026i - Problem Solving with Formulas

Lesson 26 — Problem Solving and Consolidation: Formulas and Substitution

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

Learning Intentions

• To consolidate writing, substituting into and reversing formulas.
• To solve multi-step problems requiring more than one use of a formula.

Success Criteria

I can:

1. Write a formula from a described situation, defining variables clearly.
2. Substitute forwards and work backwards within the same problem.
3. Compare two situations by applying their formulas.
4. Explain my reasoning and check my answers by substitution.

Warmup

(6 minutes — “Spot the error”, pairs)

Each student has made one mistake. Find and correct each.

Student	Task	Their work
Ana	, find when
Ben	Write “$40 fee plus $12 per hour”
Cleo	, find when
Dai	, find when

Answers: Ana — , not . Ben — the 40C = 12h + 40(-6)^2 = 365n = 50n = 10$.

Activities

Activity 1 — Multi-step Problems (16 min)

Pairs. Each problem requires writing, substituting and reversing.

Problem 1 — Event catering. A caterer charges a 250$38$ per guest.

(a) Write a formula for the total cost for guests.

(b) Find the cost for guests.

(c) A budget of 3100$ is available. What is the greatest number of guests?

(d) A rival caterer charges a flat 526020$ guests?

Problem 2 — Water tank. A tank contains L and is filled at L per minute.

(a) Write a formula for the volume after minutes.

(b) How much water after minutes?

(c) The tank holds L. How long until it is full?

(d) A second tank starts empty but fills at L per minute. When do the two tanks hold the same amount?

Problem 3 — Rectangle. A rectangle has a perimeter of cm.

(a) If the length is cm, find the width.

(b) If the length is three times the width, find both dimensions.

(c) Find the area in each case. Which rectangle has the larger area?

Socratic scaffolding for Problem 1(c):

Prompt	Purpose
Understand: what is the unknown?	The greatest whole number of guests.
What is known?	is the maximum spend.
Set up the equation.	.
Undo in reverse order.	Subtract : . Then divide: .
Is exact?	Yes — exactly.
Looking back — check forwards	✓
What if the division had not been exact?	You would round down — you cannot invite a fraction of a guest, and going over budget is not allowed.

Answers:

1. (a) (b) 25307560$2530$312020$1010$1040$, still the first, but only just.
2. (a) (b) L (c) , so minutes (d) gives , so minutes (both at L).
3. (a) , so cm (b) , so cm and cm (c) areas and — the first is larger.

Discussion of 3(c): Same perimeter, different areas. Which rectangle with perimeter cm would have the largest area? (The square, .) This connects back to the optimisation work in Lesson 18.

Activity 2 — Building Formulas from Tables (10 min)

Explicit method for extracting a formula from data.

The two-step method:

1. Find the constant difference between consecutive outputs — that is the coefficient.
2. Work back to the value when the input is — that is the constant term.

I do:

The difference is each time, so the formula begins . At , , but , so the constant is .

Check at : ✓

You do: Find the formula for each table.

(a)
(b)
(c)
(d)

(Answers: (a) (b) (c) (d) .)

Note on (d): a decreasing pattern gives a negative coefficient. Ask students what real situation this could model. (A tank draining, a countdown, money being spent from a fixed amount.)

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A tutor charges 45$20Ch$ hours.

2. Using your formula, find the cost of hours.

3. Using your formula, a session cost 155$. How many hours was it?

4. Find the formula for this table:

5. Reasoning. Two plans cost and . Without a table, explain which is cheaper for a long membership, and why.

Answers: 1. ; 2. 20045h = 135h = 3y = 7n + 2$10$60m = 12$.)*

Common Misconceptions

Misconception	How to pre-empt it
Reversing operations in the wrong order (dividing before subtracting).	Require one step per line, and state “undo in reverse order” aloud each time.
Reading the constant term as the first table value.	The constant is the value when , not when . Extend every table back to as a check.
Missing that a decreasing table needs a negative coefficient.	Include at least one decreasing table in every practice set.
Comparing two plans using a single value only.	Insist on testing at least two values, and finding the crossover point.
Rounding up when a budget constraint requires rounding down.	Ask “would the answer break the constraint?” before rounding.
Never checking the answer.	Make the forward substitution check a required final line.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A sequence follows . Which term equals ?

Answer

The th term.

E2 (AMC Junior style). Two candles burn at different rates. One starts at cm and shortens by cm per hour; the other starts at cm and shortens by cm per hour. After how many hours are they the same length?

Answer

After hours, both are cm. (Check: and ✓)

E3 (Challenge). A rectangle has perimeter cm and whole-number side lengths. Which dimensions give the largest area?

Answer

gives . Testing: , , , . The square gives the largest area, .

E4 (Challenge). The formula gives the sum of the first whole numbers. (a) Find when . (b) Which gives ?

Answer

(a) . (b) Testing, gives ✓

E5 (Challenge). A hire firm charges . Hiring for days costs 1557$295ab$.

Answer

The extra days cost 140a = 140 \div 4 = $3535(3) + b = 155b = $50C = 35n + 50$.

Homework

1. A gardener charges 65$42Ch4$275$ — how long did it take?

2. A pool contains L and is filled at L per minute. (a) Write a formula for after minutes. (b) Find after minutes. (c) How long to reach L?

3. Find the formula for each table:

   (a)
   (b)
   (c)

4. A rectangle has perimeter cm and length cm. Find its width and area.

5. Two phone plans: and . (a) Compare the cost for minutes. (b) At how many minutes are they equal?

6. A triangle has area and height cm. Find its base.

7. Reasoning. Explain why a table with a constant difference of must have a formula of the form .

8. Challenge. A taxi charges . A km trip costs 1912$40ab9$ km trip.

Answers: Q1 — (a) (b) 23342h = 2105V = 18t + 12048018t = 63035y = 8n + 3y = 4n + 2y = 43 - 3n13286\ \text{cm}^2$35$250.3t + 20 = 0.5tt = 100217$21a = $33(5) + b = 19b = $493(9) + 4 = $31$.