ESCOT Summer Workshop 2000
ESCOT Summer Workshop 20000 || Math Forum Workshops
Day 2 Summary
New ESCOT Interface
Chris shared the new ESCOT interface. The plan is to use applets this year instead of the ESCOT Runner.
Ideas for New Problems:
Students will identify
themselves before using the applet by logging in, or they can
register as a guest. Only if they login, can they save their work on
the applet. They won't have to describe as much of their work in
text. As a guest they can only use the applet.
Catalog is also new.
explanation and directions will appear at the top of the page. Then
the activity loads while you wait. When you click on the button, the
applet will appear in a new window. This way, you will have one
window to work in and one window to write answers/responses. There
is some concern about students needing to juggle multiple windows.
Other concerns include machines being able to run JAVA and the size
of the monitor. This new format prints well. Suzanne likes having the
activity as a separate window. It allows them to concentrate on the
activity without getting caught up in the print. A computer need to
have Swing in order to run the activities on the browser. This setup
only happens once.
The catalog contains components,
description of the activities, background information, and a link to
the Math Forum.
Components include Sketchpad, graphing,
box plots, histograms, world, tables, a spinner (for probability
problems), a calculator, Image/J, widgets, a recorder, and
The project will use Bean Builder 2000 to create
the ePOWs. It lists the components and allows developers to modify
the components. If you click on "sample", the program will record
data from the activity like a snapshot.
Each strand in the NCTM Standards is described
and linked to specific problems that address the
Individual participants contributed ideas for the
problems list. Some conversation ensued to clarify the problems. Ten of the problems were discussed in two break-out sessions.
Session 1: Choosing problems for collaborative teams to elaborate.
1. Fraction Tetras -- Nathalie, Bill, Gerri, Seth, Alex
The group brainstormed many ideas. Two ideas that we like:
2. 3-D -- Steve, Chris
- The first is kind of freeform. Students build a mosaic by
coloring pieces and setting fraction parts. They can build a wall or
stained glass window with the shapes. This may be something like tile
- The computer would seed a row with a number of
tiles. Then kids would be given a format for their answer
(percentage, fraction or decimal). Maybe they would want to get as
many rows as possible. Rows could be numbered, you'd be told the
pattern (1 block, 2 block, etc.) They might choose blocks from a
"box". (Leslie might have ideas to add since she had the original
seed idea.) Instead of working with the time component of the
original Tetras game, set a challenge to see how high a wall can be
built with different rows -- combinatorics.
3. Chances of making a triangle from a stick --
Dave, Vicki, Mark, Orit
- The group spent a fair amount of time
defining the problem and making sketches.
- It can be 2-D or
3-D. What happens if you double or triple the dimensions. What happens
to surface area? Three main directions:
- Try making it a
unique problem by having students work on the screen and then print
out the net. Compare volume change to surface area. They would
actually build the shapes. Perhaps use Poly for this.
- Do it more graphically. Would we provide a GSP
opportunity? Could plot data to show changes in area, perimeter,
- A 2-D question would be: How does the perimeter change
when you change the configuration of tiles? How many different
"perimeters" can you get with a certain number of tiles? This could
be a fixed area problem. Ex. take a set area, how does the perimeter
change? Or set a fixed perimeter and investigate different
The group quickly realized how difficult
this problem was. Combining math and standards: first they needed
to figure out where to make the breaks. There is a site on the Math
Forum with ideas for this. Nathalie showed some programs that could
already help with this. To define the problem, they used counting
numbers as lengths (up to 12) and listed the
4. Taxi Cabs and Euclidian Circle Geometry --
Nick, Jim, Wenming
They want to do more research, need more time
and a computer. Jim found a Math Forum site that already addresses
this problem. This could be a reason to discount it from the
project. Nick: However, many problems students get in class are
available somewhere else. Instead we should think about the on-line
experiences. Can the technology add something to
5. Zooming the graphs -- Jeremy, Holly
brainstormed many approaches to the problem and then worked to narrow
their focus to two questions. This was difficult due to the richness
of the possibilities. They have two leads.
