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# Rules from real situations

Linear relations · Solving and modelling with lines

<!-- Speaker note: Students turn a situation that is a starting amount plus a rate into a rule y = mx + c, read m and c back into words, and fit lines through real values (AC9M8A03, node 1). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** many real costs and amounts are a starting amount plus a rate. The starting amount is c and the rate is m, so the rule is y = mx + c.

- I can write a rule like T = 2d + 4 from a situation described in words.
- I can explain what m and c mean in a situation (the rate and the starting amount).
- I can use a rule to work out a value, and check it in words.

<!-- Speaker note: Before class: Do the Run and Modify boards yourself first. Students should already know what m and c do to a line (the first two lessons of this course). The new move is translation: the starting amount (paid once, the value when nothing has happened) is c; the rate ("per", "each", "every") is m. The order of the numbers in the words does not matter, and Zoe's working shows the mistake of assuming it does. The boards use the situation's own letters (T for total, d for distance), labelled on the axes. The graphs still behave exactly like y = mx + c. The next lesson uses these rules: two plans, where their lines cross, and when a rule stops being a good model. -->

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## Why it matters

> Taxi fares, phone plans, gym memberships and a phone battery running down all follow the same pattern: where you start, and how much changes each time.

<!-- Speaker note: So far the rules had $x$ and $y$. Real situations have their own letters: $T$ for a total cost, $d$ for a distance, $h$ for hours. The graph works the same way: $m$ is still how much the line goes up for each step right, and $c$ is still where it starts. -->

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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 taxi costs 4 dollars to get in, plus 2 dollars for each km. Which rule gives the fare $T$ for a trip of $d$ km?**

- $T = 4d + 2$
- $T = 2d$
- $T = 6d$
- $T = 2d + 4$

<!-- Speaker note: What happens: T = 2d + 4. The 4 dollars is paid once, before you travel: it is where the line starts (c). The 2 dollars is paid for each km: it is the rate (m). Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. 'T = 4d + 2' is the common wrong answer: the numbers taken in the order they appear. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Flinear-relations%2Frules-from-situations%2Fstep-2&code=widget%3A+graph%0Aaxes%3A+d+%28km%29%2C+T+%28%24%29%0Ax%3A+-0.5%2C+6%0Ay%3A+-1%2C+18%0Aslider+m%3A+0%2C+4%2C+1%0Aslider+c%3A+0%2C+8%2C+0%0Aplot%3A+y+%3D+mx+%2B+c%0Apoint+P%3A+1%2C+6%0Apoint+Q%3A+3%2C+10%0Apoint+R%3A+5%2C+14%0A&fresh=1}}

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

<!-- Speaker note: The points are three real fares: 6 dollars for 1 km, 10 dollars for 3 km and 14 dollars for 5 km. Set $m$ and $c$ so the line passes through all three. Was your rule right? -->

<!-- Speaker note: Teacher: before students touch the sliders, ask: what would a 0 km trip cost? (4.) Where is that on the graph? -->

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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?

* T = 2d + 4. The 4 dollars is paid once, before you travel: it is where the line starts (c). The 2 dollars is paid for each km: it is the rate (m).

<!-- 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 phone plan

Here is how to turn words into a rule.

$$\text{Fee } 20, \text{ then } 5 \text{ per GB}$$

1) $g = 0: \; T = 20$ — With no data used, you still pay the fee. That is the value when $g$ is 0.
2) $c = 20$ — The starting amount is $c$.
3) $m = 5$ — Each extra GB adds 5 dollars. "Per", "each" or "every" signals the rate, $m$.
4) $T = 5g + 20$ — Put them together.
5) $g = 3: \; T = 5(3) + 20 = 35$ — Check with a value you can work out in words: 3 GB costs 20 + 15 = 35 dollars.

<!-- 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 Fee 20, then 5 per GB and say "A phone plan costs 20 dollars a month, plus 5 dollars for each GB used. Find a rule for the cost T." Line 1: write g = 0: T = 20 and say "With no data used, you still pay the fee. That is the value when g is 0." Line 2: write c = 20 and say "The starting amount is c." Line 3: write m = 5 and say "Each extra GB adds 5 dollars. "Per", "each" or "every" signals the rate, m." Line 4: write T = 5g + 20 and say "Put them together." Line 5: write g = 3: T = 5(3) + 20 = 35 and say "Check with a value you can work out in words: 3 GB costs 20 + 15 = 35 dollars." -->

<!-- Speaker note: Teacher: model the worked example live. Underline 'a month' (once) and 'for each GB' (the rate) in the words, then write c and m. -->

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## A phone plan

**A gym charges a 30 dollar joining fee and 15 dollars a week. Write a rule for the total cost $T$ after $w$ weeks.**

* Answer: 15w + 30

<!-- Speaker note: T = 15w + 30: 30 dollars once (c), then 15 dollars for each week (m). -->

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## Find the mistake

**Zoe writes a rule for a taxi fare. Click the line where her reasoning goes wrong.**

```
 1  A taxi costs 5 dollars to get in, plus 3 dollars per km.
 2  m is the first number in the question, so m = 5.
 3  c is the other number, so c = 3.
 4  The rule is T = 5d + 3.
```

