Monday, September 15, 2008

Slope and Slant

Tonight's HW is HW8.

In class today we worked on "Slope and Slant". We talked about the relationship between the slope of a line and the angle of inclination of a line. Here are some websites that talk about today's topic.

http://www.mathwords.com/a/angle_inclination.htm
http://en.wikipedia.org/wiki/Grade_(slope)
http://math.about.com/library/weekly/aa120502a.htm

How does the slope relate to the angle of inclination?


"As the slope of the line gets steeper, then the angle of inclination......."


What general principle (using trigonometry) relates the slope of a line to its angle of inclination?



Sunday, September 14, 2008

Bored?

Here are some copies of Final Exams from various math courses at Southern Illinois University.

Math 107 -- Intermediate Algebra
Math 108 -- College Algebra
Math 111 -- Precalculus

Are there any problems from the Math 107 final that we should talk about?

#18 relates to what we are currently doing. #30, 32, and 33 are good ones as well. For #33, what dimensions would create a box with the largest volume?

I also like #27 from the College Algebra final.

Solving for Slope


The other day we worked on the worksheet, "Solving for Slope". You were given linear equations in Standard Form (Ax + By = C) and asked to convert the equations into Slope-Intercept Form and to determine the slope.

Here is an interactive website that shows you the steps to convert Standard Form linear equations.
http://faculty.ivytech.edu/~wmiley/Stand2Slope/StandToSlope.html

Here is a video demonstration by a teacher converting a Standard Form equation.
http://revver.com/video/521094/converting-to-slope-intercept-form-and-graphing/

Here is a video for those that would like to review Slope-Intercept Form.
http://revver.com/video/248264/slope-intercept-form/

"Solving for Slope" asked some interesting questions regarding the general equation Ax + By = C

  • How could you tell if the line is has a positive or negative slope just by looking at A, B, and C?
  • How could you tell if the line has a positive slope and is steeper than y = x just by looking at A, B, and C?
  • How could you tell if the line has a negative slope and is steeper than y = -x just by looking at A, B, and C?
And finally, could you convert the other way around -- given an equation in Slope-Intercept Form, convert it into Standard-Form?

Friday, September 12, 2008

Help with Engineering Supplies


Help....I need somebody....Help....Not just anybody....

My engineering class is building mousetrap cars and we need your help. If you have toy cars that you don't play with anymore, old CD's, an extra dremel, or your father's drill that you can sneak out in the morning sun, then could you think about donating them to our class? We are in need of supplies to complete our cars. Visit here to see the list. Maybe we could put your name on the hood of our cars as sponsors?

How do you feel about the class?

Now that its been two weeks, how do you feel about the class? Hate it, love it, still can't tell?

Thursday, September 11, 2008

Formative Quiz 1


Okay, so we had our first quiz. It was a formative quiz, meaning that it was for feedback and not for a score. We had an opportunity to discuss the results of the quiz with our group. Tomorrow, we will have a summative quiz, which is for a score.


What concerns, questions, or comments do you have about the formative quiz?

Do you think the results of the formative quiz will help you to perform well on the summative quiz?

Was the feedback you got from me effective? Were you able to understand what you needed to do?

Out of curiousity, do other teachers also give formative/practice quizzes?

Average rate of change



Is the average rate of change for the green function from 0 to 12 the same as the average rate of change for the red function from 0 to 12?

That was part of our discussion during today's class...does the average rate of change vary if the endpoints remain the same?

We then were able to rediscover the "forumla" for the slope of a line given two points.

Finally, we were able to find the slope of linear equations in Standard Form.

Any comments, questions, or observations from today's class?

Wednesday, September 10, 2008

How fast?


Today in class we worked on this warmup problem:
Calculate the average speed if you travel



  1. one hour for 10mph and one hour for 20mph

  2. one hour for 10mph and two hours for 20mph

We discovered that we could not just add (10+20) and divide by 3 hours for the second problem. But that we have to take into consideration the number of hours we traveled at each speed. Someone mentioned that we had to take a "weighted average". Another student mentioned that we had to calculate the "total distance traveled" and divide that by the "total elapsed time ".

