Linear Models

10 min

Teacher Prep
Setup
Groups of 2. 3–5 minutes of quiet work time then partner discussion. Follow with whole-class discussion.
Required Preparation
For the digital version of the activity, acquire devices that can run the applet.

Narrative

In this Warm-up, students work with data to determine if the situation represented by the data could be modeled by a linear function (MP4). Students are given three different data points and use what they know about linear relationships to estimate when the candle will burn out. 

In the digital version of the activity, students use an applet to visualize the height of the candle at different times. The applet allows students to plot points quickly and accurately without having to set up the axes from scratch.

Launch

Arrange students in groups of 2. Display the problem stem for all to see. Give students 30 seconds to make a guess at when the candle will burn out completely, then poll the class, displaying their responses for all to see.

Students should work with their partner on the questions. If they don't agree, partners should work to understand each other’s thinking. If any students attempt to guess a linear equation that fits the data, ask them to share during the discussion. Follow with a whole-class discussion.

Student Task

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.

  1. When do you think the candle will burn out completely?
  2. Is the height of the candle a function of time? If yes, is it a linear function? Explain your thinking.

Sample Response

  1. Sample response: Since it burns about 2 inches every hour, it will burn out between 5 and 6 hours after it was lit.

  2. The height of a candle is a function of time because at any given time, the candle will have one and only one height. Sample reasoning: It is not exactly linear, although it looks close enough to use a linear function since the rate of burning is almost constant (2 inches per hour).
Activity Synthesis (Teacher Notes)

The purpose of this discussion is for students to justify how this situation can be modeled by a linear equation. Select students who answered yes to the last question, and ask:

  • “Was the data exactly linear? If not, what made you decide that you could treat it as such?” (I couldn’t draw a line exactly through all three points, so it isn’t linear, but my line is so close to going through all three that it seems like a decent estimate.)
  • “What was the slope between the first two data points? What was the slope between the last two data points? What does it mean that their slopes are different?” (The slope from 0 to 1 hour is 2 inches per hour, while the slope from 1 hour to 3 hours is 2.25 inches per hour. Different values mean all three points are not on a straight line.)
  • “Did your estimate of when the candle would burn out increase, decrease, or stay the same?” (I first guessed 6 hours since the candle starts at 12 inches and loses 2 inches of height in the first hour. The height after 3 hours makes me think it will burn out sooner, so my guess decreased.)

Tell students that although the data is not precisely linear, it does make sense to model the data with a linear function because the points resemble a line when graphed. We can then use different data points to help predict when the candle would burn out. Answers might vary slightly, but it results in a close approximation.

Conclude the discussion by asking students to reconsider the range of values posted earlier for the first question, and ask if they think that range is acceptable or if it needs to change (for example, students may now think the range should be smaller after considering the different slopes).

Standards
Addressing
  • 8.F.4·Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
  • 8.F.B.4·Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two <span class="math">\((x, y)\)</span> values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

10 min

10 min