--- title: "Rules from real situations" subtitle: "Linear relations · Solving and modelling with lines · Lesson 10" --- # Rules from real situations **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Taxi fares, phone plans, gym memberships and a phone battery running down all follow the same pattern: where you start, and how much changes each time. :::teacher-note **In one sentence:** Students turn a situation that is a starting amount plus a rate into a rule y = mx + c, read m and c back into words, and fit lines through real values (AC9M8A03, node 1). **Students should be able to say:** 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. **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 Think about a trip of 0 km, then 1 km, then 2 km. What does each one cost? :::keep-together **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?** ☐ $T = 4d + 2$ ☐ $T = 2d$ ☐ $T = 6d$ ☐ $T = 2d + 4$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 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). ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## From words to a rule :::keep-together ### A phone plan Here is how to turn words into a rule. 1. $\text{Fee } 20, \text{ then } 5 \text{ per GB}$ — A phone plan costs 20 dollars a month, plus 5 dollars for each GB used. Find a rule for the cost $T$. 2. $g = 0: \; T = 20$ — With no data used, you still pay the fee. That is the value when $g$ is 0. 3. $c = 20$ — The starting amount is $c$. 4. $m = 5$ — Each extra GB adds 5 dollars. "Per", "each" or "every" signals the rate, $m$. 5. $T = 5g + 20$ — Put them together. 6. $g = 3: \; T = 5(3) + 20 = 35$ — Check with a value you can work out in words: 3 GB costs 20 + 15 = 35 dollars. **A gym charges a 30 dollar joining fee and 15 dollars a week. Write a rule for the total cost $T$ after $w$ weeks.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 15w + 30 T = 15w + 30: 30 dollars once (c), then 15 dollars for each week (m). ::: :::keep-together ### Find the mistake **Zoe writes a rule for a taxi fare. Click the line where her reasoning goes wrong.** ``` 1 A taxi costs 5 dollars to get in, plus 3 dollars per km. 2 m is the first number in the question, so m = 5. 3 c is the other number, so c = 3. 4 The rule is T = 5d + 3. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** 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. ::: :::keep-together ### 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 GB of data are included ☐ The monthly fee is 5 dollars ::: :::teacher-note **Answer:** 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. ::: :::typical ## New situations **No starting amount.** Each task gives some real values as points. Set $m$ and $c$ so the line goes through them all. **Going down.** A phone battery runs down at the same rate each hour. **Less than a dollar.** Printing costs a fee plus a little for each page. **Only the numbers.** No words this time: just two fares. Work out the rate first. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Rules and meanings :::keep-together ### 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.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12n + 6 T = 12n + 6: 12 dollars for each ticket (m) and 6 dollars once (c). ::: :::keep-together ### What the 60 means **A plumber charges $T = 80h + 60$ dollars for a job of $h$ hours. What does the 60 mean?** ☐ The cost of a 60-hour job ☐ 60 dollars for each hour ☐ A 60 dollar call-out fee, paid once ::: :::teacher-note **Answer:** 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. ::: :::keep-together ### Using the rule **Using $T = 80h + 60$, what does a 3-hour job cost, in dollars?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 300 80 × 3 + 60 = 240 + 60 = 300 dollars. ::: :::keep-together ### Review **Review: $y = 2x + 1$ and $y = 7$ meet at $(3, 7)$. Solve $2x + 1 = 7$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3 x = 3. Check: 2 × 3 + 1 = 7. ::: ## Make: your own situation Invent a situation that is a starting amount plus a rate: a club with a joining fee and a weekly cost, a game with a starting score and points per level. Set $c$ and $m$, then drag $P$ to the total for 4 units. Stretch: tell a partner your situation in words. Can they write your rule? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **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.** {{lines-3}} ::: :::teacher-note **Answer:** T = 4v + 50. 4 is the cost per visit (m); 50 is the membership, paid once (c). :::