I like the idea of a blog, but the fact we had both a class forum and a blog was frustrating. The most aggravating part of the blog assignments was the fact that my comments on other students’ blogs could have been deleted or not approved and thus never show up online. I felt that my grade was dependent on others’ actions. I was also frustrated at the difficulty at uploading a document to the blog while posting a document to the course forum was very easy. I did feel that most of the assignments that were completed on the blog could have been effectively completed in the course forum. I probably won’t continue to use the blog. It is a good exercise to learn how to do it, but I find my school district website more effective.

• What did you learn about yourself and your abilities or interests in Math or Algebra?

I learned that while I have a good understanding of the material that I teach, I must be thoughtful about how I present the material to students. Often times, in providing definitions or explaining procedures, I will use words that require students to have another level of mathematical understanding, which may not be necessary for the concept that is being covered.

• Did you learn or discover anything you found particularly interesting through your course actives or your own internet research? Describe one interesting discovery and why you found it fascinating.

The one thing I remember from internet research that we had to do in this course were the macroscopic fractal images from the webpage describing non-linear patterns in nature. I think that these are great to show students fractals in nature, and as I described previously, they show that we all may be part of a fractal on a large scale.

• Do you think you will use journals with your students? Do you think you will use blogs? Why or why not?

I like the idea of journals and blogs for the students, but again I think that it is essential that the blogs can be moderated by the teacher. We do have the ability to have class forums in our district, that can be moderated, but I think that I would be more likely to use written journals. Requiring students to complete assignments online often seems to invite a plethora of excuses from them as to why they could complete the assignment (internet wasn’t working, computer wasn’t working, they don’t have a computer, etc.) I could ask them to complete blogs in school, but it would be nice to be able to expect them to do it at any time. I do think that I will use a journal in my classes.

1. Look at the third term, look at the factors of 8. (1×8, 2×4)

Paraphrased: Find all the sets of two numbers that equal “c” when multiplied together. Make sure that you identify whether “c” is positive or negative and choose number appropriately.

2. Which set of factors has the sum of b? 2 and 4

Paraphrased: Identify which set of numbers from step #1, equal “b” when added together, Let one of these numbers be represented by “d” and the other be represented by “e”

3. Look at the first term and factor it is (x * x) if so both binomials will start with x

Paraphrased: Write x² as a product of x and x. Each of these x’s represent the first terms in each of the binomials.

4. In one binomial, we will add 2 and in the other we will add 4. Therefore, we will have (x+2)(x+4)

Paraphrased: In one binomial we add “d” and the other we add “e”. The factored form of this trinomial will be (x+d)(x+e)

• Did paraphrasing the words help you internalize the concepts more?

Yes, just using words such as, “numbers whose product is…” or “numbers that multiply together…” makes the directions easier for me to follow. The term “factor” requires readers to have an understanding of the corresponding definition, as well as an understanding of the relation that the term factor has to the operation of multiplication.

• How can you apply this type of exercise in a lesson for your own students?

I think that it was a good exercise paraphrasing the steps, which I think students would benefit from also. I like the idea of giving notes on how to factor quadratic equations one day, without actually writing the numbered steps out. The next day I would have the students work in pairs to come up with what the actual steps to factoring are, as we had to do in this activity.

In class I generally show students graphical and algebraic examples of functions and equations (not on the same day) as well as non-examples of each. I have them generate a good definition. If a student provides an incomplete definition, I provide a counter example and they revise the definition. For the past few years that I have tried this, the students have generated definitions pretty close to the ones that I posted. I did want to post a diagram of a function and non-function that I drew in a document that has arrows from input values to output values that indicate the rule, but I was not able to post this. (Though I was able to post something from a website-not exactly what I was looking for, but it suffices.)

I did debate as to whether to include the f(x) function notation or not as an example, but since students will certainly see it later, I thought that they may as well be introduced to it now. I did see that others included this notation as well.

This applet illustrates the rate of change of two bike riders as they are riding from their school towards an emergency shelter. They are riding at different rates and the graphs corresponding to their distances traveled in terms of minutes passed, are shown. The distance that they need to go can be varied by the user.

I think that this would be an excellent applet to use during a unit on graphing linear equations and looking at constant rates of change. Similar to the story problems that we were assigned to generate earlier in our course, this demonstrates an application of linear functions in a very visual way. To use this in my class, I would have student explore how moving the distance that the bikers need to travel effects the difference in the time that it takes them to get to the destination. I think that this would, again, really give students an opportunity to see how different rates of change have an impact on other variables in the problem, that are not as obvious from the graph. (That is calculating the DIFFERENCE in time that it takes them and not just the time it takes them to get to the destination.) I would also have students perform the Bonus part of the assignment, which is to come up with an expression that represents the distance each biker travels in terms of the rate at which they travel and the time they travel. This would be a good exercise for students to complete after being introduced to the slope-intercept form of a linear equation. Students should be able to determine the slope of each of the lines from the graph (and hopefully relate this to the rate of travel) and recognize that each biker is starting from an initial position of zero.

