More knitting and some algebra
I didn’t have a great deal of time to hash this out, but I thought I would create a few images that describe a knitting project I started. I’m knitting a seamless yoke sweater. It’s seamless because the body and sleeves are attached and then knit together to form the yoke. Other than the sewing in of ends to make things neat there’s absolutely no sewing involved with this project.
Here’s an example of what a yoke sweater looks like click on the link to enlarge:
Here are the slides where I thought out the math for starting my yoke pattern:
BTW: I may seem like I spent a lot of time creating the images, but not really. It was easy to reproduce those little dots. I just selected a whole row of them and then copied them over to the next section of the grid. I’m finding that by using shortcuts and pasting repeats of patterns makes creating illustrations much more easy.
I forgot to post my finished sweater:
My finished yoke sweater
Math writing
Arghhhh… finding a math language editor that’s easy to work is hard.
I can’t wait until they start applying handwriting recognition technology for math/science applications (and make it available to the general public at free or little cost).
I did find this, a nifty tool call DragMath:
http://www.dragmath.bham.ac.uk/
If you click on the “Demonstration Link” and download the Java applet you can play with a demo of the tool. I’m including an image/screen shot of the tool here.
You can select copy expressions to a clipboard however it converts them to the Math Markup language. In order to paste them on to a webpage like this blog post, you need to paste the code into the HTML code of your page. Actually, I’m still fiddling with this to get it to work the right way.
2+ \sqrt{3+13} =x
I’m actually pretty stoked that this application can be used with the Moodle Learning Management system. This means that courses housed in Moodle can use this tool to allow students to build their own mathematical expressions.
Graphing linear equations
I found a pretty good applet that allows you to graph and view changes in equations/function:
I found some drawbacks with this program.
You cannot copy/cut and paste former expressions from the application itself. I tried typing the the expressions in a word document and then pasting them, and this didn’t work either.
Also, you can’t place lines from different functions on the same graph to compare.
I’m sure there are other graphing tools out there that do this. I just have to find them.
Click on the thumbnail to see a larger version of the image.
The importance of building our own visual examples
Years ago, I worked with a science teacher to teach a which encouraged students to examine and then draw magnified images of natural objects such as sea urchin shells, pinecones, leaves, etc. I introduced the students to drawing by referencing lessons I adapted from Mona Brookes’ Teaching Children How to Draw.
I discovered this book when I was in teacher training in New York, and it was funny after reading the book I found myself drawing all sorts of mundane things… the seat back in front of me on the bus, the window view outside my apartment window. I suddenly felt empowered to draw just about anything because the Brookes method taught me how to visualize the basic shapes in everything I saw. I suspect this is the general method taught for drawing anyway.
I believe that we really should teach and encourage children to draw anything and everything because it allows them to both internalize and then begin to analyze what they see. The same holds true for constructing visuals for understanding Mathematics and Math concepts. As a student, I think I was on the tail end of educational pedagogy that made us sit in our seats and listen rather than experiment and explore. I was forced to memorize formulas without really understanding how they worked. Perhaps this is why I loved Geometry once I discovered it because here was a Math where we had to ‘prove things’
As a student in a middle school Algebra or Pre Algebra course I think I would have been too impatient and impulsive to see the value in proving simple things such as the Multiplicative Inverse property, but perhaps proving why rule like this worked would have helped me internalize these crucial concepts more so that I could apply them more readily in the more complex Math courses.
Pretty nifty applet: Pascal’s Triangle
I found this great applet today on Pascal’s Triangle
You can actually take a look at the patterning on the triangle when prime numbers are selected.
Click on the image to view a larger version.
Representing algebraic equations visually
I’ve been experimenting with the use of Algebra Tiles. I like the idea of visually representing values and variables. When I was learning Algebra in middle school and high school many of the things we learned seemed so abstract. I always felt like we were learning rules about abstract things, but in Catholic school we had no exposure to mathematical concepts such as the Properties of Numbers, the Properties of Rational numbers. We were drilled on how to do the ‘mechanics’ of math, addition, subtraction, multiplication and division plus a thing or two on how to use the same operations on fractions. By the time I got to high school I didn’t know what hit me. The math teacher was speaking a muddle of a language that I could barely understand, and considering my upbringing and former educational experience, I focused only on memorizing equation types. It was a very short sighted and limited way of learning. I know it was awful, but I really do think that the words of that fifth grade teacher (I wrote about her briefly in a previous post) really stuck with me. She sent this message that basically told me that it was okay to give up. I should understand now that she was only human and she had her own fears when it came to math (she was a Baby Boomer teacher and most likely grew up during the Donna Reed age where understanding and mastering higher math wasn’t a goal expected of most women).
