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Suzanne's Math Lessons || The Math Forum

Introduction
  3 methods

My background
  why I teach

AVID Math 7
  curriculum

Teacher's Role
  changing

Instruction
  ideas

Assessment
  rubrics

Equipment
  one computer
  ... or a lab


Notes
  from others


One of the pleasant side benefits of putting up Web pages is receiving notes from people who have used them and who then take the time to write to me. Since the summer of 1995, when I wrote my first tessellation tutorial on using HyperCard, I have received an average of three messages a week referring to various pages. The following messages are a representative sample of those notes. I linked to the page(s) to which they refer in their notes so that the pages on which they are commenting can be viewed.

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Hi Suzanne,

Thanks for the link. I went to it looking for math ideas and found so much more. We are currently studying China as part of our social studies curriculum and I always do an art unit on dragons and incorporate Chinese folktales (reading and writing) as the language arts part of this theme. Also Flight is one of the grade 6 units that I lead into next starting with kites. When I got to your site I thought I had died and gone to heaven. It is incredible! I can hardly wait to share some of the links with my fellow teachers. I am very impressed with the way your school must organize to make all this happen. Thanks so much for sharing. I will certainly share stories with you once we start to use some of the sites and try out the tessellations.

Nancy Weber
bweber@telusplanet.net

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I want to thank you so much for putting all the information on fractals in your web site! I really appreciate it! I had to do a fractal report and your web site helped me the most. Thank you so much for putting so much effort into your web site!
Thanks again.
Bye, Kalin

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Wow! I have just been nosing around in some of your math sites - do your 7th graders really do this stuff? I would love to teach this to my 7th grade classes, but we can't get past the basics. I have to drill them on fractions, decimals, and just old fashioned multiplication and division every few days or they lose what they have. I have never seen kids fight learning like these kids do. They won't read directions, or take initiative - they want everything spelled out for them 1, 2, 3, or they just won't try. If they can't finish their work in 10 minutes, they put down an answer - any answer. The parents are not very supportive either. They would rather Johnny be given the honor roll than earn it. I have really had to be creative to come up with ways to get these kids to learn. I have also been reprimanded for it because a parent complained about my methods. I am on the verge of giving up. I am looking for a teaching job elsewhere. I love teaching math. I even like 7th graders--call me crazy. Teaching kids with the materials you have here would be exciting and enjoyable. Keep up the good work.
Janiece


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Dear Janiece,

Wow! I have just been nosing around in some of your math sites--do your 7th graders really do this stuff?

Yes. :)

I would love to teach this to my 7th grade classes, but we can't get past the basics. I have to drill them on fractions, decimals, and just old fashioned multiplication and division every few days or they lose what they have.

I think the last part of your statement says a lot! If they "lose what they have" after a few days then they are not learning through drill.

I have never seen kids fight learning like these kids do. They won't read directions, or take initiative--they want everything spelled out for them 1, 2, 3, or they just won't try.

I find that my students will try many things just because the "basic skills" are embedded in activities. Seventh grade students love to work in groups, they love new challenges (IF they are attainable) and they love technology and manipulatives. The lessons that I continue to design have all of these components.

If they can't finish their work in 10 minutes, they put down an answer - any answer. The parents are not very supportive either. They would rather Johnny be given the honor roll than earn it. I have really had to be creative to come up with ways to get these kids to learn. I have also been reprimanded for it because a parent complained about my methods.

I must say that in my situation I am continuously encouraged and that certainly helps tremendously.

I am on the verge of giving up. I am looking for a teaching job elsewhere. I love teaching math. I even like 7th graders--call me crazy. Teaching kids with the materials you have here would be exciting and enjoyable. Keep up the good work.

Thank you and I hope that you find a situation that is more tolerable for you.

Sincerely,
Suzanne

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Dear Suzanne,

I am the Gifted/Talented teacher for two elementary schools in Highland Park. I am in the process of constructing a web site about Tessellations. Your web site is fabulous and I would like permission to include it as a link. Please send me your response as soon as possible. The school year is drawing to a close and I'm trying to finish my web site.

