"Small World" Portfolio and Unit Assessment

Any questions about the portfolio? The unit assessment?

Here are the requirements and the rubric for the unit portfolio.

Here is a pdf of supplemental materials from a Precalculus course at Arkansas Tech University. Sections 1-20 are all applicable to our Small World Unit.
For Example:

• Page 11 has examples on Rate of Change
• Page 15 discusses Linear Functions
• Page 18 goes into formulas of Linear Functions
• Page 30 show how to find the Inputs and Outputs of Functions
• Page 51 discusses Exponential Functions
• Page 55 compares Exponential Functions and Linear Functions

If you want to "read" the book that the Arkansas Tech University Precalc class uses, here is a preview. If you want to actually buy this book, buy it used from a used textbook site like alibris. I think it is a really good reference book.

Here is anther pdf comparing Expnential and Linear Functions.

Strategies for solving Small World Unit Problem

Yesterday we said the general form for exponential functions is:

Today in class we discussed several strategies to help us solve our Unit Problem for Small World.

Recall that we need to find out when the population will reach the point that we are all "squashed up". Earlier we said that this will happen when our population reaches 1.6x10^15 people. We decided that population growth was best modeled by an exponential function.

What strategies did we discuss?

1. One was to input our data from the tables into the calculator. Here is a sheet of instructions to enter data into the calculator. Then we could guess equations that fit our data the best -- like "Tweaking this Function".





2. Another way was to do something similar to HW25 and HW29. Use two of the data points and solve for the variables k and c. Some people said that in order for this strategy to work, we would have to set one of our years to 0, just like HW25 and HW29.





3. An even different approach is to use the calculator's regression feature. I am not advocating this approach, but it is a valid approach. However, you may need to do some calculations to convert the equation into the form we want the final answer to be in.

Ok.Ok.Ok. Say we use one of those strategies. And say we get an exponential equation that models our data really well. Then what? Are we done? Have we solved the unit problem?

Oct 28

In class today we went over HW28: The Limit of Their Generosity.

Yesterday, Adam deposited $1000 in the bank. At first the bank was going to double his money in 20 years.

Adam persuaded the bank to instead of giving him 100% interest at the end of 20 years to give him 5% interest each year for 20 years.

Adam them persuaded the bank to calculate the interest every 6 months and then persuaded them to calculate the interest every 3 months.

Adam noticed that he was getting more money the more often the bank was calculating the interest.

In HW28, Adam tries to persuade the bank to calculate the interest on a daily basis and then on an hourly basis. In class we even calculated on a minute basis.

What discoveries did we make? Please share your results (several students were absent from class b/c of PSAE testing and would benefit from sharing results).

Lastly, we tried to solve this equation: 2^x = e^?
Can you find the value of the question mark?

History of e

Today in class some questions about the history of e came up.

Here is link that descibes how e came about and several different ways to determine the value of e.

Here is another link about e.

And That is Why I Succeed

One of my favorite commercials



Air Jordan Air Jordan Air Jordan


Do you know Do you know Do you know


Maybe...

Small World Portfolio

We are a couple of days away from finishing our Small World Unit. With that, we will have a Unit Assessment and we will create a Unit Portfolio.

Remember this sheet I passed out at the beginning of the unit?

Well, now is a good time to go through the sheet and identify homework assignments, classwork assignments, and quiz questions that relate to the outcomes/goals.

For your portfolio, you will be required to select either a homework or classwork for each outcome/goal of the unit. The assignment that you select for the goal must have a fully explained and correct solution. It doesn't matter how well you understood the assignment at the time it was assigned. Now is your opportunity to demonstrate what you currently understand.

More specific information about the portfolio will be given in class in the near future.

Leave a comment to compare your opinions about some of the assignments and their learning outcomes.

Compound Interest

Today in class we discussed compound interest. Would anyone like to share their general formula?




If you were Adam, how would you want the bank to calculate your interest? Yearly? Quarterly? Daily? Instantly?




We then looked at the supplemental problem -- trying to figure out the percentage rate that will exactly double Adam's money in 20 years. How do we solve this?




Some classes got 3.526% by guess and check...How else could we solve this? Scroll down if you want to know.....

CLEP Precalculus Exam

Here are some sample problems for the CLEP Precalculus Exam.

Look at problems 7, 9, and 10 -- logarithms.

UIUC Engineering OPEN HOUSE

Would anyone like to attend the Engineering Open House at UIUC?
Watch this video from the 2007 open house.

Thank you for your support

The Jones Math Department wants to thank everyone that participated in our Research Lesson. This summer our department joined the Lesson Study Group. We worked collaboratively on the research lesson and invited guests from across the city to come observe the lesson and give our department feedback on the lesson.

We got some really good feedback -- things that were well done in the lesson, things that were confusing about the lesson, and things that were not done well in the lesson. Even though it "hurts" to receive candid feedback, we are excited by it, as it will improve our instruction.

The volunteers should be very proud of themselves. The observers were very impressed by the way your groups were able to evaluate the different graphs and determine which graphs met the criteria and which graphs did not meet the criteria. They mentioned that typically teachers hold the authority in class -- that teachers validate correct answers to students. But in this lesson, students were the authority and validated the correct answers. The observers also made several comments on how well groups worked collaboratively by sharing and discussing ideas.

Again, the math department cannot thank you enough for your participation in our research lesson.

SPECIAL THANKS TO:
Tom McDougal
Dr. Takahashi
Dr. Powers
Margie Pligge
Salik Mukarram
The Jones staff
Edwardo's

Progress Reports.

Today we passed out updated progress reports. The report can be alittle confusing.

