Complete the equation
a) 67 + 38 – 67 = 
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This is how I got my answer:
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b) 37 – 37 + 86 + 19 – 19 =
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This is how I got my answer:
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c) 71 – 53 + 53 = 28 + – 28
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This is how I got my answer:
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d) 95 + 58 + 23 – – 95 = 68 + 23 – 68
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This is how I got my answer:
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For more information see https://nzcurriculum.tki.org.nz/Key-competencies
| Y6 (05/2006) | ||
| a) |
38 |
easy |
| b) |
86 |
easy |
| c) |
71 |
very difficult |
| d) |
58 |
very difficult |
Students who calculated their answers were applying Number strategies. While this strategy is technically correct, those who were able to "find the zero" were using a more algebraic approach in being able to identify and exploit the additive property of zero. The intention of this resource is to encourage this type of thinking.
| Common response | Likely calculation | Likely misconception | |
|
a) b) |
172 198 |
67 + 38 + 67 37 + 37 + 86 + 19 + 19 |
Calculates left to right, ignoring operators. |
| c) | 0 | 28 + 0 = 28 |
Ignores left hand side, and misinterprets the – sign as = so adds zero to make an equation on the right hand side. Does not understand that the equals sign means that both sides of the equation need to represent the same quantity. |
| c) | 28 |
28 + 28 = 56 56 – 28 = 28 or 71 – 53 + 53 = 28 28 – 28 = 0 28 + 0 = 28 |
Assumes 71 – 53 + 53 = 28 so makes the rest of the right hand side = 0 (28 - 28). |
| c) | 53 |
71 – 53 = 28 28 + 53 = 71 71 – 28 = 53 |
Calculates left to right but ignores the position of the equals sign and subtracts the first 28, putting that total in the empty box and ignoring the final – 28. Does not understand that the equals sign means that both sides of the equation need to represent the same quantity. |
| c) | 99 |
71 – 53 + 53 = 71 71 + 28 = 99 |
Calculates left hand side of equation correctly but ignores = sign in the middle and adds on 28. Does not realise that the equals sign can come in places other than at the end of an equation and means that both sides of an equation need to represent the same quantity. |
| d) | 13 |
95 + 58 + 23 = 176 176 – 95 = 81 81 – 13 = 68 |
Calculates left to right, finding a number that will make the left hand side of the equation equal 68 - the number immediately to the right of the equals sign. Does not understand that the equals sign means that both sides of the equation need to represent the same quantity. |
| d) | 23 | 23 – 23 |
Subtracts 23 to make the left hand side of the equation equal 58. Does not understand that the equals sign means that both sides of the equation need to represent the same quantity. |
| d) | 27 |
 = 95 – 68 |
Uses only the numbers immediately surrounding the equals sign to calculate an answer. Ignores operators. |
| d) | 176 | 95 + 58 + 23 | Calculates left to right up to the empty box only. Ignores equals sign and does not understand that it means that both sides of the equation need to represent the same quantity. |
This resource explores the concept of the additive identity property. The additive identity can be expressed in four different ways, with each way being able to be written in two equivalent forms:
a + 0 = a or a = a + 0
0 + a = a or a = 0 + a
a – 0 = a or a = a – 0
a – a = 0 or 0 = a – a
It is effective for students to see each rule written in its alternative form to help them realise that the equals sign does not have to be the penultimate symbol in an equation, but can be near the beginning.
The resource is designed to explore the fourth rule a – a = 0 or 0 = a – a within the context of equation solving. This is a particularly important concept for future algebraic problem solving. Students need to be able to recognise and apply this rule in order to solve equations algebraically.
Students found it harder to "find the zero" when the numbers being added and subtracted were not next to each other, e.g., 47 + – 47 = 23 rather than 47 – 47 + = 23.
5 + 500 000 - 500 000 =
254 268 + 59 - 59 =
= 79 + 11 - 79 + 0 + 79 - 79 If students are struggling to see the additive identity embedded in a problem, have them complete open number sentences which only use one number and zero. Resource Number sentences II is a Level 3 resource which explores this idea and Number sentences III is a parallel Level 2 resource exploring the same concept.
Although students seem to have an intuitive understanding of what happens when zero is added to, or subtracted from, a number, it is important to have them articulate, share and critique these understandings as conjectures or rules about zero. Agreed-upon conjectures or rules can then be recorded on separate pieces of paper for display. The student who comes up with the rule may even have their name attached to it (e.g, "Rebecca's Rule").
One way to elicit these rules is to discuss number sentences and decide whether they are True or False. Resource Looking at zero explores this idea. For further information and examples of student generated number sentences, refer to the Algebraic thinking concept map.
Many of the common errors listed above for parts c) and d) come from a misunderstanding of the equals sign. Students 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.
Students who hold a purely arithmetical view of the equals sign meaning "and the answer is" will often ignore operators and numbers in a number sentence, focusing only on numbers preceding the equals sign. It is important for students to realise that the equals sign does not always need to come towards the end of a number sentence. Equal number sentences II and Equality are Level 3 resources which explore the concept of equality through True/False number sentences and completing open number sentences.
For further information on the ideas surrounding equality, refer to the Algebraic thinking concept map