4. Linear Modeling Projects

Linear Modeling Projects

A linear modeling project uses real or collected data to investigate whether two variables have an approximately linear relationship. Instead of being given an equation, you begin with observations or measurements and develop a mathematical model from the evidence.

A complete project usually follows this process:

question → variables → data collection → table → graph → linear model → predictions → evaluation → conclusion

This brings together many important mathematical skills:

• collecting and organizing data
• identifying variables
• constructing tables and graphs
• calculating rates of change
• writing linear equations
• making predictions
• interpreting slope and intercept
• evaluating model accuracy
• communicating conclusions using mathematical evidence

The goal is not simply to find an equation. A strong project explains what the equation means, how well it represents the data, and when it should or should not be trusted.

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Beginning with a Research Question

A good modeling project begins with a clear, measurable question involving two variables.

Examples include:

• How does walking distance change with time?
• How does the height of a burning candle change with time?
• How does the amount of water in a container change as water is added?
• How does taxi cost change with distance?
• How does the circumference of a circular object relate to its diameter?
• How does the total mass of identical objects change with the number of objects?
• How does the length of a spring change as load is added within its elastic range?

A useful question identifies quantities that can actually be measured or recorded.

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Choosing a Question Suitable for Linear Modeling

Not every investigation should produce a linear relationship.

A good candidate for linear modeling usually involves a quantity changing at an approximately constant rate.

For example:

How does distance travelled change with time when a cart moves at approximately constant speed?

If the speed remains approximately constant:

distance = speed × time

The relationship should be linear.

In contrast:

How does the area of a square change with side length?

is not linear because:

A = s²

A linear model would not appropriately represent the complete relationship.

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Defining the Variables

Every project should clearly define its variables.

Suppose the investigation asks:

How does the amount of water in a container change as water is added?

Possible variables are:

x = number of cups of water added

y = total volume of water in mL

The variables should include:

• a clear name
• a symbol
• appropriate units

This makes later graphs and equations much easier to interpret.

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Independent and Dependent Variables

The independent variable is the quantity deliberately changed, selected, or used as the input.

The dependent variable is the quantity measured or observed in response.

For a walking experiment:

Independent variable: time (s)

Dependent variable: distance travelled (m)

Distance depends on how much time has passed.

In most graphs:

• independent variable → x-axis
• dependent variable → y-axis

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Planning Data Collection

Before collecting data, decide:

• what will be measured
• which units will be used
• how measurements will be made
• how many observations will be collected
• what range of values will be investigated
• how results will be recorded
• whether repeated measurements are needed

Good planning improves the reliability of the resulting model.

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Collecting Enough Data

A linear model should not normally be based on only two measurements.

Two points always determine a straight line mathematically, but they do not provide much evidence that the real relationship is actually linear.

A project should usually collect several data points across a useful range.

For example:

The data does not need to be perfectly linear.

Real measurements commonly contain variation.

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Why Real Data Is Often Imperfect

Even when the underlying relationship is linear, measurements may not lie perfectly on one line.

Variation can result from:

• measurement uncertainty
• reaction time
• instrument precision
• environmental changes
• inconsistent technique
• natural variation
• recording mistakes

A mathematical model attempts to represent the overall pattern, not necessarily every individual measurement exactly.

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Organizing Data in a Table

Data should be organized clearly before it is graphed.

A good table should contain:

• variable names
• units
• consistent precision where appropriate
• observations in a logical order

For example:

A reader should be able to understand the data without needing additional explanation.

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Looking for a Linear Pattern in a Table

One way to investigate linearity is to compare rates of change.

Using:

(2, 3.0)

and:

(4, 6.2)

the approximate rate is:

m = (6.2 − 3.0)/(4 − 2)

m = 3.2/2

m = 1.6 m/s

Using:

(6, 9.1)

and:

(8, 12.3)

m = (12.3 − 9.1)/(8 − 6)

m = 3.2/2

m = 1.6 m/s

Similar rates provide evidence that the relationship may be approximately linear.

