052i - Choosing Representations for Linear Models

Lesson 52 — Choosing Representations for Linear Models

Strand: Algebra | Descriptor: AC9M8A03 | Duration: 45 minutes

Learning Intentions

• To represent a linear model as a table, graph, equation and verbal description.
• To select and justify the representation best suited to a given purpose within the modelling cycle.

Success Criteria

I can:

1. Construct a table of values, a graph and an equation for the same linear relationship.
2. Translate fluently between table, graph, equation and words.
3. Justify why a particular representation is most useful for a specific purpose.
4. Represent two competing financial models together so they can be compared.

Warmup

(5 minutes — quick-fire, mini whiteboards)

Given (a delivery fee model): build a table for ; state the -intercept and gradient; describe in one sentence what each means in context.

Discussion: “We’ve just produced three different representations of the same model in under two minutes. Today’s lesson is about knowing when to reach for each one.”

Activities

Activity 1 — Explicit Instruction: Four Representations, One Relationship (12 min)

Recall Stage 2 of the modelling cycle: Represent — choosing a table, graph, equation or verbal description to suit the purpose.

I do: Bolt Mobile: .

Table:

(MB)	0	200	400	600
($)	20	30	40	50

Graph: plot the points, join with a straight line since is continuous, label axes with units, mark the -intercept at .

Words: “The monthly cost starts at 205$ cents for every extra megabyte used.”

We do: Together build all three representations for FitZone gym: , for .

You do: For the plumber model , produce a table for , sketch a graph, and write a one-sentence description.

Activity 2 — Comparing Two Models on Shared Axes (12 min)

I do: Compare two rideshare companies. Company A: . Company B: . Build a combined table:

(km)	0	5	10	15
Company A	3.50	9.50	15.50	21.50
Company B	6.00	10.50	15.00	19.50

Plot both lines on the same axes. Identify (without solving algebraically yet) that the lines appear to cross between and — this crossing point is the break-even distance, where both companies charge the same fare.

We do: Together compare two phone plans: and , building a shared table and graph.

You do: Compare car hire options: Option A (no flat fee); Option B . Build a table for and sketch both lines on one set of axes.

Activity 3 — Inquiry: Which Representation Wins? (11 min)

Pairs, then whole-class share.

For each purpose below, decide which representation (table, graph, equation or words) is most useful, and produce it.

1. The gym manager wants to instantly calculate the exact cost for any random number of weeks a member asks about, including large numbers like weeks.
2. A customer wants to see visually, at a glance, roughly when Plan B becomes cheaper than Plan A.
3. A new staff member needs a plain-English summary to read out to customers over the phone.
4. A spreadsheet needs to be built showing costs for the first weeks only.

Socratic scaffolding:

Prompt	Purpose
Understand: what does the audience actually need to do with this?	Different tasks need different tools — calculating, comparing, explaining, or listing.
Does the task involve one specific value or a general rule?	A specific large value (Q1) favours the equation; a general trend (Q2) favours the graph.
Does the task involve a small, fixed list of values?	Q4 favours a table, since only rows are needed and no algebra is required of the reader.
Does the task involve someone with no mathematical background?	Q3 favours a verbal description — no symbols required.
Look back — could two representations both work?	Yes, often — but the best choice minimises effort for that specific audience and purpose.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Given , complete a table for .
2. Which representation would you choose to find the cost for exactly , and why?
3. Which representation would you choose to show a customer, at a glance, how two plans compare over the first year?
4. Write as a one-sentence verbal description.

Answers: 1. ; 2. The equation — substituting one large value is faster than extending a table or reading a graph precisely; 3. A graph — visual comparison of trend and crossing point is immediate; 4. “The cost starts at 12$3n$.”

Common Misconceptions

Misconception	How to pre-empt it
A table must always start at and go up by .	Show tables with sensible larger steps (e.g. by s for data) to suit the context.
A graph is the “real” version and the equation is just a shortcut.	Emphasise all four representations describe the exact same relationship — none is more “true” than another.
Two lines that look close together on a graph must be “about the same”.	Zoom in / rescale axes to show that visual closeness can be misleading; the equation gives exact values.
Every relationship should be graphed with a continuous line.	Discuss when the independent variable should be discrete (e.g. whole weeks, whole cars) versus continuous (e.g. distance, data).
Choosing a representation is arbitrary — any one will do.	Return explicitly to “who is using this, and what for?” before selecting.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Two lines are given by and . Using a table of values from to , identify between which two integers the lines cross.

Answer

	0	1	2	3	4
	3	5	7	9	11
	12	11	10	9	8

The values become equal at (both give ).

E2 (Kangaroo style). A graph shows a straight line passing through and . Write the equation of the line.

Answer

E3 (Challenge). A table shows a linear relationship: give . A student claims the rule is . Use the table to check, and correct the rule if needed.

Answer

Check : ✓. Check : ✓. Check : ✓. The claimed rule is correct.

Homework

1. Given , build a table for .
2. Sketch a graph of using your table from Q1, labelling both axes with units.
3. Write as a verbal description suitable for a customer.
4. Two market stalls charge: Stall A and Stall B , where is the number of items sold. Build a combined table for .
5. Reasoning. Using your table from Q4, explain which representation (table, graph or equation) would most efficiently tell you the exact break-even point, and why.
6. Challenge. A line passes through and . Find its equation, then state which representation you would use to check whether the point lies on it, and why.

Answers: 1. ; 2. Straight line, -intercept , rising per unit; 3. “The cost starts at 25$6n0, 10, 20, 3015, 19, 23, 275s = 15+2sm = \dfrac{27-9}{8-2} = 3y = 3x+3x=1003(100)+3 = 303 \ne 309$, so the point does not lie on the line.