Setup: \((7x^2 + 2x - 6) - (4x^2 - 3x + 5)\)
Distribute \(-1\): \(7x^2 + 2x - 6 - 4x^2 + 3x - 5\)
Combine: \(3x^2 + 5x - 11\)
Final: \(3x^2 + 5x - 11\)
Item 3.2 Trinomial
When \(3a^2 - 8a + 4\) is subtracted from \(5a^2 - 2a - 9\)
Setup: \((5a^2 - 2a - 9) - (3a^2 - 8a + 4)\)
Distribute \(-1\): \(5a^2 - 2a - 9 - 3a^2 + 8a - 4\)
Combine: \(2a^2 + 6a - 13\)
Final: \(2a^2 + 6a - 13\)
Item 3.3 Difference
\((-6y^2 + 5y - 2) - (2y^2 - 7y + 3)\)
Distribute \(-1\): \(-6y^2 + 5y - 2 - 2y^2 + 7y - 3\)
Combine \(y^2\): \(-6 - 2 = -8y^2\)
Linear & Const: \(+12y - 5\)
Final: \(-8y^2 + 12y - 5\)
Regents Rubric Alert: Writing \((4x^2 - 3x + 5) - (7x^2 + 2x - 6)\) in 3.1 is an immediate order error resulting in 0/2 or 1/2 points on Regents Part II!
DAY 4 KEY
Standards: NYS A-APR.A.1
Item 4.1 Minus Outside
\(3(4x - 1) - (7x + 5)\)
Distribute 3: \(12x - 3\)
Distribute \(-1\): \(-7x - 5\)
Combine: \((12 - 7)x + (-3 - 5) = 5x - 8\)
Final: \(5x - 8\)
Item 4.2 Middle Factor
\(8m - 2(3m - 4) - 9\)
Distribute \(-2\): \(8m - 6m + 8 - 9\)
Combine \(m\): \((8 - 6)m = 2m\)
Constants: \(8 - 9 = -1\)
Final: \(2m - 1\)
Item 4.3 Choice (1)
\(2(5g - 3) - 3(2g - 4)\)
Distribute 2: \(10g - 6\)
Distribute \(-3\): \(-6g + 12\)
Combine: \(4g + 6\)
Correct: (1) \(4g + 6\)
Testing Tip: In 4.3, Choice (2) \(4g - 18\) is the distractor for students who fail to multiply \(-3 \times -4 = +12\). Choice (3) is for students who add coefficients instead of subtracting.
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TEACHER WORKED SOLUTIONS
DAY 5 KEY
Standards: NYS A-CED.A.1, A-APR.A.1
Item 5.1 Rectangle Perimeter
\(l = 4x^2 - 2x + 3\), \(w = x^2 + 5x - 2\)
Formula: \(P = 2l + 2w\)
Add \(l + w\): \(5x^2 + 3x + 1\)
Double: \(2(5x^2 + 3x + 1) = 10x^2 + 6x + 2\)
Final: \(10x^2 + 6x + 2\)
Item 5.2 Missing Side
\(P = 12x^2 + 7x - 4\); 2 sides given
Sum 2 sides: \((5x^2 - 2x + 1) + (3x^2 + 4x - 6) = 8x^2 + 2x - 5\)
Subtract: \((12x^2 + 7x - 4) - (8x^2 + 2x - 5)\)
Result: \(4x^2 + 5x + 1\)
Final: \(4x^2 + 5x + 1\)
Item 5.3 Pentagon Perimeter
Sum of 5 sides
Linear terms: \((2 + 3 + 1 + 4 + 2)x = 12x\)
Constants: \(3 - 1 + 4 - 5 + 7 = 8\)
Simplified: \(12x + 8\)
Final: \(12x + 8\)
Pedagogy Tip: For 5.1, many students only calculate \(l + w = 5x^2 + 3x + 1\) and forget that a rectangle has 4 sides! Remind them to double or add all 4 sides.
DAY 6 KEY
Standards: NYS A-APR.A.1
Item 6.1 FOIL / Box
\((3x + 2)(x - 4)\)
Multiply: \(3x^2 - 12x + 2x - 8\)
Combine middle: \((-12 + 2)x = -10x\)
Standard form: \(3x^2 - 10x - 8\)
Final: \(3x^2 - 10x - 8\)
Item 6.2 Binomial Squared
\((2x - 5)^2\)
Expand: \((2x - 5)(2x - 5)\)
Multiply: \(4x^2 - 10x - 10x + 25\)
Combine: \(-10x - 10x = -20x\)
Final: \(4x^2 - 20x + 25\)
Item 6.3 Product & Subtraction
\((x + 3)(x + 5) - (x^2 - 7)\)
Expand product: \(x^2 + 8x + 15\)
Subtract: \(- x^2 + 7\)
Combine: \(x^2 - x^2 = 0\); \(8x + 22\)
Final: \(8x + 22\)
Freshman Fallacy Alert: Watch for \((2x - 5)^2 = 4x^2 + 25\). Insist that students write out \((2x - 5)(2x - 5)\) to guarantee the \(-20x\) middle term!
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TEACHER WORKED SOLUTIONS
DAY 7 KEY
Standards: NYS A-APR.A.1
Item 7.1 Distribution & Subtr.
