To the journey…..

June 8, 2010

Thank you all for a wonderful year filled with laughter, challenges, and adventures. I truly feel that I have learned so much from all of you and hope that you are embarking on a journey in a new direction well equipped with all that you need to succeed!

tyffaniphotography-079Picture contributed by Tyffani, 2010, Photography 102.

Besides Math Samples, this is the last assignment of the year! The packet was passed out on Thursday (June 3rd) and we will go over problems together on Tuesday (June 8th). The entire packet is due completed on Thursday (June 10th).

These are sites that can offer example problems and explanations (these will be described in detail on Tuesday):

Trigonometry: What is it good for?

The Triangle Calculator (to help check your work!)

Cos, Sin, Tan

Trouble finding the angle of an “oblique” triangle? An oblique triangle has no right angles.

The law of cosines

This is a simply stated equation, which we overviewed on Tuesday:

c2a2b2 – 2ab cos C.

It looks like the Pythagorean theorem except for the last term, and if C happens to be a right angle, that last term disappears (since the cosine of 90° is 0), so the law of cosines is actually a generalization of the Pythagorean theorem.

Note that each triangle gives three equations for the law of cosines since you can permute the letters as you like. The other two versions are then a2b2c2 – 2bc cos A, and b2c2a2 – 2ca cos B. However, these two versions (just as we previously talked about), are each the same equation solving for a different side (a or b, instead of c).

If you know all three sides of a triangle, then you can use it to find any angle. For instance, if the three sides are a = 5, b = 6, and c = 7, then the law of cosines says 49 = 25 + 36 – 60 cos C, so cos C = 12/60 = 0.2, and, with the use of a calculator, C = 1.3734 radians = 78.69°.

Note: When triangle is obtuse, the cos C is negative. This is ok – just take the cos-1 of your answer to find the angle of C.

Example: Suppose the three sides are a = 5, b = 6, and c = 10. Then the law of cosines says 100 = 25 + 36 – 60 cos C, so cos C = - 49/60 = - 0.81667. The cosine of an obtuse angle is negative. This is fine, and your calculator will compute the arccosine (inverse of cosine, or cos-1) properly. You’ll get C = 2.2556 radians = 129.237°.

To convert radians to degrees, use the calculator below if you are unfamiliar with your Ti-calculator:

http://www.analyzemath.com/Calculators_2/convert_radians_degrees.html

The law of sines

The law of sines is also a simply stated equation:

sin A

a
= sin B

b
= sin C

c

It is a ratio, one side divides it’s opposite angle is proportionate to another side dividing it’s opposite angle. We discussed this briefly on Tuesday also.

Packet: Finish the packet today, then take the Math Work Sample prepared (it is one word problem). The project we were going to do today is canceled due to the rain outside (boo).

Your task is to use your area equations for trapezoids, triangles, hexagons, and other shapes to design a rooftop garden. Tomorrow, we will be viewing different rooftops gardens downtown as examples!

images-1 images-4 images-5

Your report and design must include the following to receive credit:

Due date: TBD (in a few weeks)

1. A thorough explanation of why rooftop gardens are important and how they help conserve energy or provide an example of sustainability.

2. Descriptions and specifics about what local plants are used on rooftop gardens. Also define what kind of plant bedding is needed and how water will be distributed on the roof to water the plants.

3. A rough draft of your roof design (drawn). You must include at least 4 different shapes (can be of any size) and describe the area of gravel or other plant bedding you would need total for each shape. You must also include what percentage of your rooftop is used for plant bedding. Include your work on a separate sheet of paper with each step to earn credit. Then include a list of the plants you will be using on another separate sheet of paper.

Feel free to have paths, pools, fountains, etc. on your rooftop garden. Your building can also be any size that you want!

4. Your final draft should include a drawing of the bedding design and a second drawing of what your roof will look like when the plants mature (please include color, especially if your plants bloom). An example is shown below:

LA2-1

You will need to turn in ALL of your drafts, a bibliography of ALL your sources, and separate sheets of paper of ALL your calculations.

Watch Beach Walk-a-bout .

