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One of the foundational concepts students need to understand for strong multiplication skills is distributive property and teaching arrays is the perfect way to do this.
Let’s go through exactly how to teach distributive property with arrays as the foundational step to build effective multiplication strategies.
What is distributive property?
The Distributive Property of Multiplication allows us to break a large multiplication problem into smaller, easier pieces by splitting one of the factors into two parts and multiplying each separately, then adding the pieces back together.
Distributive property, and arrays, help students decompose numbers into more manageable parts, making multiplication more visual, flexible, and intuitive.
How Does the Distributive Property Work?
2(a+4)
. . . which equals . . .
2(a) + 2(4)
. . . which equals . . .
2a + 8
Teaching Multiplication with Arrays
As with almost all teaching, I like to work through the progression of concrete materials and manipulatives, then diagrams, pictures and models, before finally moving onto abstract concepts using numbers and symbols.
This is perfect for an inquiry-style lesson approach with direct teaching coming after to consolidate their learning.
1. Building Arrays with Manipulatives
Start with foam squares, cubes or something similar and an inquiry or investigation type of activity.
Note: There is a strong connection between arrays and repeated addition. This allows those students who aren’t quite ready to multiplicative thinking to still understand and have success with these questions.
INQUIRY QUESTIONs
Give students a simple multiplication question such as 12 x 8 to model. Ask them how they might break it apart to make it easier to find the answer.
Give them time to work in partners or threes to explore possible options, decide on their ‘best’ strategy and be ready to explain their thinking to the class.
Direct Teaching
As a class, share the different ways students chose to break down their array and discuss the advantages of each. Be sure to share a variety of strategies used (especially a couple of strategies that are less efficient).
Look at various ways to break the array. The 12 could be divided into two 6s. Is this efficient? Students that have a solid grasp of their 6 times tables may say yes, but most will agree that it’s not that easy. How about 10 and 2? Most students should be confident multiplying by 10s and 2s so will agree that it is a logical way to break up the 12.
How should arrays be split? Using examples and discussions, we look at all the options when breaking apart an array. Ultimately, there are two ‘best’ ways to divide an array: to use multiplication facts you know well, or according to place value.
More Complex Arrays
Gradually increase the numbers. As numbers get larger, arrays can be broken into more sections. For example, both numbers can be broken down creating 4 quadrants to the array. Alternatively, one number can be broken down into 3 or more smaller numbers such as 254 can become 200 + 50 + 4.
2. Using Models of Arrays
As numbers get larger, and students gain confidence, they’ll start to realize it’s a lot of work to build physical arrays with blocks.
Now it’s time to move from the concrete, hands-on approach to using a model or representation to teaching the distributive property of multiplication.
Drawing Arrays or Area Models on Grid Paper
I like to start with grid paper as it naturally transitions from the manipulatives. Students can use the squares to create a model of the array they would have built.
They label the sides and draw a thicker line to show where they break it apart.
This step also begins to bring in the traditional mathematical representations of the multiplication that is happening.
A multiplication sentence should be written out for each section of the array as well as the addition of those parts at the end.
3. Connect Arrays to Equations
After modeling visually, transition students into using purely numbers. Depending on your students’ age and ability, you can choose to introduce brackets or keep each step as its own equation.
Even in grades that haven’t learned order of operations yet, I do still show the brackets and explain they are just separating out each part (think of each set of brackets like one pile of cubes or section of the array).
I just wouldn’t expect students to necessarily use them independently but exposure is a good thing!
Here are two possible ways to write 23 x 12 using numbers:
When Should Students Use the Distributive Property?
✔ When a number can be easily broken into friendly parts – Example: 42 → 40 + 2
✔ When multiplying a large number by a single-digit number – Example: 6 × 38
✔ When they struggle with direct multiplication – Breaking numbers down makes it easier!
✔ When solving word problems – Especially useful for real-life applications.
As with all multiplication strategies, it is important to emphasize to students that they don’t need to use this strategy all the time. The goal is for them to recognize when it helps and use it as needed.
Common Mistakes Students Make
🅇 Forgetting to distribute both parts. Have students underline both parts before solving. Alternatively, drawing an array will help ensure all parts of each number is included – a great way to check answers.
🅇 Forgetting to add the products together at the end. Do not move on from concrete models and visuals until students truly grasp why they multiply the decomposed numbers and why the add the parts together. Showing the one array being broken into multiple smaller arrays, then ‘put back together’ helps.
🅇 Choosing inefficient breakdowns. Teach them to look for friendlier numbers. These are great Minds On activities and Math Talks sprinkled throughout the year.
Activities & Practice to Reinforce the Distributive Property
✓ Distributive Property Scavenger Hunt – Place task cards with various arrays and decomposed problems around the room (or outdoor space) and challenge students to match each one to the multiplication problems on their handout.
✓ Distributive Property Task Cards – Give multiplication problem flash cards and have students rewrite them using the distributive property or draw an array (or both).
✓ Interactive Area Model Worksheets – Give students blank grids and let them break numbers apart visually. Have them try different ways and decide on the easier.
✓ Distributive Property & Array Anchor Charts – Work with students to create an anchor chart that can be displayed in class for reference (and copied into notebooks as a study guide). Check out this ready-to-use option here.
✓ Double Checks – Use real-world, meaningful multiplication problems and have students solve it in two different ways. This is good practice for all math problems, to check their answers.
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Why should you Teach Distributive Property with Arrays?
Teaching the Distributive Property isn’t about memorizing a formula—it’s about helping students think flexibly and develop strong number sense.
When students see multiplication as breaking numbers apart and putting them back together, they gain confidence and fluency in math.
Multiplication arrays are the ideal visual and hands-on representation of this strategy that really help students comprehend what is happening behind the math jargon.
Do you have a favourite activity to teach the distributive property? I’d love to hear it below!
Teaching more Multiplication Strategies
→ Halving & Doubling Multiplication Strategy
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