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


(Posted for Keith Grove: DoverSherborn 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 34, 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 selfevaluate 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 68 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 selfevaluate 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 selfassessment and selfreflection.
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 "handson" 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 wellorganized 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 selfconfidence 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: 91793 Math Folder Entry #2 For the week of September 1317, 1993, I've selected Assignment #12 to be included in my math folder because it demonstrates that my written work is wellorganized 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: DoverSherborn High School <doversherhs@igc.apc.org> Newsgroups: geometry.precollege Date: 25 Sep 93 11:30 PDT Subject: DSHS JournalStudents'Questions MessageID: <1800700030@igc.apc.org> XAPCHostID: 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: DoverSherborn High School <doversherhs@igc.apc.org> Newsgroups: geometry.precollege Date: 25 Sep 93 11:33 PDT Subject: DSHS JournalDog Pen Week MessageID: <1800700031@igc.apc.org> XAPCHostID: 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 34, 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 68 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 selfevaluate 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 selfassessment and selfreflection.
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 "handson" 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 wellorganized 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 selfconfidence 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: 91793
Math Folder Entry #2
For the week of September 1317, 1993, I've selected Assignment # 12 to be included in my math folder because it demonstrates that my written work is wellorganized 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: DoverSherborn High School <doversherhs@igc.apc.org> Newsgroups: geometry.precollege Date: 26 Sep 93 16:01 PDT Subject: DSHS JournalPOW#2 MessageID: <1800700032@igc.apc.org> XAPCHostID: 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 nonregular 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).



