A student computes 47 × 3 by thinking: '40 × 3 = 120, and 7 × 3 = 21, so 120 + 21 = 141.' Which property makes this approach valid?
ACommutative property — swapping the order of 47 and 3
BDistributive property — splitting 47 into (40 + 7) and multiplying each part by 3
CAssociative property — regrouping the factors in a different order
DNo property — this only works by coincidence for this particular problem
The distributive property states that a × (b + c) = (a × b) + (a × c). Here, 3 × 47 = 3 × (40 + 7) = (3 × 40) + (3 × 7) = 120 + 21 = 141. This property is what makes the decomposition-by-place-value strategy reliable — it guarantees that splitting the two-digit number and multiplying each part gives the same result as multiplying the whole number.
Question 2 Multiple Choice
A student computes 25 × 4 as follows: '2 × 4 = 8, then 5 × 4 = 20, so the answer is 820.' What mistake did the student make?
AThey should have multiplied the ones digit before the tens digit
BThey treated the digit 2 as just 2, but it is in the tens place and represents 20 — so 20 × 4 = 80, not 8, giving (80 + 20) = 100
CThey cannot use this strategy unless they first draw an area model
DThe answer 820 is actually correct
This is the most common mistake when applying the distributive strategy: forgetting place value. In 25, the digit 2 is in the tens place, representing 20 — not 2. So 25 × 4 = (20 + 5) × 4 = (20 × 4) + (5 × 4) = 80 + 20 = 100. The area model makes this concrete: the left rectangle is 20 units wide, not 2 units wide.
Question 3 True / False
To multiply 63 × 4, you can break it apart as (60 × 4) + (3 × 4) and then add the results.
TTrue
FFalse
Answer: True
Yes — this is a direct application of the distributive property with place-value decomposition. 63 = 60 + 3, so 63 × 4 = (60 × 4) + (3 × 4) = 240 + 12 = 252. Breaking into tens and ones always works for any two-digit number because every two-digit number equals (tens value) + (ones value).
Question 4 True / False
In the area model for 34 × 5, the total area equals 34 + 5 = 39.
TTrue
FFalse
Answer: False
Adding the two dimensions gives the sum, not the product. The area of a rectangle = length × width. The area model for 34 × 5 shows a rectangle divided into a 30 × 5 section (area = 150) and a 4 × 5 section (area = 20). Total area = 150 + 20 = 170. Area represents multiplication, not addition.
Question 5 Short Answer
Explain in your own words why you split a two-digit number into tens and ones before multiplying, and what mathematical property makes this valid.
Think about your answer, then reveal below.
Model answer: You split a two-digit number into tens and ones because every two-digit number is a sum of its tens value and ones value (e.g., 37 = 30 + 7). The distributive property guarantees that you can multiply each part separately and add the results: 37 × 6 = (30 × 6) + (7 × 6) = 180 + 42 = 222. Each individual multiplication then involves a fact you already know (like 7 × 6) or a simple tens calculation (like 30 × 6 = 180).
The distributive property is what makes all multi-digit multiplication possible — it is also the foundation of the standard algorithm. When you 'carry' in vertical multiplication, you are performing the same decomposition in a compressed form. Understanding the area-model method first makes the algorithm understandable rather than a memorized procedure.