ESCOT Summer Workshop 2000
ESCOT Summer Workshop 20000 || Math Forum Workshops

Day 4 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 thought as a possible discussion topic.

What would it look like if we had problems that are not mentored? Is ESCOT about looking about ePoWs or is that just a part of ESCOT?

These questions were prompted by a conversation yesterday about types of questions that could lead to a more open-ended approach. It would be interesting to look at different models of ESCOT without the mentoring or with open-ended mentoring. We decided should be an ongoing conversation possibly for lunch and later in the day.
Group Presentations (continued)
Fish (formerly Marbles & Spinners)

Concepts: Ratio/proportion; visual representation of data; data sampling.

Activity 1
Students will be shown a fishtank with a specified number of guppies and goldfish, as well as four smaller fishbowls (which may or may not contain fish). Near each bowl will be a numeric ratio, as well as visual representations (e.g., bar graph, pie chart) dynamically showing the ratio of guppies to goldfish in that bowl.
  1. How can all the fish from the tank be allotted to the bowls so that each bowl contains its specified ratio?
  2. Which type of visual representation was the most helpful in arriving at that solution, and why?
Activity 2
A cloudy fishbowl is displayed -- the fish inside can't be seen. The student can sample its fish population by scooping a fish out and then returning it; this can be done as many times as desired. The results of the sample to date will be displayed in visual charts and/or tables.
  1. Which of the four existing fishbowls has a guppy:goldfish ratio that most closely matches the one in the cloudy bowl?
  2. What did you have to do to be confident that you had found the best match? [We're looking for them to discover that increased sampling creates a better idea of the hidden "data."] (There may instead be a more open-ended question, such as "What can you determine about the contents of the cloudy bowl, and how"?)
Holly added that as they are putting the fish in the bowl there is a dynamic movement of ratios and fractions. Nick said that an interesting extension might be if some of the bowls had it already. Chris added a collaborative activity might be that each kid in the group has to be responsible for one bowl.

After Dave explained the second part of the PoW, Steve added that there might be a way to combine problem 1 and problem 2. It could be made to be more open-ended by stating: Here are your tools. Tell us as much as you can.

Nathalie said that you might not even need the small fish bowls there. Leslie said that a cool extension might be to go out and find some actual data. (Go to environment and figure out how many sharks there are in the Pacific). Jeremy added that he liked the black bowl vs. white bowl. It may be just simply a place the teacher goes. He likes the image of one under your control and one not.

Nick asked if giving them the ratio in four different ways would eliminate some of the thinking that we want them to be doing? Jody said that there are two ways of doing it: multiple representation might be a nice scaffold. Nick said that it seems that if you ask what is the ratio?, you are giving them the answer.

Steve W. said that he would be inclined to give them numerical fields where they could drag one variable to various charts, so that they could make their own charts. They could make ratios of their own. They could make their own charts. Jeremy said that we could have a scatter plot.

Leslie said that then you start looking at a rubric, with types of responses and answers. There may be some value in coming up with a suggested rubric because her response and Suzanne's may be different. Suzanne agreed. Leslie volunteered that she and Suzanne make up a template for a rubric and she outlined some preliminary thoughts:

If we had a template for a rubric, then the teacher could use it when creating the expected solution. Categories to consider include: completeness, on task, mathematical thought and correctness. The process standards could be used as a starting point.

Concepts: Sequencing;logic and reasoning.

Activity 1
We recognized that it shouldn't be pirates and diamonds but it should be a liquid so that you can't just put a pile of diamonds on the table and pick them up when you are ready to go. Using some type of liquid may help keep the kids on the problem with the containers. Time was spent analyzing the previous problem and we asked ourselves what the valuable math was. What did we really want the students to learn?

The most important lessons are:
If you combine even numbers you get another even number.
If you combine odd and even numbers you get either an odd or even number.
Once we agreed that that was the mathematical focus then we could approach the design of the problem. Next we asked, what kind of support could we add to the problem that used technology usefully? We are thinking that the student version would have minimum text emphasizing an applet button that will take the student to the interactive window. There will be a space to record observations.
  1. Which amounts in the table can you get?
  2. What patterns do you notice when you look at the results of the table?
  3. What do your results say about combinations of even and odd numbers?
We weren't interested in how they were measured or the steps used to measure. We tried to stick to our goal of "You can't measure an odd number from two even containers."
Nathalie suggested starting with odd and even containers. She suggested that there is a strategy of viewing even and odd to approach problem solving.

There was a discussion of whether this is a one week or two week problem. One week of mentoring but teacher support could extend the time the students work on related activities.

Context - sometimes the context is natural and genuine but we don't necessarily have to invent a context. So, this problem might be an example of not needing a context.

