A rate compares two quantities with different units: 60 km per hour, $4 per kg, 80 beats per minute. The unit rate is how much for 1 of the second quantity: divide the first by the second. A ratio compares quantities in the same units. Speeds convert with 1 km/h = 1000 m ÷ 3600 s.
Speed limits, the price of petrol, your heart rate and how fast your phone downloads: all rates. Each one says 'this much per one of that'.
A rate compares two quantities measured in different units: 60 km per hour (km/h), 4.50perkilogram(/kg), 80 beats per minute. A ratio compares quantities in the same units: 2 cups of flour to 3 cups of milk, 2 : 3.
The unit rate is the amount for one of the second quantity. Divide the first by the second, and keep the units:
150 km in 2.5 h⇒150÷2.5=60 km per h
Rates convert like units. For speed: 1 km = 1000 m and 1 h = 3600 s, so 54 km/h=54×1000÷3600=15 m/s (divide by 3.6).
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
A car's speed
TryTry it on the board and compare the result with your prediction.
Set the rate
The top line is the distance, the bottom line the time. The dashed line marks 1 hour.
Try it yourselfYour edits are saved in this browser.
Set the rate and the time so that 2 hours matches 150 km. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Rate or ratio
Here is the routine.
Worked exampleBefore each step, say what you think comes next.
distance÷timeDivide what you want "per" by the "one" quantity.
150÷2.5=60150 km in 2.5 h.
60 km/hKeep both units in the answer.
60÷3.6≈16.7 m/sTo convert km/h to m/s, × 1000 ÷ 3600, which is ÷ 3.6.
Read Mia's working. Dollars per kilogram: which number goes first?
Rate or ratio?
60 km per hour
$3.50 per kg
2 : 3 cordial to water
80 beats per minute
3 boys to 4 girls
7 L per 100 km
Mia works out the price of apples per kilogram. Click the line where her working goes wrong.
Convert 90 km/h to metres per second.
ModifyChange the problem on purpose and see what happens.
Convert and compare
Need a hint?
The board shows m/s in its notes.
Another hint
25 m/s × 3.6 = 90 km/h.
Try it yourselfYour edits are saved in this browser.
Set the rate r (in km/h) so the speed is 25 m/s.
Use the board. This ticks when you get there.
Cheese costs $0.80 per 100 g. What is that in dollars per kilogram?
PractiseA few on your own, to check you have it.
Rates
A train travels 240 km in 3 hours. What is its speed in km per hour?
Convert 36 km/h to metres per second.
A 2 kg bag of flour costs $3.40. What is the price in dollars per kilogram?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A long drive
Then give the speed in metres per second, to 1 decimal place.
Try it yourselfYour edits are saved in this browser.
Need a hint?
280 ÷ 3.5 = ?
Another hint
The time is 3 and a half hours.
A drive covers 280 km in 3.5 hours at a steady speed. Set the rate and the time to match.
Use the board. This ticks when you get there.
For teachers
Students match rates to distances and times on a double number line, tell rates from ratios, find unit rates and convert between rate units such as km/h and m/s (AC9M8M05, node 1). About 35 minutes.
I can find a unit rate by dividing in the right order.
I can convert a speed between km/h and m/s, and tell a rate from a ratio.
Before class
Do the Run board yourself first. Students need ratios (Number Foundations, Ratios).
The one idea: a rate is "this much per one of that", with two different units; divide the first by the second. The common mistake is dividing the wrong way round (Mia's mistake).
The board's double number line lines up the two quantities; the dashed line marks the unit rate, and the notes give km/h in m/s.
Wider: compare several units for the same speed (km/h, m/s, knots); rates in First Nations land management, such as how fast a cool burn spreads in different conditions (teach with local knowledge where you can).
Running the lesson
Predict: ask what '1 hour's worth' is.
Run: ask what happens to the distance when the time doubles.
Investigate: ask Mia to say her rate in words.
Modify: ask why the cheaper bag is not the better buy.
Make: ask for the speed both ways.
The hard part (on purpose)
Dividing in the right order. Ask students to say the rate in words ('dollars per kilogram') before they divide: the 'per' quantity goes second.
When students get stuck
A student divides the wrong way. Ask: What is it per? That goes second. Fix: Dollars per kilogram = dollars ÷ kilograms.
A student drops the units. Ask: 60 what, per what? Fix: A rate always has two units.
A student multiplies by 3.6 to get m/s. Ask: Is a metre per second faster or slower than a kilometre per hour? Fix: m/s is the smaller number: divide by 3.6.
Stretch and support
Stretch: Express the same speed in km/h, m/s and knots (1 knot = 1.852 km/h). Which unit do ships use, and why?
Support: A card: 'rate = first quantity ÷ second quantity, units: first per second'.
Where it leads
This starts AC9M8M05.
Where it leads: using rates to solve problems (next lesson), rate models, and gradient as a rate (Linear relations).
What each step is for
a car's speed: Committing to a speed from a distance and a time shows whether they divide distance by time, and keep the units.
set the rate: Matching a rate to a distance and time on the double number line shows the unit rate as 'one hour's worth'.
rate or ratio: Rates have two different units, ratios one; the unit rate divides in the right order (Mia divides the wrong way); speeds convert by × 1000 ÷ 3600.
convert and compare: Converting speeds and prices and comparing two rates with the same unit shows unit rates make comparisons fair.
rates: Unit rates, conversions and rate or ratio, mixed; dividing the wrong way is the planted slip; the simplest-ratio review keeps ratios separate from rates.
a long drive: Choosing the rate and the time to match a distance makes students use the unit rate both ways.
Now you can
Write a rate with its two units.
Find a unit rate by dividing in the right order.
Convert a speed between km/h and m/s, and tell a rate from a ratio.