11 units · 64 station pages · 256 problems · 180 auto-checked
quizgen/snippet-overrides.json and rebuilding.
kbase/mom/guides/unit-*-teaching.html by
extract_items.py and this page is the audit trail: prompt, key, level, source row, and
exactly what the auto-checker will accept. Auto items grade themselves in the student quiz;
talk items are discussion/graph prompts where the group types nothing useful and the key sits
behind one tap. Station numbers show which sections made the 7-question group-work cut
(greyed rows were dropped by the cap).
Use this page as the answer key when you walk the room.
Jump to: 1.01 · 1.02 · 1.03 · 1.04 · 1.05 · 1.06 · 1.07 · 1.08 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner substitute | Evaluate x + 9 for x = 4. ★ | 13 13, x=13 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate substitute | Evaluate 10 − 2m for m = 3. | 10 − 6 = 4 4, m=4 | auto | #Just the number — no units, no x=. |
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| Hot Advanced substitute | Evaluate p² + 6 for p = 5. ★ | 25 + 6 = 31 31, p=31 | auto | #Just the number — no units, no x=. |
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| Spicy Challenge translate | A streaming service charges $12 per month plus a one-time $30 setup fee. Write an expression for the total cost of m months, then find the total cost for 8 months. ★ | 12m + 30; the total is $126 12m+30 | auto | #m+#Type the expression with no spaces; use ^ for exponents and / for fractions. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | −3 + (−4) ★ | −7 — | talk | discussion prompt |
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| Medium Intermediate discuss | 9 − 14 | −5 — | talk | discussion prompt |
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| Hot Advanced discuss | −20 + 13 − 5 | −12 — | talk | discussion prompt |
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| Spicy Challenge discuss | A submarine is at an elevation of -250 feet relative to sea level. It then ascends 120 feet, descends 75 feet, and finally ascends another 40 feet. What is the submarine's final elevation? ★ | -165 feet — | talk | discussion prompt |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner arith | −8 × (−6) ★ | 48 48 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate discuss | −100 ÷ 25 | −4 — | talk | discussion prompt |
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| Hot Advanced scale | A diver starts at sea level and descends at a rate of 12 feet per minute. What is the diver's depth after 7 minutes? (Express your answer as an integer representing feet relative to sea level.) | -84 -84, s=-84 | auto | −#Just the number — no units, no x=. |
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| Spicy Challenge scale | A deep-sea submersible is exploring a trench. It descends at a rate of 40 feet per minute for 15 minutes. Then, it ascends at a rate of 25 feet per minute for 8 minutes. What is the submersible's final depth relative to its starting point? (Express your answer as an integer.) ★ | -400 -400, s=-400 | auto | −#Just the number — no units, no x=. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner arith | 5² ★ | 25 25 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate arith | (−6)² | 36 36 | auto | #Just the number — no units, no x=. |
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| Hot Advanced arith | 2⁵ − 3² | 32 − 9 = 23 23 | auto | #Just the number — no units, no x=. |
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| Spicy Challenge arith | Evaluate (−2)⁴ − (−3)² and explain why one result is positive and the other is negative. ★ | 16 − 9 = 7; an even exponent makes the negative base positive, and squaring (−3) also gives 9, so the difference is 7 7 | auto | #Just the number — no units, no x=. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner rationalexp | √49 ★ | 7 7 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate discuss | Between which integers is √33? Approx. | 5 and 6 (≈ 5.74) — | talk | discussion prompt |
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| Hot Advanced discuss | Classify √16, √12, 0.6 (repeating), −3. ★ | rational, irrational, rational, rational (integer) — | talk | discussion prompt |
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| Spicy Challenge discuss | A square has an area of 120 square units. Another square has a side length of 10.5 units. Which square has the greater perimeter? Justify your answer without using a calculator for square roots. | The square with area 120 has a side length of \( \sqrt{120} \) units. Since \( 10^2 = 100 \) and \( 11^2 = 121 \), we know that \( \sqrt{120} \) is between 10 and 11. Specifically, \( \sqrt{120} \) is very close to 11. The perimeter of this square is \( 4 \times \sqrt{120} \). The second square has a side length of 10.5 units, so its perimeter is \( 4 \times 10.5 = 42 \) units. Since \( \sqrt{120} \) is less than 11, \( 4 \times \sqrt{120} \) is less than \( 4 \times 11 = 44 \). To compare \( 4 \times \sqrt{120} \) with 42, we can compare \( \sqrt{120} \) with 10.5. We know that \( 10.5^2 = (10 + 0.5)^2 = 100 + 2 \times 10 \times 0.5 + 0.25 = 100 + 10 + 0.25 = 110.25 \). Since \( 120 > 110.25 \), we have \( \sqrt{120} > \sqrt{110.25} \), which means \( \sqrt{120} > 10.5 \). Therefore, \( 4 \times \sqrt{120} > 4 \times 10.5 \). The square with an area of 120 square units has the greater perimeter. — | talk | discussion prompt |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner arith | 12 − 3 × 2 ★ | 12 − 6 = 6 6 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate arith | 18 ÷ 3 + 2² | 6 + 4 = 10 10 | auto | #Just the number — no units, no x=. |
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| Hot Advanced arith | 8 + 2(6 − 4)³ | 8 + 2(8) = 8 + 16 = 24 24 | auto | #Just the number — no units, no x=. |
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| Spicy Challenge discuss | Consider the expression $24 \div 4 \times 2 + 3 - 1$. a) What is the value of the expression if you follow the standard order of operations? b) Place **exactly one pair** of parentheses in the expression to make its value equal to 2. Write out the new expression. ★ | a) 14, b) 24 \div (4 \times 2 + 3 - 1) — | talk | discussion prompt |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner simplify | Combine 6m + 2m. ★ | 8m 8m | auto | #mType the expression with no spaces; use ^ for exponents and / for fractions. |
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| Medium Intermediate simplify | Simplify 7 + 3(x − 2). | 3x + 1 3x+1 | auto | #x+#Type the expression with no spaces; use ^ for exponents and / for fractions. |
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| Hot Advanced discuss | cost = 5w + 2p (walnuts w, peanuts p). Cost for 2 lb w + 3 lb p? ★ | 5(2) + 2(3) = $16 — | talk | discussion prompt |
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| Spicy Challenge scale | A rectangular garden has a length that is 5 feet more than twice its width. If the width of the garden is 'w' feet, write an expression for the perimeter of the garden in its simplest form. Then, if the width is 8 feet, what is the perimeter? ★ | 6w + 10, 58 6w+10,58 | auto | #w+#,#Type the expression with no spaces; use ^ for exponents and / for fractions. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner substitute | Evaluate 6 − b for b = 2. | 4 4, b=4 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate arith | Simplify 10 − 2(3 − 1) + 4. | 10 − 4 + 4 = 10 10 | auto | #Just the number — no units, no x=. |
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| Hot Advanced simplify | A small business sells custom t-shirts. They charge a flat fee of $25 for design setup and then $12 per t-shirt. If a customer orders 15 t-shirts, how much will their total bill be? | 205 205, t=205 | auto | #Just the number — no units, no x=. |
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| Spicy Challenge discuss | A store sells a jacket for $80. It is marked 25% off, and at the register another 10% is taken off the reduced price. What is the final price, and what single percent discount gives the same result? ★ | Final price $54; a single 32.5% discount — | talk | discussion prompt |
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| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 1.01 | More hooks | “Arithmetic is where numbers answer; algebra is where numbers ask.” — give them a teacher's sentence and translate it. | — not shown — | It's a quote without context or explanation for a 7th grader. |
| 1.01 | Real-world spark | The vanishing-money question: A festival charges $x admission. You and 3 friends each buy a $8 shirt; parking is $15. | A festival charges $x admission. You and 3 friends each buy a $8 shirt; parking is $15. How much total? | It's an incomplete question, missing what to do with the information. |
| 1.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 1.02 | More hooks | “A football team LOSES 6 yards then gains 15 — did they advance or get anchored?” | A football team LOSES 6 yards then gains 15, meaning they advanced 9 yards. | It's a question without an answer or explanation. |
| 1.02 | Real-world spark | The weather record: It swings +18° from −7°. What’s the new high? The record −4° then drops 11°. Which day’s swing is bigger, and by how muc | The weather record swings +18° from −7° to 11°. The record −4° then drops 11° to −15°. | It asks questions without providing the answers or how to find them. |
| 1.02 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 1.03 | More hooks | “A stock drops 5%, then rises 5% — are you back to start?” (It’s not; you’re down.) | If a stock drops 5%, then rises 5%, you are not back to start; you are down. | It's a question without enough context to understand the answer. |
| 1.03 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 1.04 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 1.05 | More hooks | Can an 11×11 square tile neatly with 7×7s and 4×4s? (Think remainders — and stay surprised.) | Can an 11x11 square be tiled with 7x7s and 4x4s? (Think remainders.) | It's a question without enough context to understand the answer. |
| 1.05 | Real-world spark | The pizza vs area shock: an 8-inch pizza and a 6-inch pizza — how much more pizza is the 8-inch? | An 8-inch pizza has more area than a 6-inch pizza, but how much more? | It's a question without enough context to understand the answer. |
| 1.05 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 1.06 | More hooks | “Phone vs desktop calculators disagree on −2² — who’s right?” | Some calculators disagree on −2² because of order of operations. | It's a question, not a fact, and lacks context for the disagreement. |
| 1.06 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 1.07 | More hooks | “Same value, different costume” — a masked ball of expressions. | Expressions can have the “same value, different costume,” like 2x + 3 and x + x + 3. | This is a metaphor without a concrete example or explanation. |
| 1.07 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 1.08 | More hooks | Run the 1¢-doubling replay from 1.04 to crown the unit. | — not shown — | Context for '1¢-doubling replay' and 'crown the unit' is missing. |
