# Getting the point

Purpose

This unit explores the relationships between decimals and whole numbers and fractions. A number of representations are used including double number lines, ratio tables and place value tables. The main objective is to link students’ knowledge of fractions with the decimal system.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
Specific Learning Outcomes
• Represent fractions as decimals and vice versa.
• Explain why recurring decimals occur.
• Apply decimals to measurement contexts such as finding capacities.
Description of Mathematics

Fractions can be thought of as either as numbers or as operators. Decimals are a special set of fractions used to represent parts of a whole unit. The denominators of decimals are powers of ten, tenths, hundredths, thousandths, etc. The most common situations in which decimals are used involve measurement. Creators of the metric system used base units like the metre and litre, then created part units for greater precision. For example, 1 millilitre equals 1/1000 of 1 litre, 1cm equals 1/100 of 1 metre.

In this unit a ratio interpretation of the whole is used. One way to express the approach is to consider fractions of a set. The unit begins with a uniform one in the form of paper strips divided into different numbers of equal parts. In the first session students can choose to connect division of the number of parts with their preferred way to find common fractions of a fixed length. The use of a fixed one allows for fractions to be ordered by size on a number line. Therefore, the fractions are expressed as measures of that one unit.

The next sessions involve variable units of one as the set size changes. Recognising fractions as a relationship between part/s and a whole is critical to understanding. Decimals are introduced as part-whole relationships when the whole is constrained to ten, one hundred, one thousand, etc. Ratios are used to develop the concept of decimals as both numbers and expressions of a part-whole relationship. For example, the colour ratio 2:3 is two fifths red and three fifths yellow. Making two copies of the ratio gives 4:6 which is four tenths (0.4) and six tenths (0.6). More complex ratios require copying until a whole of 100, 1000, …can be found.

In the units student come to see that some fractions have a decimal representation that terminates, and some fractions have a decimal representation that repeats forever. For instance, 1/2 = 0.5 and this decimal terminates. 1/3 = 0.333… and the 3s continue without end, so the decimal is non-terminating.

Students are shown how to go from a fractional representation of a number to its decimal representation and vice versa. There is more than one way of finding these representations. A double number line (with fractions on one side and decimals on the other) and ratio tables are used as supportive representations.

A similar unit to this, which develops the fraction-decimal ideas further, can be found in Getting Percentible, Level 4.

The learning opportunities in this unit can be differentiated by providing or removing support to students, by varying the task requirements. Ways to support students include:

• providing physical materials so that students can anticipate actions, and justify their solutions
• connecting multiple representations, particularly double number lines, ratio tables and equations
• using important mathematical vocabulary to discuss concepts, in particular words for fractions (numerator, denominator) and decimals (tenths, hundredths, thousandths, etc.), and equality/equivalence
• encouraging students to collaborate in small groups and to share, and justify, their ideas.

Tasks can be varied in many ways including:

• altering the complexity of the fractions and decimals that are used. Fractions that can be expressed as tenths and hundredths are easier than those requiring thousandths or further decimal places
• maintaining a fixed whole (Sessions One and Five) is much easier conceptually than altering the whole (Sessions Two, Three and Four)

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Decimals arise through measurement. Therefore, you might choose measurement situations that are significant to your learners rather than rely on the generic contexts presented in the unit. For example, the length of pathway (te ara) or river (awa) may be culturally significant, or the length of fish (ika) or eels (tuna) may be a more approriate food to share. The contexts for ratios can also be varied. Ratios made from people, plants or other creatures may be more motivating than the use of replaceable leads in pencils. The recurring patterns in tapa cloth, or other linear designs may provide a more appropriate context for copying a ratio.

Required Resource Materials
• Sticky labels
• Cardboard packets
• Uni-fix cubes
• Calculators
• Paper strips made from Copymaster One
• One litre plastic drink bottles
• Large plastic drinking glasses (around 500 mL)
• Capacity measurement containers
• Rubber bands
• Permanent markers
Activity

Session 1

In this session we use paper strips to help us solve fraction problems.

