Factoring Quadratics In Your Own Words

x^2 + 6x + 8

First, look at the last term (8). Determine the factors of the numbers.

The factors of 8 are 1 and 8 or 2 and 4.

Next, look at the second term (6x). Use the factors from above (1 and 8 or 2 and 4) and add or subtract them until you get the number in the second term (6).

1 + 8 = 9

1 – 8 = -7

8 + 1 = 9

8 – 1 = 7

2 + 4 = 6

Look at the first term (x^2) and write its factors out.

The factors are x and x.

Finally, let’s put it all together. Make two sets of parentheses and put a “x” in each so that it looks like:

(x     )(x     )

The factors that we previously found (2 and 4) are what will go inside the parenthesis. They were both positive numbers so we will put plus signs in the parentheses as well.

Our final answer will look like.

(x + 2)(x + 4)

Did paraphrasing the words help you internalize the concepts more?

Honestly, it was a huge help. I’ve never directly taught quadratic equations and I’ve personally hated the way that some of the math teachers try to teach it so I never fully understood how to do it and explain it to students. Thanks to this activity, I now totally remember how to do it and feel confident in teaching it to others.

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

My students really struggle with paraphrasing how to solve problems so I could have students paraphrase every concept once they had practiced it enough. They could then write it out in an easy to read visual and present it to the class.

Reflections On Blogging

I’ve always known about blogging and have thought about doing it but never got around to it. I’ve never been a huge fan of blogging so I was hesitant when I saw that there were a lot of blogging assignments with this class. Honestly, I don’t think that I will continue with blogging due to time constraints and the fact that I use other methods to keep my thoughts and share content with my students.

When I saw this class listed under the list of classes that I could take, I was ecstatic because I love teaching math at the middle levels. At the end of this school year, my principal informed me that I would be teaching English exclusively next year as I am the only teacher that is highly qualified to do so. I was devastated because I’ve primarily taught math for the past four years. Thankfully, I found and accepted a new job where I will be primarily teaching math and doing what I love.

There wasn’t one particular discovery that I had during this course but one general discovery instead. While I always try to make real world connections to content, this course really pushed me to take real world connections with more difficult concepts. My students are always asking me why they need to know how to do something and I now feel confident in being able to answer that question for every concept that I teach.

While I found the module on journals very interesting, I’m not 100% sure if I’m ready to dive in head first and take them on this year. I would also say the same about blogs. I work with students with disabilities and math is difficult enough so combining it with writing makes it an extremely difficult task. However, I do see the value in both of them and how they could help my students gain a deeper understanding of specific concepts. I think that I may try each method a few times throughout the year and see how my students do with the activities.

5-D-2: Applets

Algebra Balance Scales

http://nlvm.usu.edu/en/nav/category_g_3_t_2.html

The algebra balance scales are exactly what I do with my students when I introduce equations except that I demonstrate it using my Smart Board. It’s a great activity with the Smart Board because the students can physically touch and move the algebra tiles around to solve the problem. 

Instead of incorporating it directly into my lesson plans, I think that I would use the applet as a homework or review assignment. If students had Internet access at home, they could go home and practice 5-10 problems that night to review the material. If the students were tech savvy and had the available tools, they could record themselves solving problems with the applet so that their peers and I could watch them in action. However, I could also make this an in class activity. Either way, the students are practicing the skill and are learning it on a deeper level by explaining their thought process when solving the problems. It also allows me to see exactly where students may be struggling so that I can quickly identify the issue, provide feedback, and fix it.

Evaluating Our Definitions: Equations and Functions

After reviewing your classmates post, would you alter your definition? Why or why not? Would you provide different examples?

I decided to review Justine’s post specifically because we’ve been coworkers for four years so I wanted to see how similar ours were. I was surprised but not shocked to see that we had the same definition for an equation because there’s only so many ways that you can word it and we’ve both taught the same math program and types of students so we’re on the same page. Our definition for a function is slightly different and I think that she explained it with a bit more detail than I did. However, I like the simplicity of mine so I don’t think that I would change it. I really like how Justine gave two examples of expressions and then combined those expressions into an equation to show how they’re related. I really wish that I had done that in my post. Overall, I’m happy with my examples and think that they serve their purpose.

