## Dilations and Linear Equations

Aaron Kollar, Science/Math, Bailey Middle School

(PDFUse PDF for accurate view of all images/formulas.

Abstract

My unit blends two typically abstract objectives of 8th grade math, Dilations and Linear Equations.  The students are to design a website, using a theme of sports, fashion, gaming, or music.  They must figure out a way to make money from their website and write to an investor to try and persuade them to invest money to get their website on the internet.  As they design the cover page of their website, they will have to dilate a pre-determined image as part of their website.  Once the cover page of their website is complete, they will have to choose from one of three website publishers that will launch their website.  Each publishing company has a different start up cost, and each company will generate a different, pre-determined monthly income for the student.  As the students track the profit that they make from their website, they should discover how a y-intercept relates to their start-up fee, and how the slope of their line relates to their monthly income.  By learning these concepts through a project with which the students could be creative and take ownership of the profit that they are making with their website, the goal is to make the learning more authentic for the students, so they will remember these concepts better.

School and Classroom Background

Bailey Road Middle School is an urban public middle school located in Cornelius, North Carolina. There are approximately 1,400 students at Bailey Road Middle School, which is part of the Charlotte-Mecklenburg School District. The school is made up of mostly Caucasian students, with the following breakdown of non-Caucasians; 13% African American, 7% Hispanic, and 2% Asian. The surrounding communities are mostly middle and upper class families, which makes Bailey Road Middle School different from the average middle school in Charlotte-Mecklenburg Schools. We do, however, serve communities that are impoverished, and approximately 25% of our students are on free or reduced lunch. 7% of our students have Limited English Proficiency, and another 7% of our students are on Individualized Education Plans for a specific learning or behavioral disability [1].

At Bailey, we have two levels of math classes. There is Honors math, which in the 8th grade is Algebra I, and there is Standard Plus math. The honors classes have a little more consistency as far as the achievement level of the students. In the Standard Plus classes, there are a variety of different levels of learners, ranging from far below grade level to slightly above grade level. Every year I have a couple students that make me wonder why they aren’t in Algebra as an eighth grader. Each year that I have been at Bailey Road Middle School, I have taught both Algebra and Standard Plus classes. This year in the 8th grade, there are 7 Algebra classes (honors) and 5 Standard Plus classes, which speaks to the extraordinarily high number of high achieving students at my school. At some schools in CMS with the same number of students as Bailey, there is only one 8th grade Algebra class.

I teach from the objectives stated in the North Carolina Standard Course of Study (NCSCOS), and next year, CMS will be adopting the Common Core Standards, which are a national set of learning standards. At present, the NCSCOS for 8th grade math states two objectives that relate to dilations in the coordinate plane and the slope-intercept form of a line. They are Objective 3.03, “Identify, predict, and describe dilations in the coordinate plane,”, and Objective 5.01C, “Find, identify, and interpret the slope (rate of change) and intercepts of a linear relation.”[2]

This unit is designed for the implementation in 7th or 8th grade, as students begin to explore the concepts of graphing, slope, intercepts, the slope-intercept form of a line (y = mx + b), and dilations in the coordinate plane. It can also be used with Algebra students, as most of these topics translate to both the 8th Grade Standard Course of Study and the standards that represent Algebra I. CMS has recently adopted the “Math Investigations” textbooks and online resources, and it is important to note that this unit does not draw from these textbooks or resources.

Rationale and Objective

In my opinion, one of the biggest hindrances for the middle school student when it comes to learning math is motivation. Most 8th grade students see learning math as a chore, much like washing the dishes or cleaning their room. In my experience, very few students come into my classroom proclaiming that math is their favorite subject. One objective of this unit is to allow students to be creative, while at the same time taking ownership of the unit. Studies have shown that students working on open-ended projects can outperform their counterparts that learn in a more traditional, direct instruction driven classroom. In a study by Jo Boaler, associate professor of education at Stanford University, students that learned through an open-ended project were shown not only to score better on math questions requiring analytic and or conceptual thought, but also math questions that involved rote learning[3] . I feel that giving my students a quasi open-ended project will give them more motivation to want to learn about the mathematics behind their project.

