Competition · AMC preparation · step 4 of 4
AMC 8 · 2015 · #18
Grade 4 algebrapattern
Pick an answer.
AMC 8 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Filling in 25 unknown cells at once is too much. Tool #7 (Identify Subproblems) lets us focus on only three rows/columns and ignore the rest: find the middle of Row 1 (between 1 and 25), find the middle of Row 5 (between 17 and 81), then find the middle of Column 3 from those two values. Tool #5 (Look for a Pattern) gives the trick that does each subproblem in one line — in any arithmetic sequence with an odd number of terms, the middle term equals the average of the first and last terms. So we never need to compute the common difference at all.
Spot the middle-is-average rule
In any arithmetic sequence with an odd number of terms, the middle term equals the average of the first and last terms.
Generating a few terms of an arithmetic pattern and noticing the middle = average is exactly the Grade 4 "generate and analyze patterns" standard.
In an arithmetic sequence with an odd number of terms, the middle term equals the average of the first and last terms.
▸ Why?
The first and last terms together add up to exactly twice the middle term.
▸ Why?
The last term is the first term plus four equal steps of the common difference, and the middle term is the first term plus two of those steps, so first-plus-last comes to two first-terms plus four steps.
▸ Why?
Every term is built by adding the same fixed step to the one before, so moving a set number of places along just adds that step that many times over.
▸ Why?
Twice the middle term is two copies of (first term plus two steps), which spreads out into two first-terms plus four steps — the very same total the two ends make.
▸ Why?
Doubling a sum doubles each piece inside it, so two of (first term plus two steps) is two first-terms together with four steps.
▸ Why?
First-plus-last and twice-the-middle both reduce to the identical amount, two first-terms plus four steps, so they must equal each other.
▸ Why?
Since the two ends together make exactly twice the middle, sharing that total into two equal parts gives the middle back, because dividing by two undoes multiplying by two.
Find the middle of Row 1
Row 1 runs from 1 to 25, so its middle cell (Column 3) is = 13.
Adding 1 + 25 = 26 is the Grade 4 fluent add/subtract skill for multi-digit whole numbers.
4.NBT.B.4Identify SubproblemsFind the middle of Row 5
Row 5 runs from 17 to 81, so its middle cell is = 49.
Dividing 98 ÷ 2 = 49 is the Grade 3 "interpret whole-number quotients" skill — splitting 98 into 2 equal groups.
3.OA.A.2Identify SubproblemsAverage those two for X
Column 3 runs from 13 down to 49, so X = = 31 → (B).
The whole problem becomes three repetitions of the same Grade 4 "add two numbers and split in half" subproblem.
4.NBT.B.4Identify SubproblemsBig 5 × 5 grid, but you only need one Grade 4 idea: the middle of an arithmetic sequence is just the average of its first and last terms — used three times!
- Spot the middle-is-average rule
- Find the middle of Row 1
- Find the middle of Row 5
- Average those two for X
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