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Constant Differences

If x increases by equal amounts, linear data should have approximately constant changes in y.

For example:

The first differences are:

+3, +3, +3, +3

This is exactly linear.

Real experimental data might instead produce:

+3.1, +2.8, +3.2, +2.9

The relationship could still be approximately linear.

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Unequal x-Intervals

Constant differences in y should only be compared directly when the changes in x are equal.

Consider:

The changes in y are:

+6 and +12

They are not equal.

But the x changes are:

+2 and +4

Calculate rates:

6/2 = 3

12/4 = 3

Therefore:

m = 3

The relationship is linear despite the unequal changes in y.

Always use:

m = Δy/Δx

when x-intervals differ.

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Creating a Scatter Plot

A scatter plot displays collected data as individual points.

Each point represents one observation:

(independent variable, dependent variable)

Scatter plots are especially useful for deciding whether data follows an approximately linear pattern.

Look for:

• overall direction
• how closely points follow a straight-line pattern
• unusual points
• clusters
• possible curvature

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Positive Linear Relationships

A positive linear relationship generally rises from left to right.

As x increases:

y tends to increase

Examples might include:

• time and distance at constant positive speed
• number of identical items and total cost
• hours worked and earnings
• water added and total volume

The slope is positive.

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Negative Linear Relationships

A negative linear relationship generally falls from left to right.

As x increases:

y tends to decrease

Examples might include:

• time and water remaining in a draining tank
• time and candle height while burning steadily
• distance travelled and fuel remaining under simplified conditions

The slope is negative.

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Weak or Non-Linear Relationships

If the data points do not form an approximately straight pattern, a linear model may not be appropriate.

For example, the data might:

• curve upward
• curve downward
• level off
• fluctuate randomly
• form separate groups

A project should not force a linear equation onto clearly non-linear data simply because the assignment involves linear modeling.

Recognizing that a linear model is not appropriate can itself be a valid mathematical conclusion.

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Creating a Line of Best Fit

Real data often does not lie exactly on one line.

A line of best fit is a straight line that represents the overall trend.

A reasonable line should generally:

• pass through the middle of the data
• follow the overall direction
• have points distributed on both sides
• avoid being controlled by one unusual point

The line does not need to pass through every data point.

In fact, it may pass through very few actual points.

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Estimating a Line of Best Fit

Suppose a scatter plot suggests that a reasonable line passes near:

(2, 4)

and:

(10, 20)

Estimate the slope:

m = (20 − 4)/(10 − 2)

m = 16/8

m = 2

If the line crosses the y-axis near:

0

then a possible model is:

y = 2x

This model describes the overall trend rather than every individual measurement.

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Choosing Points for the Model

When estimating an equation from a line of best fit, choose two clear points on the fitted line.

They do not necessarily need to be original data points.

Using points far apart on the fitted line often reduces the effect of small reading errors when calculating slope.

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Writing the Linear Model

Once the slope and intercept are estimated, use:

y = mx + b

Suppose:

m ≈ 2.4

b ≈ 5

Then:

y ≈ 2.4x + 5

The symbol ≈ is useful when the equation is an approximate model rather than an exact relationship.

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Interpreting the Slope

The slope should always be explained in context.

Suppose:

d ≈ 1.5t + 0.2

where:

d = distance in metres

t = time in seconds

The slope is:

1.5 m/s

A useful interpretation is:

The object's distance increases by approximately 1.5 m for every additional second.

This is much more informative than simply stating:

m = 1.5

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Interpreting the Intercept

For:

d ≈ 1.5t + 0.2

the intercept is:

0.2 m

This means the model predicts that at:

t = 0

the distance is approximately:

0.2 m

Depending on the experiment, this might represent:

• an actual starting position
• measurement uncertainty
• an offset in the experimental setup
• limitations in the fitted model

The intercept should be interpreted rather than simply reported.

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When the Intercept Has Little Practical Meaning

Suppose a model relating height to age is:

h = 4.8a + 75

Mathematically, the intercept is:

75 cm

But if the data was collected only from children aged 8–12 years, using the equation at age zero may not be appropriate.

