Blogging Tools

Graphs of Functions:
If you want to include graphs in your blogs, go to http://www.walterzorn.com/grapher/grapher_e.htm . Here, you can enter the function you want to graph, and click on print preview on the menu to the right. You should be able to copy the image with the Print Screen key on the top right of your keyboard (It should be above the insert key). Go to paint or any other image progam, paste the graph and and save the it as a JPG image, and you're ready to upload. The grapher in the link above allows you to graph several functions at the same time, and adjust the graph window.

Bubble Won't Share

Bubbleshare is down so today's slides will be posted Tuesday morning.
Yes, I will be working while most of you are vegetating.

Anyone else working?

Scribe List

You must pick someone whose name is not crossed out.

Luis DP's Comment to Mac

I believe optical illusions have more to do with biology than mathematics. The human eye is an astonishing structure. Perhaps we can apply these illusions to our daily life. Take another look at the circles in the third illusion of your post. I believe I would see the left middle circle smaller even if it was actually a little bit larger than the right one. This makes me think that I have been fooled many times in my life by optical illusions, since the brain rarely doubts about information coming from sight. It feels creepy. Advertisement companies might be using them. Doesn't this make you think that standing by small people will make you look taller than you thought it would? How would you use optical illusions to fool the human eye and gain an advantage in life? One could definitely write a book about this.

Curiosities

Remember Harry Potter?
Remember ´The Prisoner of Azkaban´?
Harry´s godfather is named Sirius Black. He is an animagus, which is a wizard/witch that can turn into an animal. In his case he turns into a dog, a big black dog. The first time Harry sees him in this novel he appears in the "black alleyway" and when he lights his wand and what he sees is "the hulking outline of something very big, with wide, gleaming eyes." That image makes me think of bright stars on a dark, moonless sky.
Also, in the novel, J.K. Rowling lets us know that Sirius´dad was called "Orion Black". Sirius turns out to be one of the most important characters in Harry´s life.
J.K Rowling once said in an interview that she liked to give her characters distinctive and special names, most of which she had written down and collected at any point in her life when she had found them interesting.
**Sirius,(the star) is the brightest in the sky and is in the dog that follows Orion around.**
This kind of well thought details make me enjoy even more Rowling´s series and other pieces of literature I've read. I encourage you all to pay more atention to details like this one. You never know when you will find something really interesting or that relates a lot to you.

(Sirius as a dog)

Sequences Quiz

1. How do I find explicit definitions with a few terms in a sequence? When do I have to use a definition to prove a sequence?
2. I will probably make mistakes when creating a definition and with the fibonacci's sequence since I don't feel I fully understand it.
3. 4.9, -0.9, -5.7...

• Say what type of sequence it is.
• Find the difference or the ratio (d or r).
• Create a formula.
• Find the 11th term.

To my classmates and Mr. Alcantara

Hi everyone!
I would just like you to know that I'm not going to be attending to class on Monday, so I would like one of you to help me and explain to me what you did in class Monday. Mr. Alcantara, I would also like to know if there is going to be a remedial class on Wednesday morning so I can attend and catch up on my work.
I will really appreciate your help!
Thanks,
Bee

Scribe Post Instructions

You and a partner will be responsible for posting a summary of the day's lesson. Your post must appear on the blog by 8:00 pm on the day that you are the scribe.

Scribe Post Requirements:

1. Briefly explain the major concepts or skills.
2. Provide copies of the problems & their solutions.
3. Highlight any important issues associated with the problems. (These might include key steps, likely errors, connections to previous material, tips for understanding or memorizing, etc.)
4. List new homework assignments, and provide reminders for any upcoming assignments, quizzes, or class schedule changes.
5. Pick the next scribe(s) from the scribe list.

Tips for success:

• COME TO CLASS EVERYDAY!
• Check the blog the night before to see if you are the next scribe.
• Read and practice the night before you are the scribe. Make sure that you understand and have practiced the most recent skills that we have learned.

Quiz Reminder & Blog Assignment

Hello lost juniors.

Reminding you that we have a quiz this Wednesday on:

• Arithmetic & Geometric Sequences
• Explicit and Recursive Definitions

Skills required:

• Finding the next term of a sequence
• Finding the nth term
• Understanding sequences as discrete functions
• Finding explicit (function) definitions given a few terms of a sequence
• Finding recursive definitions given a few terms of a sequence

Blog Prompt:

Review your notes 1st, otherwise you won't get much out of this.

Then respond to the following:

• What questions do you have on this topic?
• What mistakes are you most likely to make? (Since no one has earned a 100% on all of the quizzes, I expect thoughtful answers from all of you)
• Create a potential quiz problem on one of these topics. You will benefit most if you make this question on the topic for which you feel the least prepared. Add your solution, or attempted solution, as a comment so that others may compare their thoughts with yours. (If I get some good questions, then I will likely use some of them on the quiz)
  

Complete this assignment as soon as possible! The seniors and your classmates are waiting to help you but time runs out on Tuesday night.

Optical Illusions

I love optical illusions, it is fun, in my opinion, to see something, or think you are seeing something, at first and then realize the truth, that it is something different, or that you have just been tricked to believe something.

A definition for an optical illusion is: A visually perceived image that is deceptive or misleading.

I wanted to share this with all of you, to find out more about these illusions. Why do we get tricked by them? I suppose there is a scientific explanation to all of these illusions....
I hope you like them...

These are some of my favourite optical illusions:

These lines are all straight...

How many black dots do you see? There are none.

Which middle circle is bigger? They are exactly the same.

Which line is the longest? They are the same...