| 1.08 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | n games cost $16 each, plus a one-time $3 fee. Write the total cost. key: 16n + 3 | A streaming service charges $12 per month plus a one-time $30 setup fee. Write an expression for the total cost of m months, then find the total cost for 8 months. key: 12m + 30; the total is $126 | 5 translate the monthly charge into 12m → add the one-time $30 fee → substitute m = 8 → multiply 12 x 8 → add th | ✅ key checks out |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | Write a context where −15 + 8 = −7 makes sense. key: owed $15, paid $8 - still owes $7 | A submarine is at an elevation of -250 feet relative to sea level. It then ascends 120 feet, descends 75 feet, and finally ascends another 40 feet. What is the submarine's final elevation? key: -165 feet model: -250 + 120 - 75 + 40 = x | 4 translate 'ascends' to addition → translate 'descends' to subtraction → perform first addition/subtraction → p | 🗣 talk-through leap 3 |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | A submarine descends 25 ft/min for 6 min. Depth change? key: −25 × 6 = −150 ft | A diver starts at sea level and descends at a rate of 12 feet per minute. What is the diver's depth after 7 minutes? (Express your answer as an integer representing feet relative to sea level.) key: -84 model: -12 * 7 = d | 3 identify rate as negative → multiply rate by time → state final depth | ✅ key checks out |
| Spicy | no acceptable candidate passed the gate yet | |||
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | For which whole-number exponents n is (−2)n positive? key: n even | Evaluate (−2)⁴ − (−3)² and explain why one result is positive and the other is negative. key: 16 − 9 = 7; an even exponent makes the negative base positive, and squaring (−3) also gives 9, so the difference is 7 | 5 apply the exponent to the grouped base (−2) → apply the exponent to the grouped base (−3) → subtract the two v | ✅ key checks out |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | A square garden has an area of 75 square feet. A circular fountain with a radius of 2 feet is placed in the center of the garden. Is there enough space around the fountain to plant a border of flowers that is at least 1 foot wide on all sides? Justify your answer without using a calculator for square roots. key: No, there is not enough space. The side length of the garden is \( \sqrt{75} \) feet. Since \( 8^2 = 64 \) and \( 9^2 = 81 \), \( \sqrt{75} \) is between 8 and 9. Specifically, \( \sqrt{75} \approx 8.66 \). The fountain has a diameter of \( 2 \times 2 = 4 \) feet. For a 1-foot border on all sides, the garden would need to be at least \( 4 + 1 + 1 = 6 \) feet wide. Since \( \sqrt{75} \approx 8.66 \) is greater than 6, there is enough space for the border. However, the prompt asks if there is enough space for a border that is *at least* 1 foot wide on *all sides*. This implies the distance from the edge of the fountain to the edge of the garden must be at least 1 foot. The total width needed for the fountain and two 1-foot borders is \( 2 \times 1 + 2 \times 2 + 2 \times 1 = 2 + 4 + 2 = 8 \) feet. Since \( \sqrt{75} \approx 8.66 \) is greater than 8, there is enough space. Wait, re-reading the prompt. 'Is there enough space around the fountain to plant a border of flowers that is at least 1 foot wide on all sides?' This means the distance from the edge of the fountain to the edge of the garden must be at least 1 foot. The side length of the garden is \( \sqrt{75} \). The diameter of the fountain is 4 feet. The remaining space on each side of the fountain is \( (\sqrt{75} - 4) / 2 \). We need to check if \( (\sqrt{75} - 4) / 2 \ge 1 \). This simplifies to \( \sqrt{75} - 4 \ge 2 \), or \( \sqrt{75} \ge 6 \). Since \( 6^2 = 36 \) and \( 75 > 36 \), \( \sqrt{75} > 6 \). So yes, there is enough space. My previous reasoning was flawed. The answer is YES. | A square has an area of 120 square units. Another square has a side length of 10.5 units. Which square has the greater perimeter? Justify your answer without using a calculator for square roots. key: The square with area 120 has a side length of \( \sqrt{120} \) units. Since \( 10^2 = 100 \) and \( 11^2 = 121 \), we know that \( \sqrt{120} \) is between 10 and 11. Specifically, \( \sqrt{120} \) is very close to 11. The perimeter of this square is \( 4 \times \sqrt{120} \). The second square has a side length of 10.5 units, so its perimeter is \( 4 \times 10.5 = 42 \) units. Since \( \sqrt{120} \) is less than 11, \( 4 \times \sqrt{120} \) is less than \( 4 \times 11 = 44 \). To compare \( 4 \times \sqrt{120} \) with 42, we can compare \( \sqrt{120} \) with 10.5. We know that \( 10.5^2 = (10 + 0.5)^2 = 100 + 2 \times 10 \times 0.5 + 0.25 = 100 + 10 + 0.25 = 110.25 \). Since \( 120 > 110.25 \), we have \( \sqrt{120} > \sqrt{110.25} \), which means \( \sqrt{120} > 10.5 \). Therefore, \( 4 \times \sqrt{120} > 4 \times 10.5 \). The square with an area of 120 square units has the greater perimeter. | 6 find side length of first square → approximate square root → find perimeter of first square → find perimeter o | 🗣 talk-through leap 3 |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | Consider the expression $12 \div 3 \times 2 + 5 - 1$. a) What is the value of the expression if you follow the standard order of operations? b) Place **one pair** of parentheses in the expression to make its value equal to 3. Write out the new expression. key: a) 12, b) 12 \div (3 \times 2 + 5 - 1) | Consider the expression $24 \div 4 \times 2 + 3 - 1$.
a) What is the value of the expression if you follow the standard order of operations?
b) Place **exactly one pair** of parentheses in the expression to make its value equal to 2. Write out the new expression. key: a) 14, b) 24 \div (4 \times 2 + 3 - 1) | 6 evaluate standard order of operations → identify target value → experiment with parenthesis placement (case 1) | 🗣 talk-through leap 3 |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | Is 2(x+3) equivalent to 2x+6? Name the property. key: yes - distributive: 2×x + 2×3 | A rectangular garden has a length that is 5 feet more than twice its width. If the width of the garden is 'w' feet, write an expression for the perimeter of the garden in its simplest form. Then, if the width is 8 feet, what is the perimeter? key: 6w + 10, 58 | 5 translate length to expression (2w + 5) → write perimeter formula (2L + 2W) → substitute expressions for L and | 🗣 talk-through leap 3 |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | Compare: Job A $15/hr; Job B $12/hr + $40 bonus for 60 hr. How much more does Job A pay than Job B? key: 140 | A small business sells custom t-shirts. They charge a flat fee of $25 for design setup and then $12 per t-shirt. If a customer orders 15 t-shirts, how much will their total bill be? key: 205 model: 25 + 12 * 15 = x | 3 Identify fixed cost → Calculate variable cost (per t-shirt) → Add fixed and variable costs | ✅ key checks out |
| Spicy | m games at $14 each, each 1/4 off. New total? key: 14m × 3/4 = 10.5m | A store sells a jacket for $80. It is marked 25% off, and at the register another 10% is taken off the reduced price. What is the final price, and what single percent discount gives the same result? key: Final price $54; a single 32.5% discount | 5 take 25% off $80 → take 10% off the reduced price → state the final price → combine the two rates (0.75 x 0.90 | ✅ key checks out |
| Where | Prompt | Guide key | What looks off | Suggested |
|---|---|---|---|---|
| 1.05 Challenge | Order √2 and 1.5 without a calculator; justify. | √2 ≈ 1.41 < 1.5 | Key does not justify the ordering. | √2 ≈ 1.41 < 1.5 because 1.41² = 1.9881 which is less than 1.5² = 2.25. So √2 < 1.5. |
| 1.06 Advanced | 8 + 2(6 − 4)³ | 8 + 2(8) = 24 | Arithmetic error in key: 2(6-4)^3 is 2(2)^3 = 2(8) = 16, not 2(8)=24. | 8 + 2(2)³ = 8 + 2(8) = 8 + 16 = 24 |
| 1.06 Advanced | 2 + 3(4 − 2)² | (2); ²=4; 2+12 = 14 | Key is not a final answer. | 14 |
| 1.08 Advanced | Compare: Job A $15/hr; Job B $12/hr + $40 bonus for 60 hr. Which pays more? | A=900, B=760 - A | Key does not state which job pays more. | Job A pays more. |
Jump to: 2.01 · 2.02 · 2.03 · 2.04 · 2.05 · 2.06 · 2.07 · 2.08 · 2.09 · 2.10 · 2.11 · 2.12 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner solve | x + 8 = 15. ★ | 7 7, x=7 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate literal | Solve for x: x / 7 = 12 ★ | 84 84, x=84 | auto | #Just the number — no units, no x=. |
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| Hot Advanced literal | Solve for x: 2(x + 3) = 18 | 6 6, x=6 | auto | #Just the number — no units, no x=. |
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| Spicy Challenge scale | A rectangular garden has a perimeter of 50 feet. If the length of the garden is 5 feet more than its width, what is the width of the garden in feet? ★ | 10 10 | auto | #Just the number — no units, no x=. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner solve | x + 3 = 10. ★ | 7 7, x=7 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate literal | Solve for y: 4y - 7 = 21 ★ | 7 7, y=7 | auto | #Just the number — no units, no x=. |
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| Hot Advanced literal | Solve for x: 3(x + 2) - 5 = 16 | 5 5, x=5 | auto | #Just the number — no units, no x=. |
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| Spicy Challenge scale | The perimeter of a rectangular garden is 54 feet. The length of the garden is 3 feet more than twice its width. What is the width of the garden in feet? | 8 8 | auto | #Just the number — no units, no x=. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | 2x + 1 = x + 4. ★ | x = 3 3, x=3 | auto | x=#Just the value, or write it as x=# — both count. |
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| Medium Intermediate twosided | 5x − 2 = 3x + 6. ★ | 2x = 8 → x = 4 4, x=4 | auto | x=#Just the value, or write it as x=# — both count. |
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| Hot Advanced literal | Solve for x: 7(x - 3) = 2x + 9 | x = 6 6, x=6 | auto | x=#Just the value, or write it as x=# — both count. |
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| Spicy Challenge arith | A rectangle has a perimeter of 50 units. Its length is 3 units more than twice its width. If you were to increase both the length and the width by 2 units, what would be the new perimeter? | 58 units 58, 58units | auto | #unitsType the expression with no spaces; use ^ for exponents and / for fractions. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner literal | Solve x + y = 5 for y. ★ | y = 5 − x 5-x, y=5-x | auto | y=#−xWrite y= then the expression — no spaces, ^ for exponents, / for fractions. |
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| Medium Intermediate literal | Solve A = 1/2 bh for h. ★ | h = 2A/b 2a/b, h=2a/b | auto | h=#a/bWrite h= then the expression — no spaces, ^ for exponents, / for fractions. |
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| Hot Advanced literal | Solve for b: P = 2a + 2b | b = (P - 2a)/2 (p-2a)/2, b=(p-2a)/2 | auto | b=(p−#a)/#Write b= then the expression — no spaces, ^ for exponents, / for fractions. |
