At a glance. One lesson of about 60 minutes. Students read a rate from two points as rise over run, say its units, compare lines by steepness, and use the rate to find values and times.
Calculator: no calculator. Every rise and run divides exactly or to a simple decimal, and the work is reading the graph and naming the unit.
What students will learn
Know
- The rate of a straight line is rise ÷ run between any two points.
- Its unit is the unit up the side per the unit across.
- On the same scales, a steeper line is a greater rate.
Understand
- The rate is the same at every point on a line, so any pair of points will do.
- The height of the end point is a total, not a rate.
- A change of units on either axis changes the number and the unit of the rate.
Do
- Find a rate from two points on a line.
- Compare two lines and say which has the greater rate.
- Use the rate to find a value or a time.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding and using a rate from two points (questions 1 to 7).
- Criterion B. Investigating patterns: the figure. Slide the run and ask what changes (the rise) and what does not (rise ÷ run); ask why.
- Criterion C. Communicating: justifying in words which line has the greater rate (questions 4 and 9).
- Criterion D. Applying mathematics in real-life contexts: journeys, tank filling, speeds in a different unit (the hook and questions 6 to 8).
The one thing to do with the figure
Ask for a prediction before the sliders move. “Kofi is 36 km out after 3 hours and Ana is 27 km out. Which line is steeper?” Take a sketch from each pair on mini whiteboards before the figure appears.
Then set the run to 1, 2, 3 and 4 and ask for the rise each time, before the readout. Ask students to predict rise ÷ run, and what that says about the rate at different points on the line.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: Kofi and Ana, with sketches on whiteboards. The figure on the board. |
| 10 min | Key idea; rise and run named on the sketch; the unit of the rate. |
| 10 min | Worked examples; students finish the faded example in pairs. |
| 5 min | Question 1, the check. If many choose B, C or D, return to rise and run on the figure. |
| 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 | 6; 60 km per hour; (4, 20). |
|---|---|
| Hook | Kofi 12 km per hour, Ana 9 km per hour. |
| Worked example 1 | 4.5 litres per minute. |
| Worked example 2 | 63 litres after 14 minutes; 8 minutes for 36 litres. |
| Faded example | 7 litres per minute. |
| 1. Rate from a point | A, 14 km per hour. B, C and D combine the coordinates by adding, multiplying or subtracting. |
| 2. Rate from a point | 13. 78 ÷ 6 = 13 litres per minute. |
| 3. The unit of a rate | A. B has the units inverted; C subtracts; D multiplies. |
| 4. Which line is steeper | A. P has rate 9 and Q has rate 6. B and D reward the bigger x; C looks at the height only. |
| 5. Rate from two points | 17. (102 − 34) ÷ (6 − 2) = 68 ÷ 4 = 17. |
| 6. Time to reach an amount | 9. 24 ÷ 3 = 8 litres per minute, and 72 ÷ 8 = 9 minutes. |
| 7. A less steep line | 25. Rate of A is 8, rate of B is 5, and 5 × 5 = 25. |
| 8. Reading with unit changes | 18. 300 m per minute × 60 = 18000 m per hour = 18 km per hour. |
| 9. Where the line ends | B. A: reads the end height as the rate. C: both lines through the origin are not equal. D: the right answer for the wrong reason. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1; 84 or 468 in question 2 | Added, multiplied or subtracted the coordinates. | Ask what the rise is and what the run is, then write rise ÷ run. |
| B, C or D in question 3 | Mixed up the units, or combined them with the wrong operation. | Write the units next to the numbers: 18 m, 6 s. |
| B or D in question 4 | Took the larger x as steeper. | Ask how far up each line goes in one step across. |
| 68, 34 or 4 in question 5 | Gave the rise only, divided by the wrong x, or gave the run only. | Ask for a rise and a run in a table, then divide. |
| 576 or 64 in question 6 | Multiplied or subtracted the amount and the rate. | Say “how many 8s make 72?”. |
| 37 or 15 in question 7 | Applied the difference to the height, or gave the gap. | Ask students to write the rate of each line first. |
| 300, 0.3 or 7500 in question 8 | Stopped at metres per minute, or at km per minute, or multiplied. | Underline the unit asked for, and change one unit at a time. |
| A or D in question 9 | Used the end height or the end x as the rate. | Divide each end height by its x value and compare. |
Questions worth asking
- Does the rate change as you slide the second point? No: rise changes but rise ÷ run does not. Listen for “steady”.
- What does it mean if one line is steeper? A greater rate, on the same scales. Ask what would happen if the scales were different.
- What are the units of the rate here? Kilometres per hour, or litres per minute. Listen for the “per”.
Differentiation
A way in for anyone stuck
- Give a table with rows for x, y, rise and run, and let students complete it from two points.
- Mark squares on the graph for the rise and run before asking them to divide.
- Start with points whose run is 1, so the rise is the rate.
Stretch
- The what-if: if line B has twice the rate of line A, when does it reach the height line A reaches at x = 6? (Half the time, x = 3.)
- Redraw the two friends' graph with the vertical axis in metres and the horizontal in minutes. What are the two rates now? (200 m per minute and 150 m per minute.)
Students learning in English as an additional language
Several words have everyday meanings. Show “rise” and “run” with a hand going up and then across on a staircase.
| Word | Plain meaning | Example |
|---|---|---|
| rate of change | how much one quantity changes for one step of another | 20 litres per minute |
| steep | rising quickly for each step across | the steeper line is the faster |
| rise | how far a line goes up between two points | 15 litres |
| run | how far a line goes across between two points | 3 minutes |
| constant | staying the same | a constant speed |
| origin | the point (0, 0) | where the axes meet |
Sentence starters for explaining aloud or in writing:
- The rise is … and the run is …, so the rate is … per …
- Line … is steeper, so its rate is …
- The line ends at …, but the rate is … because …
Two things not to say
“Slope is rise over run, learn it.” Start from kilometres in each hour; the formula is the shorthand.
“The line that is higher is the faster.” Only on the same time scale, and only at the end of the same time.
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.
- ReadingA line through the origin passes through (4, 28). What is its rate?
7 per unit across: 28 ÷ 4 = 7. - UsingA line through the origin has a rate of 6 km per hour. How far is it at 5 hours?
30 km.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Two pointsA graph through the origin passes through (2, 30) and (5, 75). What is the rate?
(75 − 30) ÷ (5 − 2) = 45 ÷ 3 = 15. - UnitsA plant's height in cm against days is a line through the origin and (10, 35). Give the rate in cm per day and in mm per day.
3.5 cm per day, which is 35 mm per day. - Earlier work (Lesson 3)Change 126 km per hour to metres per second.
35 metres per second: 126 ÷ 3.6 = 35.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA line through the origin has a rate of r, and another has 2r. Where is the second line at x = 3 compared with the first at x = 6?
The same height: 2r × 3 = 6r. - SpottingA graph of cost in rupees against kilograms passes through (2, 5) and (6, 15). Is it a line through the origin? How can you tell?
Yes: the rate is 10 ÷ 4 = 2.5, and 2.5 × 2 = 5, so the line passes through (0, 0).
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All three sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Plot the two journeys on squared paper and mark the rise and the run with a ruler.