--- title: "Comparing two plans" subtitle: "Linear relations · Solving and modelling with lines · Lesson 11" --- # Comparing two plans **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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. :::teacher-note **In one sentence:** 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). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Work out what each plan costs for 6 GB. Then think: is there an amount of data where they cost the same? :::keep-together **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 My prediction, and why: {{lines-3}} ::: :::teacher-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. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Answering in words :::keep-together ### Comparing the plans Here is how to compare two plans and answer in words. 1. $T = 5g + 20, \quad T = 40$ — Plan A and Plan B, as rules. Which is cheaper, and when? 2. $5g + 20 = 40$ — They cost the same where the two rules are equal: where the lines cross. 3. $g = 4, \; T = 40$ — The lines cross at $(4, 40)$. Check: $5 \times 4 + 20 = 40$. 4. $g < 4: \text{ A is cheaper}$ — Left of the crossing, Plan A's line is lower. 5. $g > 4: \text{ B is cheaper}$ — Right of the crossing, Plan B's line is lower. 6. $g \ge 0$ — 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." **For 2 GB a month, which plan is cheaper?** ☐ Plan A ☐ Plan B ☐ They cost the same ::: :::teacher-note **Answer:** 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. ::: :::keep-together ### 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. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** 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. ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** 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. ::: :::typical ## 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. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Compare and check :::keep-together ### 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: {{field-15}} ::: :::teacher-note **Answer:** 3 10 + 8g = 34, so 8g = 24 and g = 3. Check: 10 + 24 = 34. ::: :::keep-together ### Which is cheaper? **Using Plans C and D, which is cheaper for 5 GB?** ☐ Plan C ☐ They cost the same ☐ Plan D ::: :::teacher-note **Answer:** Plan D Plan C: 10 + 8 × 5 = 50 dollars. Plan D: 34 dollars. Above 3 GB, D is cheaper. ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** 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. ::: :::keep-together ### 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: {{field-15}} ::: :::teacher-note **Answer:** 9h + 25 T = 9h + 25: 9 dollars for each hour (m) and 25 dollars once (c). ::: ## 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? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **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?** {{lines-3}} ::: :::teacher-note **Answer:** 2 GB: X (27 dollars vs 33). 5 GB: Y (45 vs 33). They cost the same at 3 GB (33 dollars). :::