The shape of the lesson
By the end of the lesson, learners can
- Speed, distance and time: Open the lab on Average speed and change the distance and time sliders for each leg.
- Everyday rates: Pay, flow and fuel are all rates.
- Density, pressure and population: For these rates the formula is given in the question.
- Units first: km/h means km ÷ h; that tells you whether to multiply or divide.
- Time in hours: minutes ÷ 60, so 15 min \(= 0.25\) h.
- Average speed \(=\) total distance ÷ total time, including stops.
- Given formulas (density, pressure): substitute with consistent units.
- Check that the answer is sensible and has units.
One picture every idea returns to
A rate is one quantity per one unit of another. Read the units: km/h means km ÷ h, so distance \(=\) speed \(\times\) time and time \(=\) distance ÷ speed. Average speed \(= \dfrac{\text{total distance}}{\text{total time}}\).
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, So much per something.. A rate compares two different quantities: kilometres per hour, dollars per hour, litres per minute, grams per cubic centimetre. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Speed, distance and time | Open the lab on Average speed and change the distance and time sliders for each leg. Project tab 2 of the lab (Ratio, rates and percentages lab, rates) and let a student drive it while the class predicts. Model the worked example Average speed on the board, one line at a time. Your turn: Convert 72 km/h into metres per second. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Everyday rates | Pay, flow and fuel are all rates. Model the worked example Pay with overtime on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Density, pressure and population | For these rates the formula is given in the question. Model the worked example Density on the board, one line at a time. | Numbered Heads Together |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 13 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Density, pressure and population, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Numbered Heads Together
Teams of four number off. Put up a question, give the team time to make sure every member can answer, then call a number: that student answers for the team, so nobody can leave the thinking to someone else.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| 3 hours 45 minutes is NOT 3.45 hours. 45 minutes is \(\dfrac{45}{60} = 0.75\) of an hour, so the time is 3.75 hours. | Return to Speed, distance and time: ask a pair to state the correct version and show it on the lab before moving on. |
| Fuel consumption can be given as litres per 100 km OR km per litre. | Check which before multiplying: 6.4 litres per 100 km is NOT 6.4 km per litre. |
| If the area is in cm 2 but the answer must be in N/m 2, convert the area first: \(1\ \text{m}^2 = 10\,000\ \text{cm}^2\), not 100 cm 2. | Return to Density, pressure and population: ask a pair to state the correct version and show it on the lab before moving on. |
Rates, average speed and best value: worksheet
6 questions, 13 marks. Written in the style of the papers; no past paper question is reproduced.
Convert 15 m/s into km/h.
A runner completes 21 km in 1 hour 45 minutes. Calculate her average speed in km/h.
Water flows into a bath at 8.5 litres per minute. How long does it take to put 680 litres in the bath? Give your answer in hours and minutes.
A country has a population of 3.6 million and an area of 4500 km2. Population density \(= \dfrac{\text{population}}{\text{area}}\). Calculate the population density.
Mia is paid \(\$15.20\) per hour for her first 40 hours each week. Overtime is paid at 1.5 times the normal rate. One week she works 46 hours. Calculate her pay for the week.
Raj drives 150 km at an average speed of 75 km/h. He stops for 30 minutes, then drives a further 100 km in 1 hour 30 minutes.
Calculate his average speed for the whole journey, including the stop.
Answer key and marking notes
\(15 \times 3600 = 54\,000\) m per hour \(= 54\) km/h.
- 1 hour 45 minutes \(= 1.75\) h.
- \(21 \div 1.75 = 12\) km/h.
- \(680 \div 8.5 = 80\) minutes.
- 80 minutes \(= 1\) hour 20 minutes.
- \(3\,600\,000 \div 4500 = 800\).
- 800 people per km2.
- Normal pay: \(40 \times 15.20 = \$608\).
- Overtime rate: \(1.5 \times 15.20 = \$22.80\); overtime pay: \(6 \times 22.80 = \$136.80\).
- Total: \(608 + 136.80 = \$744.80\).
- First part: \(150 \div 75 = 2\) h.
