Questions: Distributive Property of Multiplication
5 questions to test your understanding
Score: 0 / 5
Question 1 Multiple Choice
A student uses the distributive property to solve 6 × 7 by writing: 6 × (3 + 4) = (6 × 3) + (6 × 4) = 18 + 24 = 42. Why does splitting 7 into 3 + 4 not change the answer?
AIt works only because 3 and 4 happen to be easy numbers to multiply by 6
BThe 6 rows apply to every column in both parts — splitting the columns doesn't change the total number of dots in the array
CMultiplication distributes only when the two parts add up to an even number
DIt works as a coincidence for small numbers but would fail for larger ones
The distributive property works because of what multiplication means in an array: 6 × 7 is 6 rows of 7 columns. Drawing a line after column 3 gives two rectangles — a 6×3 and a 6×4 — but all 6 rows still span both sections. The total dots don't change when you draw the dividing line, so (6×3) + (6×4) = 6×7 exactly. This works for any numbers and any split.
Question 2 Multiple Choice
A student is unsure of 8 × 9 but knows 8 × 5 = 40 and 8 × 4 = 32. Which expression correctly applies the distributive property?
The distributive property splits one factor into a sum: 9 = 5 + 4, so 8 × 9 = 8 × (5 + 4). Then 8 multiplies each part separately: (8 × 5) + (8 × 4) = 40 + 32 = 72. Option A incorrectly adds to both factors. Option C incorrectly multiplies the partial products together. Option D has no valid structure. The key move is always: outer factor × (part₁ + part₂) = (outer × part₁) + (outer × part₂).
Question 3 True / False
You can split a factor any way you like when using the distributive property, and you will always get the same final answer.
TTrue
FFalse
Answer: True
This is one of the most powerful features of the property. To find 7 × 8, you can split 8 as (5+3), (4+4), (2+6), or (7+1) — all give 56. The choice of split affects which partial facts you use, but the answer is always the same. A flexible student chooses the split that uses facts they know best.
Question 4 True / False
The distributive property is a special trick that mainly applies to multiplication in 3rd grade and doesn't connect to anything in later math.
TTrue
FFalse
Answer: False
The distributive property is one of the most foundational ideas in all of mathematics. It reappears when multiplying two-digit numbers (24 × 3 = (20+4) × 3 = 60+12 = 72), in algebra when expanding expressions like (x+5)(x+2), and throughout higher mathematics. The rectangle-splitting model learned in 3rd grade is exactly the same geometric intuition that underlies multiplication at every level.
Question 5 Short Answer
Describe a 4 × 7 array split into two rectangles to show why 4 × 7 = (4 × 3) + (4 × 4). What does the split make visible about why the property works?
Think about your answer, then reveal below.
Model answer: Imagine a rectangle with 4 rows and 7 columns (28 dots). Draw a vertical line after the 3rd column, creating a 4×3 rectangle (12 dots) and a 4×4 rectangle (16 dots). Together: 12 + 16 = 28 = 4 × 7. The split makes visible that all 4 rows still span both sections — the '4' is not split, only the '7' is. The multiplier applies equally to every part.
The array model shows why the property is not a trick: the number of rows (the outer factor) remains constant across both halves. Splitting the columns never changes how many rows there are, so the total count stays the same. This is the geometric proof of the distributive property, and it works for any rectangular array.