Question

Dear Mr. Mitch (Poppy),

Bet you can't guess who this is!

OK, remember the other day when we were talking about multiplying you said that 6x5 is the same as 5x6? I mean equals, right? And I said, no, how can that be when you already said that times means "groups of" -- so how could five groups of six be the same amount as six groups of five? Remember? You explained and I got it, but this morning I was thinking about it again and I couldn't remember how we did it. So could you tell me again on my computer so I won't lose it?

(Give up? It's me, Sawyer!)

Answer

Of course, Sawyer, my little mathematician. Look at these three guys:

Now look at their feet.

A lot of people might think of them as three groups of two feet because each person has one group of two feet. But if you label them like this:

Then you can think of them as three left feet and three right feet, and that gives you two different groups, each group has three feet in it; one group is left, left, left, and the other is right, right, right.

So two groups of three still adds to the same amount of three groups of two, which is six feet. OK?

One more example: five cars is five groups of four tires, which adds up to 20 tires (since each car has one group of four tires, and there are five cars). But if you line the cars up like this:

You can say there are four different kinds of tires: left front ones, right front ones, left back ones, right back ones. That's four different kinds of groups, and if you walk all the way around the row of cars, you can count five in the group of left front tires, five in the group of right fronts, five in the group of left back tires, and five in the group of right back. So, you now have four groups with five tires each, and that will also give you a total of 20 tires in all, EXACTLY the same ("equal") to five groups of four.

And if you ever feel like trying a new one like this on your own, think of snowmen.

They each have three main parts: bottom, middle, and head. So there are six groups of three parts. But you could also say that are only three groups, and...?

Love,
Mr. Mitch -- Poppy

Question

Dear Poppy,

The other night at the restaurant you gave me a really cool way to remember some of the math things that I kept forgetting, and I wanted to see if you'd let them stay on your website if I said them right, OK?

Here it goes:

There are three types of triangles, and their names are isosceles, equilateral, and scalene (mommy helped with the spelling of "isosceles" but even she said we should check it in a dictionary because it's tricky even for people as old as you!). Anyway, I always used to forget which is which, so you said to try this:

Spell isosceles WRONG on PURPOSE; spell it, ISOXELES or even I-SOCKS-ELES, and say it that way in your head, so you think of a pair of socks... a pair of socks is two matching socks, and i-socks-eles triangles have a pair of matching sides. So if a triangle has two sides that are the same size, BUT NOT THREE, then it's an isosceles 'cause it has one pair of matching sides. Like this:

Then, for scalene think of the scales on a fish. Those are the crunchy, shiny bits of skin that fish have. Even though most fish have thousands of little scales, and even though there are millions of fish in the ocean, when you examine them under a microscope, no two scales are exactly the same. So, scalene triangles are triangles that don't have any sides that are the same. Like this:

Finally, equilateral triangles have three sides that are equal size. Like this:

Hope that's right, because that's how I remember it.

Love,
Abby

Answer

Dear Abby,

That's right, because that's how I remember it, too.

Good job! Soon I'll answer the questions that you asked about angles.

Love,
Mr. Mitch (Poppy)

(Triangle Mnemonic Techniques are Mitch Adler Original Work)