Equal number sentences
Y4 (11/05) | ||
a) b) c) d) e) |
3 5 6 16 5 Total score |
very difficult very difficult very difficult very difficult difficult very difficult very difficult difficult |
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. |
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).
- What's the same as ...
- Equality
- Equal number sentences II
- Cuisenaire number sentences
- Balance pans
- What is equal?
- Matching equations
- 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