool of 36 tiles and a die: Player 1 rolls (say 5), runs 36 [ENTER] 5 [g] [÷R], returns the quotient (7) to the supply depot, swaps with [x⇄y] to see the remainder (1), and banks that 1 leftover tile as their score while play continues with the remaining tiles. If a student accidentally presses regular division ([÷]) on \(23 \div 4\) and gets 5.75, ask whether we can deal 0.75 of a solid plastic tile into a tray—reinforcing when a problem calls for whole-number remainders ([g] [÷R]) versus decimal slices ([÷]).

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Station 3.3: Introduction to Unit Fractions with Fraction Studio

Why does a larger denominator produce a smaller unit fraction (\(\frac{1}{8} < \frac{1}{2}\)) when \(8\) is bigger than \(2\)? Standing the \(84\text{ mm}\) Fraction Studio towers upright on the desk settles this third-grade question in seconds (CCSS 3.NF.A.1; TEKS 3.3.A). Begin the activity by having students stand the \(1\text{-Whole}\) tower (\(84.0\text{ mm}\)) next to two stacked \(\frac{1}{2}\) blocks (\(42.0\text{ mm}\) each) to feel their flush tops.

Now ask partners to stand a single \(\frac{1}{2}\) block next to a single \(\frac{1}{8}\) block (\(10.5\text{ mm}\)) and explain why the \(\frac{1}{8}\) block is so much shorter even though \(8\) is larger than \(2\). Seeing the blocks side by side makes the reason obvious: slicing the same \(84\text{ mm}\) bar into 8 equal shares makes each share four times thinner than slicing it into 2 shares. Have students measure the height gap between \(\frac{1}{2}\) and \(\frac{1}{8}\) on the SC-6 by entering \(\frac{1}{2}\) (0 [.] 1 [.] 2 [ENTER]) and subtracting \(\frac{1}{8}\) (0 [.] 1 [.] 8 [-]) to see 3/8 on Shelf \(X\), then stack three \(\frac{1}{8}\) blocks on top of the \(\frac{1}{8}\) block so it stands flush with \(\frac{1}{2}\) (\(\frac{4}{8} = \frac{1}{2}\)).

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
0 [.] 1 [.] 2	0	0	0	1/2	Enter fraction \(1/2\) onto Shelf \(X\)
[ENTER]	0	0	1/2	1/2	Copy \(1/2\) up onto Shelf \(Y\)
0 [.] 1 [.] 8	0	0	1/2	1/8	Enter fraction \(1/8\) onto Shelf \(X\)
[-]	0	0	0	3/8	Subtract: \(4/8 - 1/8 =\) 3/8 height difference

During partner practice, students verify three relationships with blocks and keys: stacking three \(\frac{1}{3}\) blocks to equal 1 whole (0 [.] 1 [.] 3 [ENTER] 3 [×] \(\to 1\)), measuring the height difference between \(\frac{1}{2}\) and \(\frac{1}{4}\) (0 [.] 1 [.] 2 [ENTER] followed by 0 [.] 1 [.] 4 [-] \(\to 1/4\)), and standing four \(\frac{1}{8}\) blocks beside \(\frac{1}{2}\) (0 [.] 1 [.] 8 [ENTER] 4 [×] \(\to 1/2\)).

Whenever a student slips into whole-number thinking and claims \(\frac{1}{8} > \frac{1}{4}\) because \(8 > 4\), simply ask them to stand the \(\frac{1}{8}\) block (\(10.5\text{ mm}\)) beside the \(\frac{1}{4}\) block (\(21.0\text{ mm}\)) in the comparator lid—their fingertips will correct the misconception in seconds.

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Station 3.4: Perimeter of Polygons vs. Area in the Wall Basket

Tracing the outer walls of a multi-sided polygon is easy until a student loses count of which side they started on! Combining Counting & Array Studio tiles, the Algebra Factoring L-Frame Stage, and a millimeter ruler with the SC-6 running-total basket ([Σ+]) keeps every wall accounted for (CCSS 3.MD.D.8; TEKS 3.7.B). To open the investigation, students arrange tiles inside the \(90^\circ\) L-Frame Stage to form a stepped 5-sided polygon (pentagon) with side lengths \(6\text{ cm}, 4\text{ cm}, 4.2\text{ cm}, 4.2\text{ cm},\) and \(4\text{ cm}\).

Explain that finding perimeter means walking all the way around the outer fence and tossing each side length into the SC-6 statistics basket ([Σ+]). Have students first clear any old data from the basket with [g] [CLΣ], then trace each wall of the polygon with a finger as they key in Side 1 (6 [Σ+]), Side 2 (4 [Σ+]), Side 3 (4.2 [Σ+]), Side 4 (4.2 [Σ+]), and Side 5 (4 [Σ+]).

Because the screen confirms Count = 5 after the fifth tap, students know they haven't skipped or double-counted a wall before pressing [f] [Σx] to read the total perimeter of 22.4 cm.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
[g] [CLΣ]	0	0	0	0	Clear the statistics basket (Count = 0)
6 [Σ+]	0	0	0	1	Add Side 1 (\(6\text{ cm}\)) to the basket (Count = 1)
4 [Σ+]	0	0	0	2	Add Side 2 (\(4\text{ cm}\)) to the basket (Count = 2)
4.2 [Σ+]	0	0	0	3	Add Side 3 (\(4.2\text{ cm}\)) to the basket (Count = 3)
4.2 [Σ+]	0	0	0	4	Add Side 4 (\(4.2\text{ cm}\)) to the basket (Count = 4)
4 [Σ+]	0	0	0	5	Add Side 5 (\(4\text{ cm}\)) to the basket (Count = 5)
[f] [Σx]	0	0	0	22.4	Sum all 5 sides: 22.4 cm perimeter

Have partners use [g] [CLΣ], [Σ+], and [f] [Σx] to find the perimeter of a triangle with sides \(5, 7, 8\) (Count = 3, sum 20), a rectangle with sides \(8, 5, 8, 5\) (Count = 4, sum 26), and a hexagon with sides \(3, 3, 4, 4, 5, 5\) (Count = 6, sum 24).

If a student's perimeter on the second or third shape comes out unexpectedly large (say 46 instead of 26 on the rectangle), ask them to tap [N] to check the item count: seeing Count = 7 immediately reveals that they forgot to tap [g] [CLΣ] to empty the basket before starting the new polygon!

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B.3 Grade 4 Lesson Plans (Ages 9–10)

Grade 4 stations extend division to multi-digit remainders ([g] [÷R]), build equivalent fractions and one-tap GCF reductions ([g] [SIMP]), add and subtract unlike fractions, and measure angles with the \(360^\circ\) Volvelle.

Station 4.1: Long Division with Remainders

What happens in real life when a division problem leaves a remainder—do you leave the leftover campers outside in the rain, or order one extra cabin? Using the Place Value Studio (or the shared 144-tile Counting Studio bin) and [g] [÷R], students learn how to interpret quotients and remainders in real-world packing problems (CCSS 4.OA.A.3; TEKS 4.4.F). Start with a smaller hands-on trial using a 24-tile student kit—packing 20 trail-mix bars into bags of 6 (\(20 \div 6 = 3\text{ bags}\) with \(2\text{ bars}\) left over)—and then pose the main station challenge: distributing \(89\text{ fluid ounces}\) of water into \(16\text{-ounce}\) hiking flasks.

