A student multiplies 0 × 9 and answers '9.' What error did she make, and what is the correct answer?
AShe confused multiplication with addition; 0 × 9 = 9 is actually correct
BShe confused 0 × 9 with 1 × 9 = 9; the correct answer is 0, because zero groups of nine contains nothing
CShe forgot to add the exponent; the answer is 90
DShe is correct for even numbers but not for 9, which is odd
0 × 9 means 'zero groups of nine' — no groups at all, so the total is 0. The common error is confusing 0 × 9 with 1 × 9 = 9. The equal-groups meaning resolves the confusion: one group of nine has something in it; zero groups have nothing.
Question 2 Multiple Choice
You have memorized that 7 × 6 = 42. Which other multiplication fact do you automatically know because of this?
A7 × 7 = 49, because multiplication grows by 7 each step
B6 × 7 = 42, because multiplication is commutative — swapping factors gives the same product
C7 + 6 = 13, because multiplication facts relate to addition
D14 × 3 = 42, because you can always double one factor and halve the other
The commutative property means 7 × 6 = 6 × 7. Swapping the factors gives the same product — an array of 7 rows of 6 contains the same number of squares as 6 rows of 7. This single property cuts the work of learning the times table roughly in half: every fact you learn gives you a second fact for free.
Question 3 True / False
The product of 3 × 4 is different from the product of 4 × 3.
TTrue
FFalse
Answer: False
3 × 4 = 4 × 3 = 12. Multiplication is commutative — the order of the factors does not change the product. An array shows why: a 3-by-4 grid and a 4-by-3 grid contain the same number of squares, just oriented differently. This property halves the number of distinct facts that need to be learned.
Question 4 True / False
Understanding why any number times zero equals zero is possible from the equal-groups meaning of multiplication.
TTrue
FFalse
Answer: True
Multiplication means 'this many groups of this size.' Zero groups of any size means you have no groups at all — so the total is zero. Students who understand this don't need to memorize zero facts as a special rule; they can derive them from the meaning of multiplication. This is why understanding-based learning is more durable than rote memorization.
Question 5 Short Answer
How can you use a 'near fact' strategy to figure out 7 × 8 if you don't remember it?
Think about your answer, then reveal below.
Model answer: Use a fact you know — like 7 × 7 = 49 — and add one more group of 7: 49 + 7 = 56. Or use 8 × 8 = 64 and subtract one group of 8: 64 − 8 = 56. You build from a known fact to an unknown one by adding or subtracting exactly one group.
Near-fact strategies are more reliable than trying to recall an arbitrary memorized fact under pressure. They also reinforce the meaning of multiplication — each step up in one factor adds one more group. Students who use these strategies eventually internalize the fact through repeated derivation, which is more durable than rote memorization.