Unit 2

1. Unit 2 Analyzing Graphs

2. 2.5 Two Graph Stories Consider the story: Maya and Earl live at opposite ends of the hallway in their apartment building. Their doors are 50 feet apart. They each start at their door and walk at a steady pace towards each other and stop when they meet. What would their graphing stories look like if we put them on the same graph? When the two people meet in the hallway, what would be happening on the graph? Sketch a graph that shows their distance from Maya’s door.

3. 2.5 Two Graph Stories Maya and Earl live at opposite ends of the hallway in their apartment building. Their doors are 50 feet apart. They each start at their door and walk at a steady pace towards each other and stop when they meet. What would their graphing stories look like if we put them on the same graph? When the two people meet in the hallway, what would be happening on the graph? Sketch a graph that shows their distance from Maya’s door.

4. Example 1 1. Graph the man's elevation on the stairway versus time in seconds. 2. Add the girl’s elevation to the same graph. How did you account for the fact that the two people did not start at the same time? 3. Suppose the two graphs intersect at the point 𝑷(𝟐𝟒,𝟒). What is the meaning of this point in this situation? 4. Is it possible for two people, walking in stairwells, to produce the same graphs you have been using and NOT pass each other at time 𝟏𝟐 seconds? Explain your reasoning.

5. Example 1 1. Graph the man's elevation on the stairway versus time in seconds. 2. Add the girl’s elevation to the same graph. How did you account for the fact that the two people did not start at the same time?

6. Example 1 3. Suppose the two graphs intersect at the point 𝑷(𝟐𝟒,𝟒). What is the meaning of this point in this situation? Many students will respond that 𝑷 is where the two people pass each other on the stairway. 4. Is it possible for two people, walking in stairwells, to produce the same graphs you have been using and NOT pass each other at time 𝟏𝟐 seconds? Explain your reasoning. Yes, they could be walking in separate stairwells. They would still have the same elevation of 𝟒 feet at time 𝟐𝟒 seconds but in different locations.

7. Example 2 Consider the story: Duke starts at the base of a ramp and walks up it at a constant rate. His elevation increases by three feet every second. Just as Duke starts walking up the ramp, Shirley starts at the top of the same 25 foot high ramp and begins walking down the ramp at a constant rate. Her elevation decreases two feet every second.

8. Example 2 1. Sketch two graphs on the same set of elevation-versus-time axes to represent Duke’s and Shirley’s motions. 2. What are the coordinates of the point of intersection of the two graphs? At what time do Duke and Shirley pass each other? 3. Write down the equation of the line that represents Duke’s motion as he moves up the ramp and the equation of the line that represents Shirley’s motion as she moves down. Show that the coordinates of the point you found in question above satisfy both equations.

9. Example 2 1. Sketch two graphs on the same set of elevation-versus-time axes to represent Duke’s and Shirley’s motions. 2. What are the coordinates of the point of intersection of the two graphs? At what time do Duke and Shirley pass each other? (𝟓,𝟏𝟓) 𝒕 = 𝟓 3. Write down the equation of the line that represents Duke’s motion as he moves up the ramp and the equation of the line that represents Shirley’s motion as she moves down. Show that the coordinates of the point you found in question above satisfy both equations. If 𝒚 represents elevation in feet and 𝒕 represents time in seconds, then Duke’s elevation satisfies 𝒚 = 𝟑𝒕 and Shirley’s 𝒚 = 𝟐𝟓−𝟐𝒕. The lines intersect at (𝟓,𝟏𝟓), and this point does indeed lie on both lines. Duke: 𝟏𝟓 = 𝟑(𝟓) Shirley: 𝟏𝟓 = 𝟐𝟓−𝟐(𝟓)

10. Analyzing graphs Scenario #1 • Can you see features of this information appearing in the graph? • Is it possible to deduce the time of lunch at this school?

11. Analyzing graphs • Can you see features of this information appearing in the graph? Bathroom use before and after activities. • Is it possible to deduce the time of lunch at this school? Around 10:00 a.m. the graph indicates a peak of 80 units.

12. Analyzing graphs • Around 10:00 a.m. the graph indicates a peak of 80 units. What is the number 80 representing?

13. Analyzing graphs • Around 10:00 a.m. the graph indicates a peak of 80 units. What is the number 80 representing? Gallons of water used or the amount “used” is measured by the volume of water that drains through the pipes and leaves the school.

14. Analyzing graphs • How does one actually measure the amount of water flowing out through the pipes precisely at 10:00 a.m.? What does “80 units of water leaving the school” right at 10:00 a.m. mean? (Think about when the bell rings and students “flow” through the door to leave)

15. Analyzing graphs • How does one actually measure the amount of water flowing out through the pipes precisely at 10:00 a.m.? What does “80 units of water leaving the school” right at 10:00 a.m. mean? The researchers who collected this data watched the school’s water meter during a 12-hour period. The meter shows the total amount of water (in gallons) that has left the school since the time the meter was last set to zero. Since the researchers did not know when this resetting last occurred, they decided, at each minute mark during the day, to measure how much the meter reading increased over the next minute of time. Thus the value “80” at the 10:00 a.m. mark on the graph means that 80 gallons of water flowed through the meter and thus left the school during the period from 10:00 a.m. to 10:01 a.m.

16. Analyzing graphs • What are the units for the numbers on the vertical axis?

17. Analyzing graphs • What are the units for the numbers on the vertical axis? Gallons per minute

18. Analyzing graphs • Ignoring the large spikes in the graph, what seems to be the typical range of values for water use during the school day?

19. Analyzing graphs • Ignoring the large spikes in the graph, what seems to be the typical range of values for water use during the school day? 15 gallons per minute

20. BellWork • Suppose the researchers collecting data for water consumption during a typical school day collected data through the night too. • a. For the period between the time the last person leaves the building for the evening and the time of the arrival of the first person the next morning, how should the graph of water consumption appear? • b. Suppose the researchers see instead, from the time 1:21 a.m. onwards, the graph shows a horizontal line of constant value, 4. What might have happened during the night?

21. Bell work • Suppose the researchers collecting data for water consumption during a typical school day collected data through the night too. • a. For the period between the time the last person leaves the building for the evening and the time of the arrival of the first person the next morning, how should the graph of water consumption appear? Since no toilets are being used, the graph should be a horizontal line of constant value zero. • b. Suppose the researchers see instead, from the time 1:21 a.m. onwards, the graph shows a horizontal line of constant value, 4. What might have happened during the night? Perhaps one toilet started to leak at 𝟏:𝟐𝟏 a.m., draining water at a rate of 𝟒 gallons per minute.