So easy a seventh grader could do it

Sorry for the dead air around here over the past few days. Things have been either crazy or lethargic…or both.

The funy part is, I had this post lying in my head, just waiting to spring itself on the world.

So…in the middle of my kids preparing for finals last week, I got to teach a skill that I love. Today’s tidbit is a review of that lesson.

Last fall, I was teaching an SAT prep class. One of the students was a seventh grader. I panicked. I knew I was going to have to teach him both FOIL and rational functions. He ended up dropping before we hit rational functions, but I successfully taught this child how to FOIL. Now it’s your turn!

FOIL is an acronym that stands for: First, Outer, Inner, Last. It’s the formula for multiplying polynomials.

Let’s try it out on the following problem.

(x + 3)(x + 2)

According to our acronym, we start by multiplying the first terms in both polynomials.

(x + 3)(x + 2) = x²

Now we’ll multiply the outer terms and put the result behind the result of multplying the first terms.

(x + 3)(x+ 2) = x² + 2x

Next up, the inner terms!

(x +3)(x + 2) = x² + 2x +3x

The last part of our acronym tells us to multiply the last terms of the polynomials together.

(x + 3)(x +2) = x² + 2x + 3x + 6

We’ve multplied everything, but we aren’t quite done yet. See those two middle terms? We can simplify the expression by combining these two terms. The sleek new equation looks like this:

x² + 5x + 6

And now we’re done!

The Distributive Property

Tonight’s tidbit hails from a week filled with finals or finals preparation for nearly all of my students. One of the skills I found myself reminding many of them of is the distributive property.

The distributive property says that you multiply each term of a polynomial by the same monomial and get the same answer you would have if you’d solved the polynomial first and then multiplied the result by the monomial.

That was pretty technical. Let’s look at it symbolically:

a(b + c) = ab + ac

Just to help it make even more sense, let’s throw in some numbers to see what happens on both sides of the equal sign:

3(2 +5) = 3*2 + 3*5

3(7) = 6 +15

21 = 21

Check it out! Both sides simplified to 21! Pretty neat, right? But I can already hear you asking where you would really use this, since it’s obvious from order of operations that we would just add the 2 and 5 before multiplying by the 3.

But what if we didn’t know the value of what was inside the parentheses?

12(x + 3) =96

We can’t just simplify the left side thanks to that variable. We can, however, distribute the 12 and then solve for x.

12x + 12*3 = 96 (Distribute the 12.)

12x + 36 = 96 (Multiply the 12 and 3.)

12x = 60 (Subtract 36 from both sides of the equation.)

x = 5 (Divide both sides by 12.)

When we distribute, it’s important to remember to distribute across all of the terms in the polynomial.

Correct: 12(x² + x + 2) = 12x² + 12x + 24 Incorrect: 3(2x +5) = 6x + 5

And Dead Bunny reminds us all to double check our work for minor errors. Distributing is one of most common places to make silly math mistakes.

Sometimes…it’s okay to be myself

To my great surprise, I was just named Teacher of the Year for 2006 at work.

I’ve been giggling (and hiding behind my hand) and blushing and saying, “Oh my god,” for over half an hour now. I can’t help it. It really was unexpected.

And it came on the heels of the directors telling everyone else to stop making my life hell. Which was also cool.

Eep! Oh my god! *giggles and blushes*

Two laws of exponents

Tonight’s tidbit, as usual, comes care of my students. One of my students was inducted into the sacred cult of the sticky. When a student starts working on the laws of exponents, I create a sticky note for them with all the main rules on it. You’ll have to wait until I figure out how to display complicated math equations on here to see them all, but for now enjoy these two.

• x^a * x^b = x^{a + b}

Let’s look at it with a and b replaced with 2 and 3. The new equation reads: x² * x³ = x^{2 + 3}, or x⁵.

Expanded, this equation basically says the following: (x * x) * (x * x * x) = x * x * x * x * x

We can visually see that the equation is true.

• (x^a)^b = x ^{a * b}

Again, we can substitute 2 and 3 for a and b. The equation now reads: (x²)³ = x ^{2 * 3}, or x⁶.

When we expand this equation, it looks like this: (x *x) * (x * x) * (x * x) = x * x * x * x * x * x

Again, we can visually see this one is true.

If you’re patiently waiting for the first bit of the number line article, I thank you for your continued patience. My computer had a minor meltdown and is still recovering from it. I did just get a hold of a demo of Illustrator. I can hardly wait to bring Dead Bunny to life!

Why does that work?

When I’m teaching my students, I try to not only give them the formulas and rules they need to get through an assignment, I also try to give them the reasoning behind the formula or the rule. They usually think I’m crazy for dumping useless information on them, but it’s amazing how many of them come back to me at some point later and say, “I couldn’t remember this rule, but then I remembered you said it worked because this, and I was able to solve the problem!”

