Equal number sentences

Equal number sentences

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Overview
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Marking Student Responses
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Further Resources
This task is about finding numbers that make number sentences true.

Question 1Change answer

Write a number in each box to make the number sentence true.

a)   6  +  2  =    +  5
 
b)   7  +    =  10  +  2
 
c)     +  1  =  3  +  4
 
d)   15  +  10  =    +  9
 
e)   8  +  3 =  6  + 
Task administration: 
This task can be completed with pencil and paper or online (with auto marking displayed to students).
Level:
2
Curriculum info: 
Maths, Number and Algebra, Equations and expressions
Keywords: 
equality, addition
Description of task: 
Students complete open number sentences to show their understanding of equality.
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 (11/05)
a)
b)
c)
d)
e)		 3
5
6
16
5

Total score
All 5 correct
4 correct
2-3 correct

 very difficult
very difficult
very difficult
very difficult
difficult
 
 
very difficult
very difficult
difficult
Based on a representative sample of 183 Y4 students.
Diagnostic and formative information: 
  Common error Likely calculation Likely misconception
a)
b)
c)
d)		 8
3
2
25		 6 + 2
10 – 7
3 – 1
15 + 10		 Students do not see that both sides of the equation need to be equal. They solve the problem using the left-hand expression but only the first number or box after the equals sign.  They interpret "=" as meaning "and the answer is."  This error was made by 60-70% of the students in the sample.
b)
d)
e)		 12
1
3 (or -3)		 10 + 3
10 – 9
6 – 3 (or 3 – 6)

Students do not see that both sides of the equation need to be equal. They ignore the first number or box given and solve the problem that includes the number or box before the equals sign, and all of the right-hand expression. Students with this misunderstanding may have got the correct answer for part a) for the wrong reasons.
a)
b)
c)
d)
e)		 8
12
7
25
11		 6 + 2
10 + 2
3 + 4
15 + 10
8 + 3

Students do not see that both sides of the equation need to be equal. They solve the only complete arithmetic expression that they see.
a)
b)
c)
d)
e)		 13
19
8
34
17		 6 + 2 + 5
7 + 10 + 2
1 + 3 + 4
15 + 10 + 9
8 + 3 + 6

Students do not see that both sides of the equation need to be equal. They ignore the equals sign and add all available numbers together.
Next steps: 

Students whose answers fit into the above common error categories have an arithmetical view of the meaning of the equals sign.  To them equals only means "and the answer is". This is particularly true of the first group of common errors. Students need to understand that 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, or that = means "the same as".  Initiating a discussion about the meaning of the equals sign is a starting point in coming to understand this concept.  Resource What is equal? uses true/false number sentences to generate discussion about the meaning of the equals sign.

Exposing students to a variety of ways in which number sentences can be written, e.g., 5 = 4 + 1,
6 = 6, 9 = 1 + 3 + 5 etc. can help students see that the equals sign does not always come at the end of the number sentence.

The concept of equals as meaning both sides of the equation represent the same quantity can be introduced through the concept of balancing equations.  Use simple balance pan scales and coloured multi-blocks to visually represent simple open number sentences/equations.  See resource Balance pans for further examples.
The Algebraic thinking concept map has further information on the concept of equality and the idea of closure.

For students who have grasped the idea of equality, try more challenging number sentences using larger numbers that promote relational thinking.  Use numbers on each side of the equation that are only a few digits away from each other and encourage the students to explore the relationships between the numbers rather than perform calculations, e.g.,

  • 58 + 13 =  + 15              15 is 2 more than 13, so the answer in the box is 2 less than 58 (i.e., 56)
  • 12 + 78 = 16 +              16 is 4 more than 12, so the answer in the box is 4 less than 78 (i.e., 74)
  • 213 + 62 = 65 +                   65 is 3 more than 62, so the answer in the box is 3 less than 213 (i.e., 210)

Equal number sentences II is a parallel Level 3 resource, administered to a sample of Year 6 students.  This group performed significantly better on this task, finding the parallel items "easy".

Numeracy Links
Equality as balance: refer to A Balancing Act (Book 5: Teaching Addition, Subtraction, and Place Value) and The Equals Sign Again (Book 8: Teaching Number Sense and Algebraic Thinking, page 12).