Proportional Reasoning Strategies progression, 4th step
The purpose of the activity
In this activity, the learners develop an understanding of how to find a fraction of a whole number where the answer is also a whole number.
The teaching points
The use of the language “3 parts of 7 equal parts” is the key to establishing meaning for the fraction 3/7.
The bottom number (denominator) of a fraction names the number of equal parts, while the top number (numerator) of a fraction tells how many of these parts.
The learners will understand that if you multiply a number by a fraction greater than 1, you will get a result that is greater than the number (4 x 4/3 = 16/3 = 5 1/3). Alternatively if you multiply a number by a fraction less than 1, you will get a result that is smaller than the number (4 x 2/3 = 8/3 = 2 2/3)
Discuss with the learners any misconceptions they may have had about the terms used.
Discuss with the learners relevant or authentic situations where fractions are used.
The guided teaching and learning sequence
1. Draw a bar (rectangle) on the board and ask a volunteer to place a mark to indicate where 3/4 of the bar is. Ask:
“How do you know that it is 3/4?” (Listen to ensure the learners know that finding a quarter involves splitting the bar into 4 equal parts.)
2. Draw a line below the bar and on it record 3/4 . Ask the learners to tell you what numbers to place at each end of the bar (0, 1). Discuss with the learners the fact that if the bar represents 1 unit, then its end points are 0 and 1. Record 0 and 1 on the bar.
3. Write 28 at the right end above the bar and ask the learners if they can work out what number is 3/4 of 28. Check the learners understand that they must find the number that lies 3/4 of the way along the bar. Ask the learners to share their answers and the ways they worked it out:
“What is 3/4 of 28?”
“How did you work that out?”
4. Ensure that all the learners understand how they can use the unit fraction (1/4) of a number to help them work out a non-unit fraction (3/4) of that number. For example 3/4 of 28 = 3 x 1/4 of 28.
5. Depending on the learners’ level of understanding, you may want to work through one or two more examples as a class before asking them to solve problems independently. Possible examples for further guided learning or independent work are:
4/5 x 30
5/7 x 42
2/3 x 18
4/9 x 27
The rest of this guided teaching and learning sequence repeats the above steps with fractions greater than 1. Depending on the learners’ confidence with fractions less than 1, you may decide to complete the rest of this teaching and learning sequence in a future session.
6. Write 6/5 on the board and ask the learners to explain this fraction. Encourage them to note that the 5 tells that there are 5 equal parts (fifths) and that there are 6 of them. This means that the fraction is larger than 1.
7. Write 6/5 x 30 on the board. Ask the learners to think about whether 6/5 of 30 (6/5 x 30) will be a number that is greater than or less than 30.
'“Is 6/5 of 30 larger or smaller than 30?”
“Why is it larger than 30?”
8. Discuss the similarities between this problem and the problems the learners have solved with fractions less than 1.
“Is this problem harder?”
‘Can we solve it the same way?”
“What is another name for 6/5?” (1 1/5)
9. The learners should be able to transfer their understanding from the easier problems to this problem and see that 6/5 of 30 is the same as 6 x 1/5 of 30 or 36. If not, work through the following representations.
10. Draw a bar on the board and label the end 30. Ask the learners to indicate where 6/5 of the bar would be. From the previous discussion, the learners should understand why the answer is going to be larger than the 30.
11. Draw a line below the bar that extends at least to where the learners indicated 6/5 of the bar would be and ask the learners to indicate where 0 and 1 are on the line. Ensure they understand that the 1 or unit in this problem is the 30.
12. Ask the learners if they think the previous mark for 6/5 of 30 is in the correct position and why.
13. Once more, ensure all the learners understand how they can use the unit fraction (1/5) of a number to help them work out a non-unit fraction (6/5) of that number. For example, 6/5 of 30 = 6 x 1/5 of 30 = 36.
14. Depending on your learners’ level of understanding, you may want to work through one or two more examples as a group before allowing the learners to work independently.
15. Pose problems for the learners to solve. Encourage them to share their solutions with others:
6/4 x 32
8/3 x 9
5/2 x 14
7/10 x 30
Finding fractions of numbers where the answer is also a fraction is covered in the “Fractions of numbers 2” activity.
If your learners have access to the Internet, they could practise the ideas developed in this activity by using the Fraction Bar learning object that is available from here:
This learning object poses problems that involve fractions less than 1 and problems that involve fractions less than 2. The learning object tracks the number of problems answered correctly, so you could challenge the learners to answer five problems that involve fractions less than 1 and then five problems that involve fractions less than 2.
If you are using the learning object, check the learners understand the bar represents the number that they are trying to find the fraction of and that the line below represents an ‘amount’ of that number. Learners at this stage should be familiar with the concept of a unit, and it is important that they realise that, for the purposes of this learning object, the number line from 0–1 represents the unit. Explain that for every problem, if they ‘just know’ the answer, they can always enter it, but otherwise, the steps they need to follow to solve the answer are similar to the ones presented in the guided lesson sequence above.
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