39 thoughts on “Math Corner

  1. 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?

      1. 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?

      2. 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?

        1. 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.

  2. 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.

    More later.

  3. 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.

    1. 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. 🙂

  4. I’m not sure you want me to complicate things,…

    No, that’s cool.

    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?”

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

    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

    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

    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!

    1. 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.

    2. 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.

      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.

      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.

  6. 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.

  7. 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.)

    1. 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.

    2. -7-(+9)

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

      I’m quite sure I went in a direction that’s of interest only to me.

      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.

  8. 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.

    Super cool!

      1. 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…”

  9. 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.)

      1. 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

        1. 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.

      1. 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.

  10. 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?

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

      And wouldn’t that give me the cents/ounce–i.e,. 1 cent would get me X ounces.

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

      1. 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.

        1. “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.”

          1. 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.

          2. 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.

    2. 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?)

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