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# Comparing two plans

Linear relations · Solving and modelling with lines

<!-- Speaker note: Students put two linear rules on one graph, find where they are equal, decide which is cheaper on each side in the situation's words, and say where a model makes sense (AC9M8A03, node 2). 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** put two rules on one graph. Where the lines cross they cost the same; on either side, the lower line is the cheaper plan. Then check where the model makes sense.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> Phone plans, gyms and streaming services all ask you to choose between a bigger fee and a lower rate. Two lines on one graph answer it for any amount.

<!-- Speaker note: In the last lesson you wrote one rule for one situation. Choosing between two plans needs two rules on the same graph, and a question answered in the situation's own words: which plan, for how much data, and does the answer make sense? -->

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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

**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
- They cost the same
- Plan A: it has the lower fee

<!-- Speaker note: What happens: 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. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: 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. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Flinear-relations%2Fcomparing-plans%2Fstep-2&code=widget%3A+graph%0Aaxes%3A+g+%28GB%29%2C+T+%28%24%29%0Ax%3A+-0.5%2C+8%0Ay%3A+-2%2C+64%0Agrid%3A+off%0Asnap%3A+1%2C+2%0Aplot%3A+y+%3D+5x+%2B+20%2C+from+0%0Aplot%3A+y+%3D+40%2C+from+0%0Apoint+P%3A+7%2C+6%2C+drag%0A&fresh=1}}

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

<!-- Speaker note: Plan A is $T = 5g + 20$; Plan B is $T = 40$. Drag $P$ to where the lines cross. Then look at 6 GB: which line is lower there? Was your prediction right? -->

<!-- Speaker note: Teacher: once P is on the crossing, point left and right of it: which line is lower on each side? -->

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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?

* 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.

<!-- 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>

## Comparing the plans

Here is how to compare two plans and answer in words.

Answer in words: "Up to 4 GB, Plan A is cheaper. At 4 GB they both cost 40 dollars. Above 4 GB, Plan B is cheaper."

$$T = 5g + 20, \quad T = 40$$

1) $5g + 20 = 40$ — They cost the same where the two rules are equal: where the lines cross.
2) $g = 4, \; T = 40$ — The lines cross at $(4, 40)$. Check: $5 \times 4 + 20 = 40$.
3) $g < 4: \text{ A is cheaper}$ — Left of the crossing, Plan A's line is lower.
4) $g > 4: \text{ B is cheaper}$ — Right of the crossing, Plan B's line is lower.
5) $g \ge 0$ — The model only makes sense from 0 GB: you cannot use negative data.

<!-- 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 T = 5g + 20, T = 40 and say "Plan A and Plan B, as rules. Which is cheaper, and when?" Line 1: write 5g + 20 = 40 and say "They cost the same where the two rules are equal: where the lines cross." Line 2: write g = 4, T = 40 and say "The lines cross at (4, 40). Check: 5 × 4 + 20 = 40." Line 3: write g < 4: A is cheaper and say "Left of the crossing, Plan A's line is lower." Line 4: write g > 4: B is cheaper and say "Right of the crossing, Plan B's line is lower." Line 5: write g 0 and say "The model only makes sense from 0 GB: you cannot use negative data." -->

<!-- Speaker note: Teacher: model the worked example live and finish with the sentence in words. Insist on the sentence: 'Up to 4 GB, Plan A is cheaper...'. -->

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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>

## Comparing the plans

**For 2 GB a month, which plan is cheaper?**

- Plan A
- Plan B
- They cost the same

* Answer: Plan A

<!-- Speaker note: 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. -->

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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

**Leo compares the plans. Click the line where his reasoning goes wrong.**

```
 1  Plan A: 20 dollars a month plus 5 dollars per GB.
 2  Plan B: 40 dollars a month, any amount of data.
 3  Plan A has the lower monthly fee.
 4  So Plan A is always the cheaper plan.
```

* Answer: line 4

<!-- Speaker note: 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. -->

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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>

## Does it make sense?

**The rule $T = 5g + 20$ gives $T = 10$ when $g = -2$. Why does that answer not make sense?**

- Phone plans can cost less than their fee
- You cannot use negative data, so the rule only works for g of 0 or more
- The rule is wrong: it should be T = 5g − 20

* Answer: You cannot use negative data, so the rule only works for g of 0 or more

<!-- Speaker note: A model only fits the situation for some values. Data used cannot be negative, so the rule (and its line) starts at g = 0. -->

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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>

## New comparisons



- **Gyms:** Each task puts a new pair of rules on the board. Drag $P$ to where they cost the same.
- **Catching up:** Two people saving at different rates, from different starting amounts.
- **Never crossing:** Two electricians with the same hourly rate.
- **When the model stops:** A phone battery running down.

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

<!-- Speaker note: What this step is for: 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. Hints, in order: Gym A: 50 dollars to join, then 10 a week. Gym B: no joining fee, 15 a week. Follow both lines until they meet. / Sam starts with 30 dollars and saves 5 a week. Ana starts with nothing and saves 8 a week. When has Ana caught up? For students who need more support: Give a table with g = 0 to 8 for both plans to fill in first; the crossing is where the two columns match. For early finishers: 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? When students get stuck: A student says the plan with the lower fee is always cheaper (Leo's mistake). Ask: Work out both plans for 10 GB. Which is cheaper now? Fix: The fee is only where the line starts. The rate decides who wins for large amounts: the lines cross, and the cheaper plan changes there. / A student gives the crossing as a number with no meaning. Ask: What are the 4 and the 40 in this situation? Fix: Say it with units: '4 GB costs 40 dollars on either plan'. / A student accepts a negative or impossible answer. Ask: Could this happen in real life? Fix: List the values that make sense (0 or more GB, whole tickets, 0 to 100%) before answering. -->

<!-- Speaker note: Teacher: before the electricians, ask: what did the c lesson say about lines with the same m? -->

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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>

## 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?**

* Answer: 3

<!-- Speaker note: 10 + 8g = 34, so 8g = 24 and g = 3. Check: 10 + 24 = 34. -->

<!-- Speaker note: Teacher: for each answer, ask one student to say it as a sentence about the situation. -->

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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 is cheaper?

**Using Plans C and D, which is cheaper for 5 GB?**

- Plan C
- They cost the same
- Plan D

* Answer: Plan D

<!-- Speaker note: Plan C: 10 + 8 × 5 = 50 dollars. Plan D: 34 dollars. Above 3 GB, D is cheaper. -->

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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>

## 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
- Any number, including 2.5
- Any number, including negatives

* Answer: Whole numbers from 1 up

<!-- Speaker note: You buy whole tickets, and at least one. The line joins the points, but only the whole numbers are real orders. -->

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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: an electrician charges a 25 dollar call-out fee plus 9 dollars an hour. Write a rule for the total $T$ for $h$ hours.**

* Answer: 9h + 25

<!-- Speaker note: T = 9h + 25: 9 dollars for each hour (m) and 25 dollars once (c). -->

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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: design a rival plan

Plan A is $T = 5g + 20$. Set the rate ($m$) and fee ($c$) of your own plan. The green point shows where the two plans cost the same.

Stretch: write an advert for your plan that is honest: who should choose it, and who should not?

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

<!-- Speaker note: What this step is for: 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. Hints, in order: Plan A costs 50 dollars at 6 GB. Your line must pass through (6, 50). / To be cheaper for heavy users, your rate m must be less than 5. Then choose c so 6m + c = 50. -->

<!-- Speaker note: Teacher: compare two students' rival plans. Both pass through (6, 50); which would you advertise to whom? -->

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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

**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).

<!-- 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: 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. -->