Creating facts to solve quotative division problems
The purpose of this activity is to support students to apply division as an operation to solve quotative division problems, using a full range of basic facts. Quotative division is the same as repeated subtraction as the size of sets is known but the number of equal sets is unknown.
About this resource
New Zealand Curriculum: Level 3
Learning Progression Frameworks: Multiplicative thinking, Signpost 4 to Signpost 5
These activities are intended for students who understand multiplication as the repeated addition of equal sets, and who have some knowledge of basic multiplication facts.
Creating facts to solve quotative division problems
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 materials
- Individual counters of interest and relevance to students.
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.
5.
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?
6.
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?
7.
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.
8.
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
What patterns do you see?
9.
Extend the problem to dividing 28 people into teams of four.
10.
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?
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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