A student draws a 4×6 array (4 rows, 6 columns) to solve 4 × 6. Her classmate says '6 × 4 must be a different answer because 6 rows of 4 looks different.' Who is correct?
AThe classmate — the two arrays look different, so they represent different totals
BThe student — both describe the same rectangle rotated 90°, and all 24 items are still there
CNeither — you must count both arrays separately to verify they match
DThe classmate — but only when the rows are longer than the columns
Rotating an array 90° changes its orientation but not the number of items in it. A 4×6 array and a 6×4 array are the same rectangle viewed from different angles. No new items appear; none disappear. This physical demonstration is exactly why multiplication is commutative: 4 × 6 = 6 × 4 = 24. The most tempting wrong answer (A) confuses 'looks different' with 'has a different count.'
Question 2 Multiple Choice
A classroom has 5 rows of desks with 4 desks in each row. Which answer correctly represents the total number of desks?
A5 + 4 = 9
B4 × 5 = 20, reading it as 4 groups of 5
C5 × 4 = 20, reading rows × columns
DBoth B and C, since 4 × 5 and 5 × 4 both equal 20
Both 5 × 4 and 4 × 5 correctly represent the situation — and both equal 20. This is precisely the point of commutativity: you can read the array as 5 rows of 4 (5 × 4) or 4 columns of 5 (4 × 5), and the product is the same. Option A (addition) finds how many total rows and columns exist, not how many desks.
Question 3 True / False
In a 3×5 array (3 rows, 5 columns), there are exactly 3 items in nearly every row.
TTrue
FFalse
Answer: False
In a 3×5 array, there are 3 rows and 5 columns. Each row contains 5 items; each column contains 3 items. So rows have 5 items each, not 3. The misconception comes from confusing the row count with the row size — '3 rows' tells you how many rows there are, not how many items are in each one.
Question 4 True / False
You can read a rectangular array by counting rows or by counting columns, and either way gives you the same total number of items.
TTrue
FFalse
Answer: True
This is the visual proof of commutativity. Whether you count 3 rows of 4 (reading across) or 4 columns of 3 (reading down), the total is always 12. The array is the same arrangement of dots — the total doesn't change based on which direction you count.
Question 5 Short Answer
How does a rectangular array prove that multiplication is commutative — that 3 × 4 equals 4 × 3?
Think about your answer, then reveal below.
Model answer: An array can be read in two directions: 3 rows of 4 gives 3 × 4, and 4 columns of 3 gives 4 × 3. Because it's the same physical arrangement of objects, both calculations count the same dots. Rotating the array 90° turns a 3×4 array into a 4×3 array, but the total number of dots hasn't changed — proving the two products must be equal.
The key insight is that commutativity isn't just a rule to memorize — it has a geometric explanation. The array makes it visible: you can count the same objects in two directions and always arrive at the same number. This understanding matters because it cuts the number of multiplication facts students must memorize nearly in half.