Sensible Slopes and Intriguing Intercepts

 

Introduction

The applications of linear relationships are many in the natural and social sciences.  When studied in the context of their applications, the slopes and intercepts of lines reveal important information about the quantities they involve.

 

Task

In this project you will be asked to create linear relationships to describe situations from physics, medicine, and everyday life.  You will then determine the information that is revealed by the slopes and intercepts of those lines.

 

Instructions

Work with at least one other person to complete the following exercises.  Solve each problem in order and save your work along the way, as you will create a presentation to be given to your class.

 

·         First Problem:  Fractions are one of the most important inventions of humankind, because they allow us to use a single number to compare two quantities.  In the following examples, you will use given information to find the slope of a line and then examine what that slope tells us.

 

o     Fahrenheit to Centigrade:  Temperature can be measured on the Fahrenheit scale or the Centigrade scale.  The relationship between the two is linear.  The Centigrade scale was chosen so that the boiling point of water is 100 and the freezing point of water is 0.  Given that water boils at 212 and freezes at 32, we know two points of the form ( ___, ___) that will be on the line.

 

·         List these points below and use them to calculate the slope of a line where y is the degrees Fahrenheit and x is degrees Centigrade.  (Write the slope as a reduced fraction).

 

First Point

Second Point

Slope

( ___, ___)

 

 

 

 

 

( ___, ___)

 

 

 

 

 

 

 

·         The numerator of the fraction represents a change in , and the denominator represents the corresponding change in .  If either of the numbers is negative, the change is a decrease; otherwise it is an increase.  Interpret this slope by circling the correct word in parentheses and filling in the blanks in the following sentence: Every (increase/decrease) of ___ corresponds to a(n) (increase/decrease) of ___.  In which scale are the degree units closer together?  Explain.  [Hint: Which degree scale goes through more degrees than the other when the temperature changes?]

 

·         Now, use the information given to find the intercept b in your line.  Finish by writing the equation of the line y=mx+b in which x is the temperature in degrees Centigrade and y is the temperature in degrees Fahrenheit.  Explain what the intercept means; that is, what information is it giving us about the relationship between Fahrenheit and Centigrade?

 

o     Centigrade to Kelvin:  Many times scientists use another temperature scale known as the Kelvin scale.  In this scale, the point at which water freezes corresponds to 273, and the temperature at which water boils is 373. 

 

·         Use this information to find the slope of a line y=mx+b where y is the temperature in  and x is the temperature in .  (Again, write the slope as a reduced fraction).

 

First Point

Second Point

Slope

( ___, ___)

 

 

 

 

 

( ___, ___)

 

 

 

 

 

 

 

According to your slope, every (increase/decrease) of ___ corresponds to a(n) (increase/decrease) of ___.  Explain what this means in terms of how the scales compare to each other.  Which one has degree units that are closer together?

 

·         Now, use the information given to find the intercept b in your line.  Finish by writing the equation of the line y=mx+b in which x is the temperature in  and y is the temperature in .  Explain what the intercept means; that is, what information is it giving us about the relationship between Fahrenheit and Centigrade?

 

·         Finally, find the value of x that will correspond to y=0 (this is called the x-intercept), and explain what this tells us.  Google the Kelvin scale on the web to find out why scientists chose this seemingly strange number as the x-intercept for this line.

 

o     Everything Goes Downhill with Age!:  Another relationship that is linear is a person’s age and their peak heart rate.  According to the American Heart Association website, an average 20 year old person has a peak heart rate of 200, and an average 45 year old person has a peak heart rate of 175.

 

·         Use this information to find the slope of a line y=mx+b, where y is the peak heart rate and x is the person’s age.  Interpret the slope: Every (increase/decrease) of ___ years corresponds to a(n) (increase/decrease) of ___ beats per minute in their peak heart rate.

 

·         Write the equation of the line, y=mx+b, and explain what the y-intercept is telling us in this case.

 

o     Consistent Crickets:  In their June 1995 issue, Outside magazine summarizes some research done on a particular type of cricket known as the “snowy tree cricket, Oecanthus fultoni.”  Apparently the rate at which this cricket chirps and the temperature in degrees Fahrenheit are related in a linear fashion.

 

·         Use the data in the following table to find the slope of a line where y is the temperature in degrees Fahrenheit and x is the number of chirps.

 

Number of Chirps*

Temperature in

28

54

48

64

84

82

*These numbers refer to the number of chirps that the crickets made in 26 seconds.

 

Interpret the slope of this line in the same way that you did in the previous problems:  Every…

 

·         Write the equation of the line, y=mx+b, and explain what the y-intercept it telling us. [Hint:  The meaning of the y-intercept in this case may not be as obvious as the others.]

