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# Describing a relationship

Linear relations · Rules, tables and graphs

<!-- Speaker note: Students describe how one measurement changes as the other increases, section by section, and tell the story a graph shows without reading it as a picture (AC9M7A04, node 2). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 14 min · Practise and Make 11 min). -->

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## Today

**We are learning to** describe how one measurement changes as the other increases, one section of the graph at a time: it increases, decreases or stays constant.

- I can describe a section of a graph as increasing, decreasing or constant.
- I can tell the story of a graph in order, with the measurements named.
- I can draw a graph that matches a story.

<!-- Speaker note: Before class: Do the Run, Modify (bath) and Make boards yourself first. In Make, the points move up and down only, so students choose each value and the times stay fixed. The data are plausible examples, not real measurements. The new move is describing a section by comparing its start and end: as time increases, the other measurement increases, decreases or stays constant. The common slip is reading the graph as a picture of the journey, so that a line going down means a hill. Omar's working addresses it. A flat section is a constant measurement while time still passes: the cyclist is stopped, not going slowly. The cooling drink adds a decrease that slows down: the line is always going down but gets less steep. -->

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## Why it matters

> A graph tells a story about how two things change together. Telling that story in words, section by section, is how scientists and journalists use graphs.

<!-- Speaker note: A graph does not just give values. It shows how one measurement **changes** as the other changes. Look at one section at a time, and ask: as the time goes on, does the other measurement **increase**, **decrease**, or stay **constant** (not change)? 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 a cyclist's distance from home against time. The line goes up, then is flat, then goes down. Which story fits?**

- Rides away, then home, then away again
- Rides up a hill, along the flat, then down a hill
- Rides away, then faster, then slower
- Rides away, stops for a rest, rides home

<!-- Speaker note: What happens: Rides away, stops for a rest, rides home. Up means the distance from home is increasing, flat means it is not changing (stopped), and down means it is decreasing (riding back). The graph shows distance, not the shape of the road. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Ask a student who chose the hill story what the vertical axis measures. -->

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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%2Fdescribing-relationships%2Fstep-2&code=widget%3A+graph%0Aaxes%3A+Time+%28h%29%2C+Distance+from+home+%28km%29%0Ax%3A+0%2C+5%0Ay%3A+0%2C+80%0Agrid%3A+1%2C+20%0Aseries+Data%3A+0%2C0%3B+1%2C30%3B+2%2C60%3B+3%2C60%3B+4%2C30%3B+5%2C0%0Acursor+R%3A+1%0A&fresh=1}}

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

<!-- Speaker note: Move the cursor a step at a time. For each section, say what the distance from home does as time goes on. -->

<!-- Speaker note: Teacher: ask students to narrate the ride aloud as you move 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?

* Rides away, stops for a rest, rides home. Up means the distance from home is increasing, flat means it is not changing (stopped), and down means it is decreasing (riding back). The graph shows distance, not the shape of the road.

<!-- 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>

## A rest

Here is how to describe a graph, one section at a time.

$$\text{0 to 2 h: } 0 \to 60 \text{ km}$$

1) $\text{2 to 3 h: } 60 \to 60 \text{ km}$ — The distance does not change. It is constant.
2) $\text{3 to 5 h: } 60 \to 0 \text{ km}$ — The distance decreases.
3) $\text{increases} \to \text{constant} \to \text{decreases}$ — Put the sections in order. Say it: "As time increases, the distance from home increases, then stays constant, then decreases."

<!-- 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 0 to 2 h: 0 60 km and say "Take one section at a time and compare its start and end. The distance increases." Line 1: write 2 to 3 h: 60 60 km and say "The distance does not change. It is constant." Line 2: write 3 to 5 h: 60 0 km and say "The distance decreases." Line 3: write increases constant decreases and say "Put the sections in order. Say it: "As time increases, the distance from home increases, then stays constant, then decreases."." -->

<!-- Speaker note: Teacher: model the worked example live: section by section, start value, end value, then the word. -->

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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>

## A rest

**Between 2 and 3 hours, which sentence is true?**

- As time increases, the distance from home stays constant
- As time increases, the distance from home increases
- As time increases, the distance from home decreases