This is a link to see more optical illusions: http://www.michaelbach.de/ot/

Sine and Cosine Graphs Concerns

Some of you have expressed having difficulties reading the Sine and Cosine graphs. However, it wasn't clear to me what it is you're having trouble with. If you still want help, it would be good if you could comment about what your concerns are.

Synthetic Division

If you already understand Polynomial Long Division perfectly, you might want to try Synthetic Long Division. It is a different method through which you can divide polynomials. However, it is much shorter than Polynomial Long Division. I’m not sure if you have seen it or will see it with Mr. Alcantara, but it's still a good method to know.

Differences between Synthetic and Polynomial Long Division

Synthetic Division only applies to polynomials that can be divided into polynomials of the form x-a, or monic 1st-degree polynomials.
For instance,
(x^2-2x-3) ÷ (x+1) can be done by synthetic division because the divisor is x+1, a polynomial in the form x-a, in which a = -1.
(x^3-3x^2+x-3) ÷ (x^2+1) cannot be solved through synthetic division because the divisor is x^2+1, which is not in the form of x-a.

Synthetic Division only shows the coefficients of terms, while Polynomial Long Division shows the entire polynomial.

Synthetic Division

Setup

*If you have been taught this by an American teacher, you might setup your problem different to the way I explain here. I learned it the Colombian way with Mr. Troncoso.

As with Polynomial Long Division, I will use a problem to show you the process.

(x^2-2x-3) ÷ (x+1)

Our setup will only show the coefficients of the terms. This means that instead of showing x^2, it will show 1; and instead of showing -2x, it will show -2.

In this sense, the first thing we must do to setup our problem, is write down the coefficients of the terms of the dividend, in order of decreasing degree.

1 -2 -3

*Just like in Polynomial Long Division, we must also include coefficients of all terms. For instance, if we were to write the coefficients of x^3+4x, we would write 1 0 4 0. In our problem, however, there are no coefficients of zero.

The general setup for synthetic division looks like this:

where a is found in the divisor, which is in the form of x-a

Therefore, the setup for our problem should look like this.
After we’ve setup our problem, we’re ready to begin.

1. The first thing we must do is bring down the first coefficient in the list. By bringing down, we mean copying it below the horizontal line, like shown below.
2. Then, we multiply the number we just brought down by a, or, in this case, -1.

1*(-1) = -1

3. We will now write the product below the next coefficient in line, but above the horizontal line.
4. Now, we will add the product we just wrote down and the number directly above it.

-2+(-1) = -3

5. We now write the sum below the numbers we added, below the horizontal line.
6. Repeat steps 2-5 until we run out of numbers.

-3+3 = 0

7. Our answer is the one written below the horizontal line. However, like we saw before, the method only shows coefficients. This means that the numbers below the horizontal line are the coefficients of our answer. To write our answer, we will start from the right.

The first number on the right is the remainder, which in this case is 0. The remainder is always written as a fraction, with the right-most number as the numerator and the divisor as your denominator. In this case, it will look like this:

0 ÷ (x+1) = 0, which makes sense since we already said there was no remainder for this problem.

The number directly next to the left (in this case, -3) is the coefficient of x^0; the number that follows to the left (in this case, 1) is the coefficient of x^1; and it goes on like that always increasing in degree as you move towards the left.

We can now write our answer:

1*x^1-3*X^0+0 =
x-3

This means that

(x^2-2x-3) ÷ (x+1) = x-3

Note: In the expression (x-a) explained above, a may also be thought as the solution to the function, or the zeros in the graph of the function.

For example,

(x^2-2x-3) ÷ (x+1) = x-3, where a = -1, can also be written like this:

x^2-2x-3 =
(x+1)(x-3)

Looking for the solution to this problem is the same as looking for the values of x that make it zero.

(x+1)(x-3) = 0

If we want (x+1)(x-3) to equal zero, either x+1 or x-3 must equal zero because anything multiplied by zero equals zero. Therefore, the solutions are:

x+1 = 0
x = -1

and

x-3 = 0
x = 3

In our problem, a = -1, which is one of our solutions.

Polynomial Long Division

Polynomial long division involves the same concepts and process as arithmetic long division.

I’ll try to explain the process with the following division.

*Whatever is different from the previous step has been written in red.

1. The setup for polynomial long division is the same as for arithmetic long division. You have a dividend to the right, a divisor to the left and a quotient on the top.
For the problem above, it will look like this.
Notice that the terms are organized in order of decreasing degree (x^3 goes first, then x^2, and then -42).
However, we need to include all the terms from x^0 to the biggest term in the dividend for the dividend (in this case, from x^3 to x^0) and in the divisor for the divisor (in this case, from x^1 to x^0) even if their coefficient is 0.

For instance, in the dividend you will find that there is no x^2. That is because its coefficient is 0. However, you still have to write it out, what means that the problem will actually look like this.
2. Now, we can start dividing.

We will start by dividing the biggest term in the dividend by the biggest term in the divisor. In this case, we will divide x^3 by x, which gives us x^2. This result, we will write on top of the dividend like we will do in a regular division.
You might want to place the term above its corresponding term for better organization. In this case, the x^2 in our quotient above the -12x^2 in our dividend.

3. Now, just like in regular long division, we will multiply the term we just wrote down (x^2) by the divisor (x-3).

x^2*(x-3) = x^3-3x^2

4. We will now subtract this product from the dividend like we normally do in arithmetic long division.
*Remember to properly distribute the negative sign.
5. Now, like in a regular division, we will bring the next term down.
6. We now have to repeat steps 2-5, only this time we will use the new polynomial we have on the bottom of our problem, like in a regular division, until we have no more terms to bring down. *Here we can see why we must also include the terms whose coefficients are 0. If not, we would end up subtracting -27x from -42, which is not possible.

