Tag Archives: number

Things that come in pairs

Talking Math with Your Kids week continues.

I was doing the dishes one morning while Tabitha (who was four, nearly five) drew in the other room. She came in for help.

Tabitha: Where are the scissors?

Me: I don’t know. How many pairs of scissors do we have anyway? We have a lot, but we can never find them. Why is that?

T: I just need one pair of scissors.

Me: Isn’t it weird that scissors come in pairs?

T: Yeah.

Me: What else comes in pairs?

T: Pants do. And shoes.

Me: Oooh. Good. What else?

T: Legs. And ears. And noses [giggles].

Me: Noses?!? Noses don’t come in pairs, silly!

T: Eyes do. And glasses.

Me: Nice! Eyes. You know, it’s not just people who have eyes that come in pairs. Fish do, too.

T: Of course!

Me: How many pairs of eyes are in our aquarium?

T: Seven.

Me: So how many eyes is that?

T: Nine?

[Griffin (who was seven years old) wandered in from the living room.]

Me: Griff, how many pairs of eyes are in our aquarium?

G: Seven.

Me: So how many eyes is that?

G: Fourteen.

Me: How did you know that so fast?

G: Seven plus seven is fourteen.

Me: Right. But that’s two sevens. Don’t we need seven twos?

G: Yeah, but it’s the same answer either way.

If kids are going to understand place value, they’ll need to be able to think about different units. Sometimes a unit is a thing (an eye); sometimes a unit is a group (a pair of eyes). Giving them practice counting groups and individual things supports their mathematical development. Helping them notice that some things usually do come in groups supports it too.

Correcting Tabitha when she added 7 and 2 to get 9? That wasn’t nearly so important.

But you’d better believe I talked with Griffin later about things that are commutative.

Can a number be bigger than anything you can count?

It’s Talking Math with Your Kids week here on OMT.

We’ll get started with a favorite topic: large numbers.

My son Griffin was thinking about large numbers in the car the other day. He was trying to figure out what good it is to have a number (here, googolplex, which for the record is 10^{10^{100}}) that is larger than anything you can count.

Griffin: If you put all the things [in the world] together, would that make googolplex?

Me: No.

G: Even if it’s nanoinches?

Me: Nope. Still not googolplex.

G: Even if it’s half-nanoinches?

G: Even if it’s all of the seconds of the world being alive?

Me: Nope.

G: Even if all the seconds of the universe existing?

Me: No.

I love the developing proportional reasoning embedded in Griffin’s questions.

For each example, he scales it up when his first try doesn’t do it.

If nanoinches don’t work, surely half-nanoninches will! Plenty still to learn about orders of magnitude, I’m afraid.

Composed units, continued…

Class today was an unholy mess.

Truly a mess.

You know that feeling you had as a kid on the playground? When you thought you were jumping from the first level and then remembered that you had actually started on the second? Or the first time you jumped off the swing and you realized only after doing so that your landing wouldn’t be like falling from the height of the swing, but like falling from quite a bit higher because you were headed UP when you left the swing?

I had that feeling in class today.

But you’re never gonna know how high you can jump from if you don’t go ahead and jump.

I wrote last week about my composed units assignment in the math for future elementary teachers course:

In this exercise, we had already dealt with two of three major components of the number concept, which are (with a shout out to Karen Fuson):

  1. Quantity (how many things there are)
  2. Numeration (how we write how many things there are), and
  3. Number language (how we say how many things there are).

In particular, we dealt with 1 and 3. A pair of socks is number language that indicates we have 2 socks. A dozen eggs is number language that indicates we have 12 eggs.

And we had already done other activities to deal with quantity, numeration and number language. But I wanted to bring the composed unit activity full circle by considering numeration.

So I returned to this example:

And I asked, How many eggs are there?

There are 12.

But imagine a world in which our number system weren’t based on making groups of ten. Imagine that it were based on making groups of a dozen.

Then we could say that there is one group of eggs, and that there are no ungrouped eggs.

In that alternate world, we might write the number of eggs this way:

This means that we have 1 group, no leftovers and that the size of our groups is a dozen. We will agree to read this as “1 dozen”, not as “one-zero-base-dozen” although the latter may be necessary to facilitate communication later on, just it is sometimes necessary to spell out my last name…”de-eh-en-i-ee-el-ess-OH-en”. And we probably shouldn’t read it as “ten, base dozen” because we’ll get confused.

If we have two dozen, we write this:

Next, we’ll write an equation to show some different ways we can notate a dozen.

We talked about the conventions of notation. The little dozen on the 10 there indicates to us the size of the groups we are making in this particular numeration example. We call that the base. The middle part of the equation is a typical, well-formed English phrase, and we have added the little ten as a subscript on 12 in order to make clear the size of the groups we’re making. But that 12 on the right should feel like home-that’s how we write how many there are in our usual number system.

We did another example together.

How many tires? Well, let’s call the group of tires a set. Then there are:

Then I turned them loose to consider what How many question to ask, and how we should notate it in a number system built especially for counting things according the composed unit in the picture.

Interesting examples to consider include the ones that were interesting the first time around:

How many shoes?

And this one:

How many Pop-Tarts?

