Review

I hope this will help everyone who forgot their books...

An arithmetic sequence is the difference (as in subtracting), between any two consecutive terms, and it's a constant.

A geometric sequence is the ratio (multiplication) of any two consecutive terms, and it's a constant.

A recursive definition of a sequence gives the FIRST TERM and then a RULE for getting the nth term form the previous term(s).


Finding the nth term of a:
Geometric sequence-> Tn= T1 x r^(n-1)
Arithmetic sequence-> Tn= T1 + (n-1) d

Assigment

Well Rumidog, seems like you are not the only one working today, or atleast right now.
As an answer to the assigment,
1. a question I have about the topic would be, how do you write an explicit definition? Today in class I saw the formula (Arithmetic: {Tn-1+(n-1)d}, Geometric: {(T1)r^(n-1)}), but I don't understand what does it mean.
2. A mistake I'm most likely to do would be, how to get the function of a sequence. I know how to do it, but sometimes it is easy to make a mistake (Example: today during the POD).
3. What is the explicit definition for the following sequence 3, 13, 23 (3: 2nd term, 13: 9th term, 23: 16th term)
another question: will the quiz include graphs? I HOPE NOT!

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.

Prompt Answer

What questions do you have on this topic?
I only have one question actually, when we use the first form of the explicit formula. Y=mx+b to find the nth term, does it wrok for arithmetic, geometric, complex or all kind of sequences.

What mistakes are you most likely to make?
The most common mistakes I make, are never big mistakes. I tend to miss very small things, like decimals, negatives or small algebraic mistakes. But, as long as I know the theoretic form of the problem, solving it comes pretty easy. This works as a huge part of the points given per problem.

Example problem:
Assuming an arithmetic sequence has 3rd term of π, and a 6th term of 5π/2.

• Find the explicit definition of the formula.

Solution

1. Find # of jumps? 6th term - 3rd term = 3 jumps

2. Find the the total difference.

(5π/2) - π =3π/2

3. Now that you have the # of jumps and the total difference. Divide to get the difference per jump.

(3π/2) /(3/1) =π/2

4. No that you have the difference you have to find T1 in order to find th explicit definition.

The 3rd term is π and the difference, so if you subtract the differnce 2/π twice, then you will get the first term.

π - 2(π/2) = 0

5. Given the arithmetic formula. Tn= t1 + (n-1)d

T1= 0

Tn= (n-1)π/2

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.

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.

Question

I just wanted to know what are the topics for Wednesday's quiz?
Thank you..

Finding Solution to Parabolas

Examples:
1.
x^2 + x = 0
To find the solution to this equation, you need to factor out the greates common factor which in this case is x.
x(x+1) = 0
Now you need to equal each of the terms to zero and solve for x
x = 0
x+1 = 0
x = -1
Therefore the solutions to these equations are 0 and -1

2.
x^2 + 2x + 1
In this example you can't factor out any term, so you have to use factoring.
(x+1)(x+1) = 0
Like before, we need to equal these terms to 0
x+1 = 0
x = -1

3.
3x^2 + 12x + 12 = 0
At first glance it may seem very hard to factor this equation, but if you factor a 3 first it will be very simple to factor.
3(x^2+4x+4) = 0
3(x+2)(x+2) = 0
x+2=0
x=-2

4.
4x^2 + 7x - 3 = 0

In this case it seems really hard to factor and we can't factor out any term. Basically, we need to use the quadratic formula:

In the formula, a is the first constant, b the second one and c the third one (ax^2 + bx + c)Solving for x we obtain:

x = (-7 + 9.8488)/8 and x = (-7 - 9.8488)/8

x = .356, -2.106

Complex Sequences

Currently in Mr.A's class you have studied the behavior and formulas of geometric and arithmetic sequences. Many sequences however don't follow either of those two patterns, in this post I will be trying to help you understand how to come up with the formula of some sequences that don't follow those patterns.

Example 1
1, 3, 6, 10, 15, 21, 28, 36, 45, 55....

As you may see, this sequence doesnt have a common number being added to or multiplied to get the next number in the sequence.