- Total time \(= 2 + 0.5 + 1.5 = 4\) h; total distance \(= 150 + 100 = 250\) km.
- Average speed \(= 250 \div 4 = 62.5\) km/h.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A runner covers 52 km in 1 hour 20 minutes. What is her average speed?
- A39 km/h
- B43.3 km/h
- C69.3 km/h
- D0.65 km/h
A car travels 30 km at 90 km/h and then another 30 km at 45 km/h. What is its average speed for the whole 60 km?
- A60 km/h
- B67.5 km/h
- C45 km/h
- D90 km/h
Rosa is paid \(\$12.80\) per hour for a 35-hour week, and time-and-a-half for any overtime. One week she works 41 hours. How much is she paid?
- A\(\$563.20\)
- B\(\$524.80\)
- C\(\$486.40\)
- D\(\$787.20\)
A car travels 14 km on each litre of fuel. Fuel costs \(\$1.60\) per litre. Find the cost of the fuel for a 210 km journey.
- A\(\$24.00\)
- B\(\$47.04\)
- C\(\$15.00\)
- D\(\$4704.00\)
A crate of weight 840 N rests on a base measuring 35 cm by 40 cm. Pressure \(= \dfrac{\text{force}}{\text{area}}\). Find the pressure in N/m2.
- A6000 N/m2
- B0.6 N/m2
- C60 N/m2
- D600 N/m2
A tank is a cuboid 1.2 m long, 0.8 m wide and 0.5 m high. Water flows in at 18 litres per minute. How long does it take to fill the empty tank? (\(1\ \text{m}^3 = 1000\) litres)
- A26 minutes 40 seconds
- B26 minutes 67 seconds
- C2 minutes 40 seconds
- D144 hours
Quiz key
1 hour 20 minutes \(= 1\dfrac{20}{60} = \dfrac{4}{3}\) hours. Speed \(= 52 \div \dfrac{4}{3} = 39\) km/h. The tempting 43.3 km/h uses 1.2 hours, treating 20 minutes as 0.2 of an hour. 69.3 multiplies by the time instead of dividing, and 0.65 divides by 80 minutes, giving km per minute.
Time for part 1 \(= \dfrac{30}{90} = \dfrac{1}{3}\) h; time for part 2 \(= \dfrac{30}{45} = \dfrac{2}{3}\) h. Total time \(= 1\) h, so average speed \(= \dfrac{60}{1} = 60\) km/h. The tempting 67.5 km/h is the mean of the two speeds, which is wrong because the car spends longer at the slower speed.
Normal pay \(= 35 \times 12.80 = \$448\). Overtime rate \(= 1.5 \times 12.80 = \$19.20\), and \(6 \times 19.20 = \$115.20\). Total \(= \$563.20\). The tempting \(\$524.80\) pays all 41 hours at the normal rate. \(\$486.40\) adds only the extra half (\(\$6.40\)) for the overtime hours, and \(\$787.20\) pays every hour at time-and-a-half.
Fuel used \(= 210 \div 14 = 15\) litres. Cost \(= 15 \times 1.60 = \$24.00\). The tempting \(\$47.04\) treats 14 as litres per 100 km (\(2.1 \times 14 \times 1.60\)). \(\$15.00\) stops at the number of litres, and \(\$4704.00\) multiplies everything together.
Area \(= 0.35 \times 0.40 = 0.14\ \text{m}^2\). Pressure \(= 840 \div 0.14 = 6000\) N/m2. The tempting 0.6 uses the area in cm2 (1400) without converting. 60 uses \(1\ \text{m}^2 = 100\ \text{cm}^2\) and 600 uses 1000; the correct factor is \(1\ \text{m}^2 = 10\,000\ \text{cm}^2\).
Volume \(= 1.2 \times 0.8 \times 0.5 = 0.48\ \text{m}^3 = 480\) litres. Time \(= 480 \div 18 = 26.67\) minutes, and \(0.67\) of a minute \(= \dfrac{2}{3} \times 60 = 40\) seconds. The tempting 26 minutes 67 seconds reads the decimal as seconds. 2 minutes 40 seconds uses 100 litres per m3, and 144 hours multiplies 480 by 18.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.