In this unit we explore fractions of regions as well as fractions of sets. We look for and develop understanding of the connection between fractions and division.
- Find fractions of regions.
- Find fractions of sets.
- Identify equivalent fractions.
- Locate fractions on a number line.
This unit introduces a number of important concepts relating to fractions. The first of these is that fractions represent parts of regions and sets and so are useful in a large variety of situations. Whole numbers are not enough by themselves to enable us to cover all situations that involve number. Fractions are probably the first departure from whole numbers that students will see. They give a numerical way of representing parts of lengths, regions, areas, and any set of discrete objects including money transactions.
The second useful concept is that a given number can be represented as a fraction in many ways. Knowing that fractions such as ½ can be disguised as 2/4 or 3/6, etc. is important both for recognition purposes and for use in calculations.
Finally students should know that fractions can be represented both as one whole number divided by another whole number and as points on the number line. Having a knowledge of the different representations of fractions provides connections across mathematics for students and so increases their level of understanding.
This unit includes a wide range of experiences and a number of important concepts. It would be expected that these experiences may be developed over a number of days or even several weeks as a teacher moves at a pace appropriate for his or her students.
In this unit we also introduce the idea of a fraction of 100 and so lay the groundwork for the decimal representation of fractions at Level 3 and percentages at Level 4. These ideas are developed further in the units Getting the Point, Level 3 and Getting Percentible, Level 4. But facility with fractions is also an important precursor for algebra. Algebraic fractions have a wide range of uses. Without a good knowledge of how ‘ordinary’ fractions work, students will be restricted in their work at higher levels of the secondary school when fractions occur in algebraic settings.
Here we look at different representations of 1/2.
- Write the fraction 1/2 on the board. Ask students what the number is and what they think it means. Put them into groups of three or four to brainstorm ideas they have about one-half. Ensure that they record their ideas as words, numerals or diagrams to share with the whole class.
- Get each group to report back the class on their favourite idea about one-half. Use this reporting back session to develop a class chart. Expect many of the students to have regions ideas such as cut pies and apples, half of a length, and possibly half time.
- Set up the following quick challenges around the room as stations that each group of students must attempt. Introduce the challenges briefly. Allow the students three minutes on each station. It is critical that they record how they solved each challenge.
- The stations cards are included as Copymaster one. Ensure that the following materials are available for each challenge:
- a plastic jar with twelve beads or counters in it
- two clear plastic cups and a small plastic bottle of water
- paper circles marked with ten divisions
- a sand or oil timer
- a stack of 16 Uni-fix or multilink cubes
- twenty toy 10-cent coins in a plastic jar
- a 400 gram blob of plasticine, kitchen scales, 30 cm ruler
- spinner (Copymaster two), paper-clip, pencil
- a trapezium-shaped pattern block and a set of blocks
- a toothpaste packet and multilink or Uni-fix cubes
- Get the students to report their answers and the strategies they used to find them, back to the class. Highlight the equal sharing aspect of finding one half. Tell them that you want them to try each challenge again only instead of finding one-half they need to find three-quarters. Write 3/4 on the whiteboard and discuss what it means (four equal parts and three chosen). For challenge number 8 the students have to think of what will happen to the spinner three-quarters of the time.
- Check the students recording to see how many of them have generalised three-quarters from one-half. Look for connections like two quarters make one-half so three-quarters is one-half and one-quarter.
Here we look at fractions other than 1/2 and consider ways to represent these fractions that involve 100.
- Remind the students how they found a half and three-quarters of a circle in Session 1. Discuss how many marks around the ant walked to get half way around the circle and how this could be used to divide the circle in half. Similarly the circle could be divided into quarters by marking spaces two and a half marks around and connecting the marks to the centre.
- Give the students several circles marked with one hundred spaces around (Copymaster three). Tell them that they can use any method they like to fold one circle into quarters, one into fifths, and another into tenths. Allow them to solve this challenge in groups.
- Share the results of their investigations. Some students will use geometry to fold the circles while others will use measurement (dividing the number of spaces around the outside). Either method is valid as one informs the other. Use the folding to make equivalent fraction statements, like 1/4 =25/100 . Challenge the students to write other equivalence statements, particularly with fractions that have other than one as their numerator (top line), e.g. 2/5 =40/100.
