Division by measurement

1. Introduce the context of chair stacking by showing students a stack of five chairs.
   Why do they make chairs that can stack?
   How many chairs are in one stack?
   Here are another 20 chairs. How many stacks for five can I make?
    
2. Ask students to work out the answer in their own way. If students become confused using a strategy, model, or have students model, using the strategy with chairs. The following strategies might be used:
   • Skip counting with tracking, possibly on fingers, i.e., 5, 10, 15, 20.
   • Addition facts, e.g., 5 + 5 = 10 (2 stacks), 10 + 5 = 15 (3 stacks), 15 + 5 = 20 (4 stacks).
   • Multiplication facts, ideally 4 x 5 = 10.
   • Division by halving into two stacks of ten, then four stacks of five.
      
3. Together, discuss the use of different strategies. Encourage students to use words, symbols, and diagrams or materials (e.g. cubes) to explain their thinking. 
    
4. Discuss how the problem might be solved using a calculator. Let students experiment with finding the answer. Students might repeatedly add fives until 20 is reached, try various multiplications of five until 20 is reached, or know that 20 ÷ 5 = 4.
    
5. Share the solutions reached and confirm the division equation: 20 ÷ 5 = 4
   What do 20, 5 and 4 mean in this story?
   What does the ÷ symbol mean? (Measured in stacks of …).
   You might introduce the te reo Māori kupu whakawehe, meaning divide/division.
    
6. Present another scenario involving stacks of chairs.
   Let’s get 15 chairs. Stacks of five are a bit heavy. Let’s try stacks of three chairs this time. 
   How many stacks can we make?
   Ask students to anticipate the number of stacks using their preferred strategy. Discuss their answers before moving any chairs. Check the answer using division with a calculator (15 ÷ 3 = 5). When the students agree on the quotient (answer) act out making stacks of three chairs until the collection of 15 chairs is exhausted.
    
7. Provide students with the Copymaster to work from. Allow students to work in groups comprised of students with a range of mathematical understandings, encouraging tuakana-teina and productive learning conversations. Let them use connecting cubes to act as stacks of chairs. Ask students to work out the answers with their own strategies, and explain them using symbols, words, and diagrams/materials, before using a calculator. Early finishers can be challenged to make up their own chair stacking problems for classmates to solve.
    
8. Gather the group and share the strategies used. Encourage students to consider what facts are useful for each problem. For example:
   • Problem 1. Do you know 2 x 10? How might that help?
   • Problem 2. Do you know 20 ÷ 10, how many tens are in 20? How might that help?
   • Problem 3. Do you know 4 x 5, four stacks of five? Do you know 5 x 4? How might that help?
   • Problem 4. Do you know 10 x 3, ten stacks of three? How might that help?

Next steps

1. Continue to increase students’ repertoire of known multiplication facts and associated division facts. Pay particular attention to the multiples of two, five and ten. See if students can answer multiplication facts without skip counting, e.g., say 6 x 5 = 30 with counting 5, 10, 15, 20, 25, 30.
   Connect the meaning of multiplication as “___ equal sets of ___ is the same as ___”
   Ask students to give the related division meaning as “Given ___  items, ___ equal sets of ___ can be made.”
    
2. Use the stacking chairs scenario to pose problems that have remainders. For example:
   • You have 17 chairs. How many stacks of five can you make?
   • You have 13 chairs. How many stacks of three can you make?
   • You have 27 chairs. How many stacks of four can you make?
      
3. Discuss how the remainder might be expressed. At this stage it is sufficient to write 13 ÷ 3 = 4 r 1, meaning there is an extra chair after all the possible stacks are made. You could introduce the idea of the remainder as a fraction of a stack, e.g., 13 ÷ 3 = 4 ⅓.