Here’s a real problem I recently faced. I played a game with a group of kids that required three groups. I had 56 kids. Here’s the breakdown of each group:

A: 20
B: 26
C: 10

I wanted to every participants to be in each group above, at least once. (I could do the activity in three rounds.) What’s the most efficient and least confusing way to go about this?

Sorry, I wasn’t clear; let me try again (although I’m not promising this will be clearer). I was going to play a game three times or rounds. I’m asking how I can rotate the participants into new groups so that each participants gets to play the game in A, B, and C groups. For example,

Round 1
A: 20
B: 26
C: 10

In round 2, I could move the 20 from A into group B, and 6 from group C. Now, in Round 2, group B would have an entirely new group of participants.

I could then move 20 from Group B from Round 1 to Group A (for Round 2), and 6 to Group C (for Round 2).

And then I would to shift around the participants again for Round 3. See what I’m saying?

If the maximum number of people in group C is 10, you can’t do it in three rounds because you have 56 people. Are you allowed to take as many rounds as necessary to ensure everyone gets to be in each group?

If the maximum number of people in group C is 10, you can’t do it in three rounds because you have 56 people.

You mean, you would need six rounds? I think that would be too many….I guess I was OK if not everyone got to be in group C, which I didn’t explain.

I also forgot to specify that I wanted the easiest way to break up the groups and transition to the next rounds (or maybe I conveyed that). I did this on the fly, and I drew up table, and tried show how I could move people around into different groups for subsequent rounds. It was a mess.

Thinking out loud–about the number one, fractions, and percentages

The number 1 is equal to 100% (if “equal” is the right word). Or maybe it’s more appropriate to say that the number 1 can be thought of as 100%. There’s definitely a connection, right? What is exactly is the nature of that connection? It seems like the number one can represent at least two related concepts–

A) The number 1 refers to one thing, one item. For example, if I’m counting jelly beans, the number one would refer to one specific jelly bean;

B) The number one can refer to the whole or totality of several items. (I’m not sure “items” is the best word choice, here, but I’m going to use it for now. What would be a better word?)–e.g., 1 bag of 25 jelly beans. In this case, the number 1 refers to one specific item (or quantity)–namely the bag of jelly beans–but it also involves more than one item or quantity at the same time (i.e., 25 jelly beans). Mathematically, this sense of oneness (?) can be best represented by the fraction 25/25. What can be hard to grasp (especially for kids) is the way 25/25 is both one and many at the same time.

(Aside: Is there another way of thinking about the number 1? Is there another variation or definition of the number 1?)

The B definition of the number 1 relates (obviously) to fractions and numbers of decimals. (What’s the technical name for those type of numbers? “Mixed numbers?”…Tangent: a mixed number combines the concepts of A and B above, right? Maybe this could be seen in the following example: 7.5 bags of jelly beans, where 1 bag contains 25 jelly beans. When “1” refers to the bag (of jelly beans), the number 1 is functioning in the mode B….Or is that wrong? Whereas each jelly bean can be thought operates in mode A? That doesn’t sound right–not completely. The jelly beans can be thought of as operating in mode A and B, right? Or not? (My thinking is muddled on this point.)

Understanding the different senses of the number 1 seems critical to gaining a good grasp of fractions and percentages.

I’m not sure you want me to complicate things, but to answer one of your questions, 1 can also be thought of as “on” while 0 is “off.” This is strictly a binary concept (hence the name of the base 2 number system where 11011100 equals 220 in base ten).

Incidentally, it’s also why when you’re looking at electronic equipment, the ON switch is marked with a | while the OFF switch is marked with a O.

Another way to think of 1 is as a specific identifier. If you have ten apples and ask someone to give you 1 more, if someone hands you a orange, even if it’s ONE orange, he hasn’t done what you’ve requested. This may not be a mathematical concept so much as a philosophical one (I honestly don’t know), but I suspect it’s an important idea we pick up without it ever really being taught us.

Oh I just thought of one more concept of 1 that may already be covered by some of these things we’ve listed, but 1 also means “not none.” In logic, the opposite of “none” is “one,” because the presence of 1 of anything negates the presence of “none.” Does this make sense?

It’s one reason “any” is grammatically treated as singular in American English. “Does any of my friends have ten bucks I can borrow?” is supposed to be the default but that’s been fluid for some time and it’s changing. So much that when I wrote a sentence beginning with “Does any of my friends…” I was corrected by a friend who’s an English major and a professional writer. She was wrong and I corrected her. 🙂

but to answer one of your questions, 1 can also be thought of as “on” while 0 is “off.” This is strictly a binary concept…

Is the binary concept a mathematical concept? Or are people just using “1” an “0” in an artificial way (if that’s the right way to put it)? For example, could you use something something other than “1” and “0”–e.g., “A” and “B.” Also, what exactly does “on” and “off” mean?

(hence the name of the base 2 number system where 11011100 equals 220 in base ten).

How you get from the first number to the second?

Another way to think of 1 is as a specific identifier. If you have ten apples and ask someone to give you 1 more, if someone hands you a orange, even if it’s ONE orange, he hasn’t done what you’ve requested. This may not be a mathematical concept so much as a philosophical one (I honestly don’t know), but I suspect it’s an important idea we pick up without it ever really being taught us.

Hmm, I think is a matter of semantics, not so much philosophy. In the example, “1 more” implies 1 more apple. The person being asked to get 1 more should know the requestor means 1 more apple. Suppose the person being asked doesn’t see any apples at all. When the person asks for 1 more, the person will likely respond, “1 more what?”

Oh I just thought of one more concept of 1 that may already be covered by some of these things we’ve listed, but 1 also means “not none.” In logic, the opposite of “none” is “one,” because the presence of 1 of anything negates the presence of “none.” Does this make sense?

I think so. Using 1 to mean “not none” makes sense. You could use other number greater than 1, but that would be confusing. But does this concept occur in mathematical contexts? The equating of 0 to none and 1 to not none seems more of a designation to be used in logic, but it’s not really mathematical, if you know what I mean. Or is that wrong?

It’s one reason “any” is grammatically treated as singular in American English.

Really? You’re saying “any” is treated as singular because of the number 1 in logic refers to “not none?”

Is the binary concept a mathematical concept? Or are people just using “1” an “0” in an artificial way (if that’s the right way to put it)? For example, could you use something something other than “1” and “0”–e.g., “A” and “B.” Also, what exactly does “on” and “off” mean?

Binary literally means a number system based only on the numerals 0 and 1, so I’m not sure how to answer the question. The etymology indicates that it originally means “something made of two things or parts” going back to the mid-1500s, but the base-2 mathematical system surely was conceived of long before that, perhaps by thousands of years.

To answer the rest of your questions, I guess I first need to answer

How you get from the first number to the second?

It helps to know that you have a firm grip on the base-10 system, because that’s critical here.
124 means 1 x 100 plus 2 x 10 plus 4 x 1. Each digit in its assigned place means [digit] times [value of place]. The values in each place only range between 0 and 9 because once you go to 10 times the value, the counter in that place resets to 0 and you add 1 to (or “carry”) the value to the next column.

In base 2 (I’ll say binary from here on), the values can only range from 0 to 1. When the value of a place goes to 2, you have to “carry” the value to the next column and reset the counter to 0.

Quick illustration to show the place values.

In base 10:
Ten thousands —- thousands —- hundreds —- tens —- ones

So in binary, 10111 means
(1 x 16) plus (0 x 8) plus (1 x 4) plus (1 x 2) plus (1 x 1) = 23.

In the example I give in my response above, I said

11011100 equals 220

(1 x 128) + (1 x 64) + (0 x 32) + (1 x 16) + (1 x 8) + (1 x 4) + (0 x 2) + (0 x 1) = 220

Going back to your original question about whether people are using “binary” in an artificial (I think maybe you also mean metaphorical) way, I would lean toward no, because in a binary concept, each value place either contains a value (1) or no value (0). Sure you could use A or B, but you’d be slightly changing the concept to something like “thing A” or “thing B” (or, to use a current use of “binary,” “male” or “female”).

In my response above, I was thinking more like “true” or “false,” or “thing A” or “not thing A.” I think of 1 to mean “true,” “thing A,” and “yes,” while 0 means “false,” “not thing A,” and “no.”

Also, what exactly does “on” and “off” mean?

This goes to the way computers operate. The language computers know is binary. Everything we do on a computer and everything it does for us goes (ultimately) down to values. I’m oversimplifying but mostly out of necessity because I don’t understand it more deeply. I think Grace does, though, so you could ask her. Essentially, computers store data by placing a value in a spot or not placing a value in a spot — a 1 or a 0.

In a one-dimensional imagining of this idea, it looks at 11011100 as a series of values and non-values: one 128, one 64, zero 32s, and so on. Since each column either contains a value (1) or doesn’t (0), each place is either on or off.

There may be an actual, physical toggle for 1 and 0: on and off. I don’t know ZIP about how data is actually stored and manipulated electronically, but it would make sense if there’s a physical turning ON and turning OFF when there is and isn’t a value in a designated spot.

You didn’t ask but if you wanted to take this concept and make it two-dimensional, you can get an idea of how computers take this basic concept and turn it into things we actually recognize.

When we were in 9th and 10th grade, I learned that you can reprogram the letters on a computer to look however you wanted, so if you typed “A” the monitor would display something else, while still understanding the value as “A.” Nowadays we take it for granted: we can switch to an endless number of fonts, and out computers understand the actual meaning of the letter no matter what it looks like. But remember how the computers in high school only had one font? You couldn’t change fonts back then unless you could teach the computer to do it.

I told my computer at home that when I typed the A key, I didn’t want it to type the A it had as a default. I told the computer to use this data instead:

60, 66, 66, 66, 126, 66, 66, 0

And this is how the computer read it:

You can look at that first row in the graphic visually, as humans do, or you could look at it as

off off on on on on off off

or

00111100

or (0 x 128) + (0 x 64) + (1 x 32) + (1 x 16) + (1 x 8) + (1 x 4) + (0 x 2) + (0 x 1) = 60.

That would just be the top line in the graphic above, but I gave the computer 8 values, which it interpreted as eight rows. Now when I hit the A key on my keyboard, it displayed THIS letter A, not the letter A it came with.

I bothered to explain all this so you could see that the “on-off” idea of 1 and 0 is like this super fundamental concept on which some huge ideas rest. But yeah: it’s still kind of a manufactured concept. We’ve put human-centered thinking into the way we think of 1 and 0; it’s not necessarily a universal truth. At least I don’t think it is!

Binary literally means a number system based only on the numerals 0 and 1, so I’m not sure how to answer the question.

The question I asked wasn’t very clear–especially the notion of a “mathematical concept.” I really don’t know how to articulate what I mean, and that phrasing was the first thing that came to mind. I think I mean something closer to what people would use when doing “normal” math–versus math this is more theoretical, speculative or imaginative. By the latter, I have this notion that one could take mathematical principles and construct all sorts of odd processes–ones that wouldn’t be used or even thought of as (normal) mathematics. This explanation is still not clear, but I hope it helps a little.

124 means 1 x 100 plus 2 x 10 plus 4 x 1. Each digit in its assigned place means [digit] times [value of place]. The values in each place only range between 0 and 9 because once you go to 10 times the value, the counter in that place resets to 0 and you add 1 to (or “carry”) the value to the next column.

Wait, when you say, “The values in each place only range between 0 and 9,” don’t you mean the value of each digit (in its assigned place)? “Place” refers to 1s, 10s, 100s, etc., right? The value of each digit in a specific place can only be 0 to 9. If so, I understand.

So in binary, 10111 means
(1 x 16) plus (0 x 8) plus (1 x 4) plus (1 x 2) plus (1 x 1) = 23.

OK, I think I understand. My reaction to this is, why would anyone devise a system like this? It seems very cumbersome and difficult. What context, besides computing, would be useful for the binary approach?

In my response above, I was thinking more like “true” or “false,” or “thing A” or “not thing A.” I think of 1 to mean “true,” “thing A,” and “yes,” while 0 means “false,” “not thing A,” and “no.”

OK, this explanation (and the paragraph before it) was helpful. I think I understand how “1” means yes or on and “0” means no or off. And I think it makes more sense in the context of computers. It’s especially clear in relation to creating images on a computer screen. (I’m less clear how it would apply in other computer contexts.) To create an image, a computer utilizes a grid, and images can be mad by coloring in squares or leaving them uncolored (i.e., blank). But how do you “tell” the computer which squares to color in or leave blank? The binary system provides a language that computers can “understand,” to do, right? Essentially, “1” and “0” mean color or leave blank, respectively, and the place value gives a specific location of the box. Is that right?

