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# Reading a graph

Linear relations · Rules, tables and graphs

<!-- Speaker note: Students read values from graphs of real data, starting from either axis, and say what a point means in a sentence with units (AC9M7A04, node 1). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 13 min). -->

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## Today

**We are learning to** a point on a graph of real data pairs two measurements. Find one value on its axis, go to the line, and read across to the other axis.

- I can say what a point on a graph of real data means, with units.
- I can find a value on one axis and read the matching value from the other.
- I can tell the graph's variables from its axis labels.

<!-- Speaker note: Before class: Do the Run board and the Make board yourself first. The cursor reads both values off the line; students can click the graph, use the slider, or use the − and + buttons. The data are plausible examples, not real measurements. If you have real data (a school weather station, a class experiment), swap in a graph of your own for the Investigate questions. Students know how to plot points (the last lesson). The new move is reading a point as two measurements with units: the first from the bottom axis, the second up the side. Axis labels now carry the names and units, such as Time (h). The Modify tasks read the other way (from the up-axis to the across-axis), meet a flat section, and meet negative values below the axis. -->

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## Why it matters

> Weather maps, phone battery meters and sports stats all use graphs. Before you can use one, you need to read it: which number is on which axis?

<!-- Speaker note: A graph of real data pairs two measurements. Each point says: **when this one was measured, that one was measured**. To read a graph, first check the **axis labels**: what is measured across, and what is measured up. The data on these graphs are examples. -->

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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

**A graph shows temperature (°C) up the side and time (h) along the bottom. One point on it is $(15, 24)$. What does that point say?**

- At 24 hours the temperature is 15 °C
- At 15 hours the temperature is 24 °C
- It is 15 °C for 24 hours
- The temperature went up by 15, then by 24

<!-- Speaker note: What happens: At 15 hours the temperature is 24 °C. The first number is read along the bottom axis (Time), the second up the side (Temperature), just as when you plotted points. (At 24 hours the temperature is 15 °C swaps the axes.) Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Ask a student who chose 'At 24 hours the temperature is 15 °C' which axis they read first. -->

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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%2Flinear-relations%2Freading-graphs%2Fstep-2&code=widget%3A+graph%0Aaxes%3A+Time+%28h%29%2C+Temperature+%28%C2%B0C%29%0Ax%3A+0%2C+24%0Ay%3A+0%2C+30%0Agrid%3A+3%2C+5%0Aseries+Data%3A+0%2C12%3B+3%2C10%3B+6%2C9%3B+9%2C14%3B+12%2C20%3B+15%2C24%3B+18%2C21%3B+21%2C16%3B+24%2C13%0Acursor+R%3A+3%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/reading-graphs/?full=step-2#step-2)

<!-- Speaker note: Use the slider, the − and + buttons, or click on the graph. The readout names both values with their units. At 15 hours, how hot is it? -->

<!-- Speaker note: Teacher: ask students to say each reading in a sentence with units before moving the cursor. -->

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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?

* At 15 hours the temperature is 24 °C. The first number is read along the bottom axis (Time), the second up the side (Temperature), just as when you plotted points. (At 24 hours the temperature is 15 °C swaps the axes.)

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

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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>

## Temperature

Here is how to read a graph, step by step.

$$\text{Axes: Time (h) across, Temperature (°C) up}$$

1) $\text{Time} = 9 \text{ h}$ — Find the value you know on its own axis.
2) $(9, \; ?)$ — Go up from 9 to the line.
3) $\text{Temperature} = 14 \text{ °C}$ — Read across to the other axis.
4) $\text{At 9 h the temperature is 14 °C.}$ — Say it in a sentence, with the units.

