8-B-1: Factoring Quadratics
Instructions for Factoring a Quadratic Equation: Ax2 + Bx + C
Factoring a quadratic is like solving a riddle. First you need to multiply the first and third terms (“A” and “C”), and then the riddle begins. You must find a pair of numbers that multiply to give you the product of AC and add to give to the value of B.
To do this, make a list of all of the factor pairs of AC. Determine which of these factor pairs also have a sum of B.
Next, set up the following “target”:
If the trinomial does not have a coefficient in the “A” position, then place an x in slot #1 and slot #3 and place the terms from the factor pair above (including their appropriate +/- sign) in slot #2 and slot #4 respectively.
Paraphrasing the instructions caused me to think a little deeper about the steps. (It was also challenging for me to simply paraphrase the instructions without accounting for how to factor the trinomial when it does have a coefficient in the “A” spot.) It is often easier to inherently understand a process than it is to verbalize it, so I feel this is a valuable activity to use with students as it encourages them to think carefully about each step. Reading their answers also allows me, as the teacher, to see where they may have a gap in their understanding. I think this can easily be applied in the classroom as a quick writing prompt for a journal or even a short answer question on a worksheet.
8-B-2: Reflection on Blogging
I enjoyed the blogging experience. The blogging assignments challenged me to think a little deeper about some of concepts that we discussed. I also enjoyed seeing the opinions and ideas of my classmates. They likewise caused to look at things differently and gave me some new ideas to incorporate in to my own teaching style.
The greatest learning experience for me was realizing how much my own math learning experience has impacted my teaching style. I also found great value in the various online resources we were given the opportunity to experience. They gave me ideas for how to incorporate technology in to my lessons.
This class caused me to reconsider the importance of writing in a math class. I was surprised to see how many different ways I can easily incorporate writing prompts as well as helpful they can be in solidifying students’ understanding of the material. I will definitely be incorporating math journals in to the curriculum in some capacity.
5-D-2: Applets
I really enjoyed the Algebra Balance Scales activity on the National Library of Virtual Mathematics website. I thought it did an excellent job of giving students a good visual for how to solve linear equations. It takes them step by step and allows them to see how doing the opposite to BOTH sides maintains the balance in the equation. I think this would be an excellent activity to do as an introductory activity when beginning a lesson on solving linear equations. It would also be a good activity to offer to students who are struggling and may need a little extra practice.
5-A-4: Evaluating Our Definitions: Equations and Functions
After reviewing my classmates definitions of equations and functions, I would revise my own post to include the definition for Algebraic Expressions, as well. Several of my classmates included this term and I found it to be a valuable addition to the post. I would probably also add the definition for Relations, as well, as I feel that understanding relations is an important piece to gaining a full understanding of functions.
To assess the students’ understanding of both terms, I would give students examples of both and have them explore the differences between the two. I would also introduce the students to mapping and graphing to allow them to use different visual tools (such as the vertical line tests) to evaluate whether or not it is a function.
5-A-3: My Definition of Equations and Functions
My defintion of equations and functions are as follows:
Equation: A mathematical statement in which the value on each side of the equal sign are exactly the same
Function: A relation between two sets of elements in which one set is dependent upon the other; each member of one set (domain) will be paired with exactly one member of the other set (range)
Example: The Student Council is selling tickets to the Homecoming Dance. Each ticket costs $4 and the Dance Committee spent $300 on a DJ and decorations.
**In this case, the total amount of money raised by the Student Council will depend directly on how many students purchase tickets to the dance. An input/output chart can be used to show the relationship between the number of tickets sold (input) and the amount of money raised (output).
y = 4x – 300 can be used to represent this function. The input, x, represents the number of students who purchased tickets and the output, y, represents the total revenue generated. Notice that the revenue depends on the number of students who attend!
x – input y - output
50 $-100 (Uh Oh! They’re losing money, let’s hope more people come!)
75 $0 (They just broke even here)
100 $100
150 $300
200 $500
300 $900
Magic of Proportion
5-B-1: The Magic of Proportions
Note: This is a multi-step problem serving as both my first and second example of using proportions in every day life. After all, life doesn’t always provide us with quick and easy one step problems!!
Problem: If a 150lb individual running at 5mph burns 91 calories every ten minutes, how many calories do they burn if they run three miles?
Step One: Determine how long it takes to run 3 miles at 5 mph
5 miles = 3 miles
60 min x min
Cross Multiply: 5x = 3(60)
Solve: 5x = 180 ; x = 36 minutes
Step Two: Determine how many calories are burned in 36 minutes.
91 calories = x calories
10 min 36 minutes
Cross Multiply: 91(36) = 10x
Solve: 3276 = 10x; x = 327.6 calories
Non-Linear Pattern Web Quest
Were there ideas or concepts you were not familiar with? What were they?
While I was familiar with the concept that nonlinear appear throughout nature (ex. fern leaves, snow flakes, pine cones), I was definitely in need of refreshing my skills on the specific details of a few of the terms we researched. Specifically, phyllotaxis and Fermat’s Theorem. I remember learning them, but it has been quite a while since I’ve seen or used them so I found the web quest to be a good opportunity to refresh my knowledge.
What images did you find particularly striking?
I found the picture of Romanesco (a cross between broccoli and cauliflower) particularly interesting not only because I was unfamiliar with the plant in general, but because the spiral patterns are so pronounced.
I also enjoyed the Lichtenberg figures. After seeing this picture, I recalled learning that the electrical discharge of lightening strikes can form fractal patterns, but I had long since forgotten that fun fact! I found it fascinating that science has enabled us to capture an image of these pattens such as the one I found below.
