My students are having some trouble with changing slope y-intercept form of a line to standard form and vice versa. You must first understand solving equations before this will make sense. Having a good grasp of variables on both sides of the equation will help since it involves moving a variable term from one side of the equation to the other.
Here is an example:
Rewrite y = 2x - 5 (slope y-intercept form) as standard form.
Move the 2x to the left side of the equation with the y. -2x + y = -5
That is all that is required for this one. It is now in standard form (Ax + By = C).
Another part of the assignment that we worked on required that the standard form be written with integers (no fractions).
Rewrite 3/5y = 2/3x + 2 in standard form using integers.
Move the x term to the left side of the equation. -2/3x + 3/5y = 2
Now to get rid of the fractions, you will multiply both sides of the equation! by 15 (the least common multiple of 3 and 5). You could multiply by any multiple of 3 and 5 but generally we use the least common multiple. As long as this multiplication is done to BOTH sides, the equation will stay balanced. Hint: Remember that you must use the distributive property on the left side of the equation.
15[-2/3x + 3/5y] = 2(15)
-30/3x + 45/5y = 30
-10x + 9y = 30 (standard form using integers)
find the y intercept
Tuesday, August 31, 2010
Remainder Theorem and The Rational Roots Theorem
Today in class, we went over one more time how to do synthetic division and how to find the roots of a function using the Remainder Theorem.
After that, we learned how to find the missing coefficient in a polynomial if we already know what the remainder is. Here's how you would do this:
Let's say you have the question:
The first thing you want to do is write it out in a way that is easy to understand:
Then you plug the root of the denominator into the function:
Since we already know the remainder we can rewrite it this way:
Now all we do is isolate K with a little algebra, and solve it:
And there you have it!
Near the end of the class we managed to quickly learn the Rational Roots Theorem. This theorem allow! s us to find any rational roots of a polynomial function. Here's an example:
So you're given the equation:
The first thing to do is the find all the possible positive and negative factors of the constant term:
Now we find all the positive factors of the leading coefficient:
We then list all the possible rational roots, eliminating any duplicates:
We can then test out these roots by using synthetic division and the factor theorem to turn the function into a quadratic (Remember: If the remainder is 0, then it is a root):
This then gives you:
![]()
Now you just factor the equation and find the roots:
And there you have it! That's about all we did for today, tomorrow's scribe will be...Niwatori-san
factor theorem calculator
After that, we learned how to find the missing coefficient in a polynomial if we already know what the remainder is. Here's how you would do this:
Let's say you have the question:
The first thing you want to do is write it out in a way that is easy to understand:
Then you plug the root of the denominator into the function:
Since we already know the remainder we can rewrite it this way:
Now all we do is isolate K with a little algebra, and solve it:
And there you have it!
Near the end of the class we managed to quickly learn the Rational Roots Theorem. This theorem allow! s us to find any rational roots of a polynomial function. Here's an example:
So you're given the equation:
The first thing to do is the find all the possible positive and negative factors of the constant term:
Now we find all the positive factors of the leading coefficient:
We then list all the possible rational roots, eliminating any duplicates:
We can then test out these roots by using synthetic division and the factor theorem to turn the function into a quadratic (Remember: If the remainder is 0, then it is a root):
This then gives you:
Now you just factor the equation and find the roots:
And there you have it! That's about all we did for today, tomorrow's scribe will be...Niwatori-san
factor theorem calculator
Canvas size and proportion
Franz Xavier Winterhalter 1805-1873 Madame Rimsky Korsikov
I am illustrating tonight's posts with the art of Franz Winterhalter, who was a fashionable portrait painter to the royalty of 19th century Europe. He has been derided as superficial and overly flattering to his sitters, but I have always enjoyed his technical virtuosity. He paints flawless textures and I like his color. I can't claim he has anything to do within tonight's subject but I like to show art in nearly eve! ry post.The images were, of course, provided by our friends over at art renewal .org.
