Finding fractions II
Answer the four questions below and show how you worked it out. Do not use a calculator to answer the questions.
a) At a fancy dress party \(1 \over 4\) of the 80 people had costumes. Show how to work out how many people had costumes at the party
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__________ answer |
b) \(2 \over 3\) of the class of 24 students owned cell phones. Show how to work out how many students in the class owned cell phones.
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__________ answer |
c) There are 180 students. \(6 \over 10\) of them walk to school. Show how to work out how many students walked to school.
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__________ answer |
d) There are 150 students at the school. \(4 \over 5\) of the students went to the school camp. Show how to work out how student went to camp.
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__________ answer |
- Using counting or grouping strategies to correctly solve the fraction problems.
- Using additive strategies such as repeated addition to correctly solve the fraction problems.
- Attempting to use multiplicative strategies, but made a simple miscalculation or operator error.
- Correctly using reverse multiplication or division strategies to solve these fraction problems, or uses just 1 strategy.
- Correctly using a range of multiplicative strategies (e.g., partwhole splitting, equivalence) that indicate understanding.
| Y8 (10/2010) | ||
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a)
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20 Working involving:
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very easy very easy |
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b)
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16 Working involving:
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easy easy |
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c)
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108 Working involving:
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moderate moderate |
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d)
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120 Working involving:
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moderate moderate |
Based on a representative sample of 184 Y8 students.
NOTE:
- Of the students who used Multiplying by numerator divided by the denominator, almost all students worked out the division first then multiplied.
- The main focus for the working part of these questions is on the use of an appropriate strategy, i.e., the communication of the attempted calculation as well as a stepwise sequence that could lead to a correct answer.
This resource is about fractions as operators which can be seen as a different "personality" of fractions. This involves an understanding that a fraction can act as an operation upon a quantity (in this case a whole number). The quantity is treated as "the whole" which the fraction needs to be found. Ideally, this idea follows on from the understanding that fractions represent part-whole relationships (see part-whole understanding of fractions). Fraction as operator questions can be of two forms: find the whole given the part and find the part given the whole. These questions were all of the type: find the part given the whole.
Prior knowledge
Students should have an understanding of fractions as part-whole relationships for regions, sets and whole numbers, and have explored finding simple fractions of smaller numbers.
Students solved these fractions as operator questions in a variety of ways: some used the classical method of multiplying the number by the numerator and dividing by the denominator, others used reverse multiplication, repeated addition or diagrams.
| Common response | Likely misconception | |
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a) b) c) d) |
76 (80 – 1 × 4) 18 (24 – 2 × 3) 120 (180 – 6 × 10) 130 (150 – 4 × 5) |
Misconception about the part-whole nature of fractions Students recognise that the problem is about reducing the quantity by a fraction but show a misconception about what the fraction notation means and that it is a multiplicative reduction not additive (subtraction). |
| b) |
6 (1/4 of 24) 4 (half of 1/3 of 24) 12 (half of 24) |
Students worked out some other fraction of the whole Students work out some other unit fraction of the quantity |
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a) b) c) d) |
40 8 (1/3 of 24) 18 (1/10 of 180) 30 (1/5 of 150) |
Incomplete calculation (correct first step) Students work out the first step (finding the unit), but do not increment the unit to find the fraction of the quantity, e.g., for question b) they find 1/3 of 24, but not 2/3 . |
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b)
c) |
4 [24 ÷ 3 ÷ 2 or 24 ÷ (3 × 2)] 3 [180 ÷ 10 ÷ 6 or |
Incorrect fraction calculation (uses incorrect operator in working) Students work out that this problem involves division but either use division on both numerator and denominator or perform their calculation in the incorrect order. |
Students worked out some other fraction of the whole
Students who worked out other incorrect solutions to the problems may have a misconception about the part-whole nature of fractions and need to develop their understanding about what a fraction is before looking at fractions as operator. They could als look at simpler unit fraction problems and smaller quantities (e.g., 3/4 of 12, 1/5 of 15, and If 2/3 is 6) or problems involving sets of counters, e.g.,Shading fractions of sets and shapes, Fraction soup (animation) or Partitioning sets to explore fractions of sets. The resource, Farm fractions, explores simpler fractions of smaller quantities.
Insufficient or incomplete or no working shown working
Students who showed no working or showed their working as repeating the stem or summarising the operation (e.g., 1/4 × 80) may have not experienced the need for showing their working or not believe it is important to show working if they know the answer. Developing the ability to make a mathematical statement (such as how they worked it out) is a valid part of the mathematics learning. It can also provide significant formative information about their learning needs. Students may need to discuss what constitutes working (or an argument), or even that working is their rationale for their answer. Students could share their working and identify if it is sufficiently
explicit to be a justification for the answer they have given. As the use of multiple strategies is an important aspect of contemporary mathematics pedagogy (as well as a key indicator of an advanced stage in the Number Framework), value needs to be placed on encouraging students to communicate their strategies, both orally (see Mathematical classroom discourse for more information about this), and in writing (see Writing an explanation).
Incomplete calculation
Many of these students completed the first step of a strategy that could have worked to give them a correct result if they had continued. These students could be asked to explain what the question asked, and how they solved it, to a peer. This process of sharing can highlight the discrepancy between what the question asked and what they have done (so far). Students who find it difficult to make the next steps could work with problems that work in two steps: the first to find a unit fraction; and the second to increment the unit fraction. The second step requires students to add or multiply fractions (by whole numbers). If they have difficulty with this step they may need to explore counting up in fractions (e.g., 1/5 , 2/5 , 3/5 , etc) and look at the similarities and differences with Natural (counting) numbers. This will start to develop their understanding of how fractions increment and how they can be added. These students could also look at the resource Farm fractions which explores simpler fractions of smaller quantities.
Incorrect fraction calculation
Students who do not apply the operation of the fraction correctly (divide twice or calculate in an incorrect order) could first look at sharing (explaining) their working with a peer. This can serve as a check to ensure they self check. Students may need to develop some idea about the part-whole nature of fractions which could be eplxored through the construction of fractions in partitioning exercises [e.g., Sharing shapes, Partitioning pizza & fruit loaf, and Sharing cake and pizza]. This can help them to recognise the division relationship of fraction notation. Students should make sure they are aware of the order of operations and how to solve expressions that involve both division and multiplication [e.g., 24 ÷ 3 × 2] and how to prioritise the operations. They could explore some simple whole number examples and compare what happens when order of operations is upheld to when it is not (e.g., resources: Subtraction & division boxes,Using brackets, or Using brackets II).