Continuing with our problem…

7. Once we have no more terms to bring down from our dividend, we are done. Whatever we have left over on the bottom of our problem is the remainder, as in regular division.

To properly express our answer, we must write the quotient we have on the top + the remainder over the divisor. In this case, our answer will be:

(x^2-9x-27) + [(-123) ÷ (x-3)]

We are done!

If you still want more practice, you might want to do one of the following problems (Answers on the bottom):

Answers:
1) x-5; 2) x^2+4x+3; 3) -x^3+x^2+5x-5 + [7 ÷ (x-5)]

Solving Word Problems

For: Susy L, Caro I, Mau, Lili, and Fio

To properly understand a word problem you might want to take a look at the following steps:

1. Read it (obviously)
2. Rephrase it in your own words so that you make sure you understand it.
3. Identify its parts. You will always be asked to find something using other information they do give you. So you must identify what you’re being asked for and what you are given.
4. Identify the relationship between what you’re being asked for and what you’re given. Many times, the same problem will give you the relationship. In other cases, you have to set it yourself.
5. Assign variables to information you don’t have.
6. Translate the relationship between your unknown information and known information into a math expression. There are times in which the relationship will already be expressed as a math expression so that you only have to put in proper variables.
7. Carry out the proper operations.
8. Check your work

For steps 3-5, it sometimes helps to draw a picture.

The first example is very easy, just to illustrate the process. The second one gets a bit harder. If you have any particular word problem you would like help with, please let me know.

PROBLEM 1:

Step 1: Read it

Pedro has 5 more apples than Laura. Laura and Pedro have 27 apples in total. How many apples does Laura have?

Step 2: Understand it

Step 3: Identify its Parts

What are you being asked for: The amount of apples Laura has
What you know: Laura and Pedro have 27 apples. Pedro has 5 more apples than Laura.
What you don’t know: Number of apples Pedro has. Number of apples Laura has

Step 4: Identify a Relationship

In this case, the relationship is given to you in the problem. It is what you know: that Laura and Pedro have 27 apples, and that Pedro has 5 more apples than Laura.

Step 5: Assign Variables

P=Number of apples Pedro has
L=Number of apples Laura has
Variables help you replace words with letters

Step 6: Translation

Relate what you know to your variables (what you don't know)

You now have two equations that relate what you know to what you don’t know:
L+P=27
P=5+L

Step 7: Solve

You just have to solve for L by substituting P in the first equation

L+P=27
L+(5+L)=27
2L+5=27
2L=27-5
2L=22
L=22/2
L=11 (This is your answer)

Step 8: Check your work

L+P=27 and P=5+L must be equal to each other for the value of L=11. Replace L for 11 in both equations. You should get the same value for P.

L+P=27
11+P=27
P=27-11
P=16

P=5+11
P=16

PROBLEM 2 (This one I got off the internet):

Step 1: Read it

A piece of Wire 46 inches long is bent into the shape of a rectangle having length x and width y. Express the area A of the rectangle as a function of x.

Step 2: Understand it

You have a 46 inch wire that you must bend in order to make a rectangle. The rectangle will have sides of length x and width y. Come up with a function for the area of the rectangle using only the variable x. (Making a picture sometimes helps)

Step 3: Identify its Parts

What are you being asked for: Area in terms of x, or A(x)
What you know: Perimeter equals 46 inches since it must equal the length of the wire. The shape is a rectangle. The length of the rectangle equals x and the width equals y
What you don’t know: A(x)

Step 4: Identify a Relationship

You’re looking for the area of a rectangle, which is length times width.

A=l*w

The perimeter of a rectangle equals the sum of its sides.

P=l+l+w+w
P=2l+2w

Step 5: Assign Variables

P=Perimeter=46
A=Area
The variables for the sides have already been assigned, x and y.
l=x and w=y

Step 6: Translation

In this case, translation consists in replacing variables

l=x w=y A=l*w
A=x*y

l=x w=y P=2l+2w P=46
46=2x+2y

Step 7: Solve

As said before, they ask for Area (A) in terms of x. Right now, you have Area in terms of x and y.

A=x*y

You need to replace y for an expression that contains only x. The only relationship you can use for this is 46=2x+2y.

Solve for y

46=2x+2y
46=2(x+y)
46/2=x+y
23=x+y
23-x=y

Replace y for 23-x on the function for Area.

A=x*y
A=x*(23-x)
A(x)=23x-x^2 (This is your answer)

More Help: If you still need help, you can always google “Translating Word Problems”, or “Solving Word Problems”. There are hundreds of ways to approach a word problem. You just have to find the one that works best for you.

Pre-calc

Thanks to the seniors helping us out with our problems. I definately found the posts by Luis DP and Lina very useful.

Now, going back to the current topic we're studying: I'm liking it because it's obvious that it will be very helpful in the future, because sequences can be found anywhere in any moment. If you are having problems with it, or you want to be ahead of the class then you can check this webpage. I hope you find it useful.

• http://tutorial.math.lamar.edu/AllBrowsers/2414/Sequences.asp

Key Words in Word Problems

There are some key words you must be associated with when solving word problems in order to establish relationships between your known and unknown information. Here is a list I got off the internet with words and their meanings, but there are many others you can come accross by.

Addition
increased by, more than, combined, together, total of, sum, added to
Subtraction
decreased by, minus, less, difference between/of, less than, fewer than
Multiplication
of, times, multiplied by, product of, increased/decreased by a factor of (this type can involve both addition or subtraction and multiplication!)
Division
per, a, out of, ratio of, quotient of, percent (divide by 100)
Equals
is, are, was, were, will be, gives, yields, sold for

How to Learn Math - For: Chewy

The best method to learn any math concept is by understanding so that you can do it and by practicing so that you can remember it. Are you having trouble understanding or remembering?

