Kelly Cole, Math, Bailey Middle School
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Abstract
“Slope and the Equation of a Name” uses a created line font to teach about slope and linear equations. The unit is designed to reach students who are first learning about slope, therefore it is specifically written for 8^{th} grade students, but may be used in 9^{th} grade algebra with some extension. Students first design their own linear font, creating initials on the coordinate plane. Using this, they then explore the different types of slope and how to determine the numerical value of the slopes used in their initials. Once slope has been determined, students will find the linear equations for the lines in their initials. The unit may be extended to include word problems and dilations. Students use various scale factors to reduce, enlarge, and flip their font. The game “Chutes and Ladders” has been modified for the coordinate plane in order to accommodate slope. Students will play the game to practice graphing slope. Fonts and “Chutes and Ladders” were chosen for my unit to get students interested in learning about slope and linear equations. I also hope that it encourages students to look for other places that slope and lines are found and used in the world around them.
Rationale
The slope of a line is a large part of the curriculum in 8th grade and a springboard for many of the concepts that are associated with Algebra. Yet it is a concept that many students struggle to understand. The comprehension, application, and analysis of this crucial concept aids in transitioning successfully into Algebra and beyond. Without a solid understanding of what slope is and how to find it, students are lost in interpreting linear word problems, graphing the equation of a line, and even understanding correlation and line of best fit. This unit is designed to help answer the following essential questions: What is the true meaning of slope? Is slope more than its numerical representation?
Many of my students are not math lovers. They do okay at math, some of them even do quite well. But do they really understand it? Do they perceive its beauty? Do they understand its relevance in their own lives? Does it relate to the world in which they live? In this unit I aim to catch their attention and ignite their imagination. I want them to learn about slope and lines, linear equations and inequalities in a fresh way. I want to link their world to the world of math and show them that the two can coexist.
I like the technical side of mathematics; the black and white of it; the way that math is objective. I like that there is a right answer and I like the mechanics of finding that answer. I like puzzles and the challenge of solving puzzles. I teach the way I like mathematics. I teach mechanics and technique. I teach logical process and how to get from a to b. I do a good job teaching students how to graph equations and how to find slope using a formula. However I don’t think this is enough. I want my students to know why they need to know how to graph a line and find slope. I want them to be able to interpret word problems and graphs. I want them to be able to talk logically about what slope represents in any given situation. Can they give me a real world example of a linear equation? It is not enough for a student to leave my class knowing that y = mx + b. This is a great place to start, but I ought to have higher expectations for my students. This unit was birthed from my desire to do a better job of teaching slope in a more relevant and interesting way. Students will learn not only the mechanics and understanding, but also application, analysis and synthesis.
Objective
I teach in a middle school with a population of approximately 1,400 students. The school is located just outside of Charlotte, North Carolina. It is part of the Charlotte-Mecklenburg school district. The school is in an affluent area and populated mostly by students whose parents are well-educated and have high-paying jobs. Our community is also growing rapidly, with many newcomers moving from Northern states such as New York, Michigan, and Ohio. While the majority of our student population is white (78%), we have a growing number of minority groups. The makeup of the school’s minorities is as follows: 13% African American, 7% Hispanic, and 2% Asian.^{ [1] }As stated above, our school is located in an affluent community; even so, 25% of our students receive free or reduced lunch. Six percent of our population is limited in English proficiency. Many of our LEP students come from countries like Mexico, Brazil, Guatemala, Honduras, Germany, and Sweden. Seven percent of our students are identified as having some type of learning disability and receive services from our EC department.^{ [2]}
Middle school math is broken into two levels at my school. Standard Plus classes contain students who may be well-below grade level, below grade level, or at grade level. Honors classes contain students who are at, or above grade level. Standard Plus classes teach on-grade level objectives. Honors classes teach objectives that are a year in advance, whilst maintaining objectives for the students’ current grade level. In eighth grade, that means that Standard Plus classes learn objectives from the 8^{th} grade North Carolina Standard Course of Study (NCSCOS), and Honors classes take Algebra 1 (traditionally a ninth grade course) receiving high school credit upon completion of the End of Course test at the end of the year. The Common Core standards are being adopted for North Carolina schools, and as such are being integrated into our curriculum gradually over the next couple of years. Standard Plus classes tend to have more diversity of race and socio-economic status than Honors classes. Students in Honors classes are more likely to be self-motivated. Standard Plus classes, because of their innate diversity, allow for a greater range of learning.
This unit is designed specifically for middle school students. I have included extensions that I plan to use with my eighth grade Algebra students, as well as remediation and differentiation that can be used in an inclusion setting. Students will explore slope through a variety of activities involving fonts and games.
Content Background
Prior Knowledge
It is important that students have prior knowledge of adding and subtracting integers, how to graph points on the coordinate plane, how to graph from a table, and how to solve an equation for a given variable.
Adding and Subtracting Integers
Integers are the set of whole numbers and their opposites {…-3, -2, -1, 0, 1, 2, 3 …}. When adding a positive number to a positive number the solution is a larger positive number (see example 1). When adding a negative number to a negative number the solution is a larger negative number (see example 2). When adding a positive number and a negative number the solution is the difference of the absolute values of each and will take the sign of the larger number (see examples 3 and 4).
Ex. 1 2 + 5 = 7
Ex. 2 -2 + (-5) = -7
Ex. 3 2 + (-5) = -3 Take the difference of the absolute values: |-5| – |2| = 3. Since -5 is a larger negative value than 2 is a positive value, the solution will be negative.
Ex. 4 -2 + 5 = 3 Take the difference of the absolute values: |5| – |-2| = 3. Since 5 is a larger positive value than -2 is a negative value, the solution will be positive.