If it is, I guess this is what I mean by “artificial.” People used the binary system in computers as a language, because they couldn’t use human language like English to give commands…Wait, that can’t be write, because we do use some language when we do programming.

Using 1 to mean “not none” makes sense. You could use other number greater than 1, but that would be confusing. But does this concept occur in mathematical contexts? The equating of 0 to none and 1 to not none seems more of a designation to be used in logic, but it’s not really mathematical, if you know what I mean. Or is that wrong?

It depends on whether or not you think of logic as math. It was my favorite Math League event and it’s often taught as a math course. At UH, it’s taught as a philosophy course.

Really? You’re saying “any” is treated as singular because of the number 1 in logic refers to “not none?”

No, I’m saying “any” is singular for the same reason 1 means “not none.” When you’re trying to prove a statement like “Nobody loves me,” all you’re looking for is one person who loves “me.” If “anybody” loves “me,” the assertion is false.

Here’s a word problem from my son’s homework I couldn’t solve:

Mary has a gift certificate. She spends 1/3 on lunch. She spends 1/4 of the balance on dessert. After paying $3.50 for a tip, she has $4 leftover.

I called Mitchell to help me, and he came up with this:

x-(x-1/3x)-1/4 (x-1/3x)-3.5=4

(I liked how he explained this to me.) But given the previous questions, which had to do with multiplying and dividing fractions and mixed numbers, I was pretty sure this wasn’t the way to solve the question. After reading some of the previous questions, Mitchell agreed. In the chapter, there was a section on using grids to solve problems and I suspected the solution would involve using them.

I emailed Pat Ota, and she came up with the solution:

Draw a box representing the gift certificate, divide it into 3 = strips. Label 1 strip lunch. Take the other two strips and draw a line horizontally, so now you have 4 smaller boxes. Label one of them dessert. Add the tip and leftover to get $7.50. This amount can be divided into 3 to be placed in the remaining 3 boxes. So each small box is $2.50, the dessert is $2.50, and the big lunch box is $5.
Adding the box values up results in a certificate value of $15.

OK, I won’t tell you I didn’t send that solution. Hahaha. Seriously, I think I did, I’m not sure. I just went by memory (on both times). I prefaced the equation by saying, “The equation was something like…”

Write an algebraic expression the nth term in the table below:

A: 0 1 2 3 4 5 n
B: 3 5 7 9 11 13 ?

According to Zane, the teacher said the answer is 2n+3. If this is correct, I don’t get it. Can you guys explain this? (There’s a decent chance Zane didn’t write the right equation.)

First you have to figure out the rule. 0 in, 3 out can be an infinite number of rules. 1 in, 5 out restricts it a lot. But 2 in, 9 out pretty much seals it. There are a lot of ways to come up with the formula, but at your son’s level, probably the fastest way is trial and error.

Well the first step is usually to see if the formula is simple addition. You look at 0 and then 3, and say, “Oh, maybe it’s add 3.”

But then you look at 1 and see 5. 1 plus 3 isn’t 5, so the rule cannot be add 3. When we see 2 and 7, we can see simple addition is way off, and the difference between the A and B terms gets bigger with each item in the series.

So let’s try multiplication. Normally you might have to go through a few items to try a simple multiplication rule. But in this case, the first term is 0, and nothing you multiply by 0 is ever ever ever going to give you 3.

So can we multiply the A number then add something to get the B number?

0 times ANY number is 0. Then you need to add 3 to get the B term, 3. Maybe the rule is multiply by something and then add 3?

Let’s look at the next A term: 1.

What can we multiply the next A term by? Multiplying by 0 doesn’t help (but always start with 0 because it can make your life easier). Multiplying by 1 doesn’t help. How about 2?

1 x 2 + 3 = 5. Let’s try it with the other terms. Does it work?

Tada.

Again, there are algebraic ways of figuring it out, but that’s usually algebra 2 stuff. I’m guessing that’s not what your son’s teacher wants.

Don’t take this the wrong way, but I would expect this kind of problem to appeal to you much more than more mathy excercises. It’s more a thinking problem than a math problem, and there’s nothing very abstract here except the “in terms of N” part.

Find each unit price and then determine the better buy.

1 lbs. for $1.29
12 oz. for $.95

I’m embarrassed to say that I have a hard time knowing whether to divide by 1.29 by 16 oz or 16 oz by 1.29. I mean, I can eventually figure it out by comparing the answers, but I’m having trouble knowing the right way prior to doing that. I also don’t know how to explain why 1.29 would be divided by 16 versus dividing 16 by 1.29….

…Well, if I divided 16 oz. by 1.29 that would give me ounces/cent–isn’t that what I want to know? But don’t I have to divide 1.29 by 16? And wouldn’t that give me the cents/ounce–i.e,. 1 cent would get me X ounces. But don’t I want to know the cost of 1 ounce?

If 100 people are in a room and 25 of them are wearing red shirts, 25 for each 100 are wearing red shirts. Or 12.5 for each 50. Or 10 for each 40. Or 1 for each 4. Ta-da.

I think I might be getting thrown off because “for each” refers to multiple situations. If I said 25 people out of a room of 100 people are wearing red shirts, that refers to that one specific room. For each 100 means that for every 100, I will have 25 people wearing red shirts.

The key here, I think, is that we’re talking about rates, not just a fraction. Rates are a fraction, but not all fractions are rates. I guess that’s what I’m thinking.

…and don’t be embarrassed. I have to speak words aloud almost every time I do this problem in the supermarket.

What I’m having trouble with–the language–(quantity) for (another quantity) and seeing how that gets to the mathematical equation. Why does 1 lbs for $1.29 mean 1.29 divided by 1 lbs.? Getting from the former to the latter is the part I’m struggling with–specifically, how to think about and articulate what’s going on.

And I always thought “/” was another form of the line between the numerator and denominator–basically a sign of division. (It essentially means this, right?)

What’s the explanation for dividing a numerator by the denominator , when converting a fraction to a decimal or percentage?

I’m going to try to answer this, thinking out loud. I’ll start with definitions of denominator and numerator. The denominator is the number of equal(?) pieces that make up a whole (1?)*. For example, the 4 in 1/4 means that a whole is cut up into four equal pieces. The numerator is the number of pieces in relation to denominator. If we divide a whole into four pieces, 1/4 means 1 piece out of the 4 pieces.

So why do we divide the numerator by the denominator to convert the fraction into a decimal or a percentage?

Here’s the first thing that comes to mind. The denominator basically represents the whole–i.e., 1 or 100% in relation to the numerator….But why divide, not multiply, the numerator by the denominator?

To be continued…

(*Could you create a fraction if the pieces are not equal? Off the top of my head, I would say the answer is no. Also, could a numerator or denominator be less than 1–i.e., a fraction or decimal? What would that mean?)

You absolutely can have a numberator that’s a decimal or a fraction. It’s just not considered “simplified” in that form. You can say you own 2/1 cars, but you wouldn’t because it’s not the simplest form of the answer.

Simplifying is important for a few reasons but one reason is to make sure you’re not inadvertently dividing by zerio.

Simplifying kind of standardizes things a bit: we might not know what (1/4) / (1/8) even means by looking at it, but (1/4) / (1/8) is 2, and we know what that is.

Hmm, “standardizing” isn’t a word that came to mind, but it’s something worth considering. However, your example suggests that simplifying means putting a number in the form that is most easy to understand or grasp. I/4 is easier to grasp than 13/52. Is that what simplifying comes down to?

…I guess the standardized” part refers to what most people would consider the form that is easiest to grasp?

Re: the division by zero thing. I didn’t really understand your explanation.

I don’t want to assume anything, but you do understand that you can’t divide anything by zero, right? One important thing simplifying accomplishes is making sure there isn’t a zero in a denominator somewhere — you might not be able to see it in a non-simplified numerical expression.

If you see 1/0 it’s obvious.

If you see 1/(10-7-3) it’s less obvious.

If you see 1/(tan(45)-1) it’s even less obvious. Simplifying this expression, you get 1/0.

Why wouldn’t a fraction with a decimal in the numerator not be the most simplified form? Or to ask it another way, what does simplifying really mean? I always thought it simply referred to reducing numerator to the smallest number possible, if that’s even the right way to put it.

How should one think if the numerator (or denominator?) is a decimal? What does it mean when this happens?

Simplifying is important for a few reasons but one reason is to make sure you’re not inadvertently dividing by zero.

Basically a decimal in a fraction is a fraction in a fraction, so it’s not considered simplified. Simplifying kind of standardizes things a bit: we might not know what (1/4) / (1/8) even means by looking at it, but (1/4) / (1/8) is 2, and we know what that is.

For the division by zero thing, I can’t think of a specific example but something like (1/4) / (x/8) would be 1/4 times 8/x, and if that’s an answer we have to clarify with something like “where x is not equal to 0.” This isn’t the best example but in very complicated fractions you just have to keep an eye out.

edit: reducing a fraction to its lowest terms is one part of simplifying, but it’s not all there is to it. If you ask someone how much you owe him, and he says, “Twenty dollars minus six percent,” that’s not very simple. Similarly, you aren’t supposed to leave radical signs in denominators (although the rationale for this isn’t as convincing as it once was), or a negative sign in both the numerator and denominator.

So why do we divide the numerator by the denominator to convert the fraction into a decimal or a percentage?

Here’s the first thing that comes to mind. The denominator basically represents the whole–i.e., 1 or 100% in relation to the numerator….But why divide, not multiply, the numerator by the denominator?

I’m still trying to find a way to understand this.

For 1/4, why do I divide 1 by 4? I know that a fraction basically means a fragment of the 1 (whole). If I take a deck of cards (which is 1 whole thing), one card is a fragment or fraction of the whole deck. So I know that when I see a fraction like 1/4 (i.e., where the numerator is less than the denominator) if I were to divide this, the answer has to be a decimal. But that doesn’t explain why the numerator is divided by the denominator….

What is happening when the numerator is being divided by the denominator?

How should one think if the numerator (or denominator?) is a decimal? What does it mean when this happens?

At the risk of confusing you, let me take one step back and look at what multiplication of fractions means.

1/4 times 1/2 means “one fourth of one half.”

I give you half a pie. You eat three fourths of that one half and leave only one fourth for the rest of your family. What fraction of the original pie did your family get?

One fourth of one half is what’s left over. Multipling 1/4 by 1/2 you get 1/8. I ate half the pie. You ate three eighths. Your family was left with one eighth.

So one way to think of multiplying fractions is increasing the fractiousness of a thing.

If you see (for example) (1/4) / 2 you have a fourth of a half (don’t forget, that 2 in the denominator means half). So this is one way to think of it when you see a decimal or a fration in a numerator or a denominator. (.2)/2 is .2 of a half, or 1/5 of a half, or 1/10.

I’m bringing this down here because it’s getting confusing up there.

What is happening when you divide a numerator by a denominator AND why dividing by 0 and dividing by 1 are not sorta the same thing. In one short answer!

Division by zero

“10 divided by 2” sorta means “ten things divided into two groups.” If you have ten bananas and divide them into two equal groups, there are five bananas in each group. 10 / 2 = 5.

“10 divided by 1” sorta means “ten things divided into one group.” If you have ten bowling balls and divide them into one group, there are ten bowling balls in that one group. 10 / 1 = 10.

“10 divided by 0” sorta means “ten things divided into no groups at all.” If you have ten jawbreakers, if you try to divide them into zero groups, what happens to the ten jawbreakers? They would essentially have to be zapped into nonexistence. You cannot take 10 things and put them in zero groups. And for some reason, you can neither do it in real-world terms such as these jawbreakers or in abstract terms. I don’t know why you can’t even do it abstractedly when there are such things as negative numbers.

You asked, “If ended up with 0 in the denominator, that essentially means you made a mistake somewhere, right?” Probably. But not necessarily. Sometimes you get 1/0 and it just gives you an idea of how an equation’s pieces relate to each other. Go here (https://www.desmos.com/calculator) and type “1/x” in the equation bar. You see how the line never touches the X or Y axis? It’s because values for X don’t exist (1/0) in this equation. And it’s because nothing you put in for X will ever give you 0 for Y. Summary: sometimes if you end up with division by zero, it tells you important information about the way an equation behaves.

