# 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.

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.

# Open ended investigation: Mathematical sequences as musical scores

Imagine you numbered each note of a scale, and then played the mathematical sequences on the notes like they were music. What would 1, 2, 3, 4, 5, 6, 7, 8, 9,… sound like? What would it sound like if you automatically jumped back down an octave every time you passed a multiple of 7? You may find this tool useful for actually listening to the sequence of numbered notes you generate.

What would the sequence of square numbers sound like? What about prime numbers? What if you kept the tone of the notes the same, but varied the length of the notes? How long would the sequence of notes that started with a half-note, but then halved the length of each subsequent note, take to play?

What would π sound like?

# Ten-frames to explore decimals

Chris Hunter writes on his blog about a student explaining how they would express 0.500 using ten-frames:

One student expressed this as 500/1000 and 0.500. I assumed he was just extending the pattern(s). “Yeahbut where do you see the 500 and 1000?” I asked challenged. “I imagine that inside every one of these *points to a dot* there is one of these *holds up a full ten-frame*,” he explained. As his teacher and I listened to his ideas, our jaws hit the floor. Showing 500/1000 or 0.500 using ten-frames

Have you seen examples where students come up with innovative ways of representing numbers?

# Open ended investigation: Exploring fractions in different bases

This project is related to the repeating decimals project I’ve outlined before (see this post) but with a twist.

What do fractions look like in bases other than 10 (for example: base 7)?

Instead of decimals, what would septimals look like? When we write 10.356, what we really mean is $10 + \frac{3}{10} + \frac{5}{100} + \frac{6}{1000}$. In septimal notation, $7 + \frac{4}{7}$  would be written 10.4. Can you extend this example?

Here’s a trickier question: how is 0.5 in decimal notation represented in septimal notation?

Is there a relationship between the number of digits it takes a fraction to repeat when it is written in expanded (in base ten, we call these decimals) form and the base in which it is being represented? Is it possible to write numbers in any rational base, for example 1.5? What about base $\pi$?

# Decomposing fractions

“Daddy, I’m full. I had 1 and a half…no, one and a quarter slices of pizza which is the same as five quarters of pizza,” said my son at dinner tonight.

“Okay,” I said, “you can go play if you like.”

“You know you can do that right, have five quarters of pizza, or even 10 quarters of pizza,” he continued.

“How many slices of pizza is 10 quarters of pizza?” I asked.

“Hrmmm…,” My son thought for about 10 seconds, and then said, “Two and a half slices of pizza,” and then he went off to play.

My son can do this kind of manipulation with wholes, halves, and quarters of pizza because the idea of dividing food into smaller fractions is familiar to him, and because he has seen halves and quarters used in many different contexts quite a bit during his life. Simple fractions are familiar to him, and so he can manipulate them as needed. Note that I have never once taught my son how to convert between an improper fraction to a mixed fraction, or even what those words mean. I’ve just slowly and deliberately introduced him to the ideas of fractions as they naturally fit our daily lives.

# Investigating repeating decimals

A number of years ago, I had a 9th grade math class where I decided we would investigate when the decimal representation of a fraction repeats, and when it terminates. We also decided to investigate to see if there was a pattern in the number of repeating digits when the decimal representation of a fraction does repeat. It took us about two weeks to conclude that we weren’t going to find a pattern without a lot more work, and so we abandoned the investigation, but I still remember the process quite keenly.

If you decide to try this investigation with your own students, you may find this arbitrary precision division calculator (we quickly ran into the limits of our computer’s calculator when doing this investigation) useful: http://davidwees.com/divider/