Author: Bon Crowder

  • What’s the Range of a Function?

    What’s the Range of a Function?

    You’ve taught what a function is. And the kids are starting to understand what the domain is all about.

    But then they ask, “What’s the point in the range?”

    As I wrote in a previous post, a function is a question with only one answer to a valid question. The domain is the set of all valid questions. The range of a function is the set of all answers you can get.

    Simple? Sounds like it – but kids the world over still struggle with the question, what’s the point of the range?

    To be or not to be a function.

    Why is it important to know all the answers of an equation? It has to do with the equation being or not being a function.

    If you have an equation like

    y=xy=\sqrt{x}

    you have more than one answer per question.

    Here are some valid questions associated with this equation:

    1. What is the square root of a number, specifically the number 1?
    2. What is the square root of a number, specifically the number 1.69?
    3. What is the square root of a number, specifically the number 4?
    4. What is the square root of a number, specifically the number 9?

    The answers to these questions are:

    1. 1 or -1
    2. 1.3 or -1.3
    3. 2 or -2
    4. 3 or -3

    Notice that there is not “only one” answer to each question. So this equation isn’t a function!

    But that’s no fun at all!

    You can force an equation to be a function by limiting the answers.

    By limiting the answers (AKA limiting the range of a function) you can force an equation to be a function. So if we write

    y=+xy= + \sqrt{x}

    We have just limited the range of answers to be only the positive square roots of numbers.

    The practical application for kids is the graphing.

    In this image above it’s the blue curve:

    You can see that we get only  the “upper half” of the curve. If you look at “squishing” a function (like the garbage compactor in the movie Star Wars) you can see the range of a function (all y-values) becomes the vertical line:

    The line starts at zero and goes up forever. (In the video it stops, but that’s only because I have a hard time displaying forever on a computer screen.)

    The handy thing about knowing the range of a function before you graph is that you know how much space on the paper you need – or how small to make your units!

    Does this help? Share your range of experiences with this in the comments! (And pardon the very bad pun.)

  • Is Zero Positive or Negative?

    Is Zero Positive or Negative?

    When we teach math at home, we start with the counting numbers. Then we move to either the homeschool or classroom setting and build on those with negatives. And zero is sometimes left behind.

    Zero seems to be an understood concept because it’s so much a part of our lives, as grownups. But zero is a very unusual bug.

    Is Zero Positive?

    We think of the numbers as either positive or negative.

    But actually, all real numbers can be divided into three categories: positive, negative and zero.

    This is one of the reasons zero can be forgotten in all this mix. With so many other numbers to be positive and negative, zero’s truly in a class by himself.

    Notice I wrote “All real numbers can be divided into three categories.” Imaginary and complex numbers can’t be classified this way. You can order imaginary and complex numbers, but you have to do it without labeling them as positive or negative.

    Why can’t we just say zero is positive?

    I guess we could. But you might not want to when you see this…

    In math, sometimes, we do things because if we didn’t, they’d mess everything else up.

    This is one of those situations.

    So let’s say 0 is positive, just for the minute.

    Take the rule, “When multiplying, if the signs are different, the result is negative.” So then (positive) zero times negative 3 is (negative) zero. Like this:

    0 x -3 = +0 x -3 = -0

    But we determined that zero was positive!

    So this messes things up. Therefore, zero needs to be in a class by himself.

    But zero feels positive and negative, sometimes!

    Indeed if a kid doesn’t owe his sister anything, that’s a positive feeling.

    If you owe someone $3, then you can say you have negative 3 dollars. So if you owe them nothing, don’t you have a negative zero dollars?

    This is more of a direction than a positive or negative sign. The “vector” (arrow in a direction) that has length zero can technically be pointing in any direction. It doesn’t matter. And that’s what’s going on when you think of zero as positive or negative. You think of owing (money going in a direction) nothing.

    But remember, that’s only thinking of zero as positive or negative. He really isn’t either.

    He’s just zero.

    Check out the classic Schoolhouse Rock Multiplication Rock video and let me know what you think of all this zero talk.