The y-intercept is mathematically part of the line but may lie outside the meaningful domain of the data.

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Using the Model to Make Predictions

Suppose a model is:

d = 1.5t + 0.2

Predict the distance after:

t = 12 s

Substitute:

d = 1.5(12) + 0.2

d = 18 + 0.2

d = 18.2 m

The model predicts a distance of approximately:

18.2 m

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Interpolation

An interpolation is a prediction within the range of collected data.

Suppose measurements were collected from:

0 to 20 seconds.

Predicting the distance at:

13 seconds

is interpolation.

Because the prediction lies within the measured range, it is often relatively reliable if the model fits the data well.

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Extrapolation

An extrapolation is a prediction outside the measured range.

If data was collected from:

0 to 20 seconds

and the model is used to predict the value at:

60 seconds,

that is extrapolation.

The prediction assumes the relationship continues beyond the region where it was observed.

This assumption may be incorrect.

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Evaluating Prediction Reliability

Ask:

• Is the prediction inside the data range?
• How far outside the data range is it?
• Does the linear trend have a reason to continue?
• Are there physical limits?
• Could conditions change?
• Is the predicted value realistic?

A calculated prediction should never automatically be accepted simply because it came from an equation.

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Comparing Predicted and Observed Values

Suppose the model predicts:

24.5

but an actual measurement is:

25.2

The difference is:

25.2 − 24.5 = 0.7

This difference helps us evaluate the model.

A model does not need to predict every measurement exactly to be useful.

The important question is whether the errors are reasonably small for the purpose of the investigation.

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Residuals

A residual measures the difference between an observed value and the model's prediction.

Use:

residual = observed value − predicted value

Suppose:

observed = 18.7

predicted = 18.2

Then:

residual = 18.7 − 18.2

residual = 0.5

The positive residual means the observed value lies 0.5 units above the model's prediction.

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Interpreting Residuals

If:

residual > 0

the observed value is above the prediction.

If:

residual < 0

the observed value is below the prediction.

If:

residual = 0

the model predicts the observation exactly.

Smaller residual magnitudes generally indicate that the model fits the observations more closely.

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Residual Patterns

Residuals should not merely be examined individually.

Look for patterns.

If residuals appear randomly above and below zero, a linear model may be reasonable.

If residuals show a clear pattern such as:

positive → negative → positive

or steadily become larger,

the relationship may not be truly linear.

A curved pattern in residuals is evidence that another model may fit better.

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Outliers

An outlier is a data point that lies unusually far from the general pattern.

Possible causes include:

• measurement error
• recording error
• unusual experimental conditions
• equipment problems
• natural variation
• a genuinely unusual observation

Do not automatically remove an outlier simply because it does not fit the desired model.

Investigate it first.

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How Outliers Affect a Model

An outlier can change:

• the estimated slope
• the estimated intercept
• the apparent strength of the relationship
• predictions made from the model

A strong project should mention important outliers and discuss their possible effect.

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Evaluating the Strength of a Linear Relationship

A linear relationship appears stronger when points lie close to a straight-line pattern.

A relationship appears weaker when points are widely scattered.

However, visual appearance depends partly on graph scale.

Therefore, conclusions should use several forms of evidence where possible:

• scatter plot
• consistency of rates of change
• residuals
• contextual reasoning

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Correlation Does Not Prove Causation

Suppose two variables show a strong linear relationship.

This does not automatically prove that one causes the other.

For example, during summer:

• ice cream sales may increase
• electricity use for air conditioning may increase

The variables may be correlated because both are influenced by temperature.

A modeling project should distinguish between:

association

and:

causation

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Collecting Reliable Data

Good models depend on good data.

To improve reliability:

• use appropriate measuring instruments
• use consistent methods
• collect enough observations
• use a useful range of values
• repeat measurements when appropriate
• record data carefully
• use consistent units
• avoid changing multiple conditions unnecessarily

Poor data can produce a poor model even when the mathematics is performed correctly.

