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D2.5 · Assess, report, and use a mathematical model

Learn to assess, report, and use a mathematical model through clear examples and targeted practice.

Ontario Grade 9 Mathematics

Data

Ontario MTH1W — Data Strand, Expectation D2.5

You have already built mathematical models — equations or graphs that describe a pattern in data. But building a model is only the first step. The real question is: how well does the model actually match the data? In this lesson you will learn how to assess (judge) a model, report what it tells you, and use it to make predictions — all while staying honest about what the model can and cannot do.

What you will learn

Explain what it means to assess a mathematical model against real data.
Calculate and interpret the residual (difference) between a model's prediction and an actual value.
Decide whether a model is a good fit by looking at how small and consistent the residuals are.
Report what a model says about a situation using proper units and context.
Use a model to make a reasonable prediction and recognise its limits.

What Is a Mathematical Model? (Quick Review)

A mathematical model is an equation or graph that represents a real-world relationship. For example, if a student earns CAD 12 per hour at a part-time job, the model for total earnings is

E = 12h

, where

E

is earnings in dollars and

h

is hours worked. The model lets us predict earnings for any number of hours.

Models are built from data — measurements or observations collected in the real world. Because real data always contains some variation (small errors, rounding, or random differences), no model fits every data point perfectly. That is completely normal. The goal is not a perfect fit; the goal is a model that is close enough to be useful.

Before you assess a model, make sure you can identify two things: (1) the predicted value — what the model equation gives you when you substitute a number, and (2) the actual value — the real measurement from the data. You will compare these two values throughout this lesson.

Predicted value = value given by the model equation

A model is an equation or graph representing a real relationship.
Predicted value: what the model calculates.
Actual value: the real-world measurement.
No model fits every data point perfectly, and that is expected.

Assessing a Model: Residuals and Fit

To assess a model means to judge how well it matches real data. The key measurement you use is called the residual. A residual is the difference between the actual value and the predicted value for the same input. In plain language, it tells you how far off the model is at one specific point.

A positive residual means the actual value is higher than the model predicted — the model underestimated. A negative residual means the actual value is lower — the model overestimated. A residual of zero means the model hit that point exactly.

To judge the overall fit of a model, look at all the residuals together. A model is considered a good fit when the residuals are small in size and do not show a clear pattern (for example, they should not all be positive for small inputs and all negative for large inputs, because that would mean the model is consistently off in a predictable direction). When residuals are large, or when they follow a pattern, the model may not be the right type for the data.

At the Grade 9 level, you assess fit visually — by looking at how close the points are to the model line or curve on a graph — and numerically, by calculating individual residuals and comparing their sizes.

r = y_{\text{actual}} - y_{\text{predicted}}

Residual = actual value − predicted value.
A positive residual means the model underestimated; negative means overestimated.
Small residuals with no clear pattern → good fit.
Large residuals or a pattern in residuals → the model may need to be changed.
Assess fit both visually (graph) and numerically (residual calculations).

Reporting What a Model Tells You

Once you have assessed the model and decided it is a reasonable fit, you can report what it says. Reporting a model means explaining the meaning of its equation in plain language, using proper units and context — not just listing numbers.

Consider the equation

E = 12h

. You would not report this as 'E equals 12h.' Instead, you would say: 'According to the model, total earnings increase by CAD 12 for every additional hour worked.' That explanation connects the math to the real situation.

When reporting, always include: (a) what the variables represent, (b) the units for each variable, and (c) a plain-language sentence describing what the model says about how the quantities are related. If the model has a starting value (a constant that is not multiplied by a variable), explain what that starting value means in context too.

Reporting also means being honest about the model's limits. A model built from data in a certain range is reliable only within or close to that range. Reporting a prediction far outside the data range without a warning is misleading.

E = 12h

Report the meaning of each variable with correct units.
Describe the relationship in plain language, not just symbols.
Explain any starting value (constant term) in context.
State the range of data the model was built from.

Using a Model to Make Predictions (and Knowing Its Limits)

Using a model means substituting a value into the equation to make a prediction. There are two types of predictions: interpolation and extrapolation. Interpolation means predicting within the range of the original data — this is generally reliable. Extrapolation means predicting outside that range — this can be unreliable, because the pattern may not continue.

For example, suppose a model for plant growth was built using data collected over 10 days. Using the model to predict growth on day 7 is interpolation — reasonable. Using it to predict growth on day 50 is extrapolation — the model might no longer apply because plants stop growing at some point.

