A student doesn't know 7 × 6. She draws a 7-by-6 array, splits it into a 7-by-5 and a 7-by-1, and computes 35 + 7 = 42. How does this strategy work?
AShe guessed that the two parts would be close enough to the correct answer
BShe doubled the easier fact 7 × 3 = 21 to get 42
CShe used the break-apart strategy: the two smaller arrays contain all the same tiles as the original, so their products sum to the correct answer
DShe applied commutativity to rewrite 7 × 6 as 6 × 7 and then used a known fact
Splitting the array does not change the total number of tiles — the 42 tiles in the original 7×6 array are now in two groups: 35 and 7. Adding the partial products recovers the full total. The break-apart strategy works because the split is a partition of the same tiles, not a change to the problem. This is the concrete foundation of the distributive property.
Question 2 Multiple Choice
What does rotating a 3-by-8 array 90 degrees to produce an 8-by-3 array demonstrate?
A3 × 8 produces a different answer than 8 × 3, because the rows and columns switched
B3 × 8 = 8 × 3, because rotation preserves the total number of tiles
CArrays can only be used for facts up to 5 × 5 before rotation causes errors
DRotating an array doubles its area
Rotating the array changes its orientation but not its tile count. The exact same tiles that were arranged in 3 rows of 8 are now in 8 rows of 3 — still 24 tiles. This is a visual proof of the commutative property of multiplication: the order of the factors doesn't change the product.
Question 3 True / False
You can find 6 × 9 by splitting a 6-by-9 array into a 6-by-5 and a 6-by-4 array, then adding the two partial products.
TTrue
FFalse
Answer: True
6 × 5 = 30 and 6 × 4 = 24; 30 + 24 = 54 = 6 × 9. The split is valid because 5 + 4 = 9 — the two parts account for all nine columns of the original array. Any split that adds back to the original factor produces the correct answer.
Question 4 True / False
When you split an array into two smaller arrays to find a product, the total number of tiles changes — that is why you need to add the two partial products at the end.
TTrue
FFalse
Answer: False
The total number of tiles does not change. Splitting the array partitions the existing tiles into two groups — no tiles are added or removed. You add the partial products because the same total has been divided into two parts, and adding the parts recovers the whole. The split is a reorganization, not a change in quantity.
Question 5 Short Answer
Explain how splitting an array into two smaller arrays helps you find a multiplication fact you haven't memorized. Why does this strategy always give the correct answer?
Think about your answer, then reveal below.
Model answer: When you don't know 4 × 7, you can split a 4-by-7 array at any column, say after column 5, creating a 4-by-5 (= 20) and a 4-by-2 (= 8). Adding 20 + 8 = 28 gives the correct answer. The strategy always works because the split partitions the original tiles into two groups without adding or removing any. The two partial products together account for every tile in the original array, so their sum equals the original product.
This strategy is the concrete, visual form of the distributive property: 4 × 7 = 4 × (5 + 2) = 4 × 5 + 4 × 2. Students who understand why this works — not just how to do it — are ready to apply the same reasoning to multi-digit multiplication and eventually to algebraic expressions.