What we’re doing when we divide a numerator by a denominator

Keeping in mind the stuff we just went through with bananas, bowling balls, and jawbreakers:

“1 divided by 4” means “one thing divided into four equal groups.” If I have one apple and divide it into four equal groups, each group has a fourth of an apple. 1 divided by 4 = 1/4, or .25 or 25%.

“2 divided by 4” means “two things divided into four equal groups.” Starting with two apples, splitting them into four equal groups, I have half an apple in each group. 2 divided by 4 = 2/4 or .5 or 50%.

So to answer your question: the reason you divide the numerator by the denominator is because it’s literally what 1/4 means. Yes, 1/4 means “one fourth” but “one fourth” literally means one thing divided by four.

Going to skip the other comments for now and get to the most important issue for me:

“1 divided by 4” means “one thing divided into four equal groups.” If I have one apple and divide it into four equal groups, each group has a fourth of an apple. 1 divided by 4 = 1/4, or .25 or 25%.

Here’s where I get lost. If I were to use a concrete example of 1/4, I would take a pie, for example, and cut it into 4 equal pieces. I would take one piece and say that piece represents 1/4 (of the pie). Using your example would be taking that one piece and dividing it into four equal pieces. I think this is what’s confusing me. Do you see what I mean?

I don’t think it matters which real-world object or substance you use in your example. Apples, pies, or thingies. 1/4 represents taking the thing (or group of things) and splitting it evenly four ways. The 1 represents one of those four groups, whether it’s 5 out of 20 apples, 1 quarter of a pie, half of pie out of two pies, or 3 quarters of a pie out of 3 pies.

So is this what’s happening when we say 1/4: We take one pie, cut it into four equal pieces. One of those pieces represents 1/4. But doesn’t the numerator (1) represent 1 piece of the pie (cut into 4 equal pieces)–not the whole (1) pie. I’m confused, if that’s not already obvious.

Yes, the 1 represents the one piece of the pie. You could think of the 4 as representing the whole pie if you like. But only if that doesn’t confuse you more.

But if the “1” in 1/4 represents 1 pieces of a pie that is cut into 4 equal pieces, then wouldn’t we then be dividing that one piece into 4 equal pieces? That can’t be right, but that’s what it seems like is happening.

the 1 in 1/4 doesn’t exactly represent one quarter of a pie. It represents ONE one-fourth of a pie.

ONE one fourth of the pie is the one piece out of a pie that was cut into four equal pieces, right? But 1/4 is 1 divided into 4 equal pieces. I don’t get it.

1/4 does mean one pie cut into four pieces, but it represents the VALUE of each of those pieces. HOW MUCH pie is in one piece of that one pie cut into four pieces.

1/4 doesn’t exactly represent the act of cutting a pie in four pieces, nor does it exactly represent the state of the pie, as one pie cut into four pieces. It represents the amount of pie in each piece when one pie is cut into four pieces.

So I think if you see it this way, it’s exactly the same thing in both your scenarios.

1/4 doesn’t exactly represent the act of cutting a pie in four pieces, nor does it exactly represent the state of the pie, as one pie cut into four pieces. It represents the amount of pie in each piece when one pie is cut into four pieces.

I don’t think I’m really understanding or appreciating the distinction you’re making, but let me focus on your last sentence, especially “amount of pie in each piece.” How do you get from that to dividing the numerator 1 into the denominator, 4? If the numerator–1–represents the whole (pie) that makes sense on some level. But it doesn’t if it represents a slice of the pie.

More on dividing by 1 and 0

Mitchell said,

“10 divided by 1” sorta means “ten things divided into one group.” If you have ten bowling balls and divide them into one group, there are ten bowling balls in that one group. 10 / 1 = 10.

“10 divided by 0” sorta means “ten things divided into no groups at all.” If you have ten jawbreakers, if you try to divide them into zero groups, what happens to the ten jawbreakers? They would essentially have to be zapped into nonexistence.

It may seem obvious, but I like the description of dividing as an act of dividing something in groups. So, division by 1 means dividing something into one group, and division by 0 means dividing by no groups.

Here’s why I think the dividing by 1 and 0 seem very similar. Generally, division is an action, breaking a whole into parts or groups, as you say. By when you divide by 1, you’re really not doing any action–you’re not cutting a whole into parts or breaking it into groups. The whole remains the whole. You did nothing–or you “did 0” so to speak.

I think when you use “grouping” that is helpful because if you say group all the bananas into one group, that is an action. But in this case, the word “division” is a bit misleading. You’re not really dividing anything.

But then grouping doesn’t work well when you’re using division on something like a banana pie. If I divide 1 banana pie into one group that doesn’t make as much sense, or it’s at least very awkward wording. In this case, I’m neither dividing or grouping.

Now, when we divide by 0, that seems like saying don’t do any dividing, breaking into parts, or grouping. That is, do nothing. And that seems like the same thing as dividing something by 1.

To be clear, I understand the difference between dividing by 1 and 0, and your explanation helped make this clearer–I’m not saying they’re the same thing. But I’m trying to explain why the two actions seem like the same thing. My sense is that this an instance where language is inadequate, or being able to describe and distinguish the two processes isn’t so easy.

–For some reason, It recently hit me that “percent” is really “per cent”–“per” being the Latin word for “for each” and “cent” being the Latin root for 100. So, 60% is literally 60/100. In my defense, I think writing “per cent” as one word and using the symbol “%” has obscured this meaning for me. (More on this later.)

–This got me thinking about something else: Why 100? The concept of percentage is just a way of thinking about the relationship between some part to a specific whole–e.g., how much of the total budget went to salaries. In a percentage, “100” is used to represent the whole, but we don’t have to use this number (or a percentage) to think about parts to whole relationships. For example, we could 10 to represent the whole, and speak in terms of a “perdecimage;” or we could choose a 1,000 to represent the whole and speak in terms of a “permillenia.” I’ll offer thoughts on why I think we use percentages when talking about the relationship between parts to a whole, but first I want to make the point that percentage is ultimately a way of thinking about the relationship between a part(s) to a whole. Basically, it’s just a fraction. (Right?)

–So why choose 100? My guess is that this is a “Goldilocks” number in the vast majority of cases when we’d want to understand a quantitative relationship between part(s) and a whole. 10 wouldn’t afford enough precision, while 1,000 would often afford too much. 100 is “just right.” Additionally, 100 (or a factor of 10) is easier to work with an understand. We could choose 77 to represent, but doing so would be so unwieldy.

–Now, I want to go back to the way the word “percent” and the percentage symbol have obscured the meaning of the term and concept. Suppose we chose 10 to represent the whole and we used the term “perdecemage” and also a symbol “+/+.” For example, I could say, “8 +/+ of the budget went to employee salaries.” At some point, I and others might begin to lose sight of the fact that A) perdecimage, using 10 to represent the relationship between the part and whole is useful, but ultimately arbitrary convention, and B) it’s ultimately a fraction or ratio.

…OK, I feel like there was something else I wanted to say, but I can’t remember any more.

Using 100 to represent a whole is certainly kind of arbitrary; we could use anything else, and in fact we do. In some settings.

100 works really well because of our base-10 number system, but of course our base-10 system is also an arbitrary decision. If we grew up in a society that for some reason used base-8, our concept of numbers, numerals, and percentage (which of course we’d call something else) would seem as natural to us as base-10.

One second is 1/60 of a minute. One minute is 1/60 of an hour. One hour is 1/24 of a day. One day is either 1/28, 1/29, 1/30, or 1/31 of a month. One month is 1/12 of a year. These are all parts of wholes that don’t have anything to do with hundreds.

Also arbitrary, but those seconds, minutes, and hours have to do with a circle (and thus the planet) being 360 degrees. But that 360 is arbitrary too.

Another context: when you see that a pitcher pitched 7.2 innings, that’s not 7 innings and 2 tenths of an inning. That’s 7 and 2/3 innings, and that 2/3 doesn’t have anything to do with time, distance, or anything static: it has to do with the number of outs he gets. How crazy is that? 🙂

These are all parts of wholes that don’t have anything to do with hundreds.

OK, but the situation is really different in my opinion. Using a 100 is a general way to measure wholeness, when wholeness is already predetermined. 60 seconds equals 1 (whole) minute. 11 represents the whole of a football team that’s on the field. I don’t think I’m explaining this well, but do you see what I mean? 100 is a generalized standard.

But that 360 is arbitrary too.

What? I didn’t know that.

That’s 7 and 2/3 innings, and that 2/3 doesn’t have anything to do with time, distance, or anything static: it has to do with the number of outs he gets. How crazy is that?

Crazy because .2 normally represents 2/10? Otherwise, it makes sense to me, because 1 whole inning is based on 3 outs.

Ugggggh sorry. I think I complicated matters because I thought you were asking “Why out of a hundred as opposed to out of, say, 43?” But you were asking “Why out of a hundred as opposed to out of 10, which would make as much sense?”

I was answering the first question. Needlessly, and in fact in opposition to this conversation going wherever it was going.

That is hard to imagine and hard to believe.

Perhaps difficult to imagine but not at all difficult to believe. Can you think of a single natural reason human civilization should have evolved to use base-10 as opposed to base-anything-else? The only one I can think of is that we have 10 fingers.

Also not hard to imagine, really. Computer programmers are quite literate in binary, as we’ve discussed, and in hexadecimal (base-16). I’m not fluent, but I’m conversant in both.

60 seconds equals 1 (whole) minute. 11 represents the whole of a football team that’s on the field. I don’t think I’m explaining this well, but do you see what I mean? 100 is a generalized standard.

Yes, but I was answering another question, and in that alternate conversation, 11 players making up one football team is not the same thing and here’s why:

10 hours, 59 minutes, and 57 seconds.
10 hours, 59 minutes, and 58 seconds.
10 hours, 59 minutes, and 59 seconds.
11 hours.

Our time system works a lot like base-10: in counting, add one second to the 59th second, and you reset the seconds place and “carry” the whole minute to the minutes place. If in doing so, the minutes value hits 60, you reset the minutes place and “carry” the whole hour to the hours place.

Adding a 12th player to the guys on the team doesn’t then create a second team. I was talking about percentages as a whole working in our base-10 system because adding one percent to .99 works so smoothly in our decimal system, if that makes any sense.

You were really asking the question about precision: why measure things in hundredths when we can measure them in tenths, and yeah, it goes to precision. That’s one reason our standardized test scores always gave us stanines and percentiles. In some situations people didn’t need the precision of percentiles — there’s no practical difference in knowledge or ability between all the people in the 8th stanine, but in cases (such as super-selective schools) there is a difference between someone in the 89th percentile and someone in the 81st.

What? I didn’t know that.

There may be a more complicated explanation, but 360 degrees in a circle seems utterly arbitrary to me, and it was explained this way to me when we were in high school. We could easily have made a circle be 100 degrees.

Another reason I was using the seconds-minutes-hours thing as an example of how “out of a hundred” isn’t necessarily the best or accepted way to think of parts of a whole in every context has to do with the 360 degrees (which is why I mention it).

Seconds and minutes relate to positions on the planet in relation to the earth’s rotation. It takes the earth 24 hours to rotate; this is why there are 24 time zones. And when we cite a specific location on the planet in longitude and lattitude, we give something like: 21° 18′ 25″ N. That’s 21 degrees, 18 minutes, and 25 seconds north of the equator. Those degrees and minutes have nothing directly to do with time; they have to do with fractions of the planet’s 360 degrees. We use 60 seconds and 60 minutes (pretty much) because it’s so easy to divide a 360 circle to 60 minutes.

If our ancestors had agreed to make a circle be 100 degrees, we would probably have 10 hours in a day and 10 minutes in an hour and 10 seconds in a minute, or something like that.

Anyway, sorry. All this to answer a question you didn’t ask.

Crazy because .2 normally represents 2/10? Otherwise, it makes sense to me, because 1 whole inning is based on 3 outs.

No, crazy because how long is a third of an inning? It’s a measurement of progress as if progress is all that matters to a pitcher. It doesn’t measure time, energy, or effort. Man, if you bring in a relief pitcher and he walks two sluggers but gets the next guy to ground out to end the inning, the stat line only says he pitched a third of an inning and gave up two walks, when really he probably threw 25 pitches that two monster hitters couldn’t put into play so he could end the inning facing the weaker hitter. Anyway I’m digressing. I was talking about how we don’t use “out of a hundred” to talk about progress through a baseball game — we use thirds of an inning and whole innings, but this was to answer the question you weren’t asking. Ugh.

