# Changing shapes with matrices - in the Math Clubs!

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### Maria Droujkova

Apr 26, 2009, 10:29:06 AM4/26/09
This is an activity designed by Don Cohen the Mathman, in his book "Changing Shapes with Matrices." You can find some sample book problems here, and follow links to other Don's materials: http://www.mathman.biz/html/probcswm.html

The general idea is to start with a simple "dot to dot" picture on a coordinate plane, and then apply a matrix transformation to coordinates of every dot. For example, suppose your dot is at (2,3) and your matrix is:
1, 4
-2, 5

You find the new dot by multiplying the rows of the matrix by your dot's coordinates, one by one: 1*2+4*3=14, and -2*2+5*3=11, so the new, transformed dot is at (14, 11)

Like many activities involving massive number crunching, it works much better on computers. You can experiment with this applet, transforming a doggie, on Don's site: http://www.mathman.biz/html/dogtrans6/changing_shapes_with_matrices%20ies6.html

We had certain difficulties with the activity in the math clubs. Hopefully, descriptions of these (preventable) difficulties will help others who run activities. The activity was popular in both math clubs.

- I started by telling kids to make a coordinate plane and then dot-to-dot the first letter of their name on it. Some kids used about 20 dots to make fancy font letters. In the second math club (with different kids), I suggested they use as few dots as they can - otherwise, number crunching is too much to ever finish the activity. Also, it's better to tell kids to put dots at grid line intersections (whole number coordinates), unless you want them to practice fraction operations for a couple of hours.

- Some kids made huge letters, and then could not complete their transformations. Others picked large numbers for the transformation matrix, and could not fit the results onto the paper. The second time around, I told kids to fit their letters into a 4 by 4 square near the origin, and use numbers up to 4 for their matrices. This way, everybody's transformed shape fit.

- Retrospectively, each kid could grab a couple of points in a more complex shape, do the computations for them, and then plot "their" points on a collective transformation graph. I have not tried it with kids, but it should be a neat collaborative activity, with kids helping each other resolve difficulties when points are obviously in wrong places.

- After kids did one example on paper, which took some forty minutes all told, we moved to programming an Excel spreadsheet for transformations. We only have one computer with a projector in the math club, so everybody shouted some programming suggestions as we went along. The good thing about working with software is a massive amount of pattern observation it supports. Kids could quickly test conjectures, such as: "What makes the shape flip? How can you stretch the shape more? What happens if you put opposite numbers in the matrix? Reciprocal fractions? Zeroes?" This was some excellent math by kids - the reason I love this activity so much. The problem with Excel in particular, though, is that it automatically stretches graphs to fit the screen, so the ORIGINAL picture keeps resizing as you experiment with transformations. When I do the activity again, I will modify the doggie program Don has to allow people input their own dots.

- After experiencing the activity, we looked at some videos of 2d and 3d transformations involved in computer animations. The main point: it took us a Very Long Time (TM) to transform just a few points by hand. Using mesh models for animation, thousands of points are being transformed many times a second. I think kids could appreciate the complexity of the task, and why it's so demanding computationally. Here is an example of a video: http://www.youtube.com/watch?v=J1mExXURsWk&feature=related However, I could not find videos that would particularly follow the activity. If anyone knows some cool videos, please share!

We did some very meaningful math and had a lot of fun with the activity. Don, thank you very much for your wonderful books, full of great activity ideas.

--
Cheers,

Make math your own, to make your own math.

http://www.naturalmath.com social math site
http://www.phenixsolutions.com empowering our innovations

### Don Cohen-The Mathman

Apr 27, 2009, 12:09:45 PM4/27/09
to NaturalMath
Thank you Maria, for your kind words.
And thank you for sending people to the IES applet Changing Shapes
With Matrices that they made for my website; it took 2 years before
they understood what I wanted, mostly because of the language problem.
They do a great job. Have you seen these other applets IES made with
my suggestion: 6 trig functions , (a+bi)^(a+bi)... (from chapter 11 in
my worksheet book) , a^3-b^3 identity that my student figured out and
linked from my web page .

Maria, have you seen the two applets on my website (www.mathman.biz)
that Lori Johnson Morse (you know her on twitter) and I (mostly Lori)
did using Geogebra - 1. The Nautilus Shell and 2. my showing that the
area of a triangle equals the sum of an infinite geometric series .

I'm writing up what a 3rd grader did the infinite series for 1/3 +
(1/3)^2+...very interesting! and the problem started by a 10th grader
cos(cos(cos(2/5))). He was "just playing" with a calculator, but it
ended up he and 2 others and I, discovered a new number (the Dottie
number)..very exciting!!!

Keep up the fine work you are doing Maria. I think listening to what
students say about a problem is crucial. I've been teaching math for
55 years now and doing the tutoring for the last 33 years- it's been a
blast! These are the good old days.