Have students dock 8 in TENS and 9 in ONES on the Place Value Tray to represent the \(89\text{ oz}\) supply, then key 89 [ENTER] 16 [g] [÷R] on the SC-6. Shelf \(X\) reports 5 completely filled flasks, and tapping [x⇄y] brings Shelf \(Y\) to the front to show 9 leftover fluid ounces still waiting in the pitcher. Ask students to verify the division by running the inverse check right on the stack: \(5\text{ flasks} \times 16\text{ oz} = 80\text{ oz}\), plus the \(9\text{ leftover oz} = 89\text{ oz}\).

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
89	0	0	0	89	Enter 89 fluid ounces of water
[ENTER]	0	0	89	89	Copy 89 up onto Shelf \(Y\)
16	0	0	89	16	Enter 16 ounces per flask
[g] [÷R]	0	0	9 (Rem)	5 (Quot)	5 full flasks in \(X\); 9 leftover oz in \(Y\)
[x⇄y]	0	0	5 (Quot)	9 (Rem)	Swap to inspect the 9 leftover ounces

During station practice, challenge teams to find both the quotient and remainder—and explain what you must do with the remainder in real life—for 145 campers assigned to cabins of 12 (145 [ENTER] 12 [g] [÷R] \(\to 12\text{ cabins, R }1\text{ camper}\), requiring a 13th cabin so no camper sleeps outside!), 250 eggs packed into cartons of 12 (250 [ENTER] 12 [g] [÷R] \(\to 20\text{ full cartons, R }10\text{ eggs}\)), and 500 pencils boxed in packs of 24 (500 [ENTER] 24 [g] [÷R] \(\to 20\text{ packs, R }20\text{ pencils}\)).

When coaching at tables, watch for students who read the remainder 9 on Shelf \(Y\) as 5.9 flasks. Ask them what unit the starting 89 was measured in (fluid ounces): because the remainder is simply the unpoured water left in the pitcher, 9 means 9 whole fluid ounces (\(\frac{9}{16}\) of a flask), not 9 tenths!

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Station 4.2: Equivalent Fractions & One-Tap GCF Simplification

Stack six \(\frac{1}{8}\) blocks (\(6 \times 10.5\text{ mm} = 63.0\text{ mm}\)) in the comparator well of the \(84\text{ mm}\) Fraction Studio and ask partners how many \(\frac{1}{4}\) blocks (\(21.0\text{ mm}\) each) match that exact height (CCSS 4.NF.A.1; TEKS 4.3.C). Pairing the physical fraction towers with the SC-6 Simplify key ([g] [SIMP]) proves that simplifying a fraction fuses smaller slices into the largest shared slice size (GCF) without changing the tower's height.

Students see that three \(\frac{1}{4}\) blocks stand flush with six \(\frac{1}{8}\) blocks because every pair of \(\frac{1}{8}\) blocks fuses into one \(\frac{1}{4}\) block. Now have them enter six-eighths on the SC-6 (0 [.] 6 [.] 8 [ENTER]) and press [g] [SIMP] ([g] + [a ᵇ/c]): Shelf \(X\) reduces 6/8 to 3/4 by dividing both the top and bottom numbers by their Greatest Common Factor (2).

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
0 [.] 6 [.] 8	0	0	0	6/8	Enter fraction \(6/8\) onto Shelf \(X\)
[ENTER]	0	0	6/8	6/8	Copy \(6/8\) up onto Shelf \(Y\)
[g] [SIMP]	0	0	6/8	3/4	Divide top and bottom by \(\text{GCF}(6,8) = 2\): 3/4

Have partners build the matching towers, predict the Greatest Common Factor with [GCF], and confirm with [g] [SIMP] for \(\frac{8}{12}\) (0 [.] 8 [.] 12 [g] [SIMP] \(\to \frac{2}{3}\), dividing by \(\text{GCF} = 4\)), \(\frac{10}{25}\) (0 [.] 10 [.] 25 [g] [SIMP] \(\to \frac{2}{5}\), dividing by \(\text{GCF} = 5\)), and \(\frac{18}{24}\) (0 [.] 18 [.] 24 [g] [SIMP] \(\to \frac{3}{4}\), dividing by \(\text{GCF} = 6\)), recording the starting tower, shared GCF group size, and simplified tower in their notebooks.

One quick teacher check before the rotation starts: if a student wants to hold an unreduced denominator on screen (such as keeping twelfths visible while comparing \(\frac{8}{12}\) and \(\frac{9}{12}\)), they can lock a fixed denominator with 12 [g] [/c] (Flag 8), and then tap [g] [SIMP] when ready to reduce to lowest terms (2/3 and 3/4).

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Station 4.3: Adding & Subtracting Fractions with Unlike Denominators

Before students ever compute a common denominator on paper, let them feel why unlike fractions must trade into a shared slice size using the \(84\text{ mm}\) Fraction Studio towers (CCSS 4.NF.B.3; TEKS 4.3.E). Open the activity by having partners stack one \(\frac{1}{2}\) block (\(42.0\text{ mm}\)) and one \(\frac{1}{4}\) block (\(21.0\text{ mm}\)) together and measure the combined \(63.0\text{ mm}\) tower against their \(\frac{1}{4}\) blocks.

Because the \(\frac{1}{2}\) block trades cleanly for two \(\frac{1}{4}\) blocks, the combined tower equals three \(\frac{1}{4}\) blocks (\(\frac{2}{4} + \frac{1}{4} = \frac{3}{4}\)). Have students verify this on the SC-6 by keying \(\frac{1}{2}\) (0 [.] 1 [.] 2 [ENTER]) and adding \(\frac{1}{4}\) (0 [.] 1 [.] 4 [+]) to see 3/4 on Shelf \(X\). Next, challenge them to model \(\frac{5}{6} - \frac{1}{3}\) by trading the \(\frac{1}{3}\) block (\(28.0\text{ mm}\)) for two \(\frac{1}{6}\) blocks (\(14.0\text{ mm}\) each): removing \(\frac{2}{6}\) from \(\frac{5}{6}\) leaves \(\frac{3}{6}\), which stands flush with the \(\frac{1}{2}\) block and matches \(\frac{5}{6}\) (0 [.] 5 [.] 6 [ENTER]) minus \(\frac{1}{3}\) (0 [.] 1 [.] 3 [-] \(\to\) 3/6, then [g] [SIMP] \(\to\) 1/2) on the calculator.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
0 [.] 1 [.] 2	0	0	0	1/2	Enter fraction \(1/2\) onto Shelf \(X\)
[ENTER]	0	0	1/2	1/2	Copy \(1/2\) up onto Shelf \(Y\)
0 [.] 1 [.] 4	0	0	1/2	1/4	Enter fraction \(1/4\) onto Shelf \(X\)
[+]	0	0	0	3/4	Trade \(1/2\) for \(2/4\) and add: 3/4

From there, partners build with blocks and verify three practice sums and differences on the stack: adding \(\frac{1}{3} + \frac{1}{6}\) (0 [.] 1 [.] 3 [ENTER] and 0 [.] 1 [.] 6 [+] \(\to \frac{1}{2}\)), subtracting \(\frac{3}{4} - \frac{1}{8}\) (0 [.] 3 [.] 4 [ENTER] and 0 [.] 1 [.] 8 [-] \(\to \frac{5}{8}\)), and combining \(\frac{2}{5} + \frac{3}{10}\) (0 [.] 2 [.] 5 [ENTER] and 0 [.] 3 [.] 10 [+] \(\to \frac{7}{10}\)).