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| Spicy Challenge fraction | The formula for the area of a trapezoid is A = (1/2)h(b₁ + b₂), where A is the area, h is the height, and b₁ and b₂ are the lengths of the two parallel bases. If a trapezoid has an area of 45 square units, a height of 6 units, and one base (b₁) is 10 units long, what is the length of the other base (b₂)? | 5 5, h=5 | auto | #Just the number — no units, no x=. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner solve | |x| = 6. ★ | x = 6, −6 (-6,6), (6,-6), -6,6, 6,-6, x=-6,6, x=6,-6, x=±6, ±6 | auto | x=#,−#Both answers, comma between them — either order works (x=4,-4 and x=-4,4 both count), and x=±# is fine too. |
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| Medium Intermediate solve | |x + 3| = 5. ★ | x = 2, −8 (-8,2), (2,-8), -8,2, 2,-8, x=-8,2, x=2,-8 | auto | x=#,−#Both answers, comma between them — either order works (x=4,-4 and x=-4,4 both count), and x=±# is fine too. |
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| Hot Advanced literal | Solve for x: 3|x - 5| - 2 = 10 | x = 9, x = 1 1, x=1 | auto | #Just the number — no units, no x=. |
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| Spicy Challenge discuss | A company manufactures widgets with a target weight of 120 grams. The quality control department states that a widget is acceptable if its actual weight, `w`, is within 5% of the target weight. Write and solve an absolute value equation to find the minimum and maximum acceptable weights for a widget. | Minimum: 114 grams, Maximum: 126 grams — | talk | discussion prompt |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner solve | |x + 1| = 3. ★ | x = 2, −4 (-4,2), (2,-4), -4,2, 2,-4, x=-4,2, x=2,-4 | auto | x=#,−#Both answers, comma between them — either order works (x=4,-4 and x=-4,4 both count), and x=±# is fine too. |
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| Medium Intermediate literal | Solve C = 2πr for r. ★ | r = C/(2π) c/(2pi), r=c/(2pi), r=c/2pi | auto | r=c/(#pi)Write r= then the expression — no spaces, ^ for exponents, / for fractions. |
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| Hot Advanced literal | Solve for y: 2|3y - 6| + 7 = 19 | y = 4, 0 (0,4), (4,0), 0,4, 4,0, y=0,4, y=4,0 | auto | y=#,#Both answers, comma between them — either order works (x=4,-4 and x=-4,4 both count), and x=±# is fine too. |
|
| Spicy Challenge fraction | The formula for the area of a trapezoid is A = (1/2)h(b₁ + b₂). If a trapezoid has an area of 45 square units and a height of 6 units, and one base is 3 units longer than the other, what are the lengths of the two bases? | b₁ = 6, b₂ = 9 (or vice versa) 9, h=9 | auto | #Just the number — no units, no x=. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner solve | x + 6 = 11. ★ | 5 5, x=5 | auto | #Just the number — no units, no x=. |
|
| Medium Intermediate literal | Solve for x: 3x + 8 = 5x - 2 ★ | x = 5 5, x=5 | auto | x=#Just the value, or write it as x=# — both count. |
|
| Hot Advanced literal | Solve for x: 4(x + 3) - 7 = 2x + 11 | 3 3, x=3 | auto | #Just the number — no units, no x=. |
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| Spicy Challenge scale | The perimeter of a rectangular garden is 72 feet. The length of the garden is 6 feet more than twice its width. What is the width of the garden in feet? | 10 10 | auto | #Just the number — no units, no x=. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner fraction | x/3 = 4/6. ★ | x = 2 2, x=2 | auto | x=#Just the value, or write it as x=# — both count. |
|
| Medium Intermediate literal | Solve for x: (x + 2) / 9 = 15 / 27 ★ | 3 3, x=3 | auto | #Just the number — no units, no x=. |
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| Hot Advanced arith | A car travels 210 miles on 7 gallons of gasoline. At this rate, how many miles can the car travel on 12 gallons of gasoline? | 360 360 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge discuss | A car travels 180 miles in 3 hours. At this rate, how long (in minutes) would it take to travel 250 miles? If the car maintains this speed, how many miles would it travel in 50 minutes? | 250 minutes; 250/3 miles — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | What is 10% of 90? ★ | 9 — | talk | discussion prompt |
|
| Medium Intermediate percent | After a 20% discount, a shirt costs $24. What was the original price of the shirt? ★ | 30 30 | auto | #Just the number — no units, no x=. |
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| Hot Advanced percent | A store is having a sale where all items are 15% off their original price. If a pair of shoes is on sale for $85, what was its original price? | 100 100 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge percent | A store is having a two-day sale. On the first day, all items are discounted by 20% off their original price. On the second day, the sale price from the first day is further discounted by an additional 10%. If a pair of shoes costs $72 after both discounts, what was its original price? | 100 100 | auto | #Just the number — no units, no x=. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | From 20 to 25: % increase? ★ | 25% — | talk | discussion prompt |
|
| Medium Intermediate percent | A store is having a sale where all items are 25% off. If a pair of shoes originally cost $80, what is the sale price? ★ | 60 60 | auto | #Just the number — no units, no x=. |
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| Hot Advanced percent | A company's sales decreased by 15% last quarter. If their sales for this quarter were $102,000, what were their sales last quarter? | 120000 120000, s=120000 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge percent | A company's stock price increased by 25% in January. In February, it decreased by 20% from the January-end price. In March, it increased by 10% from the February-end price. If the stock price at the end of March was $110, what was the original stock price at the beginning of January? | 100 100, s=100 | auto | #Just the number — no units, no x=. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner scale | 1 cm = 10 km; 4 cm = ? ★ | 40 km 40, 40km | auto | #kmType the expression with no spaces; use ^ for exponents and / for fractions. |
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| Medium Intermediate arith | A recipe calls for 3 cups of flour for every 2 cups of sugar. If you want to use 5 cups of sugar, how much flour do you need? ★ | 7.5 7.5 | auto | #Just the number — no units, no x=. |
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| Hot Advanced twosided | A map uses the scale 1 cm = 8 km. Town A and Town B are 5 cm apart on the map, and Town C is 3 cm from Town A. How far apart are Town B and Town C? | 16 16 | auto | #Just the number — no units, no x=. |
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| Spicy Challenge discuss | A store takes 20% off, then takes another 20% off the reduced price. A shopper says 'that is 40% off'. Explain why the shopper is wrong, and give the actual total percent off. | The second 20% applies to the already-reduced price, so the discounts multiply: 0.8 x 0.8 = 0.64, which is a 36% total discount, not 40%. — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner solve | x − 2 = 7. ★ | 9 9, x=9 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate literal | Solve for x: 3x - 7 = 14 ★ | x = 7 7, x=7 | auto | x=#Just the value, or write it as x=# — both count. |
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| Hot Advanced literal | Solve for x: 2(x + 3) - 5 = 11 | x = 5 5, x=5 | auto | x=#Just the value, or write it as x=# — both count. |
|
| Spicy Challenge discuss | A recipe calls for 2 cups of flour for every 3 cups of sugar. If you want to make a larger batch using 8 cups of flour, how many cups of sugar will you need? If you only have 10 cups of sugar, what is the maximum amount of flour you can use? | 12 cups of sugar; 6.67 cups of flour — | talk | discussion prompt |
|
| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 2.01 | Fascinating history | Solving equations is one of the oldest arts: the Babylonians solved for unknown amounts on clay tablets 4,000 years ago, and the Egyptian Rh | Solving equations is one of the oldest arts: Babylonians solved for unknowns 4,000 years ago. | The sentence is incomplete. |
| 2.01 | More hooks | The 'invisible number' in a receipt total | — not shown — | It's an incomplete thought. |
| 2.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.02 | More hooks | Filling a tank with a constant plus a top-up | — not shown — | It's too vague; needs numbers or a specific example. |
| 2.02 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.03 | More hooks | Two store cards crossing at some spend | — not shown — | It's too vague; needs numbers or a specific example. |
| 2.03 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.04 | More hooks | Solving the area formula for a side | — not shown — | It's too vague; needs a specific formula or example. |
| 2.04 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.05 | More hooks | A thermostat's comfortable range | — not shown — | It's an incomplete thought. |
| 2.05 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.06 | Fascinating history | Review blends two great tools - literal (formula-solving) and absolute value (two-case splitting). Masters of both can rework a physics form | — not shown — | It's too abstract and uses jargon. |
| 2.06 | More hooks | Formula rewriting for a hobby build | — not shown — | It's an incomplete thought. |
| 2.06 | Real-world spark | Rearranging a savings formula and checking 'within range' limits - both skills meet in budgeting. | — not shown — | It's too abstract and vague. |
| 2.06 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.07 | More hooks | The 'pre-flight check' before harder math | Pre-algebra is like a 'pre-flight check' before harder math. | It's a metaphor, not a fact. |
| 2.07 | Real-world spark | Prepping a bank balance check: a quick calculation drill ensures every later real-world value is trusted. | A quick calculation drill helps ensure real-world values are trusted, like a bank balance. | It's an instruction, not a fact. |
| 2.07 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.08 | More hooks | Mixing two ingredients in a fixed ratio | — not shown — | It's an incomplete example. |
| 2.08 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.09 | More hooks | How a 90% test score is a proportion | A 90% test score is a proportion, meaning 90 out of 100. | It's a question, not a fact. |
| 2.09 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.10 | More hooks | Rent/salary increases each year | — not shown — | It's an incomplete thought. |
| 2.10 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.11 | Fascinating history | Applications are where proportions and percents become trade: surveyors scaled maps, bakers scaled recipes, banks scaled interest. | — not shown — | It uses abstract terms like 'trade' and 'scaled'. |
| 2.11 | More hooks | Reading a road at a map scale | — not shown — | It's an incomplete thought. |
| 2.11 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 2.12 | Fascinating history | The cumulative review closes the loop: equations, proportions, and percents are the three pillars of applied math, and this review cements t | — not shown — | It's an incomplete thought and uses abstract terms. |
| 2.12 | More hooks | Reading any formula, ratio, or percent | — not shown — | It's an incomplete thought. |
| 2.12 | Real-world spark | A real 'math in life' final: budget a shopping trip, scale a plan, solve for a needed rate - all Unit 2 in one afternoon. | — not shown — | It describes an activity, not a fact. |