I am going to write tonight about selecting canvas sizes on which to paint .
I suggest that you paint only about six different sizes and stick to stock sizes when you paint. The advantage of stock or common sizes is that you don't necessarily have to have all of your frames custom made, but can instead buy them off the rack. Here are some of the most common stock sizes for frames. Artists specify frames using the height first by the width, they would say for example, 24 by 36. A 36 by 24 is a vertical. My friend from Maine, painter Scott Moore insists "if God had meant for us to paint verticals he would have placed one of our eyes above the other".
Here are the smaller sizes;
- 5 x 7
- 8 x1 0
- 9 x 12
- 1 1x 14
- 12 x 16
- 16 x 20
- 18 x2 4
- 20 x2 4
- 24 x 30
- 24 x 36
- 30 x! 40
- 14 x 18
- 22 x 28, Sargents usual non-portrait size
- 20 x 30
- 30 x 36
- Sometimes artists use what are called double squares such as 12 x 24, or 24 x 48, etc. There isnt really a standard size double square, but they are nice for panoramic pictures.
There's another Winterhalter, isn't that lovely? Shes a countess, looks real high maintenance to me.Artists I know often paint these sizes but you can't generally expect to find a premade frame for these sizes. You will save a lot of headaches by limiting yourself to six sizes. Having interchangeable framing is real handy. I have actually considered this summer only painting three sizes. 16 x 20, 18 x 24, 24 x 30.
Often times t! he proportion you choose for a painting is a function of how d! eep the view you are painting is. In the woods I am liable to choose a more square shape. Along the shore where there is a great expanse of distance I am likely to choose a wider shape.
When you hear hucksters on the radio advertising "over the sofa" sized oils, they mean a 24x36. That's a great landscape size just the same..
There's a story I heard concerning painting sizes. A well known artist visited one of his galleries unannounced and his painting wasn't hanging. But his frame was. The unscrupulous dealer had taken the frame he had bought for his own painting and put it on a painting by another artist. After that he often sent galleries odd sizes, instead of a 24 x 36 he would send them a 24 x 34, and that solved that problem!
I should tell you, that I don't quite obey this six size rule myself, I paint 26x29 canvasses. I have done lots of them. That's a size I got from Willard Metcalf. I find it designs really well for me and it has a delicacy of siz! e and shape that I like. I usually have to special order those 29 inch stretchers. I have never seen a 26x29 from anybody else except for Metcalf. Wetcalf was raised in a spiritualist family. That was a common religion in the late 19th century. Often they would hold seances and try to communicate with the dead. They were interested in mystical numerology too. I think this size may have come from Willards interest in numerology, but I don't really know that. But 29 is a prime number and 26 is twice 13, another prime number. I just like the way I can design the shape, maybe it is magic.
Empress Marie Alexandrova of Russia. Love the dress. I wish women dressed that way today. I don't guess its ever coming back though. What fun it must have been for portrait painters then.I make between 40 and 70 paintings a year, I throw about a third as many more away unfinished because they have some sort of an irredeemable flaw. So if I paint too many sizes it really gets complicated and expensive having many dedicated frames that only fit one painting.
Tomorrow I will talk more about things to paint on.
direct square proportion
Coat of Arms.
Make Your Coat of Arms allows you to easily create your own family coat of arms or family crest based on your family ancestry or on the values that are important to you and your family today.