I think I complicated matters because I thought you were asking “Why out of a hundred as opposed to out of, say, 43?” But you were asking “Why out of a hundred as opposed to out of 10, which would make as much sense?”

For what it’s worth, I had questions about both.

Can you think of a single natural reason human civilization should have evolved to use base-10 as opposed to base-anything-else? The only one I can think of is that we have 10 fingers.

From a visual standpoint, base 10 seems easier to understand and utilize, don’t you think–e.g., 10, 100, 1,000 versus 8, 16, 32, etc. But is this the case because we’re used to base 10, or is base 10 inherently more user friendly? It’s hard to answer this.

The fact that we have ten fingers seems like another compelling reason humans are more comfortable with base 10 system as well.

I was talking about percentages as a whole working in our base-10 system…

I think this is what made the conversation confusing for me. I wasn’t thinking about percentages working in our base-10 system. I was thinking about why we use percentages as a general measure for wholeness. This is not to say the way percentages and base-10 system works elegantly is not interesting–rather, I’m pointing out that this made the conversation confusing for me.

That’s one reason our standardized test scores always gave us stanines and percentiles.

Tangent: I didn’t realize stanines referred at 9 point system. Do you know why they chose 9 points?

Another reason I was using the seconds-minutes-hours thing as an example of how “out of a hundred” isn’t necessarily the best or accepted way to think of parts of a whole in every context has to do with the 360 degrees (which is why I mention it).

OK, I think this is an important point. I understand why you bring up 360 degrees, measurements of time–and this is a valid point. Percentages/100 isn’t always used as a way to measure wholeness. However, it does seem to be a generalized standard. When wholeness isn’t predetermined by a specific number why do we use percentages?

If our ancestors had agreed to make a circle be 100 degrees, we would probably have 10 hours in a day and 10 minutes in an hour and 10 seconds in a minute, or something like that.

Anyway, sorry. All this to answer a question you didn’t ask.

What you’re saying is interesting, but I think it’s harder to discuss when I’m still confused about the initial issues I brought up. (I would think the base-10 system you describe about time would be more user friendly.)

No, crazy because how long is a third of an inning? It’s a measurement of progress as if progress is all that matters to a pitcher. It doesn’t measure time, energy, or effort.

You mean, the measuring an inning by outs excludes a lot of important information? If so, I agree.

More on why we divide the numerator (N) by the denominator (D)

Maybe the way to think about it is that the D represents the whole—all the parts forming the whole. So dividing the N (a fragment or piece) by the D (all the pieces equaling the whole), you get the value of the fragment…Actually this wording is wrong, isn’t it? Dividing the N by D or N/D is not the value of N, but a numerical value of the relationship between N/D. That is, the decimal represents (is? captures?) the ratio. (What’s the proper language here?)

If N=D then the ratio or relationship is 1–the whole thing. If N

I am almost sure the problem is that dividing the numerator by the denominator is a shortcut, and nobody’s explained to you what it’s a shortcut for. The problem for me in explaining it is that it’s a shortcut for a few different things and I have to figure out how to explain the right one.

Like, I’m 99% sure this is the issue. I learned a bunch of shortcuts I never trusted until they were explained to me (I had a patient math teacher) and it’s one reason I was successful in math. I might not have remembered all the formulas or shortcuts, but I remembered how I got there. Just let me chew on it a while. Or ask someone smarter than me what dividing 1 by 4 in 1/4 is a shortcut for.

I am almost sure the problem is that dividing the numerator by the denominator is a shortcut, and nobody’s explained to you what it’s a shortcut for.

That never occurred to me. You think that’s it, huh? I tend to think it’s just a deficiency on my part–I don’t think I have a good feel or understanding of math. It requires more work for me to grasp mathematical concepts, even simple ones.

When you say, “make sense” I assume you mean that I have an understanding of what’s happening based on the description. That is, I could know the answers are correct, but feel like the sentences or equations are not entirely clear. If so, I’m not comfortable with

“3 of 5 is 15 (not 3 out of 5, but three fives)”

But I’m more comfortable with 60% of $150 (went to food).

“one half of one half is one fourth”

I think this takes a little longer to digest and “visualize,” but I think I do get this. (And it’s clearer than “3 of 5.” “3 of 5” sounds awkward for some reason.)

“1/2 x 1/2 = 1/4”

I don’t know if this is clearer, but I know the answer is correct. I think “one half of one half is a one fourth” is clearer for some reason.

I had a conversation with Grace about what we’re talking about. At one point, she suggested that the numerator 1, in 1/2 can be thought of as both one whole and the one piece of something that is divided into two broken pieces. That didn’t make sense to me, but here’s a way it might.

Maybe the numerator should be thought of as 1 or a whole number(s), not 1 piece. The one piece is really 1/2. In other words, the numerator should be thought of a whole number (unless it’s a fraction). Where I might be messing up is thinking that the numerator 1 is the piece/fragment. When looking at the fraction 1/2, the denominator does represent the whole in a way–the whole is comprised of two equal parts, and the numerator represents one of those parts. But the one part is actually 1/2–it’s the value of 1/2. The numerator is more of representation, it is not the value of the part (1 of 1/2).

If this is correct, I think understand what’s going on. That is, if I think of the numerators that are whole numbers as whole numbers than the problem might be resolved for me.

I’m pretty sure Grace is not exactly right, but I’ll abandon my line of thinking up there ^^^ and just say that the numerator you’re dividing by the denominator isn’t really the number. I understand your confusion (I think) and it makes sense (I think).

When you and Grace say the numerator is a part of a whole and a whole at the same time, it’s shorthand for “the number that’s the fraction happens to be the same numeral as the number that’s the whole.”

I’m getting close to certain that what we’re looking at is a shortcut, so without confusing you further and explaining why, I’ll say that you’re not ACTUALLY dividing the numerator by the denominator. The numerator happens to be (always) the same number as the number you’re actually dividing.

When you and Grace say the numerator is a part of a whole and a whole at the same time, it’s shorthand for “the number that’s the fraction happens to be the same numeral as the number that’s the whole.”

I don’t know what you mean by this. Nor this:

I’ll say that you’re not ACTUALLY dividing the numerator by the denominator. The numerator happens to be (always) the same number as the number you’re actually dividing.

(By the way, I spent some time writing and thinking about this and other questions I had about fractions. I’ll try to get to it later.)

I know. But I don’t want to explain it if it’s going to confuse you, and you seem to have found some peace.

Think of it this way. You know how 6 and (1 x 6) are equivalent but not EXACTLY the same thing? I mean, they have exactly the same value, but one of the expressions is saying something different?

In doing calculations of value, the difference is meaningless. 6 and (1 x 6) calculate the same way, so we don’t even bother to write (1 x 6) when we do our figuring. Skipping to just 6 is a shortcut.

1/4 and (1/4 * 1) works the same way. When you do your calculating you just divide the numerator by the denominator, but that’s conceptually not what’s going on. You’re actually dividing that other 1 on the side by the denominator. They are different 1s, but the difference in calculation doesn’t matter, because your result is equivalent.

This actually happens all the time in math, something I wasn’t really aware of until I joined the math team. The older (better) math students were doing all these shortcuts that didn’t make sense to me, the way dividing the numerator by the denominator didn’t make sense to you. I didn’t trust the shortcuts until I could work them out. Thankfully I had the help of our coach who explained these things very well. And since I still remember the explanations, I can still go to the shortcuts in most situations. So believe me when I say I completely sympathize with your problem here.

1/4 and (1/4 * 1) works the same way. When you do your calculating you just divide the numerator by the denominator, but that’s conceptually not what’s going on. You’re actually dividing that other 1 on the side by the denominator.

Whoa, whoa, whoa–explain this to me. What do you mean “you’re actually dividing that other 1?”

I’ll do it but we’re going back up to that 3 (groups) of 5 equals 15 step above. Are you game?

Yes, I’m game.

I think I have a better idea of what you’re saying. I don’t have time now to explain, but don’t spend any energy trying to elaborate, until get back to you.

(Going to start a new sub-thread)

1/4 and (1/4 * 1) works the same way. When you do your calculating you just divide the numerator by the denominator, but that’s conceptually not what’s going on. You’re actually dividing that other 1 on the side by the denominator. They are different 1s, but the difference in calculation doesn’t matter, because your result is equivalent.

Here’s my understanding: “1/4 * 1” is basically means “cutting up” 1 whole in four equal pieces and “pointing to” one of those pieces. (Multiplying a fraction is essentially dividing or “cutting up” something into smaller fragments, and choosing one or more of those fragments.)

When you say, “They are different 1s,” are you referring to the numerator (1) and the 1 being multiplied by 1/4? If so, I understand. What’s not entirely clear is why you’re pointing out this difference.

Edit

Thought about this more: Basically, “1/4 * 1” is a conceptual expression. The numerator expresses that part and the second 1 represents the whole. I think this helps.

I want to throw another idea out there, though–namely, wholeness is something that can shift. For example, if we cut a pie into four equal pieces. One piece is a fraction of the whole. But if I want to share that piece with another person, that piece can be thought of as a whole. The 1/4 piece is not equal in quantity to the entire pie, but in can be designated as a whole. In this way, the fourth of a pie can be a fragment and a whole. My point is that wholeness has this “movable” (That’s not the right word, but I can’t think of it now) quality.

I wondered why the area of a triangle is 1/2 of the b*h. The 1/2 part is what puzzled me. But then I saw this explanation (the first one), and I think I understand now. The key point is that all triangles form half of a parallelogram. I wasn’t sure if this is true, but I guess it is.

A larger problem I’ve had is understanding, in a visual or concrete way, why multiplying length by width gives the area.

If you lay out six rows of marbles in columns of five marbles each, you have 6 x 5 marbles.

If you lay out six rows of Post-Its in columns of six Post-Its each, you have 6 x 5 Post-Its.

If each Post-It is exactly one inch square and the Post-Its are placed right next to each other, you have a larger box six one-inch Post-Its wide by five one-inch Post-Its high. The area of the box is 6 x 5 inches.

“Area” means how many of ______ fit into the bounded space. 30 Post-Its in the bounded space. Or 30 square inches in this case.

If each Post-It is exactly one inch square and the Post-Its are placed right next to each other, you have a larger box six one-inch Post-Its wide by five one-inch Post-Its high.

But this just pushes the problem to a lower level.

Let me give an explanation that I tried coming up with to illustrate the problem I’m having. Imagine me making a 1 inch column of tiny dots. Let’s also suppose that each dot really doesn’t have any length or width. And now let’s stretch out each dot to the right for an inch. That would be 1 square inch.

The post-it example is a lot clearer because we’re talking about blocks or post-its in rows and columns. But when we’re talking about a play area, for example, we don’t have blocks of rows and columns. Do you know what I mean?

No, if you stretch the tiny dot to the right one inch, you have a line. In fact, you’re pretty much saying the definitions of a point and line segment. It’s still one-dimensional. It’s just a linear inch, not a square inch.

You have to stretch the tiny dot to the right one inch and down one inch in order to make a square inch.

I do know what you mean and it’s a very abstract concept to think about. But you have to think two-dimensionally or you’ll never get anywhere. Area is about two dimensions. The actual “area” is the stuff of those two dimensions.

If mass makes sense to you (and I’m guessing it doesn’t!), area is just the two-dimensional version of the same concept. Which I am not saying makes it easy. The more you think about it the less sense it makes.

You have to stretch the tiny dot to the right one inch and down one inch in order to make a square inch.

Wait–there’s a column of dots, and each of those dots would be stretched one inch to the right. The result would be a (black) square.
The problem with the post-it example is that it just pushes the problem to a “lower level.” The dot (with no length or width) tries to get around that–but it’s not really a satisfying answer as well.

If mass makes sense to you (and I’m guessing it doesn’t!), area is just the two-dimensional version of the same concept.

Yeah, that doesn’t help, but I appreciate the effort.

Wait–there’s a column of dots, and each of those dots would be stretched one inch to the right. The result would be a (black) square.

That wouldn’t be the result at all, accepting your presumption that each dot has no width or length. A point (in geometric terms) has no width or length. So stacking one dot on top of another on top of another on top of another is adding 0 length to 0 length to 0 length to 0 length, which is still 0 length.

A line (and a line segment) has one dimension but not both. Stretching the dot to the right one inch gives it width but not length (or the other way around depending on how you name things).

