You forget 8 × 7. Using the doubling strategy, which known fact gives you the most direct path to the answer?
A4 × 7 = 28, then double it to get 56
B8 × 10 = 80, then subtract 8 + 8 + 8
C8 × 5 = 40, then add 8 + 8 + 8
DSkip-count by 8 seven times
The doubling strategy says the 8s are the 4s doubled: since 4 × 7 = 28, then 8 × 7 = 56 (double 28). This requires knowing one simpler fact and performing one doubling — far faster than skip-counting. Options C and D work but are slower and more error-prone. Option B requires subtracting three 8s, which is harder than simply doubling 28.
Question 2 Multiple Choice
A student uses the 9s digit pattern: 'The tens digit is one less than the factor, and the digits add up to 9.' She is solving 9 × 6. What is her answer?
A54, because the tens digit is 5 (one less than 6) and 5 + 4 = 9
B63, because the tens digit is 6 and 6 + 3 = 9
C45, because the tens digit is 4 and 4 + 5 = 9
D56, because 9 is close to 10 and 10 × 6 = 60
For 9 × 6: the tens digit is one less than the factor 6, which is 5. The ones digit must make the sum equal 9, so 9 − 5 = 4. Answer: 54. Option B applies the pattern incorrectly (using the factor itself as the tens digit). This pattern works for all 9× facts up to 9 × 9 — it's a reliable shortcut that doesn't require any other fact.
Question 3 True / False
Knowing that 3 × 8 = 24 is enough to immediately find 6 × 8 using the doubling strategy.
TTrue
FFalse
Answer: True
The 6s are the 3s doubled: 6 × 8 = 2 × (3 × 8) = 2 × 24 = 48. This works because multiplication distributes: 6 groups of 8 is the same as two sets of 3 groups of 8. The doubling strategy turns 6s into 3s, 4s into 2s, and 8s into 4s — meaning you can derive many unfamiliar facts from ones you already know.
Question 4 True / False
A student who can usually reconstruct 7 × 8 by skip-counting has achieved fluency with that fact.
TTrue
FFalse
Answer: False
Fluency means automatic retrieval in 3–5 seconds — not reconstruction. Skip-counting by 7s to reach 56 takes considerably longer and demands significant working memory. Fluency matters because multiplication facts are used constantly inside larger procedures. If a student must reconstruct a basic fact mid-problem (e.g., during long division), it consumes cognitive resources that should go toward understanding the larger concept. The goal is instant recall, not the ability to derive.
Question 5 Short Answer
What is 7 x 8? Show how you could figure it out if you forgot the answer.
Think about your answer, then reveal below.
Model answer: 7 x 8 = 56. One way to figure it out: if you know 7 x 7 = 49, just add one more 7 to get 56. Another way: know that 4 x 8 = 32 and double it isn't right (that gives 64 which is 8 x 8), so instead use 7 x 4 = 28 and double it to get 56, since 7 x 8 = 7 x (4 x 2) = (7 x 4) x 2 = 28 x 2 = 56.
The best approach is to just know that 7 x 8 = 56 by heart. But when you forget, strategies help: you can build from a nearby fact you do know (like 7 x 7 = 49 + 7) or use doubling (7 x 4 = 28, doubled = 56). The goal is eventually to recall all multiplication facts automatically so you do not have to figure them out each time.