- Make a
line look like y=x by changing the scaling. The students won't need
to know a lot about equations to work with this. They could do this
with a line and with a parabola.
- Use two functions. You
have to change m in one, and b in the other. Make them intercect
within the zoom, make them not intercect within the
Session 2: Choosing problems for collaborative teams to elaborate.
Question to the group: Nick asked, What
kind of questions do we want to ask? Should they all have a specific
right answer? Or should they be more open-ended?
- What math
- What technology?
- What Standards and topics
will this cover?
- A child should be able to state an answer and support it
with a full explanation.
- Mentors have to work harder when
questions lead to multiple answers.
- In a sense all of our
problems are open-ended. Our real questions should be: How can we use
technology to teach concepts like pi?
6. Spinner -- Mark, Gerri, Seth, Holly, Alex
Birds of a Feather
Task: Design a spinner for a carnival game. The company
pays $10 each time they hit the CD. They pay out $1 every time they
hit the yo-yo. The company can't keep more than 10% of what it
earns. Students can change number of wedges, color of wedges, and
location of wedges. Perhaps students can define their own functions
using a spread sheet or data table.
Concern: implications of
gambling. Perhaps use chores in a family instead of raffle. Or
could use a grid where people move around and then the caller yells
stop; "players" see what color square they are on.
Lots of ideas
without focusing on one best question.
7. Marbles -- Seth, Holly
They generated a lot of ideas without exactly seeing how to put
it into a problem. 1. Two bags of marbles, compare the probability
between them. 2. Make a bag match the probability of another bag
set. Look at probability as fractions. 3. We have a bag and do not
know what is in it. What kind of sampling can you do to figure out
what is in the bag? Can you make several bags of marbles that would
have the same probability ofgetting a
8. Rational/Irrational -- Chris, Jim, Suzanne, Nathalie, Vicki
There were a few issues we tried to address. How
can we show the distinction between rationals and irrationals? How
can technology help us deal with that? Also, how do we show students
what makes numbers like sqrt(2) special?
On the issue of getting sqrt(2) to have meaning, we could approximate it and see a visible representation of the difference between our estimates and the actual number. Can we zoom in on the number line where each interval is
broken into 10 pieces and each successive decimal amounts to finer
and finer break outs. We could color code the intervals, and zoom in
by orders of ten to get closer to the irrational number. Problem: We
weren't convinced that students at the end of the period wouldn't
believe that if they just had more time they would be able to "get
Another possibility is to create the area of a circle
square so that the radius or diagonal of it is a rational number.
9. Sprinkler -- Jody, Orit, Steve (Nick - this is related to
a parking lot problem I received yesterday)
Task: Watering the
lawn. It can be any size. Cover the lawn with sprinklers, selecting
from 45's 90's and 180's. Minimize water (overlap) and cost of
sprinklers. They could change the radii and figure out the best way
to cover the yard. This problem addresses the Standards. It does not
need to be a specific mathematical skill, but focus more on real
world understanding. This is like the Grazing Goat
Vocabulary to include: Transformations, rotations and
10. Pool Table -- Jeremy, Nick, Bill,
Pool tables and angles were too easy/boring. This led to
other ideas. Question: What if there were a round pool table. Using
GSP, what are the angles so that when you hit the ball it will come
back to you. Questions for middle school are appropriate because
they will use common angle measurments (90, 60, 45) and look for
patterns. The group has an intuitive feeling that there is a good
question in there. What are the shapes? There are three kinds of
shapes that are formed: lines, polygons, stars. What is the measure
of the angles for the polygons? 180/s. Is there a relation between
the star and the polygons. Standards? observing number of interior
angles of polygons, interior/exterior angles,
Alex led a
workshop using AgentSheets
Tuesday, August 15, 2000
Others played with math.
Written by Kristina Lasher