Anticipating the result of equal subtraction (quotative division)

1. Pose problems that involve equal sharing using facts that are readily accessible, for example, multiples of two, five and ten.
   Here are 16 people.
   They need to pair up for folk dancing.
   How many pairs will there be?        
        
2. Expect students to anticipate the result of the equal sharing before any physical modelling.
   How can we work this out without moving the people?
   Students might suggest additive ways to solve the problem, such as 5 + 5 = 10, so making five pairs will require ten people. That leaves six people who will make three pairs, since 3 + 3 = 6. Altogether that is 5 + 3 = 8 pairs.
   Others may have sufficient doubles knowledge to fluently anticipate the outcome of forming pairs.
    
3. Organise the people counters into pairs to check the predictions. Ask progressive questions like:
   How many dancing pairs have we got so far?
   How many people are in pairs so far?
   How many people are left to pair up?
   How many more pairs can we make?
    
4. Show the students how the problem could be recorded using multiplication equations (8 x 2 = 16). Discuss what role the symbols play in the context.
   What does the 2 mean (represent) in the problem? (The number of people in a pair)
   What does the 8 mean in the problem? (The number of pairs that are made)
   What does the x symbol mean? (It means ‘of’ in the sense that the total number of people is made from 8 pairs of 2 people.)
    
5. Record the relevant division equation (16 ÷ 2 = 8). Discuss the meaning of the symbols, drawing attention to the meaning of ÷ as “put into sets of.” Be aware that students may think that division only involves equal sharing, as in “16 things shared equally among 2 people.” 
    
6. Pose similar problems that can be solved with multiples of two, five and ten. Allow students to work in groupings that will encourage peer scaffolding and extension. Some students might benefit from working independently, whilst others might need further support from the teacher. Consider also the different means of action and expression (e.g. verbal, written, digital, physical) that your students might use to demonstrate their thinking.
   Good examples are:

• 20 people put into teams of five people.
• 50 people put into teams of ten people.
• 18 people put into teams of two people.
• 30 people put into teams of five people
• 100 people put into teams of ten people
  Arranging the people counters into lines to support students seeing the link between repeated subtraction and multiplication.
   

7. Pose further “put into sets of” problems and focus students’ attention on predicting how many sets will be made, before using materials. 
   Can we use multiplication and division to solve this problem instead of adding or counting?
    
8. Progress to providing problems in which the story is given without physical objects and students are asked to anticipate the equal shares, such as:
   25 people are put into teams of five. How many teams are made?

• Record multiplication and division equations to represent the situation, such as 5 x 5 = 25 and 25 ÷ 5 = 5.

Next steps 

1. Increase the level of abstraction by covering the materials or only providing the story, asking anticipatory questions, and working with more complex facts.
   A suggested sequence for extending the difficulty of the equal subtractions is:

• Making pairs with even numbers up to 20.
• Making fives with numbers up to 30.
• Making tens with decades up to 100.
• Making pairs with even numbers beyond 20.
• Making fives with all multiples of five up to 50.