A farmer plants tomatoes in an array with 3 rows and 5 plants in each row. A neighbor says she can describe the same garden as '5 rows of 3.' Who is right?
AOnly the farmer — you must count rows first, then plants in each row
BOnly the neighbor — 5 is bigger so it should be the number of rows
CBoth are right — you can describe the same array by its rows or its columns and get the same total
DNeither — the array has 3 rows, so it can only be called a 3-by-5 array
An array can be described by its rows OR by its columns — both descriptions are valid and both give the same total. 3 rows of 5 and 5 rows of 3 both equal 15. This is the commutative property hiding inside a picture.
Question 2 Multiple Choice
You see 12 apples arranged in an array. Which of the following is NOT a valid way to arrange them?
A2 rows with 6 in each row
B3 rows with 4 in each row
C4 rows with 3 in each row
D5 rows with 2 in each row
5 × 2 = 10, not 12 — so 5 rows of 2 cannot be an array of 12. The valid arrangements (2×6, 3×4, 4×3) all produce 12. This tests whether students understand that an array requires equal groups that multiply to the correct total.
Question 3 True / False
A 4-by-6 array and a 6-by-4 array contain different numbers of objects.
TTrue
FFalse
Answer: False
Flipping rows and columns never changes the total. 4 rows of 6 = 24, and 6 rows of 4 = 24. This is the commutative property in visual form — the key insight that makes arrays so powerful for understanding multiplication.
Question 4 True / False
An array is any rectangular group of objects, even if some rows have different numbers of objects.
TTrue
FFalse
Answer: False
An array requires all rows to have the same number of objects (equal groups). A rectangle of randomly placed objects is not an array. The equal-group structure is what makes arrays useful for multiplication and what lets you describe them two different ways.
Question 5 Short Answer
Why can you describe the same array as '3 rows of 4' and also as '4 rows of 3'? What does this reveal about multiplication?
Think about your answer, then reveal below.
Model answer: Because each row of the array lines up perfectly into columns. When you look across, you see rows; when you look down, you see columns. Rotating your perspective doesn't change how many objects there are. This reveals that multiplication is commutative — the order of the two numbers doesn't change the product.
The visual nature of an array makes the commutative property concrete rather than abstract. Students who only memorize '3 × 4 = 4 × 3' without this visual anchor often forget it or fail to apply it when learning their times tables.