ESCOT Summer Workshop 2000

ESCOT Summer Workshop 20000 || Math Forum Workshops

## Day 5 Summary

Connections/Opening Discussion
The group started the day following a Math Forum tradition of sharing thoughts in a quiet setting. After Connections Nick shared a technical idea related to Seth's comment during Connections of students helping to create problems. We should look at using the Web in some way to have students create a version of the problem tailored to their thoughts. Jeremy said that this is a possibility.

Jody pointed out to the group that a workshop site has been made during the week. She gave a tour of the ESCOT Summer Workshop 2000 pages. Jeremy said that one thing they have been working on is how to have an online site to help facilitate online work between members of an integration team. The ESCOT ePOW Project page is being developed to serve this need. Various features were pointed out by Jeremy.
Group Presentations (continued)
Pool Table Problem

Concept: Angle of incidence equals angle of exit.

What happens when you knock a pool ball around on a circular pool table. We spent the entire meeting learning about star polygons but did not come up with an activity. It's a rich idea but if we are going to work on it, it would require more studying of geometry.

Discussion:
Leslie asked why not start with a rectangular pool table? Nick responded that this has been done too often and he would prefer exploring something a little different.
Scale N' Pop

The idea has been changed from the bowling scenario to that of a balloon traveling upward through a gap and going up toward some sharp objects that will pop the balloon.

Questions:
1. What fractions popped the balloon in booths 1, 2, 3?
2. Review the sequence of fraction you used in booth 3. Explain your strategy for picking fractions in this sequence.
3. Last year a student playing this game in a different booth found that 3/2 was too small and 5/3 was too big. Recommend a strategy the student might use to find the fraction that pops the balloon.
Discussion:
Vicki commented that the students need the original representation and the fraction used for scaling and the resulting size. When they put in 3/4 then they should also see the new size - a visual picture of this would be helpful.

Original diameter times the scaling factor equals the new diameter.
Fraction Darts

This is a game that has a line with two green pegs. The student is typing in a denominator and a numerator and then a button to press to throw the dart. The first throw has to be between the two green pegs. You can continue to play if you throw your dart between your last dart and the green peg.

Questions:
1. What was your longest sequence of fractions?
2. Explain the strategy you used to pick the fractions in your longest sequence:
3. Describe a strategy for playing a very long game.
Discussion:
Overall reaction was positive. Bill commented that many of his students' families are from countries who do not use the American system of measurement. He pointed out that reading rulers cannot be assumed because of culture. Jody suggested a Bonus question might be appropriate for this problem because for some students this would be simple. Is there a place where you can put a peg that you can't throw a dart? (an irrational number)
Olympic Graphing (Shoelaces)

Leslie said that there was a lot of discussion on Shoelaces yesterday but they really wrote down their thoughts this morning after having thought about it overnight. There is a graphing tool and they can plot points and/or show line. Data will be provided in a data table. Links to marathon history and this year's Olympic race would add to the problem. The introduction will be the men's data all graphed including the line of best fit. The students will be asked to use the tools to graph the women's data.

Questions:
1. Use your line to predict the women's time in 2050.
2. Predict when the women's time will catch up to the men's time.
3. Look at the x and y intercepts - (where the lines cross the axes) - What's the real world meaning of these points? Does this make sense? Why or why not?
Discussion:
Dave commented that he had seen this problem before and he said that he would see if he could find it again and point them to the site for reference. On a small scale the data makes sense but on a large scale it doesn't. Seth commented that for it to work the data set might need to be restricted. The question could be changed to look at earlier data, predict the 2000 results and then link to the real data to compare.
PolyRhythm

In Agents Sheets music is possible. There is a graphical display of musical notes including a measure broken into 4 equal parts and a second measure broken into thirds. Another button that says "shift thirds" and the sound is again different. There will be a link to a 10 or 20 second QuickTime music video.

To Do:
Listen to the rhythm. Listen for 12 pulses and 4 main beats.
Click on add 3rds and listen again.
Edit...so it plays 3's in a different pattern. See how many different sounding patterns you can get.

Questions:
1. Why are they called 3's?
2. If you added 8's to the beat, how many pulses before the pattern would repeat?
Discussion:
Steve A. said that it would be important for the student to see what they were actually hearing. The wording is important. The nomenclature was not clear. Jeremy said that it might work best to avoid standard musical notation and use something specific to this problem.

Jeremy suggested a downbeat for the beginning of each measure to orient the student to what they are listening to. He said that it is easy to become disoriented. Jeremy asked if Agent Sheets could handle this technologically and Alex responded that the tricky part is to play different tracks at one time. They discussed ways to get around that problem.
Galactic Exchange

Jody displayed a mockup of the work that the group did on this problem.

Scenario: It's the year 2075. You take a vacation to the planet Orange. When you arrive, you notice coins of three different shapes: stars, circles, triangles. Unfortunately, no one speaks your language, so you have to figure out the values of the coins on your own.

## Vending Machine

The prices are worn off, so you have to try different combinations of coins to get food.
1. What exact combination of coins do you need to purchase each item?
2. Which coin is worth the least, and how many of them does it take to equal each of the other coins?
3. Your friend(?) arrives in the next shuttle. You need to explain the coin system to your friend. You have to explain how you figured out the relationship between each coin because he doesnąt trust you.
Bonus: Explain how you figured out the relationship between the coins in another way because your friend didnąt understand it the first way.

