Author: Bon Crowder

  • How Calculators Inhibit Learning the Distributive Property in Algebra

    How Calculators Inhibit Learning the Distributive Property in Algebra

    Do you wonder if your children should be using a calculator “at their age”? Are you a fan of calculators, but have friends who aren’t? Are your friends “into” calculators while you oppose them?

    I often hear people say that children 50 years ago understood math concepts more quickly. Although our parents weren’t taking classes called algebra in the 7th grade, they were doing algebra in the 7th grade.

    Algebra is arithmetic.

    There are two fundamental and rarely understood facts about algebra:

    1. Algebra is arithmetic with one or more numbers in disguise.
    2. Algebra has exactly the same  rules as arithmetic.

    Which means if you can do arithmetic you already know how to do algebra!

    Our parents or grandparents, 50 or even 30 years ago, weren’t using calculators. They had to apply all the rules of arithmetic to get the job done. Which means that they had to apply all the rules of algebra.

    Teaching them a class called “Algebra” was much easier because of this.

    What are the rules?

    The basic rules that non-calculator users must apply are the distributive property and the order of operations. The distributive property is the thing that calculator use eliminates.

    Children could get practice mentally multiplying things like 3 x 86 and do 3(80+6) = 240+18=268. With this practice, they are ready for 4x(3y+2z) = 12xy+8xz.

    If they never have to multiply 3 x 86 in their head, they never get the experience of the distributive property. Which means teaching them 4x(3y+2z) = 12xy+8xz will cause anxiety and frustration. They see it as “magic” or “something you made up just to confuse me.”

    Give them the tools they need.

    Refuse to let students have the calculator. Let them have the tool of the distributive property for algebra before you teach them “Algebra”. Give them the benefit our parents and grandparents had!

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  • How to Tell if Your Child Is a Top-Down Learner

    How to Tell if Your Child Is a Top-Down Learner

    Does your child struggle when you put him down in front of his math book? Are you frustrated in your attempts to get him to do math classwork or homework?

    Maybe he’s a top-down learner. If so, you might not know based on his current “regular” work. It will help in his education if you know he needs the big picture before the details – or the big theories before the steps.

    I remember learning to do derivatives when watching the foster kids that lived with us. I was eight. I’m a top-down learner. Here’s how to find out if your child is one too:

    How the “green beans” con works.

    My mom used to leave green beans open in a can on the table. We would walk by and eat them. If she put them on our plate, we would refuse them. So she got us to eat vegetables without asking us to, by just making them available.

    You can use the “green beans” con as a test.

    For math, put out the harder stuff. Find some books at Half Price Books or someplace cheap in your area. Open up the book. Sit and do some of the math yourself while mumbling aloud. Then walk away.

    If he sees his folks (or older sibs) working through those problems, he might be interested. Watch to see if he goes up to the book to check it out (steal a green bean). Be available to answer questions if he asks.

    If he can grasp some of  that “higher level” stuff, he’s probably be a top-down learner. He won’t want the building blocks until he sees the plans for the whole house. This could be the cause of some of the struggle and frustration – he’s been given the building blocks instead.

    Let him have the big stuff – start “allowing” him to do more of the advanced books. He’ll back up on his own to learn the “lesser” stuff so he can understand the big stuff better. You won’t have to force the work on him anymore.

    Share your experience with your top-down or bottom-up learning in the comments!

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  • Teaching Subtraction Using a Balance

    Teaching Subtraction Using a Balance

    I love finding nifty ways to use tools for teaching math. Especially tools that aren’t supposed to teach math. Or at least the math I’m trying to get it to teach.

    I have this very cool balance that I got from Discovery Toys that would normally be a science toy. But, alas, I’m a mathematician, Jim, not a doctor. So I’ve taken the fancy science toy and turned it into a way to teach subtraction.

    You can, of course, use it to teach addition and later I’ll do a post on using it to teach multiplication and division.

    If you have children who struggle with math concepts, teaching them with hands on bits (manipulatives) sometimes helps. Here’s how to teach subtraction using a balance:

    This nifty trick can be done with any balance as long as you have weights appropriately sized. Sometimes that’s not so easy to find. Order a colorful balance that’s similar to the Discovery Toys one in video here.

    Did it work? How did your children receive this method of learning arithmetic? Please share your experience with it in the comments!

  • 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?