A while ago I received an email out of the blue from Texas Instruments (TI). One of their marketing people discovered this blog and offered to send me a TI-Nspire calculator to review. I quickly accepted, after all, who would turn down a free fancy calculator, right? Once I received the calculator I realized that this was no ordinary calculator; it was a visual Math learning system. I did nothing with it for a couple of months until I finally realized that I was not the best person to review it as it would take me quite a bit of time and effort to learn and appreciate its power. Sure, I could read the manual and run some demos but I didn't think that would give me enough experience to write a very in-depth review.
In discussing my challenge with TI, I learned of some teachers who were successfully using the TI-Nspire in the classroom. One person in particular, Eric Butterbaugh, was teaching Math in Harlem, New York. It occurred to us in that conversation that readers of this blog would appreciate hearing about Mr. Butterbaugh's success with the Ti-Nspire system. I created some interview questions and received back the interview you're about to read.
Earlier this month UC Berkeley professor emeritus of mathematics David Gale passed away. Gale made a number of significant contributions to mathematics and he loved puzzles, games, and finding beauty in mathematics. Gale's daughter had this to say:
In my neck of the woods in northern New Mexico, the Fractal Foundation lives the mission of inspiring interest in science, Math, and Art through the beauty of fractals. The Foundation puts on fractal-related programs in schools and takes them on the road. The Foundation also sponsors a very popular First Friday Fractals night, on the first Friday of the month, for locals and tourists at the planetarium of the New Mexico Museum of Natural History and Science in Albuquerque.
At the Foundation's website are some nice fractal-related resources, many of them free.
- The Fractal Software page has a great fractal viewer (runs on Windows and Mac) that lets you pick from 23 different fractal patterns and zoom in. This one is great fun.
- On the same page there is also a program called "Electric Sheep" that seems to be like the SETI At Home project, but for fractals. For those of you who don't know, SETI At Home is a collaborative effort among millions of PCs to find extraterrestrial life by using some of the compute power in each cooperating PC to analyze signals from space looking for patterns. The idea is that when your computer is idle, like at night, it can give some computer time to the SETI project. Well, Electric Sheep does something similar for fractals. I've not downloaded it but the idea of collaborative fractal building sounds fascinating.
- And, there's "Fractal Grower" (same page) that lets you create certain kinds of fractal patterns.
- There are a half dozen fractal videos at the Fractal Video page. I downloaded the first one, named Glomey, which is 3 minutes 20 seconds long and found the video plus accompanying music to be absolutely mesmerizing.
- The Fractal Art page has some beautiful fractals.
Check out the site. There's a fractal store, a page on chaos theory, and more.
I imagine that many of you are familiar with this remarkable mathematical equation that incorporates what are arguably the five most important mathematical constants into one equation. Yes, phi is missing from this equation. I've been reading this fascinating book, Where Mathematics Comes From. Wikipedia has an article about the book. The book seeks to found a cognitive science of mathematics. While the book is very philosophical and abstract in many places, what fascinated me were the very rich metaphors discussed for a number of common mathematical operations and concepts. These metaphors really helped me to see the conceptual basis for some mathematical processes and operations I took for granted and really helped to ground and deepen my understanding.
Some examples of metaphors:
1. Addition and subtraction. Moving to the right or left along the number line
2. Multiplication of a positive number by a positive number. Moving to the right along the number line by a factor.
3. Multiplication by -1. Rotation to the symmetry point of n.
4. A simple fraction (1/n). "Starting at 1, find a distance d such that by moving distance d toward the origin repeatedly n times, you will reach the origin. 1/n is the point-location at distance d from the origin."
5. Exponential function. The mapping of sums onto products. This explains what it means to raise numbers to non-integer and to negative powers.
I know Maria Miller through her Homeschool Math Blog. As a fellow Math blogger I like to know what others in my community are up to so I follow her blog along with others. I'm also aware that Maria has a series of Math worktexts (workbook + textbook) and worksheets that she offers online through her Math Mammoth business.
I was curious about Maria's offerings and thought that others might be as well, especially homeschool parents, so I asked Maria for a review copy of one of her books. What follows is an unpaid review. I am not currently reselling Maria's books although I might in the future. Beyond a free copy of the book I am reviewing I have received no other form of payment.
I chose the Math Mammoth Geometry 1 Elementary Math Workbook to review. It's a 113 page book, filled with great explorations, clear explanations, and nice illustrations. And it sells for all of $5 as an electronically downloadable PDF file. This is a great value and the deal is even better for folks ordering a number of different books as a set to download or on CD.
