Questions: Multiplying Two-Digit by One-Digit Numbers

Question 1

To compute 36 × 4, a student thinks: '30 × 4 = 120, and 6 × 4 = 24, so 36 × 4 = 144.' Why does this method always work?

• A. It works because you can always split the second number in multiplication
• B. It works because multiplying by 10 is easy
• C. It works because 36 = 30 + 6, and multiplying each part by 4 then adding gives the same result as multiplying 36 × 4 directly — this is the distributive property
• D. It works by coincidence for this particular problem

Answer: (C)

This is the distributive property: a × (b + c) = (a × b) + (a × c). Because 36 = 30 + 6, computing 4 × 30 and 4 × 6 separately then adding gives the correct total. This is not a trick — it works for any two-digit multiplication because any two-digit number can be decomposed into its tens and ones. Understanding why it works (not just how) is what makes it extendable to larger numbers.

Question 2

A student computes 47 × 6 by thinking '40 × 6 = 240' but writes 240 as her final answer, forgetting to add anything else. What error did she make?

• A. She multiplied by the wrong number
• B. She forgot to also multiply the ones digit (7 × 6 = 42) and add it to the tens product
• C. She should have multiplied 47 by 60 instead of 6
• D. She rounded 47 incorrectly

Answer: (B)

When decomposing 47 into 40 + 7, you must multiply BOTH parts by 6: 40 × 6 = 240 AND 7 × 6 = 42. The final answer is 240 + 42 = 282. Computing only the tens partial product (240) and stopping is the most common error with this strategy — the ones partial product is always required.

Question 3

24 × 3 = (20 × 3) + (4 × 3) because any two-digit number can be broken into its tens and ones, and each part can be multiplied independently.

• T. True
• F. False

Answer: True

Correct. 24 = 20 + 4, so 24 × 3 = (20 + 4) × 3 = (20 × 3) + (4 × 3) = 60 + 12 = 72. This decomposition works because our number system is built on place value, and the distributive property guarantees that multiplying each part then adding equals multiplying the whole.

Question 4

Decomposing a two-digit number into tens and ones before multiplying is mainly necessary when you can seldom remember the answer.

• T. True
• F. False

Answer: False

Decomposing is not a memory workaround — it is the fundamental strategy that scales to all larger multiplications. Understanding why it works (place value + distributive property) is more important than any specific answer, because the identical method extends to two-digit by two-digit, three-digit by two-digit, and beyond. The strategy IS the mathematical understanding.

Question 5

Explain why multiplying 34 × 7 gives the same answer whether you compute it directly or by decomposing as (30 × 7) + (4 × 7).

Model answer: 34 = 30 + 4, and the distributive property states that a × (b + c) = (a × b) + (a × c). So 34 × 7 = (30 + 4) × 7 = (30 × 7) + (4 × 7) = 210 + 28 = 238. Both methods multiply the same total by 7 — the decomposition just breaks it into two simpler pieces using 34's place-value structure. Because the parts add up to the whole number, their products add up to the whole product.

The distributive property is the mathematical guarantee that this always works. You are not approximating — you are computing the exact same multiplication in two steps instead of one. This is why the decomposition strategy is not just a shortcut but a genuine method that produces exact results for any two-digit multiplication.