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# Fractions and decimals on the line

Number Foundations · Fractions and decimals

<!-- Speaker note: Students place positive and negative fractions, decimals and mixed numerals on number lines split into halves, quarters and tenths, including lines without zero in the middle, and order them (AC9M7N04, node 3). About 40 minutes (Predict and Run 6 min · Investigate 12 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** split each unit into equal parts and count from zero. A negative fraction or decimal counts to the left, so −1.5 sits halfway between −2 and −1.

- I can place positive and negative fractions, decimals and mixed numerals on a number line.
- I can explain why −1.5 is between −2 and −1, not between −1 and 0.
- I can put a mixed set of fractions and decimals in order.

<!-- Speaker note: Before class: Do the Run board and the Make task yourself first. Students need Comparing integers (further left is less) and converting between fractions and decimals, which this lesson assumes was taught in class (node 2 of the descriptor); the Investigate step opens with a quick check of it. The one idea: split each unit into equal parts (the denominator) and count from zero, to the left for a negative. The common mistake is counting the part of a negative number back towards zero (−1.5 at −0.5); it is planted after the method, in Jo's working. On the board, a point's readout gives the fraction and the decimal (−1 1/4 (−1.25)). -->

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## Why it matters

> A tide of −0.6 m, a share price down 1¼ dollars, a recipe with ¾ of a cup: numbers between the whole numbers, above and below zero.

<!-- Speaker note: Between any two whole numbers on a number line there are fractions and decimals: $\tfrac{1}{2}$, $0.3$, $1\tfrac{3}{4}$. Below zero there are negative ones too: $-\tfrac{1}{2}$, $-0.3$, $-1\tfrac{3}{4}$. In this lesson you find out how to place them: positive and negative, as fractions, decimals and mixed numerals. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Where does $-1.5$ go on a number line?**

- Halfway between $-1$ and $0$
- Just to the left of $-15$
- Halfway between $-2$ and $-1$
- Halfway between $1$ and $2$

<!-- Speaker note: What happens: Halfway between −2 and −1. −1.5 is one and a half units below zero: count 1 unit left from 0, then half a unit further left. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose 'between −1 and 0' are reading −1.5 as −1 then +0.5. Do not correct it yet. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Ffractions-on-the-line%2Fstep-2&code=widget%3A+numberline%0Arange%3A+-3%2C+3%0Asnap%3A+0.5%0Apoint+P%3A+0%2C+drag%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/fractions-on-the-line/?full=step-2#step-2)

<!-- Speaker note: Drag $P$ along the line, or pick it under **Move** and use the arrows. It lands on the halves. The board says where it is under the line. -->

<!-- Speaker note: Teacher: ask students to say where −1.5 will land before they let go of P. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* Halfway between −2 and −1. −1.5 is one and a half units below zero: count 1 unit left from 0, then half a unit further left.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Quick check

**Quick check: write $\tfrac{3}{4}$ as a decimal.**

* Answer: 0.75

<!-- Speaker note: One quarter is 0.25, so three quarters is 0.75. -->

<!-- Speaker note: Teacher: if many students miss the quick check, do the quarters (0.25, 0.5, 0.75) together first. Then model the worked example live: draw a line from −2 to 1, mark the quarters, and count aloud from zero for −3/4 and −1 1/4. Reveal the prepared lines afterwards to compare. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Placing −3/4 and −1 1/4

Here is how to place a negative fraction and a negative mixed numeral.

Then: which is less, $-\tfrac{3}{4}$ or $-\tfrac{1}{4}$?

$$-\tfrac{3}{4}$$

1) $\text{count 3 quarters left from } 0$ — Negative means left of zero.
2) $-\tfrac{3}{4} \text{ is between } -1 \text{ and } 0$ — It is 3 of the 4 quarters from 0 to $-1$.
3) $-1\tfrac{1}{4} = -1 - \tfrac{1}{4}$ — A negative mixed numeral is a whole number and a part, both below zero.
4) $\text{count 1 unit left, then 1 quarter further left}$ — The quarter is further from zero, not back towards it.
5) $-1\tfrac{1}{4} \text{ is between } -2 \text{ and } -1$ — Just past $-1$, a quarter of the way to $-2$.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write -3 ÷ 4 and say "The denominator is 4, so split each unit into 4 equal parts (quarters)." Line 1: write count 3 quarters left from 0 and say "Negative means left of zero." Line 2: write -3 ÷ 4 is between -1 and 0 and say "It is 3 of the 4 quarters from 0 to -1." Line 3: write -11 ÷ 4 = -1 - 1 ÷ 4 and say "A negative mixed numeral is a whole number and a part, both below zero." Line 4: write count 1 unit left, then 1 quarter further left and say "The quarter is further from zero, not back towards it." Line 5: write -11 ÷ 4 is between -2 and -1 and say "Just past -1, a quarter of the way to -2." -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Placing −3/4 and −1 1/4