What long division are you having trouble with?

For: MAC and Daniel Rubio

What long division are you talking about? Arithmetic long division like 1025/36? or Polynomial long division like (4x^2+3x-2)/(2x+1)? What exactly are you struggling with?

Looking for Domain and Range using a Graph - For: Maria Andrea Guillen

One of the easiest ways of solving for the domain and range of a function is by looking at its graph. The domain would be the span of the X values; and the range, the span of the Y values. However, you must be careful since the graph will not cover all numbers until infinity, so you also have to apply some logic. For instance,

Take a look at the graph of f(x)=2^x

Domain: The domain will be all real numbers since every x-value has a y-value, or the function passes through every x-value. Even though, in the graph, the red line does not go beyond x=2.25, you should be able to tell that the domain goes beyond that by picturing a graph that covers more values. If you're not sure about this, you can always graph the function in your graphing calculator and zoom out as much as you want.

Range: The range is only all positive numbers. f(x) never equals a negative value for y. You can see it on the graph. The function never passes through a negative y-value. It never goes below the x-axis, and it never touches zero.

Square Roots - For: Susy Del Castillo

There is also a way to solve for square roots without using a calculator and without approximating. It is a long and complex process, but if you want to take a close look at it, you can go to the following link:
http://www.qnet.fi/abehr/Achim/Calculators_SquareRoots.html

Bee's Web Page Help

For anyone that needs help in understanding about domain and ranges, here are two webpages that might help you clear your mind about the topic. I hope this might be helpful for some of you.

• http://www.analyzemath.com/DomainRange/DomainRange.html
• http://jwbales.home.mindspring.com/precal/part1/part1.4.html

Bee Bustillo

Friday's Slides 4/20/07

Today we worked on a math problem from the year 1202:

Suppose a newly-born pair of rabbits, one male, one female, are put in a field. The rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?

The answer was 144. The trick was to develop a good system for keeping track of newborn rabbits, maturing rabbits, and breeding rabbits. Once we did that (see 1st slide or better yet, try it yourself) for the first few months, we generated a sequence of numbers. The domain of the sequence was the month number and the range was the number of pairs of rabbits. Daniel & DC explained the pattern in the sequence and we found the answer to the question (144) without keeping track of all the rabbits.

The difficult thing about the rabbit sequence is that it is not arithmetic or geometric. We could not use the simple rules that we had learned. Instead we needed a new tool:

The recursive definition of a sequence.


See the slides for more info on this topic.

The answer to the question on the 3rd slide is 511. The 4th slide shows an arithmetic sequence. We could write a simple formula using the general arithmetic sequence formula (do you know it?) but we can also write a recursive formula. Sometimes we won't be so lucky and the recursive definition will be the only one we can find.

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New Homework: Read pg. 470
On pg. 481 do problems 1-19 odd

Thursday's Slides 4/19/07

We started class by working on a geometric sequence where the given terms were the 2nd and 3rd and we had to find the 6th(see the 1st slide). The easiest way to solve the problem was to treat the 2nd term as the 1st and then find the 5th term of that new sequence instead of the 6th term of the original. (Think and count)

After finishing that problem, we created sequence problems for each other and everyone seemed to be doing pretty well.

The second slide shows another geometric sequence where we needed good algebra skills to get an exact answer.

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A Wee Spot O' Nuthin'

A scientist friend of mine sent me this link. It gives an interesting perspective on planet Earth. I have not checked it for accuracy.

FYI:

• Sirius is the brightest star in the sky and is in the dog that follows Orion around.
• Arcturus is in the constellation Bootes. Bootes looks like an ice cream cone and Arcturus in at the point of the cone.
• Betelgeuse and Rigel are both in the constellation Orion. Rigel is brighter.
• Antares is in the constellation Scorpius.

Have you seen these stars or is Bocagrande too bright? Go to the mountains of Colorado (where I am from) if you want to see a bazillion stars.

Anybody have an estimate for how many Earths might fit inside Antares?

Tiling Response

Hello Precalculus Students!
I was reading your class blog and found the discussion of the "Degradado Azul" pattern interesting. I had an idea about how I would create the pattern using tools of my profession. I'm a cartographer - which means I make maps. I use computers almost entirely to make maps and use many different programs. (You can see some of my maps here...)

Here's how I would make the pattern for the tiles. My example is a little different looking, but the concept is the same.

First I would start with this picture:




and take a small sample of it (like the area shown in red).
If you take this small sample and paste it into a new image and stretch it you end up with something like this:


Basically this is just an easy way to get the color gradient that you want. You can do other things to create the gradient also.

Next I used the pixelate filter to create a pixelated image (made of up large squares) and the image looks like this (I also rotated the image to be more like Mr. Alcantara's example):



At this point you can use it as a guide to create a tiled pattern. So my answer to Mr. Alcantara's question

"How was this tile pattern generated? "

is... from nature!

Thanks for letting me join in on your discussion.

Ann
from Bellingham, Washington, USA

Blogging Tip: Tutorials

People have expressed a desire to become stronger on such topics as factoring, long division of polynomials, interpreting graphs of trigonomteric functions, etc.

A useful and interesting post (and one incidentally that would weigh quite favorably in the grading scheme) would be to post a mini-tutorial that you have created.

