Building an Understanding for Linear Functions

1. Building an Understanding for Linear Functions Jim Rahn LL Teach, Inc. www.jamesrahn.com james.rahn@verizon.net

2. Mathematical Ideas These ideas will be integrated throughout the development of linear functions: • Proportions • Engage students in activities that build concepts before procedures are taught • Use Real data to help build connections • Use both Hand Graphing and Graphing Calculators • Build Connection between Constant Rate of Change and Linear Functions • Use Shifting to understand the form y=b+mx

3. This chart shows the lengths of several lakes in both miles and kilometers.

4. Predict the two other lengths. Create a hand graph of miles (x) vs. km. (y) What pattern or shape do you see in your graph? Connect points to illustrate your pattern.

5. L1 L2 Create a L3=L2/L1 What is the meaning of this value? Use this value to predict the length of the other two lakes.

6. L1 L2 Create a table that fits your equation. Predict the two lengths in the using the table. If x represents miles and y represents Km, what equation connects the two variables? Graph the equation in your calculator. Trace on the graph to predict the two lengths in the chart.

7. We just found out that graphs and the graphing calculator can be used to • Study more than one representation of a set of data: tables and graphs • See that graphs tell a story or follow a pattern • Read missing values • On paper graph by approximating by hand • On the graphing calculator graph by tracing • On the graphing calculator by using a table of values • By calculating and using a rate of change • By creating an equation and substituting into the equation.

8. In this relationship • As x increased, the y values increased • As x decreased, the y value decreased • How are these ideas pictured or illustrated by the graph? • We noticed that the x and y values were related by or

9. Thinking About Recursive Sequences

10. The Empire State Building has 102 floors and is 1250 feet high. How high are you when you are reach the 80th floor? Explain your reasoning.

11. A 25-story building has floors at the described heights. What recursive sequence can describe the heights? • Find the height of the 4th and 10th floors? • Which floor is 215 feet above ground? • How high is the 25th floor? • Explain your reasoning

12. How can we model this on the graphing calculator? Method 1 Method 2

13. Building a Connection between Rate of Change and Straight Lines

14. A van leaves High Point and heads south for Cape May. At the same time a pickup truck leaves Atlantic City and a sports car leaves Cape May and head toward High Point. The van is traveling at 72 mph, the pickup truck is traveling 66 mph, and the sports care is traveling at 48 mph. When and where will they pass each other?

15. Change the rates of change to miles per minute so we can study the problem in smaller increments.

16. Write a recursive sequence would model each car’s distance from Cape May?

17. Make a table to record the distance from Cape May for each vehicle every minute. • After completing the first couple of rows, change the intervals to 10 minute intervals until you have covered 4 hours.

18. Define what x and y will represent. • Make a paper graph for this traveling situation. • What do you notice about the points that represent each vehicle? • What is the starting position of each vehicle? Where is this on the table? Where is it on the graph? • How does the vehicle’s speed effect the graph?

19. How can you tell which line represents the van? • Where are the vehicles when the van meets the first vehicle heading north? • How can you tell if the pickup truck or sports car is traveling faster from the graph? • Which vehicle arrives at its destination first? How much later do the other vehicles reach their destination? • Are you making any assumptions about each of the vehicles as you answer the questions?

20. Write an equation that represents each vehicles distance from Cape May by referring to your recursive sequence. • Enter these equations in your graphing calculator. How do these graphs compare to your paper graph?

21. Time-Distance Relationships Explore time-distance relationship Write walking instructions or act out walks for a given graph Sketch graphs based on given walking instructions or table data Use an electronic device, motion sensor, and graphing calculator to collect and graph data

22. The time-distance graphs at the right provide a lot of information about the walks they picture. • Because the lines are straight and increasing means that both walkers are moving away from the motion sensor at a steady rate. • The first walker starts 0.5 meters from the sensor, whereas the second walker starts 1 meter from the sensor. • The first graph pictures a walker moving 3-1= 2 meters in 4-0 =4 seconds or 1/2 meter per second. • The second walker covers 3.5-0.5=3 meters in 3-0=3 seconds or 1 meter per second. • In the next activity you will analyze time-distance graphs and create your own graphs.

23. Walk the Line Study one of the pictures above. Each describes a 6 second walk. The vertical axis shows 0 to 4 meters. Write a set of walking instructions for each graph. Tell where the walk begins, how fast the person walks, and whether the person walks toward or away from the motion sensor. Make a paper graph of a 6 second walk based on your instructions.

24. Obtain a CBR (motion detector) to use with your graphing calculator to complete set of instructions. Be ready to discuss the results of your investigation.

25. Building Understanding for the form y = b+mx

26. Jose’s Savings • On Jose’s 16th birthday he collected all the quarters in his family’s pockets and placed them in a large jar. He decided to continue collecting quarters on his own. He counted the number of quarters in the jar periodically and recorded the data in a chart.

27. 1. Make a scatter plot of the data on your calculator. Describe any patterns you see in the table and/or graph. • 2. Select two points that you believe represents the steepness of the line that would pass through the data. (________, ________) and (________, ________) • Find the slope of the line between these two points.

28. Give a real world meaning to this slope. • Use the slope you found to write an equation of the form y = mx. • Graph this equation with your scatter plot. • Describe how the line you graphed is related to the scatter plot. • What do you need to do with the line to have the line fit the data better?

29. Run the APPS TRANFRM on your graphing calculator. Change your equation to y=B+mx. Press WINDOW and move up to Settings. Change B to start at 0 and increase by steps of 10. Press GRAPH and notice that B=0 is printed on the screen. Use the right arrow to increase the value of B. What happens to the graph as you increase the value of B. • Continue to increase or decrease the value of B until you have a line that fits the data. Write the equation for your line.Y = _____________________ • What is the real world meaning for the y-intercept you located?

30. Use your equation to predict the number of quarters Jose will have on his 21st birthday. Explain how you predicted the number of quarters. • Use your equation to predict when Jose will have collected 1000 quarters. Explain how you found your answer.

31. Graphing Linear Equations and Their Transformations Jim Rahn LL Teach, Inc. www.jamesrahn.com james.rahn@verizon.net