# 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.

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

The basis for this unit is the fact that some numbers can be represented by both fractions and decimals. In addition, we see that some fractions have a decimal representation that terminates and some fractions have a decimal representation that repeats. For instance 1/2 = 0.5 and this terminates, while 1/3 = 0.333… and the 3s repeat indefinitely.

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. Here we use a double number line (with fractions on one side and decimals on the other) and ratio tables. It is important for students to see more than one way to solve a problem. Learning is strengthened when connections are made with other knowledge. When students see more than one way to solve a problem they see connections with twice as many other ideas. Hopefully this leads to deeper knowledge and understanding.

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

Required Resource Materials

sticky labels

cardboard packets

Uni-fix cubes

calculators

paper strips made from Copymaster One

one litre plastic drink bottles

capacity measurement containers

rubber bands

Activity

#### Session 1

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

1. Use Copymaster One to produce paper strips with various divisions across the bottom.  Be sure to cut the strips off at the beginning of the number line (0) and at the right end of the number line.  Introduce the activity by saying that a new lolly company is producing liquorice straps with marks along the bottom.  This is to make equal sharing of the straps between friends much easier.  Here is a 12-mark strap.  Ask, "How could these marks be used to find half of the strap?"
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. 1. Write the equation for this problem as 1/2 x 12 = 6.  Ask, "How would the 12 strap be shared if you wanted to eat three-quarters of it? How could we show this on a double number line and as an equation?"  Confirm their solutions by folding the strip. Equation: 3/4 x 12 =  9.

1. Set the students several problems to solve using the liquorice (paper) strips:
1. 1/3 x 24 = ?;
2. 2/5 x 15 = ?;
3. 2/3 x 18 = ?;
4. 1/2 x 28 = ?;
5. 4/10 x 30 = ?;
6. 2/5 x 25 = ?.

Record each solution as a double number line and as an equation.

1. Tell the students that an employee at the lolly company made an unfortunate mistake by inventing the 17 strip.  Ask why they think it was a mistake.  (Seventeen is a prime number so the only fraction the strip can be divided evenly into is seventeenths!)

#### Session 2

In this lesson we continue to solve fraction problems by making sense of the problem using our understanding of division.

1. 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).
2. Ask, "What fraction of the leads (cubes) in this box are yellow? (three-quarters)There are 24 leads altogether.  How many of them are yellow?" 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: 1/4 x 24 =  18. 24 6 12 18 1 1/4 1/2 3/4
1. Prepare other examples of packets of endless pencils.
1. eight pencils (stacks of cubes) with two red and three black leads (box labelled,  "40 leads.  How many black?")
2. ten pencils with two green and one yellow leads (box labelled, "30 leads.  How many green?")
3. seven pencils with five blue and three red leads (box labelled,  "56 leads.  How many blue?")

With each packet take two pencils out to show the students the kind of pencil in each.  Tell them to answer the question on each packet using whatever strategies they wish.  Use equations, double number lines, and ratio tables to record the strategies as they are reported back.

2. Provide packets, labels, and cubes so that groups of students can make up similar pencil problems for other groups to solve.  Groups can then cycle these problems around and work out and discuss their solutions.

#### Session 3

In this lesson we help make sense of tenths, hundredths, thousandths and ten-thousandths.

1. 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 may know it is the tenths but encourage them to justify if (division by ten with each next place to the right).  Record the places tenths, hundredths, thousandths and ten thousands.
2. Make some stacks of Uni-fix cubes that have one yellow and one red cube (a model for halves).  Ask, "What fraction of the stacks are yellow?"  Say that in order to express one-half as a decimal we need to find an equivalent fraction for it that is so many tenths, or so many hundredths, or so many thousandths, .  (the places in the decimal system).  Ask, "Can we show one-half as so many tenths?" Student will know that five stacks will show one-half as five-tenths.  Ask, "How do we write one-half as a decimal?" Most will know that the answer is 0.5 but will not recognise the connection to tenths.  Record this as:

1. Look at the quarters by producing stacks that have one black cube and three blue cubes.  Set up a double number line to find out if one quarter can be expressed as tenths. Students will realise that one-quarter does not have an equivalent fraction of tenths.  This can shown using the stacks if need be.  Two-eighths and Three-twelfths are as close as they can get.  Ask if one-quarter can be shown as so many hundredths.  Some students will recognise that if twenty-five stacks are put together there will be one hundred cubes altogether and twenty-five will be black (one-quarter).  Ask, "So one-quarter is twenty-five hundredths.  How do we write that as a decimal? How would we write three-quarters?"
Record their solutions using both the double number line and the place value table.

An interesting point is that twenty-five hundreds is the same as two-tenths plus five-hundredths.  Ask students why they think this is so.  If necessary it can be shown with the stacks that there are enough black cubes to make two stacks of ten and have five left.  Each stack of ten is one-tenth of the hundred cubes in total (the whole).

1. Tell the students that you are going to give them some commonly used fractions and that they have to find the decimal for them.  They may use cubes, number lines, and tables to find them but they may not use a calculator.  Write up the following fractions: ; ; ; ; ; ; .

1. After a suitable period of investigation discuss their answers.  Important points to bring out are:
• decimalnumbers can be greater than one, e.g. 1.5, 2.25;
• decimal numbers can involve the thousandths and other places to the right, e.g. 1/8 = 0.125;
• decimal numbers can repeat, e.g. <1/3 = 0.3333….

#### Session 4

Today we use our 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.
2. The decimal for one-half can be found by entering 1 ÷ 2 =  .  The decimal for one-quarter can be found by entering 1 ÷ 4 =  .  The decimal for three-quarters can be found by entering 3 ÷ 4 =  .  etc.
3. Tell the students that you are now going to go the other way round and give them some decimal numbers to turn into their equivalent fractions.  Their task is to use the calculator to find the fraction that these decimals represent.  Add that they may use any of the strategies used before like the double number line, ratio table, or cube model.
4. Write up the following decimals for them to explore:

0.6;       0.75;     4.5;       1.25;     0.325;   0.875;   0.6666….;         6.2;       0.1111….

Allow the students to work in groups and to discuss their findings.

1. 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, eg. 1/5 = 0.2 so 0.6 = 3/5 ;
• recognising 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 (ones where a section of the numbers repeat) indicate a fraction that cannot be expressed as an exact number of tenths, hundredths, thousandths, etc.
1. If students get stuck on some decimals suggest the use of strategies.  For example: using a double number line progression to find the fraction for a recurring decimal:   Successive progressions make it easier to estimate the fraction accurately, in this case two-thirds, then confirm the fraction using mental calculation or the calculator.
Similarly a ratio table might be used to reduce a decimal to its simplest fractional form.  For example, 0.875 by knowing this is . #### Session 5

Today we apply our knowledge of fractions and their decimal equivalents to solve problems involving litres and millilitres.

1. Ask the students to bring along 1 litre or 1.5 litre, empty plastic drink bottles.  Fruit juice often comes in 1 litre bottles.  Ensure that a solid mark is shown on each bottle at the 1 litre point.  Get a set of measurement jugs for the students to use to do this accurately.
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.
3. Instruct them to use fraction symbols to make the marks, e.g. 3/8  .  Put the students in mixed ability groups to solve the problem.  After a suitable period of exploration bring the class together to share their strategies.
4. 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 a 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 rather than measuring. 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 achieve that degree of accuracy by pouring.

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