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Building Multiplication from Known Facts
Multiplication becomes much more flexible when students learn to build new facts from facts they already know. Instead of treating multiplication as a long list of facts to memorize, students can use patterns, visual models, and number relationships to make problems easier.
In this topic, we look at several strategies for multiplication. Arrays help students see multiplication as rows and columns, making it easier to connect multiplication to equal groups, area, and the properties of operations. For example, an array can show why 4 × 6 and 6 × 4 have the same total, even though the groups are arranged differently.
We will also explore strategies such as halving and doubling and using the properties of multiplication. For example, a student might solve 16 × 5 by thinking, “Half of 16 is 8, and double 5 is 10, so 16 × 5 is the same as 8 × 10.” Students can also use the distributive property to break apart a difficult fact, such as thinking of 7 × 8 as 5 × 8 plus 2 × 8.
As future teachers, it is important to help students see multiplication as something they can reason about. These strategies support number sense and help students understand why multiplication facts work, not just memorize answers.
Student Learning Goals
By the end of this topic, students should be able to:
Use arrays to represent multiplication problems.
Explain how known facts can help solve unknown facts.
Use halving and doubling to create easier multiplication problems.
Apply multiplication properties to transform problems.
Break apart multiplication problems using the distributive property.
Compare different strategies for solving the same multiplication fact.
Explain how flexible multiplication strategies support student understanding.
Key Vocabulary
Array - An arrangement of objects in equal rows and columns.
Known Fact - A multiplication fact a student already knows and can use to solve another fact.
Halving and Doubling - A strategy where one factor is cut in half and the other is doubled to make an easier problem.
Commutative Property - Changing the order of the factors does not change the product.
Associative Property - Changing how factors are grouped does not change the product.
Distributive Property - Breaking apart a factor to make multiplication easier.
Factor - A number being multiplied.
Product - The answer to a multiplication problem.
Example Strategies
Array - 4 × 6 - “I can draw 4 rows of 6 and count the total.”
Commutative Property - 3 × 8 = 8 × 3 - “Changing the order does not change the total.”
Distributive Property - 7 × 6 = 5 × 6 + 2 × 6 - “I can break 7 into 5 and 2.”
Halving and Doubling - 16 × 5 = 8 × 10 - “Half of 16 is 8, and double 5 is 10.”
Using a Known Fact - 6 × 7 - “I know 5 × 7 is 35, so one more group of 7 makes 42.”
Teacher Connection
Flexible multiplication strategies help students move from counting everything to reasoning with structure. A child who solves 9 × 6 by thinking “10 × 6 is 60, so 9 × 6 is 6 less” is using place value, known facts, and compensation. That is powerful mathematical thinking.
As a future teacher, your goal is not to make students use one specific strategy every time. Instead, help students build a toolbox of strategies and choose methods that fit the numbers in front of them.
Helpful teacher questions include:
What fact do you already know that could help?
Can you break one factor apart?
Could an array help you see the problem?
Is there a near fact that would be easier?
Why does your strategy keep the product the same?
Quick Reflection Question
Why might a student who does not know 7 × 8 still be able to reason their way to the answer? What known facts or strategies could help?