<!-- 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 Axes: Time (h) across, Temperature (°C) up and say "Check the labels first." Line 1: write Time = 9 h and say "Find the value you know on its own axis." Line 2: write (9, ?) and say "Go up from 9 to the line." Line 3: write Temperature = 14 °C and say "Read across to the other axis." Line 4: write At 9 h the temperature is 14 °C. and say "Say it in a sentence, with the units." -->

<!-- Speaker note: Teacher: model the worked example live on a projected graph: say the axes, put a finger on the known value, go to the line, then across. -->

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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>

## Temperature

**What is the temperature at 9 hours?**

* Answer: 14

<!-- Speaker note: At 9 h the line is at 14 °C, so the point is (9, 14). -->

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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>

## The highest point

**At what time (in hours) is the temperature highest?**

* Answer: 15

<!-- Speaker note: The highest point is (15, 24): the temperature is highest at 15 h. -->

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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

**Priya reads the temperature at 6 hours from the graph. Click the line where she goes wrong.**

```
 1  Read the temperature at 6 hours.
 2  Find 6 on the Time axis.
 3  Go up to the line.
 4  Read across: the line is at 9, so at 9 hours the temperature is 6 °C.
```

* Answer: line 4

<!-- Speaker note: Line 4: the line is at 9 when the time is 6, and 9 is read up the side, so it is the temperature. At 6 hours the temperature is 9 °C. Priya swapped the two numbers. -->

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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>

## Other graphs



- **Fuel:** Each task loads a new graph. Read the axis labels first.
- **A flat line:** 
- **Below zero:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/reading-graphs/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: read the other way (from the up-axis to the across-axis), a flat section, and negative values below the axis. Hints, in order: Read up the side to find 26, then follow across to the line. / A point below the Time axis has a negative temperature. For students who need more support: Give a printed graph with the gridlines labelled and a ruler, and read the first value together, naming the axis out loud. For early finishers: Using the temperature graph, find two different times when the temperature is 16 °C. Why are there two? When students get stuck: A student gives the time when asked for the temperature, or the reverse (Priya's mistake). Ask: What is this axis called? Is 9 measured in hours or in degrees? Fix: Write the units next to each number: 6 h and 9 °C. Then say the sentence: at 6 hours the temperature is 9 °C. / A student cannot find a value between gridlines. Ask: Which two gridlines is the line between? Is it closer to the lower or the upper one? Fix: Count the grid squares in fives or tens, and use the cursor's readout to check. / A student reads a negative value as positive. Ask: Is the point above or below the Time axis? Fix: Below the axis is below zero: write the minus sign. -->

<!-- Speaker note: Teacher: before the fuel graph, ask: which axis is the 26 on? -->

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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>

## The time

**What is the temperature at 21 hours?**

* Answer: 16

<!-- Speaker note: At 21 h the line is at 16 °C: the point is (21, 16). -->

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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>

## The fuel

**How much fuel is left after 300 km, in litres?**

* Answer: 14

<!-- Speaker note: At 300 km the line is at 14 L: the point is (300, 14). -->

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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 sentence?

**A graph of cost ($) against time (hours) has the point $(4, 30)$. Which sentence is true?**

- 30 hours cost $4
- 4 hours cost $30
- The cost went up by $4 over 30 hours

* Answer: 4 hours cost $30

<!-- Speaker note: The first number is the time (4 hours) and the second is the cost ($30). -->

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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: find $y = 4x + 1$ when $x = 3$.**

* Answer: 13

<!-- Speaker note: y = 4 × 3 + 1 = 12 + 1 = 13. -->

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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: tell a story with a graph

The points move up and down only. Then answer: what does your graph say about the battery at 4 hours?

Stretch: draw a graph where the battery falls quickly at first, then slowly.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/reading-graphs/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Making a graph from a story reverses reading: students choose each value to fit the words. Hints, in order: The battery starts full. What is the battery at 0 hours? / Each point must be lower than the one before it, and the last one is 20. -->

<!-- Speaker note: Teacher: compare two students' graphs. Both fall every two hours. How do they differ? -->

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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

**A graph of height (cm) against age (years) has the point (12, 150). What does it say?**

* At 12 years the height is 150 cm.

<!-- 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: describing the whole relationship in words (it increases, decreases or stays constant) instead of one point at a time. Where it leads in class: reading graphs in science experiments and in geography. -->