Lesson plan: multiplying three numbers using associative property

Multiplying three numbers can seem like a long calculation, but the associative property helps students regroup the factors in a way that makes the work easier. This lesson uses concrete materials, mental strategies and written equations to show that the product stays the same when the grouping changes.

Designed for upper-primary learners, the sequence fits an Australian classroom, tutoring session or homeschool routine. It connects multiplication facts with practical situations, including classroom collections, sporting equipment and prices in Australian dollars.

Learning intention and success criteria

Students will learn that multiplication can be regrouped without changing the product. They will compare equations such as (2 × 3) × 4 and 2 × (3 × 4), then explain why both produce the same answer.

By the end of the lesson, students should be able to multiply three whole numbers, use the associative property to choose an efficient grouping, and describe their reasoning using mathematical language. A suitable success statement is: “I can regroup three factors to make a multiplication problem easier.”

Materials and preparation

Prepare linking cubes, counters, mini-whiteboards, pencils and printed multiplication worksheets. Three colours of counters are useful because they allow students to build equal groups and see how the factors relate to one another.

Before the lesson, select examples suited to the class’s fluency with times tables. Australian teachers can align the activity with the relevant Australian Curriculum v9 content for multiplication and mathematical reasoning, while tutors in Sydney, Melbourne or regional schools can adjust the numbers to match the learners’ current level.

Warm-up with equal groups

Begin with a quick multiplication-facts review. Call out products such as 3 × 4, 5 × 6 and 2 × 8, asking students to show a related fact on a mini-whiteboard. Follow this with a prompt such as, “How could we find 3 × 4 × 2 without multiplying all three numbers in a fixed order?”

Give pairs 3 groups of 4 counters, with 2 counters in each arrangement if appropriate for the class level. Let students build the objects in different groupings and record (3 × 4) × 2 and 3 × (4 × 2). Establish that both expressions equal 24, even though the brackets show a different order of grouping.

Explicit teaching and modelling

Explain that the associative property of multiplication changes the grouping of factors, not their order. Write (a × b) × c = a × (b × c) and model an example such as (5 × 2) × 6 = 5 × (2 × 6). Show that the first grouping gives 10 × 6, while the second gives 5 × 12.

Choose examples where regrouping makes a known fact. For 4 × 25 × 3, students may calculate 4 × 25 first to make 100, then multiply by 3. Clarify that the associative property is different from the distributive property, which breaks a factor into a sum. For an optional extension involving algebraic structure, teachers may refer learners to factoring trinomials after they have secured the basic multiplication idea.

Guided practice with visual models

Work through several examples as a class: (2 × 5) × 7, 3 × (4 × 5), (6 × 2) × 5 and 8 × (5 × 2). Ask students to identify the easiest pair to calculate first and justify their choice. Encourage phrases such as “I grouped 2 and 5 because their product is 10.”

Use arrays, area models or cube towers to represent each equation. A context involving 4 trays with 6 lamingtons in each tray and 3 layers can make the idea memorable, while a football-card collection or school-house display may connect with familiar Australian classroom experiences. Students should record both groupings before calculating the product.

Independent worksheet activity

Provide a worksheet with three parts. In the first part, students match equivalent expressions, such as (3 × 2) × 5 with 3 × (2 × 5). In the second, they insert brackets to create an easier calculation in expressions such as 2 × 5 × 9 and 4 × 25 × 2.

The final part can use short word problems. For example, a community sports club has 3 teams, each with 4 crates, and each crate holds 5 balls. Students write two grouped multiplication expressions and find the total. Include a small Australian-dollar shopping problem, such as 4 packs of 5 stickers costing $2 per pack, while reminding students to distinguish the number of items from the price.

Differentiation and support

For students who need support, keep the factors below 10, provide counters and allow a multiplication chart. Colour-code each factor and use sentence frames: “I grouped ___ and ___ because ___.” A partner can build the model while the other student records the matching equation.

Students ready for a challenge can find several grouping choices for four factors, compare mental strategies and explain why multiplication can be regrouped but not freely rearranged when subtraction or division is involved. Avoid presenting the rule as a trick; ask learners to connect every symbolic equation to a visual or practical model.

Assessment and follow-up

Use observation during guided practice to check whether students understand that the product remains unchanged. Listen for explanations that mention grouping rather than statements that simply say “the answer is the same.” An exit ticket might ask students to solve 6 × 5 × 2 in two ways and circle the most efficient grouping.

Collect the worksheets and identify errors such as changing the order accidentally, adding instead of multiplying, or applying the property to an expression containing division. Store successful practice sheets in the class mathematics folder, following the site’s terms of use when sharing printable resources with families or colleagues. Copyright concerns about copied materials can be checked through the copyright notice.

Print the matching worksheets, gather counters and teach the lesson in a 45–60 minute block. Continue with multiplication games, factor puzzles and short daily fluency practice so students use the associative property confidently in new calculations.