Whenever a student falls into the add-across trap on paper—writing \(\frac{1}{2} + \frac{1}{4} = \frac{2}{6}\) by adding tops and bottoms separately—ask them to stand two \(\frac{1}{6}\) blocks (\(28.0\text{ mm}\)) next to their starting \(\frac{1}{2}\) block (\(42.0\text{ mm}\)). Seeing that \(\frac{2}{6}\) is shorter than the \(\frac{1}{2}\) block they started with before even adding \(\frac{1}{4}\) permanently cures the add-across habit!

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Station 4.4: Angles & Rotations with the \(360^\circ\) Volvelle Protractor

Set the pointer of the two-disc rotating \(360^\circ\) Unit Circle Volvelle to \(0^\circ\), rotate the top dial to match the square \(90^\circ\) corner of a removable clock quadrant sector, and let students feel how degree turns work in their hands (CCSS 4.MD.C.6; TEKS 4.7.C). Combining the Volvelle, the \(90^\circ\) clock quadrants, and the equilateral triangle faces of the \(d4\)/\(d8\) polyhedral dice makes acute, right, and obtuse angles effortless to compare.

Ask students to rotate the Volvelle pointer to the \(120^\circ\) mark (which corresponds to 4 hours on the Planetary Geared Clock face) and compare that opening against a \(90^\circ\) square corner to see that \(120^\circ\) is an obtuse angle. Ask how many degrees remain between that \(120^\circ\) turn and a \(180^\circ\) straight line (supplementary angles): keying 180 [ENTER] 120 [-] on the SC-6 yields 60° on Shelf \(X\). Have students nest the \(60^\circ\) corner of a \(d4\) or \(d8\) die right beside the \(120^\circ\) Volvelle sector to see the two angles snap together into a flat \(180^\circ\) straightedge.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
180	0	0	0	180	Enter straight-line angle (\(180^\circ\))
[ENTER]	0	0	180	180	Copy \(180^\circ\) up onto Shelf \(Y\)
120	0	0	180	120	Enter measured obtuse angle (\(120^\circ\))
[-]	0	0	0	60	Subtract for supplementary angle: 60°

Then invite partners to dial in and compute the complementary partner of \(35^\circ\) inside a \(90^\circ\) right angle (90 [ENTER] 35 [-] \(\to 55^\circ\)), the supplementary partner of \(45^\circ\) along a \(180^\circ\) straight line (180 [ENTER] 45 [-] \(\to 135^\circ\)), and the missing fourth angle when \(90^\circ, 120^\circ,\) and \(80^\circ\) meet around a full \(360^\circ\) circle (360 [ENTER] 90 [-] 120 [-] 80 [-] \(\to 70^\circ\)).

If a student misreads an ordinary protractor scale and reports \(60^\circ\) for an angle that is clearly obtuse, teach them the Square-Tile Corner Check: tucking a \(90^\circ\) square counting tile into the vertex first immediately tells their eyes whether the angle is narrower (\(<90^\circ\)) or wider (\(>90^\circ\)) than a right angle.

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B.4 Grade 5 Lesson Plans (Ages 10–11)

Grade 5 stations explore 3D prism volume and [Σ+] surface area baskets, fraction multiplication and flip-and-multiply division ([f] [1/x]), and parenthesis-free order of operations on the 4-shelf stack.

Station 5.1: Volume & Total Surface Area of Prisms

Hand each partner pair a \(6 \times 5 \times 4\text{ cm}\) rectangular prism (distinct from the \(5 \times 4 \times 3\text{ cm}\) gift box in Chapter 6 so students apply the 3-pair wall method to a fresh dimension set) and ask them to turn it in their hands to compare the space packed inside against the six flat walls wrapping the outside (CCSS 5.MD.C.5, 6.G.A.4; TEKS 5.6.B). Using the SC-6 running-total basket ([Σ+]), students can separate 3D interior volume from 2D exterior surface area without needing scratch paper. Have partners inspect all 6 faces of the box first and notice that opposite faces are identical twins: a matching top and bottom (\(6 \times 5\)), a matching front and back (\(6 \times 4\)), and a matching left and right end (\(5 \times 4\)).

First, have students find the interior volume (\(\text{Length} \times \text{Width} \times \text{Height}\)) by keying 6 [ENTER] 5 [×] 4 [×] (or 6 [ENTER] 5 [f] [RECT] 4 [f] [PRISM]) to get 120 cm³. Then have them clear the statistics basket ([g] [CLΣ]) to accumulate the 6 outer walls in three twin pairs: first the Top & Bottom Pair (\(6 \times 5 \times 2\) via 6 [ENTER] 5 [×] 2 [×] [Σ+] \(\to\) 60 cm², Count = 1), second the Front & Back Pair (\(6 \times 4 \times 2\) via 6 [ENTER] 4 [×] 2 [×] [Σ+] \(\to\) 48 cm², Count = 2), and third the Left & Right Pair (\(5 \times 4 \times 2\) via 5 [ENTER] 4 [×] 2 [×] [Σ+] \(\to\) 40 cm², Count = 3).

With Count = 3 pairs verified on screen, tapping [f] [Σx] recalls the complete 6-face surface area of 148 cm² on Shelf \(X\).

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
[g] [CLΣ]	0	0	0	0	Clear the statistics basket (Count = 0)
6 [ENTER] 5 [×] 2 [×]	0	0	0	60	Top & bottom pair: \(6 \times 5 \times 2 = 60\text{ cm}^2\)
[Σ+]	0	0	0	1	Toss Pair 1 into the basket (Count = 1)
6 [ENTER] 4 [×] 2 [×]	0	0	0	48	Front & back pair: \(6 \times 4 \times 2 = 48\text{ cm}^2\)
[Σ+]	0	0	0	2	Toss Pair 2 into the basket (Count = 2)
5 [ENTER] 4 [×] 2 [×]	0	0	0	40	Left & right pair: \(5 \times 4 \times 2 = 40\text{ cm}^2\)
[Σ+]	0	0	0	3	Toss Pair 3 into the basket (Count = 3)
[f] [Σx]	0	0	0	148	Sum all 6 faces: 148 cm² total surface area

Next, have partners compute both volume and total surface area for a \(6 \times 3 \times 2\text{ cm}\) box (\(\text{Volume} = 36\text{ cm}^3\), \(\text{Surface Area} = 72\text{ cm}^2\)), a \(4 \times 4 \times 4\text{ cm}\) cube (\(\text{Volume} = 64\text{ cm}^3\), \(\text{Surface Area} = 96\text{ cm}^2\)), and a \(10 \times 8 \times 5\text{ cm}\) storage crate (\(\text{Volume} = 400\text{ cm}^3\), \(\text{Surface Area} = 340\text{ cm}^2\)).