That wouldn’t be the result at all, accepting your presumption that each dot has no width or length.

I thought I noted the limitations of my example–specifically, that the dots couldn’t be without width or length. The example was trying to avoid pushing the problem to a “lower level.” But it’s not really a great example.

First, let me say I hate the way I’m explaining myself. OK, by pushing the problem to a lower level, using marbles or post-its doesn’t really get rid of the problem I’m having because those things have length and width. Think of those Russian dolls, where one doll just leads to another.

I understand the concept of whole numbers, especially in relation to fractions. Indeed, I feel like one can’t understand whole numbers without understanding the concept of fractions, and vice-versa.

But what about integers, which, as I understand it, are positive and negative numbers and zero.

I’m partly wondering if integers also have to be understood by another type of number, like whole numbers and fractions.

In any event, Mitchell responded by asking how many numbers there are between 1-10, including 1 and 10.

10 would be my answer, or if we count fractions as well, then the answer could be a lot more (e.g, 3 1/2, 3 18, etc.)

Right. The answer to my question is “infinity.” In regular conversation, we use the word “number” incorrectly. I think it’s important to be aware of this when you’re talking about the other kinds of “numbers” in math.

Anyway. Whole numbers are 0, 1, 2, etc. Integers are these numbers plus the negative values of whole numbers: -1, -2, -3, etc.

Right. The answer to my question is “infinity.” In regular conversation, we use the word “number” incorrectly. I think it’s important to be aware of this when you’re talking about the other kinds of “numbers” in math.

Before I ask any other question, how does this relate to my original question about integers?

Here’s a real problem I recently faced. I played a game with a group of kids that required three groups. I had 56 kids. Here’s the breakdown of each group:

A: 20

B: 26

C: 10

I wanted to every participants to be in each group above, at least once. (I could do the activity in three rounds.) What’s the most efficient and least confusing way to go about this?

I don’t understand the problem. You just rotated them through the three stations, right?

Sorry, I wasn’t clear; let me try again (although I’m not promising this will be clearer). I was going to play a game three times or rounds. I’m asking how I can rotate the participants into new groups so that each participants gets to play the game in A, B, and C groups. For example,

Round 1

A: 20

B: 26

C: 10

In round 2, I could move the 20 from A into group B, and 6 from group C. Now, in Round 2, group B would have an entirely new group of participants.

I could then move 20 from Group B from Round 1 to Group A (for Round 2), and 6 to Group C (for Round 2).

And then I would to shift around the participants again for Round 3. See what I’m saying?

If the maximum number of people in group C is 10, you can’t do it in three rounds because you have 56 people. Are you allowed to take as many rounds as necessary to ensure everyone gets to be in each group?

You mean, you would need six rounds? I think that would be too many….I guess I was OK if not everyone got to be in group C, which I didn’t explain.

I also forgot to specify that I wanted the easiest way to break up the groups and transition to the next rounds (or maybe I conveyed that). I did this on the fly, and I drew up table, and tried show how I could move people around into different groups for subsequent rounds. It was a mess.

Thinking out loud–about the number one, fractions, and percentagesThe number 1 is equal to 100% (if “equal” is the right word). Or maybe it’s more appropriate to say that the number 1 can be thought of as 100%. There’s definitely a connection, right? What is exactly is the nature of that connection? It seems like the number one can represent at least two related concepts–

A) The number 1 refers to one thing, one item. For example, if I’m counting jelly beans, the number one would refer to one specific jelly bean;

B) The number one can refer to the whole or totality of several items. (I’m not sure “items” is the best word choice, here, but I’m going to use it for now. What would be a better word?)–e.g., 1 bag of 25 jelly beans. In this case, the number 1 refers to one specific item (or quantity)–namely the bag of jelly beans–but it also involves more than one item or quantity at the same time (i.e., 25 jelly beans). Mathematically, this sense of oneness (?) can be best represented by the fraction 25/25. What can be hard to grasp (especially for kids) is the way 25/25 is both

oneandmanyat the same time.(Aside: Is there another way of thinking about the number 1? Is there another variation or definition of the number 1?)

The B definition of the number 1 relates (obviously) to fractions and numbers of decimals. (What’s the technical name for those type of numbers? “Mixed numbers?”…Tangent: a mixed number combines the concepts of A and B above, right? Maybe this could be seen in the following example: 7.5 bags of jelly beans, where 1 bag contains 25 jelly beans. When “1” refers to the bag (of jelly beans), the number 1 is functioning in the mode B….Or is that wrong? Whereas each jelly bean can be thought operates in mode A? That doesn’t sound right–not completely. The jelly beans can be thought of as operating in mode A and B, right? Or not? (My thinking is muddled on this point.)

Understanding the different senses of the number 1 seems critical to gaining a good grasp of fractions and percentages.

More later.

I’m not sure you want me to complicate things, but to answer one of your questions, 1 can also be thought of as “on” while 0 is “off.” This is strictly a binary concept (hence the name of the base 2 number system where 11011100 equals 220 in base ten).

Incidentally, it’s also why when you’re looking at electronic equipment, the ON switch is marked with a | while the OFF switch is marked with a O.

Another way to think of 1 is as a specific identifier. If you have ten apples and ask someone to give you 1 more, if someone hands you a orange, even if it’s ONE orange, he hasn’t done what you’ve requested. This may not be a mathematical concept so much as a philosophical one (I honestly don’t know), but I suspect it’s an important idea we pick up without it ever really being taught us.

Oh I just thought of one more concept of 1 that may already be covered by some of these things we’ve listed, but 1 also means “not none.” In logic, the opposite of “none” is “one,” because the presence of 1 of anything negates the presence of “none.” Does this make sense?

It’s one reason “any” is grammatically treated as singular in American English. “Does any of my friends have ten bucks I can borrow?” is supposed to be the default but that’s been fluid for some time and it’s changing. So much that when I wrote a sentence beginning with “Does any of my friends…” I was corrected by a friend who’s an English major and a professional writer. She was wrong and I corrected her. 🙂

No, that’s cool.

Is the binary concept a mathematical concept? Or are people just using “1” an “0” in an artificial way (if that’s the right way to put it)? For example, could you use something something other than “1” and “0”–e.g., “A” and “B.” Also, what exactly does “on” and “off” mean?

How you get from the first number to the second?

Hmm, I think is a matter of semantics, not so much philosophy. In the example, “1 more” implies 1 more apple. The person being asked to get 1 more should know the requestor means 1 more apple. Suppose the person being asked doesn’t see any apples at all. When the person asks for 1 more, the person will likely respond, “1 more what?”

I think so. Using 1 to mean “not none” makes sense. You could use other number greater than 1, but that would be confusing. But does this concept occur in mathematical contexts? The equating of 0 to none and 1 to not none seems more of a designation to be used in logic, but it’s not really mathematical, if you know what I mean. Or is that wrong?

Really? You’re saying “any” is treated as singular because of the number 1 in logic refers to “not none?”

I’ll get to the rest of this later, but you said

Binary literally means a number system based only on the numerals 0 and 1, so I’m not sure how to answer the question. The etymology indicates that it originally means “something made of two things or parts” going back to the mid-1500s, but the base-2 mathematical system

surelywas conceived of long before that, perhaps by thousands of years.To answer the rest of your questions, I guess I first need to answer

It helps to know that you have a firm grip on the base-10 system, because that’s critical here.

124 means 1 x 100 plus 2 x 10 plus 4 x 1. Each digit in its assigned place means [digit] times [value of place]. The values in each place only range between 0 and 9 because once you go to 10 times the value, the counter in that place resets to 0 and you add 1 to (or “carry”) the value to the next column.

In base 2 (I’ll say binary from here on), the values can only range from 0 to 1. When the value of a place goes to 2, you have to “carry” the value to the next column and reset the counter to 0.

Quick illustration to show the place values.

In base 10:

Ten thousands —- thousands —- hundreds —- tens —- ones

In binary:

sixteens —- eights —- fours —- twos —- ones

So in binary, 10111 means

(1 x 16) plus (0 x 8) plus (1 x 4) plus (1 x 2) plus (1 x 1) = 23.

In the example I give in my response above, I said

(1 x 128) + (1 x 64) + (0 x 32) + (1 x 16) + (1 x 8) + (1 x 4) + (0 x 2) + (0 x 1) = 220

Going back to your original question about whether people are using “binary” in an artificial (I think maybe you also mean metaphorical) way, I would lean toward no, because in a binary concept, each value place either contains a value (1) or no value (0). Sure you could use A or B, but you’d be slightly changing the concept to something like “thing A” or “thing B” (or, to use a current use of “binary,” “male” or “female”).

In my response above, I was thinking more like “true” or “false,” or “thing A” or “not thing A.” I think of 1 to mean “true,” “thing A,” and “yes,” while 0 means “false,” “not thing A,” and “no.”

This goes to the way computers operate. The language computers know is binary. Everything we do on a computer and everything it does for us goes (ultimately) down to values. I’m oversimplifying but mostly out of necessity because I don’t understand it more deeply. I think Grace does, though, so you could ask her. Essentially, computers store data by placing a value in a spot or not placing a value in a spot — a 1 or a 0.

In a one-dimensional imagining of this idea, it looks at 11011100 as a series of values and non-values: one 128, one 64, zero 32s, and so on. Since each column either contains a value (1) or doesn’t (0), each place is either on or off.

There may be an actual, physical toggle for 1 and 0: on and off. I don’t know ZIP about how data is actually stored and manipulated electronically, but it would make sense if there’s a physical turning ON and turning OFF when there is and isn’t a value in a designated spot.

You didn’t ask but if you wanted to take this concept and make it two-dimensional, you can get an idea of how computers take this basic concept and turn it into things we actually recognize.

When we were in 9th and 10th grade, I learned that you can reprogram the letters on a computer to look however you wanted, so if you typed “A” the monitor would display something else, while still understanding the value as “A.” Nowadays we take it for granted: we can switch to an endless number of fonts, and out computers understand the actual meaning of the letter no matter what it looks like. But remember how the computers in high school only had one font? You couldn’t change fonts back then unless you could teach the computer to do it.

I told my computer at home that when I typed the A key, I didn’t want it to type the A it had as a default. I told the computer to use this data instead:

60, 66, 66, 66, 126, 66, 66, 0

And this is how the computer read it:

You can look at that first row in the graphic visually, as humans do, or you could look at it as

off off on on on on off off

or

00111100

or (0 x 128) + (0 x 64) + (1 x 32) + (1 x 16) + (1 x 8) + (1 x 4) + (0 x 2) + (0 x 1) = 60.

That would just be the top line in the graphic above, but I gave the computer 8 values, which it interpreted as eight rows. Now when I hit the A key on my keyboard, it displayed THIS letter A, not the letter A it came with.

I bothered to explain all this so you could see that the “on-off” idea of 1 and 0 is like this super fundamental concept on which some huge ideas rest. But yeah: it’s still kind of a manufactured concept. We’ve put human-centered thinking into the way we think of 1 and 0; it’s not necessarily a universal truth. At least I don’t think it is!

That was maybe way more info than you wanted or needed. But hey, if you stayed to the end now you can laugh when you see the t-shirts that say this.

The question I asked wasn’t very clear–especially the notion of a “mathematical concept.” I really don’t know how to articulate what I mean, and that phrasing was the first thing that came to mind. I think I mean something closer to what people would use when doing “normal” math–versus math this is more theoretical, speculative or imaginative. By the latter, I have this notion that one could take mathematical principles and construct all sorts of odd processes–ones that wouldn’t be used or even thought of as (normal) mathematics. This explanation is still not clear, but I hope it helps a little.

Wait, when you say, “The values in each place only range between 0 and 9,” don’t you mean the value of each digit (in its assigned place)? “Place” refers to 1s, 10s, 100s, etc., right? The value of each digit in a specific place can only be 0 to 9. If so, I understand.

OK, I think I understand. My reaction to this is, why would anyone devise a system like this? It seems very cumbersome and difficult. What context, besides computing, would be useful for the binary approach?

OK, this explanation (and the paragraph before it) was helpful. I think I understand how “1” means yes or on and “0” means no or off. And I think it makes more sense in the context of computers. It’s especially clear in relation to creating images on a computer screen. (I’m less clear how it would apply in other computer contexts.) To create an image, a computer utilizes a grid, and images can be mad by coloring in squares or leaving them uncolored (i.e., blank). But how do you “tell” the computer which squares to color in or leave blank? The binary system provides a language that computers can “understand,” to do, right? Essentially, “1” and “0” mean color or leave blank, respectively, and the place value gives a specific location of the box. Is that right?

