AMC 10 · 2020 · #1
Grade 7 arithmeticProblem
What is the value of
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Compute the value of $1-(-2)-3-(-4)-5-(-6)$.
Givens: Six numbers strung together with subtraction signs; Three of the numbers are negative: $-2$, $-4$, $-6$; Three of the numbers are positive: $1$, $3$, $5$; Answer choices: (A) $-20$, (B) $-3$, (C) $3$, (D) $5$, (E) $21$
Unknowns: The single number this expression equals
Understand
Restated: Compute the value of $1-(-2)-3-(-4)-5-(-6)$.
Givens: Six numbers strung together with subtraction signs; Three of the numbers are negative: $-2$, $-4$, $-6$; Three of the numbers are positive: $1$, $3$, $5$; Answer choices: (A) $-20$, (B) $-3$, (C) $3$, (D) $5$, (E) $21$
Plan
Primary tool: #5 Look for a Pattern
Secondary: #15 Reorganize Information
Tool #5 (Look for a Pattern): the signs alternate in a clean rhythm — every other term is “minus a negative,” which flips to “plus.” Spotting that pattern turns the scary-looking string into a simple add/subtract problem. Tool #15 (Reorganize) helps once the signs are fixed: group the positives and negatives separately so the arithmetic is just two small sums.
Execute — Answer: D
7.NS.A.1 Step 1 Rewrite every “minus a negative” as “plus the positive.” So $-(-2)$ becomes $+2$, $-(-4)$ becomes $+4$, and $-(-6)$ becomes $+6$.
💡 Two minus signs back-to-back cancel each other — the same rule we use for $5-(-3)=8$.
3.OA.B.5 Step 2 Reorganize: collect the positive numbers and the negative numbers into two separate groups so each sum is easy.
💡 Addition can be reordered (commutative property), so we group the friendly numbers together.
7.NS.A.1 Step 3 - Add the positives: $1+2+4+6 = 13$.
- Add the negatives: $-3 + (-5) = -8$.
- Combine: $13 + (-8) = 5$.
💡 Once the signs are fixed, this is single-digit arithmetic.
7.NS.A.1 Rewrite every “minus a negative” as “plus the positive.” So $-(-2)$ becomes $+2$ 3.OA.B.5 Reorganize: collect the positive numbers and the negative numbers into two separ 7.NS.A.1 Add the positives: $1+2+4+6 = 13$. Add the negatives: $-3 + (-5) = -8$. Combine: Review
Reasonableness: Recompute left-to-right without grouping: $1-(-2)=3$; $3-3=0$; $0-(-4)=4$; $4-5=-1$; $-1-(-6)=5$. Same answer, so (D) is correct.
Alternative: Tool #1 (Draw a Diagram) — plot each step on a number line, jumping right for “minus a negative” and left for plain subtraction. The final landing spot is $5$.
CCSS standards used (min grade 7)
3.OA.B.5Apply properties of operations as strategies to multiply and divide (Using the commutative property of addition to regroup positives and negatives before summing.)7.NS.A.1Apply and extend understanding of addition and subtraction to rational numbers (Rewriting $-(-n)$ as $+n$ and adding signed numbers to reach $5$.)
⭐ This AMC 10 problem only needs Grade 7 “subtracting a negative is adding a positive” you already know — the messy string is really $1+2-3+4-5+6 = 5$.
⭐ This AMC 10 problem only needs Grade 7 “subtracting a negative is adding a positive” you already know — the messy string is really $1+2-3+4-5+6 = 5$.