Tag: division

  • Chuck E. Cheese & Math

    Chuck E. Cheese & Math

    This is a feature article by “K8” Crowder. She is the Daughter of Bon, the publisher of MathFour.com.

    MawMaw & Pawpaw take my cousins and me to Chuck E. Cheese every month. We have a great time running around, playing games and riding rides. There’s not any math lessons there, per se. But Taica (my mom) always manages to get some math in there somewhere.

    Every so often Chuck E. Cheese himself will come out and do a dance with a bunch of kids. Sometimes I go dance too, and sometimes I just hang back and watch.

    Mawmaw says that other people “paid for” Chuck E. Cheese to come out and play. This means I shouldn’t muscle my way in and dance too. I’m not sure what that means. But it seems important to her.

    Grownups “pay for” things.

    Grownups get stuff from other people if they give them stuff. Taica says that’s what Mawmaw means when she says other people paid for Chuck E. Cheese.

    Someone else gave Chuck E. Cheese something so he would dance with their kids. Not with me.

    I got a Chuck E. Cheese t-shirt today – just like my cousin’s. Taica gave the man a bunch of pieces of paper and he gave me the shirt. Taica pointed out to me the sign on the shirts. She said it was math.

    This was it:

    See those numbers on the yellow sign at the bottom? They represent what one grownup has to give another grownup in order for me (or another kid) to score that shirt.

    Taica said that she gave 1000 tickets for the shirt. And then she said, “That means that each ticket is the same as one penny.”

    Tickets are the same as pennies? Really?

    I’ve seen pennies. I like to put them in my mouth. And they do not look like tickets.

    She rambled on about 9 dollars and 99 cents being mostly 10 dollars. And then if you divide 10 dollars by 1000 tickets, you get one penny per ticket.

    I know that dividing means sharing your cookies with your friend. In particular, sharing so that nobody gets mad because the other kid got more. And since dividing is math, cookies are math.

    I’m good with that.

    But I’m not sure how sharing cookies has anything to do with tickets, pennies and shirts.

    But in the end, I got my Chuck E. Cheese t-shirt.

    But I still don’t believe a ticket is the same as a penny!

    What do you think about this tickets and pennies thing? Do you like to give people stuff to get other stuff? Do you use tickets, pennies or something else?

    Write about it below in the comments.

    K8

    K8 is a full time kid who can count to 20 and only miss the number 15. Her favorite songs are “A-B-C-D” and “Had a very shiny nose.” Connect with her in the comments or on Twitter at @KateCrowder.

    Related articles

  • PEMDAS and a Stupid Arbitrary Rule

    PEMDAS and a Stupid Arbitrary Rule

    The order of operations includes two types of rules: those that are based on the way the operations work, and those that are arbitrary. My friend @harrisonalg from the Twitter chat #mathchat and I have been discussing this.

    You can explain the truly arbitrary elements of PEMDAS (the left to right of AS and MD) through an experiment. Allow students, independently, to do these two problems any way they want, ignoring any stupid arbitrary rule they might have previously memorized:

    1. 3 – 2 + 8 – 3 + 4
    2. 2 x 7 ÷ 2 x 6 ÷ 3

    The idea is that they will come up with many different answers:

    1. 3 – 2 + 8 – 3 + 4 could be any of -14, 0, 10 or others
    2. 2 x 7 ÷ 2 x 6 ÷ 3 might be 7/18, 7/2, 14, etc.

    If they were on a team building a bridge with these calculations, things wouldn’t work so well.

    Enter the Stupid Arbitrary Rule (SAR).

    Because we need to all come up with the same answer, we need a rule to follow. Really, it can be any stupid arbitrary rule (SAR). But we agreed, at some point in history, to all follow the “left to right” thing once we were down to addition & subtraction or multiplication & division.

    It’s important to note that kids didn’t get to be part of that agreement we made. Just like they don’t get to vote in elections.

    Is it fair? Probably not. They would probably do a better job of choosing leaders as well as determining the order of operations. But that’s the way things likes SARs work.

    PEMDAS doesn’t cover it all!

