Grade 3What Multiplication Means3.OA.B.5

Grouping Factors

Multiply three numbers in any grouping, like (2 × 5) × 7.

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Sometimes you multiply three numbers, like 2 boxes, each with 5 bags, each with 7 marbles. You have the facts; now the question is which two to multiply first.

A block of cubes: each layer is 2 rows of 5, which is 10 cubes, and there are 3 layers. $(2 \times 5) \times 3 = 10 \times 3 = 30$. Drag to turn it.

Why it works

The block has the same cubes however you count them. Count by layers: $(2 \times 5) \times 3$. Count by the slices standing up: $2 \times (5 \times 3)$. Same block, same 30 cubes. So you may group the factors any way you like, and pick the easiest.

Key idea

When you multiply three numbers, you can multiply any two first: $(2 \times 5) \times 7 = 2 \times (5 \times 7)$. Look for a pair that makes 10.

Worked example: Make a ten

Find $2 \times 7 \times 5$.
  1. Group the 2 and the 5: $2 \times 5 = 10$.
  2. $10 \times 7 = 70$.

Answer: $70$

Worked example: Another grouping

Find $3 \times 2 \times 4$.
  1. $3 \times 2 = 6$, then $6 \times 4 = 24$.
  2. Or $2 \times 4 = 8$, then $3 \times 8 = 24$. Same answer.

Answer: $24$

Common mistake

Adding when you meant to group. $(2 \times 3) \times 4$ is 24, not $2 \times 3 + 4 = 10$. The parentheses only show which two to multiply first; every number is still multiplied.

Practice

Pass with 3 right answers, the last 2 in a row. Each right answer earns 2 XP.

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Sources and further reading

  • Common Core State Standards for Mathematics, Grade 3, 3.OA.B.5 (associative property of multiplication)
  • OpenStax, Prealgebra 2e, section 7.2, Commutative and Associative Properties
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