SECONDARY 1 MATHEMATICS CLASSROOM · CHAPTER 5 · RATE, SPEED AND UNIT CONVERSION · G2/G3
Rate, Speed and Unit Conversion: Keep the Quantities and Units Attached
In this classroom, you will not begin with a speed triangle. You will begin by naming the two quantities being compared, deciding what “per” means, and making the units agree before calculation.
A rate compares two quantities, usually with different units. Speed compares distance with time. Unit conversion changes how a quantity is described without changing the quantity itself. These three ideas belong together because a correct numerical answer with the wrong unit or wrong time interval is not a correct mathematical model.
Classroom rule: name the quantities → align the units → choose the relationship → calculate → check the unit, scale and interval.
The current Secondary One G2 and G3 Mathematics syllabuses include work with rates, speed and unit conversion, with exact sequencing and depth varying by subject level and school. This classroom teaches the common core first, then marks relative motion, combined rates and some area/volume conversions as extension where appropriate.
Official reference: MOE G2 and G3 Mathematics Syllabuses.
Navigate: rate meaning · unit conversion · time conversion · speed · average speed · journey tables · distance–time graphs · misconception clinic · guided practice · examination transfer · exit ticket.
Featured Answer: What Is a Rate?
A rate tells us how much of one quantity corresponds to one unit of another quantity. Examples include kilometres per hour, dollars per kilogram, litres per minute and words per minute.
The unit after “per” is part of the meaning. Eight litres per minute is different from eight minutes per litre. Sixty kilometres per hour is different from sixty metres per second.
The Simple Classroom Answer
A rate is a ratio with meaning attached to both quantities. Speed is distance per time. Units tell you what operation can produce the required answer.
- Unit rate: amount corresponding to one unit of another quantity.
- Conversion: same physical quantity, different unit description.
- Speed: distance divided by time.
- Distance: speed multiplied by time.
- Time: distance divided by speed.
- Average speed: total distance divided by total elapsed time for the stated interval.
- Journey table: a structure that keeps each stage’s distance, speed and time aligned.
How to Use This Classroom
- Write the quantity names and units before calculating.
- Convert incompatible units before using a rate.
- Attempt every Your Turn question before opening the solution.
- Use units to test whether multiplication or division makes sense.
- For journeys, state the interval: moving time only or whole departure-to-arrival time.
- Estimate the direction and scale before trusting a calculator.
- Return later and solve a changed problem without notes.
1. Start With the Word “Per”
Teacher: Write “12 pieces per minute”. Ask the student to complete the sentence:
For every one minute, 12 pieces are produced.
The sentence exposes the numerator quantity and denominator quantity.
2. A Rate Is Not a Bare Number
The number 60 alone is not a complete speed. It could mean:
- 60 km/h;
- 60 m/min;
- 60 m/s.
These represent very different rates. Keep units attached all the way through the solution.
3. A Unit Rate Has One Unit in the Denominator
A machine produces 84 pieces in 7 minutes at a stated constant rate.
84 ÷ 7 = 12 pieces per minute.
At that rate, 11 minutes produces 132 pieces.
4. Unit Rate and Ratio Are Closely Connected
Chapter 3 compared quantities by ratio. A rate also compares quantities, but unlike a same-unit ratio, it often keeps different units.
For example, $9 for 2.5 kg gives:
$9 ÷ 2.5 kg = $3.60/kg.
The dollar and kilogram units remain because the rate describes cost per mass.
5. Inverting a Rate Changes the Question
A machine working at 12 pieces per minute can also be described as 1/12 minute per piece.
These are reciprocal rates. One asks “how many pieces each minute?” The other asks “how many minutes for one piece?”
Do not invert a rate unless the new unit answers the question you actually have.
6. Constant Rate Is an Assumption or Given Condition
If a question states that a machine works at a constant 12 pieces per minute, scaling is justified.
If you only observe that 84 pieces were produced in 7 minutes, you can calculate an average of 12 pieces per minute over that interval. That alone does not prove exactly 12 were produced during every minute.
