- Find the slope/rate of change and y-intercept/initial value of a line from different representations of a function
- Use one of the four representations of a function to produce the others
- Use the equation y = mx + b when working with straight lines
- That’s Some Slippery Slope!: Using Functions to Model Relationships Between Quantities printable
- Answer Key: Think FUNCTIONally printable
- Standards Chart: Think FUNCTIONally printable
- Graph paper
- Whiteboard or large graph paper and markers
- Make a class set of the That’s Some Slippery Slope!: Using Functions to Model Relationships Between Quantities printable.
- Print a copy of the Answer Key: Think FUNCTIONally printable for your use.
Introduction to New Material
Step 1: Tell the class the following story involving a real-world, linear functional relationship:
A car gets on an interstate highway and travels at a constant rate of speed. After 2 hours, it traveled 110 miles and after 5 hours it traveled 275 miles.
- Ask if this story represents a functional relationship (yes, there is one and only one output for each input).
- Ask if this verbal representation of a function includes an independent variable (time) and a dependent variable (distance).
Step 2: Ask if the functional relationship could be represented in table format. Draw the following table with the known information from the story:
Step 3: Introduce students to the concept of rate of change. Start by calculating 5 - 2 = 3 and tell the class that the change in the independent variable (also called the input) is 3 hours. Then calculate 275 - 110 and indicate that the change in the dependent variable (also called the output) is 165 miles. Rate of change is found by dividing the change in output by the change in input. 165 miles divided by 3 hours equals 55 miles per hour, which is the rate of change. Note that this assumes a constant rate of speed for the car.
Step 4: Once we know the rate of change, we can fill in the missing cells in the table:
Step 5: Ask the class if it’s possible to determine the formula for the function just with the information given in the story. Introduce the class to the straight-line equation y = mx + b. Explain the various parts of the equation: x is the input/independent variable, y is the output/dependent variable, m is the rate of change, and b is the value of y, the dependent variable when x, the independent variable, is 0.
Apply the information in the story to the y = mx + b equation. We calculated the rate of change as 55 miles per hour earlier, and we know logically, that when the car has driven for 0 hours, it has traveled 0 miles, so b = 0. So, the formula for this situation is y = 55x + 0 or, simply, that y = 55x. Point out that in this case, y = 0 when x = 0, but that will not always be the case and we will develop other strategies to determine the value of y when x = 0. Point out that with this formula, when x equals 0, y also equals 0. Ask if this makes sense (yes, because the vehicle hasn’t traveled any distance at all before it starts moving).
Step 6: Ask if we can use the information from the story to represent the function as a graph. Draw the following grid on the board:
Step 7: Connect (2, 110) and (5, 275) with a straight line. Indicate that we can calculate the rate of change using these two points. Also indicate that, with lines, the rate of change is also called the slope of the line and is calculated by dividing the change in y (the dependent variable) by the change in x (the independent variable). Show how the change in the y value of the two points is 275 - 110 = 165 and the change in the x value is 5 - 2 = 3. The slope (rate of change) = 165/3 = 55. Point out that we would calculate the same value for the slope if we “went in the other direction,” i.e., subtracted the values of (5, 275) from (2, 110). We would have computed a change in y of -165 and a change in x of -3. -165/-3 = 55. As long as we go in the “same direction” for both the x and y coordinates to determine the change, we’ll get a correct result. Also note that change in y over change in x is sometimes called “rise over run” or Δy/Δx.
Step 8: Indicate that not every situation has a y/output value of 0 when the x/input value is 0. Tell the following story problem to the class: An automobile repair shop charges a $50 diagnostic fee plus $50 per hour for repairs. Show how to represent this situation with a table:
Make sure the class understands why the cost is $50 before any time is spent on the repair.
Step 9: Use this information to determine a formula representation. Show how the fee increases by $50 each hour and that there is an additional $50 diagnostic fee for each repair. So, C = 50T + 50, where C is the total cost in dollars and T is the time in hours.
Step 10: Have the class consider the formula y = mx + b in the context of this story. We know that m, the rate of change/slope = 50. We also know that y = 50 when x = 0, so the y-intercept, also called the initial value, is 50, so the formula for the line is y = 50x + 50. Draw the line on a grid pointing out that the y-intercept is 50. Calculate the slope, using two points on the line. Point out that the y-intercept could also be determined using just the information in the problem and the y = mx + b formula as follows:
If we know that the rate of change is 50, then plug the known values from one of the points into y = mx + b. If we know that in hour 1 the cost was $100, we can say that 100 (the y value) = 50 x 1 (the x value) + b. Solving for b, the y-intercept/initial value, gives a value of $50.
Step 11: Group students in pairs, distribute graph paper, and ask them to solve the following problem:
Molly started a dog walking business. She makes the same amount each week and puts all her earnings into her bank account. After 1 week, she had $75 in the bank. After 4 weeks, she had $150 in the bank.
a. How much does Molly make each week from her business? Hint: Find the rate of change for this function. Answer: The change in the y value is $150 - $75 = $75. The change in x value is 4 - 1 = 3. Dividing the change in y by the change in x = $75/3 = $25 per week.
b. How much money did Molly have in the bank before she started her business? Hint: Find the initial value/y-intercept for this function. Answer: $50. If we know that the rate of change is $25 and that Molly had $75 in the bank in week 1, then the value of the account in week 0, before the business started, was $75 - $25 = $50.
Another way to solve this is to use y = mx + b, using the data for the first week. So, $75 = 1 x $25 + b, where $75 is the bank balance in week 1 (y), 1 is the number of weeks (x), $25 is the rate of change, and b is the initial value. Solving for b, we find that b = $50.
c. Write a formula for this function. Answer: b = $25w + $50, where b is the bank balance and w is the week number.
d. Create a table showing the bank balance for the first five weeks of Molly’s business. Answer: See table below.
e. Graph the results. Answer: See graph below.
Step 12: Checking for Understanding: Review answers as a class and respond to any questions.
Step 13: Assign the That’s Some Slippery Slope!: Using Functions to Model Relationships Between Quantities printable for classwork or homework.
Step 14: Checking for Understanding: Review the answers to the That’s Some Slippery Slope!: Using Functions to Model Relationships Between Quantities printable, which are provided on page 1 of the Answer Key: Think FUNCTIONally printable. Make sure students explain their mathematical thinking. Address any misconceptions that may arise.
- Grade 8: Function Equations/Representations and Slope (CCSS: 8.F.3, 8.F.4)
- Grades 6–8: Making Sense of Problems, Reasoning, Modeling, and Attending to Precision (CCSS Practice MP1, MP2, MP4, and MP6); NCTM Algebra
For more information, download the comprehensive Standards Chart: Think FUNCTIONally printable.