Math Forum Discussions

Steve Weimar

Posts: 24
Registered: 12/3/04

j.e. - dog pen prob.
Posted: Sep 25, 1993 5:56 AM

Plain Text

(Posted for Keith Grove: Dover-Sherborn High School
<doversherhs@igc.apc.org>
Please send mail replies to Keith.)

It seems like a long time since I've written anything in my journal. I've
had computer problems as I have stated, but more than this I have mostly
just been enjoying the beginning of the year. There is nothing like the
first few weeks - with all the new relationships and surprises.

Thanks to some wonderful ideas which I have received through the Geometry
Forum I believe I have gotten off to a good beginning with portfolios. (I
will share the documents, which I have distributed to the students, later
in this entry.) Today I collected the first problem of the week (POW). I
used the well know dog pen problem:

You plan to build a pen with at least one gate for your dog. Fencing is
$1.00 per foot, fence posts are $2.00 each, and gates are $5.00 each. You
can afford only $40.00 total. How would you design your pen in order to get
the greatest possible area? Explain what you considered in making your
plan. I first asked each student for homework to write down his/her
understanding of the problem, possible strategies, diagrams, and
appropriate geometric concepts.

Then the next day in small groups of 3-4, the students spent 30 minutes
brainstorming while I floated around the room and took part in several
interesting conversations and heard about a very different geometry than I
know! The students had difficulty formulating strategies for determining
the "best" shape for the pen. In fact, many students, were not even
considering the possibility of choosing. They were just going to make a
rectangle. Those that did think that the shape would in some way affect the
area, were not thinking about collecting data or doing an inductive
investigation - even though we had spent most of the previous week using
Serra's inductive patterns and I had spoken extensively about inductive
thinking.

So the next day, I posed a simpler version of the problem (each day I share
my rationale for the lessons with the students - in this case I shared what
I heard the previous day and why I had decided to pursue a simpler problem
- a well worn mathematical strategy.) I asked them to intuitively predict
which shape from among a rectangle, square, triangle or circle would have
the greatest area if each had a perimeter (or circumference) of eight feet?
The intuitive responses included every shape as well as "it makes no
difference, they would all have the same area". When asked how we would
proceed in this investigation, several students were able to suggest
collecting data and drawing inductive conclusion!

We used geoboards to investigate the rectangles and squares and Sketchpad
to investigate circles and triangles. By the end of the class, the circle
emerged in most students minds as the best alternative from among our
choices. Two days later (today) I collected the problem. I had them
self-evaluate their work using the document below (sent to me by Henri
Picciotto) and I will use the same standard to offer feedback to the
students.

There is much more I could say, but time is beginning to run out on this
entry.

Questions for readers of this journal:
1. As always, any feedback?

2. I plan to ask each student to choose the number one interest or concern
in her/his life and begin a year long investigation into the relationships
between geometry and this interest. Also I will ask each student to choose
his/her favorite activity during school hours art, music, social studies,
lunch ...) and continually look for relationships between the activity and
geometry.

Does anyone have any feedback about this idea? What is the best way to use
the Geometry Forum to aid and inspire the students efforts? Experts?
Advisory councils? Student teachers?

3. What do people think about having a contest for the best student
Sketchpad demonstration? $1,000 scholarship for the winner, etc. money and
food seem to be powerful motivators!

4. Other ideas about how the Forum can be utilized at this point.
Logistics: It looks as if I will have frequent access to a computer lab
with 6-8 Macs. The restrictions include: I may have to share the room with
other classes (using IBM's), the Mac's are only capable of running the
Standard Version of Sketchpad (slow, slow) and no color of course.
I'm still very much interested in find a donor/sponsor to purchase 10 Mac's
for my classroom - color monitors, floating point processors, etc. I
haven't received much response to this posting.

5. Any ideas for funding???!!!

Evaluation of Writing Assignments

Writing assignments are an important part of your work in this class. Here
are the things I will be looking for when I read your papers.

The Math:
- Did you understand the problem?
- Did you express yourself clearly?
- Did you summarize all that you learned about the problem?
- Did you answer all the questions?
- Did you think about the problem beyond what was asked?
- Were you creative in your approach?

The Presentation:

- Do you have an introduction that states what the paper is about?
- Did you organize your paper into paragraphs?
- Did you illustrate the paper adequately?
- Did you avoid repeating yourself?
- Is the paper neat and legible? Is it beautiful?
- Were you creative in your presentation?