NOTE: I went back and reviewed other classmates’ posts and found that our definitions and examples were fairly similar as well.

How can you evaluate whether or not your students grasped the difference between the two?

The main difference between equations and functions is that equations usually only have one input while functions can have numerous inputs that will each produce a different output. In order to demonstrate this idea to students, I would have them solve an equation. Then, I would take the first half of the equation and turn it into a function. The students would then calculate the output of the function with different numbers. Finally, I would do the same with the other half of the equation. At the end of the activity, the students would see that the functions produce the same outputs when you use the same input. However, if you change the input, you change the input and therefore it’s no longer an equation because it’s no equal on both sides.

5-B-1: The Magic of Proportions

We use proportions almost every day when we are shopping because we have to calculate how much we are going to pay for an item when it is on sale. I was shopping for a new dress shirt the other day at Calvin Klein and found the perfect shirt for me but it was $65 and I only wanted to spend $40 at most. However, Calvin Klein was having a sale and everything in the store was 40% off. To set up my proportion, I put 40 over 100 on one side and x over 65 on the other side. Next, I cross multiplied to get 100x on one side and 2,600 on the other side. Finally, I divided by 100 on each side to get x by itself. The answer is $26.00. However, that is not my final answer. In order to get the final answer, I need to subtract $26 from $65 to find out the new cost of the dress shirt. When I subtract those numbers, I get $39. Is the shirt less than $40? Technically, the shirt is less than $40 but it will actually be slightly higher than that due to taxes.

Another way that we use proportions in our lives every day is when we calculate distances when traveling. My friends recently took a trip to Miami, Florida and they said that they traveling at a pace of 500 miles in 8 hours. They are planning a trip to Las Vegas and are trying to figure out how long it will take them to get out to Las Vegas from York, PA. It is 2,400 miles from York to Las Vegas. How long will it take them to get to Las Vegas if they drive at the same pace they did for the trip to Miami? To set up the proportion, I put 8 over 500 on one side and x over 2,400 on the other side. Next, I cross multiplied to get 500x on one side and 19,200 on the other side. Finally, I divided each side by 500 to get x by itself. The answer is 38.4 hours. So, it will take my friends approximately 38.5 hours to get to Las Vegas.

Non-Linear Pattern Web Quest

“Fibonacci” and “Phyllotaxis” and “Prime Numbers”

A Fibonacci Phyllotaxis Prime Number Sieve – the perfect combination!

http://www.vortexmath.com/ScotNelson-Instruction_Daisy_Number_Sieve.pdf

Prime Phyllotaxis Spirals

http://maxwelldemon.com/2012/03/18/prime-phyllotaxis-spirals/

“Fractals” and “Nature” and “Patterns”

Earth’s Most Stunning Natural Fractal Patterns

http://www.wired.com/wiredscience/2010/09/fractal-patterns-in-nature/

Interesting Patterns and Fractals from Nature

http://www.youthedesigner.com/2011/07/29/interesting-patterns-and-fractals-from-nature/

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

To be honest, I wasn’t really familiar with any of the concepts except for prime numbers and patterns. I’ve heard of Fibonacci and fractals but I don’t know anything about them.

What images did you find particularly striking?

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

My girlfriend’s plants have lots of different nonlinear patterns on them. Some of the fruits and vegetables, such as cauliflower, that my girlfriend eats have nonlinear patterns as well. Our curtains, carpet, and rugs also have patterns in them.

How can you adapt this webquest activity for your classroom?

Most of these topics are ones that I would never teach my students because they’re too complex and not appropriate for their grade level. However, I could have my students create different patterns and then compare and contrast the patterns that they created with others. From there, the students could research the different types of patterns and possibly learn a little about fractals linear patterns, and nonlinear patterns.

4-C-3

ORIGINAL DESCRIPTION

The first thing I noticed about Pascal’s Triangle is that both sides are comprised of the number one from the top to the bottom. If you add two numbers that are next to each other in one row, you can find the answer directly below and between the two numbers that were added. You can also reverse that pattern and see it as a subtraction problem by subtracting one of the top numbers from the bottom number to get the other top number. I also noticed that the sum of each horizontal line doubles from one line to the next (1, 2, 4, 8, 16, 32, 64, 128). Pascal’s Triangle has symmetry if you split it vertically and remove the middle column.