Standards: algebra (finding the relationship between 3 unknowns, develop an initial conceptual understanding of different uses of variables)

Discussion:
Leslie commented that maybe Question 2 was not necessary. Steve A. thinks that there should be that interim step to help kids continue with their thinking. As other comments were made Jody edited the mockup of the problem that she was displaying.
Exponential Growth

Using the virus simulation theme, students experiment with exponential growth. The idea is to have students understand qualitatively exponential growth including the common cold, population growth, and rumors. The group is still struggling with questions.

Questions:
1. Describe an exponential function.
2. Give another example of a real world phenonon described by an expontial function.
Discussion:
Nick said that there is a difference between looking at social situations but also to mis-represent them. Nathalie said that a comparison between a multiplier model to the exponential growth would be interesting. She thought the rumor model might work with this. Jeremy suggested using the idea of a computer virus.

Chris said that he has heard a lot of this type of activity. Vicki said that providing the different contexts would broaden the view and help. Chris would prefer that the activity focus on one particular question. Nick suggested one specific question as a focus. Someone is telling a nasty rumor, would you prefer that they tell A number of people, or B number of people? Nick asked what role the mentor played in this process? Nick suggested that a mentor could provide the student with some related URLs to use. Another example (of exponential growth) could be provided as feedback rather than just be in the problem itself.
Irrationals - update

Chris projected a mockup of the work he has done on the Irrational problem that he has made since the group presented the problem. There was a discussion of the importance of the display of the decimal that showed that the guess was not a good result and just telling the student that it was "a little too large" or "a little too small." Chris said that as the students were trying to find a ratio squared that would result in 2 they could proceed to a button that would "try a lot of good ratios." Seth asked if the students could see the numbers that were being produced.

Revisiting the List of Problems

Jody projected the list of problems so that the group could prioritize. Personal schedules and difficulty of the technology were considerations.
Problem Team Members Time
Elephant
Pirates/Larry Potter
Galactic Exchange
Olympic Graphing
Scale n Pop
Rumors
SnR-paths
SnR-critique
Sprinkler
Fractris
Fractris
Fish
Cloudy fish
Graph Zooming
Fraction darts
Sally & Suzie
Polyrhythm
Leslie, Jeremy, Bill, Vicki
Chris, Suzanne, Gerri, Seth
Bill, Mark, Jody
Leslie, Seth, Nathalie
Jeremy, Hollylynne, Dave, Nick
Wenming, Gerri, Vicki
Nick, Nathalie, Steve
Nick, Nathalie, Steve
Steve A, Jody, Wenming, Nick
Leslie, Nathalie, Seth, Alex
Leslie, Nathalie, Seth, Alex
Dave, Hollylynne, Mark, Andri
Dave, Hollylynne, Mark, Andri
Leslie, Jeremy, Bill, Vicki
Jeremy, Hollylynne, Dave, Nick
Gerri, Suzanne, Jeremy
Seth, Leslie, Alex
Oct-00
Oct-00
Nov-00
Nov-00
Dec-00
Dec-00
1-Jan
1-Jan
1-Apr
spring
spring
spring
spring
spring
spring
spring
spring

After the schedule was fixed Jody said that the roles for each group should be decided upon. The roles are:
We broke into individual groups to discuss the details.

Evaluation Thoughts

Ann Renninger said that she is interested in how things are working in the project.

Some questions:
What are the broad questions that guide this project?
Do the ESCOT PoWs provide a meaningful mathematical tool for learning?
What is the role of technology in learning mathematics?
Where is the math in the problem?
What does technology bring to the problem?
Do you need a story context, what's the role of the context?
What is involved in sustainability, scaling?
If the ESCOT problems go live this year, who is going to do the mentoring?
The ESCOT project as an exemplar of research and practice - what does that look like.
Ann explained that she is interested in having this conversation because she would like the ESCOT participants to have a say in what the evaluative piece will look like this year. Last year the evaluation centered on:
How did those problems work?
How did they compare to the MidPoW?
This year it would be great to have the group talk about what we would like to evaluate. The following ideas were shared:
• Library of ESCOT interactive tools
• Posibilities of customization
• Interest in studying what kinds of changes ESCOT participants see within themselves
• Cost analysis - how efficient is this project? How much does it cost and how many students are impacted?
• How do we create something that is productive but also so that the participants feel they are gaining something?
• Developing technology has previously involved researchers but ESCOT is different because teachers are getting involved.
• Involve students as part of the integration teams.
• What is an integration team? Is it just people coming together or is it a process? What does it mean to us for an integration team to be successful?
• FAQ development - do ePoWs have to have a definite answer?
• Does this tool integrate well with traditional teaching practices? Is it aligned with what students are/should be currently doing?
• What kinds of experiences have been significant? What really has worked?
Ann shifted focus and asked the participants what features of the integration teams are positive. The following ideas were shared:
• Face-to-face contact
• Always having somebody who sensed a need to step in - someone taking a global view.
• Normally at school when he whines about something no one responds but in the workshop environment, his comments meet a response
• Something about the context each different kind of expertise is valued.
• People share.
• There is modeling and leadership.
• The Forum's facilitation must be pointed out.
• We come in with well-defined roles.
• Each person's contribution is valued and as you give input you have that confirmed and that leads you to continue to contribute.
• By visualizing a problem, you will remember it for the rest of your life.
Friday, August 18, 2000
Written by Suzanne Alejandre