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.”

 

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Which is greater?

whichisbigger

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.

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

CreamCheeseOnCracker

“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.

Thinking about rates and times

As we were traveling back from track practice tonight, my son and I had a fairly natural conversation about driving on the highway.

Son: “How fast are we going right now, Daddy?”

Me: “According to my speedometer, we are going 2 miles over the speed limit. Oops!”

Son: “So if we were traveling for a whole hour, we would only go 2 miles??” Here my son is double-checking his reasoning with me and thinks I might be a bit crazy to suggest that we are going only 2 miles an hour. He has a strong sense that the speed he can observe outside the car does not match the calculation he has done.

Me: “No, I mean we are traveling 52 miles an hour and the speed limit is 50 miles an hour, so we are going 2 miles an hour too fast right now.”

Son: “Oh. Okay.”

Me: “I have a question for you. If we were traveling at 60 miles an hour, how far would we get in ten minutes?” Now, if we were not in a car and I was not trying to strike up a conversation with my son, I probably would not have asked this question. In the context of driving a car down a highway, I think this makes sense as a conversation starter – at least to a parent who is a mathematician and a mathematics teacher.

Son: “10.” (nearly instantly)

Me: “How do you know that?”

Son: “Well, there are 60 minutes in an hour and we are traveling at 60 miles an hour, so ten minutes must be ten miles. I don’t know how to explain this, Daddy!” Actually, you did explain it just fine son, but somehow you feel your explanation is inadequate.

Me: “I think I understand. The miles and the minutes have to be the same.” My son did not articulate his thinking in this way. I don’t know if this was helpful or not, but I felt that he wanted me to revoice what he had said using different language.

Son: “Yeah.”

Me: “I have a harder question.”

Son: “Okay, but I want it to be just slightly harder not a lot harder, okay?”

Me: “Okay. If we were traveling at 50 miles an hour, how far would we travel in ten minutes?” This question comes straight from a video I watched of Magdalene Lampert teaching during a workshop she ran over the summer. I was curious, how might my son approach this problem?

Son: “Hrmm. Maybe 9 miles an hour?” I love that my son’s first instinct is to estimate.

Me: “How do you know that?” I always ask this question, so this is normal for my son, rather than being like a quiz. I’m not quizzing him to find out how he understands, I am genuinely curious.

Son: “Well, it has to be less than 10 miles an hour. 50 is less than 60 and 9 is less than 10.”

Me: “How could we double-check that?” This is the teacher in me. I’m never satisfied with the first explanation.

Son: “We need to find nine times six. Hrmm. Nine plus nine plus nine plus nine plus nine plus nine.” My son was probably counting these nines in his head but he would have had to keep track of how many nines he was counting a couple of years ago.

Me: “What is that?”

Son: “Nine plus nine. Eighteen. And nine. Uh… I don’t know.”

Me: “Can you work it out? Also, I recommend paying attention to the answer.” This comes from MP7 in the Common Core and George Polya’s heuristics for problem solving — pay attention to the form and structure of your answer to problems as it can make other problems easier.

Son: “Okay. 27. Oooooohhh. Plus 9. 36. Plus 9. 45. I see the pattern. Plus 9. 54. Plus 9. Oh I can stop.” Meanwhile, while my son added on the nines, for each addition I pointed out the associated multiplication fact.

Me: “So does nine work?”

Son: “No, it’s too big. Let’s try six.”

Me: “Okay. Go ahead.”

Son: “6 and 6 is 12. 12 plus 6 is eighteen. Eighteen plus six is 22. No, 24. 24 plus 6 is 30. 30 plus 6. No, it doesn’t work. It’s too small.” I continue repeating the associated multiplication facts for each multiple. 

Me: “What’s next?”

Son: “Let’s try 7. No, that will probably be too small. Let’s try 8. 8 plus 8 is 16. 16 plus 8 is 24.” At this point, I said that 8 times 3 is 24. My son stopped adding, “Really? Hunh.”

Me: “Yes.”

Son: “24 plus 8 is 32. 32 plus 8 is 40. 40 plus 8 is 50, no 49, no 48! Too small. It must be between 8 and 9.” Do kids that understand numbers only as counting objects recognize that there are other types of numbers in between the ones we use for counting? It seems to me that kids need plenty of experience with situations that require other types of numbers to motivate an intellectual need for those types of numbers. We don’t go look for tools, even thinking tools, arbitrarily; without some clear kind of need for the tool.

Me: “Okay.”

Son: “Hrmm. Maybe it’s 8 and a half. No, that’s too big because 6 halves is 3.”

Me: “Right, and then we would have 48 and 3 which is 51. Too big.”

Son: “It can’t be 8 and a quarter either. That would be 49 and a half.”

Me: “So it must be between 8 and a quarter and 8 and a half.” This is a key idea that I am sorry I did not let my son develop himself.

Son: “What about 8 and a third? Let’s see. YES!! 6 thirds is two, so it works! I got it! Daddy, that problem was not a ‘little bit harder’ than the other problem. It was WAY harder.”

I find it hard not to turn off my teacher self when my son and I are discussing mathematics. I find myself really curious about how he will approach problems and I often find myself straddling the line between pushing him to articulate his thinking and just being a bit annoying.

On the other hand… the teacher part of me is now curious; which mathematical standards and practices did my son and I uncover in our conversation together? I remember Magdalene Lampert telling me once that when she teaches, she tries to hold all of the standards for the year in her head while working with students because she is never really working on one specific thing at once; she is trying to connect all of the ideas of the year together.

Exploring a digital block game

Screen-shot from one of the puzzles included in the block game.

Screen-shot from one of the puzzles included in the block game.

I wrote this puzzle/game last year with  the hope it could be used to help generate some thinking about area, multiplication, and addition.

Here are some questions you could use with the game.

Try solving the problems, if possible, in a variety of different ways. Which ways give you the most points? How much is each piece worth? What is a general strategy you can use to get as many points as possible? How do you know your general strategy is effective?

Note: This game does not currently work in Internet Explorer.