As I was getting my Mad Hatter costume ready, I realized that I needed a bowtie. Luckily I found a free pattern for one online. Unfortunately, the pattern’s size had been reduced so that it would fit on a sheet of 8.5×11 paper. I needed to enlarge the pattern and could do so using a photocopier. I just needed to determine the percent that the final pattern was in relation to the initial pattern that I downloaded. This percent is the copy ratio and is the number that I need in order to enlarge the pattern correctly. The original image is 10.5 inches long and the final pattern is 30 inches long. What is the copy ratio?

Solution:

You can set up a proportion where 10.5 is 100% of the original size and 30 is x percent of the final size.

10.5/30 = 100/x   by cross-multiplying and solving for x you find that x=285.7%

Problem #2

Suppose that you are planning to repaint most of the rooms in your house. You painted the living room, the largest room, already. It took 1.5 gallons of primer and 2.5 gallons of top-coat paint to cover the 600 square feet of walls in the living room. How many square feet of wall space could be painted with the 4 gallons of primer and 6 gallons of top-coat paint that you have remaining?

Suppose that you have 1500 square feet of wall space left to paint. Do you have enough paint? Explain.

Solution:

You must set up two proportions, one for primer, one for top-coat

Primer: 1.5/600=4/x          where x=sq. feet covered by primer

Solving this gives you x=1600sq. ft.

Top-coat: 2.5/600=6/y        where y=sq. feet covered by top-coat

Solving this gives you y=1440sq. ft.

You have enough primer to paint the walls, but not enough top-coat. You need enough for about 60sq. ft., which you could figure out using a proportion:

1.5/600=z/60   where z is the amount of primer needed (in gallons)

Here z=.15 gallons

(This definition is needed for the definition of equation.)

Examples

3

3x+7

3x²+(2x-4xy)+7xy²

2x(4-5x)+8x

Equation - a statement that equates two algebraic expressions

Examples

x=12

3x+2=7

-5x-8=y

3x+2=7-4x

3x²+2x-14=4x+9

Function – a rule that assigns every input value exactly one output value

We can think of functions as a collection of ordered pairs. The x-coordinate is the input value and the y-coordinate is the output value. (Sometimes an f(x) or g(x) is substituted in place of the y in equations that are functions.)

The input values are called the domain, the output values the range.

Examples

{(2,3), (3,4), (-5,7), (6,4)}

y=2x-5

3x²+2x-14=y

y=f(x)=4x+1

This is an illustration of a function with a inputs and outputs.

Image below is from: http://www.teacherschoice.com.au/Maths_Library/Functions/function_diagram.gif

]]> http://mathmannick42.wordpress.com/2009/11/06/5-a-3-my-definition-of-equations-and-functions/feed/ 3 mathmannick42 Function Diagram gallery link="file" http://mathmannick42.wordpress.com/2009/10/31/my-reflection-on-math-myths/ http://mathmannick42.wordpress.com/2009/10/31/my-reflection-on-math-myths/#comments Sat, 31 Oct 2009 22:44:00 +0000 mathmannick42 http://mathmannick42.wordpress.com/?p=43 ]]> Did you encounter any of these myths in your own experience with Math education as a student? If so, which ones?

I certainly encountered #5(there is one best way to do problems), #7(counting on fingers is bad), and #9(it is necessary to memorize formulas) as a student.

What has happened since to dispel or perpetuate your understanding of the myth?

I think that most of these myths have been dispelled through just practicing math and working through a plethora of different kinds of problems. I think that it wasn’t until I became a teacher that I really believed that #5 was a myth. As a teacher I have the opportunity to see all the different ways that students can answer questions correctly. Often times they come up with ways that I haven’t even thought of.

I still do feel that that counting fingers is not efficient and while it is not “wrong” there may be a better way to answer the question at hand. Along the same lines I think that it is very beneficial for students to memorize their multiplication tables so they aren’t trying to multiply 8 times 9 with their fingers. Again, it isn’t wrong, it just may take longer than the time allotted. I also think that it is somewhat ridiculous to not expect students to memorize basic formulas such as the area formulas of rectangles, circles, and triangles. If they did have to look these formulas up EVERY time they used them, it would be very difficulty to have students get much work, relating to these topics, done during class.

How can you help dispel any of these myths for your students?

I think that having students work through a variety of different kinds of problems, each school year, is the best way to dispel these myths. Problems presented, could be chosen to address some of the myths specifically. For example, to address #6(it’s important to have the exact answer), students could plot data that is roughly linear and then be instructed to calculate a line of best fit. The equation for the line of best fit could then be used to make a prediction about the data. Each student may have different predictions based on their own line of best fit, but it does not necessarily mean that any is more correct than another. To help facilitate the dispelling of these myths further it might also be a good idea to have students share with the class, the method that they followed to solve the specially selected problems. This may introduce students to different acceptable methods that may be used to solve problems.

