Deriving facts by doubling

1.

Create two stacks of five cubes using a single colour.

Mask the stacks with a sheet of paper.

• I have two stacks of five cubes. How many cubes are there altogether?
• How did you work out the answer? If students use an additive strategy (e.g., 5 + 5), validate this and encourage them to use multiplication.
• Could you use multiplication to find the product? How?

Record relevant doubles and multiplication strategies.

For example:

• 5 + 5 = 10, and 2 x 5 = 10.

2.

I am going to double the number of cubes under the paper. Watch.

Make another two stacks of five cubes. Briefly show students the cubes before masking them with paper.

Remind students of the number of cubes they previously identified and of the equations recorded.

• How many stacks did I have before? How did I write that?
• How many stacks do I have now?
• How many cubes are in each stack?
• How many cubes are under the paper in total? How will I write that?
• Can you use multiplication to work out the total number of cubes?

Look for students to apply multiplication to explain that 4 x 5 = 20. Record the new equation next to the previous one:

• 2 x 5 = 10
• 4 x 5 = 20

At this point, you might introduce the term product (the result when two numbers are multiplied together) and the te reo Māori kupu otinga (answer, product, quotient, result, solution).

3.

Ask:

• What patterns do you notice in the equation?

Look for students to notice that as the number of stacks doubles, so too does the product.

4.

Extend the pattern by doubling four stacks of five cubes to make eight stacks of five. Do this under the paper and tell students what you are doing.

• I am going to double my four stacks of five cubes. How many stacks will I have now?
• What multiplication will give the total number of cubes?
• What will the product be?

Record the new equation next to the previous ones:

• 2 x 5 = 10
• 4 x 5 = 20
• 8 x 5 = 40

Look for students to use doubling to continue the pattern and anticipate the product.

Confirm the answer by showing students the stacks of cubes, describing them in words (e.g., I have eight times 1, 2, 3, 4, 5 stacks of cubes), and by entering 8 x 5 on a calculator.

5.

Pose similar problems, masking the stacks of cubes under paper, and using repeated doubling. Make the doubling as accessible as possible by utilising the multiplication facts your students can recall quickly to encourage students to apply the strategy. This might be done in small groups to encourage scaffolding, extension, and productive learning conversations. Allow students to express their mathematical thinking in different ways (e.g., written, verbal, drawn diagrams). Good examples might be:

• 1 x 4 = 4
• 2 x 4 = 8
• 4 x 4 = 16
• 8 x 4 = [ ]
• 2 x 3 = 6
• 4 x 3 = 12
• 8 x 3 = [ ]
• 16 x 3 = [    ] (optional)
• 1 x 6 = 6
• 2 x 6 = 12
• 4 x 6 = 24
• 8 x 6 = 48

6.

Progress to just presenting pairs of equations with one product missing. Discuss the meaning of each equation and why the second product is twice (double) the first. Calculators might be used to confirm the answer once students have first predicted the missing product. Good examples might be:

• 3 x 7 = 21, so 6 x 7 = [ ]
• 5 x 7 = 35, so 10 x 7 = [ ]
• 4 x 4 = 16, so 8 x 4 = [ ]

• Explore the meaning of equations when the second factor is doubled. For example, 5 x 3 = 15 means “five sets of three.” 5 x 6 means “five sets of six.” How are the products connected? Since the size of sets is doubled, the product is also doubled. Other examples might be 4 x 4 = 16 so 4 x 8 =  [ ] ,  8 x 3 = 24 so 8 x 6 = [ ], 7 x 2 = 14 so 7 x 4 = [ ].
• Encourage students to learn their basic multiplication facts using the Number facts pathway in e-ako maths.
• For extension, go beyond the range of basic facts to include examples with one factor more than ten, such as 7 x 3 = 21 so 14 x 3 = [ ], 4 x 8 = 32 so 4 x 16 = [ ], and  8 x 9 = 72 so 16 x 9 = [ ].