Rules from real situations

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min

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.

Running the lesson

1. Predict: take a show of hands. 'T = 4d + 2' is the common wrong answer: the numbers taken in the order they appear.
2. Run: before students touch the sliders, ask: what would a 0 km trip cost? (4.) Where is that on the graph?
3. Investigate: model the worked example live. Underline 'a month' (once) and 'for each GB' (the rate) in the words, then write c and m.
4. Modify: before the battery task, ask: which way will this line go?
5. Practise: for the plumber question, ask students to say what 80 means in words too.
6. Make: pair students to swap situations and write each other's rules.

When students get stuck

Answers to the checks

1. a taxi fare: 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 = 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).
2. fit the fares: Set $m$ and $c$ so the line goes through all three fares.
   
3. 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.
   15w + 30
   T = 15w + 30: 30 dollars once (c), then 15 dollars for each week (m).
4. Find the mistake: Zoe writes a rule for a taxi fare. Click the line where her reasoning goes wrong.
   line 2
   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.
5. 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 is m, the rate: each extra GB adds 5 dollars. The 20 is c, the fee you pay even with no data.
6. No starting amount: Bike hire: 6 dollars for 1 hour, 24 dollars for 4 hours. Set $m$ and $c$ so the line goes through both.
   
   T = 6h: 6 dollars per hour and no fee, so c = 0 and the line starts at (0, 0).
7. Going down: A phone battery: 80% after 2 hours, 50% after 5 hours. Fit the line.
   
   B = −10h + 100: it starts full (c = 100) and goes down 10% each hour (m = −10).
8. Less than a dollar: Printing: 3 dollars for 2 pages, 5 dollars for 6 pages. Fit the line.
   
   T = 0.5n + 2: a 2 dollar fee and 50 cents a page.
9. Only the numbers: Two fares from a table: 11 dollars for 2 km and 17 dollars for 4 km. Find the rule.
   
   T = 3d + 5: up 6 over 2 km is 3 per km, and 11 − 2 × 3 = 5 to get in.
10. 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.
    12n + 6
    T = 12n + 6: 12 dollars for each ticket (m) and 6 dollars once (c).
11. What the 60 means: A plumber charges $T = 80h + 60$ dollars for a job of $h$ hours. What does the 60 mean?
    A 60 dollar call-out fee, paid once
    60 is c: what you pay even for 0 hours, the call-out fee. 80 is the hourly rate.
12. Using the rule: Using $T = 80h + 60$, what does a 3-hour job cost, in dollars?
    300
    80 × 3 + 60 = 240 + 60 = 300 dollars.
13. Review: Review: $y = 2x + 1$ and $y = 7$ meet at $(3, 7)$. Solve $2x + 1 = 7$.
    3
    x = 3. Check: 2 × 3 + 1 = 7.
14. your own situation: Choose your own starting amount and rate (both more than 0). Then drag $P$ to the total for 4 units.
    
    P is the total for 4 units: 4 lots of your rate, plus your starting amount.

Solutions

One possible solution for each task. Other correct programs are fine.

new situations › No starting amount

Open task: any program that does what the step asks.

new situations › Going down

Open task: any program that does what the step asks.

new situations › Less than a dollar

Open task: any program that does what the step asks.

new situations › Only the numbers

Open task: any program that does what the step asks.

your own situation

Open task: any program that does what the step asks.

Stretch and support

Stretch: 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.)

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.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Rules from real situations".

The idea: Formulating linear models: I can Students identify a linear relationship in a practical context, formulate it as a function y = mx + b, and identify the meaning of gradient and intercept in context.

What understanding looks like:
- Correctly identifies gradient and intercept from a practical context.
- Formulates the linear function in equation form.
- Interprets m and b in terms of the problem context.

Mistakes students often make here:
- A student uses the numbers in the order they appear (Zoe's mistake).
- A student adds the fee into the rate (T = 6d).
- A student cannot find c from two points (the last Modify task).

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Rules from real situations".

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).

What each step is for

• a taxi fare: Committing to a rule before fitting the line shows whether students can tell the one-off amount from the rate in words.
• fit the fares: Fitting a line through real fares makes c and m mean something: c is the fare for 0 km, m is the extra for each km.
• from words to a rule: In words, the starting amount is the value when nothing has happened yet (c); the rate is 'per', 'each' or 'every' (m).
• new situations: 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.
• rules and meanings: Writing a rule, reading its numbers in words and using it for a value are one skill; the review keeps solving with graphs fresh.
• your own situation: Choosing a starting amount and a rate for a situation of their own reverses the lesson: from the rule back to words.

Where it 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.