Pascal’s Triangle is an arrangement of particular numbers in a triangular form. Numbers are arranged in rows so that each row has one more number than the row above it. The top row has one number and is the top point of the triangle. The first and last number in each row is a one, and thus the top number (a first AND last number) is also a one. The other numbers are calculated in a particular way. The numbers in rows next to each other are not lined up vertically. Instead, they are staggered. That is, if 121 is the third row and 1331 is the fourth row in the triangle, then the first 1 in row 4 would be to the lower left of the first 1 in row 3. The first 3 in row 4 would be between the first 1 and 2 in the third row (but underneath them). The second 3 in row 4 would be between (and underneath the 2 and second 1 in row 3. Finally the second 1 in row 4 would be to the lower right of the second 1 in row 3. To generate another row of numbers on the triangle, add the two numbers that are above and to the left and above and to the right of where the new number is going to be written. Keep in mind that if there is only one number above a given location (to the right or left), then this must be the end of a row and will be a 1.

In theory, Pascal’s Triangle is endless. As more and more rows are generated, numerous patterns immerge amongst rows and diagonals that form as rows are added. Some of the diagonals formed are special in their own right. The third diagonal from the top consists of the triangular numbers.

A More Mathematical Definition

Pascal’s Triangle is an arrangement of particular numbers in a triangular form. Numbers are arranged in rows so that each row has one more number than the row above it. Each row of numbers can be derived from the algebraic expansion of (a+b)^n. Where n represents the row number (which starts with row 0). The terms in the expansion should be written such that the exponent of the a’s decrease and the exponent of b’s increase. Each row consists of the coefficients of each term of the expansion, written in this order. Each successive row is offset from the one above it so that numbers in adjacent rows do not line up vertically. There are an infinite number of rows in the triangle, though this cannot be demonstrated visually.

I really liked the site: http://www.lifeinitaly.com/heroes-villains/fibonacci.asp

This site provides a lot of background information on Fibonacci’s life, which is great for my History of Math class, but it also provided some mathematical examples of the Fibonacci sequence in action.

The other site I liked was: http://fractalfoundation.org/category/natural-fractals/

This organization promotes science, math, and art through the explorations of fractals. The site would be an excellent resource when discussing math in nature with students.

Questions

a.  Were there ideas or concepts you were not familiar with? What were they?

I was familiar with most of the background of Fibonacci, since I have taught it before, but it is nice to have all on one site. It would be a great opportunity to let students explore the site and gain an understanding on their own.
Some of the connections to the Fibonacci sequence I was not familiar with, such as the connection to the decimal expansion of 1/89 and the fact that in n+1 numbers from a set of 2n numbers there will always be one number that will divide into another.

b.  What images did you find particularly striking?

I was quite impressed by the photos at: http://fractalfoundation.org/category/natural-fractals/ They are satellite images of land forms on earth that resemble fractals. I have seen images of crystals and other things on a microscopic scale that resemble fractals, but these macroscopic images give a completely different perspective. The images got me to realize that I may be part of a fractal that is only visible on a macroscopic scale and not through our everyday perspective.

c.  Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they?

Obviously there are numerous nonlinear patterns all around, some of which are more apparent than others. I thought of a few that were different than some of the ones mentioned on the websites that I viewed. Firstly, the carpets in our house definitely exhibit a non-linear pattern.  Some of them are again, almost fractal-like with smaller parts of the design mimicking the larger parts. The second thing that I thought of is the pattern with which a heated object cools. This is can be effectively modeled with an exponential function. Data points could be collect to illustrate this nonlinear pattern with numbers.

d.  How can you adapt this webquest activity for your classroom?

As I noted above, the Fibonacci site would be perfect for my History of Math class. It would be a way for students to gain a greater understanding of this particular topic by learning it on their own. I could even require students to present a historical math concept to the class and this site could serve as a resource for one student. This webquest in general would be a great way for students to explore mathematical concepts outside of the classroom. I think an important concept for students to grasp is that math outside the class isn’t always going to be nice and neat and completely obvious, but that it is still there. The fractal photos for example obviously are not mathematically constructed fractals and yet there is definitely math to be found in them.

Linear Pattern – kid-friendly definition – a list of numbers that increases or decreases by the same amount between each number

Linear Pattern – my formal definition – sequentially arranged numerical values (or objects that can be quantified) whose absolute value difference between immediately previous and subsequent values, is equivalent.

What is different between the kid friendly and formal definitions of LINEAR PATTERN?

The fundamental difference is that the kid-friendly definition implies that a linear pattern must exist as a list of numbers. The formal definition allows a linear pattern to exist as a collection of objects, as long as they fit the given definition. In this way, the formal definition is more general and can be used in a greater variety of situations. The kid-friendly definition can be used in specific cases in which numbers are the objects in the pattern. This is most likely the type of linear pattern that they will experience in math class.  This definition does not however, necessarily cover geometric patterns that may exhibit characteristics of a linear pattern (say the number of sides of subsequent polygons increase by one each time).

How to get students to understand the formal definition.

I think that by introducing students to patterns, that exhibit a linear component (such as the examples explained above) but are not strictly numerical, they will recognize how patterns may linear but not necessarily contain numbers. If they are familiar with the kid-friendly definition, they should be able to adapt it to include these other example patterns.