The teacher version would have support pages which could be used to do introductory distance, rate, time concepts if necessary.
Jody suggested that the participants work out the following problem:
    Finding the Area of a Square

Three different methods for solving the problem were suggested by the group and then Jody showed some student solution examples:
    Some Sample Solutions

We looked at a variety of solutions and thought of possible responses the mentor might provide the different students. What would you ask them to do? What would you say to them?

We continued by looking at examples of what a mentor might respond to a submission by a student:
    Some Sample Solutions and Responses

Participants gave ideas of the quality of the responses and made suggestions for improvement.

Jody gave an overview of the PoW Office and briefly explained how it works. She mentioned a style that Annie Fetter uses in summarizing the submissions once the week is over. Annie has a unique style of "talking to the students"
Group Presentations (continued)
Designing a Sprinkler System
Students explore how to maximize coverage while minimize water usage which minimizes cost.

  1. What is the cost of water for your best sprinkler system?
  2. How did you come up with that cost?
  3. Why do you think your best system is best?
There was a discussion on the mathematics behind the problem. The question was asked if the lawn concept would be familiar for all students
Playing Fractions/Fractis (based on Tetris)

Week One
The students have different blocks that are color coded. They click on the bar to make a picture or letter that has them work with symmetry. What is the value of the bar? When they submit the answer they would like to have the design submitted as well.

Week Two
Level 1 - Rules:
Computer drops a piece. In put the size of the bar(s) you want to fill in a row.

Level 2 - Rules:
Computer drops 2 pieces
Input the size of the bar(s) you want to fill the row.

Possible other Level - Rules:
Not all fractions available (not same denominator)

  1. Given this set of fractions, how many different ways can you fill a row?
  2. Fill up the tray as high as possible with the number of pieces indicated.
See Sally and Suzie Ride

  1. Explain the mathematical meaning of the red, purple, green and moving gray lines, and the horizontal and vertical axes on the graph.
  2. What does the interesection of the red and purple line tell you? What does the intersection of the moving gray and purple lines tell you?
  3. Write an equation (a distance, rate, time function) that allow Sally to figure out how long it will take her to ride her bike from school to home.
The lines should be separated or listed separately in the first question so that it is very clear how many things should be explained.

Is it important to the layout that the animation space is too close to the x-axis? The answer to this was, no. Could time be on the vertical axis?

What we were trying to get at was not reached last year. What if the yellow line was coincident with the gray line, what would that show? What kinds of questions could you ask that make the students move the lines to a certain point and then ask about the relationship.

The group agreed that students encounter questions like, is this number irrational or rational? This problem would use a guess and check tool. The students would be asked to find a ratio that when squared is 4. They would be given a field to type the numerator and the denominator. A button named "square it" will produce an answer. After initial exploration the students would be asked to find a ratio that when squared is 2. Because this would not find an answer - the "square it" button would stop after a certain number of trials and would change to autotrials.

Another interesting question would be to find a ratio representing the circumference of a circle with radius 1. This would result in looking at pi.

Some history of rational numbers will be built into the teacher support pages.

Concluding question:
What would you name this?
Nick commented that this would be difficult to program because of rounding errors.

Make suggestions of numbers to try.

Once you have the definition in place (with a fraction I can't express an irrational number), then what do you do with them? Might this be where we tie it into history? Nick gave an example - after they have been introduced to the idea

point moving around the circle compare that length of the circumference then rolled out to a line segment. The line segment is a rational number and the circle is 2pi(r)
Dividing the Line

The idea of this problem is that you break a stick into three pieces, so you make two cuts anywhere along the stick. The question is whether your three pieces will form a triangle. There are two ways of looking at this problem, one continuous and one discrete.

In the discrete form, Dave suggested that we look at the probability that you will be able to form a triangle given breaks at integer lengths of the stick. You might look at different lengths of sticks. Steve W. also suggested that we provide a bar graph like representation of the sets of three pieces that make triangles.

In the continuous version, after having experimented with trying to put the pieces together to create a triangle, Steve W. suggested that we look at a graphical representation of the solution space to this problem. The first representation plots the distance along the stick to the first break againt the distance along the stick to the second break. The plotted points are colour coded: a blue point, say, would indicate that a triangle can be formed and a red point indicates that a triangle cannot be formed. Another option is to plot the length of one piece against the length of the second piece and colour code as above. As students manipulate the lengths of the cuts, they would be able to track positions on the graph to experiment which values result in a triangle being formed.
New Problems to work on
Discussion was open to suggest new ideas for problems which were then added to the list that was being generated.
Birds of a Feather
After dinner some of the participants returned to the lab to learn more about the mentoring process. Jody demonstrated the office side of the ESCOT PoW and participants tried it out.

Nick Jackiw presented a Geometer's Sketchpad overview and then the participants were treated to a showing of some of Nick's fancy and impressive sketches.
Thursday, August 17, 2000
Written by Suzanne Alejandre

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