Step1
The first step to finding an explicit formula for the sequence is to find the difference between each number in the sequence, andthen finding the difference of the differences, until the number you come up with is the same.
In our example it would work like this:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55
2 3 4 5 6 7 8 9 10
1 1 1 1 1 1 1 1

Step 2
The second step is to identify what the number of rows mean. The numbers on the second row weren't the same, so we have to repeat the process. On the third row the numbers are all the same (1), meaning that we have achieved our goal. Now, what does this all mean? Well, the number of rows of differences is the degree of the equation. In this case we have two rows, meaning its a second degree equation (ax^2 + bx + c, were a,b,c are constants).

Step3
Believe it or not we have all the information we need, now we just need to make some equations and solve for the variables a, b and c. On the formulas were about to create, X will represent term in the sequence, so when we plug 1 into x the answer we need to obtain is 1, when we plug 2 into X the answer we need to obtain is 3, etc.

Therefore,

ax^2 + bx + c = 1 when x=1, so a + b + c = 1 (1)
ax^2 + bx + c = 3 when x= 2, so 4a + 2b + c = 3 (2)
ax^2 + bx + c = 6 when x = 3 so 9a + 3b + c = 6 (3)

Step 4
Now we have three equations with threevraiables, we only need to solve for the variables. The method you use is completely optional, I prefer to use elimination.

a + b + c = 1 (1)
4a + 2b + c = 3 (2) Multiplying equation (1) by (-1) and adding we obtain

3a + b = 2 (1-2) We will use this equation later.

a + b + c = 1 (1)
9a + 3b + c = 6 (3) Multiplyin equation (1) by (-1) and adding we obtain

8a + 2b = 5 (1-3) Using this equation and equation (1-2) we can solve for one variable

3a + b = 2 (1-2)
8a + 2b = 5 (1-3) Multiplying equation (1-2) by (-2) and adding we obtain

2a = 1
a= 1/2

Plugging a into (1-3)
8(1/2) + 2b = 5
2b = 1
b = 1/2

Plugging a and b into (1)
1/2 + 1/2 + c = 1
c= 0

Replacing all the values for the constats our final equation for our sequence is:

(x^2)/2 + x/2 = nth term of the sequence

Simplifying:
(x^2 + x)/2 = nth term

Factoring an X we obtain the final equation:
(x)(x+1)/2 = nth term

This is the equation to find the sum of the first x integers. Which, if you refer to the original sequence, you will notice that that was the pattern the sequence was following.We can now test this equation to find the sum of the first 8 integers.
(8)(8+1)/2) = 36
1+2+3+4+5+6+7+8 = 36 so the equation works.

Example Problem:
Now that you know the steps to find an explicit equation, you should try to find the equation for the sum of the first n integers squared. The first terms of the sequence are:

1, 5, 14, 30, 55, 91...

Uses of Logarithms

Logarithms are mostly used in calculus and physics.

In organic chemistry, in twelfth grade, you will use logarithms to find pH. In physics, noise, earthquakes, star brightness, sensation, etc, are measured using logarithmic scales. Computers also use logarithms.

All of the scales mentioned above, however, use base-10 logarithms. The most common logarithm you will come across that is not base-10 is the natural log or ln, which is base-e. Ln will be used a lot when integrating, but you don’t have to worry about that until your second semester senior year.

The only time I’ve used logarithms based some other number different than 10 or e, was when looking for the inverse of functions. For instance, the inverse of 3^x is log base-3.

Still, there are different types of logarithms used in science and engineering. There even are imaginary-based logarithms, but I doubt any of us will encounter those in the years to come.

Reflecting Functions

Reflections are quite simple. When you reflect something, you are basically flipping it across a given line. There are many types of reflections, depending across the line you are reflecting across of. However, there are two types of basic reflections: about the x-axis, and about the y-axis.