I muddled my way through Algebra and then found my way into a Calculus class. Our teacher in this class was a very caring and methodical woman who really had a passion for her discipline. She even took us to see “Stand and Deliver.” Incidentally, Jaime Escalante became a sort of role model for me and his work actually inspired me to become a teacher myself. However, by the time I got to Calculus, I still had a really bad or superficial understanding of some of the number sense basics (Integers, Rational Numbers, Properties, Operations, etc.). I wonder how I was able to even manage a C in Calculus.
I actually created a set of my own “eTiles” for Powerpoint. I’m including images here.
Ooops…. the operation should read 2x2+4x+12. Ahhh the joys of using graphic tools that don’t have superscripting. Thanks to my commenter for the correction.
I wasn’t sure how to represent negative numbers so I did so by circling the depicted value with a dotted red line. It might be embarrassing to admit this, but I feel a sort of healing and relief as I worked to develop these visual images. Perhaps actually creating these pictures help re-reinforce the concepts I probably did get, but only half way.
More fabulous Fibonacci spirals
I like the idea of nature planning and designing things out systematically. I guess there’s an order when it comes to the plant world. We like to think that plants are completely passive and maybe even boring because they can’t talk or move (most of the time). They are always considered stationary or the backdrop to the world of fauna. But lately, since I’ve been reading and doing more investigation into non linear patterns and most particularly patterns of the Fibonacci sequence, I’ve developed a greater appreciation of the design of plants.
I read that there’s an estimate that 90 percent of all plant species exhibit patterning of leaves, buds, flowers, or petals based on Fibonacci numbers. A phenomenon called Phyllotaxis describe the spiral patterning of leave patterns determined by segmenting plant stems into fractions of 3, 5, 8… etc.
Here’s a terrific website that provides some images of this phenomenon in certain plants. You can actually click to view the spiral patterns laid over the images. I really liked the image of the Romanesque Cauliflower. It resembles some strange type of space fruit.
Mastering word problems
I don’t think I had any elementary school teachers who knew how to teach word problems effectively. I was just reading that mastering certain rules such as the properties of numbers (Associative, Commutative, etc.) actually help people understand how to solve word problems.
If you think about it many of use know how to do the mechanics of math in every day applications (figuring out your bill at a restaurant, estimating how many bags of lawn fertilizer or grass seeds to buy to cover your lawn, etc.). But how many of us really realize when we’re actually applying the properties of numbers to solve these problems? I’m going to try to come up with as many examples of everyday applications of all of the properties below.
Off the top of my head I can think of a number of situations:
- Figuring out the bill at the tapas restaurant
- Adding needed yardage of different colors of yarn
- Determining our grocery budget
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A + B = B + A and A*B = B*A
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(A + B) + C = A + (B + C) and (A * B) * C = A * (B * C)
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A + -A = 0, A * (1/A) = 1
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A + 0 = A, A * 1 = A
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A*(B + C) = (A*B) + (A*C)
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Translating number patterns into Algebra
I found a neat game. Unfortunately WordPress does not allow me to embed the code to play the game on my blog (at least, I haven’t figured out how to do this), but go to this link and you should be able to view the game:
http://maththinker.blogspot.com/2008/01/translating-patterns-into-algebraic_11.html
Exploring Patterns: Fibonnacci
I think I’m enjoying learning about non-linear patterns a great deal because there are so many applications of this pattern in nature. Also, since I was in the sixth grade, I’ve always held an awe for anything relating to the Golden Section.
This last Christmas I actually knit a scarf for my brother that followed the sequence of stripping for the first several numbers in the sequence.
1,1,2,3,5,8,13,21….
When I was substitute teaching several years ago I had fifth grade class who was reviewing number patterns and sequencing… I put a number of different sequences up on the board. There was a student who was moderately autistic in the class and before I completed one of the sequences he blurted out “Fibonacci! Fibonacci!”
I spent some time fiddling and I built my own tiled visual model of the Fibonacci sequence. I think I spent too much time trying to make sure that the squares were positioned correctly. I truly feel that hands on manipulation of visuals is key to helping me grasp and understand concepts fully. I hate passive learning.
You can draw a spiral around the pattern of increases and it looks something like this.
A Fibonacci Spiral is similar to a “Golden Spiral” based on the Golden Ratio. However, if you try to apply the the Fibonacci design to a nautilus shell interior you see that the ratio of the growth increases do not fit.