* Answer: As time increases, the distance from home stays constant

<!-- Speaker note: The line is flat from 2 to 3 hours: the distance is 60 km at both times, so it does not change. The cyclist has stopped. -->

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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>

## Riding home

**What is the cyclist doing between 3 and 5 hours?**

- Riding back towards home
- Riding away from home
- Riding down a hill

* Answer: Riding back towards home

<!-- Speaker note: The distance from home decreases from 60 km to 0 km: the cyclist is riding back towards home. -->

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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

**Omar describes the ride from 3 to 5 hours. Click the line where his reasoning goes wrong.**

```
 1  Describe the graph from 3 to 5 hours.
 2  The line goes down.
 3  So the cyclist rides down a hill.
 4  The distance from home is getting smaller.
```

* Answer: line 3

<!-- Speaker note: Line 3: the graph does not show hills. The line goes down because the distance from home is getting smaller: the cyclist is riding home. -->

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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 stories



- **The bath:** Each task loads a new graph. Describe the sections before you answer.
- **Two walkers:** 
- **A cooling drink:** 

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

<!-- Speaker note: What this step is for: One change at a time: a story with a turning point, two graphs on one set of axes, and a measurement that decreases at a slowing rate. Hints, in order: The water stops rising when the line goes flat. When does it start to go down? For students who need more support: Give the sentence frame: As time increases, the ___ increases / decreases / stays constant. Mark where each section starts and ends on a printed graph. For early finishers: Draw a graph of the depth of water in a bath where the water fills quickly, then slowly, then the plug is pulled. Describe it in a sentence. When students get stuck: A student says the line going down means the cyclist is going downhill (Omar's mistake). Ask: What does the vertical axis measure? What would the line do if the road were flat but the cyclist rode home? Fix: Rewrite with the axis name: the distance from home is getting smaller. / A student says a flat line means no time has passed, or no movement is happening at all. Ask: What is going along the bottom axis? Does it keep going? Fix: Time keeps going. A flat line means the other measurement is not changing. / A student describes only the start and end of the whole graph. Ask: What happens in the middle? Fix: Split the graph into sections where the direction changes, and describe each one. -->

<!-- Speaker note: Teacher: after the cooling drink, ask: does the temperature ever go up? How does the drop in the first 10 minutes compare with the last 10? -->

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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 ride

**Which sentence describes the ride from 0 to 2 hours?**

- As time increases, the distance from home stays constant
- The cyclist rides up a hill
- As time increases, the distance from home increases

* Answer: As time increases, the distance from home increases

<!-- Speaker note: The line goes up from 0 to 60 km: the distance from home increases. -->

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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>

## A tank

**Which story fits the graph?**

- The tank fills, then drains, then fills again
- The tank fills at the same rate the whole time
- The tank fills slowly, the water is left, then it fills faster

* Answer: The tank fills slowly, the water is left, then it fills faster

<!-- Speaker note: From 0 to 3 min it rises (30 L). From 3 to 6 min it is flat (nothing added). From 6 to 9 min it rises 60 L, faster than before. -->

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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: what is the temperature at 20 minutes, in °C?**

* Answer: 45

<!-- Speaker note: At 20 min the line is at 45 °C: the point is (20, 45). -->

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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 bath again

**Between 6 and 8 minutes, the amount of water...**

- decreases
- increases
- stays constant

* Answer: decreases

<!-- Speaker note: The line goes down from 80 L to 40 L: the water decreases (the plug is pulled). -->

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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: draw a bath

Drag each point up or down. Then describe your graph in a sentence.

Stretch: draw a bath that fills quickly at first, then slowly.

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

<!-- Speaker note: What this step is for: Making a graph from a story reverses describing: students choose each value to fit the words, and must keep the flat part flat. Hints, in order: The bath starts empty: what is the water at 0 minutes? / A flat part means two points at the same height. -->

<!-- Speaker note: Teacher: compare two students' baths: where is the flat part, and how do you know? -->

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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 a tap's water in a bucket (L) against time (min) goes up, then is flat. Describe it.**

* As time increases, the amount of water increases, then stays constant (the tap was turned off).

<!-- 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: the straight line gets a rule. The gradient (how steep) and the intercept (where it starts) name what these graphs have been showing. Where it leads in class: describing graphs in science (speed, temperature, growth) and in geography. -->