---
title: "Fractions and decimals on the line"
subtitle: "Number Foundations · Fractions and decimals · Lesson 11"
---

# Fractions and decimals on the line

**Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}}

> **Key question:** 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.


:::teacher-note

**In one sentence:** 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).

**Students should be able to say:** 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.

**Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet.

:::


## Predict

Think of a temperature of $-1.5$ °C. Is it colder or warmer than $-1$ °C?

:::keep-together

**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 $2$

My prediction, and why:

{{lines-3}}

:::


:::teacher-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.

:::


## Run

Try it on the board in the tutorial. Was your prediction right? What was different?

{{lines-2}}

## Count the parts


:::keep-together

### Quick check

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

Answer: {{field-15}}

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:::teacher-note

**Answer:** 0.75

One quarter is 0.25, so three quarters is 0.75.

:::


:::keep-together

### Placing −3/4 and −1 1/4

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

1. $-\tfrac{3}{4}$ — The denominator is 4, so split each unit into 4 equal parts (quarters).
2. $\text{count 3 quarters left from } 0$ — Negative means left of zero.
3. $-\tfrac{3}{4} \text{ is between } -1 \text{ and } 0$ — It is 3 of the 4 quarters from 0 to $-1$.
4. $-1\tfrac{1}{4} = -1 - \tfrac{1}{4}$ — A negative mixed numeral is a whole number and a part, both below zero.
5. $\text{count 1 unit left, then 1 quarter further left}$ — The quarter is further from zero, not back towards it.
6. $-1\tfrac{1}{4} \text{ is between } -2 \text{ and } -1$ — Just past $-1$, a quarter of the way to $-2$.

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

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

☐ $-\tfrac{1}{4}$

☐ $-\tfrac{3}{4}$

:::


:::teacher-note

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

−3/4 is three quarters left of 0, further left than −1/4, so it is less.

:::


:::keep-together

### 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.
```

Line: {{field-15}}

Why is it wrong?

{{lines-2}}

:::


:::teacher-note

**Answer:** line 4

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.

:::


:::typical

## 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.** 

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

:::


## Place and order


:::keep-together

### In order

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

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

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

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

:::


:::teacher-note

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

As decimals: −1.5, −1.2, −0.75, 0.5. Read left to right.

:::


:::keep-together

### Which is greater?

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

☐ $-0.3$

☐ $-0.25$

:::


:::teacher-note

**Answer:** $-0.25$

−0.25 is closer to zero, so it is greater: −0.3 < −0.25.

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:::keep-together

### Halfway

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

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** -2.5

−2.5, or −2 1/2.

:::


:::keep-together

### Review

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

☐ $-6$

☐ $-9$

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:::teacher-note

**Answer:** $-9$

−9 < −6: −9 is further left.

:::


## 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?

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

## Exit ticket

:::keep-together

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

{{lines-3}}

:::


:::teacher-note

**Answer:** −1 1/4, −0.8, −1/2, 0.4.

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