At a glance. One lesson of about 60 minutes. Students find distance, speed or time from the other two, convert minutes to hours, and find an average speed as total distance over total time.
Calculator: no calculator. The numbers divide cleanly (the one third, 40 minutes = 2/3 of an hour, is in the hook and is worked on the page), and the work is in matching units and choosing the operation.
What students will learn
Know
- Speed = distance ÷ time, distance = speed × time, time = distance ÷ speed.
- A speed per hour needs a time in hours: minutes ÷ 60.
- Average speed = total distance ÷ total time.
Understand
- The three forms are one relationship seen from three sides.
- Mixing hours and minutes is the commonest source of a wrong answer.
- Over equal distances, the slow part of a journey takes more of the time, so it pulls the average below the mean of the speeds.
Do
- Find any one of speed, distance and time from the other two.
- Change minutes to hours and back.
- Find the average speed of a journey in two parts.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using speed, distance and time with matching units (questions 1 to 7).
- Criterion B. Investigating patterns: the second figure. Set both legs to the same speed, then change one; ask when the average equals the mean and why.
- Criterion C. Communicating: explaining in a sentence why the mean of the speeds is the average speed in question 9 but not in question 7.
- Criterion D. Applying mathematics in real-life contexts: a ferry, a lorry, a coach, a bus (the hook, questions 2 to 8).
The one thing to do with the figure
Ask for a prediction before the first slider moves. “30 km per hour for 40 minutes: how far?” Take 1200 km from the room and ask whether a ferry could cross 1200 km in under an hour. Estimation is enough to reject it; the unit mismatch is the reason.
For the second figure, ask for the lorry's average speed before it is shown, and take 45 as a common answer. Then set the legs to 30 and 60 and let the bars of time settle it.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: the ferry, with the 1200 km prediction. First figure on the board. |
| 10 min | Key idea; the three forms written together; minutes to hours. |
| 10 min | Second figure and the lorry prediction. Worked examples; students finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask what 1.15 hours would mean. |
| 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 | 120 passengers per hour; 21 km; t = 7.5. |
|---|---|
| Worked example 1 | The lorry: 40 km per hour. |
| Worked example 2 | 2 hours 30 minutes. |
| Faded example | 24 km per hour. |
| 1. Hours and minutes | B, 100 km. A reads 1 h 15 min as 1.15; C adds minutes to kilometres; D leaves out the 15 minutes. |
| 2. Distance from speed and time | 180. 45 × 4 = 180 km. |
| 3. Speed from distance and time | 32. 96 ÷ 3 = 32 km per hour. |
| 4. Time from distance and speed | 3. 36 ÷ 12 = 3 hours. |
| 5. Answer in minutes | 30. 6 ÷ 12 = 0.5 hours = 30 minutes. |
| 6. Distance with minutes | 2. 24 ÷ 60 = 0.4 hours, and 5 × 0.4 = 2 km. |
| 7. Average speed | 60. 90 ÷ 45 = 2 h, 90 ÷ 90 = 1 h, and 180 ÷ 3 = 60 km per hour. |
| 8. How much faster | 4. 24 ÷ 1.5 = 16, and 16 − 12 = 4 km per hour faster. |
| 9. When the mean does work | A. B applies the equal-distance rule without checking the times; C adds the speeds; D uses the last speed only. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1; 120 in question 6 | Used minutes as hours, wrote 1 h 15 min as 1.15, added minutes to kilometres, or left the minutes out. | Ask how many minutes are in an hour, then how many hours is 30 minutes. |
| 49 in question 2; 93 in question 3; 24 in question 4 | Added or subtracted quantities in different units. | Ask whether kilometres can be added to hours. |
| 11.25 in question 2; 288 in question 3; 432 in question 4; 72 in question 5 | Divided when they should multiply, or multiplied when they should divide. | Write the three forms side by side, and check by multiplying back. |
| 0.5 in question 5 | Stopped at hours. | Underline the unit the question asks for. |
| 2 in question 5 | Divided speed by distance. | Ask which should be bigger: a 6 km jog at 12 km per hour takes more or less than an hour? |
| 4.8 in question 6 | Divided the time by the speed. | Say it in words: speed times time gives distance. |
| 67.5 or 135 in question 7; B, C or D in question 9 | Took the mean of the speeds when the times differ, applied “the slow part counts more” when the times are equal, added the speeds, or used one of them. | Find the time for each leg first, then total distance over total time. |
| 16 or 0.5 in question 8 | Gave the speed needed, or the time saved. | Ask what the question asks for: how much faster than 12? |
Questions worth asking
- Can a time in minutes go into speed = distance ÷ time with a speed per hour? Not without changing it. Listen for “divide by 60”.