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Repeated Measurements

Suppose a student measures travel time three times:

2.1 s

2.3 s

2.2 s

The mean is:

(2.1 + 2.3 + 2.2)/3

= 6.6/3

= 2.2 s

Repeated measurements can reduce the influence of random variation.

They also provide information about how consistent the measurements are.

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Range of Data

Suppose a student wants to investigate the relationship between mass and spring extension but only measures:

100 g

101 g

102 g

103 g

The range is extremely narrow.

Even if the measurements appear linear, they provide little evidence about the relationship over a wider range.

A well-designed investigation normally uses values spread across the meaningful domain.

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Sample Size

Using more observations can provide stronger evidence about a pattern.

A model based on:

2 data points

is much less convincing than a model based on:

15 well-collected observations

Two points always form a line, so they cannot demonstrate by themselves that a real relationship is genuinely linear.

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Precision and Accuracy

Precision refers to how closely repeated measurements agree with one another.

Accuracy refers to how close measurements are to the true or accepted value.

A dataset can be:

• precise and accurate
• precise but inaccurate
• imprecise but approximately accurate on average
• neither precise nor accurate

Measurement quality affects the model created from the data.

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Project Example: Walking Speed

Research question:

How does distance travelled relate to time when a person walks at a steady pace?

Define:

t = time in seconds

d = distance in metres

Collect distance measurements at regular time intervals.

A possible dataset might be:

The data shows an approximately constant increase.

A linear model is likely appropriate.

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Creating the Walking Model

Suppose the line of best fit has approximately:

m = 1.35

b = 0

Then:

d ≈ 1.35t

Interpretation:

The walker travels approximately 1.35 metres each second.

This model can be used to estimate distance for other time values within a reasonable range.

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Evaluating the Walking Model

Strengths might include:

• data follows a clear straight-line trend
• several measurements were collected
• residuals are small
• slope has a meaningful physical interpretation

Limitations might include:

• walking speed may not remain perfectly constant
• reaction time may affect measurements
• only a short time interval was tested
• the model should not be extrapolated indefinitely

A balanced evaluation discusses both.

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Project Example: Filling a Container

Research question:

How does water volume change with filling time when the flow rate is approximately constant?

Define:

t = filling time in seconds

V = volume in mL

Suppose a container initially contains:

100 mL

and water enters at approximately:

25 mL/s

A possible model is:

V ≈ 25t + 100

Slope:

25 mL/s

Intercept:

100 mL

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Evaluating the Container Model

The model may work well until the container reaches its capacity.

Suppose the container holds:

600 mL.

Find when the model reaches capacity:

600 = 25t + 100

500 = 25t

t = 20 s

Beyond 20 seconds, the model cannot describe the amount of water inside the container because the container is full.

This is a model limitation.

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Project Example: Candle Height

Research question:

How does candle height change with burning time?

Suppose:

h = candle height in cm

t = time in hours

A possible model is:

h ≈ −1.8t + 24

Slope:

−1.8 cm/hour

The candle becomes approximately 1.8 cm shorter each hour.

Intercept:

24 cm

The estimated initial height is 24 cm.

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Evaluating the Candle Model

Possible strengths:

• clear decreasing trend
• approximately constant burn rate
• simple interpretation

Possible limitations:

• burn rate may change
• wick conditions may change
• measurements may contain error
• the model cannot continue after the candle is consumed

The equation is useful only within a meaningful domain.

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Project Example: Circumference and Diameter

Measure the diameter and circumference of several circular objects.

Define:

d = diameter

C = circumference

Theory predicts:

C = πd

Therefore, a graph of circumference against diameter should be approximately linear.

Its slope should be approximately:

π ≈ 3.14

and the intercept should be close to:

0

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Comparing Experimental Results with Theory

Suppose the experimental model is:

C ≈ 3.09d + 0.3

The theoretical relationship is:

C = πd

The experimental slope:

3.09

is reasonably close to:

3.14

The small non-zero intercept might result from:

• measurement uncertainty
• difficulty measuring circumference
• instrument limitations
• imperfect circular objects

This comparison provides mathematical evidence for evaluating the experiment.

