Lesson Plan: Multiplying Three Numbers with the Associative Property

Multiplying three numbers becomes easier when students learn that the grouping of factors can change without changing the product. This lesson introduces the associative property of multiplication through practical models, mental strategies, and written calculations suitable for upper-primary learners.

The central idea is simple: when multiplying (a \times b \times c), students may calculate ((a \times b) \times c) or (a \times (b \times c)). The factors stay in the same order, while the brackets show which pair is calculated first. This distinction helps children choose an efficient method rather than treating every multi-step problem as a long calculation.

The activities suit Australian classrooms, tutoring sessions, and home learning. They can be adapted for students working towards the Australian Curriculum achievement standards in Years 4–5, with familiar examples involving Australian dollars, footy cards, school supplies, and shopping quantities.

Learning intention and success criteria

Tell students that they will use the associative property to multiply three numbers and explain why different groupings produce the same answer. Write the two forms on the board: ((3 \times 4) \times 5) and (3 \times (4 \times 5)).

A successful student can build or draw a multiplication model, solve both groupings accurately, and describe the result using mathematical language. Encourage phrases such as “I grouped 4 and 5 first because they make 20” and “The factors did not change, only the grouping changed.”

Materials and preparation

Prepare counters, linking cubes, grid paper, mini-whiteboards, multiplication fact cards, and pencils. A classroom display showing familiar Australian contexts, such as three packs of six footy cards with four cards in each set, can make the abstract rule more accessible.

For printable practice, teachers and families can select multiplication worksheets that match students’ recall of times tables and gradually add three-factor calculations. Choose pages with visual arrays for developing learners and number sentences for students who are ready for less scaffolding.

Warm-up with known facts

Begin with a quick times-table routine. Call out facts such as 2 × 5, 4 × 5, 3 × 6, and 7 × 10, asking students to show answers on mini-whiteboards. In an Australian classroom, this can fit neatly into a short first session before the morning break or mathematics block.

Next, present 2 × 3 × 5 without brackets. Ask students how they might solve it. Invite several methods, then compare ( (2 \times 3) \times 5 = 6 \times 5 = 30) with (2 \times (3 \times 5) = 2 \times 15 = 30). Do not rush to name the property before students notice the pattern.

Build the idea with models

Use counters to make three equal groups, then rearrange them into a more convenient structure. For example, represent 2 × 3 × 4 as two groups of three groups of four. Students can first combine the 3 and 4 to make 12, then double it. They can also combine 2 and 3 to make 6, then multiply by 4.

Explain that multiplication describes the total number of equal groups or objects, while the brackets identify the first calculation. The associative property does not mean that the order of the factors can be changed; changing order is connected with the commutative property. A short comparison prevents students from confusing the two rules.

Guided practice and discussion

Work through examples such as ((5 \times 2) \times 6), (5 \times (2 \times 6)), and ((4 \times 5) \times 3). Ask students to circle the pair they would calculate first and justify their choice. Look for pairs that make 10, 20, 25, or 100, as these often support efficient mental multiplication.

Place students in pairs and give each pair three factor cards. They form two valid bracketed equations, solve both, and check whether the products match. Use Australian dollar contexts, such as three trays holding four rows of five $1 tokens, while reminding students that the arithmetic represents equal groups rather than actual shopping advice.

Independent application and differentiation

Students complete a short set of calculations, beginning with concrete examples and moving towards word problems. One task might involve six boxes, each containing three packets of four stickers; another could describe four teams with five players holding two drink bottles each. Include spaces for a drawing, a number sentence, and a written explanation.

For students needing support, provide arrays, counters, multiplication charts, and partially completed brackets. Pairing a confident student with a learner can encourage mathematical talk, but ensure both students complete their own reasoning. For extension, ask learners to find three different grouping choices when factors repeat, or to explain why (3 \times 4 \times 5) can be solved efficiently without writing every intermediate step.

Australian classroom connections

Use a local market scenario to make the lesson authentic: a stall prepares five baskets, each with three bags containing four apples. Students can calculate the total using either grouping and discuss which pair is easier to multiply. A Melbourne market, Brisbane weekend stall, or Sydney school fundraising event can provide a familiar setting without requiring a real purchase.

For home learning, families might model the same idea with packets from Coles or Woolworths, collections of sports cards, or rows of seedlings in a backyard garden. Keep the language inclusive and focus on equal groups. If a digital activity is added to a home lesson, families may also browse a wider online resource alongside carefully selected mathematics materials.

Assessment and extension

Use an exit ticket with three prompts: solve ((2 \times 5) \times 7), solve (2 \times (5 \times 7)), and explain why the answers are equal. Check whether students understand the role of brackets, can recall the relevant facts, and can communicate the property rather than simply recording two matching products.

For a follow-up lesson during the next school term, connect the associative property with factor pairs, area models, and larger products. Students can design a classroom display featuring efficient calculations, or create a short word problem based on a local sports club, school canteen, or community event. These tasks reinforce multiplication fluency while giving the rule a meaningful purpose.

Print the lesson resources, prepare the counters and fact cards, and begin with a problem students can see and touch. With repeated opportunities to group factors, explain choices, and check products, learners can develop both confidence and flexible multiplication strategies.