FORMAL DESCRIPTION

The first thing I observed about Pascal’s Triangle is that both the far left and far right edges are comprised of the number one. This pattern repeats itself for the duration of Pascal’s Triangle. If you get the sum of two consecutive numbers from Pascal’s Triangle, you will have calculated the number that is located directly between and underneath the two consecutive numbers that were added together. This process can also be reversed and viewed as a subtraction problem as well. Simply create an upside down regular triangle around three numbers so that two numbers are on top and one number is on the bottom. Take one of the numbers on the top and subtract it from the bottom number to get the other top number. I also noticed that the sum of each horizontal line doubles for each consecutive horizontal line (1, 2, 4, 8, 16, 32, 64, 128). Once you remove the innermost middle column, it becomes clear that Pascal’s Triangle has symmetry and is an example of a reflection as the sides are mirror reflections of each other.

Working with the Definition of Linear Patterns

Formal Definition for a Non-Traditional Pattern

Patterns that do not follow a repetitive format

Kid Language Definition for a Linear Pattern

Linear means something that’s in a straight line while a pattern means that something repeats by the same amount each time. In other words, a linear pattern is just a pattern that increases or decreases by the same amount each time.

Formal Definition for a Linear Pattern

If the plotted points make a pattern, then the coordinates of each point may have the same relationship between the x and y values.  In such a case, the x and y values are connected by a certain rule.

A linear pattern is said to exist when the points examined form a straight line.

Found at http://www.mathsteacher.com.au/year8/ch15_graphs/02_linear/patterns.htm

Difference Between the Kid Language and Formal Definitions

While my definition may be easier to understand, the formal definition includes that there is a rule between the x and y values which is really important. The formal definition also says that the points will make a straight line which I didn’t mention. However, I was trying to think outside of the box and not mention graphing terms in my definition as linear patterns don’t always include graphing.

How I could help students learn the formal definition without having them memorize it

I would put large cut out numbers (-25 to +25) on the wall in a linear pattern around the classroom and label those numbers as x. I would have a line of y numbers below except that only a few numbers would be filled in for the students. I would give them just enough to figure out the pattern and then have them apply what they know about the pattern to discover what the missing numbers are. This would help them see the pattern on a large scale and get them to move around the classroom so that they’re busy. This activity also requires a lot of teamwork and cooperation from all of the students.

My Reflection on Math Myths

 

The first myth, that men are better at math than women, is the myth that I hear most often. Personally, I’ve always kind of believed it since most of the women I know don’t like math and aren’t very good at it. However, if you walked into my math class this past school year, you would only find 3 girls and 9 boys. During my 4 years as a special education math teacher, I’ve found that the boys actually struggle with math more than the girls. One of my more vocal male students yelled out in class one day that girls stink at math. I turned to him and was about to reply when one of the girls yelled out, “Then why are there more boys than girls in this class?” Everyone got a good laugh out of it and it goes to show that even our students have certain myths about math. I think that we can put an end to this myth by providing other stereotypes that aren’t true. For example, people that are good at math are supposed to be great at science as well. However, I hate science and I’m absolutely terrible at it!

The one myth that bugs me is that it’s always important to get the answer exactly right. I was always annoyed by teachers that would take off points if your answer was absolutely perfect to the nearest hundredth or thousandth. Unfortunately, I’ve seen this myth rub off on my students because they panic when they’re checking an answer key and see that the answer should be 28.26 but they have 28.2. I continuously have to explain to my students that if their answer is within a few tenths of the answer that I have, they won’t lose any points. This becomes even more of an issue during the unit with finding the area and circumference of circles and other shapes as they will get slightly different answers if they use 3.14 or calculator pi. I think we can put an end to this myth by discussing rounding decimals with them more often.

The other myths that I’ve heard of are that math is not creative, you must always know how you got the answer, there is a best way to do math problems, it’s bad to count on your fingers, and that some people have a “math mind” and some don’t. After spending 4 years teaching math, I can proudly say that these are just myths and quite far from reality.