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Communicating Mathematical Evidence

A conclusion should not simply say:

"The data was linear."

A stronger statement might say:

"The scatter plot showed a clear positive linear trend. The fitted model was d ≈ 1.35t, meaning distance increased by approximately 1.35 m each second. Most measured values were close to the model, with small residuals, so the linear model represented the collected data reasonably well over the tested interval."

This uses evidence rather than opinion.

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Evidence That Can Support a Conclusion

Useful mathematical evidence includes:

• tables
• graphs
• slope
• intercept
• equation
• calculated rates
• residuals
• comparisons between observed and predicted values
• discussion of outliers
• domain restrictions
• comparison with theoretical values

A strong report explains what the evidence means.

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Graph Requirements for a Project

A good graph should include:

• descriptive title
• independent variable on the x-axis
• dependent variable on the y-axis
• variable names
• units
• appropriate scale
• accurately plotted points
• line of best fit when appropriate

Avoid connecting measured scatter points one-by-one unless there is a specific reason to do so.

A scatter plot and fitted line usually communicate the relationship more clearly.

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Tables in a Project Report

A data table should:

• clearly identify variables
• include units in headings
• use consistent formatting
• contain all relevant observations
• avoid unnecessary information

If repeated trials were performed, the table might include:

The mean values could then be used for the primary graph.

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Strengths of a Linear Model

Possible strengths include:

• simple to calculate
• easy to graph
• easy to interpret
• useful for interpolation
• slope has a clear real-world meaning
• fits the observed data closely
• allows comparisons and predictions

A strength should be connected to the actual project rather than listed generically.

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Limitations of a Linear Model

Possible limitations include:

• relationship may only be approximately linear
• measurements contain uncertainty
• sample size may be small
• data range may be narrow
• outliers may influence the model
• conditions may change
• model may fail outside the tested range
• physical limits may restrict the domain
• other variables may affect the relationship

A good evaluation identifies limitations specific to the investigation.

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Improving the Investigation

After identifying limitations, suggest realistic improvements.

Examples include:

• collect more data points
• increase the range of measurements
• repeat measurements
• use more precise instruments
• control additional variables
• use automated timing
• improve measurement technique
• investigate unusual data points
• repeat the experiment under different conditions

An improvement should address a specific weakness.

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Weak vs Strong Evaluation

A weak evaluation might say:

"The experiment could be better."

A stronger evaluation might say:

"Only five measurements were collected over a narrow 4-second interval, making it difficult to determine whether the linear pattern continues. Collecting measurements over a longer time range would provide stronger evidence for the model."

The second statement identifies:

• the limitation
• why it matters
• how to improve it

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Project Conclusions

A good conclusion should answer the original research question.

It should include:

• whether a linear relationship was found
• direction of the relationship
• equation of the model
• interpretation of slope
• interpretation of intercept when meaningful
• evidence supporting the model
• one or more predictions
• assessment of reliability
• important limitations

The conclusion should be based on the collected evidence.

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Suggested Project Structure

A complete linear modeling project can be organized as follows:

Research Question

State the question clearly.

Variables

Define the independent and dependent variables and units.

Prediction

Describe the expected relationship and explain why.

Method

Explain how data will be collected.

Raw Data

Present measurements clearly.

Processed Data

Include means or calculated quantities if needed.

Graph

Create a scatter plot and appropriate line of best fit.

Linear Model

State the equation.

Interpretation

Explain the slope and intercept.

Prediction

Use the model to calculate a new value.

Evaluation

Discuss model accuracy, residuals, outliers, assumptions, and limitations.

Conclusion

Answer the research question using mathematical evidence.

