Wild About Math! Making Math fun and accessible

6Dec/075

The algebra of cross-multiplication

A few people were underwhelmed with my video on how to multiply together a pair of 2-digit numbers without writing down partial results. They didn't see a time savings. Fair enough, although not needing to write anything but the three (or four) digits of the answer can be a nice time savings and a source of smugness for many.

Next week I'm going to produce some more mathcasts. The first one will show how to multiply together a pair of 3-digit numbers using the cross multiplication approach. I think that will impress more people.

There's a general approach to these multiplications I want to teach you. If you understand the algebra behind this approach you can derive the steps for multiplying together numbers containing any number of digits. However, I'll warn you that beyond 5 digits this approach, and any other approach that works for any numbers and not just special cases, will really work your mental muscles.

Filed under: Algebra, Fast Arithmetic Continue reading
24Nov/078

A fun arithmetic game that sparks exploration

Here's a game that's easy and leads to a nice exploration of number theory for those so inclined. Two people play. All you need is a sheet of paper and a pencil or pen. Here's how to play:

  1. Each person thinks of a number between 1 and 50 without telling the other person what the number is. Then, each person writes their number on the sheet of paper.
  2. Decide who is going to go first, by tossing a coin or in some other mutually agreeable way.
  3. Players take turns writing down the positive difference between any two numbers on the sheet of paper.
  4. Numbers cannot appear more than once on the paper.
  5. The player who cannot write down a unique positive difference loses.

Here's an example of how a game might go between Sol and his friend Michele.

  1. Sol thinks of the number 5. Michele thinks 3.
  2. They write 5 and 3 on the paper.
  3. Sol goes first.
  4. 5 minus 3 is 2 so Sol adds 2 to the paper.
  5. The paper now has these numbers: 5 3 2
  6. Michele notices that 5 minus 2 is 3 but 3 is already on the paper.
  7. Michele also notices that 3 minus 2 is 1 so she writes 1 on the paper.
  8. The paper now has these numbers: 5 3 2 1
  9. Sol notices that 5 minus 1 is 4. He writes 4 on the paper.
  10. The paper now has these numbers: 5 3 2 1 4
  11. Sol wins as no more unique differences can be calculated.

Here's another sample game:

  1. Sol thinks 8. Michele thinks 6.
  2. The paper has: 8 6
  3. Michele goes first.
  4. Michele notices that 8-6=2. The paper now has: 8 6 2
  5. Sol notices that 6-2=4. The paper now has 8 6 2 4.
  6. The game is over and Sol wins as no more unique differences can be calculated.

Here are some interesting exploration questions:

  1. Once both numbers are written down is there a way to determine who will win?
  2. Once both numbers are written down does strategy matter, other than who goes first?
  3. For starting numbers of 5 and 3 all numbers between 1 and 5 got written down but when 6 and 8 were the starting numbers only 2, 4, 6 and 8 were possible differences. What determines whether all numbers get used and if not which ones are used and which aren't?

This game is related to Euclid's algorithm and to the greatest common divisor of two integers. At Cut the Knot there's a Java version of this game, Euclid's Game, that you can play alone against the computer. In the computer game the computer picks the two starting number but you can practice determining who should go first.

Filed under: Algebra, Fun, Game 8 Comments
21Nov/077

Help kids learn multiplication with this visual approach

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 multiplicationVisual Multiplication from Mudd Fun Math Facts 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.

Filed under: Algebra, Education, Geometry 7 Comments
17Nov/077

The amazing volume formula

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

where:

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.

Filed under: Algebra, Geometry 7 Comments
1Nov/075

A clever use of a parabola to perform multiplication

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:

  1. Plot the graph of y=x^2.
  2. Draw a line that crosses the parabola where x = -5 Parabola used for multiplicationand 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!)
  3. Note the value of y where the line crosses the y-axis.
  4. 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.

Filed under: Algebra, Geometry 5 Comments
8Oct/070

A picture is worth …

How many of you remember doing geometry proofs in High School? How many of you enjoyed writing them? I don’t know about you but I’ve always preferred pictures to words when it comes to understanding how something works.

Proofs Without Words“Proofs Without Words: Exercises in Visual Thinking” by Roger B. Nelsen is a wonderful book that provides visual insights into how one might go about proving mathematical theorems. The Pythagorean Theorem has always been a mystery to me. How are the squares of the sides of a right triangle related to its hypotenuse? “Proof Without Words” has five clever illustrations that guide readers in writing their own proofs.

If you ever doubted that algebra and geometry were related, the diagrams demonstrating how to compute sums of series will produce aha! experiences.

Writing proofs when one is guided by visual cues is a much more fulfilling endeavor than stringing together dry facts from memory. This book delivers much fulfillment in exploring theorems in geometry, algebra, trigonometry, sequences, and other aspects of Math.

Filed under: Algebra, Beauty, Book Review, Fun, Geometry, Proof Without Words No Comments
8Oct/070

Algebra help is just a click away

www.algebra.com is a great free site for students needing help with their algebra homework. People who enjoy helping others with their algebra homework sign up as volunteer tutors. Students post their homework problems and the tutors answer them, ideally providing an explanation of how they got to the solution.

Some months ago I was quite active in the algebra.com community, having solved and explained 188 problems under the moniker joyofmath. After having gotten bored solving the same kinds of problems over and over I started being much more selective, solving the more challenging problems that other tutors were ignoring. That was a very satisfying experience. And, if you think I’ve solved lots of problems, there’s someone with the handle stanbon who holds the record for most problems solved - 10,581 to date. Wow!

Algebra.com is easy for students and tutors to use and it even has a nice mechanism for formatting text so that it looks good, even when there are exponents and math symbols involved. So, if you’re needing help or wanting to help algebra.com has something for you.

Filed under: Algebra, Tutoring No Comments
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