Are these three strategies the same? Is one better than the others? Is one easier than the others? Is one more efficient than the others?


We then talked about ways that Ms. Jones and I could grade problem number 3 from HW4. We decided that:

  • the total distance traveled has to be 550 miles

  • the total elasped time has to be 11 hours

A couple of classes added this requirement as well: "the average speed has to be 50mph"

Do we "need" the last item? Will these items hold true for every valid scenario? Does it hold true for your scenario?



Tuesday, September 9, 2008

HW4


Add a comment and share your answers to HW4.

Do you agree with the answers already posted?

What questions do you have about HW4?

What questions do you have about your fellow students' answers?

What difficulties did you encounter with this homework?


Monday, September 8, 2008

HW3


We passed out the packets for Small World, Isnt it. There are still some extra copies left if you need one.

In class today we talked about the avg rate of change and how it related to slope. Many of you remembered slope, saying its the "rise over run" and could recite the formula for slope. Is slope the same thing as "rate of change"?

Our objective sheet for Rate of Change lists:



  • Evaluate the average rate of change given points on a graph.

  • Understand the relationship between the rate of change and the appearance of its graph.

Can you accomplish those goals? If not, can you write a question in the comments section that is related to where you are having difficulty with?

Tonight's homework is HW4.

Friday, September 5, 2008

How many more people?


Today's objective/outcome was to look at Rate of Change. You can view all of the outcomes for the Small World unit by looking at the learning outcomes document.

From HW2, some students mentioned that from ages 3 to 6, there was a "steady" increase in the boys' heights (problem #2). Other people mentioned the words "linear", "constant growth", "consistant growth", and "slope".

We then looked at the population graphs from Thursdays. We were able to calculate the average increase in population per year. Can you do this?

Does this "average increase in population per year" relate to "rate of change"?

Do you have any comments, suggestions, or concerns about the class so far? If so, add a comment to this post.

Homework due Monday is HW3. Our grace period is up. I will start enforcing the homework policy, the tardy policy, and the ID policy.

Thursday, September 4, 2008

Total land surface area


Our book is defining "squashed up" to be 1 square foot of space per person. Personally, I feel like that is way to close to get to someone. However, using that definition, we calculated that the population of a "squashed up" earth would be 1.6x10^15 people or 1.6 quadrillion people. That's alot of people.
We talked about using scientific notation to help us look at very big numbers, like a sextillion or a quattuordecillion.
Visit the JCP homework site for HW2.

Wednesday, September 3, 2008

Small World, Isn't it?


Today we talked about our unit problem, Small World, Isn't It?
Our goal is to figure out how long it will take for the world's population to grow to the point that we are all "squashed up" against one another.
We talked about some factors that can affect population growth including: food sources, natural resources, disease, climate, birth rates and death rates.
Some groups came up with strategies on how to solve this problem, but seemed to run into difficulties.
What difficulties did we talk about? Issues? Things that need to be clarified?
Homework tonight is HW1.

Tuesday, September 2, 2008

first day jitters...

After getting off to an alarming start, I think our classes went well.

Any comments on the "Fair, but not equal" statement?

How should mathematicians define "crowded"?

And finally, any comments on the parachute activity?

Sunday, August 31, 2008

Almost there...

I found some last minute shopping deals...

Office Depot has their own brand graph paper (quad ruled) composition books for $.99. I highly recommend using these as your precalc notebooks. Five should cover the entire year for our class.

Also, Staples has a great deal on the TI-84 plus for $69.99 after rebates.

Enjoy the rest of your holiday weekend!

Tuesday, August 19, 2008

WELCOME BACK!

I hope you enjoyed your summer. Did anyone do anything fun?

I spent my summer taking classes and reading. As a team, the math department participated in one week of IMP training and one week of Japanese Lesson Study. In addition, I attended a summer institute for National Board Certification. This school year, I will be working on my National Boards. If you have any video production experience, please let me know. Part of the requirements is to videotape our classes.

