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A conversation with my son on place value

This is an excerpt from a conversation I had with my son while we were walking from the subway to the theatre.

My son: Daddy, let’s play a number game.
Me: Okay. What’s seven billion, nine hundred and ninety-nine million, nine hundred and ninety-nine plus one?
My son: That’s too big Daddy, I can’t add those!!
Me: Okay, let’s try a simpler problem. What’s nine plus one?
Son: Ten.
Me: Ninety-nine plus one?
Son: One hundred.
Me: Nine hundred and ninety-nine plus one?
Son: One thousand.
Me: Nine thousand, nine hundred and ninety-nine plus one?
Son: I don’t know how to say the next number. Oh wait! TEN thousand (proudly).
Me: Ninety-nine thousand, nine hundred and ninety-nine plus one?
Son: One hundred thousand.
Me: Nine hundred and ninety-nine thousand, nine hundred and ninety-nine plus one?
Son: How do you say one thousand thousands?
Me: One million.
Son (laughs): Okay the last one is one million.
Me (continuing): What’s nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nineplus one?
Son: Ten million.
Me: What’s ninety-nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nine plus one?
Son: One hundred million.
Me: Nine hundred and ninety-nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nine plus one?
Son: How do you say one thousand millions?
Me: One billion.
Son: That’s the answer then, one billion.
Me: Okay, now try the first problem. What’s seven billion, nine hundred and ninety-nine million, nine hundred and ninety-nine thousand, nine hundred and ninety-nine plus one?
Son (no hesitation): Eight billion.
Son: What’s one thousand billions?
Me: One trillion.
Son: What’s one thousand trillions?
Me: One quadrillion.
Son (giggles): And then?
Me: Quintillion
Son: What’s next!
Me: Sextillions, then Septillions, then Octillions, then Nonillions, then probably Decillions.
Son: What’s next?
Me: Probably Endecillions1 and Dodecillions1, but that’s the limit of my Greek.
Me: What if we played our adding game forever?
Son: Infinity! But we’d have to play in Heaven because even if we played until the end of our lives, we still wouldn’t reach infinity.
(Leads to a long discussion on whether heaven exists and where we go when we die.)

This kind of conversation, between my son and I, is typical as we have a lot of conversations about numbers. In this case, I presented him with a challenging problem, and he was not able to do it. I then used George Pólya’s “trick” of asking my son simpler problems which led up to him seeing how to solve the more complicated problem. Does this mean that my son understands place value, or even all the numbers he was able to say? Probably not, our conversation was entirely linguistic, but it’s a start.

 

1.Here is a list of the names of the large numbers. Notice that my two guesses are actually wrong (but close!).

 

 

Bean counting and place value

One area of mathematics which I strongly suspect many students have problem understanding is place value. It is an important abstraction for students to understand, and without understanding it, it is unlikely that students will progress very far in arithmetic (and then will likely struggle in algebra later).

Here is an activity my friend David Miles told me about years ago which I would very much like to see in action some time.

A big bag of beans

Give the students a very large amount of beans (or something similarly small and dry) to count. For younger kids, give them a smaller amount, and for older kids, give them a larger amount.

Start by asking them to estimate how many beans are in the bag. Perhaps ask them to give you a number which is probably more than the number of beans, and a number which is definitely less. It doesn’t really matter how good this estimate is, the idea is that by asking students to give an estimate, and then letting them compare their estimates later with their more accurate answers, that students may improve in estimating.

Next, ask students to work in groups to count the beans. Give them LOTS of time. Give them some very small cups they can use to help them with their counting which should ideally hold about 10 beans maximum. If you need to use larger cups, ask students to restrict themselves to only putting 10 beans in at a time. While kids are counting, if they aren’t keeping track somehow of their numbers, count loudly to distract them, forcing them to keep track of their results. Don’t give them any paper or pencil, just the cups.

The idea is, the cups are too small to hold many beans each, and the students don’t have enough cups to hold all of the beans. What they will end up having to do is to choose one cup to represent ones, when this one fills up they will have to create another cup to put a bean in to represent 10 beans in the first cup, and when this cup fills up, they will have to create another cup to represent 10 beans in the 10-bean cup (or 1 bean represents 100 beans) and this leads to what place value is, at least for numbers greater than 1.