If it is, I guess this is what I mean by “artificial.” People used the binary system in computers as a language, because they couldn’t use human language like English to give commands…Wait, that can’t be write, because we do use some language when we do programming.

It depends on whether or not you think of logic as math. It was my favorite Math League event and it’s often taught as a math course. At UH, it’s taught as a philosophy course.

No, I’m saying “any” is singular for the same reason 1 means “not none.” When you’re trying to prove a statement like “Nobody loves me,” all you’re looking for is one person who loves “me.” If “anybody” loves “me,” the assertion is false.

Question about adding and subtracting positive and negative numbers:

Why isn’t -7-9, -7-(+9)?

(Mitchell, I’m not ignoring your other comments or lost interest in them; I’ll try to get back to them soon.)

Those are equivalent. Did you try putting them in a calculator?

You don’t have to comment or respond; I’m totally serious. I’m quite sure I went in a direction that’s of interest only to me.

This is equivalent because “-(+9)” =-9–i.e., a negative times a positive equals a negative? And -7-9 is essentially adding two negatives?

I’m interested in parts of it, but trying to work through it evokes some level of math anxiety, which I have to work through.

Yeah. You got it.

Here’s a word problem from my son’s homework I couldn’t solve:

Mary has a gift certificate. She spends 1/3 on lunch. She spends 1/4 of the balance on dessert. After paying $3.50 for a tip, she has $4 leftover.

I called Mitchell to help me, and he came up with this:

x-(x-1/3x)-1/4 (x-1/3x)-3.5=4

(I liked how he explained this to me.) But given the previous questions, which had to do with multiplying and dividing fractions and mixed numbers, I was pretty sure this wasn’t the way to solve the question. After reading some of the previous questions, Mitchell agreed. In the chapter, there was a section on using grids to solve problems and I suspected the solution would involve using them.

I emailed Pat Ota, and she came up with the solution:

Super cool!

What? Ms. Ota is still smarter than me…

I would ask who is surprised by this, but I don’t want to be uncool.

That’s not the equation I came up with. Please tell me you didn’t send my solution to Pat like that.

OK, I won’t tell you I didn’t send that solution. Hahaha. Seriously, I think I did, I’m not sure. I just went by memory (on both times). I prefaced the equation by saying, “The equation was something like…”

I’m going to have to email or text her.

Write an algebraic expression the nth term in the table below:

A: 0 1 2 3 4 5 n

B: 3 5 7 9 11 13 ?

According to Zane, the teacher said the answer is 2n+3. If this is correct, I don’t get it. Can you guys explain this? (There’s a decent chance Zane didn’t write the right equation.)

Yeah this is right. What are you disagreeing with?

I don’t disagree–I just don’t really understand how he got that answer.

First you have to figure out the rule. 0 in, 3 out can be an infinite number of rules. 1 in, 5 out restricts it a lot. But 2 in, 9 out pretty much seals it. There are a lot of ways to come up with the formula, but at your son’s level, probably the fastest way is trial and error.

0 x 2 + 3 =3

1 x 2 + 3 = 5

3 x 2 + 3 = 9

and so on.

So what would N be?

N x 2 + 3 = 2n + 3

Actually, I’ll be nit-picky and say that the answer is wrong, but I’m fairly sure they don’t want the nth term. They want the equation for n.

In this case

0 is the first term

1 is the second term

2 is the third term

3 is the fourth term

etc.

So the actual algebraic expression (don’t panic; just read this slowly) for the nth term is

2(n-1) + 3

But I almost guarantee that’s not what they’re asking, unless this is the super bonus question.

Solution check:

2(1-1) + 3 = 3

the FIRST TERM is 0

the SECOND TERM is 1

the THIRD TERM is 2

the FOURTH term is 3

the 100th term is 99

2(100-1) + 3 = 201

the Nth term is N-1

2(n-1) + 3 = 2n – 2 +3 = 2n +1

…but that’s probably not what the question is asking.

How did you get to multiplying by 2 and then adding by 3?

Well the first step is usually to see if the formula is simple addition. You look at 0 and then 3, and say, “Oh, maybe it’s add 3.”

But then you look at 1 and see 5. 1 plus 3 isn’t 5, so the rule cannot be add 3. When we see 2 and 7, we can see simple addition is way off, and the difference between the A and B terms gets bigger with each item in the series.

So let’s try multiplication. Normally you might have to go through a few items to try a simple multiplication rule. But in this case, the first term is 0, and nothing you multiply by 0 is ever ever ever going to give you 3.

So can we multiply the A number then add something to get the B number?

0 times ANY number is 0. Then you need to add 3 to get the B term, 3. Maybe the rule is multiply by something and then add 3?

Let’s look at the next A term: 1.

What can we multiply the next A term by? Multiplying by 0 doesn’t help (but always start with 0 because it can make your life easier). Multiplying by 1 doesn’t help. How about 2?

1 x 2 + 3 = 5. Let’s try it with the other terms. Does it work?

Tada.

Again, there are algebraic ways of figuring it out, but that’s usually algebra 2 stuff. I’m guessing that’s not what your son’s teacher wants.

Don’t take this the wrong way, but I would expect this kind of problem to appeal to you much more than more mathy excercises. It’s more a thinking problem than a math problem, and there’s nothing very abstract here except the “in terms of N” part.

Find each unit price and then determine the better buy.

1 lbs. for $1.29

12 oz. for $.95

I’m embarrassed to say that I have a hard time knowing whether to divide by 1.29 by 16 oz or 16 oz by 1.29. I mean, I can eventually figure it out by comparing the answers, but I’m having trouble knowing the right way prior to doing that. I also don’t know how to explain why 1.29 would be divided by 16 versus dividing 16 by 1.29….

…Well, if I divided 16 oz. by 1.29 that would give me ounces/cent–isn’t that what I want to know? But don’t I have to divide 1.29 by 16? And wouldn’t that give me the cents/ounce–i.e,. 1 cent would get me X ounces. But don’t I want to know the cost of 1 ounce?

What am I doing wrong?

OK, I think I figured out where I messed up.

This is supposed be X cents for every 1 once–X cents/ounce–which is the rate I want to know. Right?

Yeah I think you got it. Just remember that when you see a /, it means the word “per.”

1/4 means “one per four.”

putting money / units means “cents per unit.”

or if “per” is weird, say “for each.”

one for each four.

cents for each unit.

“One for each four” is odd and awkward. “One-fourth” means “one for each four?”

On the other hand, I have no problem with “cents for each unit.”

If 100 people are in a room and 25 of them are wearing red shirts, 25 for each 100 are wearing red shirts. Or 12.5 for each 50. Or 10 for each 40. Or 1 for each 4. Ta-da.

I think I might be getting thrown off because “for each” refers to multiple situations. If I said 25 people out of a room of 100 people are wearing red shirts, that refers to that one specific room.

For each 100means that for every 100, I will have 25 people wearing red shirts.The key here, I think, is that we’re talking about rates, not just a fraction. Rates are a fraction, but not all fractions are rates. I guess that’s what I’m thinking.

…and don’t be embarrassed. I have to speak words aloud almost every time I do this problem in the supermarket.

Thanks for throwing me a bone. I appreciate it.

What I’m having trouble with–the language–(quantity) for (another quantity) and seeing how that gets to the mathematical equation. Why does 1 lbs for $1.29 mean 1.29 divided by 1 lbs.? Getting from the former to the latter is the part I’m struggling with–specifically, how to think about and articulate what’s going on.

And I always thought “/” was another form of the line between the numerator and denominator–basically a sign of division. (It essentially means this, right?)

What’s the explanation for dividing a numerator by the denominator , when converting a fraction to a decimal or percentage?I’m going to try to answer this, thinking out loud. I’ll start with definitions of denominator and numerator. The denominator is the number of equal(?) pieces that make up a whole (1?)*. For example, the 4 in 1/4 means that a whole is cut up into four equal pieces. The numerator is the number of pieces in relation to denominator. If we divide a whole into four pieces, 1/4 means 1 piece out of the 4 pieces.

So why do we divide the numerator by the denominator to convert the fraction into a decimal or a percentage?

Here’s the first thing that comes to mind. The denominator basically represents the whole–i.e., 1 or 100%

in relation to the numerator….But why divide, not multiply, the numerator by the denominator?To be continued…

(*Could you create a fraction if the pieces are not equal? Off the top of my head, I would say the answer is no. Also, could a numerator or denominator be less than 1–i.e., a fraction or decimal? What would that mean?)

You absolutely can have a numberator that’s a decimal or a fraction. It’s just not considered “simplified” in that form. You can say you own 2/1 cars, but you wouldn’t because it’s not the simplest form of the answer.

Simplifying is important for a few reasons but one reason is to make sure you’re not inadvertently dividing by zerio.

Hmm, “standardizing” isn’t a word that came to mind, but it’s something worth considering. However, your example suggests that simplifying means putting a number in the form that is most easy to understand or grasp. I/4 is easier to grasp than 13/52. Is that what simplifying comes down to?

…I guess the standardized” part refers to what most people would consider the form that is easiest to grasp?

Re: the division by zero thing. I didn’t really understand your explanation.

I don’t want to assume anything, but you do understand that you can’t divide anything by zero, right? One important thing simplifying accomplishes is making sure there isn’t a zero in a denominator somewhere — you might not be able to see it in a non-simplified numerical expression.

If you see 1/0 it’s obvious.

If you see 1/(10-7-3) it’s less obvious.

If you see 1/(tan(45)-1) it’s even less obvious. Simplifying this expression, you get 1/0.

Yes. Although on some level isn’t dividing by 1 similar to dividing by 0?

That never occurred to me. If ended up with 0 in the denominator, that essentially means you made a mistake somewhere, right?

Why wouldn’t a fraction with a decimal in the numerator not be the most simplified form? Or to ask it another way, what does simplifying really mean? I always thought it simply referred to reducing numerator to the smallest number possible, if that’s even the right way to put it.

How should one think if the numerator (or denominator?) is a decimal? What does it mean when this happens?

What did you have in mind?

Basically a decimal in a fraction is a fraction in a fraction, so it’s not considered simplified. Simplifying kind of standardizes things a bit: we might not know what (1/4) / (1/8) even means by looking at it, but (1/4) / (1/8) is 2, and we know what that is.

For the division by zero thing, I can’t think of a specific example but something like (1/4) / (x/8) would be 1/4 times 8/x, and if that’s an answer we have to clarify with something like “where x is not equal to 0.” This isn’t the best example but in very complicated fractions you just have to keep an eye out.

edit: reducing a fraction to its lowest terms is one part of simplifying, but it’s not all there is to it. If you ask someone how much you owe him, and he says, “Twenty dollars minus six percent,” that’s not very simple. Similarly, you aren’t supposed to leave radical signs in denominators (although the rationale for this isn’t as convincing as it once was), or a negative sign in both the numerator and denominator.

I’m still trying to find a way to understand this.

For 1/4, why do I divide 1 by 4? I know that a fraction basically means a fragment of the 1 (whole). If I take a deck of cards (which is 1 whole thing), one card is a fragment or fraction of the whole deck. So I know that when I see a fraction like 1/4 (i.e., where the numerator is less than the denominator) if I were to divide this, the answer has to be a decimal. But that doesn’t explain why the numerator is divided by the denominator….

What is happening when the numerator is being divided by the denominator?

At the risk of confusing you, let me take one step back and look at what multiplication of fractions means.

1/4 times 1/2 means “one fourth of one half.”

I give you half a pie. You eat three fourths of that one half and leave only one fourth for the rest of your family. What fraction of the original pie did your family get?

One fourth of one half is what’s left over. Multipling 1/4 by 1/2 you get 1/8. I ate half the pie. You ate three eighths. Your family was left with one eighth.

So one way to think of multiplying fractions is increasing the fractiousness of a thing.

If you see (for example) (1/4) / 2 you have a fourth of a half (don’t forget, that 2 in the denominator means half). So this is one way to think of it when you see a decimal or a fration in a numerator or a denominator. (.2)/2 is .2 of a half, or 1/5 of a half, or 1/10.

I think I understand this (as well as the rest of your post). But what is happening when I divide the numerator by the denominator?

I’m bringing this down here because it’s getting confusing up there.

