6. Decimals

1. Decimals – introduction

Knowledge of decimal notation and meaning are important to most adults because they are intrinsic to everyday measurement and money calculations, whether in relation to shopping, cooking, planning finances, sewing and home maintenance, or to a range of work-related roles. This section provides a range of activities and games that assist students to explore this meaning and strengthen their 'sense of' decimal numbers through simulations with money, visual representations, pair and small group discussions and use of calculators. Some adult learners need a variety of approaches and a lot of revisiting to really grasp the significance of decimal notation.

The first activity, 'Where are the Decimal Points?' allows adult learners to begin the discussion of decimals by drawing on their existing knowledge of the decimal quantities that they see on a daily basis and lays a foundation for further exploration of meaning. The practice examples in later activities then encourage students to return to these everyday contexts in order to apply their developing decimal skills.

2. Where are the decimal points?

Most adults are exposed to decimals in money or measurements every day, without being conscious of how much they actually know and understand about them. This activity asks students to share their everyday knowledge to put the decimal point in the correct place within a very short story. It provides a means of observing their existing understanding of decimal notation and metric measurements in a non-threatening and enjoyable manner. It is therefore an ideal starting point for teaching about decimals or the metric system of measurement. It also provides practice at reading decimal numbers aloud which is helpful for people from language backgrounds other than English.

This activity is most effective if students have been exposed to the concept of fractions see 'Meaning of Fractions' and place value for whole numbers, for example, 'Multidigit', 'Back to Back Numbers', 'Talking about Numbers'. It can lead to the 'Exploring the First Decimals or metric estimation activities such as 'Matching Metrics'.

3. Exploring decimals and tenths 1

Many common numeracy errors are the result of lack of understanding of the decimal point and the meaning of the numbers that follow it. This activity is designed for students to begin exploring and consolidating the meaning of decimal places as fractions. It uses a selection of diagrams, fraction circles and the place value chart as possible means to examine the first decimal place as signifying tenths.

These methods can be used in one session or over a number of sessions depending on the learners. Some adult learners need a variety of approaches and a lot of revisiting to really grasp the significance of decimal notation. For others these representations can be a valuable clarifying exercise.

The activity can follow 'Where are the Decimal Points?' as this sequence would allow adult learners to begin with the decimals that they see around them on a daily basis and then explore the meaning of the decimal places as fractions of a whole. It could also follow directly from the 'Meaning of Fractions' in the fractions section, depending on the learners.

It should be soon followed by 'Exploring Decimals and Tenths 2: Mixed Numbers' and 'Exploring Decimals on Measuring Scales'.

4. Exploring decimals and tenths 2 - mixed numbers

This activity is designed to follow 'Exploring Decimals and Tenths 1' as it continues to explore the meaning of decimal places when they are combined with whole numbers. As in Part 1, it uses diagrams, fraction circles and place value charts to consolidate meaning.

'Dicing with Decimals - Games 1 and 2', in which students estimate and add decimal numbers, are useful to follow up or complement this activity as is 'Exploring Decimals and Measuring Scales'. 'Exploring Decimals and Hundredths' should follow soon after to consolidate the difference in significance of the first and second decimal places.

5. Dicing with decimals

This activity consists of three games designed to increase students' understanding of decimal place value and enhance their capacity to estimate with decimals. They provide a means of quickly revising or teaching addition of decimals and the significance of the decimal point. All games rely on luck as well as knowledge so can be used several times with any group. They are ideal as 'focus activities' at the beginning of a session or to provide a positive end to a session.

The first two games which use only one decimal place could be used in successive sessions, or Game 1 can used several times before advancing to Game 2 which presents a little more challenge.

Game 3 introduces the second decimal place and is much more challenging to play. It emphasises the relatively small value of numbers in the second decimal place, and can be introduced when students have become thoroughly comfortable with Games 1 and 2.

These games complement other activities related to understanding and estimating decimals, such as 'Exploring Decimals, 'About How Much?' and 'Is the Answer Reasonable?

6. Talking money with calculators

This activity is designed to assist students to use a calculator correctly for money calculations and to give and interpret oral instructions related to money. It emphasises the use of the decimal point to separate cents from dollars on the calculator and provides practice at interpreting the different ways that money amounts are spoken in our society. It also contains pair exercises that encourage students to speak and hear money amounts themselves as they practice using calculators, so is also helpful with students who are learning English.