To use a model, substitute the input value into the equation, calculate the output, and then interpret the answer in context. Always write the unit in your answer. Always ask: does this answer make sense in the real situation? If the model predicts a negative number of people at an event, something is wrong — either the model is poor, or you are extrapolating too far.

A strong response when using a model includes the substitution, the calculation, the answer with units, and one sentence explaining what the answer means in the real world.

y_{\text{predicted}} = m x + b

Interpolation (within data range) is more reliable than extrapolation (outside data range).
Always substitute carefully, calculate, and include units in the answer.
Check whether the prediction makes sense in context.
State whether you are interpolating or extrapolating.

Putting It All Together: Assess, Report, Use

When you work with a mathematical model, think of three jobs: assess, report, and use. Assess asks 'Is this model a good fit?' You check residuals and look at the graph. Report asks 'What does this model tell us about the situation?' You explain the equation in plain language with units. Use asks 'What prediction can this model make?' You substitute a value, calculate, and interpret — and you say whether the prediction is inside or outside the data range.

These three jobs are not always done in separate steps. In a written response, you might weave them together: calculate a residual, explain what it means, make a prediction, and add a sentence about reliability. The key is that all three ideas appear clearly in your work.

Remember: a model is a tool, not a perfect description of reality. A good model is honest about its limitations, supported by evidence from the data, and explained clearly so that anyone reading your work understands both what the model says and how confident you are in it.

r = y_{\text{actual}} - y_{\text{predicted}}

Assess: check residuals and graph fit.
Report: explain the equation in plain language with units.
Use: substitute, calculate, interpret, and comment on reliability.
A model is a useful tool, not a perfect truth.

Predicted vs. Actual Values and Residuals (Upload Hours Model)

Week (w)	Actual Hours	Predicted: U = 3w + 4	Residual (Actual − Predicted)
1	7	7	0
2	10	10	0
3	14	13	+1
4	16	16	0
5	19	19	0

Worked example

Assessing, Reporting, and Using a Linear Model for Streaming Data

A student tracked the number of hours of video content uploaded to a school's media server each week for 5 weeks. The data are shown below. A linear model was suggested:

U = 3w + 4

, where

U

is the total hours of content uploaded in week

w

. (a) Calculate the residual for week 3. (b) Calculate the residual for week 5. (c) Report what the model tells us about the situation. (d) Use the model to predict the upload hours in week 8, and comment on reliability.

Data: Week 1: 7 h, Week 2: 10 h, Week 3: 14 h, Week 4: 16 h, Week 5: 19 h.

Step 1 — Find the predicted value for week 3
Substitute $w = 3$ into the model equation $U = 3w + 4$ . This gives the number of hours the model expects in week 3.
$U = 3(3) + 4 = 9 + 4 = 13$
Step 2 — Calculate the residual for week 3
The actual value for week 3 is 14 h. Use the residual formula: actual minus predicted.
$r_3 = 14 - 13 = +1$
Step 3 — Find the predicted value for week 5
Substitute $w = 5$ into the model equation.
$U = 3(5) + 4 = 15 + 4 = 19$
Step 4 — Calculate the residual for week 5
The actual value for week 5 is 19 h. Actual minus predicted.
$r_5 = 19 - 19 = 0$
Step 5 — Assess the model
Both residuals are small (one is +1, one is 0). In a full assessment you would check all five weeks, but these two suggest the model fits well. The residuals do not show a large or consistent error.
$r_3 = +1, \quad r_5 = 0$
Step 6 — Report the model
Explain the equation in plain language. The number 3 is the rate of change: upload hours increase by 3 each week. The number 4 is the starting value: before any full week of uploading, the server had 4 hours of content.
$U = 3w + 4$
Step 7 — Use the model to predict week 8
Substitute $w = 8$ into the model. Week 8 is outside the original data range (weeks 1–5), so this is extrapolation.
$U = 3(8) + 4 = 24 + 4 = 28$

Answer: (a) The residual for week 3 is

+1

hour. (b) The residual for week 5 is

0

hours. (c) The model says upload hours start at 4 and increase by 3 hours each week. (d) The model predicts 28 hours in week 8. This is extrapolation (outside weeks 1–5), so the prediction may be less reliable if the upload rate changes after week 5.

