Problem
Count groups several equivalent ways
There are 28 beads in an array of 4 rows with 7 beads in each row. Among four offered counting methods, we must find the one that does NOT give the correct total of 28.
Operations
Your answer
Watch out (B) 7 x 4 and (C) 4 x 7 only swap the order of the factors, so it is tempting to mark one of them wrong — but both still equal 28.
How to solve
Strategy Draw a Diagram — Picture the 7-by-4 array of beads, then evaluate each choice against the array and list the totals to spot the one that is not 28.
1STEP 1
Read the array as the true total
4 rows of 7 beads make the true total 7 x 4 = 28 — every correct method must match it.
7 × 4 = 28
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 Diagram2STEP 2
Check the addition and product choices
(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.
7+7+7+7 = 28, 7×4 = 28, 4×7 = 28
Adding equal rows and multiplying rows by columns are just two views of the same array total.
2.OA.C.4Make A Systematic List3STEP 3
Check choice D
(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.
7 × 2 = 14, 14 × 4 = 56 ≠ 28
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.
Why?
The array truly holds 28 beads, because it is four equal rows of 7.
Why?
Four rows of 7 is four equal groups of 7, and 7 + 7 + 7 + 7 is 28.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The whole array is exactly its four rows put back together, with none left out and none counted twice.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Choice D counts each row as 7 x 2 = 14 instead of 7, so it doubles every row.
Why?
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.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Doubling all four equal rows doubles the whole, turning the true 28 into 56.
Why?
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.
🧱Associativity of multiplicationMultiplying three numbers, grouping any two first gives the same product.
Answer
D
Choices A, B, C all equal 28, the bead total, while D equals 56, twice as many, so D is clearly the wrong way to count.
Takeaway
Equal rows can be added or multiplied to the same total -- but counting each row as double overshoots, easy Grade 3 array sense!
- Read the array as the true total
- Check the addition and product choices
- Check choice D
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested