Commutative number sentences II

Commutative number sentences II

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about completing number sentences.

Question 1Change answer

a) For each problem, find the missing number and write it in the box.
 
   i)   35 + 28 = 28 + 
   ii)  148 +  = 97 + 148
   iii)  + 496 = 496 + 375
   iv)  919 + 762 =  + 919

Question 1Change answer

b) All the number sentences below are true. Explain why you know they are true without having to add up the numbers.
 

638 + 95 = 95 + 638

277 + 534 = 534 + 277

1 632 + 817 = 817 + 1 632

28 357 + 4 216 = 4 216 + 28 357

Task administration: 
This task can be completed online and with pencil and paper.
Level:
3
Curriculum info: 
Maths, Number and Algebra, Equations and expressions
Key Competencies: 
Thinking, Using language, symbols, and texts
Keywords: 
commutativity, equality, addition
Description of task: 
Students use the commutative property to find missing numbers in equations and explain the property.
Curriculum Links: 
Key competencies
This resource involves recognising and applying commutativity in simple addition sentences, and describing why it applies. These relate to the Key Competencies: Thinking and Using language, symbols and text.
For more information see https://nzcurriculum.tki.org.nz/Key-competencies
 
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 (11/2007)
a)

i)
ii)
iii)
iv)

35
97
375
762

easy
(for all 4 correct)
b)  

Description makes reference to the property of commutativity, i.e., the same numbers are on each side of the equals sign but in a different order – the order does not change the value on that side of the equation.  Also accept if students state that the numbers on each side add to the same total.
Examples of student responses: 
"Because equals means the same as and I know if 638 + 95 is 733 then no other number can fit in with 95."
"The numbers are the same but the other way around."
"The numbers add to the same total."

 difficult
Based on a representative sample of 167 students.
Teaching and learning: 
This resource is about the number property of commutativity, which states that a + b = b + a.  It can be used as a discussion starter with each question in part a) being given one at a time, leading to a conversation about a pattern or rule that the students might see. 
 
For further information on discussion in class, click on the link, Mathematical Classroom Discourse.

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 calculation Likely misconception
a) ii)
     iii)
     iv)		 245
0
1681		 245 = 97 + 148
0 + 496 = 496
919 + 762 = 1681		 Adds the two numbers immediately before or after the equals sign.  Believes the equals sign means "and the answer is".
a) i) 63
91		 35 + 28
35 + 28 + 28		 Ignores numbers and/or operators.
a) ii)
     iii)
     iv)		 0
0
0		 148 + 0 = 148
0 + 496 = 496
919 + 0 = 919		 Uses the additive identity to try and make the number sentence true, but in doing so ignores one of the numbers on one side of the equals sign.  May also indicate misconception about equality.
a) i)
     ii)
     iii)		 7
51
121		 28 + 7 = 35
148 = 51 + 97
496 – 375 = 121		 Finds the difference between the smaller and larger numbers.
b) It was a pattern.
It was symmetric.
The middle numbers are the same.		   Describes how the number sentences look rather than explaining why the number sentences are equal.

 

Next steps: 
Concept of equality
Students who added the numbers together, used 0 to try and make a true number sentence or found the difference between the smaller and larger numbers are most likely to have a purely arithmetical view of the equals sign.  When students believe that the equals sign means "and the answer is" they will often ignore operators and other numbers in a number sentence and focus on trying to find a solution by adding (or subtracting) the given numbers.  The equals sign is seen as a command to take an action rather than a representation of a relationship.  It is important for students to understand that the equals sign represents quantitative sameness – in other words, 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.  Equal Number Sentences II and Equality are Level 3 resources which explore the concept of equality through True/False number sentences and open number sentences.  For something more visual, the Level 2 resource Balance Pans uses balance pans to explore the concept of equality.

For further information on the ideas surrounding equality, refer to the Algebraic thinking concept map: equality.

Even students who were able to correctly work out what numbers went in the missing boxes in question a) may not have an understanding of equality as meaning quantitative sameness.  Similarly, those who saw the symmetric nature of the pattern in question b) are not likely to have this understanding either.

Commutativity
Once the algebraic understanding of equality is in place, True/False number sentences can be used to initiate conversations about commutativity.  Students can be presented with a range of number sentences and asked to decide whether they are true or false.  Incorrect student responses from question a) could be used to generate false number sentences.  Ask students:

"Is this number sentence true or false?"  
"Why/Why not?"
"How do you know?"

To prevent students from calculating, initially use small, easy-to-add, numbers but move quickly onto larger numbers.  It will soon become evident whether students are calculating or not.  Ask them:

"Do you need to calculate?"
"Why/Why not?"

Students can then write their own True/False number sentences.  These can form the basis of a discussion about a rule to describe what is happening in the number sentences.

Two examples of rules that students have generated following these types of discussions are listed below:

"It doesn’t matter if the numbers are swapped around on each side of the number sentence.  If the numbers are the same, the number sentence will still balance."

"When you add two numbers, you can change the order of the numbers you add, and you will still get the same number."

For a more visual way of looking at commutativity, Commutative number lines uses number lines to show how, for example, 16 + 9 = 9 + 16.

For more information about commutativity and algebraic thinking see the Algebraic Thinking Concept Map: Commutativity.
 
Numeracy
The activity Problems like  ? + 29 = 81 explores how commutativity can be used to help find an unknown. (Book 5: Teaching Addition, Subtraction, and Place Value), page 37: "I am learning to reverse problems like ?  + 29 = 81 to 29 +  ? = 81 and then use an appropriate mental method to solve the problem.").