DYou cannot — 4s facts must be memorized independently of the 2s facts
The double-doubles strategy: since 4 = 2 × 2, multiplying by 4 is the same as multiplying by 2 twice. So 4 × 9 = 2 × (2 × 9) = 2 × 18 = 36. This reflects a real mathematical relationship. Every 4s fact can be derived by doubling the corresponding 2s fact — which students already know — making the 4s the easiest new fact family to learn.
Question 2 Multiple Choice
A student says '3 × 6 = 9 because 3 plus 6 equals 9.' What is wrong with this reasoning?
ANothing — 3 + 6 = 9, and that is the same as 3 × 6
BMultiplication and addition are different: 3 × 6 means three groups of six, which totals 18, not 9
CThe student should have written 3 × 6 = 3 + 3 + 3 + 3 + 3 + 3 instead
D3 + 6 does equal 9, but 3 × 6 = 36
Addition and multiplication ask different questions. 3 + 6 = 9 means 'combine 3 and 6 into one total.' 3 × 6 = 18 means 'how much in three groups of six?' — which you can verify by counting three rows of six in an array: 6, 12, 18. Confusing the two is especially common with small numbers because the results seem close enough to be plausible. Arrays keep the distinction clear.
Question 3 True / False
Knowing 3 × 8 = 24 automatically tells you that 8 × 3 = 24, cutting the number of unique facts you need to memorize nearly in half.
TTrue
FFalse
Answer: True
This is the commutative property of multiplication: a × b = b × a. Every fact like 3 × 8 has a free twin: 8 × 3. A 10×10 multiplication table has 100 entries, but only about 55 are truly distinct once commutativity is applied. Recognizing this relationship is a significant efficiency gain and reflects a genuine mathematical truth, not just a memory trick.
Question 4 True / False
4 × 7 = 28 can be worked out by adding one more group of 7 to the result of 3 × 7.
TTrue
FFalse
Answer: True
Yes — if 3 × 7 = 21, then 4 × 7 is just one more group of 7: 21 + 7 = 28. This 'build up from a known fact' strategy works because multiplication is repeated addition: 4 groups is 3 groups plus one more. You don't need to memorize facts in isolation — you can always reach an unknown fact from a known one by adding or subtracting a group.
Question 5 Short Answer
Explain why multiplying any number by 4 is the same as multiplying it by 2 twice.
Think about your answer, then reveal below.
Model answer: Because 4 = 2 × 2. Making 4 groups of something is the same as making 2 groups of 2 groups. So you can double the number, then double again. For example, 4 × 6 = 2 × (2 × 6) = 2 × 12 = 24. This works for any number because multiplication is associative: (2 × 2) × n = 2 × (2 × n).
The double-doubles strategy is grounded in the multiplicative structure of 4: it is 2 squared. Mathematically, 4 × n = (2 × 2) × n = 2 × (2 × n). In practice: take the 2s fact you already know and double it. This transforms a potentially difficult fact into two uses of doubling — a skill students already have. Understanding why the strategy works makes it memorable and generalizable to other derived-fact strategies.