- When is the average speed equal to the mean of two different speeds? Only when equal times are spent at each speed. Listen for “same time” rather than “same distance”.
- Why does the slow leg count for more? It takes longer, so it is a bigger share of the journey.
- Which form do you use to check an answer? Any other: multiply back. Listen for a habit of checking.
Differentiation
A way in for anyone stuck
- Give a table with three columns, speed, time and distance, and ask which one is missing before any calculation.
- Keep to whole hours for the first few questions; add half hours, then minutes.
- Draw each leg as a bar before calculating, so the longer time is seen.
Stretch
- The what-if: how fast would the second 100 km need to be to average 100 km per hour? (It cannot be done: the first half already used the 2 hours allowed.)
- For equal distances at speeds a and b, show that the average speed is 2ab ÷ (a + b). Check with 30 and 60. (2 × 30 × 60 ÷ 90 = 40.)
Students learning in English as an additional language
Speed words are in everyday use with looser meanings. Draw an arrow for the distance and a clock for the time as each word is introduced.
| Word | Plain meaning | Example |
|---|---|---|
| speed | distance travelled for one unit of time | 60 km per hour |
| distance | how far something goes | 150 km |
| time | how long something takes | 3.5 hours |
| average speed | total distance divided by total time | 150 km in 3 hours is 50 km per hour |
| leg | one part of a journey | the first leg, 60 km |
Sentence starters for explaining aloud or in writing:
- The time in hours is … because … minutes is … ÷ 60.
- The average speed is … because the journey is … km in … hours.
- The mean of the speeds is wrong because …
Two things not to say
“Speed, distance, time: cover the one you want.” The triangle gives answers without the idea; start from speed as distance for one hour.
“Average speed is the average of the speeds.” For two different speeds it is only true when the times are equal.
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.
- DistanceA scooter travels at 40 km per hour for 3 hours. How far does it go?
120 km. - SpeedA train covers 300 km in 4 hours. What is its speed?
75 km per hour.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- TimeA flight from Colombo to Chennai covers 600 km at 400 km per hour. How long does it take, in hours and minutes?
1.5 hours, which is 1 hour 30 minutes. - MinutesA pupil walks 4.5 km at 6 km per hour. How many minutes does that take?
45 minutes: 4.5 ÷ 6 = 0.75 hours, and 0.75 × 60 = 45. - Earlier work (Lesson 1)A tailor sews 12 jackets in 8 hours. How many jackets per hour, and how many in 10 hours?
1.5 jackets per hour, so 15 jackets in 10 hours.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA cyclist rides to a village 30 km away at 15 km per hour and returns the same way at 10 km per hour. What is the average speed for the round trip, and why is it not 12.5?
2 h + 3 h = 5 h for 60 km, so 12 km per hour. The slower leg takes more of the time. - Working backA car drives for 2 hours at 60 km per hour. For how long must it then drive at 90 km per hour to average 75 km per hour over the whole trip?
The total distance is 120 + 90t in 2 + t hours, which must equal 75 per hour: 120 + 90t = 150 + 75t, so t = 2 hours.
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? Mark a road on the board in kilometres and have two students walk legs of equal distance at a fast and a slow pace, timing each.