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
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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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