Equal sharing III
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a)
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Show how you could equally share 24 ice creams between 4 people.
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b)
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If 30 Christmas trees are equally shared between 5 shops, show how to work out how many trees each shop gets.
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c)
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42 teddy bears are to be equally shared between 6 children. Show how to work out how many teddy bears each child gets.
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d)
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If 72 students going to the zoo are equally spread between 8 vans, show how to work out how many students are in each van.
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| Y4 (11/2006) | ||
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a)
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6 Working that involves any 1 of these strategies:
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easy easy |
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b) |
6 Working that shows any of the 6 strategies shown for part a). |
easy easy |
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c) |
7 Working that shows any of the 6 strategies shown for part a). |
moderate moderate |
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d) |
9 Working that shows any of the 6 strategies shown for part a). |
difficult moderate |
Based on a representative sample of 175 Y4 students.
NOTE: Some students may also have used a more sophisticated method in their head, but "showed their working" by using diagrams and counting or grouping visual sets.
This resource is about showing how to solve equal sharing problems.
| Common error | Likely calculation | Likely misconception | |
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a) b) c) d) |
4 5 6 8 |
Students worked out the other factor in the equation, but wrote down the factor they were already given. | |
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a) c) d) |
8 8 7 |
"8 × 4 = 24" not 32 "8 × 6 = 42" not 48 "7 × 8 = 72" not 56 |
Inadequate basic facts knowledge. Gets a wrong member of the relevant table. |
The results of the trial indicated that while many students were able to calculate the correct answer, some had difficulty showing how they had solved the problem.
This could be because they:
- believe that the purpose of maths is to find the right answer, not to show how they got there; or
- have difficulty putting their method on paper; or
- are not used to showing their working.
Student strategies
The strategies that students use can determine their stage of development within the Number Framework. Strategies that could be used for this resource include counting out, one-to-one mapping, visual grouping, (partitioning and trial and error), repeated addition/subtraction, skip counting, multiplication, and division.
Usage of strategies
Each strategy was used by approximately equal percentages of the students who described strategies in each part of the question, suggesting students largely employed one strategy throughout the four parts. However, fewer students were able to describe additive or visual strategies for the latter parts of the question where the numbers of objects involved was higher. This indicates that as the questions got more difficult, the value of additive or visual strategies dropped off because the numbers became unmanageable. Usage of multiplicative strategies remained fairly constant and it more easily accommodated larger numbers of objects.
Success rates using the strategies
- The most sophisticated strategies (division and multiplication) were used by approximately equal numbers of students (14% for each). These students consistently had the highest success rates on each of the parts of the question (87% and 91% respectively).
- About 55% of strategies described were representations or drawing of objects, groupings, or visualisations. This was the most common strategy. The success rates for this strategy was higher for the first two questions (77% and 81%) where the numbers in the problem were smaller, but dropped for the second two questions (64% and 60%), where the numbers in the problem were higher, and physical drawing and grouping became more difficult.
- Repeated addition was relatively uncommon (5% of strategies). The success rate was highest at 81% for part a) but steadily dropped to 25% for part d) as there were larger numbers of objects to share. This illustrates the importance of students moving on from additive strategies and using multiplicative strategies.
- No strategy. The percentage of students giving no strategy rose steadily from 21% for part a) to 47% for part d). The success rate of these students steadily dropped from 58% in part a) down to 12% for part d). This demonstrates that students who can communicate a strategy are more able to cope with more sophisticated problems of this type.
Students should be encouraged to develop an increasing repertoire of strategies, and move towards using a multiplicative strategy to solve these problems.
Students who show only the solution may not be aware of the importance of showing the processes they go through to find those solutions. Encourage students to:
- record all their workings, as the strategy students use to solve the problem can indicate their stage of numeracy development.
- share with others how they got their answers.
- critique other strategies and try to use the most sophisticated or effective strategy for the type of problem and the numbers in the problem.
Students who did not correctly solve the equal sharing problem may need to represent the problem using counters, and could also be further encouraged to write or draw their strategy. Students who correctly used marks to count out or map the items to each person could be encouraged to develop another way to solve the problem without one-to-one counting, e.g., skip counting, grouping, repeated addition. For students who used skip counting or repeated addition, encourage them to recognise the relationship between groups of numbers and multiplication, e.g., that 3 + 3 + 3 + 3 + 3 + 3 is "6 groups of 3" and that can be written as "6 × 3".
If students are recognising the multiplicative relationship between the total and the number of groups, and are using reverse multiplication (e.g., 6 × 3 = 18) or division (18 ÷ 6 = 3) to solve, then they could try solving similar problems using more complex numbers.