| 2.12 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | no acceptable candidate passed the gate yet | |||
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | no acceptable candidate passed the gate yet | |||
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | no acceptable candidate passed the gate yet | |||
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | no acceptable candidate passed the gate yet | |||
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | no acceptable candidate passed the gate yet | |||
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | no acceptable candidate passed the gate yet | |||
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | no acceptable candidate passed the gate yet | |||
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | no acceptable candidate passed the gate yet | |||
| Where | Prompt | Guide key | What looks off | Suggested |
|---|---|---|---|---|
| 2.02 Challenge | 5x + 3 = 2x + 12. | 3x = 9 → x = 3 | Key contains intermediate steps. | x = 3 |
| 2.02 Intermediate | 3x + 2x = 15. | 5x = 15 → x = 3 | Key contains intermediate steps. | x = 3 |
| 2.02 Advanced | 4(x + 1) = 20. | 4x + 4 = 20 → x = 4 | Key contains intermediate steps. | x = 4 |
| 2.03 Intermediate | 5x − 2 = 3x + 6. | 2x = 8 → x = 4 | Key contains intermediate steps. | x = 4 |
| 2.03 Intermediate | 3x + 1 = x + 7. | 2x = 6 → x = 3 | Key contains intermediate step. | x = 3 |
| 2.04 Beginner | Solve x − y = 3 for y. | y = x − 3 | Arithmetic error. | y = x - 3 |
| 2.06 Advanced | |x − 5| = 0. | x = 5 (single) | Key includes extra information. | x = 5 |
| 2.07 Beginner | x + 6 = 11. | 5 | Key is missing 'x ='. | x = 5 |
| 2.07 Challenge | Solve for x: 2x + x = 6. | 3x = 6 → x = 2 | Key includes intermediate steps. | x = 2 |
| 2.08 Intermediate | 12/15 = x/5. | x = 4 | Arithmetic error: 12/15 simplifies to 4/5, so x=4. | x = 4 |
| 2.11 Advanced | Recipe 2 eggs : 3 cups flour; need 5 cups →19 eggs? | proportion → 3⅓ eggs | Key is arithmetically incorrect. | 3.33 eggs |
Jump to: 3.01 · 3.02 · 3.03 · 3.04 · 3.05 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner inequality | Solve x − 2 < 5. ★ | x < 7 x<7 | auto | x<#Variable first: x<# (spaces optional, <= or ≤ both work). |
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| Medium Intermediate inequality | Solve 3x − 5 ≥ 1. | x ≥ 2 x>=2 | auto | x≥#Variable first: x<# (spaces optional, <= or ≤ both work). |
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| Hot Advanced inequality | Solve −2x > 6 (flip the sign). ★ | x < −3 x<-3 | auto | x<−#Variable first: x<# (spaces optional, <= or ≤ both work). |
|
| Spicy Challenge graph | A small business sells handmade bracelets. They want to make a profit of at least $150 this week. Each bracelet costs $3.50 to make and sells for $12.00. They also have fixed weekly expenses of $75 (for booth rental, etc.). Write and solve an inequality to find the minimum number of bracelets, `b`, they need to sell this week to reach their profit goal. Graph the solution on a number line. | b ≥ 26 b>=26 | auto | b≥#Variable first: x<# (spaces optional, <= or ≤ both work). |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner inequality | 4x − 1 < 2x + 7. | x < 4 x<4 | auto | x<#Variable first: x<# (spaces optional, <= or ≤ both work). |
|
| Medium Intermediate inequality | 6x − 4 ≥ 2x + 8. ★ | x ≥ 3 x>=3 | auto | x≥#Variable first: x<# (spaces optional, <= or ≤ both work). |
|
| Hot Advanced discuss | 3x + 5 < 3x − 1. | 5 < −1 is false → no solution — | talk | discussion prompt |
|
| Spicy Challenge discuss | 2(x+5) + 6x ≥ 8(x+5) + 6. ★ | no solution (10 ≥ 46) — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner inequality | Graph x > 1 AND x < 4. ★ | 1 < x < 4 (between, both open) 1<x<4, x<4andx>1, x>1andx<4 | auto | #<x<#Put x in the middle: #<x<# — or type both parts joined with and. |
|
| Medium Intermediate inequality | Solve x − 1 ≤ 2 AND 2x ≥ −2. | x ≤ 3 AND x ≥ −1 → −1 ≤ x ≤ 3 -1<=x<=3, x<=3andx>=-1, x>=-1andx<=3 | auto | −#≤x≤#Put x in the middle: #<x<# — or type both parts joined with and. |
|
| Hot Advanced inequality | Solve x + 2 > 6 OR x − 1 < −3. ★ | x > 4 OR x < −2 x>4orx<-2 | auto | x>#orx<−#Write both parts joined with or — either order works. |
|
| Spicy Challenge twosided | Solve the compound inequality 3x − 4 ≤ 8 AND 2x + 1 > 7, graph the solution on a number line, then decide whether x = 5 is in the solution set and explain. | 3 < x ≤ 4 (open at 3, closed at 4); x = 5 is NOT in the set because 5 > 4 3<x<=4, x<=4andx>3, x>3andx<=4 | auto | #<x≤#Put x in the middle: #<x<# — or type both parts joined with and. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner inequality | |x| < 4. | −4 < x < 4 -4<x<4, x<4andx>-4, x>-4andx<4 | auto | −#<x<#Put x in the middle: #<x<# — or type both parts joined with and. |
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| Medium Intermediate inequality | |x − 1| > 2. ★ | x < −1 OR x > 3 x<-1orx>3 | auto | x<−#orx>#Write both parts joined with or — either order works. |
|
| Hot Advanced inequality | |2x| − 3 ≤ 1. | |2x| ≤ 4 → −2 ≤ x ≤ 2 -2<=x<=2, x<=2andx>=-2, x>=-2andx<=2 | auto | −#≤x≤#Put x in the middle: #<x<# — or type both parts joined with and. |
|
| Spicy Challenge simplify | A company manufactures bolts that are supposed to be 3.5 inches long. The manufacturing process has a tolerance of 0.02 inches, meaning the actual length of a bolt can be 0.02 inches more or less than the target length. Write and solve an absolute value inequality to represent the range of acceptable lengths for a bolt. Then, state the shortest and longest acceptable lengths. | 3.48 <= L <= 3.52 3.48<=l<=3.52, l<=3.52andl>=3.48, l>=3.48andl<=3.52 | auto | #≤l≤#Put x in the middle: #<x<# — or type both parts joined with and. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner inequality | Solve x + 5 < 9. ★ | x < 4 x<4 | auto | x<#Variable first: x<# (spaces optional, <= or ≤ both work). |
|
| Medium Intermediate inequality | Solve 3x − 2 ≥ 7. | x ≥ 3 x>=3 | auto | x≥#Variable first: x<# (spaces optional, <= or ≤ both work). |
|
| Hot Advanced graph | Write the inequality for the number line shaded left of 3 (open). ★ | x < 3 x<3 | auto | x < #Type the grouped expression(s) — the parentheses are the answer. Use / for ÷ and * for ×; separate two answers with ; or the word and. |
|
| Spicy Challenge twosided | Solve and graph: 3x − 5 ≥ 2x + 1. Then state whether x = 7 is in the solution set and explain how you know. | x ≥ 6; x = 7 is in the solution set because 7 ≥ 6 x>=6 | auto | x≥#Variable first: x<# (spaces optional, <= or ≤ both work). |
|
| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 3.01 | More hooks | The 'at least this tall' ride sign | The 'at least this tall' ride sign is an example of an inequality. | It's an incomplete thought. |
| 3.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 3.02 | More hooks | Two phone plans crossing at a monthly-minute boundary | — not shown — | It's an incomplete thought. |
| 3.02 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 3.03 | Fascinating history | Compound inequalities unite two ideas: intersections (AND) and unions (OR) come straight from set theory, formalised by Georg Cantor in the | Compound inequalities use 'AND' (intersections) and 'OR' (unions), ideas from set theory formalized by Georg Cantor in the 1870s. | It introduces complex terms without explanation. |
| 3.03 | More hooks | Ticket price 'at least $5 and at most $20' | A ticket price 'at least $5 and at most $20' is an example of a compound inequality. | It's an incomplete example without context. |
| 3.03 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 3.04 | More hooks | 'Within 5 miles' / margin of error | Absolute value inequalities can describe a 'margin of error' or 'within 5 miles'. | It's an incomplete thought. |
| 3.04 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 3.05 | Fascinating history | Review is the ritual that turns scattered skills into one reliable tool - the way mariners re-checked their maps before trusting the sea. | — not shown — | It's a metaphor, not an algebra fact. |
| 3.05 | More hooks | Which skills trip you on a calculator-less test | — not shown — | It's a question, not a fact. |
| 3.05 | Real-world spark | A real contract: 'spending must stay between $X and $Y, never exceed $Z' - one compound AND an upper bound, all review. | A contract like 'spending must stay between $100 and $200' is a compound inequality. | It uses variables without defining them. |
| 3.05 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | A company produces widgets. The ideal weight for a widget is 15 grams. The quality control department allows a widget to pass if its weight is within 2 grams of the ideal weight. However, due to a new regulation, any widget weighing exactly 15 grams must be re-inspected, and any widget weighing less than 13 grams or more than 17 grams is immediately rejected. Write a compound inequality that represents the acceptable range of weights (w) for a widget that does NOT require re-inspection and is NOT immediately rejected. key: x >= 3 AND x < 3 -> no solution | Solve the compound inequality 3x − 4 ≤ 8 AND 2x + 1 > 7, graph the solution on a number line, then decide whether x = 5 is in the solution set and explain. key: 3 < x ≤ 4 (open at 3, closed at 4); x = 5 is NOT in the set because 5 > 4 model: 3x - 4 = 8 | 5 solve the first inequality → solve the second inequality → intersect the two solution sets → graph the compoun | ✅ key checks out |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | How can you tell from the inequality whether the answer is an AND or an OR? key: |x| < b → AND; |x| > b → OR | A company manufactures bolts that are supposed to be 3.5 inches long. The manufacturing process has a tolerance of 0.02 inches, meaning the actual length of a bolt can be 0.02 inches more or less than the target length. Write and solve an absolute value inequality to represent the range of acceptable lengths for a bolt. Then, state the shortest and longest acceptable lengths. key: 3.48 <= L <= 3.52 model: |L - 3.5| <= 0.02 | 5 translate 'tolerance' into an absolute value inequality → split absolute value into two inequalities → solve f | 🗣 talk-through leap 3 |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | Graph x < −2 OR x ≥ 4 and describe the union. key: two rays pointing outward | Solve and graph: 3x − 5 ≥ 2x + 1. Then state whether x = 7 is in the solution set and explain how you know. key: x ≥ 6; x = 7 is in the solution set because 7 ≥ 6 | 5 collect the variable terms on one side → collect the constants on the other → state the inequality → test x = | ✅ key checks out |
| Where | Prompt | Guide key | What looks off | Suggested |
|---|---|---|---|---|
| 3.01 Advanced | Solve 2(x − 3) < 6. | x < 6 | Arithmetic error in key. | x < 4.5 |
| 3.02 Challenge | 2(x+5) + 6x ≥ 8(x+5) + 6. | no solution (10 ≥ 46) | Arithmetic error in key. | no solution (10 >= 46 is false) |
| 3.03 Beginner | Graph x > 1 AND x < 4. | 1 < x < 4 (between, both open) | Key describes graph, but doesn't graph it. | A number line with an open circle at 1, an open circle at 4, and a line segment connecting them. |