Such a post might inlcude:

• Background theory
• Examples with Solutions
• Practice Problems
• Extra challenge problems
• A link to a help site

The sky is the limit. I think that if you undertake such a project, you will find it much more rewarding and effective than doing homework problems from the book.

"See it. Do it. Teach it."

Consider employing some interesting form of presentation. See Mr. Moyano for technology tips.

Good luck.

Factorization

Several students have mentioned about being uncomfortable with "factorization." I would like those students to please tell me to what they are referring. Factoring trinomials? Factoring by grouping? Completing the square? Factoring higher order polynomials? Removing perfect factors from within radicals? Removing common factors from expressions? Finding the greatest common factor of two or more terms?

Be specific; consider giving an example or two.

Proofs

I know now how to find any term in a sequence, and I know how to create the formulas to be able to find this, however I really do not understand very well how to prove a sequence. If it is definite, and the amount of numbers is small, it is easy because you just do them all, but what if there are infinite numbers? When do I have to write the definition down? I want to understand that before we get too far ahead.

Ex. 3, 11, 19, 27, 35, 43...

Tuesday's slides 4/17/07

Today we worked on geometric sequences. They are very similar to arithmetic sequences. The big difference is that each successive number in a geometric sequence is produced by a multiplicative jump instead of an additive jump.

Here are the slides of the problems that we did in class.

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Tiling

I took the following picture through the shop window of a business that is located just behind the physics lab.

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I have been passing the tiling for weeks (probably much longer) but I recently started thinking about it. I have some questions for you to consider. I do not know the answers but I am interested to hear some of your thoughts.

How was this tile pattern generated?

One obvious answer is that someone said something like, "Ok. In the first row I want 4 whites, 1 gray, 2 whites, 1 blue, 3 whites, 1 gray, and 1 white. For the second row, I want 1 gray, ..."? Do you think this likely? Explain.

If the pattern was not created tile by tile, how was it done? A likely answer was using a computer program. But what type of instructions would the programmer give?

The name of the pattern is "Degradado Azul." Starting at the top and going from one row to the next, the number of blue tiles sometimes increases and sometimes decreases. Overall, though, the rows start without having almost any blue tiles and then end up, in the bottom rows, with all or nearly all tiles being blue. The number of blue tiles in each row must be increasing even though it sometimes decreases.

Do you see any other patterns? Could there be some sequence formula that a computer could use to generate the number of blue tiles in each row? Given the number of tiles, in what positions should the blue tiles be placed? How would that be decided?

Do you see any other patterns?

Do other questions occur to you?

Blog Rubric

Below you will find the rubric by which you may measure your level of achievement in earning the blog grade that you desire.

Although it is not incorporated into the rubric, each level of performance requires that your posts and comments be respectful in tone, content, and language usage.

Click to see a larger image.







Please attach a comment if you any ideas that would improve the quality of the rubric.

Monday's Slides 4/16

Today we continued our focus on arithmetic sequences. We proved that sequences were arithmetic by showing that for any 2 consecutive terms, tn and tn+1 , their difference was a real number. (See the first 3 slides)

Then we discussed the three ways to find the nth term of a sequence:
1) Write out all previous terms
2) Find the explicit definition (formula) and substitute (slide 4)
3) How many jumps? (slides 5,6)

We ran out of time before exploring the third method. We will do that tomorrow.

Slide 7 shows a problem about a geometric sequence (what’s that?) to try at home. Slide 7 was supposed to be that problem but the slide shows a different problem. The problem that we were to try was the following:

{-2, 6, -18, 54, …} is a geometric sequence.
a) Find the next term
b) Find the 10th term.




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The homework on the board was: Read pgs. 473-476, problems pg. 476 1-9 odd, 17-41 odd.

Luis DP's Answer to Gabriela Guerrero

Dear Gabriela Guerrero,

You first question is: What are some easy ways to remember proofs?

The beauty of mathematics partially lies in the fact that there is very little to memorize. It is what sets it apart from other studies such as humanities. Memorizing proofs is counterproductive in the long run. You should rather develop your personal approaches in order to be ready to prove any proposition. Profoundly understanding the meaning of a mathematical proof will aid you in this process. A proof is a demonstration that some statement is necessarily true. A proof is a logical argument, not an empirical one. This means one must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. Gabriela, there are many types of proofs. Which proofs are you concerned about?

Your second question is: Are there any short-cuts to graph equations?
Let me reinterpret your question, since unless you are combining equations in your calculator, using your TI-83 to graph shouldn’t need any shortcut.
Are there any short-cuts to graph equations by hand?

You already know about connecting dots or points of a function in order to reveal its graph. If you can determine that a function follows a certain pattern, or that it is symmetric about a certain axis, then you can complete a graph without placing as much dots. Such is the case of y = │x│+ C, where C is any constant. This approach, however, is vulnerable to cause an overly inexact graph that may take points away during an examination, unless you are a really talented drawer.

Luis DP's Answer to Susana del Castillo

Dear Susana del Castillo,

Your question is: How can you solve for the square root of a given number (ex: 153) without using a calculator?

Let me reinterpret your question, since a given number could be 4, and you know its square root. How can you solve for the square root of a number that is not a perfect square* (ex: 153) without using a calculator?


This question was actually born in Egypt, where they extracted the first square roots around 1650 BC. It may also rise when you take the ICFES, since no calculators are allowed.

I will use your example: 153.
Process:
1. Determine the two perfect squares that are closest to 153. They are 144 and 169.
2. Since the square root of 144 is 12 and that of 169 is 13, the square root of 153 is a number 13>x>12. Make a guess. Let our guess be 12.5.
3. Divide the number you are taking the square root of by this guess. 153/12.5=12.24
4. Average your guess and the result of the division. (12.5+12.24)/2=12.37.This average is an approximation of the square root of 153. The exact answer rounded to the nearest thousandth is 12.369.
5. To obtain an even more accurate approximation, repeat steps 3 and 4 as many times as you wish, but each time letting your guess be the immediately preceding approximation. In the case exposed above, 12.37 would be your new guess. The division that follows holds 153 as its numerator.