Mr. Remiasz and I have been working on the syllabus. We are planning on doing several IMP units and then finishing the year off with several Precalculus topics. There is now a Precaculus CLEP exam. Senior may want to think about taking the exam, which may give them college credit. Mr. Remiasz and I are also thinking about taking it.

I strongly encourage you to purchase several quad-ruled composition books from Staples. As of yesterday, they were still $1.00 each -- Thanks Mr. Lin. You will also need a TI graphing calculator. I recommend the TI-84 Plus Silver, which is also on sale at Staples for $99 after rebates. But any of the TI products will do.

I recommend reading some of the recent posts -- college readiness and grading philosophies. Also, watch this video and leave a comment.

Saturday, August 9, 2008

Are You College Ready?

Taken from "Toward a More Comprehensive Conception of College Readiness" from the Gates Foundation.

Recent research has shed light on several key elements of college success...often described as “key cognitive strategies”...emphatically identified...as important or more important than any specific content knowledge taught in high school...include analysis, interpretation, precision and accuracy, problem solving, and reasoning. Close behind in importance is knowledge of specific types of content knowledge. Writing may be by far the single academic skill most closely associated with college success, but the “big ideas” of each content area are also very important building blocks. Similarly important are the attitudes and behavioral attributes that students must demonstrate: study skills, time management, awareness of one’s performance, persistence, and the ability to utilize study groups.

College is different from high school in many important ways...College is the first place where we expect young people to be adults...Almost all of the rules of the game that students have so carefully learned and mastered over the preceding 13 years of schooling are either discarded or modified drastically...Transition from high school to college is one of the most difficult that many people experience during a lifetime. Because college is truly different from high school, college readiness is fundamentally different than high school competence. The college instructor...emphasizes a series of key thinking skills that students, for the most part, do not develop extensively in high school. The college instructor expect students to make inferences, interpret results, analyze conflicting explanations of phenomena, support arguments with evidence, solve complex problems that have no obvious answer, reach conclusions, offer explanations, conduct research, engage in the give-and-take of ideas, and generally think deeply about what they are being taught.

Several studies of college faculty members nationwide expressed near-universal agreement that most students arrive unprepared for the intellectual demands and expectations of postsecondary education...The primary areas in which first-year students needed further development were critical thinking and problem solving.

Successful academic preparation for college is grounded in two important dimensions—key cognitive strategies and content knowledge...Two academic skill areas that have repeatedly been identified as being centrally important to college success: writing and research. Writing is the means by which students are evaluated at least to some degree in nearly every postsecondary course. Students need to know how to pre-write, how to edit, and how to re-write a piece before it is submitted and, often, after it has been submitted once and feedback has been provided. College writing requires students to present arguments clearly, substantiate each point, and utilize the basics of a style manual when constructing a paper. College courses increasingly require students to be able to identify and utilize appropriate strategies and methodologies to explore and answer problems and to conduct research on a range of questions. To do so, students must be able to evaluate the appropriateness of a variety of source material and then synthesize and incorporate the material into a paper or report.

Math: Most important for success in college math is a thorough understanding of the basic concepts, principles, and techniques of algebra. This is different than simply having been exposed to these ideas. Much of the subsequent mathematics they will encounter draw upon or utilize these principles. In addition, having learned these elements of mathematical thinking at a deep level, they understand what it means to understand mathematical concepts deeply and are more likely to do so in subsequent areas of mathematical study. College-ready students possess more than a formulaic understanding of mathematics. They have the ability to apply conceptual understandings in order to extract a problem from a context, use mathematics to solve the problem, and then interpret the solution back into the context. They know when and how to estimate to determine the reasonableness of answers and can use a calculator appropriately as a tool, not a crutch.