Sincerely,
Lynn Hansen

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Dear Suzanne,
Thank you for your tessellation lessons! I found them the other day while on a search for 'puzzles'. I was looking for the die-cut patterns used to make jigsaw puzzles. I didn't really find what I was looking for although I *did* find something better. Your excellent tessellation lessons!

Ever since I first saw his work, Escher has fascinated me. His tessellations are truly amazing. I always wondered how they were made. Your lessons gave me the basic understanding that I needed to investigate his technique further. I think I finally understand the principle behind the hexagonal 6,6,6, tessellation.

Again, I thank you for your generosity and for the excellent lessons! I was going through a little bit of a creative slump and you helped me get out of it! :)

Best Wishes,
Luisa

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Greetings from Louisiana!

My Honors Geometry classes have just finished a hard section on finding Lateral area, Total area, and Volume of prisms, cones, pyramids, etc. It is the end of our school year and we were looking for something interesting and fun to do. The class is made up of mostly 9th graders. When I stumbled upon your lessons using Hyperstudio, we rejoiced. Thanks from my entire class for these fun end-of-the-year projects. : )

I printed them from Netscape and gave the students the different rotations, glide reflections, and translations. They then had to perform these skills on the software program, Paint, as we don't have Hyperstudio available to all of the students. The rotations and glide reflections were a little challenging but my computer whizzes figured it out.

Diana Lewis
Neville High School
Monroe, Louisiana
dlewis@monroe.k12.la.us

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Dear Suzanne, thank you for your inspiring pages on the WWW. As an older math teacher with a new connection to the internet, it was a nice surprise to find pages of such a high quality. Last time I worked (the pupils worked!) with Polyhedron, I started with Luca Faciolis "The Divine proportions" (Venice 1509). This time I got all my inspiration and good advice from your (and a few other) pages. The class (5. form, 12-13 y.) build a.o. polyhedrons from drinking-straw and thin line (string), espc. those containing triangles were fine, those with squares, 5 or 6 sides were helped in the corners by pipe-cleaners put in the right angle. Once again, thank you!
S¿ren Kaa
Albertslund
Denmark

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hey
I just wanted to thank you for helping me ever so much, because I'm doing a math project and I didn't know how I should do it, so I searched for M.C. Escher and reached the word tessellations (because I didn't know how the subjekt is called in English), then eventually I FOUND YOUR SITE.
  you helped me very much so thank you!!
  I'll let you know how it went...
    p.s.
  you have a great site!

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Hi,
Your lessons are very well done. I would like permission to put these lessons into our BBS First Class email server. Our teachers access this system daily and we have mathematics conference. By using the application Web Buddy we can whack the web page and translate directly into Claris Works 4.0.

Kurt Kuhlmann
Mathematics Coordinator
San Jose Unified School District

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I came across the information you provide on the WWW. Two pieces in particular were of interest to our program: "Straightedge/Compass Constructions" and "What is a Tessellation?" I am holding a workshop next week (March 25) for secondary school teachers here at Georgetown University titled "Islamic Art and architecture," and would love to make copies of these two pieces to hand out. I expect about 125 teachers. The program is free of charge and is offered as a community service of the Center for Contemporary Arab Studies. My request is for this one-time use only.

Can you give us permission to photocopy these pieces? If yes, is there a special wording that you would like me to include on the first page? Now that I have printed these I don't see a website address - can you provide that for me? Thank you very much for your help, and I look forward to hearing from you.
Sincerely,
Zeina Azzam Seikaly, Outreach Coordinator
Center for Contemporary Arab Studies
Georgetown University
Check our our website!
http://www.georgetown.edu/sfs/programs/ccas

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Suzanne - saw your lesson plan on tessellations and thought it was great. I am a N.Y.C. art teacher and a novice on the computer. The step by step directions are terrific. I am eager to try them with a class. Thank you.

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Dear Suzanne,

I saw your page on factoring at /alejandre/factor1.html and have the following comments and questions. I am a graduate student (in biology, not education) who is currently tutoring an 11th grade student in Algebra 2. Your page isn't exactly relevant to us, but it's an interesting enough idea that i wanted to respond. If you don't have time to reply to this that's fine.

First, thank you for making this resource available on the web.

