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Archive for the ‘Coordinate Plane’ Category

“Algebra Survival” Program, v. 2.0, has just arrived!


The Second Edition of both the Algebra Survival Guide and its companion Workbook are officially here!

Check out this video for a full run-down on the new books, and see how — for a limited time — you can get them for a great discount at the Singing Turtle website.

 

Here’s the PDF with sample pages from the books: SAMPLER ASG2, ASW2.

And here’s the website where you can check out the books more fully and purchase the books.

 

 

 

 

 

 

 

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How to Convert a Linear Equation from Standard Form to Slope-Intercept Form


Suppose you’re given a linear equation in standard form and you need to convert it to slope-intercept form. You’ll be amazed how fast you can do this, if you know the “trick” I’m showing you here.

High-Octane Boost for Math

High-Octane Boost for Math Ed

First, let’s review the key info from my post: How to Transform from Standard Form to Slope-Intercept Form.

That post shows how to pull out the the slope and y-intercept from a linear equation in standard form.

Remember that standard form is Ax + By = C, where A, B, and C are constants (numbers).

Given the equation in standard form, take note of the  values of A, B, and C.
For example, in the equation, – 12x + 3y = – 9,   A = – 12, B = 3, and C = – 9

Then, based on the info in yesterday’s post, we get the slope by making the fraction:  – A/B.

And we get the y-intercept by making the fraction:  C/B

New info for today: once you have the slope and y-intercept, just plug them in for m and b in the general slope-intercept equation:  y = mx + b

Here’s the whole process, demonstrated for two examples.

Ex. 1:  Given, 8x + 2y = 12, A = 8 B = 2, C = 12.
So the slope = – A/B = – 8/2 = – 4. y-intercept =  12/2 = 6
So the slope-intercept form is this:  y = – 4x + 6

Ex. 2:  Given, – 5x + 3y = – 9, A = – 5, B = 3, C = – 9.
So the slope = – A/B = 5/3,  y-intercept =  – 9/3 = – 3
So the slope-intercept form is this:  y = 5/3x  – 3

Now “give it a roll.” Once you get the hang of this, try the process without writing down a single thing. You might get a pleasant jolt of power when you see that you can do this conversion in your head.

Conversion Problems (Answers at bottom of post)

1)   – 4x + 2y  =  14

2)    20x – 5y  =  – 15

3)  – 21x – 7x  =  35

4)  – 18x  + 6y  =  – 21

5)    17x + 11y  =  22

6)    – 7x + 11y  =  – 44

7)    36x – 13y  =  – 52

8)  – 8x  + 5y  =  – 17

Answers

1)   y  =  2x + 7

2)   y  =  4x + 3

3)  y  =  – 3x – 5

4)  y  =  3x – 7/2

5)   y  =  – 17/11x + 2

6)    y  =  7/11x – 4

7)    y  =  36/13x  +  4

8)  y  =  8/5x –  17/5

How to Transform from Standard Form to Slope-Intercept Form


Is there any point to doing something the long way when you can just as well do it in a shorter, much more efficient way? I say, Heck no! We can do things in the “Triple-F” way:  Fast, Fun & Friendly, and with deep understanding, to boot.

High-Octane Boost for Math

High-Octane Boost for Math Ed

So in that spirit, today I’ll get us started on quickly and effortlessly converting a linear equation from what’s called standard form (Ax + By = C) to what we know as the good-old slope-intercept form (y = mx + b).

To better grasp standard form, let’s replace its mysterious A, B, and C with actual numbers:  4 for A, 2 for B and 8 for C. That gives us the more typical looking equation of an actual line:  4x + 2y = 8. Do you recall seeing this kind of equation in your algebra text and class? Sure you do. You get this kind of equation in the chapter(s) on the coordinate plane and in other spots, too.

Now usually when books teach us how to convert from this “standard” form to slope-intercept form, they tell us to solve the equation for y. Of course that works, but it takes too darn long.

To understand the quicker way, let’s have a little fun with the standard form of the equation: Ax + By = C

We’re going to start with this standard form and solve that for y. And as you’ll see, we’ll learn some useful things from the result.

