Me: What’s 7999999999 plus 1?
My son: That’s too big Daddy, I can’t add those!!
Me: Okay, let’s try a simpler problem. What’s 9 + 1?
Me: 99 + 1?
Me: 999 + 1?
Me: 9999 + 1?
Son: I don’t know how to say the next number. Oh wait! TEN thousand (proudly).
Me: 99,999 + 1?
Son: 100, 000
Me: 999,999 + 1?
Son: How do you say 1000 thousands?
Me: 1 million
Son (laughs): Okay the last one is 1 million then.
Me (continuing): What’s 9,999,999 plus one?
Son: 10 million.
Me: What’s 99,999,999 + 1?
Son: 100 hundred million.
Me: 999,999,999 plus one?
Son: How do you say 1000 millions?
Me: One billion.
Son: That’s the answer then, 1 billion.
Me: Okay, now try the first problem. What’s 7,999,999,999 plus one?
Son (no hesitation): 8 billion.
Son: What’s 1000 billion?
Me: One trillion.
Son: What’s 1000 trillions?
Me: One quadrillion.
Son (giggles): And then?
Son: What’s next!
Me: Sextillions, then Septillions, then Octillions, then Nonatillions, then probably Decitillions.
Son: What’s next?
Me: Probably Endekatillions and Dodekatillions, 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.)
My son was given the following problem in his math club.
We can see that 2 + 2 = 4 and 2 × 2 = 4 and similarly 1 + 2 + 3 = 6 and 1 × 2 × 3 = 6.
Can you find four whole numbers where their sum and their products are the same? What about five whole numbers?
Spoiler alert: If you read further, you will get the solution to this problem. You may want to try it out yourself first. Scroll down when you are ready.
My son found 1 + 1 + 2 + 4 = 8, 1 × 1 × 2 × 4 = 8 and 1 + 1 + 1 + 2 + 5 = 10, 1 × 1 × 1 × 2 × 5 = 10. From this he conjectured that if you take a bunch of 1s followed by 2 and then followed by the sum of the 1s and the 2, that this list of numbers has the property that it multiplies and adds to give twice the last number in the list.
I asked him if he could prove that his conjecture is always true. He said, “The number of numbers minus two is the number of ones we need. Then plus two, that’s now the same value as the number of numbers at the end. If you add all of those numbers, you’ll get twice the number. And if you multiply all of those numbers you get 2 times the number, same value as the sum, because none of the 1s changes the value.”
“Or,” he continued, “If the number of numbers is n, then the number of 1s is n – 2, followed by a 2, followed by n. n – 2 + 2 is n. So the sum is 2n. The product is 1 times 1 times … and so on times 1 and then times 2 and times n which is also 2n.”
This is the equivalent problem in a different representation.
My son was comparing two fractions to decide which is bigger (a problem that appears in his Beast Academy workbook).
He looked at the fractions and noticed that if he changed the first fraction, he could get the denominators to be closer together.
Him: “Oh, now I know that is larger than !”
Me: “How do you know that?”
Him: “If we had we would know that is more than since is more than . We just have 1 more so for sure is more than .”
What I find interesting about this is that I was taught that in order to compare two fractions, one first needs to be able to find a common denominator. Obviously that’s not true.
Tonight I introduced the sad tale of Daedalus and Icarus to my son. In case you are not familiar with the story, Daedalus and Icarus are stuck in the tower in the middle of the maze with a horrible minotaur blocking their escape. Daedalus builds wings and they try and fly away.
In the mathematical version of this story, created by Gordon Hamilton, Daedalus “flies” by choosing a number. If the number is odd, he multiplies it by 3 and then subtracts 1. If it is even, he divides it by 2. For Icarus if the number is odd, he multiplies it by 3 and adds 1. If it is even, he also divides it by 2. The objective is to pick a number that does not eventually end up as a 1 because this means that Daedalus or Icarus just crashed into the ground.
My son and I played this game (based on the Collatz Conjecture) and quickly found a value of Daedalus that allows him to fly away and escape the maze. We then started trying to find values that allow Icarus to escape (we haven’t succeeded yet).
At one point my noted the following:
You know, Dad. There are an infinite number of solutions to the Daedalus problem. If you have a solution, to find another starting number that works, just double it, and so on. We can keep finding new solutions by doubling whatever number we have. Since we have one answer that works, we know there are an infinite number of ones that do.
This is essentially inductive reasoning. He’s reasoned that if we know case n works, we can find a larger case by doubling it. Since we know at least one solution for Daedalus works, and that we can double as many times as we like, then we must have an infinite number of solutions.
While driving on the highway, my wife had the following exchange with my three year old son, Timon.
“Mom, what’s three plus three?”
“Mom, what’s four plus four?”
“So… three plus four is seven.”
My wife was so stunned, she told me later, she almost stopped the car to tell me.
Timon asking us questions about addition is nothing new. He’s been trying to participate in a similar game my older son plays with me. What’s remarkable here is that Timon has derived the addition fact for three plus four.
Timon also recently noticed that one plus three and two plus two both equal four and showed me on his fingers why he knew it was true. He’s playing with addition of small numbers, sometimes asking me what the sum is, sometimes showing me by counting out on his fingers. He has yet to count on from a number he knows (he starts counting at one each time) but he does recognize groups of objects up to four and sometimes five in size without counting directly.
My point here is to pay attention when your children discover ideas on their own. They might just surprise you.
My son was reading one of his novels recently which had Roman Numerals as the chapter titles. He’d learned about Roman numerals in kindergarten but had forgotten what L meant. Here’s how he figured it out again.
I must be 1 because it was the first chapter.
I forgot what L meant and when I looked at the last chapter, which was LII, I thought L was 100. Then I looked at the page number, which was 504, and decided that was impossible because that would mean each chapter would be only five pages long which made no sense. So I decided L must mean fifty.
Knowing a little bit about Roman numerals was probably helpful here, but so was knowing something about how books work and what kind of answer would be realistic. When children don’t know what kind of answer is reasonable, it makes doing the mathematics harder.
“I want a knife. Can you give me a knife?”, Timon asked.
“Why do you want a knife?” I responded.
“I want to spread my cream cheese. On my cracker.”
“Why do you want to spread your cream cheese?”
“Because I want more cream cheese,” replied Timon.
I gave Timon a knife and he proceeded to take the cream cheese that was on his cracker and spread it out until it covered much more of his cracker. Then he smiled and ate the cracker.
Timon doesn’t yet know that no matter how he manipulates his cream cheese, there’s always going to be the same amount. He has yet to learn the Conservation of Cream Cheese Theorem.
But he from his actions we can infer what he does understand that the more area of cream cheese there is on his cracker, the more cream cheese there is. This is true, provided the thickness of the cream cheese is constant.
I wonder how much longer he’ll think that he can get more of something by spreading it out.