 

·         Second Problem – Prediction:

 

o     Fahrenheit to Centigrade:  Use the equation of your line to plot some points and graph your line.  As you read your graph from left to right, is the line uphill or downhill?  Explain the significance of this.

 

o     Everything Goes Downhill with Age!:  Use the equation of your line to plot some points and graph your line.  As you read your graph from left to right, is the line uphill or downhill.  Explain the significance of this.  At some point, this relationship ceases to be valid.  For example, determine the age at which your peak heart rate would be below 70, which is the average resting pulse rate for an adult.  Do we need to worry about this problem when actually applying the formula?  Explain.

 

o     Consistent Crickets:  Use the equation of your line to plot some points and graph your line.  For what values of x does this graph cease to be meaningful?  Explain.

 

·         Third Problem – Access Ramps:  Wheelchair access ramps must be built according to very precise specifications so that the ramps are not too steep to maneuver in a wheel chair.  If one is standing at the top of the ramp, and y stands for the height above the ground and x stands for the horizontal distance away from the top of the ramp (see the figure below), then the ramp must follow the equation .

o        Describe the meaning of the y-intercept, b, in this particular situation.

o        Suppose you wish to build a wheelchair accessible ramp into a building, and the ground floor is 4 feet above the ground.  Determine the value of b, write the corresponding equation, and graph it.  Then, use the graph to determine how far out horizontally from the building the ramp should begin.  [Hint: When you get to the end of the ramp, what will the value of y be?  Sketching a careful graph of the line will help you determine the answer.]

o        In this problem, describe what the x-intercept tells us.

 

 

Compare your computations with those from another group.  Discuss the meanings of the slopes, intercepts, and graphs and your solution to the third problem.  Work to make sure that your explanation is clear and concise. 

 

Prepare a presentation which:

1.      Explains the meaning of slope, including the significance of the sign of the slope.

2.      Explains the meaning of y-intercept the x-intercept (where applicable) in the context of the problems you worked.

3.      Explains how you obtained the predictions for Problem #3.

 

Finally, present your solution to the class.

Grade

Your project should correctly identify and justify mathematically:  (1) explains the meaning of slope, including the significance of the sign of the slope, (2) explains the meaning of y-intercept and x-intercept in the context of the problems you worked, and (3) explains how you obtained the predictions for Problem #3.

 

 

Score

Content

Presentation/Communication

4

•       The solution shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts and the information necessary for its solution.

•       The solution completely addresses all mathematical components presented in the task.

•       The solution puts to use the underlying mathematical concepts upon which the task is designed and applies procedures accurately to correctly solve the problem and verify the results.

•       Mathematically relevant observations and/or connections are made.

•       There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made.

•       Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem.

•       There is precise and appropriate use of mathematical terminology and notation.

•       Your project is professional looking with graphics and effective use of color. 

3

•       The solution shows that the student has a broad understanding of the problem and the major concepts necessary for its solution.

•       The solution addresses all of the mathematical components presented in the task.

•       The student uses a strategy that includes mathematical procedures and some mathematical reasoning that leads to a solution of the problem.

•       Most parts of the project are correct with only minor mathematical errors.

•       There is a clear explanation.

•       There is appropriate use of accurate mathematical representation.

•       There is effective use of mathematical terminology and notation.

•       Your project is neat with graphics and effective use of color. 

2

•       The solution is not complete indicating that parts of the problem are not understood.

•       The solution addresses some, but not all of the mathematical components presented in the task.

•       The student uses a strategy that is partially useful, and demonstrates some evidence of mathematical reasoning.

•       Some parts of the project may be correct, but major errors are noted and the student could not completely carry out mathematical procedures.

•       Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics.

•       There is some use of appropriate mathematical representation.

•       There is some use of mathematical terminology and notation appropriate for the problem.

•       Your project contains low quality graphics and colors that do not add interest to the project.

1

•       There is no solution, or the solution has no relationship to the task.

•       No evidence of a strategy, procedure, or mathematical reasoning and/or uses a strategy that does not help solve the problem.

•       The solution addresses none of the mathematical components presented in the task.

•       There were so many errors in mathematical procedures that the problem could not be solved.

•       There is no explanation of the solution, the explanation cannot be understood or it is unrelated to the problem.

•       There is no use, or inappropriate use, of mathematical representations (e.g. figures, diagrams, graphs, tables, etc.).

•       There is no use, or mostly inappropriate use, of mathematical terminology and notation.

•       Your project is missing graphics and uses little to no color.

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