Kiss those Math Headaches GOODBYE!

Archive for the ‘Algebra’ Category

How to Remove Math Terms from Parentheses


How do you get math terms out of parentheses? And what happens to those terms when you remove the parentheses?

It seems like the process should be simple. But this issue often plagues students; they keep getting points off on tests, quizzes, homework assignments.  What’s the deal?

The deal is that there’s a specific process you need to follow when taking terms out of parentheses, and what you do hinges on whether there’s a positive sign (+) or a negative sign (–) in front of the parentheses.

But not to worry. This video on this page settles the question once and for all. Not only that, but the video provides a story-based approach that you can teach (if you’re an instructor) or learn (if you’re a student) and remember (no matter who you are). Why? Because stories are FUN and MEMORABLE.

So kick back and relax (yes, it’s math, but you have a right to relax) and let the video show you how this process is done.

And in customary style, I present practice problems (along with the answers, too) at the end of the video so you can be sure you understand what you believe you understand.

 

 

 

 

 

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Algebra Mistake #3: How to Work Out (x + y)^2 without Confusion


Ever thought this after you got back a math test … ?

“Why did I do that? I used a rule where it doesn’t apply!”

Yep, that’s exactly what we’re looking at in Algebra Mistake #3, a case of “overgeneralizing.”

The situation we’re dealing with involves over-generalizing everyone’s “favorite” property, the distributive property!

How’s that? Well, you’re supposed to use the distributive property when a number multiplies terms inside parentheses.

But sometimes students get a little bit — shall we say — “carried away” — and use the distributive property principle in other situations, too. The results are a tad bit comic, if you’re the teacher, but not so funny if you’re the student and you’ve made the mistake 19 times on a test with 20 problems.

Anyhow, after you watch the following video you shouldn’t have to worry about this again because we’ll get the two wires in your mind untangled so you never make this mistake again. So just relax, watch and learn.

And oh yes, don’t forget that we’ve provided some practice problems at the end of the video to help you make sure you’ve got the concept nailed down.

 

 

Algebra Mistake #2: How to Understand the Difference between A x A and 2 x A without Confusion


Now that you’ve gotten a taste for the benefits of analyzing algebraic mistakes, it’s time to explore a second common mistake. This one is so common that nearly every student commits it at least once on the road to algebra success.

As you watch the video, notice how by thinking hard about two expressions, we can think this mistake through to its very root, thus discovering the core difference between two similar-looking algebraic expressions.

And along the road, we’ll learn a general strategy for decoding the meaning of algebraic expressions. What I like about this strategy is that you can use it to understand the meaning of pretty much any algebraic expression, and you’ll see that it’s not a hard thing to do. In fact, it just involves using numbers in a nifty way.

Best of all, students usually find this approach interesting, convincing and even a bit fun. So here goes, Common Algebra Mistake #2 …

 

Algebra Mistake #1: How to Understand the Difference Between -1^2 and (-1)^2 without Confusion


Welcome, welcome, welcome to my series on COMMON ALGEBRA MISTAKES!

We’re going to have some fun spotting, analyzing, dissecting, exploring, explaining and fixing those COMMON ALGEBRA MISTAKES, the ones that drive students and teachers UP THE WALL!

I’ve had so much experience tutoring that I find these mistakes fascinating, and I intend to share my (ok, bizarre) fascination in this series of videos.

Also, be aware that I’m very much OPEN to suggestions from you folks on mistakes that you’d like me to explore. I highly value the experience and wisdom of you students and educators, and I want to do all I can to work with you to un-earth the mistakes of algebra, and bring them to the light of day so we can find ways to stay out of their way!

Here’s the first video on these mesmerizing mistakes. Could any mistake be more classic than this very one? I doubt it. But watch the video and form your own opinion …

 

 

 

FUNCTIONS: Notation Confusion & the Importance of Empathy


Working with one student on functions this morning, I was reminded of how much there is that students can fail to understand.

I was trying to explain to this student that the x-value is the input, and that the f(x) value is the output. But because of the repetition of x in both of these terms, he got confused.

Stories from My Tutoring Work

Tales from the Tutoring Experience

I finally solved the problem by telling my student to view the parentheses around the x in (x) as like the slot that takes the coin in a vending machine. It sort of looks like a slot, too, right? So what goes inside it must be the “in”put. Now he at least understands clearly that what goes in the ( ) slot is the input.