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Possible Linear Modeling Project Ideas

Suitable investigations could include:

• distance travelled vs time while walking
• water volume vs filling time
• water remaining vs draining time
• candle height vs burning time
• circumference vs diameter
• total mass vs number of identical objects
• total cost vs number of identical items
• spring extension vs load within the elastic region
• temperature change over a short controlled interval
• number of stacked objects vs total height

Projects should be chosen so that measurements can be made safely and reliably.

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A Complete Mini-Project Example

Question: How does the total height of identical blocks depend on the number of blocks stacked?

Define:

n = number of blocks

H = total height in cm

Suppose the measurements are:

The data shows an approximately linear relationship.

A reasonable model is:

H ≈ 2n

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Interpreting the Block Model

The slope is approximately:

2 cm/block

This means each additional block increases the stack height by approximately 2 cm.

The intercept is approximately:

0 cm

This makes sense because a stack containing zero blocks should have approximately zero height.

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Making a Prediction

Predict the height of:

10 blocks

Using:

H ≈ 2n

H ≈ 2(10)

H ≈ 20 cm

Because 10 blocks is reasonably close to the measured range, this may be a useful prediction if the blocks remain identical and are stacked in the same way.

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Evaluating the Mini-Project

Strengths:

The measurements follow a clear linear pattern, the slope has a meaningful interpretation, and the intercept is physically reasonable.

Limitations:

Small measurement errors may occur, the blocks may not be perfectly identical, and a very tall stack could become unstable.

Possible improvement:

Measure each stack several times or use a more precise measuring instrument.

This completes the modeling cycle:

collect → represent → model → predict → evaluate

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A Reliable Linear Modeling Workflow

For a complete project:

1. Write a clear research question.
2. Identify measurable variables.
3. Define variables and units.
4. Predict the expected relationship.
5. Plan a consistent method.
6. Collect several data points.
7. Repeat measurements when appropriate.
8. Organize data in a table.
9. Create a scatter plot.
10. Look for an approximately linear pattern.
11. Identify possible outliers.
12. Create a line of best fit.
13. Determine its slope.
14. Determine its intercept.
15. Write the linear model.
16. Interpret the slope and intercept.
17. Use the model to make a prediction.
18. Identify whether the prediction is interpolation or extrapolation.
19. Compare predictions with observed values when possible.
20. Evaluate residuals and model fit.
21. Identify assumptions and limitations.
22. Suggest realistic improvements.
23. State a conclusion supported by mathematical evidence.

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Common Mistakes

Assuming the relationship must be linear

The data should provide evidence for linearity.

Using only two data points

Two points always define a line and provide weak evidence about the real relationship.

Connecting every scatter point

Measured data should usually be represented with a scatter plot and line of best fit rather than a zigzag line connecting every observation.

Forcing the line through the origin

Only use an intercept of zero when the data or context supports it.

Choosing only convenient data

All valid observations should be considered, including unexpected results.

Removing outliers without justification

Investigate unusual measurements before deciding whether they should be excluded.

Ignoring units

Slope must have meaningful units.

Reporting an equation without interpretation

Explain what the slope and intercept mean.

Assuming predictions are exact

A fitted model usually produces estimates.

Extrapolating too far

The relationship may change outside the measured range.

Ignoring residual patterns

Systematic residual patterns may indicate that a linear model is inappropriate.

Listing generic limitations

Limitations should relate directly to the investigation.

Making claims without evidence

Use measurements, calculations, graphs, equations, and residuals to support conclusions.

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Key Terms

Linear modeling project — investigation that uses data to create and evaluate a linear mathematical model.

Research question — measurable question guiding an investigation.

Independent variable — input variable placed on the x-axis.

Dependent variable — measured output variable placed on the y-axis.

Dataset — collection of observations or measurements.

Scatter plot — graph showing individual paired data values.

Linear relationship — relationship with a constant or approximately constant rate of change.

Line of best fit — line representing the overall trend of approximately linear data.

Linear model — equation used to represent or approximate a linear relationship.

Slope — rate of change represented by the model.

Intercept — predicted value of the dependent variable when the independent variable is zero.

Prediction — estimated value produced using a model.

Interpolation — prediction within the observed data range.

