At a glance. One lesson of about 60 minutes. Students find a rate and a start value from two points, work with falling quantities and negative rates, and find when a falling quantity reaches zero.
Calculator: no calculator. The numbers are whole or simple decimals, and the integer work (subtracting a larger number, a negative rate) is the skill, so a calculator would hide it.
What students will learn
Know
- A steady rate and a start value give value = start + rate × time.
- The rate is rise ÷ run between any two points, and is negative when the value falls.
- A line that does not start at zero is not proportional.
Understand
- Rise over run is the same everywhere on a line; y over x is not unless the line passes through the origin.
- The sign of the rate says which way the quantity moves.
- A falling quantity reaches zero after start ÷ rate steps.
Do
- Find a rate and a start value from two points.
- Interpret a negative rate in a context.
- Find the time at which a falling quantity reaches a given value.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using start and rate, and finding a rate from two points (questions 1 to 7).
- Criterion B. Investigating patterns: the second figure. Ask what is the same at every pair of points, and what changes; ask for a case where y ÷ x equals the rate.
- Criterion C. Communicating: explaining why a falling line has a negative rate, and why y ÷ x does not give it (question 9).
- Criterion D. Applying mathematics in real-life contexts: tanks, candles, savings, temperatures, diving (the hook and questions 2 to 8).
The one thing to do with the figure
Ask for a prediction before the sliders move. “A 60-litre tank drains at 5 litres an hour. Sketch litres against hours. Where does the line start and which way does it go?” Take sketches on mini whiteboards. Many draw a line through the origin; ask what the origin would mean.
Then drag the rate across zero in the first figure and ask what the line does at 0, and what a negative rate means in the tank. In the second figure, ask for the rate before showing it, then show y ÷ x beside it.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: the leaking tank, with sketches. The first figure on the board. |
| 10 min | Key idea; the second figure, rise over run beside y over x. |
| 10 min | Both worked examples; students finish the faded candle example. |
| 5 min | Question 1, the check. If many choose B, ask whether she is getting nearer or further. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 9; −8; n = 6. |
|---|---|
| Worked example 1 | Rate 4, start 3, value = 3 + 4x. |
| Worked example 2 | 20 litres after 8 hours; empty after 12 hours. |
| Faded example | 16 cm after 7 hours; 15 hours. |
| 1. A falling distance | A, 10 km. B adds; C uses one hour; D gives the distance ridden. |
| 2. Start plus rate | 390. 150 + 40 × 6 = 390 rupees. |
| 3. Rate from two points | 6. (32 − 8) ÷ 4 = 6. |
| 4. Time to reach a value | 13. (41 − 15) ÷ 2 = 13 weeks. |
| 5. A negative rate from a table | −3. (10 − 19) ÷ 3 = −3 degrees per hour. |
| 6. When a tank is empty | 12. 72 ÷ 6 = 12 minutes. |
| 7. A negative start value | 2. −4 + 3 × 2 = 2 m. |
| 8. Working back to the start | 25. 17 + 2 × 4 = 25. |
| 9. Rate of a falling line | B, −4. A divides y by x; C loses the sign; D gives the rise only. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B in question 1; 9 in question 8 | Added when the quantity falls, or went forward when working back. | Ask whether she is getting nearer or further, or whether the earlier value was higher or lower. |
| C in question 1; 21 in question 8 | Used one step instead of several. | Ask how many steps there are, and write each as a line of a table. |
| D in question 1 | Gave the distance ridden, not the distance left. | Ask what the question asks for: how far from the town. |
| 1140 or 240 in question 2 | Multiplied the start as well, or left the start out. | Say “start plus rate times time” and ask which part is each number. |
| 8, 2 or 24 in question 3 | Divided y by x, divided the start by the run, or gave the rise only. | Ask for the rise and the run separately. |
| 26 or 20.5 in question 4 | Gave the height still to grow, or ignored the start. | Ask how far it still has to go before dividing by the rate. |
| 3 or −9 in question 5 | Lost the sign, or did not divide by the time. | Ask whether the temperature goes up or down. |
| 432 or 66 in question 6 | Multiplied, or subtracted the rate. | Ask how many 6s make 72. |
| −10, 10 or −2 in question 7 | Went the wrong way, used 4 as positive, or used one minute. | Draw the diver on a vertical number line starting at −4. |
| A, C or D in question 9 | Divided y by x, lost the sign, or gave the rise only. | Ask whether the line goes up or down, then for the rise and the run separately. |
Questions worth asking
- What does a rate of 0 look like on a graph, and what does it mean? A flat line: the quantity is not changing. Listen for “stays the same”.
- Why can y ÷ x not be used for the rate here? The line misses the origin. Listen for the start value.
- What does the point where the line crosses the horizontal axis mean for the tank? The tank is empty. Listen for a time and a unit.
- How would you know the rate is negative before calculating? The line slopes down. Listen for “goes down as you go right”.
Differentiation
A way in for anyone stuck
- Give a table of time and value, with the first row filled and the change per step written between rows.
- Start with rising lines, then falling lines, then a line that is flat.
- Draw a vertical number line next to the diver and tank problems to show the signs.
Stretch
- The what-if: where do two candles of different starts and rates meet? (At 6 hours, 8 cm.)
- A taxi charges 50 rupees plus 15 rupees per km, and a second taxi charges 20 rupees plus 20 per km. At what distance do they cost the same? (50 + 15d = 20 + 20d, so d = 6 km, at 140 rupees.)
Students learning in English as an additional language
“Start” and “rate” have everyday meanings. Say “it starts at…, and each hour it goes up (down) by…” with a hand moving on a number line.
| Word | Plain meaning | Example |
|---|---|---|
| start value | the value when the time is zero | 30 litres at the start |
| rate of change | how much the value changes in one step | −4 litres per hour |
| negative rate | a rate for a value that is falling | −2 litres per hour |
| slope | how steeply a line goes up or down | the line slopes down |
| reaches zero | becomes 0 | the tank is empty |
| relative to | measured from | metres relative to the surface |
Sentence starters for explaining aloud or in writing:
- The start value is … and the rate is … per …, so after … the value is …
- The rate is negative because the value …
- It reaches zero when … ÷ … = …
Two things not to say
“Just use y = mx + c.” The rule is not the idea; start from the start value and the change in one step.
“Ignore the minus sign, the rate is how fast.” The sign tells you the direction, and the question often asks for it.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Start and rateA line starts at 8 and rises 3 for each step across. What is the value after 5 steps?
23: 8 + 3 × 5. - FallingA value starts at 20 and falls 4 each hour. What is it after 3 hours?
8: 20 − 12.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- From two pointsA line passes through (1, 9) and (5, 21). Find its rate and its start value.
Rate (21 − 9) ÷ (5 − 1) = 3, start 9 − 3 = 6. - TaxiA taxi charges 50 rupees plus 15 rupees per km. What does an 8 km trip cost?
170 rupees: 50 + 15 × 8. - Earlier work (Lesson 5)A line through the origin passes through (6, 45). What is its rate, and its height at x = 10?
7.5 per step, and 75 at x = 10.
3Challenge
Reasoning, generalizing and working backwards.
- ReadingA value has a start of 12 and a rate of −0.5 per minute. When does it reach 0?
12 ÷ 0.5 = 24 minutes. - MeetingOne tank starts with 60 litres and loses 5 litres per minute. A second starts empty and gains 3 litres per minute. When do they hold the same amount, and how much?
60 − 5t = 3t gives t = 7.5 minutes, when both hold 22.5 litres.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All four sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Use squared paper and a ruler; plot the tank graph from a table of hour and litres.