In my experience, when students jump into the concepts of slope, intercepts, dilations, and transformations, many of them don’t see the point. They are constantly asking when they are going to use this in real life. Through this unit, I hope to illuminate some real world examples as to why these concepts are important in some aspects of everyday life, while giving the students some chance to be creative and flexible in their learning. Teaching these concepts through direct instruction can make the material abstract and difficult to understand, and this is why this unit is focused on a project based method of teaching. As students progress through the stages of the project, they will begin to take ownership of these concepts, which will help them to better retain their understanding of them in the future, as well as provide a deeper understanding of the stated topics.

In addition, the concepts of dilation and slope-intercept form are typically taught in isolation. By using a project that incorporates both skills at different phases of the unit, I hope to make the learning of these two topics even more authentic.

History of y = mx+ b

To provide students with background information on how this puzzling equation came to appear in Algebra textbooks across the world, I will provide a brief background on the slope-intercept form of a line. The first notation of the slope-intercept form of a line appears to have come from an Italian mathematician named Vincenzo Riccati, who lived from 1707-1775. Although most of his work focused on mathematical analysis and differentiated equations, in his book Vincentii Riccati Opusculorum ad res Physica, & Mathematicas pertinentium, he refers to the equation y = mx+n. In his book, he credits a Swiss mathematician named Jacob Herman with coming up with this equation[4]. As to why the letter m is used to represent the slope of a line, the consensus is that nobody really knows. Students ask all the time why m is used to represent the slope of a line, and they are usually a little let down when, as a teacher, you have to admit that you don’t know and that nobody really knows.

Before Riccati and Herman, however, the concept of a linear equation can be traced back to ancient Egyptians and Babylonians. In the ancient Islam world, in the 9th century, an Arabic mathematician named Muhammad al-Khwarizmi wrote one of the first comprehensive documents about Algebra and equations. In Arabic, the word al-jabru means “restoration”, and it is the root word of “Algebra”. The first person to start using symbols to represent the unknown, such as “x” and “y”, was a French philosopher and mathematician named Rene Descartes. He proposed using symbols to represent the unknown in his book La geometrie, which was published in 1637[5].

Background Knowledge of Students

In order for students to successfully complete this unit, they must be proficient in several areas of math. First off, they must be able to do basic operations (multiplying, dividing, adding, subtracting). They must also be proficient multiplying with fractions, although it is not essential that they be able to do this without the use of a calculator. Students must be able to graph coordinate points, and in lesson 1, it is essential that the teacher monitor the students and ensure that they can graph coordinate points correctly. In addition, students need to be familiar with the concept of using variables to represent the unknown, as well as substituting numbers in for variables, and following a given formula, in this case, y = mx + b. Students will have to eventually plug numbers in for x in order to find y. Since a portion of this project will be done in groups, students should have experience working with groups, and the teacher should be familiar enough with their students to ensure the groups are constructed properly (more on this later).

UNIT

Lesson 1

The unit will begin by ensuring that all students review the properties of graphing coordinate points on a coordinate plane, and will continue on the first day with students making shapes by connecting coordinate points, which will be essential for the first part of this project. The teacher will first put a series of coordinate points on the board that, when connected will form a right triangle. The students should graph the points (-6 , 7) , (-6 , -5) , (4 , -5), and connect them in order. The teacher should look to ensure that all students remember how to graph these points correctly to make identical right triangles (Figure 1). The reason that it is important to ensure that all students can graph coordinate points correctly is that many times, when students have not worked with coordinate points for a while, they will have a tendency to make careless errors. Frequently they will confuse the x and y coordinate, and they also can put points in the incorrect quadrants by not paying close enough attention to the positive or negative value of each point.

Once the students have shown that they can graph basic coordinate points, the teacher will give them each a picture to draw on graph paper. This will be done by the teacher giving them a list of coordinate points, and the students will have to create the picture by connecting one point to the next. The teacher should give “Image A” to the half of the class that he/she feels are among the upper half of the class in terms of mathematical ability, and the teacher should give “Image B” to the lower half (Figures 2, 3, and 4). The teacher should assist students that need further clarification, and, of course have something extra for students to work on that finish early.

