Creating facts to solve quotative division problems

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Purpose

The purpose of this activity is to support students applying division, their extended knowledge of basic division facts, and their recognition of patterns in division facts to solve quotative division problems. Quotative division is the same as repeated subtraction in that the size of sets is known and the number of equal sets is unknown.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-2: Know basic multiplication and division facts.
Required Resource Materials
  • Individual counters of interest and relevance to students. 
Activity
  1. Pose a quotative division problem, calling on a context that is relevant to your students, that requires students to think beyond divisors of two, five or ten. For example:
    There are 32 people at the indoor football tournament. They make teams of four players. How many teams are made?
     
  2. Let students solve the problem using their own choice of strategy and representation. If needed, you might model a specific strategy or representation for some, or all, students.
     
  3. Provide time for students to share their strategies and calculations with the wider class.
    Record the expressions and diagrams used by students or invite students contribute their thinking to a shared document or display. 
     
  4. Draw attention to partitioning the dividend (number of people) as an effective strategy.
    20 ÷ 4 = 5 teams, 32 – 20 = 12, 12 ÷ 4 = 3 teams. There are 5 + 3 = 8 teams. Use the diagram to track the number of teams being made.
    Construct a schematic diagram to model the strategy.

Schematic diagram.

  1. Write a set of equations for the diagram: 20 ÷ 4 = 5, 12 ÷ 4 = 3, 32 ÷ 4 = 8.
    What patterns do you see in the equations? What stays the same? What changes?
     
  2. Extend students' understanding to working with a larger dividend: 36.
    Can you use your answer for 32 ÷ 4 = 8 to solve these problems:
    There are 36 people at the tournament. How many teams of four can they make?
    ​​​​​
  3. Let students attempt the problem in appropriate groupings. Consider how you might group students to encourage peer scaffolding and extension. Some students might benefit from working independently, whilst others might need further support from the teacher. 
     
  4. Provide time for students to share their strategies and calculations with the wider class.
    Record the expressions and diagrams used by students or invite students contribute their thinking to a shared document or display. 
    Look for students to recognise that adding four to the number of people results in one more team, meaning an increase of one to the quotient. 
    Show the equation using a schematic diagram and vertically arranged written equations. Encourage students to look for patterns in both representations. 

    32 ÷ 4 = 8
    36 ÷ 4 = 9W
    Schematic diagram.
    What patterns do you notice?
     
  5. Extend the problem to dividing 28 people into teams of four.
     
  6. Pose further problems by changing the dividend (number of people) or divisor (people in each team) to develop students' competence at using division equations. Encourage students to use known facts to get new results.
     

Examples of changing the dividend might be:

  • 12 people make teams of 6 people. How many teams are made?
    24 people make teams of 6 people. How many teams are made?
    48 people make teams of 6 people. How many teams are made?
    12 ÷ 6 = 2  and  24 ÷ 6 = 4  and  48 ÷ 6 = 8        
    What patterns do you see?
  • 20 people make teams of 4 people. How many teams are made?
    28 people make teams of 4 people. How many teams are made?
    36 people make teams of 4 people. How many teams are made?
    20 ÷ 4 = 5  and  28 ÷ 4 = 7  and  36 ÷ 4 = 9         
    What patterns do you see?
  • 21 people make teams of 7 people. How many teams are made?
    21 ÷ 7 = 3  and  42 ÷ 7 = 6  and  49 ÷ 7 = 7    
    What patterns do you see?   
    42 people make teams of 7 people. How many teams are made?
    49 people make teams of 7 people. How many teams are made?

Examples of changing the divisor might include:

  • 40 people make teams of 2 people. How many teams are made?
    40 people make teams of 4 people. How many teams are made?
    40 people make teams of 8 people. How many teams are made?
    40 ÷ 2 = 20  and  40 ÷ 4 = 10  and  40 ÷ 8 = 5         
    What patterns do you see?
  • 54 people make teams of 9 people. How many teams are made?
    54 people make teams of 3 people. How many teams are made?
    54 people make teams of 6 people. How many teams are made?
    54 ÷ 9 = 6  and  54 ÷ 3 = 18  and  54 ÷ 6 = 9    
    What patterns do you see?
  • 64 people make teams of 8 people. How many teams are made?
    64 people make teams of 4 people. How many teams are made?
    64 people make teams of 2 people. How many teams are made?
    64 ÷ 8 = 8  and  64 ÷ 4 = 16  and  64 ÷ 2 = 32    
    What patterns do you see?

Next steps 

  1. Increase the level of abstraction by using diagrams rather than physical materials, before progressing to using stories and equations only.
     
  2. Ask anticipatory questions based on a trusted result, such as, “if I know 18 ÷ 3 = 6, what other equal division problems can I solve?”
     
  3. Extend the problems to include a full range of basic facts. For example:
    90 people make teams of 9 people. How many teams are made?
    81 people make teams of 9 people. How many teams are made?
    81 people make teams of 3 people. How many teams are made?
    90 ÷ 9 = 10  and  81 ÷ 9 = 9  and  81 ÷ 3 = 27         
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Level Three