088i - Introducing Rates

Lesson 88 — Introducing Rates: Comparing Two Related Quantities

Strand: Measurement | Descriptor: AC9M8M05 | Duration: 45 minutes

Learning Intentions

• To understand that a rate compares two quantities measured in different units.
• To calculate and simplify a rate, including finding a unit rate.

Success Criteria

I can:

1. Explain what a rate is, and how it differs from a ratio.
2. Calculate a rate from two given quantities (e.g. distance and time, price and mass).
3. Simplify a rate to a unit rate — an amount “per one” of the second quantity.
4. Identify examples of rates in everyday contexts, including speed, price, pay, and flow.
5. Use a rate to find an unknown quantity.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

For each pair, decide whether it describes a ratio (same units) or a rate (different units).

1. 3 red counters to 5 blue counters.
2. 100 km travelled in 2 hours.
3. $18 for 3 kg of apples.
4. 4 boys for every 5 girls in a class.

Answers: 1. Ratio; 2. Rate; 3. Rate; 4. Ratio.

Discussion: What’s the key difference? A ratio compares quantities in the same units (both counted in “counters”, or both “people”); a rate compares quantities in different units (km and h; dollars and kg).

Activities

Activity 1 — Explicit Instruction: what is a Rate? (12 min)

A rate compares two quantities measured in different units. Unlike a ratio, a rate is always written with its units attached — e.g. km/h, not just a number.

I do: A car travels km in hours. To simplify this rate, divide both quantities by the second quantity so that it becomes “per 1 hour”:

This is called a unit rate — the amount of the first quantity for exactly one unit of the second.

We do: A tap fills L in minutes. Simplify to a unit rate.

You do: Calculate and simplify each as a unit rate:

1. km in hours
2. $18 for kg
3. words typed in minutes
4. $76 earned in hours

Answers: 1. km/h; 2. $6/kg; 3. words/min; 4. $9.50/h.

Activity 2 — Rates in Different Contexts (12 min)

Rates appear under many names: speed (distance/time), price rates or unit pricing (price/mass), pay rates (money/time), and flow rates (volume/time). The method is always the same — divide to find the “per 1” amount.

I do:

We do: Together find: a runner covering m in seconds (speed, in m/s); a factory producing items in hours (items/hour).

You do: Calculate and simplify each rate:

1. $45 for kg of apples
2. km in hours
3. $220 for hours of casual work
4. L in minutes from a tap
5. heartbeats in minutes (beats per minute)

Answers: 1. $9/kg; 2. km/h; 3. $11/h; 4. L/min; 5. bpm.

Activity 3 — Inquiry Task: Using a Rate to Find an Unknown Quantity (10 min)

Pairs, then whole-class share.

Once you know a unit rate, you can use it to find other unknown quantities — how long something takes, or how much of something is needed.

1. A printer prints at pages per minute. How long will it take to print a -page report?
2. A car uses fuel at a rate of L per km. How much fuel is needed for a km trip?
3. (Harder) A water tank is being filled by a pipe at a constant rate. After minutes, it contains L; after minutes, it contains L. The tank started empty. Find the filling rate, then predict how long a L tank will take to fill completely.

Socratic scaffolding for Problem 3:

Prompt	Purpose
Understand: what is being asked?	Find the rate first, using the two given data points, then use it to predict a later time — two separate steps.
What do you know?	Two snapshots in time ( min → L, min → L), not the rate directly.
Devise a plan	A constant rate means equal volume increases in equal time increases — find the change in volume over the change in time.
Carry out the plan	rate L/min.
Look back — does this fit the data?	Check: at min, L ✓ (confirms the tank did start empty at ).
Devise a plan for the prediction	time volume rate.
Carry out the plan	min.
Look back	Is min reasonable, given it took min to reach less than a third of the tank ( of L)? Yes — proportionally consistent.

Answers: 1. min. 2. L. 3. Rate L/min; time to fill L min.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. In your own words, what is a rate? Give an example different from those used in class.
2. Calculate and simplify to a unit rate: $54 for kg of prawns.
3. A hose fills a paddling pool at L/min. How long will it take to fill a L pool?
4. Reasoning. Explain why “3 red counters to 5 blue counters” is a ratio, not a rate.

Answers: 1. E.g. “A rate compares two quantities measured in different units, such as km/h.” 2. $9/kg. 3. min. 4. Both quantities (red counters, blue counters) are measured in the same unit — counters — so it’s a ratio; a rate requires two different units, like km and hours.

Common Misconceptions

Misconception	How to pre-empt it
Confusing a ratio with a rate.	Always ask “are the two units the same or different?” before deciding — reinforce with the Warmup sort.
Believing “rate” only ever means speed.	Deliberately mix speed, price, pay and flow-rate examples in every activity so students see the pattern generalises.
Simplifying only one part of the rate, not both consistently.	Model the division of both quantities by the same number, as with simplifying a fraction.
Dropping the units and writing a bare number as the answer (e.g. "" instead of ” L/min”).	Insist every rate answer is written with its unit attached, every time, including in working.
Treating a unit rate as a fixed, unchanging fact rather than a model of typical behaviour.	Discuss that real speeds, flows, and pay rates can vary — a unit rate is often an average or an assumption.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A machine stamps out metal parts every minutes. At this rate, how many parts does it stamp in hour?

Answer

225 parts.

E2 (AMC Junior style). Three taps, each flowing at the same constant rate, together fill a L tub in minutes. How long would it take a single tap, flowing at that same rate, to fill the tub alone?

Answer

Combined rate L/min for 3 taps, so one tap flows at L/min.

E3 (Challenge). A car’s fuel gauge shows it has used of a full L tank after travelling km. At this rate, how far can the car travel on a full tank?

Answer

Homework

1. Calculate and simplify each rate: (a) km in h (b) $36 for kg (c) words in min (d) $150 for h of work.
2. A tap fills L in minutes. Find the flow rate in L/min.
3. A car travels km using L of fuel. Find its fuel consumption rate in km/L.
4. A cyclist rides at a constant rate of km/h. How far will they travel in hours?
5. Reasoning. Explain why “$5 per 2 tickets” is not yet expressed as a unit rate, and rewrite it as one.
6. Challenge. A printer prints pages every seconds. At this rate, how many full pages can it print in seconds, and how many seconds remain unused after the last full -second cycle?

Answers: 1. (a) km/h (b) $9/kg (c) words/min (d) $12.50/h. 2. L/min. 3. km/L. 4. km. 5. It gives an amount for 2 tickets, not 1 — the unit rate is 5\div2=$2.5095\div10=91090\times 7=\mathbf{63\text{ pages}}\mathbf{5\text{ seconds}}$ left unused.