Please use these standards to self-evaluate your papers. Give yourself one
point per "yes" answer. A score of 10 or above is Excellent. If the score
is 6 or below, the paper should be redone.

Mathematics Portfolio Guidelines
--------------------------------------------------------
The purpose of compiling a mathematics portfolio is to showcase or
highlight your unique mathematical abilities, to provide a much broader
view of what you can do, and to provide an opportunity for self-assessment
and self-reflection.

Each week you will be asked to select your favorite or best work from past
writing assignments, quizzes, exams, or projects) that shows evidence of
one or more of the following criteria:
---------------------------------------------------------
Demonstrates skill in using a variety of problem solving strategies to
interpret and solve math problems:

Applying a new technique
Writing an equation or inequality
Drawing diagrams/pictures
Guessing, checking, and revising
Making an organized list or table
Using logical reasoning
Looking for patterns
Simplifying the problem
Working backward
-----------------------------------------------------------
Gives evidence of using resource material and technology sensibly as tools
in understanding math concepts:

Using "hands-on" manipulatives
Using school, community, and university library materials
Accessing computer bulletin boards to send/receive information
(telecommunication)
Using computer software to analyze, model, or simulate

-------------------------------------------------------------
Demonstrates quality and creativity in performing and documenting math
activities and investigations:

Written work is legible, and well-organized
Uses math vocabulary and terminology clearly
Approaches problems in a systematic manner
Uses creative and unique methods of solution Perseveres in solving problems
Checks work independently
Works well with others
Demonstrates self-confidence
Exercises leadership
---------------------------------------------------------------------
Each math folder selection must include a brief but specific explanation of
why you've chosen that piece for possible inclusion in your semester
mathematics portfolio.

Your reasoning should specifically refer to one or more of the guideline
criteria (above) and be stated in a positive manner. Use complete sentences
and attach your explanation to the front of our selection.
---------------------------------------------------------------------
EXAMPLE:

NAME: Bob Montgomery
LASS: Geometry
ATE: 9-17-93
Math Folder Entry #2
For the week of September 13-17, 1993, I've selected Assignment #12 to be
included in my math folder because it demonstrates that my written work is
well-organized and shows that I persevered at learning to recognize
patterns.

Good night.
Keith

From usenet Sat Sep 25 11:30 PD 1993
Path: forum.swarthmore.edu!uunet!cdp!doversherhs
From: Dover-Sherborn High School <doversherhs@igc.apc.org>
Newsgroups: geometry.pre-college
Date: 25 Sep 93 11:30 PDT
Subject: DSHS Journal-Students'Questions
Message-ID: <1800700030@igc.apc.org>
X-APC-HostID: 1
Sender: Notesfile to Usenet Gateway <notes@igc.apc.org>
Lines: 59

point quesTEXTMSWD
3(=)8(EbStudents' questions about the word POINT (after reading the
efinition in Michael Serra's book): "A point is the basic unit of
eometry. It has no size. It is infinitely small. It has only location.
very sharp pencil tip is a physical model of a point. A point
owever, is smaller than the smallest dot you can make with your
encil."
-Why wouldn't a needle be an example of a point rather than a
harp pencil tip?
Is a point simply a dot, or is it more complicated?
Is its only use location?
How big can a point get? Can it get very large?
Is the point the origin of all lines and angles?
What's a point made of?
Can you put a point anywhere besides a line?
They say a point has no size, then they say a point is smaller than
he smallest dot you make with your pencil. Which is it?
Why does a point have to be smaller than the smallest point that
ou can make with your pencil? Why can't a point be larger than
hat? Could this be a point? . If not, why not?
How do we know actually how large a point is?
How big is a point?
A point can be forever smaller, but can it be forever bigger?
It can only have a location. Why can't it have length or thickness?
How many fit in one space? Is there a limit?
How many points can be in a line, an infinite number, or can only
certain amount be in it, depending on the line's size?
How large is it?
If a physical example of a point is a sharp pencil tip, why is a dot
r point smaller than any point you can make with your pencil?
How many dots or points can you put in a circle or on the line of
he circle? Does it depend on the size?
What does it mean that a point is the basic unit of geometry?
How small can a point be?
What does it mean when they say it has only location?
How can you call a point on a worksheet (this size . ) a point if the
efinition of a point is really small, like you can't see it?
Why can a point only be "smaller than the smallest dot?"
I don't understand why a point must be small?
Where do you determine to put it?
What determines a point?
What is the actual size of a point?
Why does a point have no size?
Could you see a point?
Why do they exist?
How do people know how small the point can get?
I get what a point is, but how can it have no size? How do you
now how big or small it is then?
In addition to any comments anyone might have in response to the
uestions, you are also invited to comment on the purpose or
alue of having students write their questions.
What I would like to help the students understand is - what
akes a good question? Any help in this area would also be
ppreciated!
(Thanks to Steve Weimar for reminding me of the importance of
iving the questions.)
Keith