Watch for 74 cm² instead of 148 cm² on the \(6 \times 5 \times 4\text{ cm}\) box: that means the student added only the three visible faces (\(30 + 24 + 20 = 74\)) and forgot the opposite twin walls; have them touch both the ceiling and floor of the physical box as they multiply each face area by 2.

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Station 5.2: Fraction Multiplication & Flip-and-Multiply Division

Why does multiplying two proper fractions make a smaller piece, while dividing by a fraction flips the divisor upside down? At this station (CCSS 5.NF.B.4, 5.NF.B.7; TEKS 5.3.I, 5.3.L), students use transparent area-grid overlays (or Fraction Studio towers) alongside the SC-6 reciprocal key [f] [1/x] to see both operations in action. Begin by having students shade \(\frac{3}{4}\) of a square grid vertically in blue and \(\frac{2}{3}\) of the same square horizontally in yellow: counting the 6 green overlapping cells out of 12 total cells visually proves that \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\), which matches entering \(\frac{2}{3}\) (0 [.] 2 [.] 3 [ENTER]), multiplying by \(\frac{3}{4}\) (0 [.] 3 [.] 4 [×] \(\to\) 6/12), and tapping [g] [SIMP] (\(\to\) 1/2) on the SC-6.

Now explore why dividing by a fraction is the same as multiplying by its flipped reciprocal ([f] [1/x]). Pose the problem \(\frac{3}{4} \div \frac{2}{5}\): have students enter the starting quantity \(\frac{3}{4}\) onto Shelf \(Y\) (0 [.] 3 [.] 4 [ENTER]), enter the divisor \(\frac{2}{5}\) on Shelf \(X\) (0 [.] 2 [.] 5), and tap [f] [1/x]. Right on the bottom shelf, 2/5 flips upside down into 5/2, and pressing [×] multiplies \(\frac{3}{4} \times \frac{5}{2}\) to display 15/8 (or 1 7/8 when toggled with [a ᵇ/c]).

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
0 [.] 3 [.] 4	0	0	0	3/4	Enter starting fraction \(3/4\) onto Shelf \(X\)
[ENTER]	0	0	3/4	3/4	Copy \(3/4\) up onto Shelf \(Y\)
0 [.] 2 [.] 5	0	0	3/4	2/5	Enter divisor fraction \(2/5\) onto Shelf \(X\)
[f] [1/x]	0	0	3/4	5/2	Flip divisor \(2/5\) into its reciprocal \(5/2\)
[×]	0	0	0	15/8	Multiply \(3/4 \times 5/2 =\) 15/8 (\(1\frac{7}{8}\))

For partner practice, teams solve three fraction challenges: multiplying \(\frac{4}{5} \times \frac{5}{8}\) (0 [.] 4 [.] 5 [ENTER] and 0 [.] 5 [.] 8 [×] [g] [SIMP] \(\to \frac{1}{2}\)), dividing \(\frac{1}{2} \div \frac{1}{6}\) (0 [.] 1 [.] 2 [ENTER] and 0 [.] 1 [.] 6 [f] [1/x] [×] \(\to 3\), verified by stacking three \(\frac{1}{6}\) blocks beside one \(\frac{1}{2}\) block!), and dividing \(\frac{2}{3} \div \frac{4}{9}\) (0 [.] 2 [.] 3 [ENTER] and 0 [.] 4 [.] 9 [f] [1/x] [×] [g] [SIMP] \(\to \frac{3}{2} = 1\frac{1}{2}\)).

When a student isn't sure which of the two fractions gets flipped with [f] [1/x], remind them of the elevator roles: Shelf \(Y\) holds the starting batch of dough you have on the table, while Shelf \(X\) holds the measuring scoop you are dividing by—so [f] [1/x] always flips the divisor sitting on the bottom shelf.

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Station 5.3: Order of Operations without Parentheses (Stack Architecture)

Look closely at the SC-6 keypad and ask your fifth graders what buttons are missing: there are zero ( or ) parenthesis keys! Using printed expression cards and a 4-shelf elevator mat, students discover how the four-shelf stack handles multi-step order of operations automatically (CCSS 5.OA.A.1; TEKS 5.4.F). Display \((8 + 4) \times (9 - 3)\) to open the station and ask which parts must be solved before the multiplication can happen: both parenthetical groups must be evaluated first.

Show students how naturally the four shelves handle both groups without a single parenthesis key. When they build the first group (8 [ENTER] 4 [+]), 12 lands on Shelf \(X\). As soon as they begin typing the second group (9 [ENTER] 3), the finished 12 rides automatically up to Shelf \(Z\) to wait safely out of the way! Pressing [-] turns \(9 - 3\) into 6 on the bottom shelf while dropping 12 right back down to Shelf \(Y\), and one final tap of [×] multiplies the two groups into 72.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
8 [ENTER] 4	0	0	8	4	Stage 8 and 4 for the first group
[+]	0	0	0	12	Solve first group: \(8 + 4 = 12\)
9	0	0	12	9	Enter 9; \(12\) lifts to Shelf \(Y\)
[ENTER]	0	12	9	9	Lift \(12\) to Shelf \(Z\) and copy 9 to Shelf \(Y\)
3	0	12	9	3	Enter 3 onto Shelf \(X\)
[-]	0	0	12	6	Solve second group (\(9 - 3 = 6\)); \(12\) drops to \(Y\)
[×]	0	0	0	72	Multiply both groups: \(12 \times 6 = \mathbf{72}\)

Have partners trace the shelf movements and evaluate three multi-step expressions: \((14 + 6) \div (7 - 2)\) via 14 [ENTER] 6 [+] then 7 [ENTER] 2 [-] [÷] (\(\to 4\)), \(50 - (4 \times 8) + 6\) via 50 [ENTER] 4 [ENTER] 8 [×] [-] 6 [+] (\(\to 24\)), and \((3 + 5)^2 - 10\) via 3 [ENTER] 5 [+] [x²] 10 [-] (\(\to 54\)).

When a student evaluating \(50 - (4 \times 8)\) tries to subtract \(50 - 4\) first just because 50 is written on the left, show them how 50 [ENTER] parks 50 safely upstairs on Shelf \(Z\) while 4 [ENTER] 8 [×] computes 32 on Shelf \(X\) so [-] subtracts \(50 - 32 = 18\).

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B.5 Grade 6 Lesson Plans (Ages 11–12)

Grade 6 stations bridge into pre-algebra with two-color integer zero-pairs ([+/-]), prime factor towers and GCF/LCM intersections ([PRIME], [g] [FACT]), and unit rates with [f] [LASTX].

