Times less

1. Create two stacks of cubes, one made of 4 cubes of one colour, and one made of 12 cubes of a different colour. Briefly show students the cubes, before masking both stacks under a piece of paper.
   I have two different stacks under the paper.
   One stack is three times less than the other stack.
   What does “three times less” mean?
   Look for students to identify that “three times less” means that three lots of the smaller stack will fit into the larger stack. This is the opposite of “three time more.” 
    
2. Reach under the paper and take out the 12-stack.
   This is the large stack. The small stack is “three times less” than this.
   How many cubes are in the small stack?
   Encourage the students to work out the size of the small stack and explain their thinking to a peer. You might record a few key statements on the board for the class to consider. Remove the paper and confirm that a 4-stack fits into the 12-stack three times.
    
3. Have students reconsider their answers in relation to the cube model. You might find it beneficial for students to use cubes and build what they thought was hidden under the paper, before comparing these models to the actual cube stacks. 
    
4. Record equations showing the relationship between the 12-stack and the 4-stack. Discuss the meaning of the symbols in each equation and how the numbers relate to the amounts. Draw attention to the meaning of 1/3 x 12 as “one third of 12.” Draw attention to relevant vocabulary and te reo māori kupu, such as multiply (whakarea), divide (whakawehe), and fraction (hautau).
   
    
5. Pose similar problems, masking the cube stacks under paper, and using “times less” statements. Alter the numbers included in each problem to cater to the knowledge of your students.
   Have students work in pairs or small groups, with either students with similar mathematical ideas (i.e. to consolidate thinking), or different mathematical ideas (i.e. to scaffold and extend each other). Encourage the students to work out the size of the smaller stack from seeing the larger stack. Ask groups to record their thinking, equations and discuss the meaning of the symbols. Provide time for groups to share their ideas once all of the problems have been solved. 
   Good examples might include:
   1. 15-stack and 3-stack. The small stack is five times less than the large stack.
      
   2. 10-stack and 5-stack. The small stack is two times less than the large stack.
      
   3. 20-stack and 2-stack. The small stack is ten times less than the large stack.
      
       
6. Challenge students to independently create solution sets to “less than” comparison situations. Put two stacks under the paper to mask them.
   Under the paper I have two stacks. The small stack is four times less than the large stack. How many cubes might be in each stack?
   Look for students to offer a variety of solutions, such as 2-stack and 8-stack, 5-stack and 20-stack, 10-stack and 40-stack, etc. 
   
   It is interesting to ask: If the large stack had zero cubes, how many cubes would be in the small stack?
   Conclude the problem by producing the large stack from under the paper.
   How many cubes are in the small stack?
   Write a multiplication equation for these stacks.
   Write a division equation for these stacks.

Next steps 

• Increase the level of abstraction by covering the materials or using diagrams without visible numbers of cubes.
• Ask anticipatory questions like, “How many times less is the 4-stack compared to the 20-stack?”
• Extend the difficulty of the comparisons so students apply a full range of their basic multiplication facts. Give students a multiplication or division equation. Ask them to create stacks to match the equation. What other equations can they write comparing the two stacks?
• Encourage the application of students' knowledge 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).