From usenet Sat Sep 25 11:33 PD 1993
Path: forum.swarthmore.edu!uunet!cdp!doversherhs
From: Dover-Sherborn High School <doversherhs@igc.apc.org>
Newsgroups: geometry.pre-college
Date: 25 Sep 93 11:33 PDT
Subject: DSHS Journal-Dog Pen Week
Message-ID: <1800700031@igc.apc.org>
X-APC-HostID: 1
Sender: Notesfile to Usenet Gateway <notes@igc.apc.org>
Lines: 205

Sept. 24, 1993

It seems like a long time since I've written anything in my journal.
I've had computer problems as I have stated, but more than this I
have mostly just been enjoying the beginning of the year. There is
nothing like the first few weeks - with all the new relationships
and surprises.

Thanks to some wonderful ideas which I have received through the
Geometry Forum I believe I have gotten off to a good beginning
with portfolios. (I will share the documents, which I have
distributed to the students, later in this entry.) Today I collected
the first problem of the week (POW). I used the well know dog
pen problem:

You plan to build a pen with at least one gate for your dog.
Fencing is $1.00 per foot, fence posts are $2.00 each, and gates
are $5.00 each. You can afford only $40.00 total. How would
you design your pen in order to get the greatest possible area?
Explain what you considered in making your plan.

I first asked each student for homework to write down his/her
understanding of the problem, possible strategies, diagrams, and
appropriate geometric concepts.

Then the next day in small groups of 3-4, the students spent 30
minutes brainstorming while I floated around the room and took
part in several interesting conversation and heard about a very
different geometry than I know! The students had difficulty
formulating strategies for determining the "best" shape for the pen.
In fact, many students, were not even considering the possibility of
choosing. They were just going to make a rectangle. Those that did
think that the shape would in some way affect the area, were not
thinking about collecting data or doing an inductive investigation -
even though we had spent most of the previous week using Serra's
inductive patterns and I had spoken extensively about inductive
thinking.

So the next day, I posed a simpler version of the problem (each
day I share my rationale for the lessons with the students - in this
case I shared what I heard the previous day and why I had
decided to pursue a simpler problem - a well worn mathematical
strategy.) I asked them to intuitively predict which shape from
among a rectangle, square, triangle or circle would have the
greatest area if each had a perimeter (or circumference) of eight
feet? The intuitive responses included every shape as well as "it
makes no difference, they would all have the same area".

When asked how we would proceed in this investigation, several
students were able to suggest collecting data and drawing
inductive conclusion!

We used geoboards to investigate the rectangles and squares and
Sketchpad to investigate circles and triangles.

By the end of the class, the circle emerged in most students minds
as the best alternative from among our choices.

Two days later (today) I collected the problem. I had them self-
evaluate their work using the document below (send to me by
Henri Picciotto) and I will use the same standard to offer feedback
to the students.

There is much more I could say, but time is beginning to run out on
this entry.

Questions:

1. As always, any feedback?

2. I plan to ask each student to choose the number one interest or
concern in her/his life and begin a year long investigation into the
relationships among geometry and this interest. Also I will ask
each student to choose his/her favorite activity during school hours
(art, music, social studies, lunch ...) and continually look for
relationship among the activity and geometry.
Does anyone have any feedback about this idea?
How is the best way to use the Geometry Forum to aid and
inspire the students efforts? Experts? Advisory councils? Student
teachers?

3. What do people think about having a contest for the best student
sketchpad demonstration? $1,000 scholarship for the winner, etc.
Money and food seem to be powerful motivators!

4. Other ideas about how the Forum can be utilized at this point.

Logistics: It looks as if I will have frequent access to a computer
lab with 6-8 Macs. The restrictions include: I may have to share
the room with other classes (using IBM's), the Mac's are only
capable of running the Standard Version of Sketchpad (slow, slow)
and no color of course.