* Perfect square: an integer which is the square of some other integer, i.e. can be written in the form n^2 for some integer n.

Forgetting Curve

1. Since I was a little girl Math has been my weakest subject in school, I have a hard time understanding Math topics. My study habits are not that good, I hardly take any time to study or check on my notes at home. For my tests or quizzes I only open my notebook and skim skam the information I need for the evaluation, but for Math test and quizzes a take a lot of time at home to study and do my homework to try to get a good grade but I hardly ever pass a test. I have a tutor that goes home twice a week to study with me a try to get me on a better shape to do well in Math. I know I have to change my study habits in order to be a better student in Math class.

2. I still have a hard time with factorization I don't know what method is more easy for me to use in order to learn this topic. The new topic that we are learning now about sine and cosine is very hardo for me to understand.

Jose's Study Habits

The truth is, that my study habits aren't the best. Although my grades are not low, they could be much higher than what they are. They're totally the opposite to what the article recomends they should be. The numbers in the article seem quite accurrate, for there are many things that you forget because you don't review, and most of the time you forget most of it. I think the article is right; students should review for a short amount of time, so the topics given at school stick more in their heads.

What I could do to improve my math grades, although I consider myself a good math student, is to review the topics in my house for a short amount of time. That will help me remember the topics. Also, to study in vacations; at least for ten or 20 minutes, for many things that I study at school just fly away when I'm in vacation.

Study Habits

1. This article seems true related to my habits. I'm the typical person that knows the topic during class, but I don't review at home. After several days If I get asked again about the same topic, most of the time I can't remember everything. Presonaly I considered my short memory weak, and Its very hard for me. While reading this article, I figured out that this could be a great method to apply, and that definitely will help me for future test, and quizess. I like math, eventhough some times is hard for me to learn. By applying this method I think that my grades on finals will be higher.

2. Some of my math past topics that I had a really hard time where, proofs, and graphing equations.
- What are some easy ways to remember proofs?

- Are there any short-cuts to graph equations?

...G.DelMar

the curve

1) The only way that I can study for a quiz or test, is in my house, queit, without music or tv.My studying habit are horible but I try to learn and with the help of god ¨A¨s the tests.Those numbers are not precise but are in that range of mostif not all students in 11 grade.Math can be easier for me if we go slow ,basically advance to the other terms when everybody has understood the topic.I know that the teacher couldent to that, but at leats take more time on what he explains.


2)I have heard that the ACT is pure geometry and trigonometry .In 9 grade we had 2 math teacher that leaft the school in the first semester.Then came Mr .Troncoso that ended being the high school Dean(what a change).As you can see we didnt learn we just played dominoes outside the class.

study curve

Well , to be honest i hate to study for math, not only because its hard for me but because it is a subject that i have never liked. After reading this arcticle i realized that maybe there was a chance that when it came to study for a math test my life wasnt going to be that miserable. I cant say i will change imediatly , but i will try cause to be good at matho would be really nice.

Mau´s Habits

1. To be honest, I don’t have the habit of studying. I rarely open my books when I get home. Sometimes I study for tests at home or at the last minute in school. If I get used to study in my house what I learn in class, my grades would be higher. After reading the article I realized that reviewing the information would help me a lot.

2. I have struggle with most of all the topics. But I really don’t understand how to convert word problems into mathematical equations. The last topic about Trig functions and finding angles is not that clear to me.

Study Habits

1. In the past, mathematics was never such a problem in my life, although my study skills coincided with the Curve of Forgetting. In eleventh grade I started to see mathematical concepts which were not that easy to learn in one minute, so my grades lowered and I was concerned and interested to increase my studies but eventually I didn’t do any of what I planed. After reading this article, I notice that I should’ve change my skills in high school and I didn’t, so what is happening is that what I learn in an hour is completely eliminate of my mind in one week. I don’t review the subject we learn daily and for exams I don’t expend that much time studying, but I do like to do homework because it usually lets me a clear understanding of the material studying. If I follow these advices I know for sure that my grades will have a huge improvement and I will understand in a better way.

2. Throughout the year we’ve learned a lot of things, but what I’ve struggle the most with has been the interpretation of word problems to mathematical equations. At the beginning, I remember that I couldn’t figure out the reason of how those kinds of problems were solved. Inclusive, one of the word problems of the Pods that had to do with a cable across a river I wasn’t able to understand it properly.

Curve of Forgetting

1. It is hard to say if the numbers are actually real for I do not know the quality of the source, however they do look pretty accurate to me. I agree that practice makes perfection, as it happens with a sport, the more you play it the better you become, the brain is a muscle and it functions the same way. I really do not practice a lot of the pre-calculus skills at home, I only do my homework the day before the quiz to make sure I understand the topic. I think that the reason I have not needed to put any more effort is because I like the subject, I enjoy doing mathematics and therefore it comes easily to me. The class itself is very dinamic, and I participate and pay attention, which I believe is why I understand what I'm doing.