Station 6.1: Negative Numbers & Two-Color Zero Pairs

Slide one \(+1\) yellow counter and one \(-1\) red counter together on the desk until they cancel out as a zero pair, and signed integer arithmetic loses all its mystery (CCSS 6.NS.C.5, 6.NS.C.6; TEKS 6.2.B). Equip each pair with 30 two-color counters (15 yellow for \(+1\) and 15 red for \(-1\), or the \(+1\) and \(-1\) unit tiles from the Algebra Factoring Studio) alongside the SC-6 sign-flip key [+/-]. Start by having partners place 7 yellow counters (\(+7\)) and 4 red counters (\(-4\)) on the desk, slide each \(+1/-1\) pair together to cancel as a zero pair, and see that 3 yellow counters (\(+3\)) remain.

Now reverse the balance by having students lay out 8 red counters (\(-8\)) and 5 yellow counters (\(+5\)): after 5 zero pairs cancel out, 3 red counters (\(-3\)) remain on the desk. To enter \(-8\) on the SC-6 (after toggling [NUM] Whole Number Mode off or tapping [f] [ALL] so signed integers are unlocked), students type 8 first and tap [+/-] to flip its sign to -8 on Shelf \(X\), press [ENTER] to lift -8 to Shelf \(Y\), and then type 5 [+] to confirm -3 on the bottom shelf.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
8	0	0	0	8	Enter 8 onto Shelf \(X\)
[+/-]	0	0	0	-8	Flip sign to \(-8\) (8 red counters)
[ENTER]	0	0	-8	-8	Copy \(-8\) up onto Shelf \(Y\)
5	0	0	-8	5	Enter \(+5\) (5 yellow counters)
[+]	0	0	0	-3	Cancel 5 zero-pairs: \(-8 + 5 = \mathbf{-3}\)

During station practice, partners model with counters and verify three signed challenges on the stack: combining two groups of red counters in \(-12 + (-7)\) (12 [+/-] [ENTER] 7 [+/-] [+] \(\to -19\)), removing 6 red debt counters in \(15 - (-6)\) (15 [ENTER] 6 [+/-] [-] \(\to 21\)), and flipping four groups of 8 red counters in \((-4) \times (-8)\) (4 [+/-] [ENTER] 8 [+/-] [×] \(\to 32\)).

The most common calculator slip-up in sixth grade is pressing the subtraction operation key [-] instead of the sign-flip key [+/-] when entering a negative number. Remind students that [-] immediately subtracts Shelf \(X\) from Shelf \(Y\), whereas [+/-] flips the sign of the single number currently sitting on the bottom shelf.

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Station 6.2: Prime Factorization, GCF & LCM

Every whole number greater than 1 is built from a unique team of prime building blocks—and in the Prime Factor Studio, students can literally stack those prime factors into physical towers whose heights add up to the product (CCSS 6.NS.B.4; TEKS 6.7.A). Pairing the Counting & Array Studio and the logarithmic prime blocks (2 through 29 with the comparator scale box) alongside [PRIME], [g] [FACT], [GCF], and [f] [LCM], begin with a two-part hands-on warm-up: 1. The Rectangle Tile Test: Have students arrange 24 counting tiles into multi-row rectangles (\(2 \times 12, 3 \times 8,\) and \(4 \times 6\)), then remove 1 tile and try to arrange 23 tiles into any rectangle at least 2 rows wide. Because 23 leaves a remainder when divided by 2, 3, or 4, it can only form a single \(1 \times 23\) strip—physically proving that 23 is prime. 2. Logarithmic Factor Stacking: Have students stack a Green 3-block (\(11.09\text{ mm}\)) on top of a Blue 2-block (\(7.00\text{ mm}\)) on the Prime Factor Studio comparator stage and see the top edge align flush with the 6 mark (\(18.09\text{ mm}\)).

Invite students to test 23 [PRIME] on the SC-6 to see PRIME flash on Shelf \(X\), and then test 60 [PRIME] (COMPOSITE) followed by [g] [FACT] to reveal 2² × 3 × 5. When students pull two Blue 2-blocks, one Green 3-block, and one Yellow 5-block from the studio and stack them in any order on the comparator stage, the tower always reaches the exact 60 mark at \(41.34\text{ mm}\)—which 2 [x²] 3 [×] 5 [×] \(\to 60\) confirms on the stack.

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
60	0	0	0	60	Enter composite candidate \(60\) onto Shelf \(X\)
[PRIME]	0	0	0	60	Screen flashes COMPOSITE while keeping \(60\) on Shelf \(X\)
[g] [FACT]	0	0	0	60	Screen displays 2² × 3 × 5 while keeping \(60\) on Shelf \(X\)

For station practice, have partners factor 84 (84 [g] [FACT] \(\to\) 2² × 3 × 7) and 180 (180 [g] [FACT] \(\to\) 2² × 3² × 5), build both prime towers side by side in the comparator stage to pull out their shared blocks (\(\text{GCF} = 2^2 \times 3 = 12\), verified via 84 [ENTER] 180 [GCF]), and verify their Least Common Multiple (84 [ENTER] 180 [f] [LCM] \(\to 1,260\)).

Whenever a student asks why 1 is not considered a prime number, have them key 1 [PRIME] (INVALID DATA) and look at the Prime Factor Studio tray: if 1 were a prime block of height \(0\text{ mm}\), you could stack five, ten, or a hundred 1 blocks onto 60 (1 × 1 × 2² × 3 × 5) and destroy the rule that every composite number has one unique team of prime building blocks!

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Station 6.3: Ratios, Unit Rates & Nature's Spiral Ratio

From grocery store price tags to the spiral of a pinecone, ratios and unit rates tell us how two quantities grow together (CCSS 6.RP.A.2, 6.RP.A.3; TEKS 6.4.B). Using double number lines, tape measures, and the SC-6 [f] [LASTX] key, students explore everyday unit rates and the Golden Ratio (\(\approx 1.618\)) hidden inside Fibonacci numbers. Warm up by asking how much 1 pack of gum costs if 3 packs cost \(\$ 4.50\) (4.50 [ENTER] 3 [÷] \(\to\) \(\$1.50\) per pack).

Walk students through finding a highway fuel-efficiency unit rate on the stack: if a family car travels \(372\text{ miles}\) on \(12\text{ gallons}\) of gas, entering miles first (372 [ENTER]), gallons second (12), and pressing [÷] yields 31 miles per gallon on Shelf \(X\). Then pivot to nature's famous pattern (\(1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89\)): have students divide consecutive terms \(55 \div 34\) (55 [ENTER] 34 [÷] \(\to 1.617647\)), and then generate the next Fibonacci number on the stack (34 [ENTER] 55 [+] \(\to 89\)). Because 55 was automatically preserved inside [f] [LASTX], tapping [f] [LASTX] [÷] immediately divides \(89 \div 55 = 1.618182\) without retyping 55!