Once all students have completed their drawings based on the coordinate points they have been given, the teacher should pair up the students in carefully selected, pre-made pairings. One strategy that the teacher could use to ensure the quick transition into partner work is to write the name of each student’s partner on each worksheet before they are passed out. This way, the teacher is not spending time trying to put each student with their partner. All that the teacher has to say is, “Please pair up with the student whose name is on your worksheet.”

There are many things that should be considered when the teacher makes these pairings. The dyads should have one student that drew Image A, and one student that drew Image B. In addition, the teacher may want to consider pairing up students of the opposite gender, pairing up strong egos with students that are shy, as well as considering which students are the highest achieving with which students are the lowest achieving. Once the students have been paired up, they should describe the image they have drawn by reading the coordinate points, one at a time, and having their partner draw the image as they go. Making students have a mathematical discourse about the coordinate points, even in something as basic as this, will help the students retain information about graphing a picture using coordinate points, particularly the lower achieving students.

Although this lesson seems basic, it is critical that students can correctly work with ordered pairs in the coordinate plane, as well as explain their drawing to another student. During this lesson, the teacher should assist as much as necessary, as it is important that all students feel successful during this first phase of the unit. Research has shown that when students feel successful, it increases their motivation and reduces their anxiety [6].

Figure 1

FIGURE 2

Coordinates for Image A – Connect each pair of points in order. When you see the word STOP start a new line.

(8 , 5) à (6 , 3) à (-13 , 3) à (-13 , 9) à (-11 , 7) à (6 , 7) à (8 , 5) à (5 , 5) à (5 , 7) STOP

(2 , 5) à (-3 , 1) à (-8 , 5) à (2 , 5) STOP

(-12 , 2) à (-17 , 2) STOP

(-14 , 5) à (-17 , 5) STOP

(-14 , 7) à (-17 , 7) STOP

(-11 , 10) à (-16 , 10) STOP

(7 , 9) à (9 , 9) à (9 , 11) à (7 , 11) à (7, 9) STOP

(7 , 9) à (6 , 8) STOP

(8 , 9) à (8 , 8) STOP

(9 , 9) à (10 , 8) STOP

(9 , 10) à (10 , 10) STOP

(9 , 11) à (10 , 12) STOP

(8 , 11) à (8 , 12) STOP

(7 , 11) à (6 , 12) STOP

(7 , 10) à (6, 10) STOP

Coordinates for Image B – Connect each pair of points in order.

(3 , 10) à (5 , 8) à (4 , 8) à (6 , 6) à (5 , 6) à (7 , 4) à (6 , 4) à (8, 2) à (4 , 2) à (4 , -1) à (2, -1) à (2 , 2) à (-2 , 2) à (0 , 4) à (-1 , 4) à (1 , 6) à (0 , 6) à (2 , 8) à(1 , 8) à (3 , 10)

FIGURE 3

FIGURE 4

Lesson 2

In this lesson, the teacher will inform the students that they are going to design the homepage of a brand new website. The teacher can tell the story of Mark Zuckerberg and Facebook if they are familiar with it. A good website that sums up his life and rise to fame can be found at http://www.biography.com/people/mark-zuckerberg-507402[7]. The teacher can read this article with the students, or read it him/herself and summarize how, by creating a website, he rose to fame and fortune.