Reflecting about the x-axis means you are rotating the function across the x-axis. This is done by multiplying the function by -1, so that you are left with –f(x).
For example,
The reflection of f(x)=x/2-3 will be the same as multiplying f(x)=x/2-3 by -1, which is –f(x)=-x/2+3
Graphically, it would look like this:

f(x)=x/2-3

–f(x)=-x/2+3Another example:

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

To understand this type of reflection better, think of it the following way. By reflecting about the x-axis, you are multiplying each value of y by -1. That means that the reflection about the x-axis of the point (1,1) would be (1,-1). Using variables, this is the same as saying that the reflection about the x-axis of the point (x,y) is (x,-y). You can prove this using the graph above. Each value of y gets multiplied by -1 while x stays the same. Therefore, if the original function is f(x), the reflected function about the x-axis will be -f(x)

Reflecting about the y-axis means you are rotating the function across the y-axis. This is done by replacing x by -x, so that you are left with f(-x).
For example,
The reflection of f(x)=x/2-3 will be the same as solving for f(-x), which is f(-x)=-x/2-3
Graphically, it would look like this:

f(x)=x/2-3

f(-x)=-x/2-3

Another example:

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

To understand this type of reflection better, think of it the following way. By reflecting about the y-axis, you are multiplying each value of x by -1. That means that the reflection about the y-axis of the point (3,0) would be (-3,0). Using variables, this is the same as saying that the reflection about the y-axis of the point (x,y) is (-x,y). You can prove this using the graph above. Each value of x gets multiplied by -1 while y stays the same. Therefore, if the original function is f(x), the reflected function about the y-axis will be f(-x)

Review:
If f(x) is a function, then
-f(x) will give you its reflection about the x-axis, and
f(-x) will give you its reflection about the y-axis

The following link contains other examples: http://www.themathpage.com/aPreCalc/reflections.htm

If you would like some help on other types of reflections (about the line y=x, y=-x, etc), please let me know.

Factorizing Parabolas

Factoring parabolas comes especially useful later on in Calculus. Right now, factoring parabolas is used, primarily, to look for the solutions of the parabola, or where the parabola intercepts the x-axis. There are certain word problems that will require you to look for these values. However, this process will become more useful in Calculus when you learn optimization. Factorization, in general, is extremely helpful in calculus in order to simplify problems.

Trig Functions in the Real Life

There are many concepts you learn during Pre-Calculus that you will not know what they are used for at first. Pre-Calculus is taught so that you learn different concepts that will be useful later on in Calculus and Physics.

Trig functions, like any other math concept, are used to explain the world around us. I don’t know if Mr Alcantara gave you a problem in which you had to find the distance, with respect to time, that a toy train going around a circular track was from its starting point, or another problem in which you had to find the distance, with respect to time, Mr Alcantara was from Julian while he swung him on a swing. This was all done using trig functions.

Trig functions are used in physics to describe sound, frequency, vibrations, waves, tides, light, pendulums, and more. Anything that has a repeated pattern of motion can be described using a trigonometric function. Also, they are used to find coefficients of friction between two bodies. Trig functions are also important in geometry when working with triangles. Economists also use modified trigonometric functions to describe how different variables like offer and demand, behave in a given market. Trigonometric functions are used in all fields of study.

The following link has a video of a funny looking professor giving some examples of how trig functions are applied to real life. I didn’t watch it all but you should check it out:
http://www.coolschool.ca/lor/PMA12/unit4/U04L05.htm

Geometric Sequences

Mpooh, you asked in the problem of the geometric sequence if square root of two over two was the ratio between 2 and square root of 2. To figure out this you must do one simple step. If you have a geometric sequence you must always be able to multiply the n term times the ratio and get n+1. n *r=n+1. If you are given both n and n+1 you just have to solve for the variable. So, 2 * r = negative square root of 2. The answer will be negative quare root of two over two. This way you will know it is the ratio, only if you have been told it is a geometric sequence. On the other hand you asked for what is a ratio. It is aA relationship between two quantities, normally expressed as the quotient of one divided by the other. For example the ratio of 8 and 5 in 8/5.

