091i - Rates Consolidation and Check

Lesson 91 — Consolidation and Check: Rates

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

Learning Intentions

• To consolidate the use of rates to compare two related quantities measured in different units.
• To apply rate calculations confidently and efficiently to a range of practical problems.

Success Criteria

I can:

1. Calculate and simplify a rate from given quantities, including unit rates.
2. Convert a rate to different units to enable direct comparison.
3. Use rates to solve multi-step practical problems, including combined and changing rates.
4. Justify which of two rates represents the “better” value or performance in context.
5. Communicate my reasoning and working clearly, including appropriate units.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. A rate can be simplified in the same way as a fraction.
2. If Item A costs more in total than Item B, Item A is always the worse buy.
3. A speed of km/h means an object travels exactly km every hour, no more and no less.
4. Converting a rate to different units changes its value but not what it represents.

Answers: 1. Always — a rate like simplifies by dividing both parts by their HCF, just like . 2. Never — total price alone doesn’t reveal value; you must compare unit rates. 3. Sometimes — this is only exactly true if the speed is genuinely constant; km/h is often an average speed over uneven motion. 4. Always — e.g. km/h km/min: the numeral changes but the underlying relationship is identical.

Activities

Activity 1 — Fluency Review: Calculating, Converting and Comparing Rates (12 min)

Explicit recap, then practice.

I do: Compare two supermarket items to find the better buy.

Brand Y is the better buy. Then convert a speed between units:

We do: Together convert m/s to km/h, and compare g for 3.80750$5.55$.

You do: Students complete 4 mixed rate calculation/conversion/comparison questions independently.

Activity 2 — Applied Problems, including a Harder Multi-stage Rate Problem (21 min)

Pairs. Every answer must carry correct units and a one-sentence justification.

Problem 1. Two hoses fill a pool. Hose A alone fills it in hours; Hose B alone fills it in hours. Working together, how long will it take to fill the pool?

Problem 2 (harder). On a road trip, Priya drives the first km at an average speed of km/h, then the next km at an average speed of km/h. What is her average speed for the whole km trip?

Socratic scaffolding for Problem 2:

Prompt	Purpose
Understand: what is being asked?	Average speed for the whole trip, not the average of the two speeds.
What do you know?	Two km legs, driven at km/h and km/h.
Have you seen a related problem?	Average speed , not the mean of the speeds.
Devise a plan: what formula connects speed, distance and time?	, applied to each leg separately.
Carry out the plan	Find and , add them, then divide total distance by total time.
Looking back	Is km/h between and ? Yes — and it is closer to , because more time was spent travelling at the slower speed.

Answers: Problem 1 — combined rate of the pool per hour, so time hours ( h min). Problem 2 — km/h.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A tap fills a tank at L per minutes. Express this as a unit rate in L/min.
2. Brand X rice: kg for 7.003.5$11.90$. Which is better value? Show working.
3. A cyclist rides km in minutes. Convert this to km/h.
4. Printer A prints pages in min; Printer B prints pages in min. Working together, how many pages could they print in minutes?
5. Reasoning. Explain why you cannot simply average two speeds to find the average speed of a journey unless equal time (not equal distance) was spent at each speed.

Answers: 1. L/min. 2. Brand X: 3.50$3.4018 \div 0.75 = 2487.515.5\times 10 = 155=\frac{\text{total distance}}{\text{total time}}$; if unequal times are spent at each speed, a simple mean of the speeds weights them incorrectly — it only equals the true average when equal time was spent at each.

Common Misconceptions

Misconception	How to pre-empt it
Adding two rates without converting to common units first.	Always convert to matching units before combining or comparing.
Believing average speed equals the mean of the two speeds.	Work through Problem 2 explicitly and contrast with the naive mean of .
Assuming a bigger pack is always better value.	Insist on computing the unit rate for every comparison, never judge by total price or total mass alone.
Solving combined work-rate problems by adding times instead of rates.	Show explicitly why ; rates add, times do not.
Dropping or mixing units (e.g. km/h with m/s) mid-solution.	Require an explicit conversion line whenever units change.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A car travels from A to B at km/h and returns the same route at km/h. What is its average speed for the round trip?

Answer

Let the one-way distance be .

E2 (AMC Junior style). Three taps A, B and C can fill a tank alone in , and hours respectively. If all three are open together, how long does it take to fill the tank?

Answer

Time .

E3 (Challenge). A tank is filled by pipe A in hours. A full tank is drained by pipe B in hours. If both are open on an empty tank, how long does it take to fill?

Answer

Homework

1. Express as a unit rate: (a) km in hours (b) 184962$ minutes.
2. Convert: (a) km/h to m/s (b) m/s to km/h (c) mL/min to L/h.
3. Compare and state which is better value: (a) g for 4.00750$5.706$3.3010$5.20$.
4. A hose fills a paddling pool in minutes alone. A second hose fills it in minutes alone. How long will it take working together?
5. Reasoning. A train travels km at km/h then km at km/h. Explain, without calculating, whether the average speed for the whole journey will be closer to km/h or km/h. Then calculate to check.
6. Challenge. Two cyclists start km apart and ride toward each other, one at km/h and the other at km/h. How long until they meet?

Answers: 1(a) km/h (b) 4.5048201854$8.00$7.60$0.55$0.52\frac{1}{20}+\frac{1}{30}=\frac{1}{12}1250t_1=1.5t_2=34.5=300\div4.5\approx66.750100=40=60\div40=1.5$ hours.