Or, as a motivator, the teacher can show this clip: http://www.youtube.com/watch?v=RyhI8NuxNyY&feature=related , which is a 5-minute video discussing the career of web design [8]. The reason for showing this video is to introduce to the students the concept of web design. As mentioned previously, getting students motivated is one of the key challenges facing any middle school math teacher, and this video should help motivate students to want to stay focused throughout the course of this unit. Once the video has ended, instruct students that they are going to come up with an idea to design their own website. The students will need to choose from the following choices as the theme for their website; Sports, Fashion, Gaming, or Music. Once the students have chosen which theme they are going to go with, they should brainstorm ideas of what specifically their website is going to do. Is it going to provide a service? Is it going to be informative in nature? Are they going to try and connect with people that share their love of a common theme? The teacher may want to do a starter activity prior to this lesson to help generate some ideas as a class, since many students may feel overwhelmed by the amount creativity that this part of the unit needs. For example, at the beginning of class, the teacher may want to write on the board, “What makes a website good?” As a whole class, the students can brainstorm reasons that some websites are considered good, while other websites are not so good. This starter should help students wrap their heads around some of the things that go into starting a website.

As a Language Arts connection, the teacher should tell the students that they are going to need investors to fund their website, and they must write a proposal to investors describing, in detail, what the purpose of their website is, and, just as importantly, how they plan to make money from their website. The teacher should remind all students that advertising through their website is one way to make money, as long as they get enough traffic on their site. Other ways to make money through their website may include a charging membership fee, selling items on their website, collecting donations, etc.

Lesson 3

In this lesson, students will have to recall the activity from lesson 1. Their job is to create the cover page of their website on a piece of 20-unit by 20-unit graph paper with the following guidelines:

1) They must use one image from the choices that follow (Figures 5 and 6) – As they re-draw their image of choice, they must correctly label each “dot” with the coordinate points that they have placed it on.

2) The title of their website must appear on their cover page.

Aside from that, the students can be as creative and colorful as they wish. This part of the unit may seem a little slow moving, and perhaps a bit “artsy” for some teachers, but it is an essential part of the students claiming ownership over their website, and thus, the learning of the mathematical concepts at hand. Once students finish designing their cover page, the teacher should carefully pair them up, making sure not to pair up two students that have chosen the same image to put on their website. Tell the students that, one at a time, they are to pretend that one of them is describing their image to a web designer, and that the web designer can only create the image by using coordinate points. Since the students have done a similar activity already, this should run pretty smoothly with little need for teacher support.

Lesson 4

In this lesson, students will explore the concept of dilation. The teacher should instruct all of the students that their web designer has made a mistake. The web designer was using out-dated software, where everything had to be described on a 20 by 20 grid. If the web designer wants to use that old software, he will have to charge the students double. The newer software, which is half the price as the old software, uses a 10 by 10 grid. The teacher should ask the students, How much are we going to have to scale down our designs? The students should respond that their designs need to be half as large or that their designs need to be cut in half. The teacher should then ask students to partner up and try and develop a way to make their chosen image half as large. If the students are having trouble, tell them to focus their attention on the coordinate points. The students should come up with the idea (with the teacher’s help) that if all of the coordinate points are cut in half, then the image will be cut in half. The teacher should eventually elicit the response out of the students that they can multiply each point by one-half to cut all of the coordinate points in half. The teacher should then introduce the concept of a scale factor.

FIGURE 5

Directions: Based on the theme of your website, create an image using the coordinate points below. Plot the points and connect them as indicated. DO NOT connect the last point to the first point, unless indicated. Whenever you see the word STOP, start a new line.

MUSIC

(-5 , 5) à (-7, 5) à (-7 , 3) à(-5 , 3) à (-5 , 9) à (-11 , 9) à (-11 , 3) à(-13 , 3) à (-13 , 5) à (-11 , 5)

FASHION

(5 , 10) à(6 , 9) à (7 , 10) à (7 , 7) à (9 , 4) à (3 , 4) à (5 , 7) à (5 , 10)

VIDEO GAMES

(-3 , -2) à (0 , -6) à (-3 , -6) à (-4 , -5) à (-10 , -5) à (-11 , -6) à (-14 , -6) à (-11 , -2) à (-3 , -2) STOP

(-3 , -3) à (-4 , -3) à (-4 , -4) à (-3 , -4) à (-3 , -3) STOP

(-5 , -3) à (-6 , -3) à (-6 , -4) à (-5 , -4) à (-5 , -3) STOP

(-7 , -5) à (-7 , -6) à (-9 , -8)