Friday's Slides 4/20/07

We started with another application problem about a laptop that loses 25% of its value every year (see 1st slide). Jaime pointed out that this was the same kind of problem that we used to solve using exponential functions. Now we can solve it as a geometric sequence problem. That makes sense since the general formula for a geometric sequence (do you know it?) is an exponential function.

Then we explored some of the hidden nature of Fibonacci's sequence by drawing Fibonacci rectangles and then making a Fibonacci spiral (see slide 2 for a somewhat messy version of the spiral). It seems that many organisms, such as the nautilus, sunflowers, pine cones, and cauliflower have Fibonacci properties. They exhibit clockwise and counterclockwise spirals like the ones we made. The number of spirals were always consecutive numbers from the sequence. Kris pointed out that, at least for the ones we looked at, the number of counterclockwise spirals was always the greater of the two Fibonacci numbers.

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Larisa asked how a problem about rabbits from the year 1202 turns up in the seeds of sunflowers. I'm wondering what other people think about this strange Fibonacci pattern turning up in the structure of plants and animals. Coincidence? Something more?

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, and the math to back it up, for how many Earths might fit inside Antares?

Thursday's Slides 4/19/07

Today we continued working with geometric sequences and recursive definitions. The first slide shows an application of a geometric sequence to a real world problem. The trick is to turn the word problem into a simple geometric sequence problem. Finding n was trickier than expected and was the key to the problem. Once we made a table then it was pretty obvious that n = 11.

The rest of class was spent making recursive definitions. One problem missing from the slides was finding the recursive formula for the Fibonacci sequence. One formula could be
t(1) = 1, t(2) = 1, and t(n) = t(n-1) + t(n-2), for n > 2.

Another way to write the Fibonacci function is:
F(0) = 0, F(1) = 1, F(n) = F(n-1) + F(n-2), for n > 1

The last slide shows a problem to try at home.

Make the table!


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Get bloggin'!

Blog Tutorials

Here is to those who are having problems with factoring. Factoring polynomials is divided into different steps. I am just going post a Factoring Strategy, some B.K. ( background knowledge) on the GCF and further practice problems.

1) GCF- the largest monomial that divides into each of the terms in a polynomial .
- That means is that to find a number (monomial) that divides ( is a factor) into all of the terms in the equation (Polynomial).

For example with 14x<5-4x<3+2x,>
((NOTE: <>

Practice:
Factor out the GCF: .x(3x-1)+5(3x-1)

The Factoring Strategy :

ALWAYS check for th GCF first.
II. Binomials:
a. x < 2- a < 2 = (x + a)(x - a)
b. x<3+a<3 = (x + a) (x < 2 - ax + a < 2)
c. x < 3 =" (x">
III. Trinomials:
a. x < 2 + bx + c
Perfect square trinomial:
a < 2 =" (a">
a < 2 =" (a">
IV. Polynomials with four terms:
Factor by grouping

PRACTICE:
Factor 27a < 2 + 36a + 12 completely.
Factor y < 4 – 16 completely.
Factor x < 3 + 64y < 3 completely.
Factor 6x < 2 – 17x + 5 completely.
Factor x <>

Tiling Response

Hello Calculus 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.

Wednesday's slides 4/18/07

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 someones notes 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. Sofia 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 first slide is 511. The second 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

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.

Luis DP's Answer to JBarrika

Dear JBarrika,

Your question is: What are trigonometric functions like sine, cosine, tangent, and the inverses really useful for?

Trigonometry is essential for the human knowledge of space. Trigonometry has applications in almost all natural and social sciences. Engineering, physics, astronomy would be in huge trouble without the existence trigonometry or trigonometric functions. Let me give you an example. As you may have noticed before, curves in roads or highways are slightly slanted in the direction in which they are concave down. Civil engineers determine this slant angle depending on the acuteness of the curve. The sharper the curve, the greater will be the angle formed by the horizontal and the road. This is a complex trigonometric relationship that civil engineers take into account during road and highway construction. This slant angle generates a horizontal component of the normal force that counteracts the centrifugal force. As you may have noticed, many roads in Colombia lack a significant slant angle, making vehicles traveling at high speeds vulnerable to fly out of the road in the direction in which the curve is convex up.