\((6x^2 - 5x + 8) - 3(x^2 - 2x + 4)\)
Distribute \(-3\): \(-3x^2 + 6x - 12\)
Combine \(x^2\): \(6x^2 - 3x^2 = 3x^2\)
Combine \(x\) & const: \((-5+6)x + (8-12) = x - 4\)
Final: \(3x^2 + x - 4\)
Item 7.2 Variable Substitution
\(A = 4x^2 - 3x + 2\), \(B = -x^2 + 5x - 7\)
Find \(2A\): \(8x^2 - 6x + 4\)
Find \(-B\): \(+x^2 - 5x + 7\)
Combine: \(9x^2 - 11x + 11\)
Final: \(9x^2 - 11x + 11\)
Item 7.3 Choice (1)
\(4(x - 2)^2 - (2x^2 - 3x + 5)\)
Square: \(4(x^2 - 4x + 4) = 4x^2 - 16x + 16\)
Subtract: \(- 2x^2 + 3x - 5\)
Combine: \(2x^2 - 13x + 11\)
Correct: (1) \(2x^2 - 13x + 11\)
Regents Connection: Problem 7.2 mimics a common Regents Part II style item where polynomial expressions are defined as capital letters \(A\), \(B\), \(C\).
DAY 8 KEY
Standards: NYS A-SSE.A.2
Item 8.1 Choice (1)
\(3x(2x^2 - 4x + 5) - 2x^2(x - 3)\)
Distribute \(3x\): \(6x^3 - 12x^2 + 15x\)
Distribute \(-2x^2\): \(-2x^3 + 6x^2\)
Combine: \(4x^3 - 6x^2 + 15x\)
Correct: (1) \(4x^3 - 6x^2 + 15x\)
Item 8.2 Justify Equivalence
Is \(3(2x + 4) - 2(x - 1) \equiv 4x + 14\)?
Expand left: \(6x + 12 - 2x + 2\)
Combine: \((6 - 2)x + (12 + 2) = 4x + 14\)
Conclusion: Yes, both simplify to \(4x + 14\).
Verdict: Yes, Equivalent
Item 8.3 Leading Coefficient
\((5x^2 - 7x + 2) + 2(3x^2 + 4x - 9)\)
Distribute 2: \(6x^2 + 8x - 18\)
Standard form: \(11x^2 + x - 16\)
Leading term: \(11x^2 \rightarrow\) Coeff = \(11\)
Final: \(11\)
Regents Vocabulary Check: Remind students: Degree = highest exponent (here 2); Leading Coefficient = number multiplying the highest degree term (here 11); Constant = \(-16\).
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TEACHER SCORING GUIDE
DAY 9 GRAND FINALE KEY
Full Credit Criteria
Item 9.1 Part II (2 pts)
Subtract \(5x^2 - 8x + 11\) from \(8x^2 - 3x - 4\)
1. Setup: \((8x^2 - 3x - 4) - (5x^2 - 8x + 11)\)
2. Distribute: \(8x^2 - 3x - 4 - 5x^2 + 8x - 11\)
3. Combine: \(3x^2 + 5x - 15\)
4. Degree: \(2\)
5. Leading Coeff: \(3\)
Trinomial: \(3x^2 + 5x - 15\)
Degree: 2 | Leading Coeff: 3
Item 9.2 Part II (2 pts)
Trapezoid \(P = 15x^2 + 4x - 9\); 3 sides given
1. Sum sides: \((6+4+2)x^2 + (-2+1+3)x + (3-5-1)\)
2. Three sides sum: \(12x^2 + 2x - 3\)
3. Subtract from P: \((15x^2 + 4x - 9) - (12x^2 + 2x - 3)\)
4. Distribute: \(15x^2 + 4x - 9 - 12x^2 - 2x + 3\)
5. Combine: \(3x^2 + 2x - 6\)
Fourth Side:
\(3x^2 + 2x - 6\)
Item 9.3 Choice (1)
\(2(x + 3)^2 - 3(x - 1)(x + 4)\)
1. Square: \(2(x^2 + 6x + 9) = 2x^2 + 12x + 18\)
2. Product: \((x - 1)(x + 4) = x^2 + 3x - 4\)
3. Distribute \(-3\): \(-3x^2 - 9x + 12\)
4. Combine: \(-x^2 + 3x + 30\)
Correct Choice:
(1) \(-x^2 + 3x + 30\)
| Score | Part II Criteria (Items 9.1 & 9.2) | Common Student Error Scenarios |
|---|---|---|
| 2 Points | ||
| (Full Credit) | Complete, correct solution with all intermediate algebraic steps clearly shown. Final expression in standard form; all requested values (degree, leading coefficient) stated correctly. | Correct standard form, proper order of subtraction, correct sign distribution across every term. |
| 1 Point | ||
| (Partial Credit) | One computational error with appropriate subsequent work, OR correct algebraic simplification but fails to state the degree and/or leading coefficient. | • Signs in \(- (5x^2 - 8x + 11)\) not fully flipped. |
| • Correct trinomial found, but degree / coeff omitted. | ||
| 0 Points | ||
| (No Credit) | A completely incorrect response, two or more computational errors, conceptual errors (e.g., adding instead of subtracting), or no work shown. | • Subtracted in reverse order. |
| • Added polynomials instead of subtracting. |
Recommended 5-Minute Daily Routine:
Min 0–3: Silent Solo Work Students attempt all 3 daily items independently upon entering the classroom.
Min 3–4: Peer Check Quick turn-and-talk to compare signs, particularly negative distribution.
Min 4–5: Debrief & Stamp Cold-call one student step, display final answer, log score in Day Tracker.
Terms Takeover Worked Solutions • Algebra 1 Regents Page 5 of 5