Keep “average over the interval” separate from “constant throughout”.
Your Turn 1
- 30 litres flow in 5 minutes. Find the average rate in litres per minute.
- $18 buys 4 kg under a constant unit-price model. Find the unit price.
- 150 words are typed in 2 minutes. Find the average rate in words per minute.
Answers
6 L/min. $4.50/kg. 75 words/min.
7. Unit Conversion Changes the Description, Not the Quantity
The same length can be written as:
2.4 m = 240 cm = 2400 mm.
The physical length has not changed. Only the unit size and numerical count have changed.
8. Smaller Units Usually Need Larger Numerical Counts
One kilometre contains 1000 metres. Therefore:
3.2 km = 3200 m.
The unit becomes smaller, so more of those units are needed to describe the same distance.
9. Build Conversions From Equal Quantities
Use the equality:
1 km = 1000 m.
Then 4.6 km = 4.6×1000 m = 4600 m.
This keeps the conversion tied to a known unit relationship rather than an unexplained “move the decimal point” instruction.
10. Length Conversions
- 1 km = 1000 m;
- 1 m = 100 cm;
- 1 cm = 10 mm.
For 0.75 km:
0.75 km = 750 m.
11. Mass Conversions
- 1 kg = 1000 g;
- 1 g = 1000 mg.
Therefore 0.65 kg = 650 g.
12. Capacity Conversions
- 1 L = 1000 mL.
Therefore 2.75 L = 2750 mL.
13. Do Not Convert Between Different Quantity Types Without a Formula
Metres measure length. Square metres measure area. Litres measure capacity. Kilograms measure mass.
You cannot convert 5 m directly into 5 m². A formula involving another length is needed to create an area.
14. Extension: Area Units Square the Length Conversion
Since 1 m = 100 cm:
1 m² = 100² cm² = 10,000 cm².
Use this when area conversion has been taught in your class.
15. Extension: Volume Units Cube the Length Conversion
Since 1 m = 100 cm:
1 m³ = 100³ cm³ = 1,000,000 cm³.
The exponent on the unit is mathematical information.
Your Turn 2
- Convert 4.8 km to metres.
- Convert 350 cm to metres.
- Convert 2.4 kg to grams.
- Convert 1750 mL to litres.
- Extension: convert 0.25 m² to cm².
Answers
4800 m. 3.5 m. 2400 g. 1.75 L. Extension: 2500 cm².
16. Clock Time Uses Base 60, Not Base 100
One hour contains 60 minutes.
Therefore 1 hour 30 minutes is:
1.5 hours,
not 1.30 hours.
17. Convert Minutes to Hours by Dividing by 60
45 minutes:
45/60 = 0.75 hour.
18 minutes:
18/60 = 0.3 hour.
18. Convert Decimal Hours to Minutes by Multiplying the Fractional Part by 60
2.35 hours means 2 hours plus 0.35 hour.
0.35×60 = 21 minutes.
Therefore 2.35 hours = 2 hours 21 minutes.
19. 1 Hour 12 Minutes Is 1.2 Hours
Twelve minutes is one fifth of an hour:
12/60 = 0.2.
So 1 h 12 min = 1.2 h.
The decimal 1.12 h is only 1 h 7.2 min.
20. Elapsed Time Is a Duration
A journey starts at 14:35 and ends at 16:10 on the same day.
- 14:35 to 15:00 = 25 min;
- 15:00 to 16:00 = 60 min;
- 16:00 to 16:10 = 10 min.
Total elapsed time = 95 min = 1 h 35 min.
21. Clock Labels Should Not Be Subtracted as Ordinary Decimals
10:15 − 9:50 is not 0.65 hour by ordinary decimal subtraction.
The elapsed time is 25 minutes.
Clock notation and decimal-hour notation use different bases.
Your Turn 3
- Convert 54 minutes to hours.
- Convert 1 h 48 min to decimal hours.