What is happening when you divide a numerator by a denominator AND why dividing by 0 and dividing by 1 are not sorta the same thing. In one short answer!

Division by zero“10 divided by 2” sorta means “ten things divided into two groups.” If you have ten bananas and divide them into two equal groups, there are five bananas in each group. 10 / 2 = 5.

“10 divided by 1” sorta means “ten things divided into one group.” If you have ten bowling balls and divide them into one group, there are ten bowling balls in that one group. 10 / 1 = 10.

“10 divided by 0” sorta means “ten things divided into no groups at all.” If you have ten jawbreakers, if you try to divide them into zero groups, what happens to the ten jawbreakers? They would essentially have to be zapped into nonexistence. You cannot take 10 things and put them in zero groups. And for some reason, you can neither do it in real-world terms such as these jawbreakers or in abstract terms. I don’t know why you can’t even do it abstractedly when there are such things as negative numbers.

You asked, “If ended up with 0 in the denominator, that essentially means you made a mistake somewhere, right?” Probably. But not necessarily. Sometimes you get 1/0 and it just gives you an idea of how an equation’s pieces relate to each other. Go here (https://www.desmos.com/calculator) and type “1/x” in the equation bar. You see how the line never touches the X or Y axis? It’s because values for X don’t exist (1/0) in this equation. And it’s because nothing you put in for X will ever give you 0 for Y. Summary: sometimes if you end up with division by zero, it tells you important information about the way an equation behaves.

What we’re doing when we divide a numerator by a denominatorKeeping in mind the stuff we just went through with bananas, bowling balls, and jawbreakers:

“1 divided by 4” means “one thing divided into four equal groups.” If I have one apple and divide it into four equal groups, each group has a fourth of an apple. 1 divided by 4 = 1/4, or .25 or 25%.

“2 divided by 4” means “two things divided into four equal groups.” Starting with two apples, splitting them into four equal groups, I have half an apple in each group. 2 divided by 4 = 2/4 or .5 or 50%.

So to answer your question: the reason you divide the numerator by the denominator is because it’s literally what 1/4 means. Yes, 1/4 means “one fourth” but “one fourth” literally means one thing divided by four.

Going to skip the other comments for now and get to the most important issue for me:

Here’s where I get lost. If I were to use a concrete example of 1/4, I would take a pie, for example, and cut it into 4 equal pieces. I would take one piece and say that piece represents 1/4 (of the pie). Using your example would be taking that one piece and dividing it into four equal pieces. I think this is what’s confusing me. Do you see what I mean?

No it wouldn’t. I don’t know why you think this.

We’re talking about 1/4, right? Not (1/4)/4?

Yes.

I don’t think it matters which real-world object or substance you use in your example. Apples, pies, or thingies. 1/4 represents taking the thing (or group of things) and splitting it evenly four ways. The 1 represents one of those four groups, whether it’s 5 out of 20 apples, 1 quarter of a pie, half of pie out of two pies, or 3 quarters of a pie out of 3 pies.

So is this what’s happening when we say 1/4: We take one pie, cut it into four equal pieces. One of those pieces represents 1/4. But doesn’t the numerator (1) represent 1 piece of the pie (cut into 4 equal pieces)–not the whole (1) pie. I’m confused, if that’s not already obvious.

Yes, the 1 represents the one piece of the pie. You could think of the 4 as representing the whole pie if you like. But only if that doesn’t confuse you more.

But if the “1” in 1/4 represents 1 pieces of a pie that is cut into 4 equal pieces, then wouldn’t we then be dividing that one piece into 4 equal pieces? That can’t be right, but that’s what it seems like is happening.

Oh. I see your confusion.

the 1 in 1/4 doesn’t exactly represent one quarter of a pie. It represents ONE one-fourth of a pie.

x/4 means some number of something divided into 4. The x represents an unknown number of fourTHS.

1/4 means one of something divided into four. The 1 represents one fourTH.

ONE one fourth of the pie is the one piece out of a pie that was cut into four equal pieces, right? But 1/4 is 1 divided into 4 equal pieces. I don’t get it.

1/4 does mean one pie cut into four pieces, but it represents the VALUE of each of those pieces. HOW MUCH pie is in one piece of that one pie cut into four pieces.

1/4 doesn’t exactly represent the act of cutting a pie in four pieces, nor does it exactly represent the state of the pie, as one pie cut into four pieces. It represents the amount of pie in each piece when one pie is cut into four pieces.

So I think if you see it this way, it’s exactly the same thing in both your scenarios.

I don’t think I’m really understanding or appreciating the distinction you’re making, but let me focus on your last sentence, especially “amount of pie in each piece.” How do you get from that to dividing the numerator 1 into the denominator, 4? If the numerator–1–represents the whole (pie) that makes sense on some level. But it doesn’t if it represents a slice of the pie.

More on dividing by 1 and 0Mitchell said,

It may seem obvious, but I like the description of dividing as an act of dividing something in groups. So, division by 1 means dividing something into one group, and division by 0 means dividing by no groups.

Here’s why I think the dividing by 1 and 0 seem very similar. Generally, division is an action, breaking a whole into parts or groups, as you say. By when you divide by 1, you’re really not doing any action–you’re not cutting a whole into parts or breaking it into groups. The whole remains the whole. You did nothing–or you “did 0” so to speak.

I think when you use “grouping” that is helpful because if you say group all the bananas into one group, that is an action. But in this case, the word “division” is a bit misleading. You’re not really dividing anything.

But then grouping doesn’t work well when you’re using division on something like a banana pie. If I divide 1 banana pie into one group that doesn’t make as much sense, or it’s at least very awkward wording. In this case, I’m neither dividing or grouping.

Now, when we divide by 0, that seems like saying don’t do any dividing, breaking into parts, or grouping. That is, do nothing. And that seems like the same thing as dividing something by 1.

To be clear, I understand the difference between dividing by 1 and 0, and your explanation helped make this clearer–I’m not saying they’re the same thing. But I’m trying to explain why the two actions seem like the same thing. My sense is that this an instance where language is inadequate, or being able to describe and distinguish the two processes isn’t so easy.

Thinking out loud about the concept of percent–For some reason, It recently hit me that “percent” is really “per cent”–“per” being the Latin word for “for each” and “cent” being the Latin root for 100. So, 60% is literally 60/100. In my defense, I think writing “per cent” as one word and using the symbol “%” has obscured this meaning for me. (More on this later.)

–This got me thinking about something else: Why 100? The concept of percentage is just a way of thinking about the relationship between some part to a specific whole–e.g., how much of the total budget went to salaries. In a percentage, “100” is used to represent the whole, but we don’t have to use this number (or a percentage) to think about parts to whole relationships. For example, we could 10 to represent the whole, and speak in terms of a “perdecimage;” or we could choose a 1,000 to represent the whole and speak in terms of a “permillenia.” I’ll offer thoughts on why I think we use percentages when talking about the relationship between parts to a whole, but first I want to make the point that percentage is ultimately a way of thinking about the relationship between a part(s) to a whole. Basically, it’s just a fraction. (Right?)

–So why choose 100? My guess is that this is a “Goldilocks” number in the vast majority of cases when we’d want to understand a quantitative relationship between part(s) and a whole. 10 wouldn’t afford enough precision, while 1,000 would often afford too much. 100 is “just right.” Additionally, 100 (or a factor of 10) is easier to work with an understand. We could choose 77 to represent, but doing so would be so unwieldy.

–Now, I want to go back to the way the word “percent” and the percentage symbol have obscured the meaning of the term and concept. Suppose we chose 10 to represent the whole and we used the term “perdecemage” and also a symbol “+/+.” For example, I could say, “8 +/+ of the budget went to employee salaries.” At some point, I and others might begin to lose sight of the fact that A) perdecimage, using 10 to represent the relationship between the part and whole is useful, but ultimately arbitrary convention, and B) it’s ultimately a fraction or ratio.

…OK, I feel like there was something else I wanted to say, but I can’t remember any more.

Using 100 to represent a whole is certainly kind of arbitrary; we could use anything else, and in fact we do. In some settings.

100 works really well because of our base-10 number system, but of course our base-10 system is also an arbitrary decision. If we grew up in a society that for some reason used base-8, our concept of numbers, numerals, and percentage (which of course we’d call something else) would seem as natural to us as base-10.

One second is 1/60 of a minute. One minute is 1/60 of an hour. One hour is 1/24 of a day. One day is either 1/28, 1/29, 1/30, or 1/31 of a month. One month is 1/12 of a year. These are all parts of wholes that don’t have anything to do with hundreds.

Also arbitrary, but those seconds, minutes, and hours have to do with a circle (and thus the planet) being 360 degrees. But that 360 is arbitrary too.

Another context: when you see that a pitcher pitched 7.2 innings, that’s not 7 innings and 2 tenths of an inning. That’s 7 and 2/3 innings, and that 2/3 doesn’t have anything to do with time, distance, or anything static: it has to do with the number of outs he gets. How crazy is that? 🙂

That is hard to imagine and hard to believe.

OK, but the situation is really different in my opinion. Using a 100 is a general way to measure wholeness, when wholeness is already predetermined. 60 seconds equals 1 (whole) minute. 11 represents the whole of a football team that’s on the field. I don’t think I’m explaining this well, but do you see what I mean? 100 is a generalized standard.

What? I didn’t know that.

Crazy because .2 normally represents 2/10? Otherwise, it makes sense to me, because 1 whole inning is based on 3 outs.

Ugggggh sorry. I think I complicated matters because I thought you were asking “Why out of a hundred as opposed to out of, say, 43?” But you were asking “Why out of a hundred as opposed to out of 10, which would make as much sense?”

I was answering the first question. Needlessly, and in fact in opposition to this conversation going wherever it was going.

Perhaps difficult to imagine but not at all difficult to believe. Can you think of a single natural reason human civilization should have evolved to use base-10 as opposed to base-anything-else? The only one I can think of is that we have 10 fingers.

Also not hard to imagine, really. Computer programmers are quite literate in binary, as we’ve discussed, and in hexadecimal (base-16). I’m not fluent, but I’m conversant in both.

Yes, but I was answering another question, and in that alternate conversation, 11 players making up one football team is not the same thing and here’s why:

10 hours, 59 minutes, and 57 seconds.

10 hours, 59 minutes, and 58 seconds.

10 hours, 59 minutes, and 59 seconds.

11 hours.

Our time system works a lot like base-10: in counting, add one second to the 59th second, and you reset the seconds place and “carry” the whole minute to the minutes place. If in doing so, the minutes value hits 60, you reset the minutes place and “carry” the whole hour to the hours place.

Adding a 12th player to the guys on the team doesn’t then create a second team. I was talking about percentages as a whole working in our base-10 system because adding one percent to .99 works so smoothly in our decimal system, if that makes any sense.

You were really asking the question about precision: why measure things in hundredths when we can measure them in tenths, and yeah, it goes to precision. That’s one reason our standardized test scores always gave us stanines and percentiles. In some situations people didn’t need the precision of percentiles — there’s no practical difference in knowledge or ability between all the people in the 8th stanine, but in cases (such as super-selective schools) there is a difference between someone in the 89th percentile and someone in the 81st.

There may be a more complicated explanation, but 360 degrees in a circle seems utterly arbitrary to me, and it was explained this way to me when we were in high school. We could easily have made a circle be 100 degrees.

Another reason I was using the seconds-minutes-hours thing as an example of how “out of a hundred” isn’t necessarily the best or accepted way to think of parts of a whole in every context has to do with the 360 degrees (which is why I mention it).

Seconds and minutes relate to positions on the planet in relation to the earth’s rotation. It takes the earth 24 hours to rotate; this is why there are 24 time zones. And when we cite a specific location on the planet in longitude and lattitude, we give something like: 21° 18′ 25″ N. That’s 21 degrees, 18 minutes, and 25 seconds north of the equator. Those degrees and minutes have nothing directly to do with time; they have to do with fractions of the planet’s 360 degrees. We use 60 seconds and 60 minutes (pretty much) because it’s so easy to divide a 360 circle to 60 minutes.

If our ancestors had agreed to make a circle be 100 degrees, we would probably have 10 hours in a day and 10 minutes in an hour and 10 seconds in a minute, or something like that.

Anyway, sorry. All this to answer a question you didn’t ask.