    To throw a few more kinks into it, check out how wikipedia explains some special cases of the order of operations. . Factorials aren’t covered in PEMDAS, neither are the order of exponents.

    Here’s another fun one: even calculators do things differently (and arbitrarily). Check out the way the the TI-92 and the TI-30XII handles exponents. Clearly you get to make up your own order of exponents!

    Let them play king: they create the rules!

    Let your students determine the order of operations for addition & subtraction and multiplication & division – just for the class. Make sure to write the rule down and tack it to the wall so everyone remembers what they decided. In your class, and your class only, they are to follow that rule.

    Remember the back of the book, as well as your answer keys or teacher edition textbook, will now be totally wrong. But it’s worth it.

    Doing it this way they might start to understand which pieces of PEMDAS are dependent upon the way the operations are constructed () and which pieces of PEMDAS are arbitrary (left to right).

    Once the class determines their special arbitrary rule, practice creating expressions whose result is some important number. Some options could be:

    • The age of one of the kids
    • Someone’s favorite number
    • Your age (be careful)
    • The age of some other grown-up they know
    • The last four digits of their phone number
    • The number part of their street address

    Create the expression using the Class Rule as well as the stupid arbitrary rule. Then translate and have a good laugh. I just did my age with my “Bon Rule.”

    Bon Rule: addition and subtraction go from left to right, and multiplication comes before division.

    My age is 120 ÷ 2 x 3.

    If I did this according to the SAR, I’d be dead. Or really famous.

    Your turn!

    Can you teach the order of operations like this with your kids? If you classroom school, how much trouble are you going to get in when you become the math teaching rebel?

    Share your experiences in the comments!

    Feature image by tim846 | Flickr.com | CC BY SA

    Related articles
  • Free Activity Packet – How Your Kids Can Ponder Numbers

    Free Activity Packet – How Your Kids Can Ponder Numbers

    Here’s a FREE Activity Packet to read and share with your children to get them thinking about how numbers got started.

    Somewhere in the past we recognized that we have these “digits” on the ends of our hands. Using these, we created numbers, adding, subtraction, multiplication, division and even fractions!

    Get it free here!

    When you get it, unzip it and you’ll have all this great stuff:

    • If You Give a Man Some Hands ebook (IfYouGiveAManSomeHandsByBonCrowderMathFourDotCom.pdf)
    • If You Give a Man Some Hands Illustrators Workbook (IfYouGiveAManSomeHands_IllustratorsWorkbook.pdf)
    • If You Give a Man Some Hands Supplemental Questions (IfYouGiveAManSomeHands_Questions.pdf)
    • A list of math resources for homeschooling and afterschooling parents (HomeschoolMathResources.pdf)
    • A reprint of the article 9 ½ Ways to Homeschool Math (WaysToHomeschoolMath.pdf)

    Have questions? Ask here or shoot me a note with the contact form.

    Related articles
  • Long Division with Sumology Number Tiles

    Long Division with Sumology Number Tiles

    A few articles back, I published a video called Long Division with Playing Cards.  I did it super-fast in response to a comment at  Living Math Forum.

    I’ve pondered this a bunch since then and decided I like the idea, but the playing cards are too cumbersome. I ran across a game called Sumology (from Simply Fun) at the Texas Home School Coalition Convention. The heavens parted and angels sang.

    Or at least my heart started beating and my head started spinning.

    So here’s the same teaching method, but with a little more pizzazz and a couple of free downloads:

    What do you think? Will your child like it? Please share your thoughts in the comments.

    Related articles
  • Long Division with Playing Cards

    Long Division with Playing Cards

    I have since shot another video showing this with number tiles instead of playing cards.

    In response to a post on the Living Math Forum, here’s a way to teach long division with playing cards.

    Whatcha think? Will it work? Let me know if it does!

    Related articles
  • The Order of Operations Explained: Multiplication and Division

    The Order of Operations Explained: Multiplication and Division

    This is the 5th in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.

    Last week’s article on the order of operations included a quick mention that division is the same as multiplication – but different. Now’s the time to explain that a bit.

    The only thing mathematicians like to do more than create, is destroy. That’s how we get imaginary numbers, dividing by zero and raising things to infinity.