Extrapolation — prediction outside the observed data range.

Observed value — actual measured value.

Predicted value — value calculated from a model.

Residual — observed value minus predicted value.

Outlier — observation unusually far from the general pattern.

Accuracy — closeness to a true or accepted value.

Precision — consistency among repeated measurements.

Limitation — factor restricting the reliability or usefulness of an investigation or model.

Assumption — condition accepted when constructing or applying a model.

Domain — input values for which a model is intended to apply.

Mathematical evidence — numerical, graphical, or algebraic information used to support a conclusion.

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Key Takeaways

• Linear modeling projects begin with data rather than simply being given an equation.
• A strong project starts with a clear, measurable research question.
• Independent and dependent variables should be clearly defined with units.
• Reliable data collection requires careful planning and consistent measurement methods.
• Several observations provide stronger evidence than only two points.
• A scatter plot helps determine whether a relationship is approximately linear.
• Constant or approximately constant rates of change provide evidence for linearity.
• When x-intervals are unequal, compare Δy/Δx rather than changes in y alone.
• Real-world data usually contains some variation.
• A line of best fit represents the overall trend rather than every individual point.
• The linear model can be written in the form y ≈ mx + b when it approximates measured data.
• Slope should be interpreted as a real-world rate with appropriate units.
• The intercept should be interpreted carefully and may not always have practical meaning.
• Models can be used to make predictions.
• Interpolation is usually more reliable than extrapolation.
• Residuals measure differences between observed and predicted values.
• Randomly distributed small residuals can support the use of a linear model.
• Patterns in residuals may indicate that a different model would be more appropriate.
• Outliers should be investigated rather than automatically removed.
• The quality of a model depends partly on the quality and range of the collected data.
• A strong evaluation discusses both strengths and limitations.
• Improvements should directly address identified weaknesses.
• Correlation does not automatically establish causation.
• Models are useful simplifications rather than perfect descriptions of reality.
• Strong conclusions use mathematical evidence rather than unsupported statements.
• A complete project should collect, organize, graph, model, predict, interpret, evaluate, and communicate.

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Check Your Understanding

1. Explain why collecting only two data points provides weak evidence that a real-world relationship is linear.
2. A student investigates how distance travelled changes with time. Identify the likely independent variable, dependent variable, and appropriate units.
3. Consider the data:

Explain why a linear model may be appropriate and estimate the rate of change.

4. Explain why a scatter plot is usually more appropriate than connecting every measured data point with straight segments in a linear modeling investigation.
5. A fitted model is:

V ≈ 18t + 120

where V is volume in mL and t is time in seconds. Interpret the slope and intercept in context.

6. A model predicts 42.6 while the measured value is 44.1. Calculate the residual using observed − predicted and interpret the result.
7. Data was collected for x-values from 0 to 20. Explain why predicting at x = 12 is generally more reliable than predicting at x = 100.
8. A scatter plot contains one point far from the general linear pattern. Give three possible explanations for the outlier and explain what should be done before removing it.
9. A model has small residuals, but every residual is positive. What might this suggest about the model?
10. Explain why a curved pattern in a scatter plot would provide evidence against using a linear model.
11. A student measures the circumference and diameter of circular objects and obtains the model:

C ≈ 3.11d + 0.2

Compare this with the theoretical relationship C = πd. Discuss what the slope and intercept suggest about the experiment.

12. A student states, "My model is good because my graph looks straight." Identify at least three additional pieces of mathematical evidence that could strengthen this conclusion.
13. Design a simple linear modeling investigation using materials available in a classroom. State the research question, independent variable, dependent variable, units, and the data you would collect.
14. A model fits measurements very closely between x = 2 and x = 12. A student claims that it must therefore remain accurate until x = 10,000. Evaluate this conclusion.
15. Imagine you have completed a project investigating how the height of a candle changes with burning time. Describe what should be included in a strong final conclusion, including the equation, slope interpretation, evidence for linearity, prediction, strengths, limitations, and possible improvements.