  • Multiplication Tricks Roundup

    Multiplication Tricks Roundup

    Should kids know their multiplication facts?

    When teaching math, whether in the home or in the classroom, having the multiplication facts memorized speeds things up. If you allow a calculator, you’re still reducing the speed of doing other math calculations or thinking.

    But is there some way to memorize these and make them fun?

    I scrounged the web and found some resources for multiplication tricks.

    Poems as Multiplication Tricks

    Inspired by @Leighbra at Hearth to Heart, I found a list of math poems that can help kiddos (and grownups) remember some of the facts. My favorite: I ate and ate and got sick on the floor. Eight times eight is 64.

    Seven

    This is an impressive list of various ways to multiply by seven. Tricks to Multiply by Seven

    Nine

    This one shows a way to multiply 9s without using your fingers. Which means you don’t have to waste time putting down and picking back up your pencil! Tricks to Multiply by 9

    Using Fingers

    This article  gives three tricks to multiply using your fingers. Not my favorite, only because I hate to put down my pencil. But if you are doing purely mental multiplication, this article’s a great resource! Three finger tricks for multiplying

    Multiplication Tricks for Other Numbers

    Here’s a couple more that give multiplication tricks for many numbers:

    The one I made up

    I created this while in class one day. I have a hard time remembering the difference between 7 x 8 and 9 x 6. So I thought about it and saw that 56 = 7 x 8, which is 5, 6, 7, 8 in order!

    Do you have one? Please share with us!

  • What It Means to Be Gifted in Math

    What It Means to Be Gifted in Math

    Galois died at age 19 but produced some of the worlds most amazing mathematics! Image via Wikipedia

    Do you ever wonder if your child is gifted in math?

    We’ve been having a discussion on gifted children at the Living Math Forum. We’ve discussed a few definitions of gifted.

    Here’s my definition (from the thread):

    All parents should treat their children like tiny bundles of brilliance. And indeed all children ARE gifted. Right out of the womb. It isn’t until the grownups get to them and try to teach them “reality” that they lose that ability. Want a gifted child? Treat them as gifted from the get-go.

    But that wasn’t the end of the story. Lots of people have responded.

    The “official” definition of gifted.

    If you’re a classroom teacher or parent, you know about that. You have to differentiate between who is gifted and who isn’t. This ensures the appropriate children get placed in the right learning environment.

    A child is “gifted” if the local testing center says he is.

    Some kids have “different brain-wiring.”

    Another definition that was discussed was of a neurological nature. Many kids just “get” math while some kids struggle.

    But that doesn’t mean the strugglers can’t do math. It merely means that they will likely be brilliant writers, politicians and historians, not mathematicians.

    This definition makes my thought of “every child is gifted in math” fly out the window. Indeed every child is “special” and everyone thinks in their own ways. But some people are wired in a way that lets them “see” things more easily than the rest of us.

    Math intuitives are one of these groups. I’ve recently encountered a couple of math intuitives – people who will “see” an answer to a math problem without having to do any of the work. One of these even can do Sudoku this way (to the frustration of her father).

    Others in this group are true child prodigies. I’m not sure if this is merely an intuitive who is nurtured or actually different. But there are children that amaze scientists and mathematicians.

    Every child is still gifted!

    Regardless of how you look at it, we all do math. Like Daughter’s pediatrician told us: “We’re all potty-trained.” Indeed – we are all potty-trained and we all do math. On some level. It’s natural. It’s intuitive.

    And I still hold that if the grownups stay out of the way, the kids will do all the math their brains are built to do!

    What’s your definition of gifted? Are your children/students gifted?

  • What’s the Domain, Why You Need It and How You Get It

    What’s the Domain, Why You Need It and How You Get It

    As you teach domain and range, do you get the question, “Why are we doing this?”

    No doubt the question, “When am I ever going to use this?” comes up too, right?

    I’ve asked myself that question my whole teaching and tutoring life. And now’s the time for an answer.

    A function is really a question.

    As I wrote in a previous post, a function is a question with only one answer to a valid question.

    When I write:

    y = 3x + 2 where x = 4

    I mean: “What is three times a number (that number is four), plus two?”

