Fractions and food
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a)
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Anna ate \(1 \over 4\), Jack ate \(3 \over 8\), and Pete ate \(1 \over 8\) of a banana cake. Show or explain how to work out how much cake they ate altogether.
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b)
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Dan and Mere were given a cupcake each. Dan ate \(4 \over 5\) of his and Mere ate \(2 \over 3\) of hers.
Show or explain how to work out how much cupcake they ate altogether.
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c) |
At a party Rosa ate 1\(1 \over 2\) and Kim ate 2\(2 \over 3\) bars of chocolate. Show or explain how to work out how many bars of chocolate they ate altogether.
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d) |
There were 3\(1 \over 3\) cakes left after a party. Paddy ate \(5 \over 6\) of a cake. Show or explain how to work out how much cake was left.
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| Y10 (09/2008) | ||
| a) |
3/4 or 6/8 or 9/12 Working that involves
e.g., 1/4 × 2 to get 2/8
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easy easy |
| b) |
17/15 or 22/15 Working that involves
e.g., 4/5 × 3 + 2/3 × 5 to get 12/15 + 10/15 ;
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moderate moderate |
| c) |
41/6 or 37/6 or 25/6
e.g., 3/2 × 3 + 8/3 × 2 to get 9/6 + 16/6 ;
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moderate moderate |
| d) |
21/2 or 23/6 or 15/6
e.g., 10/3 × 2 to get 20/6 , and 20/6 – 5/6 ;
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moderate moderate |
Based on a representative sample of 161 students.
| Common error | Likely misconception | |
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a) b) c) d) |
5/20 6/8 or 3/4 33/5 36/9 |
Whole number error – not adding numerators and denominators correctly Adds all numbers across top and bottom without attempting to develop the common denominator, e.g., 1/4 + 3/8 + 1/8 = (1 + 3 + 1) / (4 + 8 + 8). |
| b) | 11/15 or 22/30 |
Whole number error with understanding of equivalence Students found the common denominator for the equivalent fraction, but then added the denominators, i.e., 12/15 + 10/15 = (12+10)/(15+15)= 22/30 , and some students simplified it to 11/15 . |
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b) c) |
11/2 41/2, 41/3, 41/4, 41/5, 41/8 |
Approximate answer Students giving an approximate answer (from either an insufficiently detailed diagram or showing approximate amounts in their working). |
Students who showed no working at all also had a very low success rate for answering the question correctly or at all. This indicates that they were not able to answer the question. Students at this level of understanding could explore the concepts of partitioning and fractions as part-whole relationships before attempting to add (or subtract) fractions. Students could also be encouraged to draw what they understand the question to be asking without concerning themselves with fractional quantities (until appropriate).
Only working shown is information from the question
Students who repeated the information from the question may not be aware of the importance or value of showing working. Many of these students got a correct answer indicating that they were capable, but did not write their strategy. As multiple strategies is an important aspect of contemporary mathematics pedagogy, strategy sharing is also important and value needs to be placed on encouraging students to communicate in a mathematical way (see whole class discussion).
Whole number error – not adding numerators and denominators correctly
Students who have any of the whole number misconceptions identified above need to develop a part-whole understanding of fractions before trying to devise a system to add or subtract fractions. If required, students could go back to partitioning and explore constructing the parts (unit fractions), combining these parts to make non-unit fractions, naming these new fractions (part-whole fractions), and then discussing the meaning of the denominator and numerator and what happens to them when fractions are added.
Whole number error with understanding of equivalence
Students who showed this misconception used their understanding of equivalence to rename the fractions to have common denominators, and then added the renamed fractions incorrectly, indicating an incorrect understanding about addition of fractions. These students could explore what is happening when they add simple fractions, e.g., 1/2 and 1/4 , and what happens to the denominator and numerator. They could also explore counting up sequences of a unit fraction to see what happens when the same fraction is added, e.g., 1/4 , 2/4 , 3/4 , 4/4 , 5/4 – with an equivalent denominator, the numerators are added without affecting the denominator.
Using diagrams to show how they added/subtracted the fractions
For students who constructed diagrams to show how to work out these fraction problems, ask students to explain how they know the diagram they have drawn accurately shows the fraction they need to use. Then get them to share with other students how the parts can be added or subtracted (because they are the same size, we can add/subtract fractions). Encourage students to use fractional notation alongside to illustrate their understanding about what happens to the denominator and numerator when fractions are added or subtracted.
The lower success rate for students using diagrams, and the fact that it was a strategy employed overall by less mathematically capable students indicates the importance of being able to represent working using fractional notation. Diagrams may be good representations but students need to develop understanding of fractions as a type of number (rational number) with a particular difference to whole numbers.
Approximate answer
Some students estimated or gave an approximation of their answer from a diagram. Whilst the idea of fraction size through estimation is an important understanding (see NM1262), this resource is about what happens to fractions when they are added or subtracted. Students could be encouraged to explore simpler fractions or fractions sequences (see above for Whole number error with understanding of equivalence).
Strategies
The most common working that students showed were:
- Showing working out the equivalent fractions as a step (26%);
- Showing cross multiplication to create equivalent fractions (about 12%);
- Writing the equivalent fractions without any indication of how they were worked out (5%);
- Drawing a diagram (11%);
and
- Repeating information from the question (10%);
- 15% did not show any working.
NOTE: percentages are averaged across all four questions.
The success of the strategy depends on the type of the question and the fractions being added or subtracted. Essentially the working that students showed in the first three strategies is the same "to find the equivalent fractions and add them". Although overall the success rates for these three strategies was similar, there was still variation between questions. This indicates the importance of students being able to select the most efficient for the particular mathematical question.
For example: notably fewer students used cross multiplication for question d) 31/3 – 5/6 .
Here some sort of renaming and then chunking could be a more efficient strategy to solve the problem:
31/3 becomes 32/6 and then separate out the 5/6 into 2/6 and 3/6 ,
so we have 32/6 – 2/6 – 3/6 (or a half);
-> (32/6 – 2/6) – 1/2 ;
-> 3 – 1/2
-> 21/2
This understanding of other efficient strategies can be scaffolded in whole class discussion where students share and critique their own and others' strategies to develop a fuller understanding. To promote understanding students should be able to use a range of strategies to show addition and subtraction of fractions, and work towards non illustrative strategies to consolidate this understanding.
Strategies and success rates
| a) | b) | c) | d) | |
| Showing cross multiplication to create equivalent fractions | 12 (95) | 17(74) | 13(95) | 7(75) |
| Showing the equivalent fractions as a step | 37 (98) | 22(74) | 24(92) | 22(86) |
| Writing the equivalent fractions | 9 (93) | 4(100) | 4(83) | 3(100) |
| Drawing a diagram | 16 (68) | 9(7) | 10(25) | 10(56) |
| Other correct | 2 (-) | 1(-) | 1(74) | 2(-) |
Based on a representative sample of 161 Y10 students.
NOTE: the numbers given are percentages of students that used that strategy and in brackets: (percentage with correct answer for that question). For example 12% of students used a showing cross multiplication to create equivalent fractions strategy and of those 95% also gave a correct solution.