Session 473: Stretching the Truth and Shifting the Blame Page | 1
CMC –South 2016
Session 473
Stretching the Truth
and Shifting the Blame
Liem Tran
| @liemttran |
Math for America, Los Angeles
Los Angeles Unified School District
Nate Goza
| @thegozaway |
Math for America, Los Angeles
Los Angeles Unified School District
Sample Tasks
1)2.1: Shifting The Blame
2)2.5: Rumor Has It
3)2.3: Stretching The Truth
2.1: Shifting the Blame
Mrs. White has been murdered!
It happened at the Library at 4:30 PM on Saturday. She was found stabbed with a knife and hit with a candlestick in the Mystery section.The townspeople are furious!
Detectives Tran and Goza were put on the case and immediately they put together a list of important characters and usual suspects.
They also created a map of the town highlighting some important landmarks.
Tran and Goza decided to start by bringing Colonel Mustard in for questioning.
Mustard says that he and Professor Plum left from Plum Palace at noon (12:00 PM) and walked toward town at a leisurely pace. After an hour of walking Mustard stopped at the Lounge for a cup of coffee and Plum continued on towards town. Mustard says he stayed at the Lounge for about two hours because he ran into Miss Scarlet and ended up talking to her for a long while. (Mustard had a thing for Scarlet - that can be sure!) Then he left the Lounge and walked to meet some friends at the Billiard Hall. He arrived there around 4:00 and stayed all evening playing pool, and enjoying food and drinks until nearly Midnight.
He continued by saying that Professor Plum had told him his plans to keep walking to the Library. Mustard estimated that Plum was walking about 2 miles per hour when they had parted ways. Furthermore, he said that Plum went to the Library every Saturday to “research” for two hours. He said Plum would usually walk home at a quicker pace (about 4 mph) in an effort to get home by 8:00. Colonel Mustard knew this because he would often see Plum walking by “Mustard Manor” around 7:15, although he hadn’t been home to confirm this on Saturday.
Next Tran and Goza brought the Reverend Mr. Green in questioning. Here was his story:
“Well Plum and I were over at his place discussing matters of the City Council. About Noon Plum said he was going to head down to the Library for his weekly research work. So I left with him. We walked toward town together, at a slow pace, I’d say about 2 miles per hour. Plum’s a slow walker during the day. He says it helps him stay cool. I decided to stop by the Lounge for a cup of coffee. Plum kept on going. Said he needed to get to the library early so he could get home before dark. But he was still walking at that same slow pace! Anyway, I stayed at the Lounge for about an hour. After that I left and walked slowly to my house. Must have got there in another hour. I just stayed at the house for the rest of the day. Lotsa good football games on TV. Later on I saw Plum heading back home. It was about 7:00. He was walking much faster. Seemed to be in a hurry, but I didn’t think much of it. Plum is a peculiar guy, you know!?”
Goza and Tran are numbers guys so they decided to create position functions to keep track of where everybody was. Since both Mustard and Plum had started from “Plum Palace” they decided to create the functions and to keep track of the two suspects’ distance from Plum’s house between 12:00 Noon and 12:00 Midnight, where was the time in hours since Noon.
1)According to Colonel Mustard what is ? How do you know this?
2)According to Mustard, what was ? What evidence supports this?
3)According to Mustard, what are the roots of ? What does they represent?
4)Detective Tran says that . Do you agree? Explain.
Use Mustard’s story to create graphs for piece-wise linear functions that the Detectives can use to help keep track of the position of each of the suspects according to Colonel Mustard’s story. In other words, graph and on the same plane below.
5)Mustard claims he is innocent. Does his story match this claim? Explain by citing graphical evidence.
6)According to Mustard’s story should Professor Plum be considered a suspect? Explain why or why not.
Just as they had done before Tran and Goza created a piece-wise linear function to model Mr. Green’s positions (relative to Plum Palace) according to Mr. Green’s story. Use the plane below to graph .
7)What is ? What does this tell us about Mr. Green?
8)Explain why Mr. Green couldn’t have killed Mrs. White according to his story.
9)Detective Goza says Mr. Green’s story gives them one more point on , Professor Plum’s position function. Which point is this? How do you know?
10)Detective Tran and Detective Goza promptly arrested Professor Plum. What evidence do they have against him?
11)Do you think Professor Plum for sure the murderer? Why or why not?
12)Is there any reason to believe Colonel Mustard or Mr. Green could still be the murderers? What evidence supports this?
13)Help Detective Tran and Detective Goza put the evidence from Mustard and Green together by creating a single graph for all three suspects and writing equations to match each man’s position hours after 12:00. Use color coding for the graph: Orange for Mustard, Green for Green and Purple for Plum.
GRAPH:
14)Help them further by writing equations for each of the functions in the graph above.
EQUATIONS:
Mustard: Green: Plum:
The following Clues were provided after Professor Plum was arrested.
CLUE 1: Colonel Mustard did not lie about the time intervals when he was walking or the rates at which he was walking, but he did lie about where he started. He actually started his day at his own house.
CLUE 2: Mr. Green did not lie about the time intervals he was walking or where he started, but at all times when he was walking he moved twice as fast as he described in his story.
CLUE 3: Professor Plum did walk slowly to the library and quickly back, just as Mustard and Green described, but they lied about the times they were with him because he actually left his home 2 hours later than usual (at 2:00 PM).
16)Miss Scarlet also provided the detectives with a statement. Let’s look into her testimony together. Site evidence from the Clues that back up her statements.
a)She was not at the Lounge at any time on Saturday.
b)She passed Mr. Green’s house on the way to the Billiard Hall around 1:30 and saw him there.
c)She spent an hour talking to Colonel Mustard at the Billiard Hall between 2:00 and 3:00.