1. Use Copymaster One to produce paper strips with various divisions across the bottom. Enlarge the copy to A3 size so the strips are easier to handle. Be sure that students cut the strips around the exact outside, and do not leave extra length. Introduce the activity by saying that a new company is producing fruit strips with cutting marks This modification is to make equal sharing of the strips between friends much easier. Here is a 12-mark strip. Ask, "How could these marks be used to find half of the strip?"

2. Students should suggest that folding would be easy and that the fold mark will line up with the 6. Draw this as a double number line like the one below. Use a paper strip to draw the number line so all numbers are located correctly to scale. 3. Ask: What other fractions are easy to fold with this strip?
Students should suggest quarters, thirds and possibly sixths. Use their responses as an opportunity to add fractions to the number line. Include unit (denominator of 1) and non-unit fractions, like three quarters. For example: 4. Write the equation for the halving problem as 1/2 x 12 = 6. Explain that the equation means “One half of 12 equals 6.”

5. Ask students to record how the other fractions of 12 might be written as equations. For example, 3/4 x 12 = 9 and 2/3 x 12 = 8.

6. Ask students to cut out the 15 parts strip.
What fractions would be easy to fold with this strip? Why would those fractions be easy?
Students might suggest that thirds and fifths are easy because 15 divides equally by 3 and 5. Ask them to work in pairs, folding one strip into thirds and one into fifths.

7. Ask students to work together to create a number line that includes thirds and fifths, including the fractions with more than one as the numerator. 8. Draw students’ attention to important features of the number line with questions like:
Where would the fractions 3/3 (three thirds) and 5/5 (five fifths) go? (These are equivalent fractions for one, just like 2/2 and 4/4)
How would you find one half of this strip? How many thirds and fifth are equal to one half? Folding in half yields 1 ½ / 3 (one and one half thirds) and 2 ½ / 5 (two and one half fifths).
Would the thirds be in the same position on this number line, with 15 parts, and the previous number line with 12 parts? Why does that happen?(Since both strips were the same length, the same fractions align)

9. Ask students to record some fraction multiplication problems for the 15 part strip. For example, 2/3 x 15 = 10 (two thirds of 15 equals 10).

10. Share the equations. Ask students to justify their equations using the strip model. Does a student record, 1/2 x 15 = 7 ½?

11. Set the students a few closed problems to solve using the fruit (paper) strips:
• 1/3 x 24 = ?
• 2/5 x 15 = ?
• 2/3 x 18 = ?
• 1/2 x 28 = ?
• 4/10 x 30 = ?
• 2/5 x 25 = ?

12. Open up the task so it is group worthy.
Choose a strip we have not used as a class. Use that strip to create a number line with at least six different fractions.

13. After a suitable time discuss their attempts. You might staple the strips that groups create on the wall, vertically aligned.

14. Eyeball some fractions and record them:
Which fraction is greater in this trio; 2/5, 4/10, 1/5? How do you know?
The discussion might provoke the concept of equivalent fractions, and help students to realise that the size of a fraction is determined by the size of both the numerator and denominator.

15. Tell the students that an employee at the fruit strip company invented the 17 strip. His boss is not convinced that the 17 part strip is useful. Is the boss correct? (Seventeen is a prime number so the only fraction the strip can be divided evenly into is seventeenths).

Session 2

1. In this session students solve fraction problems using division. The variation in this session is that the whole set is variable. That is in contrast to session one in which the whole strip length was kept constant. Students also build on the concept of fractions within ratios.

2. Tell the students that they are working for the company that makes endless pencils. These pencils have a variety of leads that can be pushed through the body of the pencil until the required colour is found. Tell six students to make each make a pencil containing three yellow leads and one blue lead. The students make a stack of cubes to represent each pencil. Put the "endless pencils" into a cardboard packet (e.g. a Milo Box).