To kick things off, we start with Ax + By = C, and we subtract the Ax term from both sides. That leaves us with this equation:

By = – Ax + C

Now take this new equation and divide both sides by B. That gives us this little gem of an equation:

y = (– A/B) x + C/B

I’m going to call this the magic equation both to give us a way to refer to it and to show us what’s so useful about it.

The big insight is that this magic equation is actually, believe-it-or-not, in slope-intercept form; we just need to SEE it that way. Here’s how.

In slope-intercept form (y = mx + b), notice that the y variable is all by its lonesome on the left side. Do we have that in the magic equation? Yes we do. So … CHECK!

In slope-intercept form, there’s a value called m (aka, the slope) that is multiplying the x variable. Do we have something in the magic equation that’s multiplying the x variable? Why yes, and it happens to be
(–A/B). So do we have the slope showing in the magic equation? Yes, the slope is:  (–A/B). So … CHECK!

Finally, in slope-intercept form, there’s a constant (i.e., a number term, not a variable term) that appears after the mx term. So do we have a constant after the mx term in the magic equation? Yes, indeed. We have C/B. Note that in any actual linear equation, B and C will be actual numbers, not variables. So the value you get when you divide B by C (the quotient B/C), also must be a real number, just as surely as the real numbers 8 and 2 gives us the real number 4 when we divide 8 by 2.

So, to address the final question, do we have a b-value in the magic equation? Yes, it’s (C/B). C/B is the y-intercept, the real number we call b in slope-intercept form. So once again … CHECK.

So all in all, do we now have the equation in slope-intercept form? Yes, indeed. You just need to realize that
(–A/B) is the slope, and (C/B) is the y-intercept.

In my next post I’ll show you how you use these results to quickly transform the equation from standard form to slope-intercept form. It will be amazing.

Reader Input on Slope Post


A longtime reader of Turtle Talk, Jeff LeMieux, of Oak Harbor, WA, sent in a suggestion based on today’s post on positive and negative slope. Jeff found a way to help students remember not only positive and negative slope, but also the infinite slope of vertical lines, and the 0 slope of horizontal lines … all using the letter “N.”

This is clearly a situation where the picture speaks more loudly than words, so I’ll just let Jeff’s submitted picture do the talking. By the way, to see this image even better, just double click it!

slopeclues

Slope Memory Trick

Thanks for putting this together and sharing it, Jeff!

Remember the Difference in LOOK between Positive and Negative Slope


Some ideas just slap you in the face.

I got slapped this morning as I was flying home from LA to Albuquerque. Those little cocktail napkins they hand out with “beverage service” often give me the urge to write. So this morning, nerdily enough, as I sipped my orange juice at 30,000 feet above the Salton Sea, I worked on figuring out a better way to help students grasp the difference in look between positive and negative slope.

That’s when I got “slapped.”

First, you must realize that I use the three-letter abbreviations of POS and NEG for positive and negative. Do some of you use these as well? I mention this because those abbreviations hold the key. You have to use the first letter of the NEG abbreviation and the last letter of the POS abbreviation.

Let’s start with NEG.

The first letter of NEG is, of course, “N.” But look what I noticed …

Visual Clue for Negative Slope

Visual Clue for Negative Slope

The trick for POS is a tad more complicated. But I’m hopeful it will work.

Visual Clue for Positive Slope

Visual Clue for Positive Slope

So what do you think? Will this work for your students?

If you test it out, please let me know what you find. I’m interested to know. Thanks!

Algebra Survival e-Workbook arrives TODAY!!


The “Algebra Survival” Program goes totally electronic!

Singing Turtle Press is delighted to announce that the companion Workbook for the Algebra Survival Guide is now available in eBook format.

Algebra Survival Workbook - electronic version

e-Version of ASG Workbook ARRIVES!

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Origins of the Coordinate Plane


A fly …

Who would think that a mere fly could play a major role in the history of human thought?

But when it comes to the development of Algebra, that’s the story. I’ll explain how this works just a bit later in this blog. But it is all related to what is happening now in algebra classes all around the world.

For it’s spring, that time of year again when we get out the graph paper and the ruler. Kids are working on the Cartesian coordinate plane.

One about I like about the coordinate plane is that there’s an interesting story about how it was discovered, or should I say, invented. [Hard to know the right word for an intellectual Invention like the coordinate plane.]
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