- Convert 2.6 h to hours and minutes.
- Find the elapsed time from 08:47 to 10:15.
Answers
0.9 h. 1.8 h. 2 h 36 min. 1 h 28 min.
22. Speed Is Distance Divided by Time
For a journey interval:
speed = distance/time.
The units tell the same story: kilometres divided by hours gives kilometres per hour.
23. Distance Equals Speed Times Time
At a constant speed of 12 km/h for 0.75 h:
distance = 12×0.75 = 9 km.
The hour unit cancels: km/h × h = km.
24. Time Equals Distance Divided by Speed
A 6 km route at 8 km/h takes:
6÷8 = 0.75 h = 45 min.
Do not read 0.75 hour as 75 minutes.
25. Teacher Model 1: Find Speed
A journey covers 36 km in 1 h 12 min.
Convert time:
1 h 12 min = 1.2 h.
Average speed:
36÷1.2 = 30 km/h.
This does not prove the actual speed was exactly 30 km/h at every instant.
26. Teacher Model 2: Find Distance
A cyclist travels at constant speed 18 km/h for 40 minutes.
40 min = 2/3 h.
distance = 18×2/3 = 12 km.
27. Teacher Model 3: Find Time
A car travels 150 km at a stated constant speed of 60 km/h.
time = 150÷60 = 2.5 h = 2 h 30 min.
28. Use Units to Decide the Operation
If you multiply 120 km by 60 km/h, the unit becomes km²/h, which is not time.
If you divide:
120 km ÷ 60 km/h = 2 h.
The unit check exposes the correct structure.
29. Speed Conversion Changes Both Distance and Time Units
One metre per second means 1 m every second.
In one hour there are 3600 seconds, so at 1 m/s the distance covered is 3600 m = 3.6 km.
1 m/s = 3.6 km/h.
30. Convert m/s to km/h by Multiplying by 3.6
5 m/s:
5×3.6 = 18 km/h.
31. Convert km/h to m/s by Dividing by 3.6
72 km/h:
72÷3.6 = 20 m/s.
The 3.6 factor comes from converting both kilometres to metres and hours to seconds.
32. Teacher Model 4: Convert 150 m/min
To convert to m/s:
150÷60 = 2.5 m/s.
To convert to km/h:
150×60 = 9000 m/h = 9 km/h.
Your Turn 4
- A runner covers 2.4 km in 12 min. Find average speed in km/h.
- A cyclist travels at 15 km/h for 32 min. Find the distance.
- Find the time for 42 km at 28 km/h.
- Convert 90 km/h to m/s.
- Convert 7.5 m/s to km/h.
Worked answers
12 min=0.2 h, so 2.4/0.2=12 km/h. 32 min=8/15 h, so distance=15×8/15=8 km. Time=42/28=1.5 h. 90/3.6=25 m/s. 7.5×3.6=27 km/h.
33. Average Speed Uses Total Distance and Total Time
average speed = total distance travelled ÷ total elapsed time.
Do not average the listed speeds automatically.
34. Equal Distances Do Not Give Equal Time Weights
A cyclist travels 12 km at 8 km/h and 12 km at 12 km/h.
- first time = 12/8 = 1.5 h;
- second time = 12/12 = 1 h.
Total distance = 24 km.
Total time = 2.5 h.
average speed = 24/2.5 = 9.6 km/h.
The simple mean 10 km/h is wrong because more time is spent at 8 km/h.
35. Equal Times Do Allow the Simple Mean of Constant Speeds
Travel 30 minutes at 12 km/h and 30 minutes at 18 km/h.
- distance 1 = 6 km;
- distance 2 = 9 km.
Total = 15 km in 1 h.
average speed = 15 km/h.
This equals (12+18)/2 because the durations are equal.
36. Stops Count If the Question Uses the Whole Journey
A cyclist travels 10 km in 30 min, rests 15 min, then travels 10 km in 30 min.
Total distance = 20 km.
