Commutative number lines

Commutative number lines

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.
 
  i)

 
       

7  +  3  = __ +   __
 
  ii)

 
  

__ + __ = __ + __
 
b)
Use the number lines below to draw in each number sentence.
 
  i)

 
 
 
        

  4 + 8 = 8 + 4   

 
  ii)

 
 
 
        

16 + 9 = 9 + 16
 
Task administration: 
This task is completed with pen and paper only.
Level:
2
Curriculum info: 
Maths, Number and Algebra, Equations and expressions
Keywords: 
commutativity, addition, number lines
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 3
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
      Y4 (06/09)

a)

i)
ii)

 3 + 7
2 + 15 = 15 + 2 or 15 + 2 = 2 + 15

moderate
moderate

b)

i)

ii)

difficult

 


difficult
Based on a representative sample of 162 students.
Teaching and learning: 
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 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. 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 error Likely misconception
a) i)   Lacks appreciation of commutativity
a) i)

b) i)

 10 and any other number
(Often 10, 3, or 7)
2 + 15 = 17 + other numbers or
15 + 2 = 17 + other numbers		 Misunderstands the equals sign
Treats the equals sign as "and the answer is" and gives the sum of the first two numbers. This is followed by a range of different numbers.

 

Next steps: 
Lacks appreciation of commutativity
Ask the students what differences they see in the jumps above the number line and the jumps below it. Draw their attention to the arrows which all point to the right towards the same number (10 or 17 respectively). This number represents the total amount on each side of the equation. 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.

Misunderstands the equals sign
Encourage students 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.

Other resources
For more information about commutativity and algebraic thinking see the Algebraic Thinking Concept Map: commutativity.

Numeracy

Problems like  + 29 = 81 explores commutativity when finding an unknown. (Book 5: Teaching addition, subtraction, and place value (pdf)).