Find the missing numbers
Y4 (11/2005) | ||
a) | 91 | difficult |
b) | 72 | difficult |
c) | 7 | difficult |
d) | 0 | very difficult |
e) | 4 | very difficult |
Based on a representative sample of 182 Y4 students.
This resource explores the application of the additive identity property, a + 0 = a, in its inverse form, a – a = 0, and the commutative property, a + b = b + a, in the context of solving problems. Understanding and applying these number properties is important for future experiences that involve the solving of algebraic equations.
- Check student work for evidence of calculation.
- This resource can be used to initiate discussion about the two number properties. Have students explain how they worked out their answer, sharing with a group or the whole class. Discuss the different strategies employed and start exploring the "rules" being applied. Each problem could be presented individually followed by discussion.
- To listen to a student using his knowledge of the additive identity to work out the answer to question d), click on the link, additive strategy video clip.
Common error | Likely calculation | Likely misconception | |
c) e) |
0 59 |
0 + 48 = 48 + 0 0 + 59 = 59 + 0 |
Students are able to apply the additive identity and have some recognition of the commutative property. However, they do not see the number that needs to be added to the right hand side of the equation to make the two sides equal. |
c) | 55 | 48 + 7 | Students are able to apply the additive identity and the commutative property. However, they continue adding after solving. |
a) | 37 | 63 – 63 + 37 = 37 | Students treat the equals sign as meaning "and the answer is", ignoring other numbers on the right hand side of the equation. |
b) | 0 | 89 + 0 = 89 | Students look only at the numbers surrounding the empty box, reinterpreting the minus sign as an equals sign. |
a) b) c) e) |
0 0 0 0 |
63 – 63 = 37 – 37 = 0 59 – 59 = 89 – 89 = 0 34 – 34 = 0 42 – 42 = 12 – 12 = 0 |
Students identify the additive identity (inverse) but do not apply it to the equation. |
b) c) |
131 14 |
59 + 72 48 – 34 |
Students perform an operation on only two numbers in the equation. |
a) b) c) d) e) |
63 (or 37) 59 or 89 34 or 48 9, 24 or 37 12, 42, 55 or 59 |
Repeats one of the numbers in the equation. |
NOTE:
- Approximately 19% of students used some computation to work out their answer.
- Most of the common errors were made due to a misconception about the meaning of the equals sign.
If students have successfully solved the problems by recognising and applying both the additive identity (inverse form) and the commutative property, give them more complex problems such as:
91 + 58 + 23 – – 91 + 5 = 66 + 5 + 23 – 66. Alternatively, have them write their own number sentences which demonstrate this understanding. Encourage them to use large numbers. These can be shared with other group or class members. Solving problems provides examples of number sentences which use only the additive identity.
For those students who were able to apply the additive identity but not the commutative properties, have them rewrite the problem, replacing the numbers that make zero with the symbol 0. Have students consider their next step with this new problem.
e.g.,
7 + 34 + 48 – 34 = 48 +
7 + 0 + 48 = 48 +
7 + 48 = 48 +
If students apply the additive identity and the commutative property appropriately but then continue adding, they may have a misconception about the meaning of the equals sign. When they see 48 + 7, they want to "close" the problem by adding the two numbers and making them equal 55. This concept of closure is explored in the Algebraic Thinking Concept Map.
Similarly, those students who ignored numbers and/or operation signs to work out their answer may also have underlying misconceptions about equality. To them the equals sign means "and the answer is" as they are used to seeing equations in the form a + b = c. Understanding that equality means 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 is a fundamental algebraic concept and needs to be firmly in place before looking at the additive identity or exploring commutativity.
Resources such as What is equal? and Equal number sentences explore the idea of equality. Balance pans further examines the concept of equality as balance.
Further information about equality, the additive identity property and commutativity can be found in the Algebraic Thinking Concept Map.