089i - Converting and Comparing Rates

Lesson 89 — Converting and Comparing Rates

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

Learning Intentions

• To convert a rate from one pair of units to another.
• To compare two rates given in different units by converting to a common unit, and determine which represents better value or performance.

Success Criteria

I can:

1. Convert a rate to different units (e.g. km/h to m/s, $/kg to $/g).
2. Explain the conversion process, including converting each part of the rate.
3. Compare two rates by first converting them to the same units.
4. Justify which of two options represents better value or performance.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

Quick unit conversions — no rates yet, just the building blocks.

1. Convert km to m.
2. Convert hours to seconds.
3. Convert kg to g.
4. Convert L to mL.

Answers: 1. m; 2. s; 3. g; 4. mL.

Discussion: Converting a rate means converting both quantities inside it — today we combine these building blocks with what you learned about rates last lesson.

Activities

Activity 1 — Explicit Instruction: Converting Rates (12 min)

I do: Convert km/h to m/s. Convert each part separately, then combine.

Shortcut: since , you can convert km/h to m/s by dividing by , and m/s to km/h by multiplying by .

I do (second example): Convert $4.50/kg to $/g.

We do: Convert m/s to km/h; convert $3.20/L to $/mL.

You do: Convert each rate:

1. km/h to m/s
2. m/s to km/h
3. $2.80/kg to $/g
4. mL/min to L/h

Answers: 1. m/s; 2. km/h; 3. $0.0028/g; 4. L/h.

Activity 2 — Explicit Instruction: Comparing Rates for “better value” (12 min)

To compare two rates fairly, first convert them to the same units.

I do: Brand A: g for $3.20. Brand B: kg for $8.40. Which is better value?

Brand B is better value.

I do (second example): A cheetah runs at m/s. A car travels at km/h. Which is faster?

The cheetah ( m/s) is faster.

We do: Job A pays $18.50/h. Job B pays $145 for a h shift. Which pays better per hour?

Job B pays better.

You do: Compare each pair and state which is better value or faster:

1. Petrol: Station A $1.65/L; Station B $158.40 for L.
2. Flow rates: Pipe A L/s; Pipe B L/h.
3. Typing speeds: Student X types words/min; Student Y types words in hours.

Answers: 1. Station B 1.65=8400\div3600\approx2.33=5000\div120\approx41.7$ words/min — Student X is faster.

Activity 3 — Inquiry Task: the Best Deal Investigation (10 min)

Pairs, then whole-class share.

A swimming pool is filled by one pump running at L per minutes on Monday. On Tuesday, a different pump is used, rated at L/s. Which pump fills faster? Then, find how much longer (in minutes) the slower pump would take to fill a L pool compared to the faster one.

Socratic scaffolding:

Prompt	Purpose
Understand: what is being compared?	Two flow rates in different units (L per 15 min vs L per second) — they must be converted to a common unit before comparing.
Devise a plan	Convert both rates to L/min, then compare directly.
Carry out the plan	Monday: L/min. Tuesday: L/min.
Interpret	Monday’s pump is twice as fast as Tuesday’s.
Devise a plan for the second part	Use time volume rate for each pump filling L, then subtract.
Carry out the plan	Monday: min. Tuesday: min. Difference min.
Look back	Does “twice as fast” imply “half the time”? Check: is indeed roughly double — consistent with Tuesday’s pump being half the rate.

Answers: Monday’s pump ( L/min) is faster than Tuesday’s ( L/min). Filling L takes Monday’s pump min and Tuesday’s pump min — a difference of minutes.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Convert km/h to m/s.
2. Convert m/s to km/h.
3. Compare: Brand A $4.80 for g; Brand B $7.50 for kg. Which is better value?
4. Reasoning. Explain why you must convert two rates to the same units before comparing them, using an example from today’s lesson.

Answers: 1. m/s. 2. km/h. 3. Brand A 8.00=$7.501.65/L directly against “$158.40 for 96 L” is meaningless until both are expressed per litre — only then can the numbers be compared fairly (Activity 2, Q1).

Common Misconceptions

Misconception	How to pre-empt it
Converting only one part of a rate (e.g. km to m) and forgetting the other (h to s).	Model both conversions explicitly, side by side, every time.
Using the wrong direction for the km/h ↔ m/s shortcut (multiplying by instead of dividing, or vice versa).	Anchor to a known fact: ” km/h is a gentle jog, about m/s” — check the shortcut against this benchmark.
Assuming the item with the larger total price is always worse value.	Insist a unit rate is calculated for every comparison — never judge by total price alone (Activity 2, Q1’s “equal value” surprise is a good check on this habit).
Comparing two rates directly without converting to the same units first.	Require a labelled conversion line before any comparison is made.
Rounding an intermediate conversion too early, especially when two rates are close in value.	Keep extra decimal places until the final comparison, particularly when values are close (as in Activity 2, Q1).

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A snail moves at cm every seconds. Convert this speed to metres per hour.

Answer

E2 (AMC Junior style). Train A travels km in hours. Train B averages m/s over its whole journey. Which train is faster, and by how many km/h?

Answer

Train A is faster, by .

E3 (Challenge). A recipe uses ingredients at a rate of g of flour for every mL of milk. If you have L of milk, how much flour (in kg) do you need to keep the same rate?

Answer

Homework

1. Convert: (a) km/h to m/s (b) m/s to km/h (c) $6.50/kg to $/g (d) mL/min to L/h.
2. Compare and state which is better value: (a) $3.60 for g vs $9.90 for kg (b) $2.25/L vs $8.80 for L.
3. A hose fills at L per seconds. Convert this to L/min.
4. Two cars: Car A averages km/h. Car B averages m/s. Which car is faster, and by how much (in km/h)?
5. Reasoning. A student converts “$5 per 250 g” to a per-kg rate by writing “$5 × 250 = $1250/kg”. Explain what has gone wrong, and give the correct per-kg rate.
6. Challenge. A conveyor belt moves boxes at a rate of boxes every seconds. How many boxes does it move in hour, and how many minutes (to the nearest minute) would it take to move exactly boxes?

Answers: 1. (a) m/s (b) km/h (c) $0.0065/g (d) L/h. 2. (a) A 7.20=$6.602.25/L vs $2.20/L — the second is better. 3. L/s L/min. 4. Car B km/h; Car A is faster, by km/h. 5. The student multiplied by (the gram amount) instead of by the conversion factor (since lots of g in kg); the correct rate is 5\times4=$20=\dfrac{3}{8}13,6003,600\div8=450\times3=1,3501,0001,000\div\dfrac{3}{8}=1,000\times\dfrac{8}{3}\approx2,666.7\approx44$ minutes.