Equality

Equality

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Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This taks is about understanding the meaning of the equals sign.

Question 1Change answer

 
equals sign.png
 
Look at each of the number sentences below. Select whether it is True or False or Not a proper number sentence.
 
a) 8 + 5 = 13 TrueFalseNot a proper number sentence
b) 15 = 6 + 9 TrueFalseNot a proper number sentence
c) 4 + 9 = 13 + 6 TrueFalseNot a proper number sentence
d) 9 + 8 = 7 + 10 TrueFalseNot a proper number sentence
e) 12 + 5 = 17 + 2 TrueFalseNot a proper number sentence
 
Task administration: 
This task can be completed with pencil and paper or online (with auto-marking).
Level:
3
Description of task: 
Students explore the concept of equality by deciding whether addition sentences are true, false or not proper number sentences.
Learning Progression Frameworks
This resource can provide evidence of learning associated with within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y6 (11/2005)
a)
b)
c)
d)
e)

 

T
T
F
T
F

Total score
All 5 correct
4 correct

very easy
moderate
easy
easy
easy


difficult
easy

Based on a representative sample of 173 Y6 students.
Diagnostic and formative information: 

This resource explores the meaning of the equals sign and the concept of equality. It can be used to initiate a discussion about what constitutes a true number sentence and the meaning of the equals sign.

  Common error Likely misconception
b) Not a proper number sentence Students are not accustomed to seeing the equals sign written near the beginning of a number sentence and therefore do not think it is a proper number sentence.
c)
e)

d)

True
True

False

Students often believe that the equals sign means "and the answer is".

Here they see only the following part of the equation:
  4 + 9 = 13
12 + 5 = 17

Students are expecting to see "17" on the right-hand side of the equals sign. They know that 7 is not the answer to 9 + 8.

c) & e) Not a proper number sentence The large number of students who stated that this was "not a proper number sentence" indicates that they are aware that the "+ 6" or "+ 2" do have some meaning but are not able to state that the number sentence is false.

NOTE: Students who get only a) correct see the equals sign as meaning "and the answer is".

Next steps: 
This resource can be used as the basis for a group or class discussion about what makes a number sentence true or false, leading to, (or stemming from) a discussion about the meaning of the equals sign.
In our research we did the following:

  1. We got students to brainstorm the meaning of the equals sign.
  2. We then presented number sentences similar to those in the resource and used Think/Pair/Share 1 to generate discussion.
  3. The number sentences were presented one at a time.
  4. We had the class share their ideas about whether each one was true or not.
  5. We challenged students to explain to the class how they knew each number sentence was "true" or "false" or "not a proper number sentence".

An alternative approach would have been to have students record their answers (T/F/N) on an individual whiteboard before showing the rest of the group/class.

1  Kagan, S. (1994). Cooperative Learning. San Clemente, CA: Kagan Cooperative Learning, pp. 11:2 - 11:3.
The purpose of the discussion is to move the students away from the arithmetical idea that the equals sign only means "and the answer is". They need to realise 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. For example 3 + 4 = 5 + 2 is a true number sentence because both the expressions (3 + 4 and 5 + 2) give the same answer, 7. The phrase we encouraged students to use was "the same as", or "gives the same answer as". 

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

It is important for students to realise that the equals sign does not always need to come towards the end of a number sentence. Once students are comfortable with the notion of equality as comparing the expressions on each side of the equals sign, have them generate their own examples of number sentences that are in a form other than a + b = c, e.g., 7 = 1 + 6 or 8 + 2 = 5 + 5

Cuisenaire number sentences is a practical activity that can be used to reinforce the ideas of equality and also representing number sentences in alternative forms.  The Algebraic thinking concept map has further examples of student generated number sentences.

A way to check on the students' understanding, following these discussions and activities about equality, is to have students record their understanding of the equals sign in a journal using a starter statement, e.g., "I think the equals sign means…" For more information about journalling refer to Journalling in Mathematics.
The idea of equality meaning balance can be introduced using simple balance pans and blocks, which can lead on to solving simple problems presented as open number sentences. See Balance pans for balance pan ideas and Equal number sentences for open number sentences.

What is equal? is a parallel Level 2 resource, administered to a sample of Year 4 students.
 
Figure It Out
Good as Gold (Number Sense and Algebraic thinking, L2-3, book 2, pages 12-13).
The Fish Hooks of Ngake (Number, L3-4, book 2, pages 20-21).
Crunch Machine (Algebra, L2-3, page 17 Activities 2 and 3).
 
Numeracy Links
Equality as balance: refer to A Balancing Act (Book 5: Teaching Addition, Subtraction, and Place Value, page 40) and
The Equals Sign Again (Book 8: Teaching Number Sense and Algebraic Thinking, page 12).