**Which is less, $-\tfrac{3}{4}$ or $-\tfrac{1}{4}$?**

- $-\tfrac{1}{4}$
- $-\tfrac{3}{4}$

* Answer: $-\tfrac{3}{4}$

<!-- Speaker note: −3/4 is three quarters left of 0, further left than −1/4, so it is less. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Jo places $-1.5$. Click the line where her working goes wrong.**

```
 1  Place −1.5 on the number line.
 2  −1.5 is one and a half below zero.
 3  Start at −1.
 4  Move half a unit to the right, because 0.5 is positive.
 5  So −1.5 is halfway between −1 and 0.
```

* Answer: line 4

<!-- Speaker note: Line 4: −1.5 is 1.5 below zero, so the 0.5 is further below too. From −1, move half a unit left, to halfway between −2 and −1. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **A negative fraction:** Each task changes one thing. First: quarters below zero.
- **A negative mixed numeral:** 
- **Tenths:** 
- **Zero is not in the middle:** 
- **Two names, one place:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/fractions-on-the-line/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a negative fraction, a negative mixed numeral, tenths, then a line that does not have zero in the middle, then the edge case of two names for one place. Hints, in order: Each unit is split into 4. Count 1 quarter left from 0. / Count 1 whole unit left from 0. / Then 3 more quarters further left. / This line is split into tenths. Count 6 tenths left from 0. / Zero is not in the middle of this line. Find it first. For students who need more support: Use the board for every question: the readout gives the fraction and the decimal. A strip of paper folded into quarters helps for the fractions. For early finishers: Find a number between −1/3 and −1/4. Then explain why there is always another number between any two. When students get stuck: A student puts −1.5 between −1 and 0 (Jo's mistake). Ask: Is −1.5 °C colder or warmer than −1 °C? Fix: Colder, so further left. Count 1 left, then half more left: −1.5 is further from zero than −1. / A student cannot place −3/4. Ask: How many equal parts is each unit split into? Which way is negative? Fix: Count the quarters aloud from zero: −1/4, −2/4, −3/4. / A student thinks −0.3 is greater than −0.25 because 3 is less than 25. Ask: Write both with two decimal places. Which is further left? Fix: −0.30 and −0.25: −0.30 is further from zero, so it is less. -->

<!-- Speaker note: Teacher: before the line from −2 to 5, ask: where is zero on this line? -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## In order

**Which list is in order from least to greatest?**

- $-1.2,\; -1\tfrac{1}{2},\; -\tfrac{3}{4},\; 0.5$
- $-\tfrac{3}{4},\; -1.2,\; -1\tfrac{1}{2},\; 0.5$
- $-1\tfrac{1}{2},\; -1.2,\; -\tfrac{3}{4},\; 0.5$

* Answer: $-1\tfrac{1}{2},\; -1.2,\; -\tfrac{3}{4},\; 0.5$

<!-- Speaker note: As decimals: −1.5, −1.2, −0.75, 0.5. Read left to right. -->

<!-- Speaker note: Teacher: in the ordering question, students can write every number as a decimal first. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Which is greater?

**Which is greater, $-0.3$ or $-0.25$?**

- $-0.3$
- $-0.25$

* Answer: $-0.25$

<!-- Speaker note: −0.25 is closer to zero, so it is greater: −0.3 < −0.25. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Halfway

**What number is halfway between $-3$ and $-2$?**

* Answer: -2.5

<!-- Speaker note: −2.5, or −2 1/2. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review

**Review: which is less, $-6$ or $-9$?**

- $-6$
- $-9$

* Answer: $-9$

<!-- Speaker note: −9 < −6: −9 is further left. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: always another number

The line now shows only $-1$ to $0$, in twentieths. Find a number between $-0.3$ and $-0.2$.

Stretch: find a number between $-\tfrac{1}{3}$ and $-\tfrac{1}{4}$. Can you always find another between any two?

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/fractions-on-the-line/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Finding a number between two close numbers reverses the lesson: from the places back to a number, and shows there is always another one in between. Hints, in order: This line is from −1 to 0, in steps of 0.05. Find −0.3 and −0.2. / What number is halfway between them? -->

<!-- Speaker note: Teacher: the board lands on −0.25; ask for other answers on paper (−0.21, −0.275). Then ask: is there a number between −0.3 and −0.25 too? -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Put in order from least to greatest: 0.4, −1 1/4, −0.8, −1/2.**

* −1 1/4, −0.8, −1/2, 0.4.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next in Number: the four operations with positive fractions, decimals and percentages (AC9M7N06), then negative fractions and decimals, with the same moves as integers. Where it leads: plotting points with fractional coordinates, reading scales in science (thermometers, measuring cylinders), and the density of the rational numbers. -->