A student cannot immediately recall 7 × 8. Which strategy best uses known facts to derive the answer?
ACount by 7 eight times from 0: 7, 14, 21, 28, 35, 42, 49, 56
BUse 7 × 7 = 49 (a known fact) and add one more group of 7: 49 + 7 = 56
CWrite 7 × 8 on a flashcard and memorize it by repetition
DSkip it for now — harder facts cannot be derived from easier ones
The near-facts strategy uses a known fact as a launching point. 7 × 7 = 49 is typically learned before 7 × 8. Since 7 × 8 is one more group of 7 than 7 × 7, add 7: 49 + 7 = 56. This is faster than counting by 7 eight times from scratch, and it builds a connected web of facts. The commutative property also gives a second route: 8 × 8 = 64, subtract one group of 8: 64 − 8 = 56.
Question 2 Multiple Choice
To find 9 × 6 quickly, a student thinks: '10 × 6 = 60, so 9 × 6 = 60 − 6 = 54.' Is this reasoning correct?
ANo — you can only derive 9-facts from other 9-times facts, not from 10-facts
BYes — subtracting one group of 6 from 10 × 6 correctly gives 9 × 6 = 54
CNo — 9 × 6 = 56, not 54
DYes, but this strategy only works when the multiplier is even
The anchor-to-tens strategy is fully valid: 9 × n = 10 × n − n. Here, 10 × 6 = 60, and removing one group of 6 gives 9 × 6 = 54. This works for any multiplier and is especially powerful for the 9s because 10 × n is always trivial. You can verify with the digit-sum check: 5 + 4 = 9, confirming the answer is a multiple of 9. (Note: 56 is 7 × 8, a common confusion.)
Question 3 True / False
The main reliable way to achieve fluency with multiplication facts for 6s through 9s is to memorize each fact individually through repetition, without using strategies.
TTrue
FFalse
Answer: False
Using strategies — near facts, anchor-to-tens, digit-sum patterns — builds more durable and flexible memory than isolated rote repetition. When you derive 8 × 7 from 8 × 8 − 8 = 56, you connect the new fact to one you already know, creating a network of relationships. If you blank on a fact, a strategy gives you a recovery path. Strategies and distributed practice work together; rote repetition alone leaves you with nothing when memory fails under pressure.
Question 4 True / False
Because multiplication is commutative, knowing 6 × 9 = 54 means you automatically also know 9 × 6 = 54.
TTrue
FFalse
Answer: True
The commutative property states that a × b = b × a for all numbers. Every multiplication fact has a 'twin' — a version with the factors swapped — that gives the same product. This cuts the number of unique facts to learn roughly in half, and it provides two different near-fact entry points for any unknown: to find 8 × 7, you can approach from 8 × 8 − 8 or from 7 × 7 + 7. Both routes are valid because of commutativity.
Question 5 Short Answer
Explain the 'near facts' strategy and show how it can be used to find 8 × 7 without recalling it directly.
Think about your answer, then reveal below.
Model answer: The near-facts strategy uses a known fact one step away from the target. For 8 × 7: since 8 × 8 = 64, and 8 × 7 is one fewer group of 8, subtract: 64 − 8 = 56. Alternatively, since 7 × 7 = 49, and 8 × 7 is one more group of 7, add: 49 + 7 = 56. Both give the correct answer.
Near facts work because multiplication is repeated addition: 8 × 7 is just 8 × 8 with one group of 8 removed, or 7 × 7 with one group of 7 added. This strategy transforms a hard-to-recall fact into a known fact plus simple addition or subtraction. Students who learn strategies are more resilient under pressure than those who rely on rote recall alone — because when memory fails, the strategy still works.