AMC 10 · 2012 · #3
Grade 6 arithmeticA bug crawls along a number line, starting at −2. It crawls to −6, then turns around and crawls to 5. How many units does the bug crawl altogether?
Pick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A bug on a number line starts at $-2$, crawls to $-6$, then turns around and crawls to $5$. Find the total number of units it travels along the way (the full path length, not how far it ends from where it began).
Givens: The bug starts at position $-2$.; It crawls left to $-6$.; It then turns around and crawls right to $5$.; Answer choices: (A) $9$, (B) $11$, (C) $13$, (D) $14$, (E) $15$.
Unknowns: The total distance crawled, adding both legs of the trip.
Understand
Restated: A bug on a number line starts at $-2$, crawls to $-6$, then turns around and crawls to $5$. Find the total number of units it travels along the way (the full path length, not how far it ends from where it began).
Givens: The bug starts at position $-2$.; It crawls left to $-6$.; It then turns around and crawls right to $5$.; Answer choices: (A) $9$, (B) $11$, (C) $13$, (D) $14$, (E) $15$.
Plan
Primary tool: #1 Draw a Diagram
Secondary: #7 Identify Subproblems, #8 Analyze the Units
The bug moves along positions on a number line, so the natural first move is Tool #1: sketch the line and mark $-2$, $-6$, and $5$. The picture instantly shows the trip has two straight legs in opposite directions. Tool #7 then splits the whole crawl into those two subproblems — leg one from $-2$ to $-6$, leg two from $-6$ to $5$ — each an easy distance on its own. Tool #8 keeps the focus on "units of distance," reminding us that a distance is always the positive gap between two points (never a negative), so the two legs simply add up.
Execute — Answer: E
6.NS.C.6 Step 1 Draw the number line
- Sketch a number line and mark the three key positions: the start at $-2$, the turning point at $-6$, and the end at $5$.
- The bug first goes left to $-6$, then reverses and goes all the way right to $5$.
💡 Seeing the points on the line makes the two separate crawls obvious at a glance.
6.NS.C.7 Step 2 Length of leg one
- Find the length of the first leg, from $-2$ to $-6$.
- Distance is the gap between the two points, which is the absolute value of their difference — always a positive number of units.
💡 From $-2$ to $-6$ you step across $4$ tick marks, so that leg is $4$ units long.
6.NS.C.7 Step 3 Length of leg two
- Find the length of the second leg, from $-6$ to $5$.
- Again take the absolute value of the difference to get the positive distance.
💡 Crossing $0$ from $-6$ up to $5$ covers $6$ units to reach $0$ and $5$ more after it, which is $11$ units.
2.OA.A.1 Step 4 Add the two legs
- The total crawl is the sum of the two leg lengths.
- Add them to get the answer in units.
💡 Total distance is just the first leg plus the second leg, so the bug crawls $15$ units in all.
6.NS.C.6 Sketch a number line and mark the three key positions: the start at $-2$, the tu 6.NS.C.7 Find the length of the first leg, from $-2$ to $-6$. Distance is the gap between 6.NS.C.7 Find the length of the second leg, from $-6$ to $5$. Again take the absolute val 2.OA.A.1 The total crawl is the sum of the two leg lengths. Add them to get the answer in Review
Reasonableness: The total must be larger than the longer single leg of $11$ units, because the bug also crawled the $4$-unit backtrack before it — and $15 > 11$, so that fits. Notice the choice $11$ is the trap for anyone who counts only the long leg and forgets the first crawl. The full trip runs from $-6$ up to $5$ (a span of $11$) plus the extra $4$ units the bug went left first, giving $15$ units, so $\textbf{(E)}$ is consistent.
Alternative: Tool #6 (Guess and Check by counting): on the drawn number line, tally tick marks one by one — $4$ ticks from $-2$ to $-6$, then $11$ ticks from $-6$ to $5$ — and read off $4 + 11 = 15$. It reaches the same answer directly from the picture without the absolute-value formula.
CCSS standards used (min grade 6)
6.NS.C.6Understand a rational number as a point on the number line (Placing $-2$, $-6$, and $5$ as points on the number line so the two crawls can be seen and measured.)6.NS.C.7Understand ordering and absolute value of rational numbers (Computing each leg's length as the absolute value of the difference of its endpoints: $|-6-(-2)|=4$ and $|5-(-6)|=11$.)2.OA.A.1Solve one- and two-step word problems using addition and subtraction within 100 (Adding the two leg lengths $4 + 11 = 15$ to get the total distance crawled.)
⭐ Distance is always the positive gap between two points, so when something turns around you add each leg separately: $4 + 11 = 15$ units.
⭐ Distance is always the positive gap between two points, so when something turns around you add each leg separately: $4 + 11 = 15$ units.
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