    The domain is all the possible questions:

    • What is three times a number (that number is five,) plus two?
    • What is three times a number (that number is six,) plus two?
    • What is three times a number (that number is seven,) plus two?
    • What is three times a number (that number is eight,) plus two?
    • <how long will I have to do this – Egad!>

    Not only do the questions go on forever, but they also have all the fractions and decimals in between. And all the negative versions of those numbers (and zero).

    So, what’s the point, you might ask. Looks like the questions go on forever and you can just pick any number.

    The domain might not include all the numbers.

    The two sticky points for the definition of “function” are bolded:

    A function is a question with only one answer to a valid question.

    The “valid question” part is where the domain comes in.

    The numbers that make  “valid” – meaning we actually can get some answer – are the numbers that aren’t negative.

    Many functions have “all real numbers” as a domain. There are no limits on the things you can put in, other than numbers that aren’t imaginary or alligators.

    For the most part, there are only two places where you have to be careful of limited domains. Those are

    • Numbers that cause a zero to turn up in the denominator
    • Numbers that cause negatives to turn up in square roots.

    Here are two videos tackling each:

    What do you think? Is this easier to teach when you consider “analyzing” the function rather than “solving” it? Share your thoughts and tips in the comments!

  • What’s a Function?

    What’s a Function?

    Other than being the most feared f-word in math teaching, a “function” is a question with only one answer.

    Take the question: “How tall are you?”

    We can change this to: “What’s the height of you?”

    And if we wanted to compare your height to other people’s heights we can ask: “What’s the height of <insert person’s name here>?” This is the question template – the formula.

    You can answer this question in inches, feet or cm, but the value of the answer remains unique, based on the person.

    And that last little piece of the sentence is what makes the difference, based on the person.

    The question changes with this little change. <cue music> This is the variable in the equation.

    And we say, “Height is a function of the person.”

    So where’s the fear come in?

    As always, the notation is the kicker when it comes to teaching math.

    Let’s change the question a little.

    What’s the height of Enrique tomorrow if he grows three inches tonight?

    Rather contrived, but work with me…

    The question template is

    What’s the height of <insert person’s name here> if he/she grows three inches tonight?

    Which becomes

    <height> = <height now> + 3

    Or

    H = N + 3

    Egad!

    And we haven’t even started with the f(x) stuff!

    What’s this “domain” thing about?

    I wrote the first sentence of this post a little too hastily. A function has only one answer if there’s a valid question.

    If you ask, “How tall is love?” someone will laugh at you. Or think you’re from California.

    Our question template included some specifics that you don’t normally get:

    What’s the height of <insert person’s name here>?

    If we instead ask, “What’s the height of x?” we would then have to ask: what kinds of things can we put in for x? Can we put concepts, like love? Or just objects? The kinds of things that you can put in for x is called the domain.

    For our question, we would need to specify that x is a person.

    What do you think? How does this feel when explaining it to your kids?

  • Counting with Ordinal Numbers

    Counting with Ordinal Numbers

    In the last post I discussed cardinal and ordinal numbers. I gave definitions but not much in the way of examples.

    Here’s an example of using ordinal numbers to count up to arrive at the final cardinal number. Oh – and I’m using the fun little critters from Discovery Toys!

    Whatcha think? Does it make you want to count some bugs?

    What other ways can you use to teach counting?

  • An Observation of Cardinals and Ordinals at the Playground

    An Observation of Cardinals and Ordinals at the Playground

    We were hanging at the playground on Mother’s day. Daughter had taken up with a cute little girl – a good playmate for the teeter-totter. I watched Daughter’s new friend do something interesting. She was “counting” some gumballs that had fallen off the sweetgum tree.

    Here’s what she said: ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT

    I looked and there were ten gumballs. I watched her more and saw that while she was reciting the terms and pointing at the gumball pile, she wasn’t associating each number word with the next gumball in the pile.

    Developmentally, I’m not sure if this is okay or not. I wonder.

    Mathematically, I know that there’s still a disconnect with this girl and what numbers mean to her.

    There are two types of numbers.

