Combining arrays and the commutative property in one lesson

Teaching multiplication through visual arrays gives young learners a concrete picture of what multiplication actually means, while the commutative property unlocks a shortcut that halves the work of memorising times tables. When these two ideas are taught side by side, students start to see multiplication as a flexible, logical system rather than a long list of facts to rote learn. For teachers in primary classrooms from Sydney to Perth, weaving both concepts into a single session creates stronger connections and faster recall.

This lesson plan is designed for upper Key Stage 1 and lower Key Stage 2 classrooms, mapping neatly to the Australian Curriculum: Mathematics content descriptors for Years 2, 3, and 4. It works equally well in homeschool settings, after-school tutoring, or maths intervention groups. The structure moves from concrete drawing to abstract reasoning, and includes printable resources that can be photocopied straight from the website.

Learning objectives and curriculum links

The Australian Curriculum expects Year 2 students to represent multiplication as repeated addition and to recognise the relationship between the number of rows and columns in an array. By Year 3, learners are extending this to the commutative property and applying it to solve problems involving two-digit numbers. By the end of Year 4, the expectation is that students fluently recall multiplication facts up to 10 times 10 and use them in real-world contexts.

For a single combined lesson, the explicit objectives are: draw an array from a given multiplication sentence, recognise that reversing the rows and columns still gives the same total, and use that recognition to reduce the load of memorising times tables. A useful success criterion is whether a student can independently say "I already know 4 times 6, so I know 6 times 4" after completing the activity.

Materials and preparation

Gather a set of counters, square tiles, or small cubes for the hands-on phase. Print an array template sheet with ten blank grids, each marked with rows on one side and columns on the other. Prepare a short list of multiplication sentences for the warm-up, ideally a mix that includes facts above and below the students' current fluency level. If teaching in Melbourne during a wet winter recess, indoor floor tape can replace outdoor space, allowing learners to physically stand inside the rows of their array.

A solid printable collection of multiplication worksheets provides a reliable starting point for the written component of the lesson. These sheets allow students to draw their own arrays before moving to abstract number sentences, and they reinforce the visual link that supports long-term retention. Photocopy enough copies so that each learner can work through at least three different fact families.

Introducing the array model

Begin by writing 3 × 4 on the board and asking students what it means. Translate the sentence into three groups of four, then challenge them to draw it. Most learners will naturally sketch three rows of four dots, and this becomes the bridge to defining an array as a rectangular arrangement of objects in equal rows and equal columns. Label the rows and columns clearly so the vocabulary is anchored from the start.

Move quickly to a second example, 5 × 2, and ask the class to draw it on grid paper. Some will draw five rows of two, others will draw two rows of five. This variation is the gateway into the next phase. Before discussing it, ask the class to count the total in both drawings to confirm they match. This quiet setup plants the seed for the commutative discovery without giving the property away too early.

Uncovering the commutative property

Now place two array drawings side by side, such as 3 × 4 and 4 × 3, and ask the class what they notice. Guide them toward the observation that the first array can be rotated 90 degrees to look like the second, and that the total number of dots stays the same. Frame the rule clearly: changing the order of the factors does not change the product. Record this on the board as a formal statement that students can refer back to during practice.

Run a quick verification challenge using counters. Each pair of students chooses two single-digit numbers between two and nine, builds both arrays on a flat surface, and confirms that the totals are equal. Australian classrooms often love a small competition element, so timing the verification activity can add energy without detracting from the maths. The hands-on proof cements the concept in a way that simply stating the rule cannot.

Practice activities and differentiation

Once the property is established, learners can apply it to a fluency-building task. Provide a list of multiplication sentences covering the six and seven times tables, with each fact paired with its flipped version. Students solve one half and record the matching fact alongside it. Year 3 students working towards NAPLAN readiness benefit from this pattern because it reduces the number of unique facts they need to memorise.

Word problems grounded in familiar Australian settings strengthen the connection between the property and real reasoning. A Year 4 problem might ask how many seats are in a Brisbane train carriage with eight rows of six seats, then extend by asking how many seats would fit if the carriage were rearranged into six rows of eight. Differentiation can include extension tasks that combine arrays with multi-digit multiplication, such as using a 12 × 15 array to introduce distributive reasoning. Finish the lesson with a quick exit ticket: each student writes one multiplication fact they already knew and one new fact the lesson helped them unlock.

Browse the full library of multiplication worksheets to find extension sheets, missing-factor puzzles, and times-table review packs that align with this lesson. Teachers across Adelaide, Hobart, and regional Queensland use these resources as ready-made follow-up work for the next session, ensuring that the combined lesson on arrays and the commutative property is reinforced rather than forgotten by Friday.