Wild About Math! Making Math fun and accessible

3Nov/071

Probability and divisibility by 30

Well, probability is on my mind - this sounds like a great song title, no?

I've got another probability problem, this one presented to me by my brother Abe as I called him from the airport half way to my awesome vacation in Hawaii. I chewed on it for a little while then called him back with the answer, got his answering machine and left the answer there to impress him. He didn't call me back but I think I got the answer right. Maybe that's why he didn't call?

Here's the problem:

What is the probability that a 10 digit number will be divisible by 30?

The problem was really stated as asking how many ten digit numbers are that are divisible by 30, which is in essence the same problem.

This is a neat problem in that it involves several aspects of solving Math problems and has a number of dimensions to it that lead to a pleasant exploration. Some ideas on approaching this problem:

  1. Once must assume that a 10 digit number has no leading zeros, i.e. that the first digit is between 1 and 9. How does this special case for the first digit affect the probability? Without this consideration someone might hastily and incorrectly conclude that the probability is real close to 999,999,999 / 30.
  2. How many 10 digit numbers are there? How would you count them?
  3. If you take the number of 10 digit numbers and divide that number by 30 do you then get the right answer? Why or why not?
  4. What is interesting about the number 30? How would you test for divisibility by 30? Hint: Factor 30 into primes.
  5. Does the fact that 30 ends in zero lead to a simplification of the problem?
  6. Can you solve this problem for numbers with fewer digits? How many single digit numbers are there divisible by 30? How many 2 digit numbers? How many such 3, 4, and 5 digit numbers are there?
  7. What if the problem asked about divisibility by some other number, maybe one that fewer or more factors? Might it then be easier to utilize an approach that didn't take into consideration the factors of the number we're testing divisibility for?
Filed under: Probability 1 Comment
3Nov/070

Fun calendar probability problem

I picked up a little gem from Math Wonders to Inspire Teachers and Students, byMath Wonders to Inspire Teachers and Students Alfred Posamentier. The book, by the way, is outstanding. Posamentier has become one of my heroes for writing books that awe and inspire students and teachers to get, well, Wild About Math! Someday I will review this, and other, outstanding books that make Math accessible and fun.

Here's the little problem:

What's the probability that 4/4, 6/6, 8/8, 10/10, and 12/12 all fall on the same day of the week?

It's fun to see what people come up with, especially if they give you an answer off the top of their heads.

A fun little exploration is why the answer is what it is. This exploration is a nice first step in doing calendar arithmetic.

Filed under: Probability No Comments
2Nov/0710

11 tips for building a strong Math foundation for kids

During my high school years I spent two summers at the Ross Math Program studying Number Theory and other more esoteric branches of mathematics. The head of the program, Professor Arnold Ross, was not only a brilliant mathematician but he knew how to teach Math to bright students in a way that we all learned some rather difficult Math and really enjoyed it. He taught using a "discovery" method that encouraged exploration and that guided students to bigger discoveries and proofs from simpler ones.

This article incorporates what I learned from Professor Ross with what I've learned from other life experiences.

1. Create a fun environment.

A favorite quote of mine is "Environment is stronger than willpower." For two summers I got to live and breathe the truth of that statement. Imagine twenty or so bright and motivated high school students going to class together, eating meals together, living in the same dorm, working on problem sets (homework) together and you have the makings of too much fun and an awesome learning environment. None of us had to exert too much willpower to thrive in that environment.

Granted, most of you don't have the luxury of this natural learning environment but there are things you can do to lead with fun. If you're in a teaching or helping role, is there a Math club your students or children can join? Can you create one? Can you get a few of the neighborhood kids together one evening or Saturday every week to learn Math together?

If you're working one-on-one with a student or doing Math on your own see my article 10 Ways to get wild about Math for ideas on creating a Math-friendly learning environment.

2. Define success.

From years of personal growth work I've realized the value of having a vision of what I want to achieve. Success for you or your student might be to learn the multiplication table or to be good at solving a variety of algebra word problems, or perhaps to attain a particular score on the Math part of the SAT. Or, success might be an increased self-esteem or greater confidence on Math tests.

Get clear on what you want to achieve, write down the outcomes as specifically as possible, review and adjust as needed, and you'll be well on your way to Math success.

3. Develop a plan for achieving success with measurable goals.

Step 2 was the vision piece. This is the action step.

How can you help your child to better learn his multiplication tables? Maybe the action steps are:

1. You draw the multiplication table, your child copies it.
2. You show him patterns in the table that can simplify learning.
3. You use flash cards to teach the basic Math facts.
4. You have your child create his own flash cards.

As in visioning, the more specific and clear you can be with your action plan the more likely you are to succeed.

Measurable goals are critically important and you can have multiple goals on the way to achieving the big goal. For learning the multiplication table intermediate goals might be:

1. Lower anxiety when working on learning the table
2. Can fill in a quarter of the table by himself using patterns you've shown him
3. Consistently gets half of the flash card problems right
4. Gets 100% of the problems right

A timetable for achieving goals can be very helpful but it can also backfire, especially if you don't have a good sense of how long it should take to learn something. I recommend setting time goals but being very willing to adjust them as needed. Remember, any progress is better than no progress even if it's not as much as you would like to see. Finally, if your child has self-esteem challenges I would de-emphasize the performance aspect altogether and focus on building his confidence.

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