What is equal?
Y4 (11/05)  
a) b) c) d) e)

T T F T F
Total score 
very easy difficult difficult difficult difficult

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. Over 40% of the students responded this way. 
c) e) d) 
True True False 
Here they see only the following part of the equation: 3 + 4 = 7 4 + 5 = 9 Students are expecting to see "9" on the righthand side of the equals sign. They know that 2 is not the answer to 6 + 3. About 40% of students had these answers. 
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 "+ 2" or "+ 1" do have some meaning but are not able to state that the number sentence is false. 
NOTE: Students who get only a) correct most likely see the equals sign as meaning "and the answer is".
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:
 We got students to brainstorm the meaning of the equals sign.
 We then presented number sentences similar to those in the resource and used Think/Pair/Share 1 to generate discussion.
 The number sentences were presented one at a time.
 We had the class share their ideas about whether each one was true or not.
 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 lefthand side of the equals sign represents the same quantity as the expression on the righthand 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.
Figure It Out
Good as Gold (Number Sense and Algebraic thinking, L23, book 2, pages 1213).
The Fish Hooks of Ngake (Number, L34, book 2, pages 2021).
Crunch Machine (Algebra, L23, 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).
Equality is a parallel Level 3 resource, administered to a sample of Year 6 students. A large improvement on questions b) to e) can be seen with this group. Success rates moved from 25% 35% to 55%  70%.
 What's the same as ...
 Equality
 Equal number sentences II
 Cuisenaire number sentences
 Balance pans
 Matching equations
 Equal number sentences
 Greater than, less than, or equal number sentences
 Multiplication boxes and triangles II
 Commutative number sentences II
 Number pairs
 Commutative number sentences I
 Different addition number sentences
 Different subtraction number sentences