I personally found the concept somewhat confusing at first and hard to visualize in my head. If I'd had tiles in front of me I guess it would have been straightforward. For me, though, thinking numerically - as in (a-b)(a+b)=a^2-b^2 - seems much more intuitive and straightforward than doing mental translations between numbers and shapes.

What I wonder is how much the tiles would help a student who has no familiarity with factoring at all.

On the one hand it seems that it would allow a student to concretely prove to herself that a "magical" formula like (a-b)(a+b)=a^2-b^2 actually works.

On the other hand it requires the student to learn an extra (and ultimately useless) set of skills; how to convert algebraic equations into their geometric interpretations.

While it seems that the geometric analog of multiplying constant integers is useful (seeing 3 * 2 geometrically) it seems that when it comes to algebra, and especially equations involving subtraction, the relation between the expressions and the geometry may become more complex than helpful.

So, I'm curious how students respond to this way of teaching in the classroom. Is there evidence that students learn better or faster? Also I'm not sure what grades these activities are suitable for.

Again though, thank you for taking the time to develop and publish these resources.

-amal.


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Dear Amal,

Thank you for writing. I found reading your thoughts on the factoring pages interesting.


I personally found the concept somewhat confusing at first and hard to visualize in my head. If I'd had tiles in front of me I guess it would have been straightforward. For me, though, thinking numerically - as in (a-b)(a+b)=a^2-b^2 - seems much more intuitive and straightforward than doing mental translations between numbers and shapes.

I have found that that is a normal reaction from a person who already knows the algebraic algorithm.

What I wonder is how much the tiles would help a student who has no familiarity with factoring at all.

Help with what? My intent in presenting factoring geometrically is so that the student understands what is behind the algebraic representation.

On the one hand it seems that it would allow a student to concretely prove to herself that a "magical" formula like (a-b)(a+b)=a^2-b^2 actually works.

I think that is important.

On the other hand it requires the student to learn an extra (and ultimately useless) set of skills; how to convert algebraic equations into their geometric interpretations.

It's not useless if it provides understanding.

While it seems that the geometric analog of multiplying constant integers is useful (seeing 3 * 2 geometrically) it seems that when it comes to algebra, and especially equations involving subtraction, the relation between the expressions and the geometry may become more complex than helpful.

I agree that once there is a foundation, continually using the geometric method might be cumbersome, but am a person who has no trouble at all with the algebraic methods for dealing with this. My guess is that it is quite easy for you, too. There are students who do not have this facility. Have you heard of Martin Gardner's Theory of Multiple Intelligences? If you scroll down a bit you will see a description of the different intelligences.

Some students need to have spatial experiences in order to fully understand concepts. They also need kinesthetic experiences, which they can get if they work with manipulatives. When I "teach" factoring this way with my students they use manipulatives called "algebra tiles."


So, I'm curious how students respond to this way of teaching in the classroom. Is there evidencte that students learn better or faster? Also I'm not sure what grades these activities are suitable for.

I am a middle school teacher but I have taught 9th grade general mathematics using these techniques also.

http://www.rialto.k12.ca.us/school/frisbie/welcome.htm
http://www.rialto.k12.ca.us/school/frisbie/coyote/coyote.html

I have no hard data that students learn better or faster using geometric representations when they are first learning algebra, but students who traditionally have not done well in mathematics respond very positively to such techniques. My students are not "gifted" students. Often they have had very mechanical experiences in mathematics. My hope is to broaden their view and give them experiences that will make mathematics more meaningful.

Have you ever read The Ascent of Man by Jacob Bronowski? In that book there is a lovely picture of a proof of the Pythagorean Theorem using tiles in the sand. That is an example of how I wish my students to view mathematics. I want them to have a visual understanding as their basis, which can then be followed by mechanical mastery.


Again though, thank you for taking the time to develop and publish these resources.

You're welcome. I have fun making them!

Sincerely,
Suzanne


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To Whom it May Concern:

I had a math report to do and I picked tessellations. I did the Yahoo search and found this page. I really appreciate this very much. This helped me very much. So I just wanted to say thank you for your help.

Justin
Huntington N.Y.
9th grader

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