    Once we create with multiplication, we want to know what happens when we destroy with the opposite of multiplication.

    Enter Division, stage right.

    Multiplication is created from the need to quickly add a bunch of numbers that are all the same. They each must be equal to make this work:

    6 + 6 + 6 + 6 + 6 is shortcutted to 5 X 6.

    Division is the breaking up into pieces that are all equal.

    Technically we can break 30 up into these 5 pieces: 4, 8, 7, 6 and 5. But “division” requires (or implies) that we are dividing equally. So 30 would have to be broken up into 5 equal pieces of 6 each.

    Of course this is in a purely mathematical world. When you get into a toddler world, things will be different.

    How division is the same as multiplication.

    We represents multiplication with little x‘s or dots or stars. Like this:

    3 X 5 or 3 • 5  or 3 * 5

    We represent division with this cute little symbol: ÷\div

    But we can write it in many more ways. Notice the cute symbol ÷\divlooks like a fraction with dots on the top and bottom. That’s not a coincidence. A fraction means division.

    7÷37 \div 3 is the same as 7×137\times\frac{1}{3}.

    Remember the poem:

    When dividing fractions
    Don’t bat an eye
    Just flip the last
    And multiply!

    Well, you can change this to:

    When dividing numbers
    Don’t bat an eye
    Just flip the last
    And multiply!

    And “flipping” the last just means taking the “assumed 1” that’s underneath it and putting it on the top.

    Now go forth and divide…

    So when you’re teaching the MD in PEMDAS, the order of operations, remember that D is the same as M. And if things get a little confusing, demand some parenthesis before doing the problem.

    Share your thoughts in the comments!

    Related articles
  • The Order of Operations Explained: Exponents, Multiplication and Addition

    The Order of Operations Explained: Exponents, Multiplication and Addition

    This is the 4th in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.

    The Order of Operations can be boiled down into three “real” operations.

    Parenthesis are merely a way to group things – they aren’t a real operation. So they doesn’t count as a real operation.

    Since division is really just multiplication turned upside down, we don’t need to include it separately, either. Likewise, subtraction is addition on its ear. So we throw him out, too.

    Now we have only three: exponents, multiplication and addition.

    Exponents are the shortcut for multiplication.

    In a previous article about remembering exponent rules, I recorded this video about exponents:

    You see that 3 x 3 x 3 x 3 is 34. The exponent is the shortcut for multiplication.

    Multiplication is the shortcut for addition.

    Likewise, when we get overwhelmed with adding up the same numbers over and over, like in the video, 4 + 4 + 4 + 4 + 4, we can use multiplication to shortcut it: 5 x 4.

    Follow the shortcut evolution.

    The shortcut evolution is like this

    1. Addition came first.
    2. Then we created multiplication to make addition easier.
    3. Then we created exponents to make multiplication easier.

    So when you do arithmetic, we should do the “recent” shortcuts first (exponents) and then the “older” shortcuts (multiplication) and then the “regular” arithmetic (addition). Remember that subtraction is addition and division is multiplication.

    And keep in mind that we need to watch the grouping or anything isolated with parenthesis, absolute value bars or in a fraction. The grouping/isolation tools always trump the other operation rules.

    What if your answer isn’t the same as the back of the book?

    Some textbooks actually have multiplication done before division. This means that the problem 9 ÷ 3 x 2 will show a different answer (3/2) in that textbook’s solution pages than what you would get following the OoO I’m describing here (6). These texts are rare, but I’ve seen them. So keep your eyes peeled.

    The Order of Operations is a set of rules that we’ve agreed on. Which means that as long as a textbook clearly defines their order that they will follow, they can do things like this.

    If you get confused as to which to do first, demand parenthesis. (Yes, you can do that.) Or put parenthesis into your child’s textbook to help them out.

    The problem sets should be there to enforce, not confuse.

    What do you think? Does this help or hinder the way you’ve always viewed the Order of Operations? Share your thoughts in the comments.

    Related articles

  • The Order of Operations Explained: Parenthesis

    The Order of Operations Explained: Parenthesis

    This is the 2nd in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.

