# Generalizing

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.”

# Which is greater?

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).

$\frac{2}{5} \circ \frac{7}{21}$

He looked at the fractions and noticed that if he changed the first fraction, he could get the denominators to be closer together.

$\frac{8}{20} \circ \frac{7}{21}$

Him: “Oh, now I know that $\frac{8}{20}$ is larger than $\frac{7}{21}$!”

Me: “How do you know that?”

Him: “If we had $\frac{7}{20}$ we would know that is more than $\frac{7}{21}$ since $\frac{1}{20}$ is more than $\frac{1}{21}$. We just have 1 more $\frac{1}{20}$ so for sure $\frac{8}{20}$ is more than $\frac{7}{21}$.”

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.

# Inductive reasoning

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.

# Three plus four is seven

While driving on the highway, my wife had the following exchange with my three year old son, Timon.

“Mom, what’s three plus three?”

“Six.”

“Mom, what’s four plus four?”

“Eight.”

“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.

# Deciphering Roman Numerals

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.

# Conservation of Cream Cheese Theorem

“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.”

“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.

# We have four people in our family

“We have two people.”

Timon, who is 3 years old, holds up 2 fingers on the same hand.

“What if Mommy comes back? Then we have three people. See? Three people.”

Timon holds up 3 fingers on the same hand and then says, “And if Thanasis comes too? Then we have four people. Four people.”

Timon holds up four fingers on one hand for emphasis. Then he giggles and holds up two fingers on one hand and two fingers on the other hand and says “See four people?”

I ask, “And what if Grandpa comes? How many people then?”

Timon says quickly, without any obvious counting with his fingers, “Five!” and then holds up all five fingers on one hand. Here, I notice him counting on rather than counting from one.

“What if Grandma comes too?”, I ask.

Timon, without speaking runs up and shows me five fingers on one hand and one finger on his other hand and then says, “Six people.”

And then he’s done with this game and moves on.

Later I ask him again how many people are in our family and he holds up five fingers, looks at them and says, “I need to take one away. Four people” and folds his thumb up next to his hand.

This is the use of three different representations of numbers from 1 through 6, specifically the oral naming of the numbers, the use of his fingers to show the numbers, and then actual number of people being represented, and Timon is moving between the three representations fluently.

But he’s also only doing this for the first six numbers and I know that he doesn’t know the symbols for these six numbers yet. I’m also not sure yet that in every instance of these numbers appearing around him that he’s as fluent as when he is counting people. And as I recall with my older son, he was fluent with counting people before he was able to count other things.

I especially noticed this interaction because this is the first time I have seen him move between number representations greater than 4 things so effortlessly.