# 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>?

# Open-ended problem solving: Make the largest possible number

Use the numbers 2 and 3 only once (but any combination of mathematical symbols you know), make the largest possible number you can. Make the smallest possible number you can. Make a number as close to zero as you can.

Bonus, use any numbers you like, and then look for patterns between the numbers. In general, which collection of operations will produce the largest number for any two given initial numbers?

Thanks to Michael Pershan for the idea.

# Lesson planning: Homeschooling my son

My wife and I are homeschooling my son, and so I work with him on his mathematics. For the first part of the year, our experiences have been fairly informal, but I feel like he and I will benefit from more structure, and so I’m planning on writing lessons and keeping track of the resources we have used, as well as reflecting on those lessons to help me plan what’s next. My son’s number sense and understanding of number operations is fairly strong. Our work here will continue informally, but for at least the next month, I’m going to focus on geometry with him.

The Common Core cluster standard in grade 2 geometry is:

2.G.A. Reason with shapes and their attributes

And the specific standards in this cluster are:

2.G.A.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces (Sizes are compared directly or visually, not compared by measuring.). Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

http://www.illustrativemathematics.org/illustrations/1506

2.G.A.2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

http://www.illustrativemathematics.org/illustrations/827
Example MARS task: Half and Half

2.G.A.3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

http://www.illustrativemathematics.org/illustrations/826
Example MARS task: Half and Half

My guess is that my son would find all of these tasks fairly easy to do, but I plan on spending our first few days together doing these tasks, and seeing if I can surface my son’s understandings and misunderstandings related to these standards. In terms of a more open-ended project, I suspect that the creation of mathematical art with polygons may be appropriate. For example, my son and I recently created as many hexagons as we could using his pattern blocks (We made one mistake. Can you find it?).

# Open-ended problem solving: Fermi problems

A Fermi problem is an estimation problem, where the objective is to answer a question that requires a number of assumptions be made, and for which the final answer can usually be best given as a range of possible values.

Some examples of Fermi problems are:

Could the forests of North America serve as the lungs of the world?
How many phone books are delivered in NYC?
How many piano tuners are there in the United States?
How many different civilizations like ours exist in our galaxy?

An activity that students could do would be to brainstorm their own Fermi problems, and then choose one of the problems to attempt to solve in a small group.

# Open ended project: Investigate facial recognition algorithms

Here is a sample outline for what a project about facial recognition algorithms might look like.

1. Investigation phase:

What is automated facial recognition? How does it work? What does a computer do to look at two different photos of someone and determine if they are the same person? What algorithms are in use? How successful are these algorithms? Where is facial recognition used?

2. Problem posing:

What factors affect the success of automated facial recognition? How efficient are these algorithms? What are the problems with these algorithms?

3. Experiment
4. Share results