    In the realm of counting numbers, there are two types. Cardinals and ordinals. A cardinal number tells us how many we have. Like this:

    And ordinal number tells us the order:

    When we count, we use both types of numbers.

    The group of gumballs has the cardinal number of 10. But to arrive at that cardinal number, to determine how many there are, the friend had to “order” them. She was trying to point to each one and assign it a position. “YOU, I label 1st, you are 2nd … and since you are 10th, I know I have 10 gumballs.”

    We grownups take advantage of this procedure. Indeed most people don’t know what cardinals and ordinals are.

    But knowing this when you teach counting is quite helpful. At home you can do something similar to the potatoes in the pictures above. At school, you can buy some great posters (I found one at Teacher’s Heaven last night) that demonstrate this.

    Will you change the way you think and talk about numbers?

  • How to Train the Brain to Understand the Transitive Property

    How to Train the Brain to Understand the Transitive Property

    Remember the ol’ “if A equals B and B equals C, then A equals C” deal? At parties it’s a great line to drop. In math, it’s officially called … cue music…

    The Transitive Property

    Saying it is fun, teaching it is curious, learning it can be weird.

    Grownups think it’s intuitive. But to a kid, it isn’t. It takes experience and experimentation to learn all the bits that we think are “common sense.”

    The transitive property is really thinking things through. Starting from one place and moving along through another and then arriving at a third place.

    There are many ways to help kids with this learning. Word problems simulate thinking stepping stones. But they can be rather stressful. If you do it through play, you reduce the stress that they face and give them skills they need to tackle advanced thinking, forever.

    This video shows a nifty “toy” from Discovery Toys that can get kiddos using those brain stepping stones.

    Notice the flow is

    1. Choose the number tile with the question number.
    2. Read and answer the question.
    3. Correspond the answer to the letter in the answer box.
    4. Put the number tile with the question number in the corresponding letter box.

    Thinking through from question number to answer letter while avoiding the pitfalls is the challenge.

    Have you played with these? How do you train your kids’ brains for the transitive property?

  • Some Fun Ways to Teach Counterexamples

    Some Fun Ways to Teach Counterexamples

    This post is inspired by a discussion with Betty Ann on the www.Teachers.net math chatboard. She writes:

    I’ve got a high school student doing a very basic geometry course. She’s having trouble with the concept of a counterexample. I’m writing a worksheet on counterexamples and would love to have some more simple conjectures for her to work with.

    A counterexample is a special kind of example that disproves a statement. We start using these in Geometry because that’s the first course that really teaches proving things.

    Counterexamples are an essential part of logic.

    They don’t really need to be associated with math (or even philosophy) to be applicable. Which is the cool thing about them.

    Suppose someone says, “I always get to school on time.” It only takes one day when he isn’t on time at school to negate this statement. That one day would be considered the counterexample.

    We do this all the time and never use the fancy math term “counterexample.” So when we teach it, it’s helpful to tap into these everyday uses.

    Counterexamples are everywhere.

    Here are some statements for which students can come up with easy counterexamples.

    In the house:

    • Any four legged piece of furniture is a table.
    • If something has a knob on it, it’s a faucet.
    • Everything in the house with hands is a clock.
    • If a living being has eyes, it’s a human.

    In the grocery store:

    • Everything that costs $2.99 is a gallon of milk.
    • Everything that’s hot is fried chicken.
    • If something is white, then it is mayonaise.

    In the classroom:

    • If it’s a book, it has words. (Make sure there are blank journal books around.)
    • All books teach arithmetic.
    • Anything on the wall is a whiteboard.
    • If it’s full of pencils, then it’s a coffee mug.

    You can make your own statements for counterexamples.

    Choose a noun. Notice a feature about it. Then put it together using this MadLibs format:

    • Everything that has <feature> is a <noun>.
    • All <plural noun> have <feature>.
    • If it <has this feature>, then it’s a <noun>.

    You can also reverse them like this:

    • Every <noun> has <feature>.

    For instance:

    • Every cow is brown.
    • Every lightbulb is 60 watt.
    • Every hammer has a wooden handle.

    Which counterexamples or counterexample building method do you use?