Extra Practice

My partner and I did an arithmetic sequence. The first four terms were: { 5, 2, -1, -4, ... } Find the tenth term.

Take the following into consideration:
Arithmeticc Sequence: The difference between any two consecutive terms is a constant.
Tn = T1 + (n - 1)(d)
Geometric Sequence: The ratio of any two consecutive terms is a constant.
Tn = T1( r^ (n - 1) )

I'de be glad to help anyone who doesn't understand.
Also, for extra practice Mr. A assigned Pgs. 473- 475 # 1-9 odd , 17-41 odd



Answer: Tn = 5+ (n- 1) (-3)
T10 = 5 + (10 - 1 (-3)
T10 = - 22

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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Then we worked in pairs to create sequences for each other to solve. Everyone seemed to get it.

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.

Tiling Question

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?

Larisa's Question

Larisa asked how to add labels to an old post.

At the bottom of each post you will find a pencil icon. Clicking on that icon allows you to edit your post. In edit mode you can add labels, correct math errors, or fix spelling and grammar. Separate labels with spaces not commas. Each label must be a single word. After adding the label(s) click on "publish" and then check to see if the label appeared. You may need to reload or refresh the page.

Luis DP's Answer to Maria Manzur

Dear Maria Manzur,

Your question is: How can I difference trigonometric graphs? I cannot differ which is a tan, cos, or sin graph. Do I have to memorize them?

Remember that a graph is described by its function’s equation. Once you understand the process behind each trigonometric function, you can reproduce their graphs. Make sure not to memorize information that you can deduce from the basic concepts. The graphs of sin(x) and cos(x) are similar. To obtain the cos(x) graph, you must shift the sin(x) graph π/2 units in the negative x direction. To obtain the sin(x) graph, you must shift the cos(x) graph π/2 units in the positive x direction. The easiest way of distinguishing between the sin(x) and cos(x) graphs through observation is by determining their value at the origin. Cos(x) “starts” at 1, but sin(x) “starts” at 0. In other words, cos(0)=1 and sin(0)=0. The graph of tan(x) is quite different. It holds characteristics such as tan(0)=0 and vertical asymptotes at x = π/2 + πn (n any integer). But again, just memorizing this sets you apart from someone who can derive it from more basic knowledge.

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)

Then Mr. A. gave us 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.

Trigonometric Inequalities

Sir, I tried to test the inegualities of the trig functions. The only way I came up to find a solution was by working with trial and error. I tested 15 different inequalities for every trig functions and I had pretty differetn results.
I found out that for cosine and secant you cant swith the inequality sign and it will work, I tested it with negative angles and second revolution angles as well, and it worked all the times but I ami still trying to find out why. On the other hand with tangent, cotangent, sine and cosecant I figured out you dont have to change the sign, It will work just the way you have the inequality from the beginning.

The Forgetting Curve

1. This information was very interesting to read about. I had no idea about the amount of information that is lost each day. That may explain why while during finals I forget about certain topics. I do think that this information depends on the person. Some people can understand something once and it stays with them much longer than it would with another person. However, most students that are willing to do the homework will understand the topic much more thoroughly.
2. I am still very confused about the graphs of the trigonomic functions. What are trigonomic function graphs used for? Also, how can people/I use first revolution angles in my daily life?
April 10, 2007 4:55 PM

Forgetting Curve

1. The numbers in the graph seem reasonable. I am not very good in math, I really try to understand in class and make notes but I don´t study at home. I will try to check my notes once in a while in order to do better.
2.What is the difference between refernce and first revolution angles? Today we talked about them in class and I didn´t understand how they are related.

Forgetting Curve

1. I had never realized how things I read took this short time to get forgotten. Normally during class I take notes in order to study for tests although my math skills aren’t very good. Even thought I have the notes to study I don’t do it as much as I should. I really want to improve in this class an in order to do it I have to acquire study skills that will refresh my mind with certain topics I don´t dominate.
2. How can I difference trigonometric graphs? I cannot differ which is a tan, cos, or sin graph. Do I have to memorize them?