Keystroke	Register T	Register Z	Register Y	Register X	What Happens on the Stack
372	0	0	0	372	Enter total distance (\(372\text{ miles}\))
[ENTER]	0	0	372	372	Copy 372 up onto Shelf \(Y\)
12	0	0	372	12	Enter fuel used (\(12\text{ gallons}\))
[÷]	0	0	0	31	Divide for unit rate: 31 miles per gallon

Finally, let partners compute the unit price of 6 pairs of socks for \(\$ 18.60\) (18.60 [ENTER] 6 [÷] \(\to\) \(\$3.10\) per pair), a typing speed of 385 words in 7 minutes (385 [ENTER] 7 [÷] \(\to 55\text{ words per minute}\)), and the next Fibonacci ratio \(144 \div 89\) (55 [ENTER] 89 [+] [f] [LASTX] [÷] \(\to 1.617978\)).

Whenever a student divides unit-rate numbers in reverse order (\(12 \div 372 = 0.032\text{ gallons per mile}\)), teach them to listen for the word "per" (miles PER gallon, dollars PER pair, words PER minute): the unit before "per" always rides up to Shelf \(Y\) first, and the unit after "per" divides it from Shelf \(X\) (or one tap of [x⇄y] before [÷] swaps them into place).

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B.6 Classroom Observation Guide & Common Student Stumbling Blocks

Across all 25 station lessons, teachers can check student progress across four friendly milestones:

Performance Band	Score	Hands-On Block Building	Visual Sketching	Calculator Stack Execution
Exemplary	4 pts	Fluidly constructs physical model with zero prompting. Explains spatial relationships and patterns clearly to peers.	Accurately sketches bar models, area arrays, or number lines with correct proportional scaling.	Keys problem into SC-6 fluently; explains shelf states (\(X, Y, Z, T\)); uses [f] [LASTX] or inverse operations to self-check.
Proficient	3 pts	Correctly builds model after initial launch. Self-corrects minor block errors without teacher help.	Draws recognizable visual models matching the problem context.	Enters correct keystrokes on SC-6; successfully identifies solution on Shelf \(X\).
Developing	2 pts	Requires teacher prompting to begin building or select the right 3D-printed math blocks.	Sketches disorganized diagrams that do not yet match the block sizes.	Occasionally forgets [ENTER] separator or mixes up [+/-] with [-]; needs step-by-step keystroke guidance.
Getting Started	1 pt	Needs a partner or teacher model to begin building with blocks; counts one by one.	Needs a starter template before sketching; writes isolated digits.	Needs guided practice with the calculator buttons; guesses without checking screen feedback.

B.6.1 Conducting a 90-Second Table Conference (The "Point-and-Prove" Check)

When you pull up a chair beside a partner pair during the 15-minute Station Investigation window, you can assess all three columns of the rubric in under 90 seconds using three quick "Point-and-Prove" prompts: 1. "Point to Shelf \(Y\) on your desk:" Ask the partners to touch the exact pile of counters, fraction tower, or rectangle side on their table that matches the number currently waiting on Shelf \(Y\) of their SC-6 screen. When a student can point directly from the upper line of the LCD to the first physical group on the desk, you know they are reading the stack as a real physical workspace rather than a black box. 2. "Predict before you press:" Before Partner B taps the operation key ([+], [-], [×], [g] [÷R], or [g] [SIMP]), cover the calculator screen with an index card and ask both partners what their physical blocks say the answer must be. Lifting the card after the key press turns every calculation into an instant self-check. 3. "Show me with [f] [LASTX]:" After an operation completes and Shelf \(Y\) and Shelf \(X\) collapse into a single result, ask the student what number was sitting on the bottom shelf just a moment ago—and have them tap [f] [LASTX] to prove it and run the inverse operation back to where they started.

B.6.2 Common Student Slip-Ups & Coaching Rescues

When a student gets stuck at a workshop station, use this quick table to spot what happened and get them rolling again:

What You Might See	Why It Happens	How to Help
Stuck-Together Digits (e.g., 64 instead of 6 [ENTER] 4)	Forgot to press [ENTER] before typing the second number	Remind them of the Elevator Model ("Lift 6 to Shelf \(Y\) first before 4 steps onto Shelf \(X\).")
Surprise Negative Answer (e.g., negative area or distance)	Typed the smaller number first before subtracting, or divided in reverse order	Point to the physical blocks: "Can \(4\) tiles have negative area? Tap [x⇄y] before you subtract!"
Counting One-by-One from \(1\)	Not yet spotting groups of \(5\) and \(10\) at a glance	Switch to Ten-Frame Mode ([▦]) and practice spotting full rows of \(5\) and \(10\)
Wild Guessing in Quiz Mode	Timer or level feels too fast right now	Drop down two difficulty levels and build one problem with physical blocks before restarting the sprint

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B.6.3 Integrating Fluency Sprints & Differentiated Pacing Across the School Week

Once students have built a concept with physical blocks during the 20-minute station investigation, short, low-stakes bursts in the SC-6 Quiz Engine ([f] [?], detailed in Chapter 8) consolidate their concrete models into effortless recall. Rather than assigning long paper worksheets at the end of a lesson, educators can weave a two-minute [f] [?] sprint into the final five minutes of the workshop block using three classroom management strategies:

Spaced 2-Minute Replay Schedule Across the Unit: Brief, spaced two-minute replays across several days build far stronger fluency than a single long drill session. When introducing a new station concept on Monday, schedule a 2-minute [f] [?] sprint at the matching level on Day 2, Day 4, Day 8, and Day 16:

Workshop Touchpoint	Recommended Classroom Format	Teacher Facilitation Focus
Day 1 (Station Launch)	Untimed Two-Way Mastery (10 problems with blocks + SC-6)	Verify every student can build the physical block model before touching [f] [?].
Day 2 (1st Replay)	Solo 2-Minute Personal-Best Sprint ([f] [?])	Students record their starting score and streak in their notebook log.
Day 4 (2nd Replay)	Partner Check with Physical Studio Blocks Nearby	Any missed fact that returns in the \(+3/+7\) review queue is built once with blocks.
Day 8 (3rd Replay)	4-Player Baton Relay (Shared Target Level 1–6)	Mixed-ability table teams pass the calculator every 2 questions (1–10).
Day 16+ (Spiral Review)	Friday Warm-Up Tournament (Shared Level + [g] [CLΣ] Reset)	Celebrate personal growth over Day 2 baseline scores across the whole class.

Synchronized Level Launches & Smart Review Coaching: During independent practice, the SC-6 shuffles questions automatically so students sitting side by side cannot copy each other's screens. When you want to run a whole-class warm-up or a 4-player table relay where every team races through the same skill tier, write the target Quiz Level (1–6) on the board. Having every student tap [f] [?] and select that same level launches every calculator in the room into the identical 3-band challenge deck (where students submit each answer with [≠] and tap [C] to advance to the next prompt).

Because the SC-6 Quiz Engine automatically recycles any missed fact 3 turns later and 7 turns later within the same round (while resting mastered facts for 30 turns), you never need to grade stacks of fluency slips to spot which facts a student finds tricky. As you circulate during the last two minutes of a sprint, glance at which problems reappear on a student's screen and keep the corresponding studio tray within reach—whether that means sliding two counters across the 5-anchor rib for a make-ten fact, turning a Counting Tile array \(90^\circ\) for a multiplication fact, or canceling \(+1/-1\) counter pairs for a signed-integer challenge.

