Group member(s): Chastity Colbert, Theresa Hogan, Danielle Ortega

 

Author of Lesson: Chastity Colbert

 

Title of lesson:   Investigation of Slope and Speed                                        

 

Date of lesson: N/A

 

Length of lesson:   (2) 50 minute class periods

 

Description of the class:

                     Name of course: Algebra II

                     Grade level: 10th or 11th grade

                     Honors or regular: N/A

 

Source of the lesson:

Acceleration and Velocity Investigation

http://www.edb.utexas.edu/insite/istetest/pbiprojects/Spring2001/Physics/content/VelAccIn.htm

 

Understanding Distance, Speed, and Time Relationships Using Simulation Software

http://standards.nctm.org/document/eexamples/chap5/5.2/index.htm#APPLET

 

           

 

TEKS addressed:                   

(5)   Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems.

(A.1.b)  gather and record data and use data sets to determine functional relationships between quantities;

 (A.5.c)  use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions

(A.6.a)  develop the concept of slope as rate of change and determine slopes from graphs, tables, and algebraic representations;

(A.6.b)  interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs;

(A.6.c)  investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b

 

 

 

NCTM standards

 

I.  Overview

            Although students will have had some exposure to this concept from the CBR activity, this lesson allows students to analyze the relationship between slope and velocity more deeply. In this lesson, students make the connection between slope and velocity by examining the slope of a position vs. time graph. Later students will apply this concept to a real world situation through the use of technology.

 

II.  Performance or learner outcomes

Students will be able to:

·       observe and record data from an object moving at constant velocity.

·       observe and record data from an object moving under constant acceleration.

·       to create, analyze, and interpret graphs of objects in linear motion.

·       understand distance, speed, and time relationships using simulation software

·       understand the relationship between slope and velocity

                       

           

III. Resources, materials and supplies needed    

Tennis ball, tracks for the tennis balls (metal with a U-shaped cross section would work best),Video cameras that records time (to the nearest second at least) on the video Meter stick to measure off lengths along the track, TV (preferably with large screen) and VCR, Overhead transparencies (one for each student), graph paper, warm-up worksheet.

 

 

Engagement: (5 min)

Teacher Does

Students Do

Handout warm-up worksheet.  Worksheet should include a table in which students must use table to make a graph of data. Students must find slope and y-int. 

 

 

Explore (60 min)

Divide students into their project groups. Each group needs a secretary, a tech person, and a person in charge setting up the track.  Note: The following “Students Does” directions should be written on a handout to give to students.  Be sure to go over handout aloud with students.

Set up the track so that it is level. Set up the video camera to record the movement of the tennis ball along the track all in one shot (without panning). Use the meter stick to clearly mark off equal lengths along the track (every decimeter should be sufficient; make sure the markings will be clearly visible on the video tape.) Set the bowling ball rolling along the track and record its motion (make sure it moves for several seconds).

 

Now students elevate one end of the track a few inches and place the tennis ball at the elevated end and letting it roll to the other end under the effects of gravity alone. Adjust the height of the elevated end to make sure the travel time is several seconds.

Assist students who are having trouble with technology.

Students will play back the recorded motion on the TV and find the position and time from the tape.  by taping a transparency to the screen and using "pause" to trace positions and record times. Each group should make a table to record their data.

 

 

From the data, they should make position vs. time graphs. Ask students to analyze graph by finding the slope and y-intercept of the graph.

 

 

If students finish early, allow them to help other students.

 

 

 

Explain (15min)

Lead a discussion of the results. Ask the students how they determined the slope from the position vs. time graphs.

 

“What do you notice about you line and how fast the ball traveled?”

 

 

 

“Now, what part of the graph relates to steepness?”

 

 

 

“So, we can say that the slope determines how fast the ball traveled or the speed. So, we can determine the velocity or speed from a position vs. time graph. What are the units of the slopes of the various graphs?”

“I used the terms speed and velocity to describe how fast the ball traveled. Do you know the difference between speed and velocity?”

 

Speed describes the how fast an object moves.  However, velocity is more precise.  It tells us the rate at which an object changes its position. Because of this, it can be either negative or positive.

 

 

 

 

 

“When the ball traveled on the higher incline, it went faster. And our graph is steeper than it was when our ball traveled on the level track.”

 

“The slope.”

 

 

 

 

Centimeters per second.

 

 

 

 

 

Elaborate/Extension (20 min)

Present Applet from NCTM website to students. 

“What does this graph show?”

 

Set a starting position for the runners by dragging their icons along the tracks.  Change the direction they face by clicking once on their icons. Set the length of the stride for each runner using the controls on the lower left.

“What do you think the race will look like? Who will win the race? Why”

Click Go to run the simulation.

 

“Did what happened match your prediction? If it did, how does the graph show what you predicted?”

 

 

 

 

 

 

 

 

 

 

Various answers will arise depending on how your runners are set up.  Make sure students you slope and speed. 

 

 

Make a change in one of your settings (e.g., the length of the girl's stride or the boy's starting position). How will this change affect the graph? Run the simulation again and see what happens. Continue making changes and predicting the result. After each run of the simulation, think about what the graph shows and think about what happened and why.

 

If time permits, use applet to discuss negative velocity. 

 

 

Evaluation

A short oral evaluation will be done at the end of class reviewing relationship between slope and velocity.