How to work out the answer
| Y4 (11/2006) | |||
|
a) |
i) |
25 |
easy |
|
b) |
Explanation includes reference to the additive identity within the equations, i.e., a – a = 0, and therefore calculations are not required to solve the problems. |
very difficult |
|
This resource explores the application of the additive identity property, a – a = 0, in the context of solving problems. Understanding and applying the additive identity is important for future experiences that involve the solving of algebraic equations.
If students use calculation, ask them if they can see an easier way to do each problem. They could write this down under the box.
Success on all, or most, of the questions in part a) does not necessarily indicate the student has employed the concept of the additive identity. If their explanation in b) is unclear, check orally how they did part a).
| Common error | Likely calculation | Likely misconception | |
|
a) i) ii) iii) iv) v) |
57 204 198 166 235 |
25 + 16 + 16 28 + 36 + 36 + 52 + 52 62 + 74 + 62 78 + 44 + 44 67 + 23 + 55 + 23 + 67 |
Ignores subtraction signs and adds all available numbers. |
| a) i) | 47, for example | Adds all available numbers but makes a calculation or place value error. | |
|
a) i) ii) iv) |
41 80 122 |
25 + 16 28 + 52 78 + 44 |
Only adds some, or all, positive numbers. |
|
a) i) ii) iii) iv) v) |
23, 24, 26 or 27 24 – 27, 29 – 32 72, 73, 75 or 76 76, 77, 79 or 80 51 – 54, 56 – 59 |
Uses counting on and counting back with some errors included. | |
|
a) ii) iii) iv) v) |
0 0 0 0 |
52 – 52 or 36 – 36 + 52 - 52 62 – 62 ¯44 + 44 67 + 23 – 23 – 67 |
Ignores the number with no additive inverse. |
|
a) i) ii) iii) iv) |
36 or 52 62 44 67 or 23 |
Gives one of the numbers that has an additive inverse. |
Use of any of the first four types of errors implies a student is not employing the concept of additive identity. Either of the latter two indicates that the student may have some concept of additive identity, but does not apply it consistently.
Student strategies
Usage and success rates using the additive identity concept
In our sample, 37% of students included at least something in part b) to indicate that they were employing the concept of the additive identity. These accounted for 43% of the strategies students described.
- 13% of students had an explanation including reference to the concept of the additive identity and no calculations were used. These students got an average of 94% of their answers in part a) correct, and maintained their success throughout the five parts of the question.
- 19% applied the additive identity rather than a calculation but were unable to clearly explain what they had done. 85% of these students' answers in part a) were correct, but they were slightly less successful with the harder questions.
- 5% had an explanation that included reference to the additive identity but they also did some calculations. Only 67% of these students' answers in part a) were correct, but they were less successful with the harder questions.
The remaining 63% of students described computational methods, other methods, or gave no explanation in part b). These accounted for 57% of the strategies students described. These students had lower success rates that varied from 0% to 68%, and their success rates generally dropped off on the harder questions.
If students are able to identify and apply the additive identity property, 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.
Students who are able to apply the additive identity but are not able to explain it in writing may be able to express their understanding verbally. Alternatively, use a resource such as Looking at zero II or Equal number sentences II to explore what happens when you use zero in a number sentence. This can lead to the development of conjectures or rules about zero which may further help students articulate their understanding.
Those students who ignored numbers and/or operation signs to work out their answer may 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. Exploring the idea of equality can be done using resources What is equal? and Equal number sentences.
Further information about the additive identity property and examples of student-generated problems can be found in the Algebraic Thinking Concept Map.
Appendix of student strategies for Solving problems (AL7123)
Table 1. Frequencies and percentages of strategies in each part of the question
| Description of strategy in part b) | Number of students | Percent of strategies1 | Percent of students2 | Success rate3 |
Additive identity strategies
Explanation includes reference to the additive identity but some calculations are also included. |
24 35 9 |
15% 22% 6% |
13% 19% 5% |
94% 85% 67% |
Computation strategy
Incorrect calculation (e.g., ignores some numbers or operators) |
47 8 2 23 4 |
29% 5% 1% 14% 2% |
25% 4% 1% 12% 2% |
49% 68% 50% 32% 0% |
| Other explanations | 10 | 6% | 5% | 54% |
| TOTAL strategies used | 162 | 100% | 86% | 62% |
| No explanation | 26 | 14% | 27% | |
| TOTAL | 188 | 100% | 100% | 57% |
Based on a representative sample of 188 students.
- Percentages based on the frequency of total strategies used (162).
- Percentages based on the frequency of total students (188).
- Averaged over 5 questions in part a) – see final column of Table 2.
Table 2. Frequencies and percentages of questions correctly answered
| Description in part b) |
a) i) Freq % |
a) ii) Freq % |
a) iii) Freq % |
a) iv) Freq % |
a) v) Freq % |
TOTAL Freq % |
Additive identity
|
23 96% 35 100% 8 89% |
24 100% 31 89% 7 78% |
23 96% 33 94% 7 78% |
20 83% 26 74% 5 56% |
23 96% 23 66% 3 33% |
113 94% 148 85% 30 67% |
Computational
|
32 68% 6 75% 1 50% 11 48% 0 0% |
28 60% 5 63% 2 100% 10 43% 0 0% |
19 40% 4 50% 1 50% 4 17% 0 0% |
28 60% 6 75% 0 0% 8 35% 0 0% |
9 19% 6 75%
1 50% |
116 49% 27 68% 5 50% 37 32% 0 0% |
| Other explanations | 8 80% | 8 80% | 3 30% | 5 50% | 3 30% | 27 54% |
|
TOTAL Strategies |
124 77% | 115 71% | 94 58% | 98 60% | 72 44% | 503 62% |
| No explanation | 11 42% | 8 31% | 5 19% | 9 35% | 2 8% | 35 27% |
| TOTAL students | 135 72% | 123 65% | 99 53% | 107 57% | 74 39% | 538 57% |
Based on a representative sample of 183 students.
1. Percentages in first five columns based on Number of students column in Table 1.
E.g., 96% = 100 × 23/24
77% = 100 × 124/162
42% = 100 × 11/26,
72% = 100 × 135/188
2. Percentages in last columns based on Number of students column in Table 1 multiplied by 5 (to reflect the total number of possible strategies).
E.g., 94% = 100 × 113/(24 × 5)
62% = 100 × 503/(162 × 5)
27% = 100 × 35/(26 × 5)
57% = 100 × 538/(188)
NEMP
Report 37: Mathematics 2005. Number Helper, p.15, Questions 4 + 5. Just over 50% of Year 4 students were able to identify 0 as the number you can add or take away from 8 and 8 will stay the same.