Analytic Geometry

Lesson

- To visualize slope in a real world situation.
- To investigate the slope of parallel lines.

- Poster board
- Markers
- Toy car
- Ruler
- Stopwatch
- 1 Cardboard paper towel roll
- Scissors
- Tape

- Draw lines to represent the $x$
`x`and $y$`y`axes. - Below the $x$
`x`-axis add numbers $0-20$0−20 in the correct places. - Do the same thing in the space to the left of the $y$
`y`-axis. - Tape the poster board to the wall so that it is secure.
- Cut the cardboard paper towel roll in half.
- Tape the two halves together end to end.
- Measure the length of the ramp that you have created and record this length.

- Use the ramp to model a line of a particular slope.
- Place the toy car at the top of the ramp.
- Release the car and start the stopwatch.
- Stop the stopwatch when the car reaches the end of the ramp.

Work with a partner or small group to answer the following questions. After creating the indicated lines release the car as described in the procedure. Be sure to record the time it takes the car to reach the end of the ramp after every trial.

- Create a line with an undefined slope. Describe the car’s movement. Would you want to be a passenger in the car traveling down such an incline? Why or why not?
- Create a line with a slope of $0$0 using the ramp. Describe the car’s movement.
- Start with your ramp at a slope of $0$0. Then, tilt the ramp very slightly forward. What would you estimate the new slope to be? Is it positive or negative?
- Create a line that has a slope of $-4$−4 using the ramp. What is the equation of the line you created?
- Create a line that has a slope of $-7$−7 with the ramp. What is the equation of the line you created?
- Create a line that passes through the points $(-5,5)$(−5,5) and $(-3,0)$(−3,0) with the ramp. What is the slope of this line? What is the equation of this line?
- Create a line that passes through the points $(4,0)$(4,0) and $(0,16)$(0,16) with the ramp. What is the slope of this line? What is the equation of this line?
- Create a line that is parallel to the line in question number $7$7. What is the slope of this line? What is the equation of this line?
- Create a line that is perpendicular to the line in question number $7$7. What is the slope of this line? What is the equation of this line?
- How is the slope of the equation in number 9 different than the other equations?
- Use the formula $speed$
`s``p``e``e``d`=$dis\tan ce$`d``i``s``t``a``n``c``e`÷$time$`t``i``m``e`to calculate the speed the car achieved while traveling down each of the ramps created in questions 1-8. - What units are the speeds in?
- How are the different speeds related to the slope of the car?
- Compare the speeds of the car on the ramps you created in questions $7$7 and $8$8. Make observations and give reasons why they happened.
- On which equation did the car travel fastest? Why?
- Compare and contrast the equation you created in question number $8$8 with another group’s answer. Are your two equations parallel? Why or why not?

Determine, through investigation, various formulas for the slope of a line segment or to determine the slope of a line segment or a line