AMC 10 · 2020 · #1
Grade 8 arithmeticPick an answer.
The expression looks heavy, but it is really four separate small calculations stacked together, so Tool #7 (Identify Subproblems) says to handle one root at a time. Once the inner sums are computed they turn out to be 1, 4, 9, 16 — Tool #5 (Look for a Pattern) recognizes these as perfect squares and explains why: adding consecutive odd numbers starting at 1 always lands on a square. That makes every root a whole number, and Tool #3 (Eliminate Possibilities) then rules out the two choices that still contain radicals.
Add inside each root first
Add inside each root first.
The root sign is a lid: finish the addition underneath it before you do anything else.
1.OA.C.6Identify SubproblemsRecognize the perfect squares
A sum of odds is always a perfect square.
Consecutive odd numbers are the growing L-shaped borders of a square, so their running totals are squares.
Consecutive odd numbers are the growing L-shaped borders of a square, so their running totals are squares.
▸ Why?
The odd numbers climb by the same fixed step, so each new one wraps neatly around the last square.
▸ Why?
Pairing the first with the last gives the same total as pairing inward, so the sum is the count times the middle.
Take each square root
Each root comes out a whole number.
The square root undoes the squaring, so a root of a perfect square is the number that was squared.
8.EE.A.2Identify SubproblemsAdd the four whole numbers
Adding the four gives 10.
Once no radicals are left, the answer has to be one of the plain whole-number choices.
2.NBT.B.5Eliminate PossibilitiesAdding odd numbers in order always builds a perfect square, so those square roots come out as plain whole numbers you can just add up.
- Add inside each root first
- Recognize the perfect squares
- Take each square root
- Add the four whole numbers