You can memorize all day long, but if you ever accidentally confuse a rule or formula, you’re in trouble. Or what happens when you flat out forget? You sit there for minutes thinking, “Okay, I know it had to do with this and that, but I can’t remember how they went together.” When you look up the formula afterwards, you discover it had nothing to do with b at all. Instead, it required c.

(Some favorite examples of these are the laws of exponents, sine-cosine-tangent, the slope formula, and just about any equation for a line.)

Knowing the reasoning behind a rule can also be instrumental in translating word problems. Once you’ve figured out what information you already have and what you’re being asked to solve, you can generally fumble your way toward the right formula to use if you know why.

You can read more about problem solving in this great article!

Multiplying the same variable

The other night, I shared the amazing fact that x + x = 2x.

Tonight, I shall dazzle you with the rule every student should put on their flashcard right under that one.

x * x = x²

If you ever find yourself confused about which function gives which answer, just put in any number great than 0, 1, or 2. It’ll clear things up right away.

If you have a math concept that confuses you, just send it to deadbunnyed@gmail.com, and I’ll try to explain it in the Math Tidbits!

What is slope?

Sorry about not posting a math tidbit yesterday. I came home and fell asleep. (I was up late two nights ago helping a student with a research paper, and it caught up with me last night. That’s a story for another post.)

Today’s tidbit, again, care of a number of my students who are studying hard for their finals, is about slope.

When I say “slope”, the first thing that should come to your mind immediately is “rise over run”. Somewhere along the way, some math teacher should have drilled this little phrase into your head.

If you can recite this when you are asked what slope is, then you’re a step ahead of the game, but do you actually understand what “rise over run” means? Some of you may say, “Sure. It’s y over x.”

All right, that now gives us two possible definitions for slope, but it still really doesn’t tell us much about why we know this.

Slope is really the change in the y-coordinate divided by the change in the x-coordinate, or (y₂ – y₁) over (x₂ – x₁).

This is also know as the slope formula. We can take any two points, label one of them Point 1 and the other Point 2, and plug them into the formula to find out what the slope for that line is. If you are just going to graph the slope, it is useful to leave your answer in the fraction so you can easily see how many spaces to move up and how many spaces to move to the right as you’re creating the points that make up the line. If you are putting it into an equation for a line, then go ahead and simplify it.

Merging like terms

Today’s math tidbit, care of one of my students:

x + x = 2x

This fact seems to confuse a lot of people, but think about it this way. We may not know what x is equal to at the moment, but we do know that every single x on the left side of the equation will have the same value.

If each x will have the same value, then this is just a case of repeated addition. We add up how many times x occurs, and end up with the right side of the equation.

 

Starting off

I’ll be starting off the articles with a brief one on the number line. I got all of my notes laid out for it, and then did some homework to make sure I hadn’t forgotten anything.

There are some very savvy number line teaching tools out there, including one that will generate number line worksheets for you teachers out there.

My favorite, though, was this cute demo game from Harcourt. You get to be an engineer, and your lion friend tells you were to pick up your passengers. It’s pretty basic, but it definitely gets across the concept of being able to move back and forth on the number line to get where you need to be.

My own article should be available by Wednesday. Stay tuned!

Welcome to Dead Bunny’s blog

It may take me a bit to figure out how to use this place to my satisfaction, but have no fear I’ll get it.

For those of you unfamiliar with the Dead Bunny, please allow me to introduce this mascot. Eventually, I’ll save this to an about page so everyone can meet him.

Let’s get the preliminaries out of the way, Dead Bunny actually started out as a live bunny. I was trying to teach a student prepositions by explaining that a preposition is a word that can describe the relationship between a bunny and a box. (I was scarred in my youth by a video teaching prepositions this way.)

We were working through a worksheet, and I was trying to help her without giving her the answers. At one point, I had the bunny-shaped hand moving off the box-shaped hand. She guessed all manner of prepositions, none of them the right one. Finally, I made the bunny hand fall off the box hand and land on its side.

The student, a nine year old with a penchant for pink frilliness, brightly exclaimed, “The bunny falls off the box and dies!” She drew pictures of the unfortunate bunny on everything, and even took to calling me “Dead Bunny”. It was quite fun to have to explain that one, but it also made me the resident expert on teaching prepositions.

The original plan was to have Dead Bunny teach grammar and writing, but I’ve finally decided that Dead Bunny should actually be the mascot for all of my teaching products I’m planning out.

Dead Bunny’s first offering is going to be a math book designed to serve as a reference for students struggling with pre-algebra and algebra. I’m writing it as a series of articles that will be smoothed into a book. The plan is to post each article here, and then it will be your job to let me know what you think about it.

To put it simply, you’re going to help me make sure my book is actually helpful!

The first articles are already underway, so keep an eye here for them.