- Hold onto the paper circles for Session 3.
This session involves fractions in problem situations.
- Pose this problem for the students, " It is Jo’s twelfth birthday party. Including herself there are four people at the party. If everyone gets one quarter of the cake, how many candles do they get?" Get the students to solve the problem with counters and their paper circles from the previous session.
- Discuss the strategies that the different students used. These might include sharing twelve counters evenly onto the sections of the quarter circle, using addition (6 + 6 is 12 so 6 is a half so 3 is a quarter), or division (12 ÷ 4 = 3).
- Give the students other sets problems (see Copymaster four). Get them to record their strategies as they solve the problems. Students who use sharing strategies should be encouraged to anticipate the result of their sharing before it is complete.
- Students may like to write their own cake problems for others to solve. These can be made into a class book of problems for independent activity.
Another way to represent numbers is the number line. Here we use the number line to show the relative positions and sizes of fractions.
Draw a number line from 0 to 10 on the board (about 1 metre long). Build up the number line by getting students to write where different numbers might be. Once the whole numbers are in place, ask students to think about where numbers like 1/2, 3/4, 3/2, and 4 2/3 might be. This will help students to realise how fractional numbers extend the existing set of whole numbers and can be represented on a number line in the same way.
Give the students several paper strips of the same length cut from scrap paper. Ask them to fold one strip in half, one into quarters, and one into eighths. This is relatively easy as they can be folded by repeated halving. Ask the students to label each strip using symbols: 1/2 ,1/4 , and 1/8 .
Take a full strip and use it to draw the number line from 0 to 1 by marking each end. Shift the strip to the right and mark 2 at the right-hand end, shift it again and mark 3, etc. (0 – 5 is sufficient). Ask the students if they can use their strips to show exactly where one-half would be. Expect students to align the strip folded in half to do this. Ask this for other fractions like one-third, three quarters, two-thirds, and extend it to fractions greater than one like five-halves, four-thirds, and three and seven-eighths.
Pose these problems for students to solve using their strips.
- Draw a number line to show 1/2, 1/4, and 1/8. Mark where you think 1/3, 1/5, and 1/10 would go on your number line. Explain where you placed them. Why do fractions with one on the top line get smaller as the number on the bottom gets larger, e.g. one-half is larger than one-third?
- Which of these fractions is closest to one, 1/2, 2/3, or 3/4 ? Why?
These problems will highlight students’ knowledge of the relative size of fractions. For example, a student might find half of the distance between 0 and 1/5 to see where 1/10 should be or half the distance between 1/2 and 1 to see the location of 3/4 . The problems will also highlight their understanding of the role of the numerator (top number) as the selector of the number of parts and the role of the denominator (bottom number) as nominating how many equal parts the whole is separated into.
Here we try to link the concepts of fractions in length and sets by dividing up a big worm.
To link the concept of fractions as they apply to lengths and sets, tell the story of the two early birds who caught a worm. Produce a stack of one hundred Uni-fix cubes or multilink cubes joined together so that sections ten cubes long are in the same colour. Tell the students that this is the worm and when the birds measured it they found that it was very long. How long? Ask, "Suppose the two birds wanted to share the worm equally. How could they do that?" Students should use the idea of half of 100 being 50. Ask, "If they caught three worms this size and shared them out, how much would each bird get?" Record their responses using equations like 1/2 of 300 is 150.
Extend the problem. Ask, "Suppose that it took four birds to pull this worm out of its hole. How much of the worm will each bird get? What if there were five birds, ten birds?" Record the students’ strategies using symbols and diagrams.
Pose a series of problems for them to solve independently, of the form:
There were three birds. The worm was 18 cubes long. How much did each bird get?
There were four birds. Each bird got six cubes of worm. How long was the worm?
The worm was 18 cubes long. Each bird got three cubes of worm. How many birds were there?
Students will enjoy making up birds and worm problems for others to solve. It is vital that they record their solutions using fraction symbols. Use their responses to these problems to assess which type of strategies (sharing, adding or dividing) each student uses. Try to extend the number of strategies that each student has.