Comparing two plans
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.
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.
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?
- Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
- Try: Try it on the board and compare the result with your prediction.
- Investigate: Work out why it happens, one step at a time.
- Modify: Change the problem on purpose and see what happens.
- Practise: A few on your own, to check you have it.
- Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
Which plan?
Work out what each plan costs for 6 GB. Then think: is there an amount of data where they cost the same?
Where they cost the same
Plan A is ; Plan B is . Drag to where the lines cross. Then look at 6 GB: which line is lower there? Was your prediction right?
Drag to where the two plans cost the same.
Use the board. This ticks when you get there.
Answering in words
Here is how to compare two plans and answer in words.
- Plan A and Plan B, as rules. Which is cheaper, and when?
- They cost the same where the two rules are equal: where the lines cross.
- The lines cross at . Check: .
- Left of the crossing, Plan A's line is lower.
- Right of the crossing, Plan B's line is lower.
- The model only makes sense from 0 GB: you cannot use negative data.
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."
Read Leo's working one line at a time. Is his last line true for every amount of data?
Some values of make sense in the situation, and some do not.
Leo compares the plans. Click the line where his reasoning goes wrong.
New comparisons
Each task puts a new pair of rules on the board. Drag to where they cost the same.
Need a hint?
Gym A: 50 dollars to join, then 10 a week. Gym B: no joining fee, 15 a week. Follow both lines until they meet.
Two people saving at different rates, from different starting amounts.
Need a hint?
Sam starts with 30 dollars and saves 5 a week. Ana starts with nothing and saves 8 a week. When has Ana caught up?
Two electricians with the same hourly rate.
A phone battery running down.
Gym A is ; Gym B is . Drag to where they cost the same.
Use the board. This ticks when you get there.
Sam's savings are ; Ana's are . Drag to when they have the same amount.
Use the board. This ticks when you get there.
Compare and check
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?
Review: an electrician charges a 25 dollar call-out fee plus 9 dollars an hour. Write a rule for the total for hours.
Design a rival plan
Plan A is . Set the rate () and fee () 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?
Need a hint?
Plan A costs 50 dollars at 6 GB. Your line must pass through (6, 50).
Another hint
To be cheaper for heavy users, your rate m must be less than 5. Then choose c so 6m + c = 50.
Design Plan B so that it costs the same as Plan A at 6 GB, and is cheaper than Plan A above 6 GB.
Use the board. This ticks when you get there.
For teachers
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.
Open the teacher guide: answers, solutions, slides and a worksheet.
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.
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
- 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.
- Run: once P is on the crossing, point left and right of it: which line is lower on each side?
- 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...'.
- Modify: before the electricians, ask: what did the c lesson say about lines with the same m?
- Practise: for each answer, ask one student to say it as a sentence about the situation.
- Make: compare two students' rival plans. Both pass through (6, 50); which would you advertise to whom?
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.
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.
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.
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.
- Find where two plans cost the same from their graphs, and say what that means.
- Decide which plan is cheaper for a given amount, and explain why in words.
- Say which values make sense for a model, and spot an answer that does not.