    I mentioned in the introductory article for this series that “the order of operations isn’t best practice for expressions involving variables.”

    When you involve a variable, you can’t just “do” the arithmetic. Like in the case of 2(3x + 4)=11. You can’t add 3x and 4 to get a result before moving on. You have to use the distributive property.

    And even the distributive property won’t work sometimes – as in the case of absolute values shown below.

    Parenthesis mean isolation.

    When we say “parenthesis” in the Order of Operations (OoO for short), we mean anything that’s grouped together and isolated. This could mean with actual parenthesis. This could mean [square brackets] or {curly brackets}.

    Although grownups seem to know that square and curly brackets are the equivalent of parenthesis, children don’t. This has to be said out loud.

    “Parenthesis” in the OoO can also mean |absolute value bars|. This one’s not so clear.

    When you try to solve the equation 4|2x+3|=20, you have to start with dividing by 4 to isolate the absolute value chunk. There’s not a “rule” for distributing the 4.

    (Although, it would be worth it, and fun, to see if your children can come up with some rules for distributing within absolute value bars. This would be some real mathematics at work for them – experimenting and discovering.)

    Tops and bottoms of fractions are implied parenthesis.

    It’s also the case that the numerator (top) and denominator (bottom) of a fraction are isolated places. These fall under the OoO as parenthesis.

    Check out this older video I did. It shows how this works with fractions:

    Parenthesis are for deviation from the other rules.

    David Chandler of Math without Borders commented this in the previous article of this series:

    The rule is to do higher level operations first. Use parentheses whenever your intention is to deviate from this rule.

    If you can focus on this instead of a mnemonic device, you can get students to internalize what’s going on with the OoO. It’s important, however, to make sure they remember about other bracket shapes as well as isolation.

    Let us know your tips and thoughts on the P in PEMDAS!

    Related articles

  • The Order of Operations Explained: Intro and Mnemonics

    The Order of Operations Explained: Intro and Mnemonics

    The Order of Operations (OoO for short) is used everywhere in mathematics because it encompasses many of the foundational rules that we’ve agreed to follow.

    Alas, students have been given the cheap and dirty version of it for years. “Here, memorize this thing about your Dear Aunt Sally!” What the heck?!

    There are subtleties in the Order of Operations that every person over the age of seven should know.

    The series begins today.

    The order of operations is a set of rules – like the drivers’ handbook for math. If everyone follows the rules, we’ll all be safe. But if someone makes a bad turn, we could be looking at a crash.

    But the Order of Operations is only a set of rules for arithmetic! It isn’t even the best practice when it comes to expressions involving a variable like x. I’ll cover what I mean in this weekly series.

    Here are the proposed articles:

    1. Intro and mnemonics
    2. Parenthesis
    3. Exponents
    4. Exponents, Multiplication and Addition
    5. Multiplication and Division
    6. Addition, Subtraction and Conclusions
    7. Exponents of Negative Numbers
    8. Another Reason to Ban PEMDAS (aka parenthesis aren’t an operation)

    Mnemonics for PEMDAS

    Well, there’s one: PEMDAS (pronounced just like it looks). That’s what the cool kids in high school always said. It was the same kids who said “soh-cah-toa” – which I thought sounded really goofy.

    And then there’s “Please Excuse My Dear Aunt Sally.” And of course “Piranhas Eat Mostly Decayed Antelope Skin.”

    What’s your way to remember it?

  • Two Reasons to Memorize Math Facts

    Two Reasons to Memorize Math Facts

    I learned my math facts by “singing” them while looking at flashcards.

    Having these facts ingrained with chanting or singing isn’t a bad idea. It might not “feel right” because we’re so into experiential learning these days. But if a kid can’t immediately access and use things like 8 x 7 = 56, he’s going to be slower than if he can.

    And if he’s slower, he might get frustrated and start to think that he’s not good at math.

    Also, knowing these cheap and dirty math facts helps with confidence. Even if a kid’s struggling with other things in math, knowing that he has this one thing (the “facts”) will help out.

    I fight this battle often. Some people feel that math facts shouldn’t be memorized. But there’s so much value in it.

    How about you? Which side of the fence are you on?