No, crazy because how long is a third of an inning? It’s a measurement of progress as if progress is all that matters to a pitcher. It doesn’t measure time, energy, or effort. Man, if you bring in a relief pitcher and he walks two sluggers but gets the next guy to ground out to end the inning, the stat line only says he pitched a third of an inning and gave up two walks, when really he probably threw 25 pitches that two monster hitters couldn’t put into play so he could end the inning facing the weaker hitter. Anyway I’m digressing. I was talking about how we don’t use “out of a hundred” to talk about progress through a baseball game — we use thirds of an inning and whole innings, but this was to answer the question you weren’t asking. Ugh.

For what it’s worth, I had questions about both.

From a visual standpoint, base 10 seems easier to understand and utilize, don’t you think–e.g., 10, 100, 1,000 versus 8, 16, 32, etc. But is this the case because we’re used to base 10, or is base 10 inherently more user friendly? It’s hard to answer this.

The fact that we have ten fingers seems like another compelling reason humans are more comfortable with base 10 system as well.

I think this is what made the conversation confusing for me. I wasn’t thinking about percentages working in our base-10 system. I was thinking about why we use percentages as a general measure for wholeness. This is not to say the way percentages and base-10 system works elegantly is not interesting–rather, I’m pointing out that this made the conversation confusing for me.

Tangent: I didn’t realize stanines referred at 9 point system. Do you know why they chose 9 points?

OK, I think this is an important point. I understand why you bring up 360 degrees, measurements of time–and this is a valid point. Percentages/100 isn’t always used as a way to measure wholeness. However, it does seem to be a generalized standard. When wholeness isn’t predetermined by a specific number why do we use percentages?

What you’re saying is interesting, but I think it’s harder to discuss when I’m still confused about the initial issues I brought up. (I would think the base-10 system you describe about time would be more user friendly.)

You mean, the measuring an inning by outs excludes a lot of important information? If so, I agree.

More on why we divide the numerator (N) by the denominator (D)Maybe the way to think about it is that the D represents the whole—all the parts forming the whole. So dividing the N (a fragment or piece) by the D (all the pieces equaling the whole), you get the value of the fragment…Actually this wording is wrong, isn’t it? Dividing the N by D or N/D is not the value of N, but a numerical value of the relationship between N/D. That is, the decimal represents (is? captures?) the ratio. (What’s the proper language here?)

If N=D then the ratio or relationship is 1–the whole thing. If N

I am almost sure the problem is that dividing the numerator by the denominator is a shortcut, and nobody’s explained to you what it’s a shortcut for. The problem for me in explaining it is that it’s a shortcut for a few different things and I have to figure out how to explain the right one.

Like, I’m 99% sure this is the issue. I learned a bunch of shortcuts I never trusted until they were explained to me (I had a patient math teacher) and it’s one reason I was successful in math. I might not have remembered all the formulas or shortcuts, but I remembered how I got there. Just let me chew on it a while. Or ask someone smarter than me what dividing 1 by 4 in 1/4 is a shortcut for.

That never occurred to me. You think that’s it, huh? I tend to think it’s just a deficiency on my part–I don’t think I have a good feel or understanding of math. It requires more work for me to grasp mathematical concepts, even simple ones.

OK thanks. (I’ll ask Pat Ota about this, too.)

Reid, do these things make sense to you?

3 of 5 is 15 (not 3 out of 5, but three fives)

4 of 5 is 20

5 of 5 is 20

3 x 5 = 15

4 x 5 = 20

5 x 5 = 25

one half of one half is one fourth

one third of one half is one sixth

one fourth of one half is one eighth

1/2 x 1/2 = 1/4

1/3 x 1/2 = 1/6

1/4 of 1/2 = 1/8

When you say, “make sense” I assume you mean that I have an understanding of what’s happening based on the description. That is, I could know the answers are correct, but feel like the sentences or equations are not entirely clear. If so, I’m not comfortable with

“3 of 5 is 15 (not 3 out of 5, but three fives)”

But I’m more comfortable with 60% of $150 (went to food).

“one half of one half is one fourth”

I think this takes a little longer to digest and “visualize,” but I think I do get this. (And it’s clearer than “3 of 5.” “3 of 5” sounds awkward for some reason.)

“1/2 x 1/2 = 1/4”

I don’t know if this is clearer, but I know the answer is correct. I think “one half of one half is a one fourth” is clearer for some reason.

I had a conversation with Grace about what we’re talking about. At one point, she suggested that the numerator 1, in 1/2 can be thought of as

bothone wholeandthe one piece of something that is divided into two broken pieces. That didn’t make sense to me, but here’s a way it might.Maybe the numerator should be thought of as 1 or a whole number(s), not 1 piece. The one piece is really 1/2. In other words, the numerator should be thought of a whole number (unless it’s a fraction). Where I might be messing up is thinking that the numerator 1 is the piece/fragment. When looking at the fraction 1/2, the denominator does represent the whole in a way–the whole is comprised of two equal parts, and the numerator represents one of those parts. But the one part is actually 1/2–it’s the value of 1/2. The numerator is more of representation, it is not the value of the part (1 of 1/2).

If this is correct, I think understand what’s going on. That is, if I think of the numerators that are whole numbers as whole numbers than the problem might be resolved for me.

That’ll work, and if it gets you where you need to be, then great. It’s the same shortcut I was getting at but without all the explainy stuff.

I’m pretty sure Grace is not exactly right, but I’ll abandon my line of thinking up there ^^^ and just say that the numerator you’re dividing by the denominator isn’t really the number. I understand your confusion (I think) and it makes sense (I think).

When you and Grace say the numerator is a part of a whole and a whole at the same time, it’s shorthand for “the number that’s the fraction happens to be the same numeral as the number that’s the whole.”

I’m getting close to certain that what we’re looking at is a shortcut, so without confusing you further and explaining why, I’ll say that you’re not ACTUALLY dividing the numerator by the denominator. The numerator happens to be (always) the same number as the number you’re actually dividing.

I don’t know what you mean by this. Nor this:

(By the way, I spent some time writing and thinking about this and other questions I had about fractions. I’ll try to get to it later.)

I know. But I don’t want to explain it if it’s going to confuse you, and you seem to have found some peace.

Think of it this way. You know how 6 and (1 x 6) are equivalent but not EXACTLY the same thing? I mean, they have exactly the same value, but one of the expressions is saying something different?

In doing calculations of value, the difference is meaningless. 6 and (1 x 6)

calculatethe same way, so we don’t even bother to write (1 x 6) when we do our figuring. Skipping to just 6 is a shortcut.1/4 and (1/4 * 1) works the same way. When you do your

calculatingyou just divide the numerator by the denominator, but that’s conceptually not what’s going on. You’re actually dividing thatother1 on the side by the denominator. They are different 1s, but the difference in calculation doesn’t matter, because your result is equivalent.This actually happens all the time in math, something I wasn’t really aware of until I joined the math team. The older (better) math students were doing all these shortcuts that didn’t make sense to me, the way dividing the numerator by the denominator didn’t make sense to you. I didn’t trust the shortcuts until I could work them out. Thankfully I had the help of our coach who explained these things very well. And since I still remember the explanations, I can still go to the shortcuts in most situations. So believe me when I say I completely sympathize with your problem here.

Whoa, whoa, whoa–explain this to me. What do you mean “you’re actually dividing that other 1?”

I’ll do it but we’re going back up to that 3 (groups) of 5 equals 15 step above. Are you game?

Yes, I’m game.

I think I have a better idea of what you’re saying. I don’t have time now to explain, but don’t spend any energy trying to elaborate, until get back to you.

(Going to start a new sub-thread)

Here’s my understanding: “1/4 * 1” is basically means “cutting up” 1 whole in four equal pieces and “pointing to” one of those pieces. (Multiplying a fraction is essentially dividing or “cutting up” something into smaller fragments, and choosing one or more of those fragments.)

When you say, “They are different 1s,” are you referring to the numerator (1) and the 1 being multiplied by 1/4? If so, I understand. What’s not entirely clear is why you’re pointing out this difference.

EditThought about this more: Basically, “1/4 * 1” is a conceptual expression. The numerator expresses that

partand the second 1 represents the whole. Ithinkthis helps.I want to throw another idea out there, though–namely, wholeness is something that can shift. For example, if we cut a pie into four equal pieces. One piece is a fraction of the whole. But if I want to share that piece with another person, that piece can be thought of as a whole. The 1/4 piece is not equal in quantity to the entire pie, but in can be designated as a whole. In this way, the fourth of a pie can be a fragment and a whole. My point is that wholeness has this “movable” (That’s not the right word, but I can’t think of it now) quality.

I wondered why the area of a triangle is 1/2 of the b*h. The 1/2 part is what puzzled me. But then I saw this explanation (the first one), and I think I understand now. The key point is that all triangles form half of a parallelogram. I wasn’t sure if this is true, but I guess it is.

A larger problem I’ve had is understanding, in a visual or concrete way, why multiplying length by width gives the area.

If you lay out six rows of marbles in columns of five marbles each, you have 6 x 5 marbles.

If you lay out six rows of Post-Its in columns of six Post-Its each, you have 6 x 5 Post-Its.

If each Post-It is exactly one inch square and the Post-Its are placed right next to each other, you have a larger box six one-inch Post-Its wide by five one-inch Post-Its high. The area of the box is 6 x 5 inches.

“Area” means how many of ______ fit into the bounded space. 30 Post-Its in the bounded space. Or 30 square inches in this case.

But this just pushes the problem to a lower level.

Let me give an explanation that I tried coming up with to illustrate the problem I’m having. Imagine me making a 1 inch column of tiny dots. Let’s also suppose that each dot really doesn’t have any length or width. And now let’s stretch out each dot to the right for an inch. That would be 1 square inch.

The post-it example is a lot clearer because we’re talking about blocks or post-its in rows and columns. But when we’re talking about a play area, for example, we don’t have blocks of rows and columns. Do you know what I mean?

No, if you stretch the tiny dot to the right one inch, you have a line. In fact, you’re pretty much saying the definitions of a point and line segment. It’s still one-dimensional. It’s just a linear inch, not a square inch.

You have to stretch the tiny dot to the right one inch and down one inch in order to make a square inch.

I do know what you mean and it’s a very abstract concept to think about. But you have to think two-dimensionally or you’ll never get anywhere. Area is about two dimensions. The actual “area” is the stuff of those two dimensions.

If mass makes sense to you (and I’m guessing it doesn’t!), area is just the two-dimensional version of the same concept. Which I am not saying makes it easy. The more you think about it the less sense it makes.

Wait–there’s a column of dots, and

eachof those dots would be stretched one inch to the right. The result would be a (black) square.The problem with the post-it example is that it just pushes the problem to a “lower level.” The dot (with no length or width) tries to get around that–but it’s not really a satisfying answer as well.

Yeah, that doesn’t help, but I appreciate the effort.

That wouldn’t be the result at all, accepting your presumption that each dot has no width or length. A point (in geometric terms) has no width or length. So stacking one dot on top of another on top of another on top of another is adding 0 length to 0 length to 0 length to 0 length, which is still 0 length.

A line (and a line segment) has one dimension but not both. Stretching the dot to the right one inch gives it width but not length (or the other way around depending on how you name things).

I thought I noted the limitations of my example–specifically, that the dots couldn’t be without width or length. The example was trying to avoid pushing the problem to a “lower level.” But it’s not really a great example.

I guess I don’t understand what lower level means. Do you mean less abstract and more concrete?

First, let me say I hate the way I’m explaining myself. OK, by pushing the problem to a lower level, using marbles or post-its doesn’t really get rid of the problem I’m having because those things have length and width. Think of those Russian dolls, where one doll just leads to another.

Question about integersI understand the concept of whole numbers, especially in relation to fractions. Indeed, I feel like one can’t understand whole numbers without understanding the concept of fractions, and vice-versa.

But what about integers, which, as I understand it, are positive and negative numbers and zero.

I’m partly wondering if integers also have to be understood by another type of number, like whole numbers and fractions.

In any event, Mitchell responded by asking how many numbers there are between 1-10, including 1 and 10.

10 would be my answer, or if we count fractions as well, then the answer could be a lot more (e.g, 3 1/2, 3 18, etc.)

Right. The answer to my question is “infinity.” In regular conversation, we use the word “number” incorrectly. I think it’s important to be aware of this when you’re talking about the other kinds of “numbers” in math.

Anyway. Whole numbers are 0, 1, 2, etc. Integers are these numbers plus the negative values of whole numbers: -1, -2, -3, etc.

So what’s a numeral?

Before I ask any other question, how does this relate to my original question about integers?