# 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.

# 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.

# 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.

# Noticing number strategies

Recently I ran a math class for a few younger students, including my son. The objective of this class were to start making connections between how students visualize numbers and early arithmetic strategies.

Here are some of the various ways students visualized a group of dots to figure out how many total dots there were.

Here are some of the strategies students used when they doubled numbers in their pattern.

I asked students to look at the visualizations for the dot patterns and the arithmetic patterns and see if they noticed anything in common between the dot visualizations and the doubling strategies. They didn’t.

So I told them to find a partner, talk to their partner about the same questions, and then told them we would discuss it again. Two minutes later, I asked students to sit back down in the group after their discussions and re-asked the same question: “What is common to the strategies you (as a group) used for the dot visualizations and the doubling strategies? What is the same about what you did?”

One student said that in both cases, the students counted in order to find the answer, which was true. In both the dot visualizations and the doubling strategies, one strategy at least some students used was counting.

Another student said that in the dot pattern and the doubling strategies, we were trying to duplicate something. Not everyone knew what he meant by duplicating, so he explained it again, and I revoiced him and used the word doubling, which everyone understood.

Finally, another student had an epiphany. “When we did the dot pattern and when we did the doubling, something in common is that we figured out a big group and left some over to figure out at the end.”

Here are the actual two strategies he linked together with his observation:

In the first one, in order to figure out 23 x 2, first we do 20 x 2 and we save the 3 to figure out later. In the second one, first we counted the perimeter and saved the inside shapes for later.

I’m excited by this observation as I had never thought of thinking of the distributive property as “saving part of the calculation” for later (although I have often seen other visualizations of the distributive properties).

# Understanding place value

Last night I had an interesting realization about my son’s understanding of place value. It is clearly incomplete.

We were continuing our made-up story about Max, the 701 hundred year old 7 year old. He was cursed at the age of 7 to never age and to never die, and now he is 701 years old and working hard with his friends to try and break the curse.

At one point Max gives an explanation of curse to the sea elf king in order to ask the king for help and Max says how many years old he is. I left the actual number of years Max had been alive out of the story though and so my son filled it in.

“Max has been alive for six hundred and four years,” my son said.

“Oh, how did you get that?” I asked.

“Six hundred and three plus seven is seven hundred, so six hundred and four plus seven is seven hundred and one,” my son replied.

He spoke so confidently and assuredly that I did not correct him. Also, I wasn’t totally clear at that time exactly at that time what he was thinking, and it was late.

I think that he was regrouping the ten and confusing a regrouped ten as a one hundred. He did the same thing when he first tried counting on from 100 at age four. I remember quite clearly him counting 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200. I also remember how stubbornly he continued with this understanding.

I know of some strategies that I can use to continue building his understanding of place value. I am not sharing this story to look for help or a diagnosis but just to point out that a child who the previous night confidently and accurately multiplied 400 by 21 in his head still has unresolved issues with place value.

I think my take-away is that I need to be cautious about what assumptions of knowledge I make about my son (and of course the same is true of my students).

# “It’s 8400”

I had a fascinating exchange with my son last night. I was telling him a made up story, and at one point this character who is cursed and very old said that his age was “Four hundred times a score and one.” My son asked what a score was, and I told him twenty.

He then tried to figure out the person’s age. He started by asking what is four times twenty, and I told him he could figure that out. He counted up by fours to get eighty, and then said that four hundred times twenty is eight hundred. He thought for a moment and said that no, it must be eight thousand, which meant the final answer is eight thousand four hundred.

This makes me curious about how he understands the number twenty. He knows apparently that four hundred times something is one hundred times whatever four times the something is, although I am not clear he would explain it like that. However, he apparently did not use the fact that twenty times something is the same as ten times two times the something.

Thoughts?

My son and I were walking back from the park, and he was trying to convince me that I had time to play Clash of Clans, and that I really should start playing it.

He said, “Daddy, if you played one third of one quarter of your time, that would not be very much. See look, if you had one minute, one quarter would be 15 seconds, and one third of that would be five seconds. And if you only had 1/3 of that time, that would be five seconds divided by three. Daddy, what is five divided by three?”

