Proportional Reasoning Strategies progression, 5th 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 may also be a fraction.
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 original 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 original number (4 x 2/3 = 8/3 = 2 2/3).
Discuss with the learners ways to remember the names and functions of the fraction parts (denominator, numerator) and how to remember them.
Discuss with the learners relevant and authentic situations where fractions are used.
The guided teaching and learning sequence
1. Draw a bar (rectangle) on the board and record a 7 above the bar.
2. Break the bar into 7 ones.
3. Ask a volunteer to draw a line on the bar that shows where 2/3 of 7 is. Note that there are two possibilities. One is a vertical line 2/3 of the way along the bar which is somewhere between the fourth and fifth break. The second is a horizontal line that is 2/3 of the way down the bar. Acknowledge both possibilities and then explain that you are going to use the horizontal line.
4. Ask the learners to show where 1/3 of 7 would be as a horizontal line and record 1/3 and 2/3 and 3/3(1) on the left-hand side of the bar.
“What is 2/3 of 7?” (Check that the learners understand that this is represented by the area above the 2/3 line. Also check that the learners understand that this is the same as 2 lots of 1/3 of 7).
Ask the learners to share how they would work out 14 thirds (14/3) or 4 and 2/3 of 7.
6. Check that all the learners understand that the ‘smallest pieces’ on the bar each represent 1/3 of one whole. Shade in one of the pieces and explain that this is 1/3 and that there are 14 thirds (14/3) in the area above the line.
7. Check all the learners understand how to convert 14 thirds (14/3) to 4 and 2 thirds (4 2/3). Ask: “Why is 14 thirds equal to 4 and 2 thirds?” (Listen for explanations that use the understanding that 3 thirds are 1. For example, 3 thirds are 1 and therefore 14 thirds are 4 (ones) and 2 thirds or 14 thirds divided by 3 gives 4 ones with 2 thirds remaining.)
8. List the different ways of recording this on the board:
2/3 x 7 = 14/3
2/3 x 7 = 4 2/3
Two-thirds of seven equals 14 thirds or four and two-thirds.
9. Depending on your learners’ level of understanding, you may want to work through one or two more examples as a class before asking the learners to solve problems independently. Possible examples for further guided learning or independent work are:
2/5 x 8
3/4 x 15
2/3 x 16
3/7 x 27.
The rest of this guided teaching and learning sequence repeats the above steps with fractions greater than 1. Depending on your learners’ confidence with fractions less than 1, you may decide to complete the rest of this teaching and learning sequence in a future session.
10. Write 5/4 on the board and ask the learners to explain this fraction. Encourage them to note that the 4 shows that there are 4 equal parts (quarters) and that there are 5 of them. This means that the fraction is larger than 1.
11. Write 5/4 x 13 on the board. Ask the learners to think about whether 5/4 of 13 will be a number that is greater than or less than 13.
“Is 5 quarters of 13 larger or smaller than 13?”
“Why is it larger than 13?”
12. 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 5 quarters?”
(1 1/4 or one and one-quarter)
13. The learners should be able to transfer their understanding from the easier problems to this problem and see that 5/4 of 13 is the same as 5 x 1/4 of 13 or 65/4 or 16 1/4 . If not, work through the following representations.
14. Draw a bar on the board and label the end 13. Ask the learners to indicate where 5/4 of the bar would be and explain their reasoning. From the previous discussion, the learners should understand why the answer is going to be larger than the 13.
15. Once more, ensure that all the learners understand how they can use the unit fraction (1/4) of a number to help them work out a nonunit fraction (5/4) of that number. For example, 1/4 of 13 = 13/4 (see shading).
16. If the learners understand that 1/4 of 13 is 13 quarters, then they should be able to understand that 5 quarters of 13 is 5 times this amount (13 x 5 = 65 quarters).
17. Depending on your learners’ levels of understanding, you may want to work through one or two more examples as a group before allowing the learners to work independently.
18. Pose problems for the learners to solve. Encourage them to share their solutions with others.
5/3 x 7
3/2 x 21
4/3 x 16.
Pose problems for the learners to solve independently or in pairs.