Total departure-to-arrival time = 75 min = 1.25 h.
whole-journey average speed = 20/1.25 = 16 km/h.
37. Moving Average and Whole-Journey Average Can Differ
In the same journey, moving time is 60 min = 1 h.
average while moving = 20 km/h.
Both 16 km/h and 20 km/h are correct, but for different time intervals.
38. A Stationary Interval Has Speed Zero
During a 15-minute rest, no distance is travelled.
Average speed for that rest interval is:
0 distance ÷ positive time = 0.
It still adds time to the whole journey.
39. Average Speed Should Fit the Stage Speeds and Stops
Without stops and with positive stage durations, an average speed should lie between the slowest and fastest stage speeds.
With a stop included, zero becomes part of the interval, so the whole-journey average can fall below the slowest moving speed.
40. Teacher Model 5: Three-Stage Journey
A cyclist travels:
- 6 km at 12 km/h;
- 9 km at 18 km/h;
- 3 km at 6 km/h.
Each stage takes 0.5 h.
Total distance = 18 km.
Total moving time = 1.5 h.
average moving speed = 18/1.5 = 12 km/h.
The simple mean works here because the stage times are equal.
41. Add a Rest and the Average Changes
Add a 15-minute rest to the three-stage journey.
Total elapsed time = 1.5+0.25 = 1.75 h.
average including rest = 18/1.75 ≈ 10.29 km/h.
The additional time lowers the average, as expected.
Your Turn 5
A traveller covers 30 km at 60 km/h, rests for 15 minutes, then covers 20 km at 40 km/h.
- Find total distance.
- Find total moving time.
- Find total elapsed time including rest.
- Find whole-journey average speed.
Worked answer
First stage time=30/60=0.5 h. Second=20/40=0.5 h. Total distance=50 km. Moving time=1 h. Rest=0.25 h, so elapsed time=1.25 h. Whole-journey average=50/1.25=40 km/h.
42. Use a Journey Table Before a Long Calculation
| Stage | Distance | Speed | Time |
|---|---|---|---|
| 1 | 48 km | 40 km/h | 1.2 h |
| 2 | 72 km | 60 km/h | 1.2 h |
| Total | 120 km | from totals | 2.4 h |
Average speed = 120/2.4 = 50 km/h.
The table prevents the distance of one stage from being paired with the time of another.
43. Put Stops in Their Own Row
A stop has:
- distance = 0;
- speed = 0;
- time = positive duration.
This makes the whole-journey time visible and reduces accidental omission.
44. Keep Exact or Fuller Times Until the Final Average
If a stage takes 7/12 hour, keep 7/12 or a full calculator value instead of rounding immediately to 0.58 h.
Several rounded stage times can accumulate enough error to change the final answer.
45. A Journey Table Can Reveal Missing Information
If a row contains only a distance and no speed or time, neither remaining quantity can be found uniquely without another relationship.
Do not invent a speed just to complete the table.
46. Read a Cumulative Distance–Time Graph by Changes
On a cumulative distance–time graph, the vertical coordinate gives distance travelled so far and the horizontal coordinate gives elapsed time.
Speed over an interval is:
change in distance ÷ change in time.
47. Teacher Model 6: Read an Interval From Two Points
At minute 2, cumulative distance is 300 m. At minute 5, it is 900 m.
Distance added during the interval:
900−300 = 600 m.
Time interval:
5−2 = 3 min.
average speed = 600/3 = 200 m/min.
48. A Horizontal Distance–Time Segment Represents a Stop
If time increases while cumulative distance remains unchanged, no additional distance is travelled during that interval.
The speed is zero there.
49. A Steeper Distance–Time Segment Means Greater Speed Only After Reading the Scale
Visual steepness can be distorted by axis scales. Compare numerical changes in distance and time before concluding that one segment represents greater speed.
50. Read the Vertical Axis Label Before Interpreting the Graph
“Cumulative distance travelled” cannot decrease.
“Distance from the starting point” can decrease if someone returns toward the start.
