Kiss those Math Headaches GOODBYE!

Archive for November 25, 2008

Tutor Tales #1: Color in Geometry


This if the first in what I hope will be a long series of brief blogs called Tutor Tales blogs.

The idea is that while I’m tutoring I get ideas or insights on how to help students, and then I write up a short blog entry on that experience, preferably on the day that the event occurred.

I hope that these Tutor Tales will give you examples of approaches to math that help students (or that do not help, depending on what I did), and that they give you a chance to reflect on your own teaching.

For the first Tutor Tale entry, I just noticed how useful it can be to use color in geometry.

The girl I was tutoring had a problem:  Find out how many diagonals can be drawn inside a regular, convex nine-sided polygon.

I’ve already noticed that this girl likes color, and she is 17 years old. So I had a hunch that she would be open to trying a color-approach.

We created the non-agon by first drawing a circle, and then marking off nine points on the circle. Then we connected the points sequentially.

To find out how many diagonals we could draw for such a figure, we chose one color for the top point, green, and drew all of the diagonals we could for that point, in green. It turned out that there were 6 diagonals, so we put a big 6 in green at this vertex. Then we tried the next vertex, which we colored pink. We found that we could create 6 additional diagonals from this vertex, and we colored these pink. So we put a big pink 6 by this vertex. We went around the circle in a clockwise way, using a different color for each vertex. All in all we found that the pattern of diagonals was:  6, 6, 5, 4, 3, 2, 1, 0, 0, for a grand total of 27 diagonals.

Here’s the image of the figure we worked on.


One thing to consider, especially if you teach geometry, is how many opportunities there are in geometry to use color to separate different concepts and to relate similar concepts. Check it out and see what you discover.

Happy Teaching!

—  Josh

How to Practice Dividing Fractions

Last week I sent out a blog describing my FRACTION SANDWICH approach to dividing a fraction by a fraction. The title of the blog was: Dividing Fractions:  From Annoying to FUN!

In today’s post I offer ten solid problems that help students practice the FRACTION SANDWICH, with answers just below.

Note:  if your students struggle with these problems, it’s very possible that some of them could brush up on the rules of divisibility. For a quick review of those rules, go to:

Try these:

a)  (2/9) / (8/12)

b)  (12/16) / (15/40)

c)  (36/21) / 63/35)

d)  (44/24) / (77/48)

e)  (15/49) / (21/56)

f)  (39/52) / (24/56)

g)  (27/45) / (33/65)

h)  (12/18) / (28/45)

i)  (17/51) / (99/121)

j)  (28/18) / (42/54)


a)  (2/9) / (8/12)  =  1/3

b)  (12/16) / (15/40)  = 2

c)  (36/21) / 63/35)  =  20/21

d)  (44/24) / (77/48)  =  1 and 1/7

e)  (15/49) / (21/56)  =  40/49

f)  (39/52) / (24/56)  =  1 and 3/4

g)  (27/45) / (33/65)  =  1 and 2/11

h)  (12/18) / (28/45)  =  1 and 1/14

i)  (17/51) / (99/121)  =  1/27

j)  (28/18) / (42/54)  =   2

Multiplication Trick #5 — How to Multiply Two-Digit Numbers by 11

This is the fifth in my series on multiplication tricks. I suggest that you make mental math “tricks” a steady part of your math instruction. Benefits students will reap include:

—  delight with the tricks themselves

—  enhanced confidence in working with numbers

—  students who otherwise don’t like math — or don’t like it much — often find the tricks irresistibly fun and interesting


WHAT THE TRICK LETS YOU DO: Multiply two-digit numbers by 11.

HOW YOU DO IT:  To multiply a two-digit number by 11, first realize that the answer will have three digits. The first (left-most) digit of the answer is the first digit of the number; the last (right-most) digit of the answer is the last digit of the number; and the middle digit is the sum of the first and last digits.

But those are just words … here’s a living, breathing example …

Example:  11 x 25


Look at 25. The first digit is 2; the last digit is 5.

First digit of answer is 2, so thus far we know the answer looks like:  2 _ _

Last digit of answer is 5, so now we know the answer looks like:  2 _ 5

Middle digit is 7, since 2 + 5 = 7.

The answer is the three-digit number:  2 7 5, more casually known as 275.

It’s that easy!


First digit of answer is 6, so thus far we know the answer looks like:  6 _ _

Last digit of answer is 3, so now we know the answer looks like:  6 _ 3

Middle digit is 9, since 6 + 3 = 9.

The answer is the three-digit number: 6 9 3, or just 693.

Try these for practice:

11 x 24

11 x 31

11 x 52

11 x 27

11 x 34

11 x 26

11 x 62


11 x 24 = 264

11 x 31 = 341

11 x 52 = 572

11 x 27 = 297

11 x 34 = 374

11 x 26 = 286

11 x 62 = 682

NOTE:  If you’re clever (and we’re sure that you are), you have probably realized that this trick, as described, works only when the digits add up to 9 or less. So what do you do when the digits add up to 10 or more? Some of you may figure this out on your own. For those who need a little help, the answer to this will be included in an upcoming blog post.

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on  Just click the links in the sidebar for more information!