Swapping arrays to show the commutative property of multiplication

Multiplication becomes less abstract when children can see equal groups and move them. A rectangular array of counters, tiles or drawn squares turns two factors into rows and columns. Turning the model changes its orientation without changing the amount being counted.

Teachers can use that physical change to make the commutative property of multiplication visible: swapping the factors gives the same product. This approach supports counting, grouping and early algebra while remaining engaging for hands-on primary classrooms.

Start with a physical rectangle

Give each pair of learners 15 counters and ask them to arrange three rows of five. Draw a similar model on squared paper and label its dimensions with 3 and 5. Invite children to count along each row, then rotate the rectangle a quarter turn or redraw it as five rows of three.

Both models contain the same 15 counters, producing the number sentences (3 \times 5 = 15) and (5 \times 3 = 15). Ask learners to explain what changed. They can say that the order of the groups changed, while the number in each group and the total stayed the same. Introduce the terms factors, product and commutative law as concise labels for ideas they have already demonstrated.

Build a predictable swap routine

Familiar objects can provide useful starting points. Rows of canned soup at Coles or Woolworths, bundles of recycled paper, school shoe racks and stacks of mail all resemble arrays. These examples connect classroom mathematics to the weekly grocery shop and other routines familiar to many Australian families.

Use the same sequence throughout the lesson: build the array, count in one direction, swap the rows and columns, count again, record the matching number sentence and explain the result. Children can use different colours for rows and columns, making the rotation easier to see. Give each learner a role such as arranger, counter, recorder or explainer so that discussion involves more than one student.

Connect rows and columns to counting

The skip-counting rationale provides a useful bridge between arrays and multiplication facts. A row of five supports counting by fives, while a column of three supports counting by threes. The two sequences are different, but both arrive at 15.

Ask learners which counting sequence matches each orientation of the model. This helps them see that swapping factors changes the counting path without changing the total. It also links multiplication to number patterns, place value and multiplication tables. Once the idea is secure, pupils can extend the activity by finding other factor pairs that produce the same product.

Match the task to the learner

Australian Curriculum Mathematics provides national expectations, while state and territory education authorities implement the curriculum through their own education legislation, policies and school resources. In cities such as Sydney and Melbourne, class groupings, access to physical materials and the pace of instruction can vary considerably, so the same mathematical idea may need different supports.

Younger learners can arrange equal-group cards before drawing an array. Pupils who need additional scaffolding can trace each row with a finger and match it to a skip-counting sequence. Older students can record several examples, include a missing factor or represent the same product as an area model. Home-learning versions can use buttons, pasta shapes, coins or squares on squared paper.

Correct common misunderstandings

A frequent error is counting only the border of the array and obtaining 8 instead of 15. Other pupils write (3 + 5 = 8), swap only the labels without redrawing the groups, or assume that changing the number of rows must change the product. Ask what the model actually shows before supplying a correction.

Use precise mathematical language. “The order of the factors does not change the product” is more useful than “turning around always works”. Distinguish multiplication from addition and subtraction, where reversing the order changes the answer. Division also needs care: (15 \div 3 = 5), but (3 \div 15) does not equal 5. Clear vocabulary helps children generalise the rule correctly.

Assess reasoning rather than guessing

Use a short sequence of checks rather than relying on a single worksheet score. Ask pupils to draw the original array, draw it with rows and columns exchanged, write the two number sentences and give a brief oral explanation. Their drawing and language reveal whether they understand the unchanged total or have simply memorised two facts.

A train carriage with three rows of five seats offers another familiar setting, while rows of market produce or bottles on a pantry shelf can provide spontaneous examples. For a rotation-themed follow-up, the stellar spins update offers a playful extension before pupils return to explaining the maths.

Print, swap and explain in one lesson

Prepare an array-swapping worksheet that includes a drawing space, two number sentences and a short explanation prompt. Gialdini Worksheets also provides printable lesson plans, quizzes, puzzles and multiplication practice that can support the same learning goal. For additional material, compare the resources available through Jamb Inc before deciding what to print.

Download an array-swapping worksheet from Gialdini Worksheets, print one for each pair and begin with a three-by-five model. Have the learners exchange the rows and columns, record both number sentences and finish by explaining why the total remains fifteen.