| 3.03 Beginner | Graph −1 ≤ x < 4. | closed at −1, open at 4, between | Key describes graph, but doesn't graph it. | A number line with a closed circle at -1, an open circle at 4, and a line segment connecting them. |
| 3.03 Advanced | Write the union for x < 0 OR x > 5. | two rays; no gap in between | Key describes union, but doesn't write it. | (-infinity, 0) U (5, infinity) |
| 3.05 Advanced | A mammal lives 10 to 15 years; write as |x − 12.5| ≤ 2.5. | 10 ≤ x ≤ 15 | Key does not answer the question. | |x - 12.5| <= 2.5 |
Jump to: 4.01 · 4.02 · 4.03 · 4.04 · 4.05 · 4.06 · 4.07 · 4.08 · 4.09 · 4.10 · 4.11 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | f(x) = x + 3; f(2)? ★ | 5 5 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate twosided | f(x) = 3x; f(4)? ★ | 12 12 | auto | #Just the number — no units, no x=. |
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| Hot Advanced twosided | A function is defined by the rule $f(x) = 3x + 7$. If the output of the function is 22, what was the input? | 5 5 | auto | #Just the number — no units, no x=. |
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| Spicy Challenge twosided | A function `f` is defined by the rule `f(x) = 3x - 7`. Another function `g` is defined by the rule `g(x) = 2(x + 1) + 5`. If `f(a) = g(a)`, what is the value of `a`? | 14 14 | auto | #Just the number — no units, no x=. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner writeeq | Slope through (1, 2) and (4, 8). ★ | m = 2 2 | auto | #Just the number — no units, no x=. |
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| Medium Intermediate twosided | Slope of y = 5x + 1. ★ | 5 5, y=5 | auto | #Just the number — no units, no x=. |
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| Hot Advanced discuss | Slope through (2, 5) and (2, 9). | undefined (vertical) — | talk | discussion prompt |
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| Spicy Challenge writeeq | A linear function passes through the points (3, 7) and (x, 19). If the slope of this function is 4, what is the value of x? | 6 6, x=6 | auto | #Just the number — no units, no x=. |
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| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | m and b of y = 3x + 2? ★ | m = 3, b = 2 2 | auto | #Just the number — no units, no x=. |
|
| Medium Intermediate writeeq | Write: slope 5, y-intercept (0, −1). ★ | y = 5x − 1 5x-1 | auto | #x−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced discuss | Graph the line y = 2x − 1 (intercept and slope). | start (0, −1), rise 2 run 1 — | talk | discussion prompt |
|
| Spicy Challenge writeeq | A line passes through the points (2, 5) and (x, 11). If the slope of the line is 3, what is the value of x? | 4 4, x=4 | auto | #Just the number — no units, no x=. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner writeeq | Through (0, 2) and (2, 6)? ★ | y = 2x + 2 2x+2 | auto | #x+#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate writeeq | Through (0, −1) and (3, 8)? ★ | y = 3x − 1 3x-1 | auto | #x−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced discuss | Which is steeper: y = −3x or y = 2x? | −3x (|−3| > 2) — | talk | discussion prompt |
|
| Spicy Challenge twosided | A line passes through the points (3, 7) and (9, y). If this line is parallel to the line represented by the equation 2x + 3y = 12, what is the value of y? | 3 3, y=3 | auto | #Just the number — no units, no x=. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | y = 7x; constant of variation? ★ | 7 7, y=7 | auto | #Just the number — no units, no x=. |
|
| Medium Intermediate simplify | The number of calories burned (C) varies directly with the time spent exercising (T). If you burn 180 calories in 20 minutes, what is the constant of variation (k)? ★ | 9 9 | auto | #Just the number — no units, no x=. |
|
| Hot Advanced arith | The number of pages a printer can print varies directly with the time it operates. If the printer can print 120 pages in 4 minutes, how many pages can it print in 7 minutes? | 210 210 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge discuss | The amount of water (W) in a swimming pool varies directly with the time (T) the hose has been running. If a hose fills 150 gallons in 3 minutes, how long will it take to fill a 275-gallon pool? Express your answer in minutes and seconds, rounded to the nearest second. | 5 minutes 30 seconds — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | y-intercept of x + y = 5? ★ | (0, 5) 0,5 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Medium Intermediate twosided | x-intercept of 2x + y = 6? ★ | (3, 0) 3,0 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Hot Advanced twosided | Put y = 2x − 6 in standard form. | −2x + y = −6 -2x+y=-6 | auto | −#x+y=−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge discuss | A line is given by 3x + 4y = 24. Find both intercepts, then write the equation of a line parallel to it that passes through (0, 1), in standard form. | Intercepts (8, 0) and (0, 6); parallel line: 3x + 4y = 4 — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | Point-slope slope 2 through (3, 4)? ★ | y − 4 = 2(x − 3) — | talk | discussion prompt |
|
| Medium Intermediate twosided | Convert y − 3 = 3(x − 1) to slope-intercept. ★ | y = 3x 3x, y=3x | auto | y=#xWrite y= then the expression — no spaces, ^ for exponents, / for fractions. |
|
| Hot Advanced writeeq | Through (2, 7) with slope 5: equation? | y = 5x − 3 5x-3 | auto | #x−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge discuss | A line passes through the points (2, 5) and (6, 17). Write the equation of this line in point-slope form using the point (2, 5). Then, convert this equation to slope-intercept form. Finally, write the equation of the same line in point-slope form using the point (6, 17). Are the slope-intercept forms of both equations identical? | y - 5 = 3(x - 2); y = 3x - 1; y - 17 = 3(x - 6); Yes — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | Convert x + y = 5 to slope-intercept. ★ | y = −x + 5 -x+5, y=-x+5 | auto | y=−x+#Write y= then the expression — no spaces, ^ for exponents, / for fractions. |
|
| Medium Intermediate twosided | Convert y = 2x − 4 to standard form. ★ | −2x + y = −4 -2x+y=-4 | auto | −#x+y=−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced twosided | Standard -> slope-int: 3x + y = 9. | y = −3x + 9 -3x+9, y=-3x+9 | auto | y=−#x+#Write y= then the expression — no spaces, ^ for exponents, / for fractions. |
|
| Spicy Challenge standardform | A line passes through the points (2, 5) and (6, 13). Write the equation of this line in standard form. Show your work. | -2x + y = 1 1 | auto | #Just the number — no units, no x=. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | Parallel to y = 4x + 1: one example. ★ | y = 4x 4x, y=4x | auto | y=#xWrite y= then the expression — no spaces, ^ for exponents, / for fractions. |
|
| Medium Intermediate discuss | Perpendicular slope to 2? ★ | −1/2 — | talk | discussion prompt |
|
| Hot Advanced discuss | Are y = 3x and y = −(1/3)x perpendicular? | yes (negative reciprocals) — | talk | discussion prompt |
|
| Spicy Challenge writeeq | Line A passes through (1, 2) and (3, 8). Write the equation of the line perpendicular to A that passes through (2, 1). | y = −1/3 x + 5/3 -1/3x+5/3 | auto | −#/#x+#/#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | As hours studied rise, grades rise: correlation? | positive — | talk | discussion prompt |
|
| Medium Intermediate discuss | As temperature rises, coats sold fall: correlation? | negative — | talk | discussion prompt |
|
| Hot Advanced writeeq | Trend slope through (1, 2) and (3, 6). ★ | m = 2 2 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge discuss | A scatterplot shows the relationship between the number of hours a student spends studying for a test and their score on the test. The line of best fit for this data is given by the equation $y = 5x + 60$, where $x$ is the hours studied and $y$ is the test score. If a student studied for 3 hours and scored 80, is this point above, below, or on the line of best fit? Explain your reasoning. | Below. The predicted score for 3 hours of studying is 75 (5*3 + 60 = 75). Since the actual score of 80 is greater than the predicted score of 75, the point (3, 80) is above the line of best fit. — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | f(x) = 2x + 1; f(3)? ★ | 7 7 | auto | #Just the number — no units, no x=. |
|
| Medium Intermediate writeeq | Slope through (0, 0) and (5, 10). ★ | m = 2 2 | auto | #Just the number — no units, no x=. |
|
| Hot Advanced writeeq | Write the equation of the line through (0, 4) and (2, 8). ★ | y = 2x + 4 y | auto | yType the grouped expression(s) — the parentheses are the answer. Use / for ÷ and * for ×; separate two answers with ; or the word and. |
|
| Spicy Challenge discuss | A line passes through the points (3, 7) and (5, 1). Another line is perpendicular to this first line and passes through the point (4, 2). What is the y-intercept of the second line? | y = (1/3)x + 2/3 — | talk | discussion prompt |
|
| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 4.01 | More hooks | A vending machine input→snack | — not shown — | too short to teach anything; no concrete number or example |
| 4.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 4.02 | More hooks | A payroll's per-hour slope | — not shown — | It's incomplete; it needs a verb or more context. |
| 4.02 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 4.03 | More hooks | A grade: base score + per-test points | A grade can be like y=mx+b: base score (b) + per-test points (m). | It's an incomplete thought, missing context for 'base score' and 'per-test points'. |
| 4.03 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 4.04 | More hooks | Predicting from 2 data points | You can predict a future value from 2 data points. | It's an incomplete thought, missing what is being predicted. |
| 4.04 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 4.05 | More hooks | Miles-per-gallon distance | — not shown — | Too vague, needs context. |
| 4.05 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 4.06 | More hooks | Reading intercepts for extremes | — not shown — | Too vague, needs context. |
| 4.06 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 4.07 | More hooks | Building a line from a single point | — not shown — | Too vague, needs context. |
| 4.07 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 4.08 | Fascinating history | Forms are like dialects of the same language - the same line, spoken three ways so different questions are easy. | Forms are like dialects: same line, different questions are easy. | It's an analogy without concrete examples of forms or questions. |
| 4.08 | More hooks | Switching to read the intercept easily | — not shown — | It lacks context about what is being switched or why. |
| 4.08 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 4.09 | More hooks | Train rails staying parallel | Train rails staying parallel is an example of parallel lines. | It's an example without the concept it illustrates. |
| 4.09 | Real-world spark | Railroad tracks, wallpaper pattern lines rising together, and a room's perpendicular walls are all you've just classified. | — not shown — | It refers to prior classification, which is missing. |
| 4.09 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 4.10 | More hooks | A weather trend over days | — not shown — | It's too vague; it doesn't explain what about a weather trend. |