[ Coat of Arms ]
Here's another [ Coat of Arms ] generator. (via Pitoche)
coat of arms generator
[ Coat of Arms ]
Here's another [ Coat of Arms ] generator. (via Pitoche)
coat of arms generator
Impact Factors - The Basics
Impact Factors - The Basics - Part 1 (Cross 2008) is an explanatory guide to Impact Factors [IFs] from by Taylor & Francis. It does in fact cover the basics, what impact factors are and how they are calculated. However, it goes into greater detail on the very important topic of Subject Variation within Impact Factors including some very helpful graphs to illustrate the point. Here are some of the key figures made:
- Average category impact factors for Economics, Nursing, Education & Educational Research and Business are all less then one. This compares with the top category of Cell Biology at c5.7
- The coverage of journals in the Social Sciences by the Journals Citation Reports [JCR] are significantly less that the Sciences. For example Economics c 42% and Business and Management c33%. This compares with the top category Physics a! t c83%
- The rates of citing in the Impact Factor Window (2 years) are higher in the Sciences, for example Cell Biology 22% compared with Economics at 8%
These figures provide a big health warning to comparing IFs even between apparently related subjects - there is also variation within subjects. The guide is just 7 pages long and worth a read especially for those researching in arts, humanities and social sciences who may wonder at the comparatively higher IFs of scientific colleagues.
References
Cross, J., 2008. Impact Factors - The Basics - Part 1. London: Taylor & Francis. Available from: http://www.tandf.co.uk/libsite/newsletter/issue9/Back_to_Basics.pdf [Accessed: 17 June 2008].
Links
Journal Citation Reports [JCR] which publishes IF's is available here. It i! s ATHENS Authenticated.
basics of factors
The problem with word problems 2
For the purpose of this post, we could divide word problems to three different categories:
1) routine word problems
2) non-routine word problems
3) algebra word problems
Actually you could divide algebra word problems to routine and non-routine as well, but I want to now talk about word problems kids encounter in school before algebra - in grades 1-8 usually.
J.D. Fisher suggested in the comments section of my previous post on word problems that kids are encouraged to think linearly, step-by-step. Then, when the word problems they encounter don't anymore follow any step-by-step recipe, they are lost. You might want to go back and read that.
Don't typical math book lessons kind of follow this recipe:
LESSON X
---------------------
Explanation and examples.
Numerical exercises.
A few word problem! s.
In other words, the word problems are usually in the end of the lesson. (That might make solving them a rush.)
Then, have you ever noticed... If the lesson is about topic X, then the word problems are about the topic X too!
For example, if the topic in the lesson is long division, then the word problems found in the lesson are extremely likely to be solved by long division.
And, typically the word problems only have two numbers in them. So, even if you didn't understand a word in the word problem, you might be able to do it. Just try: let's say that the following made-up problem is found within a long division lesson. Can you solve it?
La tienda tiene 870 sabanas en 9 colores diferentes. Hay la misma cantidad en cada color. Cuantos sabanas de cada color tiene la tienda?
My thought is that over the years, when kids are exposed to such lessons over and over again, they kind of figur! e it out that it's mentally less demanding just not even read ! the prob lem too carefully. Why bother? Just take the two numbers and divide (or multiply, or add, or subtract) them and that's it.
I'm not saying that such word problems are not needed in the end of division lessons. I'm sure they have their place. But too much of those simple 'routine' problems can be a problem... I feel kids then "learn" a rule: "Word problems found in math books are solved by some routine or rule that you find in the beginning of the corresponding lesson."
It might teach their minds to be lazy and not willing to tackle non-routine problems.
Maybe it would help to give students a bunch of short routine word problems, and NOT ask them to find answer. Instead, ask them to tell what operation(s) are needed to find the answer.
Maybe it would help to have separate lessons with mixed word problems, including some non-routine, an! d devote some time to them.
I'm curious to hear your thoughts on this.
And, lastly some (most are free) word problem resources if you need more than what's in your math book:
Word problems for kids
A great selection of word problems for grades 5-12. A hint and a complete solution available for each problem.
Aunty Math
Math challenges in a form of short stories for K-5 learners posted bi-weekly.
Problem of the Week home page
Links to 'problem of the week' websites organized by grade levels. These are excellent for finding more challenging problems and to motivate.
Primary Mathe! matics Challenging Word Problems
For grades 1-6 from ! Singapor e Math. The books include answer key, worked examples, practice problems, and challenging problems. About $8 per book.
I have some more resources listed at my own site.