SPORTS

Plot the following points and then make a CIRCLE around them, using the coordinate points as a guide. Do not use straight lines for these first four points:

(2, -5) – (6 , -1) – (10 , -5) – (6 , -9)

Connect the following points with STRAIGHT LINES

(4 , -2) à (5 , -3) à (5 , -7) à (4 , -8) STOP

(8 , -2) à (7 , -3) à (7 , -7) à (8 , -8)

Figure 6

This would be a good time to put the word scale factor on a word wall, if there is one present in the classroom. Tell the students to start with their chosen image, and re-draw it using coordinate points that are half of their original. Once they have systematically dilated their chosen image in half, (something that can easily be checked by the teacher), they can try and dilate the rest of their webpage. For 8th grade students, it is unrealistic that their entire web page will be dilated correctly, but if they can correctly dilate their image using its coordinate points, then they have been introduced to and learned the concept of dilation. Again, it is important for the teacher assist each student to make them feel successful, so that math anxiety and self doubt do not come in to the picture in the students’ minds.

The teacher should then ask the students, What could we have done if we wanted to make our designs twice as large, instead of cutting them in half? The students should eventually respond that they could multiply all of the coordinate points by 2. What would the number 2 be called in this instance? (The scale factor). The teacher should ask multiple questions like this (What would we do if we needed to make our designs a third of the size of the original?) .

At this point, the students should be given a sheet of 10 by 10 graph paper. Tell the students to start with their chosen image, and re-draw it using coordinate points that are half of their original. Remind the students that they can multiply all of the coordinate points of their image by one-half, or they can simply divide all of the points by two. They will have to estimate points that lie between two integers, like 3.5 for instance. After they have re-drawn their original design, the teacher should give the students a ruler and have them measure several lengths of their line segments on both designs and compare them. Students should come to the conclusion that, not only are the coordinate points half of what they were in the original design, but the side lengths are also half as long.

Once this process is complete, the teacher should explain that they have all used a mathematical process known as dilation. This would be a good time to put the word dilation onto the word wall if one is present. To reinforce this skill, the teacher should give a homework assignment where the students have to dilate different objects in the coordinate plane using a variety of different scale factors. Students should come to the conclusion that, any time a scale factor is less than one, the object shrinks, or gets smaller. Likewise, any time a scale factor is greater than one, the object is enlarged.

Lesson 5

Starting with this lesson, the students will transition from learning how to dilate an image to the concept of the slope intercept form of a line. The students will begin by choosing the company that will help launch their website. The teacher will give the students three choices that will provide creative advice, put their site online, and provide advertising for their website. Website group #1, which is called Web-X Designs, will cost the student \$2,000 in start-up fees, but they have an excellent reputation for promoting new websites. Website group #2, which is called Net Surfer Plus, costs \$500 in start-up fees, but they are an unknown group, and so the promotion of the students’ websites is a bit of a risk with investment group #2. The third option is to pay the computer teacher at their school \$100 to launch and promote their website. Although he swears he will do a great job, he seems a little strange, but you can’t beat \$100 to launch your website.

After each student chooses how they are going to start their website (Web-X Designs, Net Surfer Plus, or their computer teacher), they are now locked-in to a course of action that will take them through the rest of the project, learning about a linear equation and the slope-intercept form of a line along the way. The students that started with Web-X Designs should be told that their website will make a steady stream of revenue of \$250 per month. The teacher can be creative as to explain how this happened. They can even individualize the profit for each student, if they have time. For example, if a particular student wrote that their website would make money by charging a membership fee, the teacher can say that the membership fees are coming in at a rate of \$250 per month. If the student explained that their website would make money through advertising, then the teacher could tell them that Web-X Designs predicts that their advertising revenue will amount to an extra \$250 each month. The students that chose Net Surfer Plus will make a steady stream of income of \$150 per month. The students that chose to have their computer teacher launch their website will receive an income of \$50 per month. Pair up students that chose the same group (keeping in mind that factors that were discussed earlier), and have them project how much money they will make after 1month, 2 months, 3 months, etc., up to six months. Remind students that they have to subtract the amount that they paid to launch their website. It might be helpful to provide a table that looks like this:

 Month Profit Made (Don’t forget to subtract your start-up fee) 1 2 3 4 5 6

Although they won’t know it at the time, the students are already using the concepts of functions and linear equations. They are using linear equations, like y = 250x – 2,000 if they chose Web-X Designs, without really even knowing it. This makes the learning authentic to each learner.

