207. Building Models from Data

Learning Intentions

• Use data points to create simple linear or quadratic models.
• Substitute values into a model to make predictions.
• Check whether predictions are reasonable for the context.

Pre-requisite Summary

• Know that a data point can be written as an ordered pair .
• Know that a table of values can show the relationship between an input and an output.
• Know that a linear model has constant first differences and can be written in the form .
• Know that a quadratic model has changing first differences and may involve a squared term such as .
• Know that substituting means replacing a variable with a given value.
• Know that predictions should be checked against the context, including whether values such as time, length, cost or height make sense.

Worked Examples

Worked Example 1

Use the table to create a linear model.

a) Find the first differences.

b) Write a rule in the form .

c) Explain what the data point tells you.

Worked Example 2

Use the data points to create a linear model.

A delivery cost is $ when the distance is km and $ when the distance is km.

a) Write two data points.

b) Find the rate of change.

c) Write a rule for the cost after kilometres.

Worked Example 3

Use the table to decide whether a linear or quadratic model is more appropriate.

a) Find the first differences.

b) Find the second differences.

c) Choose a linear or quadratic model.

Worked Example 4

Use the data points to create a simple quadratic model of the form .

a) Use the data point where to find .

b) Write the quadratic model.

c) Check the model using .

Worked Example 5

Use the data points to create a simple quadratic model of the form .

a) Use the data point to find .

b) Write the quadratic model.

c) Check the model using .

Worked Example 6

Use the linear model to make predictions.

a) Predict the cost when .

b) Predict the cost when .

c) Interpret each prediction in context.

Worked Example 7

Use the quadratic model to make predictions.

a) Predict when .

b) Predict when .

c) Interpret each prediction in context.

Worked Example 8

Check whether the prediction is reasonable.

A ball’s height is modelled by , where is height in metres and is time in seconds.

a) Predict the height when .

b) Predict the height when .

c) Decide whether each prediction is reasonable in context.

Problems

Problem 1

Use the table to create a linear model.

a) Find the first differences.

b) Write a rule in the form .

c) Explain what the data point tells you.

Problem 2

Use the data points to create a linear model.

A taxi cost is $ when the distance is km and $ when the distance is km.

a) Write two data points.

b) Find the rate of change.

c) Write a rule for the cost after kilometres.

Problem 3

Use the table to decide whether a linear or quadratic model is more appropriate.

a) Find the first differences.

b) Find the second differences.

c) Choose a linear or quadratic model.

Problem 4

Use the data points to create a simple quadratic model of the form .

a) Use the data point where to find .

b) Write the quadratic model.

c) Check the model using .

Problem 5

Use the data points to create a simple quadratic model of the form .

a) Use the data point to find .

b) Write the quadratic model.

c) Check the model using .

Problem 6

Use the linear model to make predictions.

a) Predict the cost when .

b) Predict the cost when .

c) Interpret each prediction in context.

Problem 7

Use the quadratic model to make predictions.

a) Predict when .

b) Predict when .

c) Interpret each prediction in context.

Problem 8

Check whether the prediction is reasonable.

A ball’s height is modelled by , where is height in metres and is time in seconds.

a) Predict the height when .

b) Predict the height when .

c) Decide whether each prediction is reasonable in context.

Exercises

Understanding and Fluency

Exercise 1

Use each table to create a linear model.

a)

b)

Exercise 2

Use the data points to create a linear model.

a) A cost is $ when and $ when .

b) A savings balance is $ when and $ when .

c) A tank has L when and L when .

Exercise 3

For each table, decide whether a linear or quadratic model is more appropriate.

a)

b)

Exercise 4

Find the first differences and second differences for each table.

a)

b)

Exercise 5

Use the data points to create a quadratic model of the form .

a)

b)

Exercise 6

Use the data points to create a quadratic model of the form .

a)

b)

Exercise 7

Substitute the given value into each linear model.

a) when

b) when

c) when

d) when

Exercise 8

Substitute the given value into each quadratic model.

a) when

b) when

c) when

d) when

Exercise 9

Use each model to make a prediction and include units.

a) , where is cost in dollars and tickets.

b) , where is savings in dollars and weeks.

c) , where is area in square metres and metres.

Exercise 10

State whether each prediction is reasonable.

a) A cost model predicts $ after weeks.

b) A height model predicts a ball is metres above the ground after seconds.

c) A square area model predicts when the side length is .

d) A savings model predicts $ after weeks.

Reasoning

Exercise 11

Explain why the table below is better modelled by a linear model.

Exercise 12

Explain why the table below is better modelled by a quadratic model.

Exercise 13

A student creates the model from the table.

Explain why the model is correct.

Exercise 14

A student says the table below should use a linear model because the values increase.

Explain why this reasoning is incomplete.

Exercise 15

A model predicts that a phone bill after months will be $ .

Explain why this prediction might be mathematically correct but not useful in context.

Exercise 16

Decide whether each statement is true or false. Justify your answer.

a) Substitution means replacing a variable with a value.

b) A model that works for known data always gives reasonable predictions for every possible input.

c) A negative prediction can be unreasonable when the quantity represents height, money or distance.

Problem-solving

Exercise 17

A car hire company charges a fixed booking fee and a constant amount per day.

Days
Cost	$	$	$	$

a) Create a linear model for the cost.

b) Predict the cost for days.

c) Check whether the prediction is reasonable in context.

Exercise 18

The area of a square display is shown in the table.

Side length
Area

a) Create a quadratic model for the area.

b) Predict the area when .

c) Explain why the prediction is reasonable only if the display remains square.

Exercise 19

A ball’s height is modelled by .

a) Predict the height when .

b) Predict the height when .

c) Explain which prediction is more reasonable and why.

Exercise 20

Create your own modelling problem.

Your response must include:

• a table of at least four data points
• a decision about whether a linear or quadratic model is more appropriate
• a model rule
• a prediction using substitution
• a sentence checking whether the prediction is reasonable in context

Potential Misunderstandings

• Students may create a rule from only one data point and not check it against the rest of the table.
• Students may choose a linear model whenever the values increase, even if the first differences are not constant.
• Students may choose a quadratic model because the numbers are large, rather than because the pattern involves squared growth or constant second differences.
• Students may substitute into the wrong variable or use the output value as the input.
• Students may forget to follow order of operations when substituting into quadratic models.
• Students may calculate a prediction correctly but forget to include units.
• Students may think every prediction from a model is automatically reasonable.
• Students may ignore context restrictions such as negative time, negative height, negative length or unrealistic future values.
• Students may use a model far outside the data range without considering whether extrapolation is sensible.

Next: 208. Comparing Financial Options