CLUE 1: Colonel Mustard did not lie about the time intervals when he was walking or the rates at which he was walking, but he did lie about where he started. He actually started his day at his own house.17)Based on CLUE 1 Tran and Goza created a new function for Colonel Mustard’s updated position relative to Plum Palace. Sketch a graph of this function below.
18)Detective Tran says that is a transformation of the original function . [No pun intended.] Which equation represents this transformation?
a)
b)
c)
d)
19)Explain your choice. Site evidence and be as thorough as possible.
CLUE 2: Mr. Green did not lie about the time intervals he was walking or where he started, but at all times when he was walking he moved twice as fast as he described in his story.20)Based on CLUE 2 Tran and Goza created a new function for Mr. Green’s updated position relative to Professor Plum’s house. Sketch a graph of this function below.
21)Detectives Tran and Goza know that a transformation of the original function . Tran says that and Goza thinks . Which detective is correct? How do you know?
CLUE 3: Professor Plum did walk slowly to the library and quickly back, just as Mustard and Green described, but they lied about the times they were with him because he actually left his home 2 hours later than usual (at 2:00 PM).22)Based on CLUE 3, create a new graph for Professor Plum’s updated position. We will call this function .
23)Explain how this graph is related to the previous graph you had created for Professor Plum (the graph of the function .
24)Detective Goza says . Do you agree? Explain why this is or isn’t correct.
25)Based on the new graphs what should Detective Tran and Detective Goza do next? Explain why?
26)To help the detectives make their case they need equations for the functions , , and . Use the equations from #14 to write three new equations.
2.5: Rumor Has It
Once the townspeople discovered that Colonel Mustard and Mr. Green had been arrested for Mrs. White’s murder there was lots of chatter and rumors. One of the rumors caught the attention of Detectives Goza and Tran. Apparently Mrs. Peacock had been seen with Mr. Green and Colonel Mustard in her car on the evening of the murder!
Let’s assume that the rumor was started by just one person at noon on Sunday. If each person who know the rumor tells one more person every hour…
1)How many people will know the rumor after 1 hour?
2)How many people will know the rumor after 3 hours?
3)Create an equation for , the number of people who know the rumor hours after it began to spread?
4)Graph on the plane below.
5)What is ? What does this tell us in terms of the scenario?
Assume instead that the rumor spreads twice as fast as was originally stated.
6)How would this change the graph of ?
7)Say represents the “twice as fast” scenario. What is ? How does this compare to
8) is a transformation of . Write an equation to justify that this is true.
What if the rumor spreads half as fast?
9)Say represents the “half as fast” function. What is ?
10)Write an equation for How does the graph of compare to that of .
Assume instead that the initial witness had actually been with a group of 4 other people and all of them had started to spread the rumor.
11)What would the graph look like then? Sketch it quickly below.
12)Write an equation that matches this graph.
13)If there are 5,000 people in the town how long will it take for the rumor to spread to every resident? (Hint: Remember logs?)
14)Is this function a transformation of ? If so, write an equation for the new function in terms of . If not, explain why not.
15)Detective Goza argues that your initial graph should have been . Explain why.
16)What could cause the graph of to shift up more than 3 units?
17)Would it make sense for the graph of to shift downwards? Explain why or why not.
18)What could/would cause the graph of to shift left in terms of the scenario?
2.3: Stretching the Truth
Last weekend Mr. Goza drove to San Clemente to meet his cousins for lunch. He fought some moderate traffic on the way. Traffic was heavier on the way home so he stopped in Seal Beach to watch the UCLA Football game with Mr. Tran. After a great Bruin victory Mr. Goza was back on the road. There was still a decent amount of traffic on the way home so it took an hour to get back to LA. The graph D = G(t) below shows Mr. Goza’s distance from home t hours after he left to meet his cousins (at 12:00 PM).
1)At what time did Mr. Goza meet his cousins? How long did he spend with them? How do you know?
2)What were Mr. Goza’s velocities during his travels? How do they match the details of his day?
3)Use your graph and/or equation to fill out the following table.
t / G(t) / 2G(t) / G(2t)0 / 0 / 0
1 / 80
2 / 80 / 160
3
4
5 / 60
5.5 / 100
6 / unknown (probably 0)
7
8 / 80
9
10 / 40
11 / 0 / IDK
4)What are the roots of 2G(t)?
5)What are the roots of G(2t)? Why are they different from the roots of G(t) and G(2t)?
6)Why are there unknown values in the table?
7)On the plane below graph y = 2G(t) and y = G(2t) in different colors.
8)Explain how multiplying the output by 2, as we did to create 2G(t), changes the function G(t).
9)Explain how multiplying the output by 2 would change the story.
10)Explain how multiplying the input by 2, as we did to create G(2t) changes the function G(t).
11)Explain how multiplying the input by 2 would change the story.
12)Write a piece-wise equation to match 2G(t). (This one is easy.)
13)Write a piece-wise equation to match G(2t). (Harder. Use substitution.)
14)Using your own table and/or the graph of y = G(t), graph y = G(t) and y = Gin different colors on the plane below.
15)Explain how multiplying the output by , as we did to create G(t), changes the function G(t).
16)Explain how multiplying the output by would change the story.
17)Explain how multiplying the input by , as we did to create G,changes the function G(t).
18)Explain how multiplying the input by would change the story.
19)Write a piece-wise equation to match G(t). (This one is easy.)
20)Write a piece-wise equation to match G. Check your work with Desmos!
Created and designed by Liem Tran and Nate Goza for