The activity 'Key Words for Calculations' in the Exploring Numbers section is a useful preparation for this activity.

Recommended complementary activities are: 'Calculating Change' in the In the Head Calculations section, to discourage students from becoming overly dependent on calculators, and 'Estimate or Accurate' in the Exploring Numbers section, to the introduce the skills of estimating numbers and the associated language, which are also important for sensible use of calculators. The activity 'About How Much?' which follows in this section reinforces all of these skills.

7. About how much

'About how much will it be?' is an important question to ask when calculating with any decimal quantity, whether it is related to money or measurement. Many adults recall that there were lot of rules to remember when multiplying or dividing with decimal quantities (often associated with counting of decimal places). Trying to remember which rule to use and when often gets in the way of a sensible approach.

This activity is designed to encourage students to think of decimal quantities in terms of 'sensible' or 'friendly' whole numbers and to use them in order to estimate calculations involving decimals. The activities 'Capital Cities of Australia' and 'Estimate or Accurate', both in the 'Exploring Numbers' section introduce the idea of estimation and the language associated with it: the first uses large whole numbers only, while 'Estimate or Accurate' uses a mix of whole and decimal numbers and is a useful complement to this activity. Its Practice Sheet: 'Can I afford it?' can also be used in conjunction with this activity.

The activity 'Is the Answer Reasonable?' in this section, continues developing the skill of estimating answers when calculating with decimals, and in particular when using a calculator.

8. Exploring decimals and hundredths with money

Many adult learners who have little experience with the meaning of decimal notation have difficulty understanding the differences between numbers such as, 5, 0.5 and 0.05. This activity uses the familiarity of local currency to clarify these differences. Decimal numbers are modelled with coins, and emphasis on the name 'cents' is used to establish the second decimal place as representing one hundredth part and establish the significance of the zero in .05.

This activity may be done after students display confidence with the meaning of the first decimal place in 'Exploring Decimals and Tenths 1 & 2' and also after they have had some practice at estimating and adding decimals with one decimal place, as in the games 'Dicing with Decimals 1 and 2'.

9. Exploring decimals and hundredths further

Many adult learners who have little experience with decimal notation and its meaning, have difficulty understanding the differences between numbers such as, 5, 0.5 and 0.05. This activity uses 'Hundredths Grids' to visualise the relative sizes represented by the first and second decimal places then consolidates the concepts further using a Place Value Chart. The place value chart is also used to briefly TO introduce the third decimal place as representing thousandths.

The activity can be used either to build on the ideas first developed in 'Exploring Decimals and Hundredths with Money' or as an initial or revision activity for more advanced students who do not need the concrete materials.

It should be done after students display confidence with the meaning of the first decimal place in 'Exploring Decimals and Tenths 1 & 2' and also after they have had some practice at estimating and adding decimals with one decimal place, as in the games 'Dicing with Decimals 1 and 2'.

10. Exploring decimals on measuring scales

Being able to read mixed numbers with decimals on a measuring scale is an important numeracy skill. In fact it is one of the most important reasons that many students need to understand decimals. This activity presents an opportunity to explore decimal representations in relation to measuring scales. The activity is a valuable follow up to 'Exploring Decimals as Tenths Parts 1 and 2'. It's progressive stages may need to spread over several sessions.

11. Matching decimals with common fractions

This activity is designed for small groups of students to explore and expand their knowledge of decimal equivalents of common fractions and decimal place value and is an ideal focus activity to start the session as it encourage students to talk to each other about numeracy.

It can be used as an introductory decimal activity that allows you to observe the existing knowledge of your students and the areas that will need further attention, or as a learning activity after students have been introduced to the meaning of decimals.

A possible variation invites all members of the class to circulate in order to find their pair.

This activity is best used only after students have explored visual representations of fractions (see 'Meaning of Fractions', 'Matching Common Fractions' and 'Comparing Fractions' in the 'Fractions' section.

12. True or false fractions and decimals

This activity, ideally done in pairs or small groups, is designed to reinforce students' understanding of decimal place value, the meaning of fraction notation and the relationships between decimals and common fractions.

