091i - Formulas with Several Variables

Lesson 91 — Manipulating Formulas with Several Variables

Strand: Algebra | Descriptor: AC9M7A06 | Duration: 45 minutes

Learning Intentions

• To work with formulas containing several variables.
• To find any one variable when the others are known.

Success Criteria

I can:

1. Substitute values for all but one variable and solve for the remainder.
2. Do this for any variable in the formula, not just the subject.
3. Keep track of units through a multi-variable substitution.
4. Explain what each variable in a formula controls.

Warmup

(6 minutes — how many unknowns? mini whiteboards)

For the formula :

1. How many variables does it contain?
2. Find when , , .
3. Find when , , .
4. What is the same about Q2 and Q3? What is different?

Answers: 1. Four; 2. ; 3. ; 4. Both leave one unknown after substitution; Q2’s unknown is the subject (direct), Q3’s needs a solving step (Lesson 25’s reversal, now with more variables).

The principle: a formula with variables needs known values to pin down the last one — whichever one it is.

Activities

Activity 1 — Explicit Instruction: Any Variable Can Be the Target (14 min)

I do — the trapezium area formula, gathering the whole course’s threads:

(Four variables: the two parallel sides , the height , the area — met informally in Lesson 21’s pool.)

Forward: , , :

Backward to : , , :

Backward to : , , :

Narrate the constant strategy: substitute everything known first, then what remains is a one-variable equation — Lessons 31–36 finish the job. New formula, old skills.

We do — speed, distance, time. :

1. when km/h, h. (200 km)
2. when km, h. (90 km/h)
3. when km, km/h. (2.5 h)

Units discipline: every answer carries the unit the formula’s structure dictates — km, km/h, h. Mixed units (minutes given, hours needed) must be converted before substituting.

Activity 2 — Formula Circuit (14 min)

Pairs. Four formulas, each with a forward and a backward question. Working shown in full.

Station A — Perimeter of a rectangle, :

1. when , .
2. when , .

Station B — Simple interest, (Lesson 79 revisited):

3. when , , .
4. when , , . (As a percentage.)

Station C — Average, :

5. for scores .
6. Third score when , , .

Station D — The prism, (triangular prism, Lesson 20):

7. when , , .
8. when , , .

(Answers: 1. ; 2. ; 3. 722250r = 90 \Rightarrow r = 0.04 = 4%75c = 240 - 153 = 8712030h = 90 \Rightarrow h = 3$.)

Station C’s move deserves the board: to find a missing score from an average, multiply the average back up to a total (), then subtract the known scores. Averages hide totals; recover the total first. (This foreshadows Lesson 108’s mean work.)

Activity 3 — Inquiry: Which Variable is in Charge? (11 min)

Pairs.

The cost of a taxi journey is , where is kilometres travelled and is minutes spent waiting.

1. Interpret each of the three numbers.
2. Find for a km trip with minutes’ waiting.
3. Which changes the fare more: one extra kilometre, or one extra minute of waiting? How do you know without calculating a fare?
4. A fare came to 35.308$ minutes of waiting. How far was the trip?
5. Invent a journey where the waiting cost exceeds the distance cost.

Socratic scaffolding for Q3 and Q5:

Prompt	Purpose
What does each coefficient do?	Each extra unit of its variable adds exactly that amount.
So compare and .	A kilometre costs 2.20$0.80$ — the kilometre wins.
For Q5: what has to be true?	— lots of waiting, little driving.
Give a concrete case.	E.g. , : waiting 3.20 >$2.20$ ✓
Looking back	Coefficients are exchange rates between a variable and the output — reading them beats recalculating.

Answers: 1. 4.50$2.20$0.804.5 + 22 + 4 = $30.504.5 + 2.2k + 6.4 = 35.3 \Rightarrow 2.2k = 24.4 \Rightarrow k = 11.09\ldots \approx 11.1k = 11C = $35.100.8m > 2.2k$.

Checks for Understanding

(5 minutes — exit ticket)

1. For : find when , , .
2. For : find when , .
3. For : the average of three masses is kg and two of them are kg and kg. Find the third.
4. In (a hire cost with hours and people ): which adds more, an extra hour or an extra person? Why?
5. Reasoning. Why does a four-variable formula need three known values before the fourth is determined?

Answers: 1. ; 2. ; 3. total ; third kg; 4. The hour (); coefficients are the per-unit contributions; 5. Each known value removes one unknown; with three fixed, one equation in one unknown remains — solvable. Fewer, and many combinations fit.

Common Misconceptions

Misconception	How to pre-empt it
Believing only the subject can be found.	The warmup’s Q2/Q3 pairing; every station asks both directions.
Substituting for the target variable by mistake.	Circle the target first, substitute everything else.
Unit mixing ( in minutes into a km/h formula).	Convert before substituting — the units discipline of Activity 1.
Finding a missing value from an average without recovering the total.	Station C’s board note.
Treating coefficients as interchangeable with their variables.	The taxi inquiry: coefficients are rates, variables are amounts.
Dropping units from answers.	Every station answer carries its unit; marked.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). For with , , : find .

Answer

.

E2 (AMC Junior style). The average of five numbers is . Four of them are . Find the fifth.

Answer

Total ; known sum ; fifth .

E3 (Challenge). For the trapezium formula : a trapezium has area , height , and one parallel side twice the other. Find both parallel sides.

Answer

: sides and .

E4 (Challenge). In , a fare of 26.10k = m$). Find them.

Answer

— so km and minutes.

E5 (Challenge). The formula gives a student’s average over four tests. After three tests her average is . What must she score on the fourth to lift her overall average to ?

Answer

Three-test total ; four-test total needed ; fourth score .

Homework

1. For : (a) when (b) when (c) when .
2. For : (a) at km/h for h (b) for km at km/h (c) for km in minutes.
3. For : (a) on 12003%5t$8004%$96P5%2$55$.
4. The average of four prices is 18$14$21$19$. Find the fourth.
5. A gym charges (months , visits-with-trainer ). (a) for months with trainer visits. (b) How many trainer visits made a -month bill reach 260$? (c) Which matters more per unit, a month or a trainer visit?
6. Reasoning. For , explain why knowing only cannot determine and , and list three possibilities.
7. Challenge. For : area , height , and the parallel sides differ by . Find both.

Answers: Q1 — (a) (b) (c) . Q2 — (a) km (b) h (c) min h: km/h. Q3 — (a) 18032t = 96 \Rightarrow 30.1P = 55 \Rightarrow $55072$1820 + 90 + 30 = $14020 + 180 + 3v = 260 \Rightarrow v = 2015 > 3(l, w) = (12, 8), (15, 5), (10, 10)(a + b) \times 3 = 54 \Rightarrow a + b = 184117$.