I'm still very much interested in find a donor/sponsor to purchase
10 Mac's for my classroom - color monitors, floating point
processors, etc. I haven't received much response to this posting.

5. Any ideas for funding???!!!

Evaluation of Writing Assignments

Writing assignments are an important part of your work in this
class. Here are the things I will be looking for when I read your
papers.

The Math:

Did you understand the problem?
Did you express yourself clearly?
Did you summarize all that you learned about the problem?
Did you answer all the questions?
Did you think about the problem beyond what was asked?
Were you creative in your approach?

The Presentation:

Do you have an introduction that states what the paper is
about?
Did you organize your paper into paragraphs?
Did you illustrate the paper adequately?
Did you avoid repeating yourself?
Is the paper neat and legible? Is it beautiful?
Were you creative in your presentation?

Please use these standards to self-evaluate your papers. Give
yourself one point per "yes" answer. A score of10 or above is
excellent. If the score is 6 or below, the paper should be redone.

Mathematics Portfolio Guidelines
---------------------------------------------------------
The purpose of compiling a mathematics portfolio is to showcase or
highlight your unique mathematical abilities, to provide a much
broader view of what you can do, and to provide an opportunity
for self-assessment and self-reflection.

Each week you will be asked to select your favorite or best work
(from past writing assignments, quizzes, exams, or projects) that
shows evidence of one or more of the following criteria:
----------------------------------------------------------
Demonstrates skill in using a variety of problem solving strategies
to interpret and solve math problems:

Applying a new technique
Writing an equation or inequality
Drawing diagrams/pictures
Guessing, checking, and revising
Making an organized list or table
Using logical reasoning
Looking for patterns
Simplifying the problem
Working backward
------------------------------------------------------------
Gives evidence of using resource material and technology sensibly
as tools in understanding math concepts:

Using "hands-on" manipulatives
Using school, community, and university library materials
Accessing computer bulletin boards to send/receive information
(telecommunication)
Using computer software to analyze, model, or simulate
--------------------------------------------------------------
Demonstrates quality and creativity in performing and
documenting math activities and investigations:

Written work is legible, and well-organized
Uses math vocabulary and terminology clearly
Approaches problems in a systematic manner
Uses creative and unique methods of solution
Perseveres in solving problems
Checks work independently
Works well with others
Demonstrates self-confidence
Exercises leadership
----------------------------------------------------------------------
Each math folder selection must include a brief but specific
explanation of why you've chosen that piece for possible inclusion
in your semester mathematics portfolio.

Your reasoning should specifically refer to one or more of the
guideline criteria (above) and be stated in a positive manner. Use
complete sentences and attach your explanation to the front of
your selection.
----------------------------------------------------------------------
EXAMPLE:

NAME: Bob Montgomery
CLASS: Geometry
DATE: 9-17-93

Math Folder Entry #2

For the week of September 13-17, 1993, I've selected Assignment
# 12 to be included in my math folder because it demonstrates that
my written work is well-organized and shows that I persevered at
learning to recognize patterns.

Good night.

Keith

From usenet Sun Sep 26 16:01 PD 1993
Path: forum.swarthmore.edu!uunet!cdp!doversherhs
From: Dover-Sherborn High School <doversherhs@igc.apc.org>
Newsgroups: geometry.pre-college
Date: 26 Sep 93 16:01 PDT
Subject: DSHS Journal-POW#2
Message-ID: <1800700032@igc.apc.org>
X-APC-HostID: 1
Sender: Notesfile to Usenet Gateway <notes@igc.apc.org>
Lines: 30

Problem of the Week
September 27, 1993

Objective: Discover (hypothesize) some of the relationships among
area and perimeter.

Task: Perform inductive investigations comparing the areas of
triangles, quadrilaterals, hexagons, and circles given a fixed
perimeter or circumference of 8 inches.

1. How do the areas of regular polygons (separately consider
triangles, quadrilaterals, and hexagons) compare to the areas of
non-regular polygons with the same number of sides?

2. Of the figures under consideration, which one has (or ones have)
the greatest area(s) given a fixed perimeter or circumference?

Requirements include the twelve components of a written
problem including the following:

1. an introductory paragraph explaining both your understandings
of the problem and possible strategies for solving the problem,
2. tables of data,
3. diagrams,
4. all calculations,
5. a paragraph interpreting the data and drawing any possible
conclusions or hypotheses, and
6. a paragraph summarizing what you have learned (to include but
not limited the geometric relationships you induce to be true).