Teaching multiplication through smart grouping

The associative property helps students see that multiplication can be rearranged into useful groups without changing the answer. For example, (2 \times (3 \times 4)) has the same value as ((2 \times 3) \times 4). The factors are not being reordered; the brackets simply show which factors are multiplied first.

For elementary learners, this idea becomes easier when it is connected to arrays, equal groups and familiar multiplication facts. A carefully sequenced lesson can move from hands-on objects to drawings, number sentences and independent practice, helping children use grouping as a calculation strategy rather than memorise a rule without meaning.

Build the idea with equal groups

Begin with counters, blocks or small classroom objects. Ask students to make three groups of four, then arrange those groups into two larger groups of three groups. The total number of objects remains twelve, even though the grouping looks different. This gives learners a visual reason that (2 \times (3 \times 4)) and ((2 \times 3) \times 4) produce the same result.

Australian classrooms can use locally familiar examples, such as trays of eight mini fruit pieces, packs of six pencils or groups of footy cards. In a Year 3 class in Brisbane or Adelaide, students might describe the same collection using different layers of groups before recording the matching equations.

Separate grouping from changing order

Students often confuse the associative property with the commutative property. Changing (3 \times 5) to (5 \times 3) changes the order of the factors and demonstrates commutativity. Changing (2 \times (3 \times 5)) to ((2 \times 3) \times 5) changes the grouping while keeping the factor order the same.

Display both ideas side by side. Use one colour for the factors and another for the brackets, then ask students to explain exactly what changed. A sentence frame such as “The factors stay in the same order, but…” encourages precise mathematical language and helps prevent students from treating every multiplication shortcut as the same property.

Use arrays and area models

Arrays offer a strong bridge between concrete materials and symbolic notation. Draw a rectangle with 24 dots and partition it into groups that represent (2 \times (3 \times 4)). Students can first identify the inner group of 12, then multiply that group by 2. Redrawing the same array as ((2 \times 3) \times 4) shows why both calculations reach 24.

Area models work especially well with older primary students. A rectangle measuring 2 by 3 by 4 can be discussed as layers, rows and columns, even though the model is represented on paper. In Sydney or Melbourne, this could connect with classroom discussions about floor tiles, garden beds or rectangular sports equipment storage while keeping the focus on factors and efficient calculation.

Link grouping to known facts

The associative property is most useful when it makes a difficult calculation easier. For instance, (4 \times 25 \times 3) can be grouped as (4 \times (25 \times 3)), or as ((4 \times 25) \times 3), depending on the known facts. In many cases, students can create a multiple of ten or one hundred, then finish the calculation mentally.

Give learners examples such as (2 \times 5 \times 7), (4 \times 3 \times 5) and (8 \times 2 \times 5). Encourage them to circle pairs that create ten, twenty or one hundred. In an Australian money context, groups of five-cent and two-dollar coins can help illustrate friendly factors, though the calculation should remain mathematical rather than becoming a lesson in currency.

Design printable practice with purpose

A useful worksheet should include a progression rather than a random collection of equations. Start with matching activities in which students connect two equivalent grouped expressions. Continue with missing brackets, such as (3 \times (2 \times 6) = (3 \times \square) \times 6), and finish with word problems requiring students to choose an efficient grouping.

Include visual prompts for learners who need support and extension questions for confident mathematicians. A puzzle might ask students to find three different groupings for 2, 3 and 5, while a calculation task could involve larger factors and multi-digit multiplication. Teachers and families using downloadable resources should also review the site’s terms of use before copying or sharing materials.

Check understanding through explanation

Correct answers do not always show genuine understanding. Ask students to draw a model, circle the groups and explain why the total remains unchanged. A student who writes (6 \times (2 \times 5) = (6 \times 2) \times 5) should be able to describe the first expression as six groups of ten and the second as twelve groups of five.

Use exit slips with one calculation and one explanation prompt. For example: “Choose the easier grouping for (3 \times 4 \times 25), solve it and explain your choice.” During Australian school terms, short checks like this can fit into a regular maths warm-up, allowing teachers to identify whether students need more work with brackets, factors or multiplication facts.

Print a small set of grouping cards, gather counters and invite students to test several expressions aloud. Then browse Gialdini Worksheets for multiplication worksheets, puzzles and practice activities that can reinforce the same reasoning at school, at home or in a homeschool setting.