025i - Formulas in Real Contexts

Lesson 25 — Applying Formulas to Real-World Contexts

Strand: Algebra | Descriptor: AC9M7A01 | Duration: 45 minutes

Learning Intentions

• To apply formulas to authentic situations and interpret the results.
• To determine an unknown by substituting known values and working backwards.

Success Criteria

I can:

1. Select and apply the right formula for a practical situation.
2. Substitute known values and solve for the remaining unknown.
3. Interpret the answer in context, with correct units.
4. Judge whether an answer is reasonable for the situation.

Warmup

(6 minutes — formula matching, pairs)

Match each formula to its situation and state what each variable means.

Formula		Situation
		A. Distance travelled at constant speed
		B. Total cost of items at a fixed price
		C. Tradesperson: call-out fee plus hourly rate
		D. Area of a triangular garden bed

(Answers: C; A; D; B.)

Discussion prompt: Two of these have a “starting amount” built in and two do not. Which, and how can you tell by looking?

Activities

Activity 1 — Forward Substitution in Context (10 min)

Explicit modelling with an emphasis on interpretation.

I do — mobile phone plan. The monthly cost is , where is minutes of calls.

Interpretation questions to ask after every calculation:

• What does the represent? (The fixed monthly charge, paid even with no calls.)
• What does the represent? (The cost per minute — cents.)
• What happens to the cost if doubles? (It does not double — only the variable part does. At , 95$130$.)

This last point is worth dwelling on. Students routinely assume proportionality where a fixed term breaks it.

We do: Three contexts, each with an interpretation question attached.

1. Car hire: . Find the cost for days. What is the ?
2. Taxi: . Find the cost for a km trip.
3. Swimming pool heating: , where is hours of heating. Find after hours.

(Answers: 340$33.8031^\circ$C.)

Activity 2 — Working backwards to Find an Unknown (14 min)

The core skill of the lesson. Model the reversal explicitly.

I do — the plumber. . A job costs 400$. How long did it take?

Protocol to state aloud: undo the operations in reverse order. The formula multiplies then adds, so to reverse it you subtract then divide.

I do — geometry. A triangle has area and base cm. Find its height.

Pairs practice — contexts requiring reversal:

1. . The total is 205n$.
2. . A car travels km at km/h. Find .
3. . A rectangle has cm and cm. Find .
4. . A box has , cm, cm. Find .
5. . A bill is 54t$.

Socratic scaffolding for Q5:

Prompt	Purpose
Understand: what is known, and what is unknown?	is known; is unknown.
Which part of the bill is not from texting?	The fixed 30$.
So how much came from texts?	24$.
Devise a plan	Divide by the cost per text.
Carry it out	texts.
Looking back — check forwards	✓
Is it reasonable?	texts in a month is plausible. ✓

(Answers: ; hours; cm; cm; .)

Insist on the forward check. Substituting the answer back into the original formula catches nearly every arithmetic slip.

Activity 3 — Inquiry: Which Plan is Better? (8 min)

Pairs.

Two gyms:

• FitZone: (joining fee plus monthly)
• PowerHouse: (no joining fee)

where is the number of months.

1. Find the cost of each for months, months and months.
2. Which is cheaper for a short membership? For a long one?
3. At how many months do they cost the same?
4. What advice would you give someone joining?

Socratic scaffolding for Q3:

Prompt	Purpose
Understand: what does “cost the same” mean?	The two formulas give equal values for the same .
What have you noticed from your table?	FitZone is dearer early, cheaper later — so they cross somewhere between.
Narrow it down	At : vs — PowerHouse still cheaper. At : vs — now FitZone is.
Try between	At : and . Still PowerHouse, just.
Can you set it up algebraically?	, so and .
Interpret in context	Memberships are whole months, so from month onwards FitZone is cheaper.
Looking back	Why did the algebra give a non-whole answer? (The formulas are continuous; the real situation is not.)

Answers: 1. months: 126$606$162$12012$234$240m = 11.2512$. 4. Choose based on how long you intend to stay — under a year, PowerHouse.

Checks for Understanding

(5 minutes — exit ticket)

1. . Find the cost of a -hour job.
2. . A job costs 285$. How many hours did it take?
3. . A train travels km in hours. Find its average speed.
4. Reasoning. In , explain what each number means for a hire company.
5. A triangle has area and height m. Find its base.

Answers: 1. 22530h = 210h = 7s = 420 \div 3.5 = 120$45$18\tfrac12 b(8) = 60b = 15$ m.

Common Misconceptions

Misconception	How to pre-empt it
Assuming doubling the input doubles the output when a fixed term is present.	Address directly in Activity 1 with the phone plan. Compute both and compare.
Dividing before subtracting when reversing, e.g. then subtracting.	State the “undo in reverse order” rule and require one step per line.
Not checking the answer by substituting forwards.	Make the forward check a compulsory final line in every solution.
Giving an answer with no units or context.	Every contextual answer must end in a sentence with units.
Ignoring whether a non-whole answer makes sense in context.	Activity 3 Q3 forces the interpretation of months.
Selecting the wrong formula because two look similar.	Have students state what each variable means before substituting.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A taxi charges dollars for kilometres. A trip costs 28.50$. How far was it?

Answer

E2 (AMC Junior style). Two printers charge and dollars for copies. For how many copies do they cost the same?

Answer

E3 (Challenge). The formula converts Fahrenheit to Celsius. Convert F (normal body temperature).

Answer

E4 (Challenge). A rectangular garden has perimeter m. Its length is twice its width. Find both dimensions and the area.

Answer

With : , so m and m. Area .

E5 (Challenge). A phone plan is . Amara budgets 50$ per month. What is the greatest whole number of minutes she can use?

Answer

She can use up to minutes.

Homework

1. gives a job’s cost. Find the cost of (a) a -hour job (b) a -hour job.
2. . A job costs 375$. How many hours did it take?
3. . Find (a) when km/h and h (b) when km and km/h (c) when km and h.
4. A triangle has area and base cm. Find its height.
5. A rectangle has perimeter cm and width cm. Find its length.
6. A box has volume , length cm and height cm. Find its width.
7. Two bike hire firms charge and . (a) Compare the cost for hours and for hours. (b) At how many hours are they equal?
8. Reasoning. For , explain why tripling does not triple . Illustrate with and .
9. Challenge. A tank is filled by the rule , where is litres and is minutes. (a) How much is in the tank at the start? (b) After how long does it hold L? (c) The tank’s capacity is L — when does it overflow?

Answers: Q1 — (a) 215$335734047812176$43$33$73$886h + 25 = 11hh = 5n=2C = 150n=6C = 250450$1004015t = 210t = 1415t = 360t = 24$ min.