Suggested Learning Intentions

- To estimate and locate fractions and decimals on a number line
- To understand fractions and decimals as different representations of the same number

Sample Success Criteria

- I can locate simple fractions, improper fractions, and decimals on a number line
- I can represent fractions as decimals using different models: area models, number lines and set models
- I can demonstrate my understanding using a variety of manipulatives

This stage has been inspired by the *Fractions to Decimals (on a rope)* and *Fraction Estimation* problems on Maths300, and is reproduced with permission. Access to the problems on Maths300 requires a subscription.

This is a useful activity to review the concept of fractions being equal parts of a whole.

Invite two students to hold either end of a piece of rope. Explain to the class that one end represents zero and the other represents the number one.

Give another three students a different coloured peg and ask them to, one at a time, place their peg at the point on the rope that represents the number two thirds. Alternatively, place their name with the peg to distinguish each person’s estimate.

Provide students time to review where they have placed their peg. Once each student is satisfied with the placement of their peg, invite other students in the class to nominate who they think has made the best estimate. *Votes for each estimate could be recorded on the board. This way all students have a vested interest in finding out who is most accurate. *

Discuss strategies that could be used to determine whose estimate is the most accurate with students. Record all suggestions on the board and invite feedback on how accurate these strategies may be.

*Sample responses may include:*

*Folding the rope into three parts**Pacing from one end of the rope to the other**Using a known fraction such as ¹/₂**to estimate the position of ²/₃**.*

Ask a series of prompting questions to help develop student understanding of the position of a fraction on a number line. Sample prompts include:

- What is the whole?
- To locate the fraction ²/₃, how many equal parts do I need to divide the whole into?
- How many of these equal parts do I need so that I can represent ²/₃?

Students work in small groups to practice estimating the location of different sized fractions on a number line, using a rope and pegs. Provide a set of fraction cards for students to work with, or ask students to select their own fractions.

As students work through this task, refer them back to the prompts used earlier so that they continue to consider the size of the whole, and the number of equal parts that they need to divide the whole into. Consideration of these concepts will help students develop their understanding of the components of fractions, namely the numerator and the denominator.

**Enable **students by providing only unit fractions for them to work with. *For example, they could estimate ¹/₂**, ¹/₃**, or ¹/₄**.*

**Extend **students by encouraging them to estimate more complicated fractions such as ³/₈ or ⁶/₁₁.

Once students have completed several different estimations, ask them to reflect on their estimating skills and what they have learned from this activity and record this in their maths journal.

**1. Develop an understanding of equivalent fractions and decimals**

*The focus of this part of the stage is for students to use the ropes and pegs to create two different number lines that, when compared, can help students understand the relationship between fractions and decimals.*

Explain to students that they are going to take the same approach used previously to create two different number lines.

Ask students to form groups of at least three students. Provide each group with a rope and pegs.

Ask students to create a number line on their rope by estimating the position of the fractions ¹/₅, ²/₅, ³/₅ and ⁴/₅ and marking these with pegs.

These fractions have been deliberately selected as they work well to show the connection between fractions and their equivalent decimals. Students will most likely use the strategy of folding the rope to estimate their fractions and therefore the choice of rope/string is important as it needs to be easily folded, otherwise the pegs may fall off every time the rope is folded.

Invite each group to present their completed rope and share the strategies they used with the class.

Provide each group with a second rope of the same length and some more pegs. They will use this to create a number line by estimating the position of the decimals 0.2, 0.4, 0.6, and 0.8.

Allow students sufficient time to explore, discuss and test different strategies. Invite students to share their strategies with the class. Ask them to consider whether they have used similar or different strategies to those used when working with fractions.

*One strategy may be to divide their rope into 10 equal parts, that is, into tenths. This could be done by folding or estimating. Establishing tenths on the rope by dividing the rope into ten equal parts and exploring through estimating or folding will help students understand how common fractions convert to decimals.*

Now ask each group to lay both their ropes side by side and discuss what they notice.

By comparing both ropes, students should be able to see how the fractions, fifths, are related to tenths. This visual representation is important for students to develop their conceptual understanding of fraction to decimal equivalence. Using the rope and checking their placement of pegs also enables students to understand that a decimal is a part of the whole, just as fractions are parts of a whole.