“Position” can even become negative relative to a chosen origin.
The same-looking line can mean different things under different axis labels.
51. Returning to the Start Does Not Make Average Speed Zero
A person walks 600 m east and 600 m west in 20 min.
Total distance = 1200 m.
average speed = 1200/20 = 60 m/min.
The final position is the starting point, but the path length travelled is not zero.
52. Extension: Relative Speed Describes the Rate at Which a Gap Changes
Two walkers 1.8 km apart move directly toward each other at 4 km/h and 5 km/h.
The gap closes at:
4+5 = 9 km/h.
Meeting time:
1.8/9 = 0.2 h = 12 min.
Use this as extension where appropriate.
53. Extension: Catch-Up Speed Uses a Difference
A follower at 6 km/h starts 0.6 km behind a walker at 4 km/h in the same direction.
Gap closes at:
6−4 = 2 km/h.
Catch-up time:
0.6/2 = 0.3 h = 18 min.
54. Extension: Combined Rates Add Compatible Contributions
Tap A fills one tank in 2 h, so its rate is 1/2 tank/h.
Tap B fills one tank in 3 h, so its rate is 1/3 tank/h.
Together:
1/2+1/3 = 5/6 tank/h.
Time for one tank:
1÷(5/6) = 6/5 h = 72 min.
55. Rate Models Need Their Assumptions
A model may assume constant speed, constant flow, no queue, no rest, a fixed distance or a fixed unit price.
Those assumptions are not defects. They are the conditions under which the mathematical relationship is being studied.
Do not quietly extend a one-hour average into a prediction for ten hours unless the problem gives or permits a constant-rate model.
56. Unit Price Is Another Rate
Pack A contains 750 g for an invented $6.30.
Mass = 0.75 kg.
unit price = 6.30/0.75 = $8.40/kg.
The same reasoning used for km/h transfers to dollars per kilogram.
57. Lower Total Price and Lower Unit Price Are Different Comparisons
Pack B contains 1.2 kg for an invented $9.60.
Unit price = $8.00/kg.
Pack A has the lower total price. Pack B has the lower price per kilogram.
State which comparison the question asks for.
58. Flow Rate Is Another Rate
An inlet supplies 12 L/min and a drain removes 4 L/min under a constant-flow model.
Net rate:
12−4 = 8 L/min.
Compatible rates can be combined after units are matched and directions are interpreted.
59. Misconception Clinic: 1 h 30 min = 1.30 h
Thirty minutes is 30/60 = 0.5 h. Therefore 1 h 30 min = 1.5 h.
60. Misconception Clinic: Distance = Speed ÷ Time
Use units. km/h × h gives km. Therefore distance = speed×time.
61. Misconception Clinic: Keep the Numerical Value When Converting km/h to m/s
Both distance and time units change, so the numerical value must change. 72 km/h = 20 m/s.
62. Misconception Clinic: Average Speed Is the Mean of Listed Speeds
Use total distance divided by total time. A simple mean is valid only under special weighting conditions such as equal durations.
63. Misconception Clinic: Ignore a Rest Because No Distance Is Travelled
If the requested interval is the whole journey, rest time belongs in the denominator even though it adds no distance.
64. Misconception Clinic: Returning to the Start Means Average Speed Zero
Average speed uses total distance travelled, not final displacement. A return journey can have positive average speed.
65. Misconception Clinic: Smaller Units Always Mean a Bigger Number
That direction rule is useful for simple one-unit conversions, but compound units can behave differently because numerator and denominator both change. Use full conversion reasoning for speed.
66. Misconception Clinic: Divide Clock Labels as Decimals
09:50 and 10:15 are clock readings, not base-ten decimals. Find the elapsed duration using hours and minutes.
67. Misconception Clinic: A Constant Average Means Constant Motion
Average speed summarizes a whole interval. Actual speed may vary within it unless constant speed is stated.
68. Misconception Clinic: A Formula Makes Missing Information Appear
If only distance is known, neither speed nor time is uniquely determined without another relationship. Do not invent a value because a formula has three letters.