Check: Verify week 1:

U = 3(1) + 4 = 7

. Actual is 7 h. Residual =

7 - 7 = 0

. Verify week 2:

U = 3(2) + 4 = 10

. Actual is 10 h. Residual =

10 - 10 = 0

. All checked residuals are small, confirming the model is a strong fit for this data.

Common mistakes and how to avoid them

Writing the residual as predicted minus actual, which flips the sign.

Correction: Always write residual = actual − predicted. A positive residual means the model was too low; a negative residual means the model was too high.

Saying a model is 'perfect' just because one residual equals zero.

Correction: Check residuals for all data points (or a representative sample). One zero residual does not mean the whole model fits well.

Reporting the model only in symbols (e.g., 'U = 3w + 4') without explaining what the numbers mean.

Correction: Always describe what the rate of change and the starting value mean in the context of the problem, and include units.

Making an extrapolation prediction without any comment about reliability.

Correction: Whenever you predict outside the data range, state that the prediction is an extrapolation and may be unreliable if the pattern changes.

Confusing the actual value with the predicted value when substituting into the residual formula.

Correction: The actual value comes from the original data table; the predicted value comes from calculating the model equation. Keep them clearly labelled.

Lesson summary

A mathematical model is an equation or graph that represents a real-world pattern in data.
The residual measures how far off a model is at one point: residual = actual − predicted.
A good fit has small residuals with no consistent pattern of over- or under-estimating.
Reporting a model means explaining the equation in plain language with correct units and context.
Using a model means substituting a value, calculating, and interpreting the answer — including whether you are interpolating or extrapolating.
Always comment on a model's limitations, especially when predicting outside the original data range.

Check your understanding

Question 1

A model predicts that a plant will be 18 cm tall after 4 weeks. The plant is actually 20 cm tall. What is the residual?

−2 cm
+2 cm
+18 cm
+38 cm

Show answer and explanation

+2 cm

Residual = actual − predicted = 20 − 18 = +2 cm. The positive sign means the model underestimated the plant's height.

Question 2

A student uses the model

C = 5n + 10

to predict the cost (in CAD) of ordering

n

items. Which statement correctly reports what the number 5 means?

The total cost is always CAD 5.
There is a one-time starting cost of CAD 5.
Each additional item costs CAD 5 more.
The cost increases by CAD 10 for every 5 items.

Show answer and explanation

Each additional item costs CAD 5 more.

In the model

C = 5n + 10

, the number 5 is the rate of change — it means the cost goes up by CAD 5 for every additional item ordered. The 10 is the starting cost (the cost when zero items are ordered).

Question 3

A model was built using data from months 1 to 6. A student uses it to predict the value in month 20. What is this type of prediction called, and how reliable is it?

Interpolation; very reliable because the model was tested.
Extrapolation; less reliable because month 20 is outside the original data range.
Interpolation; less reliable because the residuals might be large.
Extrapolation; completely reliable as long as the equation is correct.

Show answer and explanation

Extrapolation; less reliable because month 20 is outside the original data range.

Month 20 is outside the original data range of months 1–6, so this is extrapolation. Extrapolation is less reliable because the pattern observed in the data may not continue beyond the range that was measured.

Question 4

A model's residuals for six data points are: +8, +9, −1, +7, +10, +8. What does this pattern of residuals most strongly suggest?

The model is a perfect fit for the data.
The model fits well because some residuals are positive and one is negative.
The model is mostly overestimating — all but one residual is large and positive.
The model needs more data points before any assessment is possible.

Show answer and explanation

The model is mostly overestimating — all but one residual is large and positive.

Almost all residuals are large and positive, which means the actual values are consistently much higher than what the model predicts. This shows the model is consistently underestimating, which is a sign of poor fit — not a good model.

Key terms

Mathematical model: An equation or graph that uses numbers and variables to represent a real-world relationship or pattern.
Predicted value: The output you get when you substitute an input into the model equation.
Actual value: The real measurement taken from data, as opposed to what the model calculates.
Residual: The difference between the actual value and the predicted value at the same input: residual = actual − predicted.
Assess (a model): To judge how well a model fits the data, using residuals and graphs.
Interpolation: Making a prediction for an input value that falls within the range of the original data.
Extrapolation: Making a prediction for an input value that falls outside the range of the original data; generally less reliable.
Rate of change: How much the output variable changes for every one-unit increase in the input variable; the slope of a linear model.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation D2.5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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