055i - Reviewing and Refining Linear Models

Lesson 55 — Reviewing and Refining the Appropriateness of a Model

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

Learning Intentions

• To evaluate whether a linear model’s predictions remain reasonable, by comparing with real data and checking the limits of its domain.
• To refine a linear model — by restricting its domain or converting it to a piecewise form — when it no longer fits the situation.

Success Criteria

I can:

1. Explain the difference between interpolation (within the data) and extrapolation (beyond it), and why extrapolation is riskier.
2. Compare a model’s predicted value with a real value and calculate the discrepancy.
3. Give a real-world reason why a linear model might stop fitting a situation (e.g. caps, discounts, thresholds).
4. Refine a model by restricting its domain or splitting it into a piecewise rule.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. A linear model that fits the data perfectly for small inputs will also fit perfectly for very large inputs.
2. Using a model to predict a value far beyond the range of data it was built from is risky.
3. If real data doesn’t exactly match a model’s prediction, the model is useless.
4. A linear model can be improved rather than thrown away entirely.

Answers: 1. Sometimes — many real situations (like postage or data plans) hold linear only up to a threshold, then change; 2. Always — this is called extrapolation, and the assumptions that held near the data may no longer hold far away; 3. Never — small discrepancies are normal; the question is whether the fit is close enough for its purpose; 4. Always — refining (restricting the domain, adjusting a constant, or adding a piece) is usually better than discarding a useful model.

Discussion: Today is Stage 5 of the modelling cycle: Evaluate — checking whether a model is reasonable, and refining it if not.

Activities

Activity 1 — Explicit Instruction: Interpolation, Extrapolation and Checking the Fit (12 min)

I do: FitZone gym data was collected for to and matched the model closely. Question: is it reasonable to use this model to predict the cost at (about years)?

Talk aloud: “Predicting within the range I have data for — say — is interpolation, and I can trust it. Predicting far outside that range — — is extrapolation. Will the weekly fee really stay at 18$ for a decade? Almost certainly not — prices change, and the gym might close, discount long-term members, or the person might stop attending. The model’s straight line doesn’t know any of that.”

We do: Together, compare the phone model against a real bill: for MB, the model predicts:

But the actual bill was 741000$ MB) after which a cheaper bulk rate applies — the model’s constant rate per MB no longer matches reality beyond the cap.

You do: For each pair of model-prediction and real-value data, calculate the discrepancy and decide: good fit, minor refinement needed, or major breakdown.

Scenario	Model prediction	Real value
Taxi fare, km	88.20$	88.20$
Market stall profit,	430$	395s=40$)
Printing cost, pages	124$	95500$ pages)

Activity 2 — Explicit Instruction: Refining the Model (12 min)

I do: Refine the phone plan to account for the data cap. For , the original rate applies; beyond that, a higher rate of cents/MB applies to the excess only.

Check at : 118$. (A different real plan’s numbers would be used to match a real bill — the key skill is the structure of a piecewise refinement, not this specific figure.)

We do: Together, refine a car hire model that originally had no discount, given that data shows every th day is charged at half price. Discuss how the model would need to become piecewise (or use a modified rate) to capture this.

You do: A market stall model (profit per item, no discount) is found to under-predict costs once bulk buyers appear: for , real data shows a discount of 140$. Write the refined piecewise model.

Activity 3 — Inquiry Task: Does the Courier’s Model Still Hold? (11 min)

Pairs, then whole-class share.

A courier’s cost model is for parcels up to kg, based on data collected in that range. The courier’s website shows real prices for larger parcels: kg costs 4515$65$.

Compare these real prices with what the original model predicts. Decide whether the linear model is still appropriate for parcels heavier than kg, and if not, suggest a refinement.