What were these students looking for? What did you see in the video? What details did you notice about the environment?

Today, we will be taking a walk around Forest Grove.

1. Collect specimens of plants and flowers in public spaces (NOT other people’s yards, please!) and bring these back to class. A minimum of two items (leaves are great) should be saved in your journal’s pocket!

2. Be on watch for any interesting living things.

3. Think of ways to describe your environment.

When you return, you will be spending 10 minutes each on journal entries below:

1. Write or draw a story incorporating (including and using) your two objects.

2. Write or draw an interaction you may have had with the living things which you witnessed.

3. Write or draw freely about your thoughts on this walk. These can be personal, random, revelation, or merely a list of what you saw.

Remember! A journal entry can be ……

… a list….. a drawing….. a poem….. a written summary….. lyrics to a song that YOU wrote….. a sketch or a bunch of doodles….. magazine clippings……or a collage of all or some of these…… you can write upside down, different directions, or even in a swirl…… draw with markers, pen, pencil, pastels, etc….

On the back of your worksheet, it asks to solve a group of problems by substitution and another group by elimination.

To solve a system by substitution, look at the examples on the Substitution page.

To solve a system by elimination, look at the examples on the Elimination page or watch the step-by-step video on Mr. Vandervest’s blog.

Check your graphing on the front page before turning in the assignment to the pocket by the door today.

Why are cells so small?

April 28, 2010

large_white-cells

We have studied Surface Area for the past few weeks, now we are going to continue to investigate Volume. Above are pictures of a human’s white blood cells. White blood cells patrol our blood stream, and consume hazardous bacteria or other intruders. Besides the given reason that “otherwise the cell won’t fit in our veins,” why are these white blood cells so small?

Answer the following questions on a separate sheet of paper and turn your finished work into the pocket by the door when finished.

1. What is a Surface Area to Volume Ratio?

Check out definitions on Wiki, Webster’s, and Google. Write down a definition from each.

2. Click on the following link to investigate how surface area and volume change.

Surface Area and Volume of Polyhedrons

Move the bars labeled “Width,” “Depth,” and “Height” and watch as the surface area and volume of your polyhedron change. List three observations.

Now your turn – In the right hand upper corner of the interactive’s screen, there are two buttons for Mode. Click on the button for “Compute.” Now it is your turn to calculate the Surface Area and Volume of your polyhedron. The width, height, and depth are already given for you. Check your solutions.

Your job is to list the height, width, depth, your work and solution, and the answers for 5 problems. Simply click on “Next Question” on the bottom of the interactive to see another problem.

3. Watch the Quicktime video Cells. Write down 5 notes in complete sentences.

4. Answer the question based on what you have found today: Why are cells so small? Use reasons from what you learned from the video and include an example from the polyhedron interactive (2) that supports your answer.

Your assignment for the week is to create a self portrait.

Look at these following examples:

Using REFLECTIONS with camera in hand:

20040804

You can use objects that you feel “define” you or your interests (above).

_mg_3347

You can use really small items such as ball bearings or marbles for an effective self portrait – depending on how you position your reflector! (above and below)

Escher Self Portrait

Use objects already positioned around you and set the timer to your camera. The trick is to avoid including the camera in your shot (above). Car mirrors are another great object to use.

Self portraits using COMPUTER SOFTWARE:

project1_0526

You can create a collage of different views (notice you cannot see her face above) or a collage with different objects of value to you. And, as above, it is alright to include shots upside-down.

about_selfPortrait

A person can paste multiple shots together within the same image. Notice you cannot see the camera in the photo above.

bl_w_portrait_nick_07

You can also hold objects or pictures up to your face to represent a different mood (see above). At Nana Cardoon’s, one student took a picture of me when I held a giant cabbage in front of my head (below).

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The woman photographer (below) created a surreal effect by holding two fish – do you notice that the eyes of the fish are right where the eyes of the photographer are?!

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_mg_8537

Be sure to create a contrast in your photo. You can do this by isolating the color and making the rest of the picture black and white (as above).

Your Assignment:

Create 2 professional quality images for your self portraits. One must be a reflection and another must be using computer software to edit. Also, one of these must have the camera hidden from view in your image. Don’t forget to check out the timer setting on your camera!!!