Quotitive division
All the questions in this resource are framed as partitive division (equal sharing) questions. If students are indicating a good understanding and use of strategies to solve these problems, give them some problems framed as quotitive division:
Questions could be asked in the form of finding the number of sets:
"If there are 27 dogs and each family is getting 3 dogs, how many families are there?"
this is often not as easily recognised as a division problem.
Student strategies
Table 1. Frequencies and percentages of strategies in each part of the question
| Strategy used |
Part a) Freq % |
Part b) Freq % |
Part c) Freq % |
Part d) Freq % |
TOTAL Freq % |
| Division | 16 12% | 17 13% | 19 17% | 16 17% | 68 14% |
| Reverse multiplication | 16 12% | 23 18% | 13 11% | 13 14% | 65 14% |
| Repeated adding (or -) | 6 4% | 6 5% | 6 5% | 4 4% | 22 5% |
| Drawing, grouping, or visual | 86 62% | 68 53% | 63 55% | 43 46% | 260 55% |
| 1-1 mapping | 4 3% | 4 3% | 2 2% | 1 1% | 11 2% |
| Halving | 4 3% | 0 0% | 0 0% | 0 0% | 4 1% |
| Other | 7 5% | 10 8% | 11 9% | 15 16% | 43 9% |
| TOTAL strategies1 | 139 100% | 128 100% | 114 100% | 92 100% | 473 100% |
| No strategy given2 | 36 21% | 47 27% | 61 35% | 83 47% | 227 32% |
| TOTAL students | 175 | 175 | 175 | 175 | 700 |
Based on a representative sample of 175 students.
NOTES:
1. All percentages listed above this line in the table are based on the Total number of strategies for that part. Examples:
- The percent of strategies for part a) that use division is 16/139 (12%).
- The percent of strategies overall that use division is 68/473 (14%).
These percentages look at the relative frequency of different strategies, compared with the total number of strategies given.
2. All percentages in this line are based on the Total number of studentsfor that part. Examples:
- The percent of students who gave no strategy for part a) is 36/175 (21%).
- The total percent of students who gave no strategy is 227/700 (32%).
These percentages look at the relative frequency of different strategies, compared with the total number of strategies given. Interpretation of the relative number of students giving each strategy can be made from the raw frequencies, as the denominator is 175 for each part (or 4 times 175 = 700 in the Total column).
Table 2. Frequencies and percentages of successful strategies in each part
| Strategy used |
Part a) Freq % |
Part b) Freq % |
Part c) Freq % |
Part d) Freq % |
TOTAL Freq % |
| Division | 13 81% | 16 94% | 16 84% | 14 88% | 59 87% |
| Reverse multiplication | 13 81% | 21 91% | 13 100% | 12 92% | 59 91% |
| Repeated adding (or -) | 5 83% | 4 67% | 3 50% | 1 25% | 13 59% |
| Drawing, grouping, or visual | 66 77% | 55 81% | 40 64% | 26 60% | 187 72% |
| 1-1 mapping | 4 100% | 4 100% | 1 50% | 0 0% | 9 82% |
| Halving | 3 75% | - - | - - | - - | 3 75% |
| Other | 5 71% | 5 50% | 4 36% | 3 20% | 17 40% |
| TOTAL strategies1 | 109 78% | 105 82% | 77 68% | 56 100% | 347 73% |
| No strategy given2 | 21 58% | 21 45% | 16 26% | 10 12% | 68 30% |
| TOTAL students | 130 74% | 126 72% | 92 53% | 66 38% | 414 59% |
All percentages are based on denominators from the frequencies in Table 1. Examples:
- In part a), 13 out of 16 (91%) students giving division strategies got the correct answer.
- In total [i.e., averaged over parts a) to d)] 59 out of 68 (87%) of students who used a division strategy got the correct answer.
- 109 of the 130 (78%) students giving any strategies in part a) got the correct answer.
- Overall, 347 of the 473 (73%) strategies mentioned in parts a) to d) led to the correct answer.
- 130 of the 175 students (74%) got a correct answer for part a).
Number Framework
Equal sharing is addressed at stages 2-4 (from Counting from one to Advanced counting) in the Number Framework (strategies) under the operational domain of Proportions and ratios. Students using repeated addition are indicating stage 5 (Early additive) strategies, and students using multiplicative strategies are indicating stage 6 (Early multiplicative).
These equal sharing (division) problems are regarded as basic facts knowledge at stage 6 (Early multiplicative).