The top portion is what is used to calculate your grade. It looks like you have taken about 16 quizzes but, each entry is actually one problem on a quiz. Quiz2.3 is really question #3 from Quiz 2. We entered the information this way so that we could generate the bottom portion.

The bottom portion is not used for calculating the grade, but to provide feedback to you. It lists all of the goals that we have studied so far. You can see which topics you could improve in. Those are topics that you may choose to review and re-assess. The calculations are not averages, but they are calculated using a power formula.

Any other questions about the progress report?

Quiz 5 Content

Can you....

*find the derivative of a function at specific points?
*graph a derivative of a function given certain conditions about the function?
*develop and use a formula for calculating interest?
*apply the principles of logarithms?

Questions, comments, or issues -- leave a comment.

The Graph of the Derivative

We have spent the last couple of days looking at the graph of the derivative.

Here is a cool applet.

This video also talks about some of the things we have covered in class.


On a side note...what would you do if this was your prom?

Quiz 4 Comments...

Any comments, questions, or concerns about Quiz 4?

If you feel that Quiz 4 didn't truely show your knowledge on the learning goals, please come see me or Ms. Jones before or after school. We can give you additional opportunities to provide evidence that you understand the goals.

The Math Department needs volunteers to star in an International Video! Imagine being seen by millions, billions, ALOT of people.

What? Where? When? You will have to come to school on Friday, October 24th from 9:00 am to 11:00 am. We will provide you with breakfast and lunch. You will have your own "green room" to prepare for your video shoot. Unfortunately, we won't be able to accommodate performer riders, but we can meet some dietary requests.

Why? Mr. Bywater, Ms. Fulton, Mr. Remiasz and I participated in a workshop this summer to improve our teaching, which ultimately improves student learning. We designed a math lesson that we want feedback on. Math teachers from different schools will observe and critic the lesson. We will gain valuable information from having these experts analyze and evaluate our performance. This lesson will have a tremendous effect on our practices and will help establish the Jones College Prep Math Department as a well respected, research-leading department.

I want to thank those individuals that volunteer, in advance. Words cannot express our thanks, appreciation, and gratitude for your committement to helping the Jones College Prep Math Department grow and improve.

HW24

The answers to HW 24 were....

1.a.i. $10,500

1.a.ii. $2521.05

1.a.iii. $423.71

1.b. V(t) = 15000 ( 1 - .30 ) ^t = 15000 (.7) ^t , where t is the number of years after 1990.

2. Clarabell is wrong. The more expensive car will NEVER be worth less than the cheaper car. We verified this by looking at the graph of the two equations, looking at the table of the two equations, and by trying to find the intersection of the two equations.

Exponential Functions

Recently, we've been talking about exponential functions. We examined various exponential functions to determine which ones have the "proportionality property".

How did we determine if an equation had the proportionality property? What did we have to do?

We finally came to the conclusion that functions in the following form would have characteristics of the proportionality property:

f ( x ) = AB ^cx

where A, B and C are constants

This happens to be the general form for exponential functions.

Do you think there are limitations on the values of A, B, or C? Can they be decimals? Negative numbers? Equal to zero? Here is an applet that could help you answer these questions -- be patient, it takes a while to load.

Since we are examining exponential functions, here is a great tutorial and goes really well with "A Basis For Disguise", the classwork that we worked on Friday.

7 ^x= 5 ^?

Can you write a general rule for writing 7 ^x as a power of 5?

Here are some YouTube videos demonstrating how to solve exponential equations.

POW10: Around King Arthur's Table

Questions or comments? Do you have an answer? What have you tried? What doesn't work?

HW20

We came up with the equation for the derivatives of two functions:
f(x) = 3x + 4
g(x) = x^2 - 9

For the f(x) function, we found that the derivative is always 3, no matter what value of x we used.
Some classes developed a conjecture for all linear equations...For ALL linear equations, the derivative IS the slope.

For the g(x) function, we found that the equation for the derivative is 2x, and changes depending on the value of x.
One class developed this conjecture for all quadratic equations...For ALL quadratic equations, the derivative = 2*A*x, where A is the coefficient of the x^2 term.

NOTE: These are conjectures, which we will accept to be true until someone can find an example where it doesn't work. Can you find an example that will prove our conjecture to be false?

Slippery Slopes

For this assignment, we had to find the derivative for the equation y = 2^x, an exponential function. Most students used the "find the slope of a secant for two very close points" method. Other students use the calculator to get the derivatives. Can you do both methods?

We then tried to examine the relationship between the y-value and the derivative. Some noticed some interesting patterns -- the y-values double and the derivatives double. What makes the y-values double? For #2, the equation is y = 10^x. What patterns for the y-values did you see or would you expect to see? What patterns for the derivatives did you see or would you expect to see?

Finally, our goal was to create an equation. For 1b, some said that the equation is:

Derivative = (y / 1.44)

Some others said:

Derivative = 0.693 * y

Which is correct?

Can you create the equation for the derivative in terms of the y-value for problem #2?

Can you generalize the derivative of any exponential function (y = A^x)?

Quiz 3

Do you have any questions, comments, or concerns about Quiz 3?

Quiz Friday. Topics include all of the outcomes we have covered so far...Rate of Change, Slope and Linear Equations, and Derivatives/Instantaneous Rate of Change.

• Can you calculate the slope of a line?
• Can you find an equation of a line given two points that the line passes through?
• Can you find the slope of different secant lines of a curve/graph?
• Can you find the average rate of change given a scenario/table/graph/equation?
• Can you find the approximate instantaneous rate of change given a scenario/table/graph/equation?
• Can you find the derivative of an equation at a specific point on a curve?
• Can you draw a tangent line at a specific point on a curve?