3. Ask: What fraction of the leads (cubes) in this box are yellow? (three-quarters).
Do students realise that each unit of four pencils is three quarters yellow so the whole set must be three quarters yellow?
There are 24 leads altogether. How many of them are yellow?"

4. Allow the students time to attempt the problem and then discuss their strategies. Some may derive the result from one-half, "One-half would be 12, so one-quarter is 6, so three-quarters is 18." Others may use a division strategy; "I knew that there are six pens. There are three yellow leads in each pen so that is six times three, that’s eighteen." Remind the students that what they have solved could be represented as an equation, as a double number line, and as a ratio table, as shown below:
• Equation: 3/4 x 24 = 18.
• Double number line: • Ratio table:  Number of parts 24 6 12 18 Fraction 1 1/4 1/2 3/4

5. Prepare other examples of packets of endless pencils.
• Eight pencils (stacks of cubes), each with two red and three black leads (box labelled: 40 leads. Two red and three black are in each pencil. How many black leads are there altogether?")
• Six pencils, each with two green and one yellow leads (box labelled: "18 leads. Two green and one yellow are in each pencil. How many green leads are there altogether?")
• Seven pencils, each with five blue and three red leads (box labelled:"56 leads. Five blue and three red are in each pencil. How many blue leads are there altogether?")
• Ten pencils, each with three blue and seven white leads (box labelled:"100 leads. Three blue and seven white are in each pencil. How many blue leads are there altogether?")

6. With each packet take two pencils out to show the students the kind of pencil inside. Encourage students to use whatever strategy they think is appropriate to solve each problem. Use equations, double number lines, and ratio tables to record the strategies as they are reported back.
For example, problem ii) might be solve like this:
• Equation: 2/3 x 18 = 12
• Double number line: • Ratio table:  Number of Parts 18 1/3 2/3 Fraction 1 6 12

7. Provide packets, labels, and cubes so that groups of students can make up similar pencil problems for other groups to solve. The problems can be exchanged. Ask the solver group to discuss their solutions with the creator group.

8. Conduct a plenary for your students where you share important connections among ways to represent fractions. Highlight these points:
• The pencil shown below has a ratio of 2 grey leads to 3 white leads. • The relationship of colours can be written as a ratio 2:3 and as fractions 2/5 grey and 3/5 white.
Why are the fractions fifths?
• If the ratio is copied this results in a range of pencils that all have the same fraction of grey and white. For example, the ratio below is 6:9 (three copies of 2:3). • The longer pencil is still 2/5 grey and 3/5 white though the fractions could be expressed as 6/15 and 9/15.
• The number of grey leads could be worked out using 2/5 x 15 = 6.
How could the number of white leads be worked out in this way?

9. Pose similar practice problems for your students.

Session 3

1. In this session students transfer their knowledge of fractions to create tenths, hundredths, thousandths and ten-thousandths. In doing so, they connect fraction and decimal notation.

2. Write 4 567 on the board and get the students to tell you what they know about the number. Some will mention the place values (thousands, hundreds, tens, ones). Ask what the next place value is to the left is and how they know that (ten thousands because that is ten times the place immediately to its right). Ask what the place to the right of the ones place is. Some students may know that the place is the tenths but encourage them to justify if (division by ten with each next place to the right).

3. Make some stacks of connecting cubes that have one yellow and one red cube (a model for halves).
What fraction of the stacks are yellow? (one half)
In order to express one-half as a decimal we need to find an equivalent fraction for it that is so many tenths.
How can we show one-half as so many tenths?
Students will know that five stacks will show one-half as five-tenths. How do we write one-half as a decimal?
Most students will know that the decimal for one half equals 0.5 but will not recognise the connection to tenths. Record the decimal as: 4. Ask several students to produce stacks of cubes that have one blue to four white.
What fractions are in this ratio? (Students should say one fifth and four fifths)
Could we copy this ratio until we have tenths? (two copies will make a collection ten cubes)
How many cubes will be blue and how many will be white?
What are the decimals for one fifth and four fifths? (0.2 and 0.8)
What are the decimals for two fifths and three fifths? (0.4 and 0.6)

