Question 1: Combine like terms (2 points)
\(8a+5a=13a\) and \(-3+7=4\), so the answer is \(13a+4\).
Scoring: 1 point for combining the \(a\) terms, 1 point for combining the constants.
Question 2: Expand (2 points)
(a) \(4\cdot 2x=8x\) and \(4\cdot 5=20\): \(8x+20\).
(b) \(-3\cdot x=-3x\) and \(-3\cdot(-6)=18\): \(-3x+18\).
Scoring: 1 point per part. In (b), award no credit for \(-3x-18\).
Question 3: Factor (2 points)
The greatest common factor of \(20\) and \(30\) is \(10\). \(20x: 10=2x\) and \(30: 10=3\), so \(10(2x-3)\).
Scoring: 1 point for finding the greatest common factor 10, 1 point for the correct factored form. Award 1 point total for \(5(4x-6)\) or \(2(10x-15)\), which are not factored completely.
Question 4: Equivalent or not (2 points)
\(3(2x+4)=6x+12\).
(a) \(6x+12\): equivalent. (b) \(6x+4\): not equivalent, because the \(4\) was not multiplied by \(3\). (c) \(2(3x+6)=6x+12\): equivalent. (d) \(5x+7\): not equivalent; at \(x=1\), \(3(6)=18\) but \(5+7=12\).
Scoring: 2 points for all four decisions with valid justification, 1 point for three correct decisions, 0 otherwise.
Question 5: Subtract expressions (2 points)
\(5x-2-2x+7=3x+5\). Answer: \(3x+5\). Check with \(x=1\): \(3-(-5)=8\) and \(3+5=8\).
Scoring: 1 point for distributing the negative sign to both terms, 1 point for the final simplified expression.
Question 6: Fraction and decimal coefficients (3 points)
\(\frac{1}{4}(8x-12)=2x-3\). Then \(2x-3+0.5x=2.5x-3\). Answer: \(2.5x-3\).
Check with \(x=2\): \(\frac{1}{4}(4)+1=2\) and \(2.5(2)-3=2\).
Scoring: 1 point for \(2x-3\), 1 point for combining \(2x+0.5x\), 1 point for the correct final expression.
Question 7: Dinner with a tip (3 points)
(a) \(c+0.18c=\)\(1.18c\).
(b) \(1.18(40)=47.20\): \(\$47.20\).
(c) \(1.18c=59\), so \(c=59: 1.18=50\): \(\$50\). Check: \(1.18(50)=59\).
Scoring: 1 point per part.
Question 8: Perimeter (2 points)
\(P=2(2x+1)+2(x+4)=4x+2+2x+8=\)\(6x+10\).
Check with \(x=2\): sides \(5\) and \(6\) give \(22\), and \(6(2)+10=22\).
Scoring: 1 point for a correct unsimplified expression, 1 point for the simplified result.
Question 9: Find the error (2 points)
Ethan multiplied only the first term by \(4\). He must multiply \(-3\) by \(4\) as well: \(4(x-3)=4x-12\). A check with \(x=5\) shows the difference: \(4(2)=8\) and \(4(5)-12=8\), but \(4(5)-3=17\).
Scoring: 1 point for explaining that the \(-3\) was not multiplied, 1 point for \(4x-12\).
20 points
See every chapter on the Grade 7 math page.