I asked him, “What do you think it is?”

He replied, “One and one third, I think.”

I asked him, “So let’s check. What would one and one third times three be?”

He said, “Let’s see. One times three is easy. That’s three. One third times three is one. Oh, that would be four total, not five. Maybe it’s one and one half?”

I asked him, “We can check that. What would one and one half times three be?”

He said, “Hrmm. Four and a half. That’s not right either. It has to be bigger. What about one and three-quarters? If I multiply that by three, that’s six and one quarter. That’s too big. So it has to be between one and one-half and one and three quarters. Is two-thirds between one half and three quarters? Let me check. Two thirds times three is the same as two thirds plus two thirds plus two thirds. That’s one and one third plus two thirds is two. So it works! Five divided by three is one and two thirds!”

I then pushed my luck and asked, “What’s half of one third?”

My son said, “One quarter! See,” while showing me his hands, “here’s one quarter and here’s half of one third. They are the same size.”

A woman passed by at this point and overheard our conversation and my question to my son. She stopped to listen.

“Okay, so how many halves of one third fit into a whole?” I responded to my son. The woman at this point started to speak, but I said to my son, “And remember the point of this is for you to figure this out yourself.” The woman looked at me and nodded and then stopped speaking.

After some thinking my son said, “Six, because three thirds is a whole. Oh, but four one-fourths fit into a whole, so they aren’t the same size.” My son then said, “One half of one third is a sixth, right? Because it fits six times into a whole.”

I push my luck again, “What about a half of a sixth?”

My son said, “One twelfth, then one twenty-fourth, then one forty-eighth. Oh those are the same numbers from Threes! Except instead of getting larger, they keep getting smaller.”

We stopped our conversation about fractions at this point as my son started talking about how in the games of Threes, the numbers aren’t actually multiples of three and the game is misnamed.

This is the first time I have heard my son talk about fractions other than thirds and halves. I saw my role here  was to find the right questions to ask at the moment that my son has an intellectual need for other fractions. These kinds of conversations are common-place between me and my son, but still every time I am amazed by how much thinking about this he must be doing when I’m not around.

# I eat five pancake!

This morning we had mini-pancakes for breakfast. I gave my youngest son (who is two) three mini-pancakes. When he was still hungry, I gave him two more.

Half-way through his final pancake, he said:

There is interesting research that suggests, through an ingenious experiment, that we are literally born with some knowledge of numbers. Not symbols, like the word “five” my son is using above, but numbers. It seems to me that while my son is developing his language, it is an excellent time to develop his innate understanding of number and connect it to the language he is learning.

So I pay attention to what my son says and even at two years old, when he is first developing language, we count together, we group objects when we play, and I help him give language to the thoughts he is already having.

# What’s 192 plus 192?

My son, “Daddy, what’s 192 plus 192?”

Me, “What’s 200 plus 200?”

My son, “400.”

Me, “What’s 192 plus 192 then?”

My son, “I don’t know.”

Me, “Try breaking it down like this. 100 plus 100 plus 90 plus 90 plus 2 plus 2. Does that help?”

My son, “Hrmm. Okay, it’s 284. No, I mean 384. Do you want to know why I want to know?”

Me, “Absolutely. Why do you want to know what 192 plus 192 is?”

My son, “I started with 1 plus 1, and now I’m at 8192, and I want to know what 8192 plus 8192 is.”

Me, “Do you mean you added 1 plus 1 to get 2, and then 2 plus 2 to get 4, and so on?”

My son, “Yes.”

What makes this exchange especially interesting to me is that I remember attempting to do exactly the same calculations my son is doing, at about the same age.

# Problem solving with exponents

This post by Michael Pershan has an excellent way to give students opportunities to play around with exponents and problem solve with them.

The basic idea is, give students a bunch of dots in an array, and ask them to find ways to change the picture so that it represents a power.

A related, and more generic, question is:

How do I change this diagram/image/equation/formula so that it represents <something else>?