5. Show how the fractions of fifths can be expressed as decimals by counting in lots of two tenths on the place value chart. 6. Follow a similar process with quarters. Ask all students in your class to make stacks of one black cube and three white cubes.
Can the ratio be repeated until there are ten parts, tenths? (No)
What is the decimals place to the right of the tenths? (Students may notice the symmetry about the ones place or realise that dividing ten parts each into ten parts give hundredths)
Can the ratio be repeated until there are one hundred parts, hundredths? (Yes)
How many copies of 1:3 are needed?
How many of the 100 cubes will be black? (25/100) How many will be white? (75/100)
What are the decimals for one quarter, and three quarters? (0.25 and 0.75) 7. You might show why the fractions are expressed as tenths and hundredths. Repackaging 25 black tubes in to two towers of ten and five cubes shows how the notation works. Each tower is one tenths of the total collection of 100 cubes. Do the same thing with the white cubes to form seven towers of ten (7 tenths) and five cubes (hundredths).
What is the decimal for two quarters? Why? (Students will know that two quarters equal one half and the decimal for one half equals 0.5)
If one quarter equals 0.25, then two quarters equals 0.50. Why is two quarters shown as 0.5?

8. Tell the students that you are going to give them some commonly used ratios. Their task is to identify the fractions within the ratios and work out the decimal for each fraction. They may use cubes, number lines, and place value tables but they may not use a calculator. Write up the following ratios:
• 1 red cube: 9 blue cubes (one tenth: nine tenths)
• 4 green cubes: 6 yellow cubes (four tenths: six tenths)
• 3 black cubes: 5 orange cubes (three eighths: five eighths)
• 6 red cubes: 2 blue cubes (six eighths: two eighths)
• 2 white cubes: 1 green cube (two thirds: one third)

9. After a suitable period of investigation discuss their answers. Important points to bring out are:
• The decimal place to the right of the hundredths is the thousandths place.
• As the places include more parts it is easiest to use multiplication to figure out the decimals, e.g. 125 copies of eight cubes make 1000.
• Equivalent fractions have the same decimals, e.g. 6/8 = ¾ = 0.75.
• Decimals for some fractions have recurring digits and do not terminate, e.g. 1/3 = 0.3333….

Session 4

In this session we use calculators to explore the links between fractions and their decimal equivalents.

1. Many students will know that calculators can be used to convert fractions into their decimal equivalents. Model this with some well-known fractions such as 1/2, 1/4, 2/5 and 3/8.
The decimal for one-half can be found by entering 1 ÷ 2 = .
Why does the operation one divided by two give the answer of one half, 0.5?
If one anything is partitioned equally into two parts, one of those parts equals one half.
The decimal for one-quarter can be found by entering 1 ÷ 4 =

2. Ask: How can the decimal for three quarters be found?
The decimal for three-quarters can be found by entering 1 ÷ 4 = then multiplying the answer by three (3/4 is three lots of one quarter). Alternatively, use the operation 3 ÷ 4 = 0.75.

3. Tell the students that you are now going to go the other way round and give them some decimals. Their job is to find the equivalent fractions. that these decimals represent. Add that they may use any of the strategies used in the previous session, like the double number line, ratio table, or cube model.

4. Write up the following decimals for them to explore:
0.6;       0.75;    1.5;       1.25 (five quarters);     0.325 (three eighths);   0.875 (seven eighths);   0.6666….(two thirds);    1.2 (six fifths or 12 tenths);       0.8 (four fifths or eight tenths) , 0.4444….  (four ninths);  0.3125 (five sixteenths)

5. Allow the students to work in pairs and to discuss their findings. It is important that they record their answers.

6. After a suitable period of investigation bring the class together to discuss their strategies. Key ideas will be:
• Use of known fraction-decimal links to get at unknowns, e.g. 1/5 = 0.2 so 0.6 = 3/5
• Recognise that when a decimal is greater than one the corresponding fraction is improper, that is its top number (numerator) is greater than its bottom number (denominator).
• Recurring decimals (decimals where a section of the numbers repeat) indicates a fraction that cannot be expressed as an exact number of tenths, hundredths, thousandths, etc.