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B.7 School Library Makerspace, Math Circle & Year-Long Pacing Guide

Beyond the daily 45-minute math classroom, school librarians, makerspace coordinators, and math coaches frequently use the StackCalc SC-6 and its 3D-printed math block studios to host drop-in library exploration corners, after-school Math Circles, and Family STEM Nights. Because every block tray snaps into a self-contained Gridfinity box where missing pieces are visible at a glance, a library or media center can circulate kits with the same ease as board games or robotics bins.

B.7.1 Setting Up Circulating Library "Math Explorer Backpacks"

School and public libraries can assemble sturdy, check-out-ready Math Explorer Backpacks for students in Grades 3–6 to borrow over a weekend with their families. Each circulating backpack pairs this book with a curated subset of 3D-printed trays so students can teach their parents and siblings at the kitchen table:

Backpack Theme	Recommended Grades	Included 3D-Printed Studio Trays	Featured Family Kitchen-Table Quests
Pack 1: Patterns, Arrays & Mind-Reading Math	Grades 3–4	Ten-Frame Studio + 24-Tile Counting & Array Studio	The 1089 Mind-Reading Trick (Ch. 2), Gauss's \(1\text{–}100\) Staircase & Four-Color Map Game (Ch. 4)
Pack 2: Fraction Towers, Musical Strings & Recipes	Grades 3–5	84 mm Fraction Studio + Twin Comparator Lid	Musical Glass Water Fractions (Ch. 5), Pharaoh's Unit-Fraction Bakery & Sourdough Scaling (Ch. 9)
Pack 3: 3D Boxes, Penguin Huddles & Clock Gears	Grades 4–6	Planetary Geared Clock + Counting Tiles & Cardstock Nets	Emperor Penguin Square-Cube Huddle, Cereal-Box Surface Area Audit & Clock-Hand Angle Hunt (Ch. 6)
Pack 4: Fair Dice, Cicada Primes & Secret Partners	Grades 5–6	Fair-Weight Dice & Spinner + Prime Factor Comparator Box	36-Roll Lucky 7 Tower Race, Impostor Prime Hunt & GCF \(\times\) LCM Secret Partner Challenge (Ch. 7)

To make check-in effortless at the circulation desk, tape a photo of the filled socket grid inside the lid of each backpack pouch: when all 24 Counting Tile sockets, 10 Ten-Frame pockets, or 14 Prime Factor slots are filled flush to the rim, the librarian knows in five seconds that every piece has come home safely.

B.7.2 Running a 60-Minute After-School Math Circle (Grades 3–6)

In an after-school Math Circle or library club, students from Grades 3 through 6 often sit together around shared tables without grade-level tests or homework grades. The goal of a Math Circle is joyful mathematical discovery—posing a surprising mystery that a 3rd grader can explore right away with physical blocks while a 6th grader uncovers why the pattern always works on the four-shelf stack. Here are four proven 60-minute Math Circle sessions built directly from this book:

1. The Square-Frame & Odd-Number Staircase Mystery (Chapter 3 & Chapter 4): Start by handing each table 25 Four-Color Counting Tiles and asking them to build a \(1 \times 1\) white square in the corner, wrap a 3-tile blue L-border around it to form a \(2 \times 2\) square (\(1 + 3 = 4\)), wrap a 5-tile gold L-border around that to form a \(3 \times 3\) square (\(4 + 5 = 9\)), and then add a 7-tile slate L-border (\(9 + 7 = 16\)) and a 9-tile white L-border (\(16 + 9 = 25\)). Challenge the circle to predict what happens if you add the first 20 odd numbers (\(1 + 3 + 5 + \dots + 39\)) before testing it on the SC-6 (20 [x²] \(\to 400\)). Then ask older students to remove an inner \(3 \times 3\) square from a \(5 \times 5\) square (\(5^2 - 3^2 = 16\)) and rearrange the leftover L-border into a single rectangle to discover why \((5 - 3) \times (5 + 3) = 2 \times 8 = 16\).

2. The Musical Monochord & Egyptian Fraction Bakery (Chapter 5): Bring four identical drinking glasses, a pitcher of water, a ruler, and a metal spoon to the library table alongside the 84 mm Fraction Studio. Have students measure the glass height, pour water to \(\frac{1}{4}\), \(\frac{1}{3}\), \(\frac{1}{2}\), and \(\frac{3}{4}\) full, and tap the glasses to hear the musical intervals rise as the air column shortens. Next, pose the ancient Egyptian scribe puzzle from the Rhind Papyrus: How can 8 bakers share 5 loaves of bread fairly without crumbling the loaves into tiny bits? Have students model \(\frac{5}{8}\) in the Twin Comparator Well by stacking one \(\frac{1}{2}\) block (\(42.0\text{ mm}\)) and one \(\frac{1}{8}\) block (\(10.5\text{ mm}\)) to reach \(52.5\text{ mm}\), then verify 0 [.] 1 [.] 2 [ENTER] 0 [.] 1 [.] 8 [+] \(\to\) 5/8 on the SC-6.

3. The Antarctic Penguin Huddle & Giant Monster Scaling Lab (Chapter 6): Ask the circle why a movie monster that is \(10\text{ times}\) taller than a human would collapse under its own weight, and why Emperor penguins huddle tightly together in \(-40^\circ\text{C}\) blizzards. Have students take 8 Unit Miniblocks and compare their exposed square faces when standing apart (\(8 \times 6 = 48\) chilly faces) versus snapped into a \(2 \times 2 \times 2\) cube (\(6 \times 4 = 24\) exposed faces—cutting heat loss in half!). Then challenge teams to use [g] [CLΣ] and [Σ+] to find which 3D rectangular prism built from 24 Counting Tiles (\(1 \times 1 \times 24\), \(1 \times 2 \times 12\), \(1 \times 3 \times 8\), \(1 \times 4 \times 6\), or \(2 \times 3 \times 4\)) has the smallest outer surface area (2 × 3 × 4 wins with just \(52\) square faces!).

4. The Settlers of Catan Dice Pyramid & Prime Factor Vault (Chapter 7): Give each team two volume-equalized \(d6\) dice and a supply of Counting Tiles arranged along a number line from 2 to 12. As teams roll the dice 36 times (and simulate rapid rolls on the SC-6 with [🎲] [🎲] [+]), they stack a tile on the matching sum for every roll and watch the tall pyramid peak over 6, 7, and 8. Follow up by switching to the Prime Factor Studio: challenge partners to race the SC-6 [g] [FACT] key by building factor towers for 36, 60, 84, and 120 in the comparator stage and discovering why stacking blocks in any order always reaches the exact same millimeter mark.