**2. Use the decimal number line to further explore relationships between fractions and decimals**

Ask students to consider what other information they could add to their decimal number lines. Establish an understanding that the values 0.1, 0.3, 0.5, 0.7 and 0.9 could be added and provide students with pegs so that they can add these values to their number line.

Ask students to use their decimal number line to:

- estimate the location of different fractions
- estimate the decimal equivalent of different fractions.

**Enable **students by providing a set of common fractions, such as halves, quarters and eighths.

**Extend** students by encouraging them to select their own fraction family using the strategies they have previously used and learned about.

Students may use a variety of strategies to undertake this task. For example, they may use fractional number lines created earlier and compare these with the decimal number line by laying them side by side. Alternatively, they may fold the decimal number line into a certain number of equal parts.

Students may naturally start to explore fractions such as ¹/₃, which do not have an exact decimal equivalent. By using either of the strategies suggested above, students should recognise that ¹/₃ lies between 0.3 and 0.4. The following think-aloud is one way of helping students understand the decimal approximation of ¹/₃:

*'We can see that ¹/₃ is somewhere between 0.3 and 0.4 on the number line. Let’s try a strategy to help us work out how we might write ¹/₃ as a decimal.*

*Imagine I had 10 objects to share between 3 people. Each person wants an equal share – they each want one third of the total. How many objects will each person get? *

*We can see that each person will get 3 out of the original 10 objects, and there will be one left over. So, the best way we have of dividing 10 into thirds is by each person getting 3 out of 10. We can think about this as ³/₁₀, which is the same as 0.3. So right now we think that ¹/₃ is 'about' the same as ³/₁₀, which we can also see on our number line. *

*Imagine I had more objects to share. This time, I have 100 objects to share equally between three people. How many will they each get?*

*We can see that each person will get 33, and there will be one left over. So, the best way of dividing 100 into thirds is by each person getting 33 out of 100. We can think about this as ³³/₁₀₀, and say it as ‘thirty-three hundredths’ which, using our knowledge of place value, we can write as 0.33. *

*So now we can estimate that the fraction ¹/₃ is pretty close to ³³/₁₀₀, or 0.33, but not quite – why not? Because we still have that one left over.*

*What happens if we have 1000 objects to share equally? Each person will get 333 objects, which can be written as ³³³/₁₀₀₀, or 0.333. So now we have a more accurate estimation for ¹/₃. It is very close to 0.333, but it’s still not exact, because we still have that one left over.*

*Will we ever get an exact decimal equivalent for ¹/₃?'*

The use of 10, 100 and 1000 in this explanation is deliberately linked to the base-10 decimal system and the place value of tenths, hundredths, and thousandths. After further exploration, students should recognise that there is no way of expressing ¹/₃* *exactly as a decimal. Students could use a calculator to verify the decimal approximation.

While the division algorithm is inadvertently used here, there is no need to introduce this process formally yet. It is important for students to understand that using the calculator is only used as a means of checking their estimates and reinforcing confidence in their estimation skills and not about getting the ‘right answer’.

As students are exploring other fraction families, encourage them to look for equivalent fractions and decimals. For example, they may notice that ³/₆ and ⁴/₈ is the same as 0.5 or ²/₆ can also be approximated by the decimal 0.333.

**3. Explore the position of other decimal numbers on the number line**

Write a decimal number line on the board that shows the tenths from 0.1 to 0.9. These should be familiar to students from their work so far in this stage.

Invite students to share some of their findings from the previous investigation. Use their contributions to generate a list of matching fractions and decimals. Add newly discovered relationships to the number line on the board, for example, ¹/₄=0.25 or ³/₂₀=0.15.

Select two decimal numbers from the number line (say, 0.6 and 0.7) and ask:

*How many numbers do you think can be placed between 0.6 and 0.7?*

Invite contributions from students and ask them to justify their responses. Provide opportunities to uncover and correct any misconceptions.* *A common misconception can relate to the relative size of decimal numbers such as 0.67 and 0.7, where students may believe that 0.67 is larger than 0.7, because 67 is larger than 7.

Provide students with a list of decimals, with different numbers of decimal places, and ask them to position these as accurately as possible on their own decimal number lines.

