Commutative number lines II

Commutative number lines II

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about using number lines to show number sentences.
a)  Write the number sentence that each number line shows in the boxes below.
 
three number lines
b)
Use the number lines below to draw in each number sentence. 
 
i) 8 + 7 = 7 + 8
 

 
 
ii) 16 + 9 = 9 + 16
 

 
 
iii) 13 + 19 = 19 + 13
 
 

 

Task administration: 
This task is completed with pen and paper only.
Level:
3
Curriculum info: 
Maths, Number and Algebra, Patterns and relationships
Keywords: 
commutativity, number lines, addition
Description of task: 
Students use number lines to show their understanding of commutativity.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Using symbols and expressions to think mathematically, set 4
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y6 (11/2006)
a)

i)
ii)
iii)

5 + 21
42 + 14 = 14 + 42 or 14 + 42 = 42 + 14
58 + 62 = 62 + 58 or 62 + 58 = 58 + 62

 moderate
moderate
moderate
b)

i)

ii)

iii)

 moderate

moderate

moderate
Based on a representative sample of 124 Y6 students.
Teaching and learning: 
Commutativity is the number property which states that a + b = b + a.  In other words, the order of the numbers can be changed without changing the value of each side of the equation.

This resource is designed to show commutativity in a visual way.  It is helpful if students have some prior experience of using number lines as a way of representing addition or subtraction.

Commutativity is an important number property which is often used in solving problems.  Students may intuitively use it without being aware of doing so.  For example, a student, given the problem 4 + 17 may count on from 17 to get to 21.  They have reversed or "commuted" the problem to read 17 + 4 in order to make it easier to solve.

Diagnostic and formative information: 
  Common response Likely misconception
b) i) - iii) Jump made from smaller to larger number, e.g. from 9 to 16.
 Represents a different relationship between the two numbers in the equation, e.g. 9 + ? = 16, so ? = 7.
b) i) - iii) Only one side of the expression correctly represented.
 May not understand the concept of equality, meaning the expression that is on one side of the equals sign is the same as the expression on the other side of the equals sign, so can only show one side of the expression.
Next steps: 
Students who made jumps on the number line from one number in the expression to the other, e.g., from 13 to 19, may need experience in using number lines to represent addition (or subtraction) equations.  To find resources that use number lines to solve problems, search the ARB maths banks using the keyword number lines. Search results for Level 3 using keyword number lines.

These same students may also be unfamiliar with seeing an equation written in a form other than a + b = c, and are therefore unsure how to represent an expression where the equals sign signifies a relationship between two equivalent expressions rather than a command to solve the problem and "find the answer". 

If students only drew in one side of the expression correctly, encourage them to look at both sides of the equals sign and discuss the idea of equality meaning quantitative sameness (i.e., the expression on the left-hand side of the equals sign represents the same quantity as the expression on the right-hand side of the equals sign).

The concept of equality is explored in the Algebraic Thinking Concept Map: equality and resources that focus on this idea can be found by using the keyword equality. Search results for Level 3 using keyword equality.

If students were successful in recognising that both sides of the expression are equivalent and could be represented on a number line, then discuss any patterns they can see. Encourage students to come up with a conjecture or a rule to describe the general pattern.  Discuss whether this rule would apply in all situations, and if so, why. 
 
For further ideas on exploring commutativity, see the Algebraic Thinking Concept Map: commutativity.
For further information see the Algebraic Thinking Concept Map: commutativity.
 
Numeracy Project
Book 5: Teaching Addition, Subtraction, and Place Value, p. 57: Problems like  + 29 = 81. This section explores commutativity when finding an unknown.