Socratic scaffolding:

Prompt	Purpose
Understand: what is being tested?	Whether the original rate (2+$5$) still applies once mass exceeds the range the model was built from.
What does the model predict for kg and kg?	29C(15) = 2(15)+5 = $35$.
How do these compare with the real prices?	Real prices (45$65$) are much higher — a large, growing discrepancy, not a small rounding difference.
Is this interpolation or extrapolation?	Extrapolation — kg and kg lie outside the – kg range the model was built from.
What might explain a steeper real rate for heavier parcels?	Oversized parcels may need special handling, larger vehicles, or attract a surcharge — a genuinely different per-kg rate.
Devise a plan to refine the model	Keep for ; fit a new, steeper linear piece to the two heavier data points for .
Carry it out	Rate for : dollars/kg. Using : , giving for .
Looking back	Does the refined piecewise model fit both data points reasonably? Check : ✓. The original model should be explicitly restricted to .

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Explain, in one sentence, the difference between interpolation and extrapolation.
2. A model predicts 62$60$. Is this a good fit, a minor refinement, or a major breakdown? Justify.
3. State one real-world reason a linear cost model might stop fitting beyond a certain point.
4. Reasoning. A gym model was built from data for to . A student uses it to predict the cost at . Explain why this prediction should not be trusted.

Answers: 1. Interpolation predicts within the range of the data used to build the model; extrapolation predicts beyond that range; 2. Good fit — a 2$60w=2003.8w=110$ data range — this is extrapolation, and real-world factors like fee changes or discounts make the straight-line prediction unreliable.

Common Misconceptions

Misconception	How to pre-empt it
A model that fits well for the data given will fit well forever.	Explicitly discuss the model’s domain and what lies outside it whenever extrapolation is proposed.
Any deviation between prediction and real data means the model is useless.	Distinguish small, acceptable discrepancies from large, systematic ones using percentage or dollar-value comparisons.
Refining a model means starting completely from scratch.	Show that refining often means restricting the domain or adding one new piece, keeping the original where it still works.
Interpolation and extrapolation are the same thing.	Anchor both terms to a visual: points between known data (interpolation) versus points beyond the known range (extrapolation).
Piecewise models are “not really linear” and therefore invalid.	Clarify that a piecewise model is built from linear pieces, each valid over its own restricted domain — a legitimate refinement, not a different kind of maths.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A model predicts a taxi fare of 5220$58$ due to a night surcharge. What is the discrepancy as a percentage of the model’s prediction?

Answer

E2 (Kangaroo style). A model fits data for to perfectly. For , the real cost is 100$75$. Suggest, without calculating a new rate, whether this is interpolation or extrapolation, and why the difference might arise.

Answer

Extrapolation, since is well beyond the – range the model was built from. A likely cause is a change in rate beyond a threshold, e.g. a bulk surcharge or new pricing tier.

E3 (Challenge). A phone plan model is for and for . Find the cost for MB, and explain why using the original rate for all MB would give the wrong answer.

Answer

Using the original rate for all MB would give 1451000$ MB cap is exceeded.

Homework

1. A model predicts 210$150$. State whether this is a good fit, a minor refinement, or a major breakdown, with reasoning.
2. Give one real-world reason a linear savings model might become inaccurate after several years.
3. A courier model was built from data for to kg. Explain why using it to predict the cost of a kg parcel is risky.
4. A café’s cost model for a catering order is for up to guests, after which a bulk discount of 430$. Write the refined piecewise model.
5. Reasoning. Explain, using the words “interpolation” and “extrapolation,” why a model built from one week’s worth of data should be used cautiously to predict costs a year later.
6. Challenge. A gym model fits real data well for to (one year). In year two, the gym introduces a loyalty discount: every week after is charged at 14$18Mww>52$.

Answers: 1. Major breakdown — a 6040%50110T=120+22g0 \le g \le 30T = 120+22(30)+18(g-30) = 780+18(g-30)g>30M=50+18w1 \le w \le 52M = 50+18(52)+14(w-52) = 986+14(w-52)w>52$.