Can you identify any manifestations of nonlinear patterns within your home or your workplace? What are they?
There are many examples of nonlinear patterns in the landscaping at both my home and my office. At home I don’t have to look much further than the mums by my front door or the circles in my own fingerprints to find a good example. At work my office window overlooks the White River, so depending on which way the wind is blowing or what boat has come through and made waves, I get to view a different nonlinear pattern just about every time I look out the window.
How can you adapt this webquest activity for your classroom?
I think this is an excellent activity to adapt for classroom use as it really shows students how prevalent math is throughout nature. The students could be given some detailed parameters to guide their search including a worksheet to record their findings. They could also be asked to do a webquest and then find examples in their home/school/etc. and take pictures of those examples to share with the class.
Translating Pattern Narrative in to Formal Language
Pascal’s triangle begins with a 1 at the vertex (row 1) following by two 1’s in row two.
1
1 1
After this, to construct the following rows you always place a one on the far left and far right ends of each row. Each row has one additional term added to it. (Row 1 has one number, row 2 has two numbers, row 3 has three numbers, etc.) The additional term is determined by finding the sum of the two numbers immediately above it. For example:
Let’s look at another row to see how the pattern continues as the triangle grows. Notice that in row four, there are two 3’s. They were both created by finding the sum of the 1 on the outside (the left or the right, respectively) to the 2 in the middle. This works because the 3 on the left (circled) sits squarely between the left 1 and the 2 (also circled). Likewise, the 3 on the right (in red) sits squarely between the 1 on the right and the 2 (both also in red.)
The pattern continues as the triangle grows. The image below illustrates that the same pattern is evident even in the center of the rows (see the two 10s). (Circles/changed font color have been used to denote the numbers that are “working together.”)
Defining Linear Patterns
Non-Linear Pattern:
Formal Definition
Mathematics not in direct proportion; describes a relationship or function that is not strictly proportional
Linear Pattern:
Informal Definition
Patterns where all of the points lie in a straight line
Formal Definition
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
I think the biggest difference in the informal (“kid language”) definition and the formal definition is that the informal definition simply states the bottom line. The formal definition includes other “math words” (such as coordinates) and refers to the x and y values. For a student who struggles with math and who may not understand the concept of x and y coordinates, this definition wouldn’t make much sense. Conversely, the informal definition that I provided simply states the outcome of a linear pattern – the straight line. They may not understand how to get there or how to manipulate the equation, but at least they can visualize what we’re talking about.
Ultimately, the total understanding of the formal definition is the outcome of a well taught lesson rather than a starting point for that lesson. The lesson can be introduced with the kid-friendly informal definition. Through teaching about coordinates in a linear pattern, the students will be able to visualize develop an understanding of the relationships between the x and y coordinate. Thus, through working with, manipulating and exploring linear patterns, they will develop a better understanding of the formal definition…. even if they can’t repeat it verbatim J
My Reflection on Math Myths
Myth: Math requires only a very logical mind.
When I read this myth, I immediately thought of my high school calculus class. I recall studying with my friend, Martin. We made great study partners because our minds worked exactly opposite. If he really “got it”, then I was confused. Likewise, if it was a concept that I grasped easily, then it was likely more difficult for him. We both did well and it didn’t mean that one of us was better than the other, but our minds did work very differently. At the end of the day, it wasn’t one personality type or thought process that made one or the other of us more successful. Our own natural predispositions may have made one of us more adept to certain topics, but by working together we were able to learn how to train ourselves to think in a way that allowed us to succeed even with more difficult topics.
As a teacher, I feel it is my job to give students the tools to learn how to use their own strengths to succeed. Even in a subject like math, it is important to teach to various learning styles. We need to reach the visual learners as well as the auditory, the kinesthetic learners as well as the logical. A very “logical” minded student may be most successful simply by memorizing the formulas and applying them, while a more creative student may prefer to use pneumonic devices to remember the formulas. It is our job as teachers to present the material using a variety of methods to ensure that we reach all of the students.
Myth: It’s wrong to count on your fingers.
Well, here is comes again, my soapbox about learning styles! I suppose I should back up first and respond to the myth before I go off on my tangent 
I don’t recall ever being told it was “wrong” to count on fingers, but I do remember it being frowned on as I got older. I honestly don’t think this myth was fully dispelled for me until I took some educational psychology classes in college. (Here comes my soapbox…) Again, it all boils down to the fact that are students aren’t robots, they are individuals which means they each come with their own individual learning style. In the lower levels or even the remedial classes we as teachers understand the need for manipulatives. Somehow as we get in to higher levels of math, this basic learning tool gets overlooked. Is a student using his or her fingers to count really all that different from a student using a number line? It is a quick and easily accessible tool for a visual learner to be able to check their work. Now, this comes with a disclaimer. There does reach a point where it is necessary for a student to have a strong mathematical foundation and a solid understanding of number sense. If their constant need to use their fingers overrides their ability to commit even the most basic math facts to memory, they may need to be given some alternative tools to help ensure that their fingers don’t become a crutch.
Again, as a teacher it is our job to teach to the different learning styles. If we realize we have a chronic “finger counter”, there is a good chance that the student is a visual learner. Using tools such as algebra tiles and number lines will help us teach to this student’s learning style. We should, however, keep a close eye on the student to ensure that their finger counting isn’t a sympton of poor number sense. If we realize that is the problem, it likewise becomes our responsibility to offer appropriate accomodations to help pull them up to speed.