2. I haven't had real problems with understanding whatever is going on in class, my only doubts are about Long Division. This may seem weird for everybody has told me it is the easiest thing to do, but I missed that class for whatever reason and I haven't been able to master it since then. I think my greatest "whole" or weakness in mathematics is actually geometry. When we began trigonometry, there were several equations that we should all have known from before, but I didn't at all.

study habits

1. Math has not been of much intrest in m life but although it is not of much interest i try to pay the best attention in the clases that i have. I lost all my interest in math when the past 3 year havent been the best because we havent had the best teachers in the world. This has been the best for me but really tough because we came to 11 grade with very low bases. Paying attention in class is the best method that i have to learn but have to confess that after class i never review or study after class i only study when there is a test coming up and i get a personal tutor to explain me the terms that i have to study other than that i nevr study.

2. There are two terms that have confused me alot and that i have never asked about it. One those are the word problems, how to solve them and how to convert them in to equations. What are the first steps? the other problem that i have are the reading of graphs and how to use them.

Forgetting Curve

1. Describe how this relates to your study habits. Do those numbers seem accurate? What specific changes could you make which would make learning math easier for you?

Well to be really honest I don´t study at home at all, its been a while since last time I opened my backpack at home. I actually believe the numbers are accurate, i mean, the best way for me to learn something is to practice it a lot; though if it´s something interesting, it makes the process easier and more nejoyable. Learning math for me is not as scary as it is to others, im not like mathophobic,but you should take a look at EXTREME cases like this one... though i have to say I hate homework. I should practice a little bit

<======== Found this picture on the net, thought it was kinda funny.

2. Think back over this past year, or even beyond, and identify one or two particular math concepts that were and maybe still are particularly troubling for you. Ask 1 or 2 questions in your post about these topics. Remember, the final still awaits. This is your chance to get your questions heard.

The first thing that comes to mind is finding roots and long division... p/q, factors, etc.. totally hated that. I missed like 3 days at school cus I was sick, and when I came back, i was severely lost. I would like to know an easier way to do all that processes.

Forgetting Curve

1. Honestly, mathematics is one of my greatest weaknesses in school. I have struggle with it since i was in seventh grade. When i get home i dont study the topics we see in class. I wait for the last minute (when there is a test) to call a tutor or a friend and explain me all we have done. Paying attention in class helps me alot but if i dont practice (which i dont do) the effort when paying attention is in vain. Its much easier that a student explains me what i dont understand because i have more time and can ask as much questions as possible. From now on i will start practicing to improve my math skills.

2. Two things that are difficult for me to understand are: first, going from word problems to equations, what is the first thing you need to do to find an equation? A second thing is reading graphs when certain information is given and how to apply information to graphs.

Fio's Study Habits

1. Scientifically, people have to repeat 13 times the information, in order to remain it in the brain. I rarely study on a daily basis. I usually study for tests, but the day before. My study habits are poorly compared to what this article recommends. Not reviewing what i learn is affecting my understanding. I think that if I acquire the habit of studying, will affect in a positive way my grades.

2. Of all the topics we have learned this year, i have struggle converting word problems into equations, and also with the last topic of cosine and sine. I don’t feel very confident with factorization.

Does the Forgetting curve helpful?

1. After reading the Forgetting curve, i realized that reviewing a topic learned in class would really help me afterwards. I usually never study or neither take extra time to check a problem or a lesson that we`ve learned that day or the day before. Study habbits are usually really important which can efficiently shows your skills in the subject. i rarely study if i dont have a test or quiz, which is not a good thing because if i did, i could be more prepared and could undertand more in the part im struggling.

2. Pre-Calculus has been a struggle for me throughout the year. I`ve noticed that i`m not very good at it, but i think at least i`ve tried to study and learn the concepts, eventhough the effort most of the time is not reflected. working with word problems and graphs have given me a hard time. I sometimes get really confused in the intersection of lines, and the ability to use the domain and range. What is the easier way to understand a word problem and apply it to an equation?

study habbits

1. When I get home, I rarely open a book or a notebook unless I have to for a test or quiz. My study habbits aren't the best because I rely on the information I learn by paying attention in class. For a test or quiz I review my notes from the class and do some problems and if I don't understand I will have some of my classmates explain the problems.

2. Of all the topics we have learn this year I have struggled the most with the latest one on Trig functions and finding the angles. I also struggle with the graphs of cosine, sine and tangent. When we are using graphs to estimate angles, how do I know where to find the x or y value?

Great info

This information about the forgetting curve is very interesting. Sincerely i have never tried to study the first or second day after a new lecture begins. The truth is that I always pay attention in class, never review, and study one or two days before the quiz. But the information really looks convincing and reliable, so from now on I will follow those tips. That way It would be easier for me and get better grades in the exams. But surely after 20 days or more of a lecture I never review it again, just like 3 days before but for final exams. Mathematics is not very hard for me, I think is just matter of review to get really good grades. But I want to ask my classmates of they really know to do geometry if the have an exam now? Beacause in 9th grade we had 4 different teachers. So please answer.

Forgetting Curve & Myself

1. Due to all the experience I have attained throughout all my school years, I believe the Forgetting Curve is pretty accurate. For instance, when one learns something by studying hard, paying attention, and making an effort ( this does not mean making an effort to cram for a test), it is more likely to hold on to all the information learned. In the past, I have learned about U.S History thoroughly, and I am proud to say I know a lot about it. It's all about studying hard everyday and not cram for tests. I can't say I haven't crammed for a test, but when I don't, I get the reward later on when I get a review test or a midterm. When I study for Pre-Calculus, I get good results when I pay attention in class and when I do my homework completely. By doing my Homework I get to notice the kinds of problems I am weak at and the one's I can't do for myself. By doing so, I am able to ask questions in class the next day so I can clear my questions on math problems. I believe my study skills are very good. When I am determined to do good in a class, I would do it by studying hard, paying attention in class, and always doing my Homework. I believe that is the key for success in the school life.