Science: College science courses emphasize scientific thinking in all their facets. In addition to utilizing all the steps in the scientific method, students learn what it means to think like a scientist. This includes the communication conventions followed by scientists, the way that empirical evidence is used to draw conclusions, and how such conclusions are then subject to challenge and interpretation. Students come to appreciate that scientific knowledge is both constant and changing at any given moment, and that the evolution of scientific knowledge does not mean that previous knowledge was necessarily “wrong.” Students grasp that scientists think in terms of models and systems as ways to comprehend complex phenomena. This helps them make sense out of the flow of ideas and concepts they encounter in entry-level college courses and the overall structure of the scientific discipline they are studying.

In addition to the key cognitive strategies and content knowledge, a range of behaviors that reflect greater student self-awareness, self-monitoring, and self-control of a series of processes and behaviors is necessary for academic success. The key academic behaviors consist largely of self-monitoring skills and study skills. Self-monitoring is a form of metacognition, the ability to think about how one is thinking. Examples of metacognitive skills include: awareness of one’s current level of mastery and understanding of a subject, including key misunderstandings and blind spots; the ability to reflect on what worked and what needed improvement in any particular academic task; the tendency to persist when presented with a novel, difficult, or ambiguous task; the tendency to identify and systematically select among and employ a range of learning strategies; and the capability to transfer learning and strategies from familiar settings and situations to new ones.

Friday, August 1, 2008

Grading Philosophies

Lately, I have really been thinking about the way I grade students.

Sometimes I feel that grading can be very arbitrary -- its entirely up to the teacher. Teachers decide the distribution of grades (ie. 25% quizzes, 25% homework, etc.). Teachers decide how many points a quiz is worth. Teachers decide how much partial credit to award.

I stumbled upon this book, and while I haven't had a chance to read it yet, I did read the study guide. The book discusses 15 fixes for broken grades. Here are 5 fixes that I am racking my brain around. I typically do all of these "Don'ts".

  • Fix 1: Don't include student behavior in grades.
  • Fix 2: Don't reduce grades on work submitted late.
  • Fix 3: Don't give points for extra credit.
  • Fix 4: Don't punish academic dishonesty/cheating with reduced grades.
  • Fix 12: Don't include zeros for missing assignments when calculating grades.
Any comments?

Thursday, July 24, 2008

Links for Parents

The Bill and Melinda Gates Foundation is committed to "preparing all students for college, career, and life."

The foundation's goal is for all students—regardless of race or family income—to graduate from high school prepared to succeed in college, career, and life. All students, all schools, everywhere.


Below are some articles based on research done by the Gates Foundation.

Improving Math Performance -- Gates Foundation
Towards a More Comprehensive Conception of College Readiness -- Gates Foundation
Math: Most important for success in college math is a thorough understanding of the basic concepts, principles, and techniques of algebra. This is different than simply having been exposed to these ideas. Much of the subsequent mathematics they will encounter draw upon or utilize these principles. In addition, having learned these elements of mathematical thinking at a deep level, they understand what it means to understand mathematical concepts deeply and are more likely to do so in subsequent areas of mathematical study. College-ready students possess more than a formulaic understanding of mathematics. They have the ability to apply conceptual understandings in order to extract a problem from a context, use mathematics to solve the problem, and then interpret the solution back into the context. They know when and how to estimate to determine the reasonableness of answers and can use a calculator appropriately as a tool, not a crutch.


Rethinking High School: Supporting All Students to be College Ready in Math -- Gates Foundation

Rethinking High School: Preparing Students for Success in College, Career, and Life -- Gates Foundation


Here are some links about assessment.

Inside the Black Box: Raising Standards Through Classroom Assessment

Five “Key Strategies” for Effective Formative Assessment

Improving the Way We Grade Science


The Transition Math Project has developed a student and parent portal with up-to-date math planning resources, tips and tools that will help you talk to your child about the importance of math in being college- and work-ready.

Here is an article describing how parents can help with math homework when the answers aren’t in the book.

Motto

I hear...I forget
I see...I remember
I do...I understand


The Goals of Mathematical Education -- George Polya (circa 1969)