Tags: math, elementary, curriculum
algebra word problem solver
Triangle Congruence Theorems
are so boring, and there is no nice way to teach them. A google search turns up a hojillion versions of "state the theorem and show an example." Here's what I did this year. It sucks and I'm looking for better ideas.
What makes triangles congruent? They're the same size and shape. This means all the sides and all the angles are congruent. The kids had to sit through one proof like this. What a pain, having to write 6 statements to show that two triangles are congruent.
In class the next day, I had them draw three segments with a ruler on scrap paper: one the length of their index finger, one the length of their ear, and one the width of their palm. I told them to construct a triangle with sides of these three lengths. I passed out compasses but didn't give them any instruction on how to do it. Some of them figured it out. A few of these guys showed everyone what they did with the giant compass and yardstick on the whiteboard.
We had a "discussion" about how there was only one triangle you could make with the three lengths. And when I say "discussion" I mean I said - did you notice that those three lengths would only make one triangle? That the angles were sort of 'locked in'? You couldn't just draw the other sides at any old angle? Then I did a lame little demo of how with 4 sticks, the angles are all wobbly and you can make a ton of different quadrilaterals, but with three sticks, you can only get them to make one triangle. I said something about bridges. Then I modeled and they practiced a bunch of SSS proofs.
Today was SAS. In previous years I passed out protractors, and asked them to draw a triangle with two specific side lengths and a specific included angle, then look around and notice how everyone ended up with the same triangle. I didn't feel this really sent the message that the angle has to be included. This year, I asked them to solve these two problems:
I don't honestly know how effective this was. Yeah "we" eventually made the point, but it took FOREVER, and I'm sure some kids got the point, but I'm also sure that half the class was just sitting politely waiting for the torture to be over. And some kids, of course, get downright indignant when you ask them to do something that turns out to be impossible. Which, whatever, but it's much harder to teach somebody who has concluded you are a crap teacher.
Then we practiced determining which other pair of corresponding things you would need in order to use SAS, then we did one example proof.
I know this sucks but I don't know anyone doing anything better. So let me hear it, hot shots.
Theorem on Area of Triangle
What makes triangles congruent? They're the same size and shape. This means all the sides and all the angles are congruent. The kids had to sit through one proof like this. What a pain, having to write 6 statements to show that two triangles are congruent.
In class the next day, I had them draw three segments with a ruler on scrap paper: one the length of their index finger, one the length of their ear, and one the width of their palm. I told them to construct a triangle with sides of these three lengths. I passed out compasses but didn't give them any instruction on how to do it. Some of them figured it out. A few of these guys showed everyone what they did with the giant compass and yardstick on the whiteboard.
We had a "discussion" about how there was only one triangle you could make with the three lengths. And when I say "discussion" I mean I said - did you notice that those three lengths would only make one triangle? That the angles were sort of 'locked in'? You couldn't just draw the other sides at any old angle? Then I did a lame little demo of how with 4 sticks, the angles are all wobbly and you can make a ton of different quadrilaterals, but with three sticks, you can only get them to make one triangle. I said something about bridges. Then I modeled and they practiced a bunch of SSS proofs.
Today was SAS. In previous years I passed out protractors, and asked them to draw a triangle with two specific side lengths and a specific included angle, then look around and notice how everyone ended up with the same triangle. I didn't feel this really sent the message that the angle has to be included. This year, I asked them to solve these two problems:
I don't honestly know how effective this was. Yeah "we" eventually made the point, but it took FOREVER, and I'm sure some kids got the point, but I'm also sure that half the class was just sitting politely waiting for the torture to be over. And some kids, of course, get downright indignant when you ask them to do something that turns out to be impossible. Which, whatever, but it's much harder to teach somebody who has concluded you are a crap teacher.
Then we practiced determining which other pair of corresponding things you would need in order to use SAS, then we did one example proof.
I know this sucks but I don't know anyone doing anything better. So let me hear it, hot shots.
Theorem on Area of Triangle
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