Lesson 6

In this lesson, students will start graphing their tables on a coordinate plane. Since the students have already plotted so many points in the earlier lessons of this project, the mistakes of the actual graphing of points should be minimal. The teacher should, as a whole class, display how to set up their graphs. (See Figure 7). The teacher should guide the students to make sure that they label the x-axis with Months, the independent variable, and the y-axis with Profit, the dependent variable. Also guide the students to understand why profit is the dependent variable, because it depends on how many months the website has been online. Working with a partner, they should individually graph the first 6 months of their profits. The graphs should show that, after 6 months, the students that chose Web-X Designs have not made a profit yet. In fact, they are still at negative \$500. Net Surfer Plus has made \$400 in profits and the third group (the computer teacher group) has made \$200. Next, have them graph the next 6 months of profits, so that they will have profit data for one full year.

After a year, Web-X Designs will have made a profit of \$1,000, Net Surfer Plus will have made a profit of \$1,300, and the computer teacher will have made the students a profit of \$500 (Figure 7). Once all students have graphed out their profits for one year, arrange students in groups of 3, so that each “company” is represented. The teacher should once again consider the factors of ability, gender, self-esteem, etc. Have each student copy the lines that represent the other groups on to his/her graph. The teacher may want to have each student color code each line. For example, the line that represents Web-X Designs could be blue, the line that represents Net Surfer Plus could be red, and the computer teacher line could be brown. Have them analyze each other’s graphs, and have them answer the following questions: Where does each graph touch the y-axis? What does this number represent? (The start up fees). Which line is the steepest, and which line is the least steep? Why is this? (Web-X Designs is the steepest because they make the most per month. The people who chose their computer teacher have a line that is not very steep at all, because they make the least per month). If these websites were on the internet for 2 years, who would make the most money? Why? (Web-X Designs would make the most money because their graph grows at a higher rate than the other two groups).

Lesson 7

In this lesson, students will make the connections between their graphs, the y-intercept of a line, the slope of a line, and the slope-intercept equation (y = mx + b). First, as a whole class, have students see if they can come up with an equation to match each of the 3 groups’ profits. Tell them that, just like on their graphs, we will let x represent the amount of months the website has been online, and we will let y represent their profit after a given month. Since the equation is going to calculate the profit of each group, the teacher should guide the students to realize that the equation should start with “y =”. Next, guide the students to come up with the 3 equations to represent the three groups. Web-X Designs will be represented by the equation y = 250x – 2000, Net Surfer Plus will be represented by the equation y = 150x – 500, and the computer teacher will be represented by the equation y = 50x – 100.

Ask the students Which line was the steepest? How does that relate to the equation for that group? Which line was the least steep? How does that relate to the equation for that group?

This is a perfect time to introduce the concept of slope to the students. By looking at how the steepness of the graph matches the amount made per month, students should see the connection between the slope of the actual line and the slope, or “m”, of the equation. The students should see that the slope is the same thing as the amount made per month. This is another opportunity to add the word slope to the word wall.

FIGURE 7

Next, ask the students How does the point at which the line crosses the y-axis relate to each equation? Students should notice that each line touches the y-axis at the same number that ends each equation, i.e. the “+ b” part. This is a great opportunity to introduce the concept of “y-intercept.” Students should also see the connection between the y-intercept and the fact that, for each group, it is their start up fee. In many linear equations, the y-intercept represents some sort of starting point. At this point, the teacher should put y-intercept on the word wall.