DUE: Monday, April 26th

Also, be sure to update your blog for Nana Cardoon’s from last week.

Homophones are words that sound the same, but have different meanings. Read “Homophones” and then scroll towards the bottom to reveal questions. Which sentences are correct for #18, #66, #67, #75, and #77? Check your answers with the button to the side and press the “voice” button to hear the word pronounced.

1. Play the Homophones game. PRINT your final score and place it in your portfolio. You can play the game again until you get a perfect score! (out of 50 points)

2. A homograph is a word that has multiple meanings and can be used in many ways. Play the Homographs game. PRINT your final score and place it in your portfolio. You can play the game again until you get a perfect score!

3. ASK for the first Homophones Crossword. This is to prepare for the second crossword. Place the sheet with your answers into your portfolio.

4. Play the Homographs Jeopardy Game with a partner or by yourself. Click on “one-player” or “2-players” to begin. PRINT your scores (if you had a partner make sure each person gets a copy) and place your scores in your portfolio.

5. Try to finish the Homophones Crossword. PRINT the final puzzle (as far as you got today). You can turn on the “easy mode” that will tell you if your answer is correct and use the homophones dictionary provided at the top of the page or look up your own!

6. Take the Homophones Quiz. PRINT your score and place it in your portfolio.

YOU SHOULD HAVE 5 ITEMS IN YOUR PORTFOLIO TODAY! This is worth a total of 100 points.

1. Homophones Game scores

2. Homographs Game scores

3. Homophones Crossword 1 answers

4. Homographs Jeopardy scores

5. Homophones Crossword (whatever you have finished)

6. Homophones Quiz score

Congrats! You have moved on to Step 2: understanding exponential equations! Your first worksheet has problems in which exponents are multiplied or divided.

See What to do with exponents! <——-

This site has step-by-step demonstrations – just click on “play” and watch what happens. You can press “play” again to see it over and over if you like. Turn this worksheet in when you are finish and we will go over it on Monday!

Exponential Decay

March 29, 2010

What is exponential decay?

Exponential decay occurs when the rate of change (the slope of the graph) decreases over time. This means that your line changes slower and slower as time goes on.

The equation we use for exponential growth is 

a = the initial amount (the amount you start with)

r = the rate of decay that is often in decimal form (100% = 1.00, 50% = 0.50, and so on)

x = amount of time that has passed

For example, each year the local county club sponsors a tennis tournament. The first round starts with 128 participants. During each round, half of the players are eliminated. How many players remain after 5 rounds?

Start with what you know.

You know there are 128 players to start with. You also know that each round, half (50%) of the players are kicked out. And you know how many rounds there are (5).

Plug what you know into the equation (below).

At the first round, you have 64 players left. When round two is over, you have 32. When round 3 is over, you have 16. When round 4 is over, you have 8 players. And when round 5 is over, you have only 4.

Notice the total is cut into half each time. This is a great example of exponential decay.

We are going to continue to look at exponential graphs. Graph the information that you find for each interactive below.

Math in the Real World

Due: At the end of the period. Draw your answers on graph paper.

The atoms on your paper, that make up the computer you are using, and that make of your very body are decaying at a slow rate exponentially. Check out this interactive that displays what happens as atoms “die” away –> Death of Nuclei

You should have seen an example of what the graph looks like. Now it is your turn to graph the following decays:

1. Look of the height of a bouncing ball (below). On the first bounce, the ball was at a height of 4 feet [and at zero seconds]. Graph each bounce of the ball shown.

2. I have placed a bowl of hot soup or water or top ramen in front of you. You know as well as I do, that the longer I let the bowl sit, the cooler the soup inside will become. The reason is that the soup is losing heat through what is called conduction – the transfer of temperature from one object to another. The heat in the soup is transferred to the air, the bowl, and to the table under the bowl.

If I set a bowl of soup in front of you that is 200 degrees F, and it looses 15% of its heat every minute… how many minutes will it take to cool the soup close to room temperature? (room temp is 72 degrees F) Graph your answers.