2. In the beginning of the year I had trouble with Pre Calculus because I wasn't used to graphing and the definition of functions. I still have weaknesses in the functions part and when it comes to finding domain and range, I always hesitate. Since I am a very visual person, is there any way I can find the domain and range of a function just by locating x and y?

p.s marig is mari guillen :)

Another Blogging Tip

When creating a new post, use the built in spell check.

We all make mistakes, but remember, what you write goes out to the entire world. You can make yourself look better with just a simple click.

DC's Study Habits

1. Never in my life have I taken time after school to review a topic learned in class. If I've had any intentions to do so in the past I, most surely, haven't had time to do so. I do regret not having such study habits yet but, honestly, I haven't done much about it. The habit I am proud of having is homework, especially math homework. That is my idea of reviewing for math, instead of reading notes over and over again. It allows you to actually "experience" what you learn and understand the "why" to it better. When I do all my math homework I don't have to rush and worry for even half an hour the day before a test because I already understood the topic well.
2. I struggle a bit with the last topic we talked about, specifically the cosine and sine graphs but I don't have any specific question to ask about any topic at the moment. I imagine that if I where to do any past problems from this year I probably would have a question but not right now. I would like to know a bit better factorization and also: how can you solve for the square root of a given number (ex: 153)without using a calculator?

my study habbits

cristifranco2 says...
mathematics is not one of my trenghts, this is why i know i should study on daily basis so I can remember what I learn in class and how to do it again by my self. eventhough i know i need to studdy, i wait until the last day to study for mu quizes and examns. this is why y do poorly on my quizes and have a hard time doing my homework and P.O.D. I think that if i start to prcatice every day what i leanr in class, it will help me a lot to remeber and to understand much better the new topics.
2. i still have some troubles and questions about the domain and range. I tend to get confused with both of them. I also have a hard time with factorization, i tend to forget the right order to solve those problems.

Study Habits Graph

1. My study habits are nowhere near what this article recommends. I do not study on a daily basis, and I seldom study before exams. Now I understand why I usually remember the conpects that I learned the day before much better than those that were learned at the beginning of the grading period. I think that If i invested more time into studying daily it would reflect ina positive way in my exams.

2. Some questions that I have are about finding information in a graph with a cosin or sin. I dont know wether the line must be vertical or horizontal, and if the answer is the number on the x axis or the one in the y.
April 9, 2007 12:21 PM

Forgetting Curve

1. I hardly ever take time to study in my house, and if I do, it would only be if I'm studying for a test. I do take time studying for tests because thats the only way I find efficient to do well in them. The numbers in the graph are not that accurate to my study habits because, even though I don't have the discipline to review in my house, in school I pay close attention to what I learn and why I'm learning it, therefore I retain information for a longer time.

2. The last topic mentioned in class, about sin and cosine graphs,is the topic that has been hardest for me to understand. How do I find the answer I'm looking for in a graph of sin and cosine in regards of finding for X or Y?

Blogging Tip

Give your response a meaningful and relevant title. That way, when the post appears in our blog archive (see sidebar), everyone will know the topic.

You can also attach a label in the bar below the post so future bloggers can more easily search for post. For instance, I attached the label "tip". Later, when more tips have been posted we can find them all by searching for "tip".

See my earlier post labelled Forgetting Curve. Later, I can go back and find it easily.

Scribe post

Today we created our own version of the Tower of Hanoi, a famous math puzzle created by a French mathematician over 100 years ago. Go here to see and try an online version.

By trial and error, we found that the minimum number of moves for 3 disks is 7, and the minimum number for 4 disks is 15.

The original problem, from the 19th century, was to move 8 disks. Supposedly, we can use our results for the 3 and 4-disk problem to find the number of moves for the 8-disk problem. Mr. A asked us to think about this problem, try to come up with an approach, and then post our solutions to this blog.

He also told us that he is no longer collecting homework. Yippeee!!

Instead, we will get a blog grade and he started explaining the whole blog process. We will learn more about this on Wednesday (bridge building all day tomorrow) . He told us to come here for our 1st blog assignment.

That's it. I'm going to go complete my blog homework right now!

Just your average talented, motivated, and happy precalculus student.

Welcome to 1st call

You found it!!!

Welcome to our mutual venture into the land of math blogging. This is the place to come when you did not quite understand that last topic from class, or you were too shy to ask your question, or you want to share an interesting and useful math website or new problem-solving strategy, or maybe just to chat about your math struggles and/or successes.


As with everything, you will get out what you put in.



So let's get started. Here is your 1st assignment.

The first 5 people that respond appropriately will receive a 105% for this assignment. (It will be your job to help others figure out how to get here and post.)

Everyone must respond by midnight, Thursday April 12.

Go here http://www.adm.uwaterloo.ca/infocs/study/curve.html
and read about The Forgetting Curve.

1. Describe how this relates to your study habits. Do those numbers seem accurate? What specific changes could you make which would make learning math easier for you?

Blogging is just one way to revisit a new piece of information. Below each post you will see a section for comments. Use these comments to help and learn from your friends but also as way to review new info. Keep that curve high!

2. Think back over this past year, or even beyond, and identify one or two particular math concepts that were and maybe still are particularly troubling for you. Ask 1 or 2 questions in your post about these topics. Remember, the final still awaits. This is your chance to get your questions heard.

Once you finish your assignment, go here

http://oos.moxiecode.com/examples/cubeoban/

to play a fun and deceptively challenging game. Level 1 is automatic. Level 2 is a quick hello. It's not until level 3 that you will appreciate the game. Remember, this is for AFTER you finish your assignment!

Happy bloggin'.