| 4.10 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 4.11 | Fascinating history | The unit review welds functions, slope, forms, parallel/perpendicular and scatterplots into one fluent toolkit - the way an athlete runs dri | — not shown — | It's an analogy for a unit review, not a fact about algebra. |
| 4.11 | More hooks | One toolkit for all line questions | — not shown — | It's too vague; it doesn't explain what the toolkit is. |
| 4.11 | Real-world spark | Model a taxi bill, predict a trend, compare rails - the whole of Unit 4 in a real day of making sense of lines. | — not shown — | It's an activity description, not a fact about algebra. |
| 4.11 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | no acceptable candidate passed the gate yet | |||
| Where | Prompt | Guide key | What looks off | Suggested |
|---|---|---|---|---|
| 4.03 Challenge | If m is negative, how does the graph fall? | falls right-to-left | Key is opposite of correct answer. | falls left-to-right |
| 4.05 Intermediate | y = 9 when x = 3; find the direct variation k. | k = 3, y = 3x | Key includes extra info. | k = 3 |
| 4.11 Advanced | Through (0, 4) and (2, 8). | y = 2x + 4 | Question asks for slope, key gives equation. | Equation of the line through (0, 4) and (2, 8). / m = 2 |
Jump to: 5.01 · 5.02 · 5.03 · 5.04 · 5.05 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | Graph to solve y = x and y = 2. | (2, 2) 2,2 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Medium Intermediate twosided | Solve by graphing: y = 2x − 1 and y = x + 2. | (3, 5) 3,5 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Hot Advanced discuss | How many solutions? y = 2x + 1 and 2x − y = 0. ★ | no solution (parallel: y = 2x) — | talk | discussion prompt |
|
| Spicy Challenge discuss | Graph the system y = 2x − 3 and y = −x + 6, state the solution, and verify it algebraically. ★ | The lines meet at (3, 3); checking: 2(3) − 3 = 3 and −(3) + 6 = 3 — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | y = 4 and 2x + y = 10. | (3, 4) 3,4 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Medium Intermediate twosided | y = 2x and 3x + y = 10. | (2, 4) 2,4 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Hot Advanced twosided | x − 2y = 2 and 2x + 3y = 11. ★ | (4, 1) 4,1 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Spicy Challenge discuss | A rectangular garden has a perimeter of 48 feet. The length of the garden is 6 feet more than its width. What are the dimensions of the garden? ★ | Length: 15 feet, Width: 9 feet — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | x + y = 6 and x − y = 2. | (4, 2) 4,2 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Medium Intermediate twosided | 2x + 3y = 8 and 2x − 3y = −4. | (1, 2) 1,2 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Hot Advanced twosided | 4x + 3y = 24, 4x − 3y = −24. ★ | (0, 8) 0,8 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Spicy Challenge discuss | A farmer has a total of 30 animals, consisting of chickens and cows. If the total number of legs among all the animals is 84, how many chickens and how many cows does the farmer have? ★ | (18 chickens, 12 cows) — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | Is (0, 3) a solution to y ≤ 2? Test it. | 3 ≤ 2 is false → no — | talk | discussion prompt |
|
| Medium Intermediate discuss | Graph y ≤ 3x − 1; is (1, 1) in the solution? | 1 ≤ 2 → yes — | talk | discussion prompt |
|
| Hot Advanced discuss | Which boundary is solid vs dotted for y > 2x + 1 and y ≤ x − 4? | dotted (strict), solid (≤) — | talk | discussion prompt |
|
| Spicy Challenge discuss | A small business sells two types of custom-printed t-shirts: basic and premium. Each basic shirt costs $5 to produce and sells for $12. Each premium shirt costs $8 to produce and sells for $20. The business has a daily production budget of $200 and can produce at most 30 shirts in total per day. Let 'x' be the number of basic shirts and 'y' be the number of premium shirts produced. Write a system of two linear inequalities that represents these constraints. Then, determine if producing 15 basic shirts and 10 premium shirts is a feasible solution. Show your work. ★ | System: 5x + 8y <= 200, x + y <= 30. Feasible: Yes. — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner twosided | Solve by any method: y = 2x and y = x + 3. | (3, 6) 3,6 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Medium Intermediate twosided | 3x + y = 7 and x − y = 1. | (2, 1) 2,1 | auto | #,#Ordered pair, x first then y: (#,#). |
|
| Hot Advanced discuss | Difference is 5, sum is 9. ★ | 7 and 2 — | talk | discussion prompt |
|
| Spicy Challenge discuss | A small business sells two types of custom-printed t-shirts: basic and premium. Basic t-shirts cost $12 each to produce and premium t-shirts cost $18 each. Last month, the business produced a total of 250 t-shirts. If the total production cost for all t-shirts was $3600, how many of each type of t-shirt did they produce? ★ | 150 basic t-shirts, 100 premium t-shirts — | talk | discussion prompt |
|
| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 5.01 | More hooks | The meeting point of two hikers on different trails | — not shown — | It's too vague; it doesn't explain what about the meeting point. |
| 5.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 5.02 | More hooks | Swapping one item for its equal value (trade) - the original algebra | — not shown — | It's too vague; 'original algebra' needs context. |
| 5.02 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 5.03 | Fascinating history | Elimination is adding two true statements to get another true statement - an idea that traces to Euclid's axioms (c. | — not shown — | The snippet is cut off and incomplete. |
| 5.03 | More hooks | Adding equal weights to both sides of a balance | Adding equal weights to both sides of a balance is like solving an equation. | It's an incomplete thought; what does this relate to? |
| 5.03 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 5.04 | More hooks | 'Feasible region' in real logistics and shipping | — not shown — | It's a term without explanation or example. |
| 5.04 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 5.05 | More hooks | A mechanic picking the right wrench for a job | Choosing the right math tool is like a mechanic picking the right wrench for a job. | It's an analogy without explanation. |
| 5.05 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
| Where | Prompt | Guide key | What looks off | Suggested |
|---|---|---|---|---|
| 5.01 Beginner | Graph y = x and y = 3. | (3, 3) | Key is the intersection, not the graph. | Graph shows intersection at (3,3) |
| 5.03 Advanced | 4x + 3y = 24, 4x − 3y = −24. | (0, 8) | Arithmetic error in key. | (0, 8) |
| 5.04 Intermediate | Graph y ≤ 3x − 1; is (1, 1) in the solution? | 1 ≤ 2 → yes | Arithmetic error in key. | 1 <= 3-1 -> 1 <= 2 -> yes |
Jump to: 6.01 · 6.02 · 6.03 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | y = 6(1.5)ˣ growth or decay? | growth (base > 1) — | talk | discussion prompt |
|
| Medium Intermediate arith | A population of bacteria doubles every hour. If you start with 50 bacteria, how many will there be after 3 hours? ★ | 400 400 | auto | #Just the number — no units, no x=. |
|
| Hot Advanced discuss | 300, 150, 75, … — nth-term equation and 6th term. ★ | aₙ = 300(0.5)ⁿ⁻¹; a₆ = 9.375 — | talk | discussion prompt |
|
| Spicy Challenge arith | A certain radioactive isotope decays such that its mass is halved every 12 hours. If you start with 400 grams, how many grams will remain after 2 days? | 25 25 | auto | #Just the number — no units, no x=. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner percent | A car worth $18,000 depreciates 12%/yr. Decay factor? ★ | 0.88 0.88 | auto | #Just the number — no units, no x=. |
|
| Medium Intermediate discuss | $8,000 saved at 6%/yr — write the equation. ★ | y = 8000(1.06)ˣ — | talk | discussion prompt |
|
| Hot Advanced discuss | 500 bacteria triple each hour — equation and count after 3 hours. ★ | y = 500(3)ˣ; 13,500 — | talk | discussion prompt |
|
| Spicy Challenge discuss | A rare coin appreciates in value by 4% every year. Another coin, starting with the same initial value, appreciates by 0.5% every month. After one year, which coin is worth more? Justify your answer by comparing their annual growth factors. | The coin appreciating by 0.5% monthly is worth more. Its annual growth factor is approximately 1.0617, which is greater than the first coin's annual growth factor of 1.04. — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | y = 7(0.5)ˣ growth or decay? | decay — | talk | discussion prompt |
|
| Medium Intermediate discuss | Decay rate if the factor is 0.85? ★ | 15% — | talk | discussion prompt |
|
| Hot Advanced word | A radioactive substance decays such that its mass halves every 15 years. If you start with 120 grams, how much will remain after 45 years? ★ | 15 15 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge percent | A new smartphone model is released. In its first week, 10,000 units are sold. Due to high demand, sales are projected to increase by 15% each week. A competing model, released at the same time, sold 15,000 units in its first week, but its sales are projected to increase by a constant 800 units each week. In which week will the new smartphone model's total cumulative sales first exceed the competing model's total cumulative sales? (Assume sales are cumulative from week 1.) | 10 10, s=10 | auto | #Just the number — no units, no x=. |
|
| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 6.01 | More hooks | The folding-paper-to-the-moon trick (≈ 42 folds) | Folding paper 42 times would reach the moon. | It's a trick without explanation. |
| 6.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 6.02 | More hooks | Phone resale value crashing each year | — not shown — | It's a fragment, not a complete thought. |
| 6.02 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 6.03 | More hooks | A news headline claiming something 'grew exponentially' - is it really? | — not shown — | It's a question without an answer or context. |
| 6.03 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | no acceptable candidate passed the gate yet | |||
Jump to: 7.01 · 7.02 · 7.03 · 7.04 · 7.05 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner zeroexp | 9⁰ ★ | 1 1 | auto | #Just the number — no units, no x=. |
|
| Medium Intermediate arith | 4⁻³ | 1/64 1/64 | auto | #/#Just the number — no units, no x=. |
|