Staying with the vending machine analogy, I told him that the f(x) is what the function machine (like a vending machine) gives you after you put the x-value in the slot.

I did need to clarify that when you’re working out the value of inputs and outputs, you must insert the x-value twice: once inside the ( ) slot, and secondly on the other side of the equation, in where the ‘x’ stands.

Obviously this student has a lot of trouble with processing the visual symbols of math. But working with him reminds me of something important. It shows how much students can get confused by math concepts and math notation. I feel that it’s important for us educators to keep this in mind as we teach. There’s so much that we take for granted in our understanding of math. But for students who struggle with notation and with the visual aspect of math, notation can be confusing.

One thing I try to do when I work with students comes from something I saw in a Great Courses class by Bruce Edwards, an excellent teacher of higher math. Mr. Edwards likes to say things like, “Now this is next part is a little bit tricky …” Just by saying this, Mr. Edwards shows that he understands that not everyone will get the concept, and that, I believe, helps students relax.

Ever since I saw Mr. Edwards use this way of talking, I’ve been using it in my tutoring work, too. And I find that it helps students. It makes them feel like no one will think badly of them for not understanding, since I, the teacher, have acknowledged that the concept is “tricky.” As a result, students relax, and that helps them be more relaxed in taking in what you’re going to tell them. A nice thing to learn from a master teacher, and another lesson in the importance of the way in which we talk to students to help them learn. There’s so much more to being a good math teacher than just being thorough and clear. The affective aspects of communicating, such as showing empathy, are very important as well.

How to find the GCF of 3+ Numbers — FAST … no prime factorizing


Suppose you need to find the GCF of three or more numbers, and you’d really prefer to avoid prime factorizing. Is there a way? Sure there is … here’s how.

 

High-Octane Boost for Math

High-Octane Boost for Math Ed

Example:  Find the GCF for  18, 42 and 96

Step 1)  Write the numbers down from left to right, like this:

………. 18     42     96

[FYI, the periods: …. are there just to indent the numbers. They have no mathematical meaning.]

Step 2)  Find any number that goes into all three numbers. You don’t need to choose the largest such number. Suppose we use the number 2. Write that number to the left of the three numbers. Then divide all three numbers by 2 and write the results below the numbers like this:

2    |  18     42     96
……..  9     21     48

Step 3)  Find another number that goes into all three remaining numbers. It could be the same number. If it is, use that. If not, use any other number that goes into the remaining numbers. In this example, 3 goes into all of them. So write down the 3 to the left and once again show the results of dividing, like this:

2    |  18     42     96
3    |    9     21     48
……… 3      7      16

Step 4)  You’ll eventually reach a stage at which there’s no other number that goes into all of the remaining numbers. Once at that stage, just multiply the numbers in the far-left column, the numbers you pulled out. In this case, those are the numbers:  2 and 3. Just multiply those numbers together, and that’s the GCF. So in this example, the GCF is 2 x 3 = 6, and that’s all there is to it.

Now try this yourself by doing these problems. Answers are below.

a)   18, 45, 108
b)   48, 80, 112
c)   32, 72, 112
d)   24, 60, 84, 132
e)   28,  42, 70, 126, 154

Answers:
a)   GCF =  9
b)   GCF =  16
c)   GCF =  8
d)   GCF =  12
e)   GCF =  14

Pi Day in 5 Digits


Happy Pi Day … in 5 digits!

 

Thanksgiving Apple, Pecan, Cherry, Caramel, Pumpkin Spice And Ch

O.K., so we all know it’s Pi Day, and we all know it’s a special Pi Day because it’s not just 3.14, it’s 3.1415 since this is the year 2015. But if you really want to get technical about it, how about adding a 5th digit, the digit 9, for 3.14159 …

How to celebrate this very special moment?

First just ask yourself how would you find .9 or 9/10 of a 24-hour day?

Well, just make a proportion:  9/10 = x/24

That gives you that x = 21.6

And that means that at 21.6 hours into this day we would be celebrating the moment of Pi Day with 5 digits. And what is that time? 21.6 hours from 12 midnight would be 9:36 pm. Where I live, that’s just about half an hour from now. So I just thought that everyone might like to know about this special moment.