Rules from real situations
Many real costs and amounts are a starting amount plus a rate. The starting amount is c and the rate is m, so the rule is y = mx + c.
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.
So far the rules had and . Real situations have their own letters: for a total cost, for a distance, for hours. The graph works the same way: is still how much the line goes up for each step right, and is still where it starts.
- 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.
A taxi fare
Think about a trip of 0 km, then 1 km, then 2 km. What does each one cost?
Fit the fares
The points are three real fares: 6 dollars for 1 km, 10 dollars for 3 km and 14 dollars for 5 km. Set and so the line passes through all three. Was your rule right?
Set and so the line goes through all three fares.
Use the board. This ticks when you get there.
From words to a rule
Here is how to turn words into a rule.
- A phone plan costs 20 dollars a month, plus 5 dollars for each GB used. Find a rule for the cost .
- With no data used, you still pay the fee. That is the value when is 0.
- The starting amount is .
- Each extra GB adds 5 dollars. "Per", "each" or "every" signals the rate, .
- Put them together.
- Check with a value you can work out in words: 3 GB costs 20 + 15 = 35 dollars.
Read Zoe's working one line at a time. How did she decide which number is ?
Read the rule back into words.
A gym charges a 30 dollar joining fee and 15 dollars a week. Write a rule for the total cost after weeks.
Zoe writes a rule for a taxi fare. Click the line where her reasoning goes wrong.
New situations
Each task gives some real values as points. Set and so the line goes through them all.
Need a hint?
What does hiring the bike for 0 hours cost?
A phone battery runs down at the same rate each hour.
Need a hint?
The battery goes from 80% to 50% in 3 hours. How much each hour?
Another hint
It goes down, so m is negative. Then work back to 0 hours for c.
Printing costs a fee plus a little for each page.
Need a hint?
From 2 pages to 6 pages the cost goes up 2 dollars. How much per page?
No words this time: just two fares. Work out the rate first.
Need a hint?
From 2 km to 4 km the fare goes up 6. How much per km?
Another hint
Work back from 2 km to 0 km to find the fee.
Bike hire: 6 dollars for 1 hour, 24 dollars for 4 hours. Set and so the line goes through both.
Use the board. This ticks when you get there.
A phone battery: 80% after 2 hours, 50% after 5 hours. Fit the line.
Use the board. This ticks when you get there.
Printing: 3 dollars for 2 pages, 5 dollars for 6 pages. Fit the line.
Use the board. This ticks when you get there.
Two fares from a table: 11 dollars for 2 km and 17 dollars for 4 km. Find the rule.
Use the board. This ticks when you get there.
Rules and meanings
Concert tickets cost 12 dollars each, plus a 6 dollar booking fee for the order. Write a rule for the total for tickets.
Using , what does a 3-hour job cost, in dollars?
Review: and meet at . Solve .
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 and , then drag to the total for 4 units.
Stretch: tell a partner your situation in words. Can they write your rule?
Need a hint?
Pick a starting amount with c and a rate with m.
Another hint
Work out the total for 4 units: 4 lots of m, plus c. Drag P there.
Choose your own starting amount and rate (both more than 0). Then drag to the total for 4 units.
Use the board. This ticks when you get there.
For teachers
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). About 45 minutes.
Open the teacher guide: answers, solutions, slides and a worksheet.
Students will be able to
- 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.
Before class
Do the Run and Modify boards yourself first. Students should already know what m and c do to a line (the first two lessons of this course).
The new move is translation: the starting amount (paid once, the value when nothing has happened) is c; the rate ("per", "each", "every") is m. The order of the numbers in the words does not matter, and Zoe's working shows the mistake of assuming it does.
The boards use the situation's own letters (T for total, d for distance), labelled on the axes. The graphs still behave exactly like y = mx + c.
The next lesson uses these rules: two plans, where their lines cross, and when a rule stops being a good model.
Running the lesson
- Predict: take a show of hands. 'T = 4d + 2' is the common wrong answer: the numbers taken in the order they appear.
- Run: before students touch the sliders, ask: what would a 0 km trip cost? (4.) Where is that on the graph?
- Investigate: model the worked example live. Underline 'a month' (once) and 'for each GB' (the rate) in the words, then write c and m.
- Modify: before the battery task, ask: which way will this line go?
- Practise: for the plumber question, ask students to say what 80 means in words too.
- Make: pair students to swap situations and write each other's rules.
When students get stuck
- A student uses the numbers in the order they appear (Zoe's mistake). Ask: Which amount is paid only once? Which is paid again for every km? Fix: Underline 'per', 'each' or 'every': that number is m. The one-off amount is c.
- A student adds the fee into the rate (T = 6d). Ask: What would a 0 km trip cost with your rule? With the real taxi? Fix: Work out 0 km and 1 km in words: 4 dollars and 6 dollars. The fee is paid once; only the 2 repeats.
- A student cannot find c from two points (the last Modify task). Ask: How much does the fare go up each km? What would 0 km cost? Fix: Work back from the first point one step at a time: 11 at 2 km, 8 at 1 km, 5 at 0 km.
Stretch and support
Stretch: A candle is 20 cm tall and burns 1.5 cm an hour. Write its rule, and work out when it burns out. (h = 20 − 1.5t; after about 13.3 hours.)
Support: Give a two-column table (0, 1, 2, 3 units) to fill in from the words before writing the rule: the 0 row gives c and the difference gives m.
Where it leads
Next: comparing two plans. Two rules on one graph; where the lines cross is where they cost the same, and the model only makes sense for some values (no negative km).
Where it leads in class: science graphs (a starting temperature and a heating rate), budgets and savings plans, and any "fee plus rate" price.
What each step is for
- a taxi fare: Committing to a rule before fitting the line shows whether students can tell the one-off amount from the rate in words.
- fit the fares: Fitting a line through real fares makes c and m mean something: c is the fare for 0 km, m is the extra for each km.
- from words to a rule: In words, the starting amount is the value when nothing has happened yet (c); the rate is 'per', 'each' or 'every' (m).
- new situations: One change at a time: no starting amount (c = 0), an amount that goes down (negative m), a rate that is not whole, and only two points with no words.
- rules and meanings: Writing a rule, reading its numbers in words and using it for a value are one skill; the review keeps solving with graphs fresh.
- your own situation: Choosing a starting amount and a rate for a situation of their own reverses the lesson: from the rule back to words.
- Write a rule like T = 2d + 4 from a situation described in words.
- Explain what m and c mean in a situation (the rate and the starting amount).
- Use a rule to work out a value, and check it in words.