093i - Systematic Variation of Variables

Lesson 93 — Describing the Effect of Systematic Variation

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

Room/equipment: devices helpful but optional — printed result tables provided for all investigations.

Learning Intentions

• To describe precisely what happens to a formula’s output when one variable is changed systematically.
• To distinguish additive effects from multiplicative effects.

Success Criteria

1. I can hold all variables but one fixed, and vary that one systematically.
2. I can describe the effect using “adds … each time” or “multiplies by …“.
3. I can predict the effect of doubling or tripling a variable before computing it.
4. I can explain why an effect is additive or multiplicative from the formula’s structure.

Warmup

(6 minutes — predict then check, mini whiteboards)

For with , (so ):

1. Predict: what happens to if increases by ? Check.
2. Predict: what if doubles to ? Does double? Check.
3. For (): what if doubles? Does double? Check.
4. Why did doubling behave differently in Q2 and Q3?

Answers: 1. : adds ; 2. — not doubled ( would be); 3. — exactly doubled; 4. In the -part is untouched by ‘s change (addition separates them); in the whole output is a multiple of (multiplication couples them).

Today’s distinction, named: additive structure (unchanged parts dilute the effect) versus multiplicative structure (the effect scales the whole output).

Activities

Activity 1 — Explicit Instruction: the Two Effect Types (12 min)

I do — the step effect (additive). : each extra hour adds exactly 25h = 15h24$5090 \to 14090 \to 18040$ never doubles.** (Lesson 25’s phone-plan point, now as structure.)

I do — the scaling effect (multiplicative). with fixed at : so . Tabulate –: . Doubling doubles — from anywhere. Tripling triples. The output is a pure multiple of .

The diagnostic question to teach: “Is there a term the changed variable doesn’t touch?” Yes → additive dilution (doubling the variable does not double the output). No → clean scaling.

We do — classify before computing:

1. , doubling . (Clean scaling — doubles.)
2. , doubling . (Diluted — the stands still.)
3. , doubling . (Clean — every term carries .)
4. Same formula, doubling only. (Diluted — ‘s share is untouched.)

Activity 2 — Investigation Stations (16 min)

Pairs. Each station provides a result table (or a device to generate one). The deliverable per station: an effect sentence and a why-sentence.

Station A — The cube. for –: .

1. Describe the steps between outputs. (Growing: — nothing like linear.)
2. What does doubling (say , ) do to ? (, : multiplies by .)
3. Predict the effect of tripling ; test with and . (.)

Station B — Interest. , , , rate –: .

4. Effect of each extra percentage point? (Adds 20$ — additive in appearance…)
5. Effect of doubling the rate? (Doubles the interest — because the step is proportional: with no constant term, additive-per-step and multiplicative-under-doubling coexist.)

Station C — The border. (Lesson 87’s pond) for –.

6. Effect of each extra unit of ? (Adds .)
7. Does doubling double ? Test . (: no — the dilutes.)

Station D — Cooling drink (data card, no formula). Temperature: at – minutes.

8. Describe the steps. (Shrinking: .)
9. Is this additive or multiplicative behaviour? (Neither pure type: each step is roughly of the last — a decay pattern. Real data escapes the two clean categories; naming that is the point.)

Socratic prompts while circulating:

Prompt	Purpose
Before touching the table: what does the formula predict?	Structure first, data second.
Your effect sentence — does it hold at every row?	”Adds ” must survive checking all steps.
Station B looks additive and doubles cleanly. Contradiction?	No constant term — the resolution is structural.
For D: find the approximate ratio of consecutive steps.	Extends the toolkit beyond the two taught types.

Activity 3 — Inquiry: Design the Experiment (11 min)

Pairs.

A landscaper’s quote formula is , where is the area in squares and the number of plants.

Design and run (by hand or sheet) the minimal set of calculations that would let someone discover all three numbers in the formula, if they could only request quotes.

1. Which quotes do you request?
2. Extract the three numbers from your results.
3. What is the smallest number of quotes that works? Why?