* Answer: line 2

<!-- Speaker note: Line 2: m is the rate, the amount for each km, which is 3. It does not matter which number comes first in the question. The rule is T = 3d + 5. -->

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## What the numbers mean

**A phone plan costs $T = 5g + 20$ dollars a month for $g$ GB of data. What does the 5 mean?**

- 5 dollars for each GB used
- 5 GB of data are included
- The monthly fee is 5 dollars

* Answer: 5 dollars for each GB used

<!-- Speaker note: 5 is m, the rate: each extra GB adds 5 dollars. The 20 is c, the fee you pay even with no data. -->

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## New situations



- **No starting amount:** Each task gives some real values as points. Set $m$ and $c$ so the line goes through them all.
- **Going down:** A phone battery runs down at the same rate each hour.
- **Less than a dollar:** Printing costs a fee plus a little for each page.
- **Only the numbers:** No words this time: just two fares. Work out the rate first.

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

<!-- Speaker note: What this step is for: One change at a time: no starting amount (c = 0), an amount that goes down (negative m), a rate that is not whole, and only two points with no words. Hints, in order: What does hiring the bike for 0 hours cost? / The battery goes from 80% to 50% in 3 hours. How much each hour? / It goes down, so m is negative. Then work back to 0 hours for c. / From 2 pages to 6 pages the cost goes up 2 dollars. How much per page? / From 2 km to 4 km the fare goes up 6. How much per km? / Work back from 2 km to 0 km to find the fee. For students who need more support: Give a two-column table (0, 1, 2, 3 units) to fill in from the words before writing the rule: the 0 row gives c and the difference gives m. For early finishers: A candle is 20 cm tall and burns 1.5 cm an hour. Write its rule, and work out when it burns out. (h = 20 − 1.5t; after about 13.3 hours.) When students get stuck: A student uses the numbers in the order they appear (Zoe's mistake). Ask: Which amount is paid only once? Which is paid again for every km? Fix: Underline 'per', 'each' or 'every': that number is m. The one-off amount is c. / A student adds the fee into the rate (T = 6d). Ask: What would a 0 km trip cost with your rule? With the real taxi? Fix: Work out 0 km and 1 km in words: 4 dollars and 6 dollars. The fee is paid once; only the 2 repeats. / A student cannot find c from two points (the last Modify task). Ask: How much does the fare go up each km? What would 0 km cost? Fix: Work back from the first point one step at a time: 11 at 2 km, 8 at 1 km, 5 at 0 km. -->

<!-- Speaker note: Teacher: before the battery task, ask: which way will this line go? -->

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## Tickets

**Concert tickets cost 12 dollars each, plus a 6 dollar booking fee for the order. Write a rule for the total $T$ for $n$ tickets.**

* Answer: 12n + 6

<!-- Speaker note: T = 12n + 6: 12 dollars for each ticket (m) and 6 dollars once (c). -->

<!-- Speaker note: Teacher: for the plumber question, ask students to say what 80 means in words 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>

## What the 60 means

**A plumber charges $T = 80h + 60$ dollars for a job of $h$ hours. What does the 60 mean?**

- The cost of a 60-hour job
- 60 dollars for each hour
- A 60 dollar call-out fee, paid once

* Answer: A 60 dollar call-out fee, paid once

<!-- Speaker note: 60 is c: what you pay even for 0 hours, the call-out fee. 80 is the hourly rate. -->

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## Using the rule

**Using $T = 80h + 60$, what does a 3-hour job cost, in dollars?**

* Answer: 300

<!-- Speaker note: 80 × 3 + 60 = 240 + 60 = 300 dollars. -->

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## Review

**Review: $y = 2x + 1$ and $y = 7$ meet at $(3, 7)$. Solve $2x + 1 = 7$.**

* Answer: 3

<!-- Speaker note: x = 3. Check: 2 × 3 + 1 = 7. -->

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## Make: your own situation

Invent a situation that is a starting amount plus a rate: a club with a joining fee and a weekly cost, a game with a starting score and points per level. Set $c$ and $m$, then drag $P$ to the total for 4 units.

Stretch: tell a partner your situation in words. Can they write your rule?

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

<!-- Speaker note: What this step is for: Choosing a starting amount and a rate for a situation of their own reverses the lesson: from the rule back to words. Hints, in order: Pick a starting amount with c and a rate with m. / Work out the total for 4 units: 4 lots of m, plus c. Drag P there. -->

<!-- Speaker note: Teacher: pair students to swap situations and write each other's rules. -->

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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 swimming pool charges a 50 dollar membership plus 4 dollars a visit. Write the rule for the total T for v visits, and say what each number means.**

* T = 4v + 50. 4 is the cost per visit (m); 50 is the membership, paid once (c).

<!-- 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: comparing two plans. Two rules on one graph; where the lines cross is where they cost the same, and the model only makes sense for some values (no negative km). Where it leads in class: science graphs (a starting temperature and a heating rate), budgets and savings plans, and any "fee plus rate" price. -->