Multiplication practice that makes sense

Multiplication describes equal groups. In 3 × 4 = 12, there are 3 groups, 4 in each group, and 12 altogether. The factors are 3 and 4; the product is 12.¹

1. Make equal groups. Name the number of groups, how many are in each group, and the total.
2. Connect the representations. Show the same idea with objects, repeated addition, an array, words, and an equation.
3. Use known facts. Build with 0, 1, 2, 5, and 10 before deriving less familiar facts.
4. Split and recombine. Break an array into facts the learner knows, then add the results.

Repeated addition and arrays help show the relationship. They are representations of multiplication, not separate rules to memorize.²

Secure 0, 1, 2, 5, and 10 first. Build 3, 4, and 6 from those anchors, then derive 7, 8, and 9. The aim is not one required trick: the learner should choose a valid route and explain why it keeps the same product.³

Ways to work out a fact

Use an anchor fact6 × 5 = 30

Make 2, 5, and 10 facts readily available; use them as parts of harder facts.

Turn it around7 × 4 = 4 × 7

Swap the factor order when the turn-around fact is easier. The array turns, but the total stays the same.

Double a known fact4 × 6 = 2 × 6 + 2 × 6

Build fours by doubling twos, and build some eights by doubling fours.

Split and recombine7 × 8 = 5 × 8 + 2 × 8

Split one factor into useful parts, solve the two smaller products, then put the whole array back together.

Rehearse every fact within 100, but do not rely on reciting one table in a fixed order. Retrieval becomes useful when the order, representation, and unknown can change.

1. Build it. Show a new or uncertain fact with equal groups or an array and name what each factor means.
2. Derive it. Use an anchor, turn-around, double, or split. Say which known fact did the work.
3. Mix it. Practise facts out of order and mix direct products with missing factors and one-step situations.
4. Return to it. Revisit the same facts in later sessions. If an error returns, rebuild the relationship before repeating the answer.

Add brief timing only after recall is accurate and the learner can still explain or rebuild a sampled fact. Speed alone is not mastery.³⁴

Treat multiplication and division as the same relationship with a different unknown. If 6 groups of 4 make 24, the related division questions ask for the group size or the number of groups.¹

Mix missing-product, missing-factor, sharing, and grouping questions. Ask what the unknown means before choosing an operation.

After facts are reliable, extend the split-and-recombine idea with place value. Larger multiplication should still show where each place-value result comes from.¹

Use an area model or expanded form before compressing the reasoning into a written algorithm.

Use one prompt, then leave the next mathematical decision with the learner.

• How many equal groups are there?
• How many are in each group?
• Can you show the same fact with an array or equation?
• Which fact you already know could help?

New to multiplication: start with equal groups and arrays. Building facts: use anchors, turn-around facts, doubles, and split arrays. Reviewing: mix products, missing factors, related division facts, and one-step situations.

1. Common Core State Standards Initiative. Mathematics Standards. 2010.
2. U.S. Institute of Education Sciences, What Works Clearinghouse. Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades. March 2021.
3. National Council of Teachers of Mathematics. Procedural Fluency in Mathematics. Position statement.
4. U.S. Institute of Education Sciences, What Works Clearinghouse. Organizing Instruction and Study to Improve Student Learning. September 2007.