| Hot Advanced exponents | Simplify (5⁻²)(5³) using properties. | 5¹ = 5 5 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge twosided | A radioactive substance decays such that its mass is halved every 5 years. If you start with 800 grams of the substance, write an expression using exponents to represent the mass remaining after 't' years. Then, use your expression to find the mass of the substance 10 years *before* the starting measurement (t=0). | 3200 3200, t=3200 | auto | #Just the number — no units, no x=. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner scientific | Write 6,700,000 in scientific notation. ★ | 6.7 × 10⁶ 6.7*10^6, 6.7e6 | auto | #e#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate scientific | Write 0.00000092 in scientific notation. | 9.2 × 10⁻⁷ 9.2*10^-7, 9.2e-7 | auto | #e−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced polyops | Divide (8 × 10⁸) ÷ (2 × 10³). ★ | 4 × 10⁵ 4*10^5, 4e5 | auto | #e#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge scientific | A microchip holds 4.5 × 10⁶ transistors. A factory produces 3.2 × 10³ chips per day. Write the total number of transistors per day in scientific notation. | 1.44 × 10¹⁰ 1.44*10^10, 1.44e10 | auto | #e#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner simplify | x³ · x². | x⁵ x^5 | auto | x^#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate simplify | (x⁵)³. ★ | x¹⁵ x^15 | auto | x^#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced exponents | Simplify x⁸/x² and (y⁷)/(y⁷). | x⁶; 1 x^6 | auto | x^#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge discuss | A scientist is comparing two bacterial cultures. Culture A starts with $10^5$ bacteria and doubles every hour. Culture B starts with $10^7$ bacteria and is reduced by a factor of $10$ every hour. After 3 hours, which culture has more bacteria, and by what factor? | Culture A has 80 times more bacteria than Culture B. — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner rationalexp | 100^(1/2). | 10 10 | auto | #Just the number — no units, no x=. |
|
| Medium Intermediate rationalexp | 64^(1/2). | 8 8 | auto | #Just the number — no units, no x=. |
|
| Hot Advanced rationalexp | Evaluate 81^(1/4). ★ | 3 3 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge twosided | A spherical balloon has a volume of $V$ cubic units. The formula for the volume of a sphere is $V = \frac{4}{3}\pi r^3$, where $r$ is the radius. Write an equation that expresses the radius $r$ in terms of the volume $V$ using a rational exponent. Then, if a balloon has a volume of $288\pi$ cubic units, what is its radius? | r = (3V / (4π))^(1/3), radius = 6 units 6, 6units | auto | #unitsType the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner exponents | Simplify x⁴ · x⁵. | x⁹ x^9 | auto | x^#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate scientific | Write 5,800,000 in scientific notation. ★ | 5.8 × 10⁶ 5.8*10^6, 5.8e6 | auto | #e#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced substitute | Evaluate x³ for x = 3. ★ | 27 27, x=27 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge discuss | A scientist is comparing two bacterial colonies. Colony A starts with $2^{10}$ bacteria and doubles every hour. Colony B starts with $4^3$ bacteria and quadruples every hour. After 3 hours, which colony has more bacteria, and by what factor? | Colony A has 2 times more bacteria than Colony B. — | talk | discussion prompt |
|
| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 7.01 | More hooks | Half-life of carbon-14 (a negative exponent) | The half-life of carbon-14 is 5,730 years, which can be described using negative exponents. | It's not clear what a 'negative exponent' has to do with half-life. |
| 7.01 | Real-world spark | A shrinking or decaying factor each period uses a negative exponent; a 'per-second' decay rate is a tiny negative power in disguise - the sa | — not shown — | It refers to 'the same rule you just wrote' which is not provided. |
| 7.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 7.02 | More hooks | The 'how many zeros in a trillion?' quick-draw | A trillion has 12 zeros, which can be written as 10¹². | It's a game description, not a fact about numbers. |
| 7.02 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 7.03 | More hooks | Doubling savings vs adding a fixed amount | Doubling savings (like 2, 4, 8) uses exponents, while adding a fixed amount (like 2, 4, 6) does not. | It's a comparison, not a fact about exponents. |
| 7.03 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 7.04 | More hooks | Quarter-frequency sound = 4th root of length | A sound's quarter-frequency is the 4th root of its length. | It's too short and lacks context for a 7th grader. |
| 7.04 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 7.05 | Fascinating history | Every great idea earned its notation over centuries. After a week of exponent properties, review cements them - like a bookkeeper rehearsing | Exponent notation, like 2^3, took centuries to develop. | It's an analogy, not a fact about exponents. |
| 7.05 | More hooks | Matching exponent rules to real curves | Exponent rules can describe real-world curves, like growth patterns. | It's too vague and lacks specific information. |
| 7.05 | Real-world spark | Choose between two savings rates or payment plans by their exponents - review decides which curve wins. | — not shown — | It describes a choice, not a fact about exponents. |
| 7.05 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
Jump to: 8.01 · 8.02 · 8.03 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner standardform | Write 4x + 3x² − 1 in standard form. ★ | 3x² + 4x − 1 3x^2+4x-1 | auto | #x^#+#x−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate polyops | Add (3x² + 2x − 1) + (x² − 3x + 4). ★ | 4x² − x + 3 4x^2-x+3 | auto | #x^#−x+#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced polyops | Subtract (5x² + x) − (2x² − 3x + 2). | 3x² + 4x − 2 3x^2+4x-2 | auto | #x^#+#x−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge discuss | Subtract (4x² − 5x + 2) − (2x² + 3x − 6), then state the degree of the result. | 2x² − 8x + 8, degree 2 — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner polyops | Multiply 2x(3x − 4). ★ | 6x² − 8x 6x^2-8x | auto | #x^#−#xType the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate polyops | Multiply (x + 4)(x + 3). ★ | x² + 7x + 12 x^2+7x+12 | auto | x^#+#x+#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced polyops | Multiply (2x − 3)(x + 2). | 2x² + x − 6 2x^2+x-6 | auto | #x^#+x−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge polyops | Multiply (x + 1)(x² − x + 1). | x³ + 1 x^3+1 | auto | x^#+#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner standardform | Write 3x + 5x² − 1 in standard form. ★ | 5x² + 3x − 1 5x^2+3x-1 | auto | #x^#+#x−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate polyops | Multiply (2x + 1)(x + 4). ★ | 2x² + 9x + 4 2x^2+9x+4 | auto | #x^#+#x+#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced standardform | Add (6x² − 4x + 1) + (2x² + 7x − 5), then write the result in standard form. | 8x² + 3x − 4 8x^2+3x-4 | auto | #x^#+#x−#Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge discuss | A rectangle's length is (3x + 2) and its width is (x − 5). Write the area as a polynomial in standard form, then find the area when x = 10. | Area = 3x² − 13x − 10; the area is 160 when x = 10 — | talk | discussion prompt |
|
| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 8.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 8.02 | More hooks | Shrinking a pattern by a strip (x)(x+... ) | — not shown — | It's incomplete and lacks context. |
| 8.02 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 8.03 | More hooks | Checking a plan's dimensions multiply out | — not shown — | It's too vague and lacks specifics. |
| 8.03 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | no acceptable candidate passed the gate yet | |||
| Spicy | Why must exponents match to combine like terms? Use 3x² + 2x as a counterexample. key: x² and x are different terms (can't combine directly) | Subtract (4x² − 5x + 2) − (2x² + 3x − 6), then state the degree of the result. key: 2x² − 8x + 8, degree 2 | 5 distribute the minus sign across all three terms → combine the x² terms → combine the x terms → combine the co | ✅ key checks out |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | Classify 7x² + 3x by degree and terms. key: quadratic binomial | Add (6x² − 4x + 1) + (2x² + 7x − 5), then write the result in standard form. key: 8x² + 3x − 4 | 4 combine the x² terms → combine the x terms → combine the constants → write the terms in descending degree | ✅ key checks out |
| Spicy | Add and multiply: (x + 3) + (2x − 1) vs (x + 3)(2x − 1) key: sum 3x + 2; product 2x² + 5x − 3 | A rectangle's length is (3x + 2) and its width is (x − 5). Write the area as a polynomial in standard form, then find the area when x = 10. key: Area = 3x² − 13x − 10; the area is 160 when x = 10 | 5 multiply the two binomials → combine the like terms → write in standard form → substitute x = 10 → compute the | ✅ key checks out |
| Where | Prompt | Guide key | What looks off | Suggested |
|---|---|---|---|---|
| 8.01 Beginner | Standard form of 7 + x². | x² + 7; degree 2 | Key includes extra information not asked for. | x² + 7 |
Jump to: 9.01 · 9.02 · 9.03 · 9.04 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner factor | Factor the GCF: 8x² + 4x. ★ | 4x(2x + 1) 4x(2x+1) | auto | #x(#x+#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate factor | Factor the GCF: 18x³ − 12x. | 6x(3x² − 2) 6x(3x^2-2) | auto | #x(#x^#−#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced factor | Factor by grouping: x³ + 3x² + 2x + 6. ★ | (x² + 2)(x + 3) (x^2+2)(x+3) | auto | (x^#+#)(x+#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge discuss | A rectangular prism has a volume given by the expression $x^3 + 5x^2 + 4x + 20$. If one of its side lengths is $(x+5)$, what are the other two side lengths? (Assume all side lengths are linear expressions with integer coefficients.) | (x+2) and (x+2) — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner factor | Factor x² + 5x + 4. ★ | (x + 1)(x + 4) (x+1)(x+4) | auto | (x+#)(x+#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate factor | Factor x² − 5x + 6. ★ | (x − 2)(x − 3) (x-2)(x-3) | auto | (x−#)(x−#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced factor | Factor x² + 2x − 15. | (x + 5)(x − 3) (x+5)(x-3) | auto | (x+#)(x−#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge discuss | A rectangular garden has an area that can be represented by the expression $x^2 + 10x + 24$ square feet. If the length of the garden is 2 feet longer than its width, what are the expressions for the length and width of the garden? | Length: x + 6, Width: x + 4 — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner factor | Factor x² − 4. ★ | (x + 2)(x − 2) (x+2)(x-2) | auto | (x+#)(x−#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate factor | Factor x² − 100. ★ | (x + 10)(x − 10) (x+10)(x-10) | auto | (x+#)(x−#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced factor | Factor completely: 2x² − 18. | 2(x + 3)(x − 3) 2(x+3)(x-3) | auto | #(x+#)(x−#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge discuss | Factor completely and name the pattern: 3x³ − 27x. | 3x(x − 3)(x + 3) — GCF then difference of squares — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner factor | Factor x² + 6x + 8. ★ | (x + 2)(x + 4) (x+2)(x+4) | auto | (x+#)(x+#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate factor | Factor x² − 49. ★ | (x + 7)(x − 7) (x+7)(x-7) | auto | (x+#)(x−#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Hot Advanced factor | Factor by grouping: x³ + 2x² + 4x + 8. | (x² + 4)(x + 2) (x^2+4)(x+2) | auto | (x^#+#)(x+#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Spicy Challenge factor | Factor completely: x³ + 3x² − 4x − 12, then state every zero of the polynomial. | (x + 3)(x − 2)(x + 2); zeros at x = −3, 2, −2 (x+3)(x-2)(x+2) | auto | (x+#)(x−#)(x+#)Type the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 9.01 | More hooks | Sorting shared ingredients out of recipes | Did you know factoring is like sorting shared ingredients out of recipes? | It's an incomplete thought. |