69. Misconception Clinic: Unit Price Alone Decides the Best Purchase
Unit price answers one mathematical comparison. It does not by itself decide suitability, required quantity or product quality. Keep conclusions within the requested mathematical question.
70. Guided Practice Set A: Unit Rates
- 96 pages are read in 8 minutes. Find the average rate.
- 45 L flow in 9 min. Find L/min.
- An invented $27 buys 6 kg. Find $/kg.
- 240 m are covered in 3 min. Find m/min.
Solutions
12 pages/min. 5 L/min. $4.50/kg. 80 m/min.
71. Guided Practice Set B: Basic Conversions
- 2.75 km to m.
- 480 cm to m.
- 3.6 kg to g.
- 850 mL to L.
Solutions
2750 m. 4.8 m. 3600 g. 0.85 L.
72. Guided Practice Set C: Time Conversion
- 36 min to hours.
- 1 h 24 min to decimal hours.
- 3.45 h to hours and minutes.
- Elapsed time from 13:38 to 15:02.
Solutions
0.6 h. 1.4 h. 3 h 27 min. 1 h 24 min.
73. Guided Practice Set D: Distance, Speed and Time
- 84 km in 1.5 h: find average speed.
- 18 km/h for 50 min: find distance.
- 75 km at 50 km/h: find time.
- 3.6 km in 18 min: find average speed in km/h.
Solutions
56 km/h. 50 min=5/6 h, so distance=15 km. Time=1.5 h. 18 min=0.3 h, so 3.6/0.3=12 km/h.
74. Guided Practice Set E: Speed Conversion
- 54 km/h to m/s.
- 108 km/h to m/s.
- 4 m/s to km/h.
- 12.5 m/s to km/h.
Solutions
15 m/s. 30 m/s. 14.4 km/h. 45 km/h.
75. Guided Practice Set F: Average Speed
A traveller covers 24 km at 48 km/h and then 36 km at 72 km/h. Find the average speed for the whole journey.
Worked solution
First time=24/48=0.5 h. Second=36/72=0.5 h. Total distance=60 km, total time=1 h. Average=60 km/h.
76. Guided Practice Set G: Average Speed With a Stop
A cyclist travels 12 km in 40 min, rests 20 min, then travels 9 km in 30 min. Find the whole-journey average speed.
Worked solution
Total distance=21 km. Total time=40+20+30=90 min=1.5 h. Average=21/1.5=14 km/h.
77. Guided Practice Set H: Unit Price
Pack A: invented $7.20 for 800 g. Pack B: invented $10.50 for 1.25 kg. Find both prices per kg.
Worked solution
Pack A: 800 g=0.8 kg, so 7.20/0.8=$9/kg. Pack B: 10.50/1.25=$8.40/kg.
78. Guided Practice Set I: Graph Interval
On a cumulative distance–time graph, distance is 500 m at minute 4 and 1400 m at minute 10. Find the average speed over that interval.
Worked solution
Change in distance=900 m. Change in time=6 min. Average speed=150 m/min.
79. Challenge Practice: Equal Distances
A traveller covers 30 km at 30 km/h and another 30 km at 60 km/h. Find the average speed.
Worked solution
Times are 1 h and 0.5 h. Total distance=60 km, total time=1.5 h, so average=40 km/h. The simple mean 45 km/h is not valid because the stage times differ.
80. Challenge Practice: Arrival Time
A journey starts at 08:40. It consists of 6 km at 12 km/h, 9 km at 18 km/h, 15 min rest, then 3 km at 6 km/h. Find the arrival time.
Worked solution
Each moving stage takes 30 min, so moving time=90 min. Add 15 min rest: total=105 min=1 h 45 min. Arrival=10:25.
81. Challenge Practice: Relative Speed
Two walkers start 2.1 km apart and move directly toward each other at 5 km/h and 2 km/h. Find the meeting time.
