Depth · one concept from first principles · step 2 of 4
Find the choice that is not a correct way to count how many beads there are in all. Write its letter.
(A) Count with .
(B) Count with .
(C) Count with .
(D) Count by adding four times.
The beads are arranged in rows with beads in each row. The round, colored beads form an array that is across and down, beads in all.
4 rows of 7 beads make the true total 7 x 4 = 28 — every correct method must match it.
An array of equal rows is exactly the product of rows and columns, so 4 rows of 7 is 28.
3.OA.A.1Draw A Diagram(A) 7+7+7+7, (B) 7 x 4, and (C) 4 x 7 all give 28 — repeated addition and either multiplication order count the same array.
Adding equal rows and multiplying rows by columns are just two views of the same array total.
2.OA.C.4Make A Systematic List(D) adds 7 x 2 = 14 four times: 14 + 14 + 14 + 14 = 56, double the real 28 — so (D) is the wrong way to count.
Each row holds 7 beads, not 7 x 2 = 14, so D double-counts every row and overshoots to 56.
3.OA.A.1Make A Systematic ListChoice D adds 7 x 2 four times and reaches 56, not the 28 beads in the array, so it is not a correct way to count.
The array truly holds 28 beads, because it is four equal rows of 7.
Four rows of 7 is four equal groups of 7, and 7 + 7 + 7 + 7 is 28.
The whole array is exactly its four rows put back together, with none left out and none counted twice.
Choice D counts each row as 7 x 2 = 14 instead of 7, so it doubles every row.
One row is a single group of 7 beads, but 7 x 2 means two groups of 7 — twice as many as are really there.
Doubling all four equal rows doubles the whole, turning the true 28 into 56.
Adding 7 x 2 four times is the four rows of 7 taken twice, since the factors 4, 7, and 2 give the same product however you group them.
Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Branching State Enumeration, Small-Domain Constrained Search. Why the shapes repeat →