Socratic scaffolding:

Prompt	Purpose
Which request isolates the callout fee?	: . (Lesson 88’s move, now in two variables.)
Then how to isolate ?	Change only by one: ; difference .
And ?	Change only : ; difference .
So the minimum?	Three quotes — one per unknown number. Fewer leaves an unknown free.
Would , , also work?	Yes: differences isolate then , then back-substitute — systematic variation means changing one thing at a time, not necessarily starting at zero.

The closing principle (board): change one variable at a time, and the formula confesses one coefficient at a time. This is the experimental method in miniature — fair testing, as science calls it.

Checks for Understanding

(5 minutes — exit ticket)

1. For : the effect of each extra ? The effect of doubling from to ?
2. For (fixed-height box): what does doubling do to ? Test with .
3. Classify: doubling in — clean scaling or diluted?
4. A quote formula gives , , . Reconstruct it.
5. Reasoning. Why does a constant term stop doubling-the-input from doubling-the-output?

Answers: 1. Adds ; fare goes , not ; 2. : ; 3. Clean — no untouched term; 4. ; 5. The constant contributes the same amount regardless of the input; the doubled part doubles, the constant does not, so the total falls short of doubling.

Common Misconceptions

Misconception	How to pre-empt it
”Double the input, double the output” as a universal law.	The warmup’s Q2/Q3 contrast is the lesson’s spine.
Reading Station B as contradicting the additive/multiplicative split.	The no-constant-term resolution, addressed head-on.
Changing two variables at once when investigating.	The design-the-experiment inquiry makes one-at-a-time the winning strategy.
Expecting all real data to fit the two clean types.	Station D’s decay pattern names the boundary honestly.
Believing the effect of in is constant.	Station A’s growing steps.
Confusing the step in a table with the coefficient when inputs step by more than 1.	Carried from L88; re-flagged at Station B if rates step by .

Enrichment — Competition-Style Problems

E1 (Kangaroo style). For : by what factor does grow when grows tenfold?

Answer

.

E2 (AMC Junior style). A formula gives and each extra unit adds . A second formula gives and doubles whenever the input doubles, with . Write both formulas.

Answer

and .

E3 (Challenge). In (preview of Lesson 95): a pizza’s radius grows from cm to cm. The menu doubles the price. Fair?

Answer

Area quadruples (); doubling the price undercharges relative to dough — the large pizza is the bargain. (Exact areas unneeded: pure scaling argument.)

E4 (Challenge). A mystery formula over inputs returns: , , . Reconstruct it.

Answer

-step: per unit. -step: per unit. Base: . Formula: . (Check all three ✓)

Homework

1. For : (a) the effect of one extra hour (b) at and — did doubling double ? (c) what would make the doubling clean?
2. For (fixed ): (a) effect of one extra unit of (b) effect of tripling (c) why are both descriptions true at once here?
3. Tabulate for –. (a) The steps between outputs. (b) The effect of doubling . (c) One sentence: how does squaring differ from a linear rule?
4. A quote system returns , , . Reconstruct the formula and predict .
5. A cooling curve gives at minutes –. (a) The steps. (b) Roughly what fraction of each step is the next? (c) Predict minute .
6. Reasoning. “Doubling any ingredient of a formula doubles the answer.” Give one formula where this is true and one where it is false, with the structural reason.
7. Challenge. For (surface area of a box): does doubling all three of double ? Test on a box and state the true factor.

Answers: Q1 — (a) 127812630+10\times 33, 5, 7, 9, 11\times 445a60/2 = 30p45/5 = 9Q = 45 + 30a + 9p(3,4) \to 45 + 90 + 36 = 171-20, -15, -11, -8\tfrac34\approx -630d = stC = 40 + 25h1\times2\times3S = 4 + 6 + 12 \to 222\times4\times6S = 16 + 24 + 48 = 88 = 4 \times 224$, because every term is a product of two doubled lengths.