Worked solution
Closing rate=7 km/h. Time=2.1/7=0.3 h=18 min.
82. Challenge Practice: Combined Flow
An inlet supplies 15 L/min while a drain removes 3 L/min. Under the stated constant-rate model, how long does the tank’s content take to increase by 240 L?
Worked solution
Net rate=12 L/min. Time=240/12=20 min.
83. Challenge Practice: Rate and Percentage
A machine’s stated constant production rate increases from 40 pieces/min to 50 pieces/min. Find the percentage increase in rate.
Worked solution
Increase=10 pieces/min. Reference=original 40 pieces/min. Percentage increase=10/40×100%=25%.
84. Examination Method: Write the Units Beside Every Given Quantity
Do not write only 36 and 1.2. Write 36 km and 1.2 h. The units often reveal the correct operation.
85. Examination Method: Convert Before Combining
Do not combine 45 minutes with 2 hours until both are expressed in compatible time units.
86. Examination Method: State the Interval for an Average
Write “including the rest” or “while moving” where relevant. A correct average depends on which time interval belongs in the denominator.
87. Examination Method: Build a Table for Multi-Stage Journeys
Keep distance, speed and time from each stage in the same row. Add only like quantities in the total row.
88. Examination Method: Estimate Before Trusting the Calculator
If a 60 km journey takes about an hour, a calculated average of 600 km/h is implausible. Use scale sense as a control layer.
89. Examination Method: Keep Exact Time Fractions When Useful
Forty minutes is exactly 2/3 h. That exact fraction may produce cleaner arithmetic than rounding to 0.67 h.
90. Examination Method: Use an Independent Check
- distance: divide your answer by time and recover speed;
- time: multiply your answer by speed and recover distance;
- speed: multiply by time and recover total distance;
- conversion: convert back to the original unit;
- average: verify total distance and total time separately.
91. Oral Classroom Check
- What does “per” mean in a rate?
- Why is 60 not a complete speed by itself?
- What is the difference between average rate and constant rate?
- Why is 1 h 30 min equal to 1.5 h?
- How do units show that distance=speed×time?
- Why does km/h to m/s change the numerical value?
- Why can you not always average two speeds directly?
- When must a stop be included in average speed?
- Why does returning to the start not make average speed zero?
- What should a journey table contain?
The student should answer with a numerical example. If the explanation becomes “because of the triangle”, return to units and meanings.
92. Exit Ticket
- Convert 1 h 18 min to decimal hours.
- Convert 72 km/h to m/s.
- Find the average speed for 42 km in 1.4 h.
- Find the distance travelled at 18 km/h for 40 min.
- Find the time for 45 km at 30 km/h.
- A traveller covers 20 km at 40 km/h, rests 15 min, then covers 30 km at 60 km/h. Find whole-journey average speed.
- At minute 3 a cumulative distance graph shows 400 m; at minute 8 it shows 1400 m. Find average speed over the interval.
Exit-ticket solution
1 h 18 min=1.3 h. 72 km/h=20 m/s. 42/1.4=30 km/h. 40 min=2/3 h, so distance=12 km. Time=45/30=1.5 h. Stage times are 0.5 h and 0.5 h; add 0.25 h rest, total 1.25 h; distance 50 km, average=40 km/h. Graph interval: distance change=1000 m, time change=5 min, average=200 m/min.
93. Homework: Retrieval, Variation and Transfer
Layer 1 — Retrieval
- Define rate and unit rate.
- Write five common unit conversions.
- Explain decimal hours versus clock minutes.
- Derive distance, speed and time relationships from units.
- Write the definition of average speed.
- Explain when a stop belongs in total time.
Layer 2 — Variation
- Four ordinary unit conversions.
- Four time conversions.
- Three distance-speed-time questions.
- Three speed-conversion questions.
- Two multi-stage average-speed questions.
- One journey with a stop.
- One graph-interval question.