B.7.3 Hosting a Five-Table Family Math & Maker Night

One of the most rewarding ways to build community excitement around mathematics is a 90-minute Family Math & Maker Night in the school library or cafeteria, where students act as the station hosts and teach their parents and grandparents how the four-shelf stack works. Set up five tables around the room with student ambassadors stationed at each table:

Family Night Table	Studio Equipment on the Table	What Student Ambassadors & Families Do Together
Table 1: "Beat the Smartphone!" Relay	SC-6 Calculators + Parent Smartphones	Students hand parents a smartphone calculator while using the SC-6 themselves to race on two-group snack-bar orders like \((14 \times 6) + (18 \times 4)\) and mental-math squares like \(65^2\) (65 [x²] \(\to 4,225\)).
Table 2: \(84\text{ mm}\) Fraction Tower Challenge	\(84\text{ mm}\) Fraction Studio + Comparator Lid	Families race to find three different fraction block combinations that stand flush with the \(84.0\text{ mm}\) 1 WHOLE tower (such as \(\frac{1}{2} + \frac{1}{3} + \frac{1}{6}\)) and verify each with Two-Dot Entry and [g] [SIMP].
Table 3: Cereal-Box Unfolding Station	Flattened Cardboard Boxes + Rulers + SC-6	Families measure the three pairs of matching walls on unfolded boxes, toss each pair into [Σ+], verify Count = 3, and read the total cardboard area with [f] [Σx].
Table 4: Tabletop Catapult & Pendulum Lab	Rubber-Band Catapults + Tape Measure + SC-6	Families launch three catapult shots across the floor, log the landing distances with [Σ+], and press [g] [x̄] and [g] [MAD] to see both the bullseye average and the consistency spread (Chapter 9).
Table 5: Live 3D-Printing Tile Station	Desktop FDM 3D Printer + Digital Scale + SC-6	A 3D printer extrudes \(19.0 \times 19.0 \times 7.0\text{ mm}\) Counting Tiles live while student ambassadors show families how the SC-6 calculates the exact volume, gram weight (3.1 g), and 6-cent cost ([$]) of each tile!

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B.7.4 Pacing the 25 Stations Across the School Year & Companion Workbook Cross-Reference

To weave these 25 station lessons across a full 36-week school year (or a summer math camp), educators can group the stations into four seasonal modules that spiral from whole-number foundations to applied STEM labs and align directly with Book 2A (Grades K–2 Elementary Companion Workbook), Book 2B (Grades 3–5 Elementary Companion Workbook), and Book 3 (Big Ideas Middle School SC-6 Workbook):

Seasonal Module	Recommended Weeks	Featured Station Rotations	Companion Workbook Units (Book 2A / Book 2B / Book 3)	Companion Book Chapters
Quarter 1: Number Sense, Place Value & Stack Foundations	Weeks 1–9	Stations F.1–F.10 (Ten-Frames, Make-a-Ten, Bundling, Regrouping & [g] [REP] Ladders)	Book 2A Units U01–U05 (PLACE-01 Ten-Frames, Number Bonds, Regrouping & Skip-Counting)	Chapters 01–03
Quarter 2: Rectangular Arrays, Remainders & Unit Fractions	Weeks 10–18	Stations F.11, 3.1–3.4, and 4.1 ([f] [RECT] Arrays, [g] [÷R] Sharing, Unit Fractions & [Σ+] Perimeters)	Book 2A U06–U08 & Book 2B U01–U03 (PLACE-01 Arrays, Division with Remainders & FRAC-01 Unit Towers)	Chapters 04–05
Quarter 3: Equivalent Fractions, 3D Prisms & Order of Operations	Weeks 19–27	Stations 4.2–4.4 and 5.1–5.3 ([g] [SIMP] Reduction, Unlike Fractions, Volvelle Angles & 3D Prism Nets)	Book 2B Units U04–U07 (FRAC-01 Equivalence, Partitive Division, 3D Prism Nets & 4-Shelf Order of Operations)	Chapters 05–06
Quarter 4: Primes, Data Detective, Zero Pairs & Maker STEM Labs	Weeks 28–36	Stations 6.1–6.3 plus STEM Maker Projects ([+/-] Zero Pairs, [g] [FACT] Prime Towers, Ratios & Sprints)	Book 2B U08–U10 & Book 3 Units U01–U06 (PRIME-01 Logarithmic Towers, GCF/LCM, 1Q/3Q/IQR & Signed Integers)	Chapters 07–09

Within any single school week, teachers can rotate five table groups through a balanced 5-Day Weekly Workshop Rhythm paired with 3-Tier Differentiation (Practice A–F) from the companion workbooks so every student experiences hands-on block building, visual screen discovery, four-shelf stack verification, and team fluency relays:

Day of the Week	45-Minute Classroom Focus	Physical Studio & SC-6 Mode	3-Tier Workbook Differentiation (Practice A–F) & Formative Check
Monday (Launch & Build)	Whole-class 5-minute anchor story + Partner A/B physical block build at Tables 1–5	Core Anchor Studio (PLACE-01, FRAC-01, or PRIME-01) + Visual Mode ([▦], [f] [LINE], [PV], [🕒])	Tier 1 (Practice A–B Concrete Launch): First +10 token placed on Quest Scoreboard for verified physical model
Tuesday (Stack Connection)	Partners translate Monday's physical block model into 4-shelf stack keystrokes (\(X, Y, Z, T\))	[NUM] or Fraction Mode ([a ᵇ/c]) + [f] [LASTX] reverse checks	Tier 2 (Practice C–D Representational Stack Bridge): Second +10 token stacked (+20 total) + 90-second "Point-and-Prove" check
Wednesday (Station Switch)	Table groups rotate clockwise to a companion station or multi-step extension puzzle	Companion Studio Tray + Multi-step Shelf \(Z\) or [Σ+] basket	Tier 2–3 (Practice D–E Multi-Step Reasoning): Students log one 4-register trace table in their math notebooks
Thursday (Maker Application)	Real-world STEM measurement lab (rulers, scales, stopwatches, or cardstock nets)	Chapter 9 Maker Lab + Unit Conversions ([f] [▸cm], [f] [▸g], [g] [x̄], [g] [MAD])	Tier 3 (Practice E–F Transfer & STEM Challenge): Partner teams compare Count, Mean, and MAD across table trials
Friday (Relay & Reflection)	Synchronized Level 4-Player Baton Relay ([f] [?]) + Math Circle discussion	Shared Target Level ([f] [?], then level 1–6, [≠] submit, and [C] next) + Studio Trays on table	All Tiers (Practice A–F Spiral Synthesis): Celebrate personal-best growth over Tuesday baseline scores

Why the Four-Shelf Stack Works in Elementary Classrooms: Across all four quarters, teachers consistently notice that students make far fewer multi-step calculation mistakes on the SC-6 than on conventional parenthesis calculators. Studies of human memory show that our brains can comfortably focus on only a few numbers at a time (Miller, 1956; Sweller, 1988), and classroom research confirms that young learners grasp multi-step problems much more naturally when they enter numbers in the exact order they build with their blocks (Hamrick, 1996). By combining parenthesis-free stack entry (Burks et al., 1954) with exact decimal and fraction arithmetic (Kahan, 1979), the SC-6 keeps every step visible on Shelves \(X, Y, Z,\) and \(T\)—so students can focus on the meaning of the math rather than hunting for missing brackets.