7. If students experience difficulty with some decimals suggest the use of scaffolding strategies. For example, when nothing in the decimal looks familiar explore the unit fractions until a useful decimal is found. In the case of 0.4444… finding that one ninth (1 ÷ 9 = ) 0.1111… is handy. Four lots of one ninth equals four ninths, 4 ÷ 9 = 0.4444…

8. Use a double number line progression to find the fraction for a recurring decimal: 9. Successive progressions make it easier to estimate the fraction accurately. In this case knowing one thirds equals about 33/100, so two thirds equals about 66/100. The fraction can be confirmed using mental calculation or the calculator, and altered to become more accurate, e.g. 333/1000 is closer to one third than 33/100.

Similarly a ratio table might be used to reduce a decimal to its simplest fractional form. For example, 0.875 by knowing this fraction equals 875/100. Session 5

In this session students apply their knowledge of fractions, and their decimal equivalents, to solve problems involving litres and millilitres. The metric system is based on decimals, since it uses a base of ten.

1. Ask the students to bring along 1 litre empty plastic drink bottles. Fruit juice often comes in 1 litre bottles which is preferable to 1L milk bottles that are not as regular. Ensure that a solid mark is shown on each bottle at the 1 litre point. Give students access to a set of measurement jugs so they to do this accurately. Discuss the relationship between litres and millilitres. The litre is the base unit of capacity and is equivalent to the volume of a cube that measures 10cm x 10cm x 10cm (large place value block cube). Milli is the prefix for 1/1000 (one thousandth) so a millilitre is one thousandth of one litre. This relationship can be written as 1mL = 0.001L, or 1000mL = 1L.

2. Tell the students that they will be given two 1 litre bottles, several rubber bands (to mark water levels), a marker, and a measurement jug. Their task is to mark one of the bottles to show where the water level would be if it was one-half full, one-quarter and three-quarters full, one-eighth, three-eighths, five-eighths, and seven-eighths full.Ask your students to use fraction symbols to make the marks, e.g. 3/8.

3. Put the students in mixed ability groups to solve the problem. Watch your students and look for the following:
• Do they recognise that decimals, thousandths can be used to accurately find the marks?
• Do they show understanding of tenth, hundredths, and thousandths?
• Do they label the marks appropriately with fractions (and possibly decimals)?

4. After a suitable period of exploration bring the class together to share their strategies. Some students may have used pouring methods to find the marks. For example, half can be found by splitting a full bottle equally between two bottles, one quarter can be found by splitting one half between two bottles, etc. Other students may have used the decimal properties of the metric measurement system. For example, to find the three eighths mark, they may realise that its decimal is 0.375 and so measure out 375 millilitres using the measurement jugs (Since there are 1000 ml in 1 litre).

5. Ask how it might be possible to find the one-third and two-third marks if they had another bottle. Get them to predict approximately where these marks will be. Students should use their knowledge of ordering fractions to do this. For example, two-thirds is slightly more than five-eighths since 0.6666... is more than 0.625.

6. Get the students to find the one-third and two-third marks by pouring between 1 litre containers rather than measuring. They might fill one container to the 1L mark, then share the contents equally among three same shaped containers (large plastic glasses are good for this task). Pouring one then two third shares into the original container gives the one third and two thirds marks. Then get them to check the accuracy of their pouring using measurement. One-third of a litre is about 333 millilitres. Recognise that students will not usually achieve that degree of accuracy by pouring.

7. Extend the activity be asking for other fraction marks:
• Tenths are lots of 100mL (0.1L)
• Fifths are collections of two tenths (0.2L)
• Sixths are halves of thirds (0.1666…L)
• Ninths are thirds of thirds (0.1111…L)
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