Lesson 9Linear Models
Learning Goal
Let’s model situations with linear functions.
Learning Targets
I can decide when a linear function is a good model for data and when it is not.
I can use data points to model a linear function.
Warm Up: Candlelight
Problem 1
A candle is burning. It starts out 12 inches long. After 1 hour, it is 10 inches long. After 3 hours, it is 5.5 inches long.
This tool is here for you to use if you choose. To plot a point, type its coordinates. For example, try typing
When do you think the candle will burn out completely?
Is the height of the candle a function of time? If yes, is it a linear function? Explain your thinking.
Print Version
A candle is burning. It starts out 12 inches long. After 1 hour, it is 10 inches long. After 3 hours, it is 5.5 inches long.
When do you think the candle will burn out completely?
Is the height of the candle a function of time? If yes, is it a linear function? Explain your thinking.
Activity 1: Shadows
Problem 1
When the sun was directly overhead, the stick had no shadow. After 20 minutes, the shadow was 10.5 cm long. After 60 minutes, it was 26 cm long.
This tool is here for you to use if you choose. To plot a point, type its coordinates. For example, try typing
Based on this information, estimate how long it will be after 95 minutes.
After 95 minutes, the shadow measured 38.5 cm. How does this compare to your estimate?
Is the length of the shadow a function of time? If so, is it linear? Explain your reasoning.
Print Version
When the sun was directly overhead, the stick had no shadow. After 20 minutes, the shadow was 10.5 cm long. After 60 minutes, it was 26 cm long.
Based on this information, estimate how long it will be after 95 minutes.
After 95 minutes, the shadow measured 38.5 cm. How does this compare to your estimate?
Is the length of the shadow a function of time? If so, is it linear? Explain your reasoning.
Activity 2: Recycling
Problem 1
In an earlier lesson, we saw this graph that shows the percentage of all garbage in the U.S. that was recycled between 1991 and 2013.
Sketch a linear function that models the change in the percentage of garbage that was recycled between 1991 and 1995. For which years is the model good at predicting the percentage of garbage that is produced? For which years is it not as good?
Pick another time period to model with a sketch of a linear function. For which years is the model good at making predictions? For which years is it not very good?
Lesson Summary
Water has different boiling points at different elevations. At 0 m above sea level, the boiling point is
This slope means that for each increase of 2,500 m, the boiling point of water decreases by
This equation is an example of a mathematical model. A mathematical model is a mathematical object like an equation, a function, or a geometric figure that we use to represent a real-life situation. Sometimes a situation can be modeled by a linear function. We have to use judgment about whether this is a reasonable thing to do based on the information we are given. We must also be aware that the model may make imprecise predictions, or may only be appropriate for certain ranges of values.
Testing our model for the boiling point of water, it accurately predicts that at an elevation of 1,000 m above sea level (when