The Forgetting Curve

1. The forgetting curve seems very close to accurate. When i tend to learn a subject in which i am interested this doens't really relate because i understand the whole basis. Nevetheless most of the time studnets including me dont really take the time to get to the bottom of why things are that way and just learn to do the visual part because of memory. When i just memorize things but dont really know why, i tend to forget them really quickly. This technique of reviewing seems really reasonable and preoductive to keep my mind functioning on the subject and i know everyone is going to say they will do this from now on but we are really lazy and need to actually learn the habbit.
2. * How can i understand the inverse function graphs without confusing the angles and the y coordinate meaning sine, cosine, etc?

forgetting curve

1. Well if i relate the Forgetting Curve to my life, I would say that it doesnt characterize me. It also depends because I consider myself to be a very organizd and responsible person. I even taught myself how to review my notes before for example going to precalculus so I could have a better comprehension of the topic and eventually long-term retention. In Readak, a technique has also been taught to us and that is of r=r which menas repetition equals retention. The more a student repets, in any way, his notes will eventually retain the information in his long-term memory. However I dont dismiss the fact that sometimes the information may seem overwhelming and the studying too exhausting, still i support the fact that if every student was to at least review or skim and skan through his notes daily, the learning process will be easier.
2. One question I have is regarding the topic about logs. I dont understand how is log, lets say with anothr base besides 10 important, beneficial or useful in our lives, or even in math itself?

the forgeting curve

1.This doesnt relate to my study habbits because i dont study enough time. And the numbers seem accurate to me, and im going to try to spend more time studing.


2 Two particular math concepts that i had problems before in the school, were getting y by its self, and also geometry. One question that i would like to ask is if people know geometry.

Forgetting Curve

1. I'm not really sure if the numbers are 100% accurate, but the truth is that when I do daily homework about a topic for atleast 20 minutes, it's really easy to remmember it in the future; it even allows me to be more fluent and fast (atleast in math).

2. I'm still confused with, how to get the 'first revolution angles'? and how to read the 'log graph'?

The Forgetting Graph

1. The numbers seem pretty accurate and it does relate with my study habits. I never open my notebook at a different time than in class. Sometimes i forget small details when im takin a quiz. Im going to try to go over notes and do more homework than usually, eventhough there is a small percent of probabilities(about .001%). Just joking!

2. I have not beem too familiar with roots this year, thank God we are over it. I think i did so many mistakes because problems were so long and had multiple solutions. Right now i have no questions, i think all the questions I had, I asked them to you right away during class.

The Forgetting Curve

1. the forgetting curve represents a very common issue in a students study life. i think it is more or less accurate. sometimes we forget what we have been taught even the day before, and the concepts you have been taught can only stay if the student more or less takes in mind those concepts not only by studying but also applying them. maybe a more effective way of studying mathematical concepts could be based on practicing some exercises or reviewing so the concept wont forget completely and obviously by trying to understand the concept as more as possible.2.this year the topic that has caused most problems is reflections. i didnt really learned all the factors that affect or that change and take place in a reflection. id also like to have a clearer view of concepts taught in geometry since i didnt learn too much that year.

Changing names

Sir, I changed my nickname to lers... it's larisa.

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.

The Forgetting Curve Graph

1.- In class, i spend all my time trying to understand the topics that we are talking about, so, at that time i know the 100% of the topic, but my biggest problem is that when the class is over, i read the notes until the next quiz, so it provokes that i failed the quizzes or exams. Whit this graph, i could understand what was my problem at the time to do my quiz, so that numbers are accurate.