It is ideal as a focus activity to start or end a session and will provide opportunity to observe students' existing knowledge as well as highlighting areas that may need more attention. Two sets of questions are provided. These can be used in different sessions rather than one after the other immediately, since revisiting of the concepts in between may be required.

13. Sorting decimals - a near thing

This is an activity that asks students to arrange a collection of decimals numbers (cut into manipulable cards) according to whether they are nearest to 0, ½, or 1. Observing students undertaking this task provides valuable insight into their understanding of decimal place value.

It can be used as a revision task or a learning task undertaken by small groups or pairs of students sharing their understanding of decimals. The two versions of the cards will allow for both uses. It can be used after activities which explore the meaning of decimals, such as Exploring the First and Second Place Decimal activities or as an introductory activity to refresh the knowledge of a more advanced group.

14. Matching decimals, scales and pictures

This activity is designed for students to review and/or expand their understanding of decimal place value through matching activities designed for pairs or small groups.

• Set 1 compares decimal numbers to fractions and diagrammatic representations.

• Set 2 requires learners to choose correct readings on decimal measurement scales.

These matching tasks are ideal as introductory activities, since they allow you to observe students' existing skills as a starting point for teaching whilst encouraging students to cooperate and discuss their numeracy understanding. They are also useful as revision tasks.

The activities 'Exploring Decimals and Tenths1 and 2', 'Exploring Decimals on Measuring Scales' and 'Exploring Decimals and Hundredths' provide detailed teaching material to support this activity.

15. Target 100

This is a game designed to develop students' awareness of decimal place value and estimation skills when multiplying with decimals. Students enjoy it because they can normally see improvement after a few rounds, but it also allows more advanced students to be challenged further by varying the initial input number.

This activity should follow activities such as 'Exploring the Second Decimal Place', which explore the meaning of decimal place value. It also compliments other decimal estimation activities such as 'Sorting Decimals: a near thing' and 'Is the Answer Reasonable?' , and the more complex estimation game 'Decimal Dilemma'.

16. What's the decimal number?

This activity contains a series of Cooperative Logic problems. These use the same rules and structure as those in Introducing Cooperative Logic in the Getting Started section and 'What's the Secret Number?' in Exploring Numbers and are designed to be done in small groups. They should reinforce what students have learned about decimals as they follow the clues to find the unknown numbers. The activity encourages learners to interpret and use the language of decimals.

The seven problems can be done over the course of two or more sessions. Do several in the first session so that students get accustomed to the structure of the problems and the language of decimals that is used, the remainder could be used in one other session or as focus activities at the beginning of several sessions.

If students have not previously solved some of the 'What's the Secret Number?' problem solving tasks it would be a good idea to use these prior to commencing the decimal problems.

17. Multiplying and dividing decimals

The ability to multiply numbers which include decimals by powers of 10 (10, 100, 1000 etc.) is important for estimating calculations and for converting metric measurements.

This activity takes a 'sensible number' approach, focusing on the leading digit (front number) to do these calculations. This extends the logic and methods for multiplying and dividing whole numbers by powers of 10, rather than introducing new rules about shifting the decimal point backwards and forwards.

18. Is the answer reasonable?

There are many ways of making mistakes when using calculators or computers to perform calculations. With calculators it is easy to put the decimal points in wrongly, press the zero button too many times, or even to enter a digit twice without realising. Every time we make calculations with a calculator or computer we need a way of checking that the answers it gives are reasonable; that they make sense. This activity presents exercises designed to introduce students to this way of thinking and give them practice at estimating simple calculations.

It draws on some of the skills presented in 'In the Head Calculations' activities 'Multiplication by Tens' and 'Dividing by Tens', as well as skills from the previous activities in this section, 'About How Much?' and 'Talking Money with Calculators'.

19. Decimal Dilemma

This is an estimation game for two players, designed to enhance students' estimation skills with decimal operations. It is a particularly effective way to provoke students' curiosity about why they get counter-intuitive results when they multiply and divide by numbers less than 1. This creates powerful teaching opportunities related to multiplication and division by fractions as well as decimals.

It is an activity for more advanced numeracy and mathematics students who have an understanding of the meaning of fractions and decimals. It can usefully follow the earlier estimation game, Target 100.

20. Answers

Answers for Decimals.