Layer 3 — Transfer
Create a three-stage journey containing two moving stages and one rest. Give all units, solve for whole-journey average speed, then rewrite the same rate structure as a non-speed context such as litres per minute or dollars per kilogram.
94. The Seven-Day Return Cycle
- Day 0: complete teacher models and guided practice.
- Day 1: convert units and solve one distance, one speed and one time question.
- Day 3: solve one journey with a stop and one speed conversion without notes.
- Day 7: repeat the exit ticket with changed values and explain every unit decision aloud.
95. A 60-Minute Teaching Lesson
- 5 minutes: unit-rate retrieval.
- 10 minutes: metric and time conversion.
- 15 minutes: distance-speed-time models.
- 10 minutes: speed conversion.
- 10 minutes: average speed and stops.
- 5 minutes: graph or journey-table transfer.
- 5 minutes: exit ticket.
96. A 90-Minute Teaching Lesson
- 10 minutes: rate and unit diagnostic.
- 15 minutes: conversion modelling.
- 20 minutes: distance-speed-time teaching and guided practice.
- 10 minutes: speed conversion.
- 15 minutes: average-speed multi-stage journeys.
- 10 minutes: journey tables and graph interpretation.
- 5 minutes: oral explanation.
- 5 minutes: exit ticket.
97. The Full Chapter Routine
For a general rate:
name numerator quantity → name denominator quantity → align units → divide to one unit → interpret.
For speed:
identify distance and time → convert units → choose unknown → use unit-compatible operation → check scale.
For average speed:
define interval → total distance → total elapsed time → divide → compare with stage speeds and stops.
For multi-stage journeys:
one row per stage → fill distance/speed/time → add like quantities → compute whole-journey result → interpret.
98. Why This Chapter Matters Beyond Chapter 5
Rate reasoning becomes a foundation for algebra, functions, graphs, geometry, science and everyday quantitative decisions. A straight-line graph can encode a constant rate. A formula can connect quantities with different units. Density, flow, unit price and speed all rely on the same discipline: identify what is being measured per unit of what.
The deeper habit is this: never let a number detach from its quantity and unit. When the units remain visible, the mathematical operation becomes easier to justify and easier to check.
99. Connect Back to Chapters 3 and 4
Chapter 3 introduced ratio and proportional scaling. Chapter 4 controlled the reference base in percentage comparisons. Chapter 5 extends these ideas to quantities measured per unit of a different quantity.
- Secondary 1 Mathematics Classroom | Chapter 3: Ratio and Proportion
- Secondary 1 Mathematics Classroom | Chapter 4: Percentages and Reverse Percentages
100. Ready for Chapter 6?
You are ready to move on when you can do all of the following without prompts:
- explain the meaning of a unit rate;
- keep numerator and denominator quantities correctly ordered;
- convert common metric units correctly;
- convert between clock time and decimal hours;
- find elapsed time;
- derive speed, distance and time relationships from units;
- convert km/h and m/s;
- calculate average speed from total distance and total time;
- include stops when the stated interval requires them;
- distinguish whole-journey average from moving average;
- organise multi-stage problems in a table;
- read rate over an interval from a cumulative distance–time graph;
- apply the same rate structure to unit price or flow;
- state assumptions such as constant rate when they matter.
If one item is weak, return to the smallest section that owns it and complete a changed example. If all are stable, continue to Algebraic Language, Expressions, Formulae and nth-Term Patterns, where quantities, units and relationships are compressed into symbolic form.
Continue the Secondary 1 Mathematics Learning Route
- Secondary 1 Mathematics Learning Guide | Rate, Speed and Unit Conversion
- Secondary 1 Mathematics Learning Guide | Ratio and Proportion
- Secondary 1 Mathematics Learning Guide | Percentages and Reverse Percentages
- Secondary 1 Mathematics Learning Guide | Numbers, Number Lines, Approximation and Estimation
- Secondary 1 Mathematics Learning Guide | Word Problems and Mathematical Representation
- Secondary Mathematics Sengkang | S1–S4 Capability Map
