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Properties of Multiplication

A 45-minute Grade 3 lesson on the three properties that turn a handful of known facts into all of them — turning round, regrouping three factors, and breaking one factor apart.

Properties of Multiplication is a free 45-minute Grade 3 maths lesson plan for Common Core standard 3.OA.B.5 — use known facts and patterns to work out harder ones. It is timed across 5 sections, gives the exact wording to use where the wording carries the mathematics, names 3 misconceptions with the teaching move that addresses each, and ends with an exit ticket and a printable worksheet with its answer key.

Grade:
Grade 3
Length:
45 minutes
Standard:
3.OA.B.5

Grade 3 · Operations & Algebraic Thinking

3.OA.B.5

Apply properties of operations as strategies to multiply and divide.

Objective

Students will be able to use the commutative, associative and distributive properties to work out a multiplication fact they do not know.

Students are successful when they can

  • Uses 4 × 7 to answer 7 × 4.
  • Reorders three factors to make the calculation easier.
  • Splits 7 × 8 into 5 × 8 and 2 × 8 and adds the results.

What you need

  • Squared paper for arrays
  • Counters
  • Scissors for cutting arrays apart
  • The printable practice sheet linked at the end of this plan

The lesson

Warm-up — which do you know?

5 min
  1. Give eight facts and ask pupils to mark which they know instantly and which they do not.
  2. The unknown ones are what the properties are for, and pupils should see the list is short.

Teach — three ways to reuse

14 min
  1. Draw a 4 by 7 array on squared paper. Count it as 28. Turn the paper ninety degrees: now it is 7 by 4, still 28.
  2. Nothing was added or removed, so the two facts are one fact. That halves the table to learn.
  3. Now 2 × 3 × 5. Do it as (2 × 3) × 5 = 30, then as 2 × (3 × 5) = 30. Same answer, and one order is easier.
  4. Now the useful one. Draw 7 × 8 and cut the array between the fifth and sixth row.
  5. The two pieces are 5 × 8 = 40 and 2 × 8 = 16. Together 56. Both pieces are facts the class already knows.
  6. Do 6 × 9 the same way, splitting as 5 × 9 and 1 × 9, and let them see it works on anything.
“You do not have to know every fact. You have to know a few and be willing to cut the array up.”

Guided practice — split the hard ones

12 min
  1. Pairs take a fact they marked as unknown in the warm-up and split it into two known facts.
  2. They draw the array and the cut, then write both facts and the sum.
  3. Ask each pair where they cut and why. Cutting at five is usually best, and hearing why beats being told.

Independent practice

11 min
  1. Students complete a properties page.
  2. A pupil who splits 7 × 8 into 7 × 4 and 7 × 4 has understood something real — accept it and ask about splitting the other factor.

Close

3 min
  1. Give 8 × 6 and ask for two ways to get it from easier facts.
  2. Collect a split and a turnaround, and stop.

What goes wrong, and why

Each of these is a reasoning error rather than carelessness. The fix addresses the reasoning.

Splits 7 × 8 into 5 × 8 and 2 × 8 and then multiplies the two results.

Why: The pieces have been produced correctly and then combined with the operation the whole problem is about.

Fix: Go back to the cut array. The two pieces of paper are joined back together, not multiplied. The picture settles it in one move.

Splits both factors: 7 × 8 as 5 × 5 and 2 × 3.

Why: The splitting rule has been over-applied by symmetry, without checking the pieces against the array.

Fix: Draw it. Splitting both ways produces four pieces, not two, and two of them have been left out. The array shows exactly what is missing.

Uses the turnaround for division: 56 ÷ 8 as 8 ÷ 56.

Why: A property of multiplication has been generalised to division because both are written the same way and both are being learned together.

Fix: Test it with counters. Eight shared between fifty-six gives less than one each, and the property visibly fails.

If they are not there yet

  • Secure the 2, 5 and 10 facts first. Every distributive split leans on them, and without them the strategy has nothing to stand on.
  • Use pre-drawn arrays that can be physically cut rather than drawn from scratch.

If they finish early

  • Split a two-digit multiplication: 12 × 6 as 10 × 6 and 2 × 6.
  • Find the easiest order for 4 × 7 × 5.
  • Explain why splitting at five is usually the best cut.

Exit ticket

Work out 6 × 7 by splitting it into two easier facts. Show both.

What to look for: 5 × 7 = 35 and 1 × 7 = 7, summing to 42. Any valid split is fine. Multiplying the two pieces instead of adding them is the error to catch.

Printable practice for 3.OA.B.5

Free to print and copy, answer key included. Each sheet can generate a fresh set of problems for a retake.

Questions about Properties of Multiplication

How long does the Properties of Multiplication lesson take?

45 minutes, split across 5 timed sections: warm-up 5 min, teach 14 min, guided practice 12 min, independent practice 11 min, close 3 min. The minutes are stated per section and add up to the stated length, so the plan can be cut or extended at a section boundary rather than abandoned halfway.

Which standard does Properties of Multiplication teach?

CCSS.MATH.CONTENT.3.OA.B.5 — 3.OA.B.5, Operations & Algebraic Thinking for Grade 3: “Apply properties of operations as strategies to multiply and divide.” In plain terms, use known facts and patterns to work out harder ones.

What do I need to teach this lesson?

Squared paper for arrays, counters, scissors for cutting arrays apart and the printable practice sheet linked at the end of this plan. Everything listed is either ordinary classroom equipment or a free printable from this site — there is nothing to buy and nothing to prepare beyond photocopying.

What mistakes should I expect in this lesson?

Three, each a reasoning error rather than carelessness: splits 7 × 8 into 5 × 8 and 2 × 8 and then multiplies the two results; splits both factors: 7 × 8 as 5 × 5 and 2 × 3 and uses the turnaround for division: 56 ÷ 8 as 8 ÷ 56. The plan gives the thinking behind each one and the teaching move that addresses the thinking rather than the symptom.

How do I know whether the lesson worked?

The exit ticket asks: “Work out 6 × 7 by splitting it into two easier facts. Show both.” 5 × 7 = 35 and 1 × 7 = 7, summing to 42. Any valid split is fine. Multiplying the two pieces instead of adding them is the error to catch. That tells you whether the class needs a reteach or a thirty-second conversation before you plan tomorrow.

Is there a worksheet to go with Properties of Multiplication?

Yes — 2 printable 3.OA.B.5 worksheets, linked at the end of the plan and free like the rest of the site. Each one comes with an answer key and 25 versions, so the practice after the lesson can be a different paper on every desk.

Is Properties of Multiplication free to use?

Yes. The plan, its printables and their answer keys are free to read, print and photocopy for your own class, with no account and no email capture. It was last reviewed on 5 September 2026. Please do not resell it or republish it as your own.

Last reviewed 2026-09-05. Aligned to 3.OA.B.5 of the Common Core State Standards for mathematics.