Considering only the silver pack lying on the table, we have:

But there are 4 packs in a box. How should we notate that? Here are three possibilities for your consideration:

Each is justifiable, and we worked through the first two. That discussion took a lot of mental effort on everyone’s part. One student asked very apologetically, “This might be a question that doesn’t go anywhere, but writing one-zero-zero doesn’t seem right, because aren’t we making groups of 2, not of 4?”

After insisting that she retract the apology, I observed what a smart question it was. After all, don’t we make groups of ten, and then ten groups of ten, and then ten groups of ten groups of ten, etc.? Wouldn’t it be a crummy number system if the size of the group kept changing on you? (Side note-the Mayan number system did this: first grouping is of 20, second is of 18, then 20’s the rest of the way.)

So why might we want to make the third place worth four times the second place? Because that’s how Pop-Tarts come (according to the photograph). The original unit is a Pop-Tart. These are composed into packs of 2. These packs are composed into boxes of 4.

We tried one more thing…what if we write (for the sake of moving forward):

And then someone comes along, opens a pack and eats one Pop-Tart. How should we write how many remain?

Yup…I jumped off that swing today. Didn’t really have a solid plan for sticking the landing. But I learned an awful lot by scraping my elbow when I landed.

And my students made some lovely connections. After class, a student wanted to check his thinking on the gallon of milk. How many ounces are there? He wrote:


See a previous post for information on our alternate base system that led to the expression at the far right.

Questions from middle schoolers VII: Proper factors

Why do we talk about “proper factors” but not “proper multiples”?

Good question.

I had never thought about this before. A colleague and I are thinking about this. My colleague’s guess is that in number theory (the mathematical field that deals with this stuff), there are some things that are true of proper factors that are not true of the number itself, and that this is not a problem with multiples.

But I don’t really know for sure. I’ll find out.

Questions from middle schoolers VI: Irrational numbers

How do they know irrational numbers never repeat?

What a lovely question. When you are told that “irrational numbers have decimal representations that never repeat,” it’s a good instinct to say, “That’s just because you haven’t looked hard enough.”

There are only 10 digits, right? So they must repeat eventually. There are only so many ways to arrange 10 digits. So it makes sense that they don’t repeat very often, but they must repeat eventually, right?

Wrong.

And the way we know they don’t repeat isn’t what you expect. You expect that mathematicians have looked, but not found patterns in the digits of irrational numbers, and that from this they conclude that the digits don’t repeat. You are correct in being suspicious of this argument.

But it’s not the one mathematicians use.

Instead, we know that all repeating (or terminating) decimals are rational. And we know that rational numbers have other properties (like that they can be simplified-or reduced-to a fraction with whole number numerator and denominator). And then we know that a number such as pi or the square root of 2 cannot be reduced to a fraction with whole number numerator and denominator. Therefore pi and the square root of 2 must have decimals that don’t repeat (or terminate…because if they did, they would be rational, which would mean they could be reduced which they cannot).

This is common in mathematics-knowing that something is true in a roundabout way.

Questions from middle schoolers II: New numbers

Are mathematicians looking for any new numbers?

They are always looking for the next biggest prime number. We know that there are infinitely many of them, and we know some really big ones. Every so often, it makes headlines when a supercomputer finds another, bigger one. The goal isn’t to prove that there is another one; we know that already. The goal really is to test the computational power of computers, since all the undiscovered primes have many, many digits (in 2008, mathematicians found one that has 13 million digits).

Instead of new numbers, mathematicians are always on the lookout for new categories of numbers. Whole numbers, rational numbers, irrational numbers, imaginary numbers, transcendental numbers, etc. These are all categories of numbers that mathematicians discovered (created?) in the process of their work. Surely there are more to come…

Questions from middle schoolers I: Pi

I travel a lot to work with middle school math teachers. In one classroom recently, I was oversold as a visiting mathematician. Visiting I was. Mathematician I am not. I do think about mathematics for a living, but I do not create original mathematics.

Nonetheless, students in the class generated a number of questions they wanted to ask a mathematician. So this series is for Ms. Otto’s eighth-grade students at KMS…

Who discovered Pi?

Can you believe that an entire book has been written about this subject? It’s called A History of Pi by Petr Beckmann.

Here’s the very short version…

In order to discover pi, people needed to understand ratios. That is, they needed to not just be able to count objects, they needed to be able to see relationships between quantities. That took a long time, until about 2000 B.C.

At that time, there were two major societies doing serious mathematics-the Babylonians and the Egyptians. Each of these societies appears to have noticed that there is a relationship between the diameter of a circle and its circumference, and both societies had decent estimates for pi (although neither called it that).

The next several thousand years involved people getting better estimates of the value of pi. There are lots of interesting mathematical advances that came about as people tried to do this.

In the 1700’s, at least two mathematicians demonstrated that pi was irrational (Lambert and Legendre). In 1882, Lindemann proved that pi is not just irrational; it has an even more profound property-it is transcendental. This means that it is not the square root (nor cube root, nor…) of any rational number. It is an even more special number than the square root of 2.

So who discovered pi? That’s a bit like asking who built the Empire State Building. Pi is an achievement of human intellect more than of a single person.