**Enable** students by providing a set of decimals that have one or two decimal places.

**Extend **students using prompts such as:

- Write down a decimal that has three decimal places and would be a little bit larger than 0.5
- Write down a decimal that has four decimal places that would lie in between 0.34 and 0.36
- Write down three decimals that would be equally spaced between 0.39 and 0.51.

Consider presenting number lines in different orientations (both horizontal and vertical) so that students can further develop their understanding of number lines in different contexts. Vertical number lines are useful when considering temperature, and students will encounter both horizontal and vertical number lines when learning about the Cartesian Plane later in their learning.

'Fractions to Decimals' *exposes students to a range of mathematical skills and concepts which you may wish to review and consolidate with students. Consider the progress made by students when determining which skills and concepts you select to review.*

Two suggested concepts for review are presented below.

**1. Number lines can be used to locate, estimate, and order fractions and decimals**

You may wish to emphasise the following concepts:

- Number lines help us see the relative size of fractions and decimals compared with whole numbers.
- Number lines help us understand that there is an infinite amount of numbers that sit between any two values.
- By annotating a number line, we can make clear links between equivalent fractions and decimals.
- Number lines can be presented in a range of formats.

You could support students to consolidate their understanding of number lines by inviting them to play Spiralling decimals, from the NRICH website. The purpose of this game is for students to take turns in positioning decimals on a curved number line. In order to win the game, students will need to understand the relative size of decimals and their location on a number line.* *Further ideas are provided in the ‘Teachers Resources’ section of the activity.

Collect students' number lines to assess their understanding.

**2. Fractions and decimals can have the same value.**

You may wish to emphasise the following concepts:

- Some fractions can be expressed as an exact decimal. These decimals are known as
**terminating decimals.** - Some fractions cannot be expressed exactly as a decimal. However, there will be a noticeable pattern in the digits of the decimal. These decimals are known as
**recurring decimals.**

You could support students to consolidate their understanding of these concepts by asking students to provide equivalent representations of a number in a variety of forms.* *For example, ask students to present the fraction ⁵/₈ as a decimal, on a number line, and as an area model.

Use a range of initial prompts including fractions, decimals, number lines and area models.

Consider using numbers both smaller and larger than one, and fractions that will convert to either terminating or recurring decimals.

*This will help students develop their understanding of the interconnections between different ways of representing a number.*

Australian Association of Mathematics Teachers, n.d. *Fraction Estimation. *[Online]

Available at: maths300.com/members/m300full/033lestf.htm

[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

Australian Association of Mathematics Teachers, n.d. *Fractions to Decimals (on a rope). *[Online]

Available at: maths300.com/members/m300full/182lfrac.htm

[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

University of Cambridge, n.d. *Spiralling Decimals. *[Online]

Available at: nrich.maths.org/10326

[Accessed 15 March 2022].

Other stages

1. Fractions as Equal Parts of a Whole

EXPLORESuggested Learning Intentions

- To understand that fractions can represent equal parts of a whole

Sample Success Criteria

- I can accurately identify the name of each fraction part
- I can correctly name fractions using the number of equal parts needed to make up the whole
- I can explain and justify my solutions using a variety of manipulatives

2. Chocolate Cake

EXPLORESuggested Learning Intentions

- To understand fractions result from division
- To compare fractions, noting how the numerator and denominator affect the size of a fraction

Sample Success Criteria

- I can identify fractions that have different names but are the same size
- I can identify which fraction is larger or smaller based on the name of the parts and how many of these parts there are
- I can justify my thinking and solutions using manipulatives or other representations

4. Colour in Fractions

EXPLORESuggested Learning Intentions

- To understand two fractions can be the same size but have different names

Sample Success Criteria

- I can identify fractions that are the same size but have different names
- I can make fractions that are equal in size using different parts
- I can use manipulatives to model and explain my thinking and solutions

5. Colour in Decimats

EXPLORESuggested Learning Intentions

- To understand place value in decimals
- To understand how place value determines the size of a decimal

Sample Success Criteria

- I can accurately colour decimals on a decimat to demonstrate place value
- I can order decimals correctly on a number line
- I can use different manipulatives to model and explain my solutions and thinking