While surfing the web I discovered this: World's Hardest Easy Geometry Problem. There's actually two geometry problems on the page. Apparently these are classic problems that have tortured students for a long time. I spent a little bit of time working on the first, and even looking at the two hints, and didn't solve it.
Give it a try.
I'm enjoying the process of discovering how a little Math can go a long way. Readers are enjoying mental Math tricks, Math magic, and other simple things that engage children of all ages to see Math with a sense of awe.
One of my very favorite Math web-sites is Antonio Gutierrez' Go Geometry, subtitled "From the Land of the Incas". I can't even begin to describe this site. It has the most beautiful illustrations of geomtric constructions I've ever seen plus a number of challenging geometry problems. One could easily get lost in this site, and spend many enjoyable hours exploring its many sections.
I'm always impressed to see a new way to do something familiar. Recently, I happened upon a fascinating video, titled Weaving Numbers at the IsAllAboutMath web-site, which has some instructional Math-related videos.
Weaving Numbers demonstrates several non-traditional ways to do multiplication. I found the Napier's bones approach depicted fascinating as well but the one I want to focus on today is the visual approach to multiplication.
The video goes a bit fast for my tastes but since I already had a sense of what visual multiplication would be like I was able to follow it. Here's a nice explanation of the approach from Mudd Math Fun Facts if you can't figure out what's going on in the video or if you want to understand why this technique works.
What I particularly like about this number weaving approach is the visual nature of it. Kids who have a hard time memorizing the multiplication table can simply count the number of points of intersection between the lines that cross. After a while the idea that 2 rows of 3 dots = 3 rows of 2 dots = 6 dots will come naturally to them. What's also wonderful about this approach is that kids can do multiplication by doing addition! So, as soon as kids are comfortable with addition, including carrying, they can learn to multiply. Also, kids can use different colors, as in the illustration above, to engage more fully with the numbers they're multiplying.
Once kids get grounded in this approach to multiplication, and as their confidence builds, they'll learn more quickly, and with better understanding, the approach most of us are taught in school.
A final point, as a Math fanatic, I am delighted whenever I see something like multiplication, which is pretty much taught as an algebraic function, seen from a geometric perspective.
More Fun With Mathematics by Jerome Meyer is a nice little book of interesting Math explorations. It's out of print but Amazon has a few very inexpensive used copies available. In the book I discovered this very odd volume formula that I've never seen before and couldn't find via Google. The author calls the formula "The Amazing Prismoidal Formula."
The formula states the following for any regular solid:
V = H*(B+4M+T)/6
V = volume
H = height
B = area of the base
M = area of the middle of the solid
T = area of the top of the solid
Take a cube with side = 2 as a simple example:
H = 2
B = 2^2 = 4
M = B = 4
T = M = B = 4
V = H*(B+4M+T)/6 = 2*(4+4*4+4)/6 = 8, which is 2^3.
Use a cone with a base of radius r as another example, ignoring for the moment that it's not a regular solid. We'll get back to that.
H is not fixed. It can be any value.
B = pi*r^2
M = (pi*r^2)/4 since the radius of the circle in the middle is 1/2 of the radius of the base
T = 0
V = H*(B+4M+T)/6 = H*((pi*r^2) + 4*(pi*r^2)/4 + 0 )) /6 = (H*pi*r^2)/3, which is the familiar formula for the volume of a cone.
Meyer claims this formula works for any regular solid. Well, I think of regular solids as the 5 platonic solids. Meyer has illustrations of a cube, cone, cylinder, sphere, and conic frustrum (truncated cone).
I tried his formula on a tetrahedron and it works. I could not get it to work for an octahedron but that might have been an algebra mistake on my part. As the number of sides of the regular polygon increases determining the height and the area of the middle becomes more difficult.
An interesting exploration would be to determine for what solids, regular or not, does this formula work.
I read lots of Math books and I've run into many many interesting Math "things" in my travels but here's something very clever I've never encountered before. Let's say you want to multiply 5 by 8. Do the following:
- Plot the graph of y=x^2.
- Draw a line that crosses the parabola where x = -5 and where x = 8 on the parabola. (Ignore the fact that x = -5 and not +5 at the left intersection point; this calculator does not do signed arithmetic!)
- Note the value of y where the line crosses the y-axis.
- The value of y is 40 and indeed 5 x 8 = 40.
Can you figure out why this trick works? Never mind that it's much more work to plot the graphs and determine where the line crosses the y-axis than it is to do the arithmetic in the first place!
This clever exploration, plus a number of other nice explorations for high school students come from the book Mathematics: A Human Endeavor by Harold Jacobs.