| 9.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 9.02 | More hooks | Finding two numbers from product & sum | Did you know factoring can be like finding two numbers from their product and sum? | It's an incomplete thought. |
| 9.02 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 9.03 | More hooks | Removing a square hole from a square sheet | Removing a square hole from a square sheet is like subtracting areas in algebra. | Lacks context for algebra. |
| 9.03 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 9.04 | Fascinating history | Review is the factor-spotting reflex: after weeks of GCF, grouping, trinomials and special cases, the eye learns to 'see' the structure inst | Factoring becomes a reflex, like an electrician seeing clean loops in wiring. | Too abstract, uses jargon. |
| 9.04 | More hooks | Reading a messy floor plan as products | Reading a messy floor plan as products is like factoring algebraic expressions. | Lacks context for algebra. |
| 9.04 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
| Where | Prompt | Guide key | What looks off | Suggested |
|---|---|---|---|---|
| 9.04 Challenge | Factor x² + (a+b)x + ab; check it equals (x + a)(x + b). | expands back by FOIL | Key is not a factorized expression. | Factor x² + (a+b)x + ab. / (x + a)(x + b) |
Jump to: 10.01 · 10.02 · 10.03 · 10.035 · 10.04 · 10.05 · 10.06 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | Is y = 5x + 2 quadratic? | no (linear) — | talk | discussion prompt |
|
| Medium Intermediate discuss | Which way does y = −2x² open? ★ | down — | talk | discussion prompt |
|
| Hot Advanced discuss | Vertex of y = (x + 3)² − 1. ★ | (−3, −1) — | talk | discussion prompt |
|
| Spicy Challenge discuss | What does the sign of a tell you when a is negative in y = ax²? | opens down; vertex is a maximum — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | Vertex of y = x² − 3. ★ | (0, −3) — | talk | discussion prompt |
|
| Medium Intermediate discuss | Vertex + direction of y = −(x + 2)² + 1. ★ | (−2, 1), down — | talk | discussion prompt |
|
| Hot Advanced discuss | From y = x², describe y = (x − 4)² + 2. | right 4, up 2, vertex (4,2) — | talk | discussion prompt |
|
| Spicy Challenge discuss | The height of a ball thrown upwards is modeled by the equation h(t) = -16t² + 64t + 80, where h(t) is the height in feet and t is the time in seconds. What is the maximum height the ball reaches, and how long does it take to reach that height? | Maximum height: 144 feet, Time: 2 seconds — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner substitute | h(1) for h = −16t² + 48t. | 32 ft 32, 32ft | auto | #ftType the expression with no spaces; use ^ for exponents and / for fractions. |
|
| Medium Intermediate twosided | When does h = −16t² + 48t hit the ground? ★ | solve → t = 0 or 3 s 0or3s, t=0or3s | auto | t=#or#sWrite t= then the expression — no spaces, ^ for exponents, / for fractions. |
|
| Hot Advanced scale | A rectangular garden has a length that is 5 feet more than its width. If the area of the garden is 84 square feet, what is the width of the garden? | 7 7 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge discuss | A rectangular dog pen is to be built using 120 feet of fencing. One side of the pen will be against the house, so no fencing is needed for that side. The owner wants to divide the pen into two equal sections using an additional fence parallel to the house. What is the maximum possible area of the entire dog pen? | 600 square feet — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner discuss | Simplify √12. ★ | 2√3 — | talk | discussion prompt |
|
| Medium Intermediate discuss | Simplify √20. ★ | 2√5 — | talk | discussion prompt |
|
| Hot Advanced discuss | Simplify √200. | 10√2 — | talk | discussion prompt |
|
| Spicy Challenge discuss | A square garden has an area of 72 square feet. A landscaper wants to put a fence around it. What is the exact length of fencing needed for one side of the garden, in simplest radical form? | 6√2 feet — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner solve | x² = 25. ★ | x = ±5 (-5,5), (5,-5), -5,5, 5,-5, x=-5,5, x=5,-5, x=±5, ±5 | auto | x=±#Write x= then the expression — no spaces, ^ for exponents, / for fractions. |
|
| Medium Intermediate twosided | x² + 5x + 6 = 0. ★ | x = −2, −3 (-2,-3), (-3,-2), -2,-3, -3,-2, x=-2,-3, x=-3,-2 | auto | x=−#,−#Both answers, comma between them — either order works (x=4,-4 and x=-4,4 both count), and x=±# is fine too. |
|
| Hot Advanced twosided | x² − 10x + 21 = 0. | x = 7, 3 (3,7), (7,3), 3,7, 7,3, x=3,7, x=7,3 | auto | x=#,#Both answers, comma between them — either order works (x=4,-4 and x=-4,4 both count), and x=±# is fine too. |
|
| Spicy Challenge discuss | A rectangular garden has an area of 108 square feet. Its length is 3 feet longer than its width. What are the dimensions of the garden? | width = 9 feet, length = 12 feet — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner simplify | Discriminant of x² + 5x + 6. ★ | 25 − 24 = 1 1 | auto | #Just the number — no units, no x=. |
|
| Medium Intermediate discuss | Discriminant of x² + 2x + 5. ★ | −16 → no real roots — | talk | discussion prompt |
|
| Hot Advanced twosided | Solve x² − 6x + 8 = 0 by formula. | x = 4, 2 (2,4), (4,2), 2,4, 4,2, x=2,4, x=4,2 | auto | x=#,#Both answers, comma between them — either order works (x=4,-4 and x=-4,4 both count), and x=±# is fine too. |
|
| Spicy Challenge discuss | A rectangular garden has a length that is 5 feet more than its width. If the area of the garden is 84 square feet, what are the dimensions of the garden? Use the quadratic formula to find your answer. Only positive dimensions are valid. | Width = 7 feet, Length = 12 feet — | talk | discussion prompt |
|
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner solve | Solve x² = 144. ★ | x = ±12 (-12,12), (12,-12), -12,12, 12,-12, x=-12,12, x=12,-12, x=±12, ±12 | auto | x=±#Write x= then the expression — no spaces, ^ for exponents, / for fractions. |
|
| Medium Intermediate twosided | x² + 3x − 10 = 0. ★ | x = 2, −5 (-5,2), (2,-5), -5,2, 2,-5, x=-5,2, x=2,-5 | auto | x=#,−#Both answers, comma between them — either order works (x=4,-4 and x=-4,4 both count), and x=±# is fine too. |
|
| Hot Advanced discuss | Simplify √45 and say why square roots come back to quadratics. | 3√5; the root method undoes the square — | talk | discussion prompt |
|
| Spicy Challenge discuss | A projectile is launched vertically upwards from the ground with an initial velocity of 48 feet per second. Its height $h$ (in feet) after $t$ seconds is given by the equation $h(t) = -16t^2 + 48t$. How long does it take for the projectile to reach a height of 32 feet? What method did you use to solve the quadratic equation, and why was it the best choice? | t = 1 second or t = 2 seconds. Factoring was the best method because the equation could be easily factored after setting it to zero and dividing by a common factor. — | talk | discussion prompt |
|
| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 10.01 | More hooks | The headlight / dish reflector (a parabola) | A headlight or dish reflector is shaped like a parabola. | It's an incomplete sentence and lacks context. |
| 10.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 10.02 | More hooks | Projectile heights at different angles | — not shown — | It's an incomplete sentence and lacks context. |
| 10.02 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 10.03 | More hooks | The drop test of a fragile phone | — not shown — | This is a fragment, not a complete thought. |
| 10.03 | Real-world spark | Kick a football, drop a phone - the quadratic tells how high and when it lands. Every launch is this unit in the real world. | — not shown — | It's unclear what 'this unit' refers to. |
| 10.03 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 10.035 | More hooks | Diagonal of a TV screen (√2 scale) | The diagonal of a TV screen often uses a √2 scale, like a 40-inch screen. | This is a fragment, not a complete thought. |
| 10.035 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 10.04 | More hooks | The zero product 'zero-knows' trick | — not shown — | It's a title, not a fact. |
| 10.04 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 10.05 | More hooks | When 'no real roots' means it never crosses | — not shown — | It's incomplete; 'it' is undefined. |
| 10.05 | Real-world spark | Even when factoring fails, the formula saves the day - from error margins to encryption. It's the universal answer-key for every quadratic. | — not shown — | It mentions 'the formula' without defining it. |
| 10.05 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact | |
| 10.06 | More hooks | A mechanic choosing the right wrench | Choosing the best method for a quadratic is like a mechanic choosing the right wrench. | It's an analogy, not a math fact. |
| 10.06 | Real-world spark | When a problem 'looks factored' use factoring; when the 'isolate the square' use roots; when messy, the formula. Choosing the tool IS the sk | — not shown — | It mentions 'the formula' without defining it. |
| 10.06 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
Jump to: 11.01 | 🖨 QR sheet for this unit ↗
| Flavor | Problem | Key & accepted spellings | Grading | Format shown to students | Hints (gated 30s / 60s) |
|---|---|---|---|---|---|
| Mild Beginner arith | Legs 5 and 12 — find the hypotenuse. ★ | 13 13 | auto | #Just the number — no units, no x=. |
|
| Medium Intermediate discuss | Find the missing leg: hypotenuse 25, leg 7. ★ | 24 — | talk | discussion prompt |
|
| Hot Advanced arith | The hypotenuse of a right triangle is 17 units long. One leg is 8 units long. What is the length of the other leg? ★ | 15 15 | auto | #Just the number — no units, no x=. |
|
| Spicy Challenge discuss | A rectangular garden has a diagonal path that is 25 feet long. If one side of the garden is 7 feet shorter than the other side, what are the dimensions of the garden? | 15 feet by 20 feet — | talk | discussion prompt |
|
| Section | Block | Shown before | Now | Why |
|---|---|---|---|---|
| 11.01 | More hooks | Measuring a room's diagonal for a rug/carpet | — not shown — | It's an incomplete thought. |
| 11.01 | Social game (minimal materials) | — not shown — | teacher-facing activity instructions, not a student fact |
| Flavor | Current (too easy) | Proposed | Steps | Verified |
|---|---|---|---|---|
| Hot | A right triangle has legs 7 and 24 — the hypotenuse is? key: 25 | The hypotenuse of a right triangle is 17 units long. One leg is 8 units long. What is the length of the other leg? key: 15 model: 8^2 + x^2 = 17^2 | 3 Set up the Pythagorean theorem with given values → Calculate squares and subtract to isolate x^2 → Take the sq | ✅ key checks out |
| Spicy | Which triangle is right: 6-8-10 or 5-6-9? Justify. key: 6-8-10 (100 = 36 + 64) | A rectangular garden has a diagonal path that is 25 feet long. If one side of the garden is 7 feet shorter than the other side, what are the dimensions of the garden? key: 15 feet by 20 feet model: x^2 + (x+7)^2 = 25^2 | 6 define variables for sides (x and x+7) → set up Pythagorean theorem equation → expand (x+7)^2 → simplify and r | 🗣 talk-through leap 3 |