Understanding rate of change
- Author:
- Mathguru, CorvairDave
Understanding the Rate of Change
4 Representations of Linear Equations
#1: Introduction to Rate of Change
Background
Suppose you have a job that pays $7/hour and your friend works with you, but gets paid $9/hour. As you work side by side, the total amount of earnings for your friend increases faster than your total earnings.
Suppose one road crew can build 3 miles of highway every 4 weeks, and another crew can build 5 miles of highway every 4 weeks. The total number of miles built will increase faster for the second group.
Suppose you and a friend are both trying to lose weight. In 10 days you lose 2 pounds and your friend loses 1 pound in the same amount of time. Your weight will decrease faster than your friend’s weight.
The amount that your earnings increase, the miles of highway increases, and your weight decreases is called a rate of change. A rate of change is symbolized by a fraction with the amount of change in the numerator (blue on the applet) and the units (red on the applet) in the denominator.
Inquiry Steps
Leave the red slider, Δx (change in x), in one place and move the blue slider Δy (change in y),to several different positions. As you move the slider, observe what happens to the line and the steps in the applet.
Question 1-1: How does making Δy a greater number affect the line and steps in the applet?
Question 1-2: Why do you think it has this affect – what does this affect represent?
Question 1-3: How does making Δy zero affect the line and steps in the applet?
Question 1-4: Why do you think it has this affect – what does this affect represent?
Question 1-5: How does making Δy a lesser number affect the line and steps in the applet?
Question 1-6: Why do you think it has this affect – what does this affect represent?
Question 1-7: What do the line and steps look like when Δy is positive?
Question 1-8: What do the line and steps look like when Δy is negative?
Leave the blue slider, Δy (change in y), in one place and move the red slider Δx (change in x),to several different positions. As you move the slider, observe what happens to the line and the steps in the applet.
Question 1-9: How does making Δx a greater number affect the line and steps in the applet?
Question 1-10: Why do you think it has this affect – what does this affect represent?
Question 1-11: How does making Δx zero affect the line and steps in the applet?
Question 1-12: Why do you think it has this affect – what does this affect represent?
Question 1-13: How does making Δx a lesser number affect the line and steps in the applet?
Question 1-14: Why do you think it has this affect – what does this affect represent?
Move both sliders and observe the equation.
Question 1-15: What part of the equation seems to be affected by the rate of change?
Question 1-16: Make Δy=6 and Δx=3. Why does the equation say “2”?
Click the reset arrows at the top right of the applet, then press the “Start/Pause” button. Notice how the points move up the line and up the steps. Also notice the numbers that appear in the spreadsheet. The two columns on the left(A and B) follow the steps, while columns C and D follow the line.
Question 1-17: If you only had the spreadsheet, how could you determine Δx and Δy?
Question 1-18: Do you find it easier to follow the steps (Columns A and B) or the line (Columns C and D), and why do you find it easier to follow?
Reflection
Question 1-19: Write a paragraph describing in your own words what you have learned through this exercise.
Question 1-20: What would you change about this exercise to make it better?
Question 1-21: I found this: 1-Very interesting, 2-Interesting, 3-OK, 4-Boring, 5-Very Boring
Question 1-22: I learned: 1-A lot, 2-Something, 3-Not much, 4-More confused than before.