Comparing two plans

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

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• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• I can find where two plans cost the same from their graphs, and say what that means.
• I can decide which plan is cheaper for a given amount, and explain why in words.
• I can say which values make sense for a model, and spot an answer that does not.

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. This lesson uses the last two: solving with graphs (where lines meet) and writing rules from situations.

The new move is interpretation. The crossing answers "when are they the same?"; the lower line answers "which is cheaper?"; and every model has values where it makes sense (no negative data, whole tickets, a battery that stops at 0%). Answers should be sentences in the situation's words, not just numbers.

The lines start at 0 on purpose: the graph itself shows where the model applies.

Running the lesson

1. Predict: take a show of hands. 'Plan A, lower fee' is the common wrong answer. Ask a student who chose it to work out 6 GB on both plans.
2. Run: once P is on the crossing, point left and right of it: which line is lower on each side?
3. Investigate: model the worked example live and finish with the sentence in words. Insist on the sentence: 'Up to 4 GB, Plan A is cheaper...'.
4. Modify: before the electricians, ask: what did the c lesson say about lines with the same m?
5. Practise: for each answer, ask one student to say it as a sentence about the situation.
6. Make: compare two students' rival plans. Both pass through (6, 50); which would you advertise to whom?

When students get stuck

Answers to the checks

1. which plan?: Plan A costs 20 dollars a month plus 5 dollars per GB. Plan B costs 40 dollars a month for any amount of data. Mia uses 6 GB a month. Which plan is cheaper for her?
   Plan B. For 6 GB, Plan A costs 20 + 5 × 6 = 50 dollars and Plan B costs 40. A's lower fee only wins for small amounts of data.
2. where they cost the same: Drag $P$ to where the two plans cost the same.
   
3. Comparing the plans: For 2 GB a month, which plan is cheaper?
   Plan A
   For 2 GB, Plan A costs 20 + 5 × 2 = 30 dollars and Plan B costs 40. Left of the crossing, Plan A's line is lower.
4. Find the mistake: Leo compares the plans. Click the line where his reasoning goes wrong.
   line 4
   Line 4: Plan A's lower fee only wins up to 4 GB. After that its 5 dollars per GB adds up, and Plan B is cheaper. Which plan is cheaper depends on how much data you use.
5. Does it make sense?: The rule $T = 5g + 20$ gives $T = 10$ when $g = -2$. Why does that answer not make sense?
   You cannot use negative data, so the rule only works for g of 0 or more
   A model only fits the situation for some values. Data used cannot be negative, so the rule (and its line) starts at g = 0.
6. Gyms: Gym A is $T = 10w + 50$; Gym B is $T = 15w$. Drag $P$ to where they cost the same.
   
   At 10 weeks both cost 150 dollars. Before that Gym B is cheaper; after it, Gym A.
7. Catching up: Sam's savings are $S = 5w + 30$; Ana's are $S = 8w$. Drag $P$ to when they have the same amount.
   
   After 10 weeks both have 80 dollars. After that, Ana has more: her rate is bigger.
8. Never crossing: Two electricians charge $T = 15h + 30$ and $T = 15h + 50$. When does the second one cost less?
   Never: the lines are parallel, so the first is always 20 dollars cheaper
   Both charge 15 dollars an hour, so the lines are parallel and never cross. The first is always 20 dollars cheaper, for any job.
9. When the model stops: A battery is modelled by $B = -10h + 100$. What does the model give after 12 hours, and does it make sense?
   −20%: it does not make sense, so the model only works from 0 to 10 hours
   −10 × 12 + 100 = −20. A battery cannot go below 0%: it is flat at 10 hours, and the model stops there.
10. When they are equal: Plan C costs 10 dollars plus 8 dollars per GB. Plan D costs 34 dollars for any amount. At how many GB do they cost the same?
    3
    10 + 8g = 34, so 8g = 24 and g = 3. Check: 10 + 24 = 34.
11. Which is cheaper?: Using Plans C and D, which is cheaper for 5 GB?
    Plan D
    Plan C: 10 + 8 × 5 = 50 dollars. Plan D: 34 dollars. Above 3 GB, D is cheaper.
12. Values that make sense: Concert tickets cost $T = 12n + 6$ dollars for $n$ tickets. Which values of $n$ make sense?
    Whole numbers from 1 up
    You buy whole tickets, and at least one. The line joins the points, but only the whole numbers are real orders.
13. Review: Review: an electrician charges a 25 dollar call-out fee plus 9 dollars an hour. Write a rule for the total $T$ for $h$ hours.
    9h + 25
    T = 9h + 25: 9 dollars for each hour (m) and 25 dollars once (c).
14. design a rival plan: Design Plan B so that it costs the same as Plan A at 6 GB, and is cheaper than Plan A above 6 GB.
    
    Your plan ties with Plan A at 6 GB (50 dollars) and has a lower rate, so it is cheaper for anyone who uses more.

Solutions

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

new comparisons › Gyms

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

new comparisons › Catching up

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

new comparisons › Never crossing

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

new comparisons › When the model stops

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

design a rival plan

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

Stretch and support

Stretch: A third plan costs 25 dollars plus 3 per GB. Where does it cross Plan A and Plan B? Which plan is cheapest for 2, 5 and 10 GB?

Support: Give a table with g = 0 to 8 for both plans to fill in first; the crossing is where the two columns match.

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 "Comparing two plans".

The idea: Solving, interpreting and reviewing the model: I can Students solve a linear model for a given value, interpret the solution in context, and evaluate whether the model is appropriate for the situation.

What understanding looks like:
- Uses the linear model correctly to answer a specific question.
- States the solution in contextually appropriate terms.
- Identifies at least one limitation or domain restriction of the model.

Mistakes students often make here:
- A student says the plan with the lower fee is always cheaper (Leo's mistake).
- A student gives the crossing as a number with no meaning.
- A student accepts a negative or impossible answer.

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.
- If I seem to be missing the idea that comes before this one (Formulating linear models), check it quickly, then come back.
- 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 "Comparing two plans".

Exit ticket

Plan X costs 15 dollars plus 6 per GB; Plan Y costs 33 dollars flat. Which is cheaper for 2 GB and for 5 GB? Where do they cost the same?

2 GB: X (27 dollars vs 33). 5 GB: Y (45 vs 33). They cost the same at 3 GB (33 dollars).

What each step is for

• which plan?: Committing to a plan before seeing the graph exposes the belief that the lower fee always wins.
• where they cost the same: On one graph the two plans' lines cross at a single point: the amount of data where they cost the same, which splits the light users from the heavy ones.
• answering in words: The crossing answers 'when do they cost the same'; the lower line answers 'which is cheaper'; and a model only makes sense for some values.
• new comparisons: One change at a time: a plan with no fee, one amount catching up with another, two plans that never cross, and a model that stops working.
• compare and check: Finding the crossing, choosing the cheaper plan for a given amount, and saying where a model makes sense, mixed; the review keeps writing rules fresh.
• design a rival plan: Designing a plan that ties with Plan A at a chosen amount and beats it for heavy users reverses the lesson: from the crossing to the rule.

Where it leads

This finishes the Linear relations course so far. Later courses use the same ideas: simultaneous equations (two lines, one crossing, solved with algebra), inequalities (which side of a line), and non-linear models when a straight line stops fitting.

Where it leads in class: any "which deal is better" question, break-even in business, and checking whether a model's answer makes sense in science.