And now, finally, the teacher can introduce the algebraic way to write the slope-intercept form of a line, the infamous y = mx + b. The students will almost certainly ask, “Why does the letter m represent slope?” And the teacher can give the honest, but frustrating answer of “Nobody knows.”

Once this part of the unit is complete, teacher can begin to extend on the concept of y = mx + b with more real word situations. Although the amount of situations that a teacher can create and have the students represent both graphically and algebraically are limitless, some examples include:

• A taxi service charges a \$3.00 flat fee and then charges that passenger \$1.25 per mile
• A leaky bucket has 10 gallons of water and is losing water at a rate of a ½ gallon every minute. (This situation is good to introduce the concept of a negative slope as opposed to a positive slope).
• Little Johnny and his mom arrange an allowance that pays Little Johnny \$5.00 a week, plus an additional \$2.00 that Johnny works around the house.
• A local gym charges an up front, \$50.00 fee, plus an additional \$30.00 per month for a membership.
• A 350 pound man starts a diet, which enables him to lose 6 pounds a week.

In summary, this unit is designed to teach students about the slope-intercept form of a line and the dilations of objects, without these objectives coming off as abstract and non-relatable. By trying to hook students in by telling them the story of Mark Zuckerburg and letting the students be creative with the design of their own website, the goal is to set a stage where each student will relate to these topics. In the end, the goal is for the students to understand that, in dilation, an object is enlarged or scaled down, and on a coordinate plane, this can be accomplished by using coordinate points, and multiplying them by a scale factor. Students should also understand the relationship between a real life situation, a table of values, an equation, and a graph. They should realize that the slope of a line is the rate of change for that line, and that the y-intercept of a line is not only where a line crosses the y-axis, but it often relates some sort of “starting point” of a linear equation.

Works Cited

http://www.greatschools.org/cgi-bin/nc/other/3383#toc (Accessed October 3, 2011).

2. North Carolina Department of Instruction. “8th Grade Math Standard Course of Study.”

3. Edutopia. “PBL Research Summary: Studies Validate Project-Based Learning.” Edutopia.org

http://www.edutopia.org/research-validates-project-based-learning (Accessed September 12, 2011).

4. Askville. “In the slope intercept formula y=mx+b, why is the letter “m” chosen to represent slope?” Askville.com. http://askville.amazon.com/slope-intercept-formula-y%3Dmx%2Bb-letter-chosen-represent/AnswerViewer.do?requestId=6061505 (Accessed September 12, 2011).

5. Derbyshire, John. Unknown Quantity: A Real and Imaginary History of Algebra. Washington, DC: Joseph Henry Press, 2006

6. Turner, J. C., et. al., “Motivating Mathematics Learning: Changes in Teachers’ Practices and Beliefs During a Nine-Month Collaboration.” American Educational Research Journal v. 48 no. 3 (June 2011): p. 718-62

7. Biography. “Mark Zuckerburg. Biography” Biography.com

http://www.biography.com/people/mark-zuckerberg-507402 (Accessed November 15, 2011).

Annotated Bibliography

Turner, J. C., et. al., “Motivating Mathematics Learning: Changes in Teachers’ Practices and Beliefs During a Nine-Month Collaboration.” American Educational Research Journal v. 48 no. 3 (June 2011): p. 718-62. This is a great research article about the research involved with motivating students to learn math. It talks specifically about how anxieties and fear of math can be a roadblock to student achievement, and it touches on how confidence is a very important factor in determining the success of a math student.

Derbyshire, John. Unknown Quantity: A Real and Imaginary History of Algebra. Washington, DC: Joseph Henry Press, 2006. This book provides a detailed history of algebra. It is full of interesting facts and stories about the mathematicians that invented what we know today as Algebra.

Edutopia. “PBL Research Summary: Studies Validate Project-Based Learning.” Edutopia.org.

http://www.edutopia.org/research-validates-project-based-learning (Accessed September 12, 2011). The Edutopia website is full of free resources for teachers and students. This article summarizes research that stresses the importance of authentic learning instead of rote learning. On this website, there are also links to other studies, as well as actual projects. I use Edutopia for my classroom all the time.