Equal sharing II

Equal sharing II

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about showing how to share amounts equally.

a) Show how to work out how to share 15 cookies equally between 3 people.

 
 
 
 
How many cookies will each person get? _____

 
b) Show how to work out how to share 18 apples equally between 6 people.

 
 
 
 
How many apples will each person get? _____

 
c) Show how to work out how to share 28 pears equally between 4 people.

 
 
 
 
How many pears will each person get? _____

 
d) Show how to work out how to share 30 sweets equally between 5 people.

 
 
 
 
How many sweets will each person get? _____
Task administration: 
This task is completed with pencil and paper only.
Level:
2
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
equal sharing, division
Description of task: 
Students show how they solve equal sharing problems.
Curriculum Links: 
This resource can help to identify students' ability to use basic facts and knowledge of place value and partitioning whole numbers to solve partitive division problems.

Key competencies

This resource involves recording the strategies students used to solve partitive division problems. This relates to the Key Competency: Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Multiplicative thinking, sets 3-4
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y4 (11/2005)
a) 5 and
Working that involves:

  • 1-1 mapping (cookies to people).
  • counting out into 3 groups
  • partitioning 15 sticks (including trial and error)
  • reverse multiplication: 3 × 5 = 15
  • division: 15 ÷ 3 = 5.
 easy
(answer and working)
b) 3 and
Working that involves:

  • 1-1 mapping (apples to people).
  • counting out into 6 groups
  • partitioning 18 sticks (including trial and error)
  • reverse multiplication: 6 × 3 = 18
  • division: 18 ÷ 6 = 3.
 moderate
(answer and working)
c) 7 and
Working that involves:

  • 1-1 mapping (pears to people).
  • counting out into 3 groups
  • partitioning 28 sticks (including trial and error)
  • reverse multiplication: 7 × 4 = 28
  • division: 28 ÷ 4 = 7.
 moderate
(answer and working)
d) 6 and
Working that involves:

  • 1-1 mapping (sweets to people).
  • counting out into 5 groups
  • partitioning 30 sticks (including trial and error)
  • reverse multiplication: 5 × 6 = 30
  • division: 30 ÷ 5 = 6.
 moderate
(answer and working)

Based on a representative sample of 183 Y4 students in November 2005.

NOTE: Most students (about two-thirds) used counting or mapping strategies, one-tenth used repeated addition (or subtraction), and one-tenth used multiplicative strategies.

Teaching and learning: 

This resource is about showing how to solve equal sharing problems. The strategies the students use can determine their level of development within the Numeracy framework.  Strategies that can be used for this resource can range from counting out, 1-1 mapping, visual grouping (partitioning involving trial and error), repeated addition, skip counting, multiplication, and division.

Diagnostic and formative information: 
  Common error Likely calculation Likely misconception
a)
b)
c)
d)		 3
6
4
5		 15 ÷ 5
18 ÷ 3
28 ÷ 7
30 ÷ 6		 Students worked out the other factor in the equation, but wrote down the factor they were already given.
c)
d)		 6
7		 6 × 4 = 28
5 × 7 = 30		 Students who were using skip counting (or repeated addition) stopped one factor early or late (miscounting by one).

The results of the trial indicate that while many students were able to calculate the correct answer, some had difficulty showing how they had solved the problem. This could be because they:

  • believe that the purpose of maths is to find the right answer not to show how they got there; or
  • have difficulty putting their method on paper, or
  • are not used showing their working.
Next steps: 

Students who show only the solution may not be aware of the importance of showing the processes they go through to find those solutions.

  • Encourage students to record all their workings, as the strategy students use to solve the problem can indicate their stage of numeracy development.
  • It is also important to encourage students to critique other strategies and try to use the most sophisticated/effective strategy for the type of problem and the numbers in the problem.

Students who did not correctly solve the equal sharing problem may need to represent the problem using counters and explore writing number sentences (starting with counting or additive) to describe the problem.  They can also be further encouraged to write or draw their strategy.

Students who correctly used marks to count out or map the items to each person could be encouraged to develop another way to solve the problem without one-to-one counting, e.g., skip counting, grouping, repeated addition.

For students who used skip counting or repeated addition, encourage them to recognise the relationship between groups of numbers and multiplication, e.g., that 3 + 3 + 3 + 3 + 3 + 3 is "6 groups of 3" and that can be written as "6 × 3".

If students are recognising the multiplicative relationship between the total and the number of groups, and are using reverse multiplication (e.g., 6 × 3 = 18) or division (18 ÷ 6 = 3) to solve, then they could try solving similar problems with more complex numbers.

Questions could be asked in the form of finding the number of sets: "If there are 27 dogs and each family is getting 3 dogs, how many families are there?" which is not as easily recognisable as a division problem.

Links to the Number Framework
Equal sharing is addressed at stages 2-4 (from Counting from one to Advanced counting) in the Numeracy framework (strategies) under the operational domain of Proportions and ratios.