Commutative property

1. Laminate several copies of the cars copymaster and cut them up into cards with a single car on each. Lay out four cars in a line. Note that this context could be adapted to better reflect to your students' interests, cultural backgrounds, and their learning from other curriculum areas.
    
2. Create a simple ‘people in car’ scenario with equal numbers of people in each car. For example:
   There are four cars, and each car has two people in it. (Build the model)
   What is the multiplication equation for this story? (4 x 2 = 8)
   If I change four and two around to write 2 x 4 = , what will the model look like now?

• Look for students to recognise that 2 x 4 means two cars with four people in each car.
  
  
  
  
   

3. Discuss: Which story has more people in it, 4 x 2 or 2 x 4?
   Students should say that both stories have a total of 8 people.
   Write 4 x 2 = 2 x 4
   What does this equation mean? The equation shows us that four cars with two people in each car, is the same number of people as two cars with four people in each car.
    
4. Let’s see how that works. Suppose the two people in each car have an argument with their fellow passenger and want to go into different cars.
   Take one person from each car to form a set of four. Repeat that with the remaining passengers.
   By taking one person from each car I make a set of four. I can do that twice.
   Put each set of four in their own car to show that 4 x 2 becomes 2 x 4.
   
   
    
5. You might do the same thing with 2 x 4. The fours have a big argument, and each person wants to be in a different car. Taking one from each car four time produces four cars with two people in each car.
    
6. Use other examples of the commutative property to illustrate that reversing the order of the factors does not affect the product. Consider your students' multiplication facts knowledge when setting these. More knowledgeable students might make up their own examples for a partner to solve. Good examples are:
   • Five cars with three people per car is the same number of people as three cars with five people per car. (5 x 3 = 3 x 5)
   • Two cars with seven people per car is the same number of people as seven cars with two people per car. (2 x 7 = 7 x 2)
   • Ten cars with four people per car is the same number of people as four cars with ten people per car. (10 x 4 = 4 x 10).
      
7. Use cars to record the equalities. Ask students to express the pattern that is true for all cars. Look for them to say something about the order of factors and to make reference to the equality. For example, show 4 x 3 = 3 x 4 using the cars and people.

Next steps

1. Explore when using the commutative property is advantageous to calculation. For example, ten cars with three people per car is more easily solved by considering it as equal to three cars with ten people per car (10 x 3 = 3 x 10).
    
2. Explore simple relational thinking that involves the commutative property and understanding of the equal sign.
   For example, 8 x □ = 3 x 8. What number is □?
   Use the cars and people model to represent the equality if necessary.
   Add to the collection of known multiplication facts on the maths wall using the commutative property. If 4 x 5 = 20 is know then 5 x 4 = 20 is known.