2.- One of the biggest topics that were difficult for me, were the cos, sin and tan graphs, and the snowman sheet; But thanks of the forgetting curve graph, i can trade to improve my grades on quizes and exams.

curve graph

1. When we study things on class and we don’t review them again, is easier to forget what we had studied. When we study for mid test or final test we have to review a lot and sometimes study it again because we don’t even remember when we study the topic. I think that the numbers of the curve are accrued. After 30 days of learning a subject we only remember 2-3 percent, but when we see it again you remember it. I think the percentage changes depending in the person. Some people have a better long term memory.
2. The graphs are very confusing for me especially the ones of sin, cos and tan. The inverse graphs of the sin,cos and tan are harder to understand and is easier for me to get confused.
(Elizabeth)

Maintaining my curve

1. The information explaining the graph relates to my study habits in some way. I am not the kind of person who tends to review things after 30 days, but, for instance, if someone asks me a homework question I did a week ago, I’d have to look at my work in order to remember. The numbers in the graph seem correct, which has made me realize one good way I can keep my curve high. It will obviously not be my first choice, but in the long-term effect it looks more practical and beneficial.
2. When we first started talking about special triangle rules, I didn’t really understand, and, not to my surprise, I got stressed out when I lost one day of class and came back understanding even less. Then, another subject I couldn’t manage completely was the snowman sheet, which relates to the special triangle rules. Thankfully, one of my friends explained it to me and I realized it wasn’t that hard. I have managed to see in me a defect that I am trying to fix, which is that when I see something that looks complicated, I freak out a bit and close my mind, but at the end, I understand that if everyone else can do it, so can I, and maybe, even better.

Forgetting Curve Graph

1. Actually this number look pretty accurate. I dont usually study on a daily basis, what I learn one day I onle use it again for the finals. When the finals come I do get to some topics taht look pretty weird. The changes that are mentioned before are pretty good, personally I wont do them. To say, study 10 minutos each day looks easy, but so much information of many things at a time is complicated. Maybe reviewing once every week might help as well.

2. I actually have two questions related to the past year. What are trigonometric functions like sine, cosine, tangent, and the inverses really useful for? Second,Id like to know why is it important to learn to factor parabolas?

graph

1. I’ve found out that when we are taught something in class we think it is just enough to learn, and that with the explanation of the teacher we be able to do well in the exam. But the truth is that we just forget about the topic if we don’t talk about it again in class and for the day of the test we have to read the material again for it to be stuck in our head. I’m not sure if the numbers given are accurate but I actually believe and agree with the fact that if we don’t review at least 10 minutes per day the information will not remain in our mind at least not as accurate.

2. Graphs at the beginning where simple and easy, but after a while when we started to see the graphs of sin, cos, and tan etc, and we started to use graphs in a different way to interpret different confusing problems. Graphs are causing me a lot of confusion and making it harder to understand many exercises.

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.

I forgot to say that it was very interesting to hear about the Forgetting Curve. Thats maybe the reason why I forget certain things and then I mess up in the quiz. Im definetely going to take the information as an advice for my study habits.

1. I always write notes in class to have a better understanding of the topic and sometimes they help me when the day of the quiz is arriving, even though I dont really have habits of study. Since I decided to get better grades at precalculus, which is the most difficult subject for me, im having extra precalculus classes two or three days per week about four months ago and they have definetely helped me a lot.

2. Something I trouble with ALWAYS are graphs. Surely, I dont know how to interpret them and I dont quite understand when they talk me about them, unless its something basic. I would like to hear ideas on how I could get better at understanding and interpreting graphs.

MARTHA ALCOCER (MORDOR)

The Forgetting Curve Answers!!

1. This number is very accurate since all the information given is true. This is the way I study, a re-read each day so I don’t forget anything but I make sure to understand instead of remembering things. Some changes I would make to learn math easier is study one or two topics per day and make some practice problems, since math is the hardest class to study because of all the equations given. But math is base in understanding why things are done, not just of learning the equations.

2. Some topics that were troubling for me: how to interpret percentages to find its results? How to make an analysis and interpret graphs of trigonometric equations?

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

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 person to respond appropriately will receive a 105% for this assignment. (It will be your job, if needed, to help others figure out how to get here and post.)

Everyone must respond by Friday at 2:30.

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? What strategies do you use that might benefit others?

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 a way to review new info. Keep that curve high!

2. Think back over last 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. 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'.