Don

On Apr 26, 9:29 am, Maria Droujkova <droujk...@gmail.com> wrote:
> This is an activity designed by Don Cohen the Mathman, in his book "Changing
> Shapes with Matrices." You can find some sample book problems here, and
> follow links to other Don's materials:http://www.mathman.biz/html/probcswm.html
>
> The general idea is to start with a simple "dot to dot" picture on a
> coordinate plane, and then apply a matrix transformation to coordinates of
> every dot. For example, suppose your dot is at (2,3) and your matrix is:
> 1, 4
> -2, 5
>
> You find the new dot by multiplying the rows of the matrix by your dot's
> coordinates, one by one: 1*2+4*3=14, and -2*2+5*3=11, so the new,
> transformed dot is at (14, 11)
>
> Like many activities involving massive number crunching, it works much
> better on computers. You can experiment with this applet, transforming a
> doggie, on Don's site:http://www.mathman.biz/html/dogtrans6/changing_shapes_with_matrices%2...
> it's so demanding computationally. Here is an example of a video:http://www.youtube.com/watch?v=J1mExXURsWk&feature=relatedHowever, I could

### kalli shevzov

Apr 28, 2009, 10:38:30 AM4/28/09
 -forgive my ignorance, folks,I did this exercise in math club, and I had difficulty with it then  and still don't "get it" now. Sometimes I need a sense of   purpose or a reason for doing something to understand what I am doing. Can someone enlighten me about what I could do with this exercise? Is this a way of enlarging a figure in a predictable fashion? Feeling Like I missed Something aka Kalli

### Maria Droujkova

Apr 30, 2009, 11:20:15 AM4/30/09
Kalli, I think you missed the second part of the activity in the Math
Club when we explored patterns in matrices with Excel. Go to the
"puppy transforming software"
http://www.mathman.biz/html/dogtrans6/changing_shapes_with_matrices%20ies6.html
and try it with these matrices:

2 0
0 2

3 0
0 3

0 1
1 0

-1 0
0 1

Try your own variations of "diagonal" matrices. You will notice what
matrices produce flips, stretches or rotations. After this,
predictability comes.

In the software, the "Init" button (for "initial") resets everything.

Let us know what you discover with matrices!

--
Cheers,

Make math your own, to make your own math.

http://www.naturalmath.com social math site

http://www.phenixsolutions.com empowering our innovations

### Maria Droujkova

May 1, 2009, 9:22:00 AM5/1/09
to natur...@googlegroups.com, Don Cohen- The Mathman
On Fri, May 1, 2009 at 8:56 AM, Don Cohen- The Mathman
<doncohe...@gmail.com> wrote:
> Hi Marie,
> I was reading your original email concerning the matrices and realized
> something- there was a reason I chose only 0's, 1's and -1's for the
> transformation matrix, and a simple picture with only 9 integral points, the
> doggie. I think once the kids can handle the transformations of the points
> on the doggie, then they can use larger numbers to dilate the shape. That
> way they don't get bogged down with all the computations and can concentrate
> on the multiplication of matrices and what the transformations matrix does
> to the shape. Of course they can eventually use a calculator to do the
> computations. I find that kids in algebra 2 use the calculator much too
> soon, before they understand what a matrix is.

Don,

The problem with 1s is that kids don't necessarily see multiplication
by 1 as multiplication! It is a good example of learning paradoxes.
Once kids are strong on multiplication, they find multiplying by 1 the
easiest thing in the world. However, when they are still
conceptualizing multiplication, or a new instance of it like
multiplication of matrices, examples involving 1s may confuse. Such
examples don't clearly express any multiplicative action (e.g.
grouping or stretching) - so they are more abstract than
multiplication by other numbers. I hope it makes sense, and if not, I
can describe it some more. In the same spirit, I try not to introduce
addition via adding 0, and can't introduce sets starting with either
empty set or the set of all objects in your space.

I eventually told kids to pick whole numbers less than three.
Interestingly, 1s were picked by the most advanced kids in the groups
who already had some algebra and figured out it will be the easiest
thing to try!

> This activity goes back to my work with Bob Davis (EXPLORATIONS IN
> MATHEMATICS - a text for teachers, by Bob Davis; Addison-Wesley 1967, out of
> print), further back to A Path to Modern Mathematics by W.W. Sawyer 1966,
> and further back to On Growth and Form, by D'Arcy Thompson, originally
> published in 1917! I have a 1961, abridged ed. published by Cambridge U
> Press; no matrices, but lots of tranformations in animals, fish, honeycombs,
> and shells (equiangular spirals-The Nautilus!). And invariants.
> I guess I try to make things simple for kids, Maria.
> Keep up the fine work you are doing!!!
> Don
>
> On Sun, Apr 26, 2009 at 9:29 AM, Maria Droujkova <drou...@gmail.com>

> wrote:
>>
>> This is an activity designed by Don Cohen the Mathman, in his book
>> "Changing Shapes with Matrices." You can find some sample book problems
>> here, and follow links to other Don's materials:
>> http://www.mathman.biz/html/probcswm.html
>>

--
Cheers,