Comparing sets using addition and multiplication

1.

Create two stacks of cubes, made of 4 cubes of one colour and 12 cubes of another colour, in front of the class. You might adapt this by using smaller numbers of cubes (e.g., 5 and 10, 2 and 4) that your students are more confident multiplying. Similarly, you might extend the problem by using larger numbers of cubes.

• What can you say about how tall one stack is compared to the other?

Students might say, “The 12-stack is 8 more than the 4-stack."

Model the difference with cubes (e.g., by partitioning, counting, and iterating the cubes) and record relevant equations: 4 + 8 = 12, 12 - 4 = 8

2.

Challenge students to compare the stacks in a different way. If necessary, prompt the student/s:

• How many times taller is the 12-stack than the 4-stack? 

If the student/s offer a multiplicative comparison, such as “The 12 stack is three times taller than the 4-stack,” challenge them to convince you that they are correct. An appropriate argument is that the 4-stack fits into (measures) the 12-stack three times. Look for the student to model how the 4-stack iterates three times and use relevant language (you might model this initially). Record relevant equations to support the language and modelling used by students (e.g., 3 x 4 = 12).

3.

Support students to make a fractional comparison.

• What fraction of the 12-stack is the 4-stack?

You might introduce the kupu, hautau (fraction), as part of this.

4.

Pose similar comparison problems, looking for students to make statements about difference and "times as many.” 

Ask students to record equations for the comparisons they make: addition and subtraction for differences, multiplication for “times as many”, and multiplication for fraction comparisons, such as 1/4 x 20 = 5.

You might find it useful to have students with different levels of mathematical knowledge working together, to allow for peer scaffolding and the extension of knowledge. 

Examples might be:

• Compare a stack of 3 cubes with a stack of 15 cubes.
• Compare a stack of 5 cubes with a stack of 20 cubes.
• Compare a stack of 2 cubes with a stack of 12 cubes.

5.

Provide time for students to share their models, equations, and language used with other students and the wider class. 

Increase the level of abstraction by covering the materials or using diagrams without visible numbers of cubes (e.g., bars or paper strips).

Ask anticipatory questions like:

• “How many times do you think a 5-stack fits into a 15-stack?”

Extend the difficulty of the comparisons so students apply a full range of their basic multiplication facts.

Encourage the application of students' basic multiplication facts by presenting them with culturally relevant problems (e.g., there are 20 boys and 5 girls in our classroom). How many times larger is the group of boys than the group of girls?), or by asking students to make up similar problems with the use of other relevant materials (e.g., stick figures, animal toys).