I use the strategy of “heaps and holes”^{ [3]} when teaching about integers. A “heap” represents a positive number. A “hole” represents a negative number. When a heap meets a hole, the heap fills the hole creating “level ground” (i.e. zero). This is the concept of zero pairs. The first four examples can be shown visually as seen in Figure 1.
Figure 1: “Heaps and Holes”
Subtraction is the inverse of addition. We can employ “add the opposite” tactics to solve subtraction problems. I use the strategy “KFC”, which stands for “Keep the first number the same, Flip the minus sign to a plus sign, and Change the sign of the second number.” Students then employ the rules of addition.
Ex. 5 2 – 5
K F C
2 + (-5) = -3
Ex. 6 -2 – (-5)
K F C
-2 + 5 = 3
Ex. 7 2 – (-5)
K F C
2 + 5 = 7
Ex. 8 -2 – 5
K F C
-2 + (-5) = -7
Graphing on the Coordinate Plane
The coordinate plane is the perpendicular intersection of two number lines at zero. Points can be graphed in this two-dimensional space by using an x-coordinate and a y-coordinate (x, y). First move left or right along the x-axis, then up or down the y-axis to place a point.
Figure 2: The coordinate plane
Graph from a Table
An (x,y/ input,output) table is another way to represent a set of ordered pairs. The table may be vertical or horizontal.
x |
y |
1 |
5 |
2 |
7 |
3 |
9 |
4 |
11 |
X |
1 |
2 |
3 |
4 |
Y |
5 |
7 |
9 |
11 |
Figure 3: Vertical and horizontal tables of points graphed on coordinate plane.
Solve for a Variable
Equations are often written in a form where the dependent variable is not isolated. In order to graph the equation of a line in slope-intercept form (y = mx + b, where x and y are coordinates, m is the slope of the line and b is the y-intercept) it is important to know how to solve for the dependent variable. Students will isolate the dependent variable by using inverse operations and keeping the equation balanced by performing the same operation to both sides of the equals sign. An example of this is listed below.
First, subtract the term with the independent variable from both sides. This is done in order to isolate the dependent variable, noted in red, as seen below:
Ax + By = C
-Ax -Ax
By = -Ax + C
Since the dependent variable, y, is still not isolated, we must divide both sides of the equation by the dependent variable’s coefficient, as seen below:
=
y = +
Slope
�f* h��Z�nator. Break the top rectangle and bottom rectangle into said number. Roll the dice again to determine the numerator and shade this number in this number. Write the fraction about. Roll the dice one more time to determine the next numerator and shade in this number. It’s important to note here, that because of the probability of what the dice will roll, it is possible that the first fraction they create will be smaller than the second fraction. Because in fourth grade we do not work with negative numbers, I always am sure to tell my students the bigger fraction will always come first. The students then subtract the fractions.
Due to the nature of the fourth grade end of grade testing in math, I like the students to relate the numbers into word problems. Eighty percent of this test is word problems, so the more students work with word problems, and create their own word problems, the better. My students love movies, and they are especially fond of Harry Potter. To make a connection, and to make word problems more enjoyable to read, I have created five addition and subtraction fraction word problems that involve Harry Potter:[vii] The nature of these word problems involve with fractions in time. So it is important to review with students the fractions on a clock: ¼ hour= 15 minutes, ½ hour= 30 minutes, ¾ hour= 45 minutes, 1 hour=60 minutes. Also when talking about word problems, it is important to review with students how to read a word problem:
- Read the problem two times. Underline the clue words that tell you what the problem is asking you to solve. (*note we always refer back to our word wall with word problem clue words-ex. Add= in all, altogether, etc.)
- Re-read the problem. Determine the numbers you need and don’t need. Circle the ones you need, cross out the information you don’t need.
- Solve the problem. Use any strategy you are comfortable using solving the problem (for example, if it is a multiplication problem, some students draw pictures, some use an algorithm, some break apart numbers).
- Reflect: does the answer to your problem make sense? Did you answer the question?
- Check your work.
Here are some word problems for practice:
- Harry Potter practiced ½ hour of magic in the morning and ¼ hours of magic in the afternoon. How long did he practice altogether?
- Harry Potter and his best friend Ron Wesley were practicing for their Quidditch match against Slytherin. They spent ¾ of an hour in the morning practicing dodging bludgers, Harry Potter spent ¼ of an hour finding the Golden Snitch, and Ron spent ½ of an hour practicing in the goal. How much time did they spend practicing?
- Hermione Granger was in the library studying for her classes. She spent ½ hour studying for Potions, 2 ½ hours studying for Defense Against the Dark Arts, and 1 ¾ hours studying for Herbology. How much time did she spend studying?
- Ron, Hermione, and Harry Potter had 12 hours to find the last Horcrux that Lord Voldemort hid. They already spent ¾ of an hour arguing on where to start looking, 3 ¼ hours travelling, and 5 hours dueling with various enemies. How much time do they have left to find the Horcrux?
- Dumbledore decided to have another Tri-Wizard Tournament. The five athletes had 2 hours to get the Golden Egg from the untamed dragons. It took Harry Potter ¼ of an hour to walk out on to the battle field, and ½ hour running away from the dragon. How long does he have left?
I think that it is important once the students have practice solving given word problems, they create their own.
Lesson 14-15: Creating your own number system for fractions
To go along with the history of fractions, why not have the students pretend they are Ancient Babylonians or Ancient Romans and create their own number system for fractions. This also creates a fun way to solve addition and subtraction problems, as well as give them another visual that when they add and subtract fractions with like denominators, the denominator always stays the same. On the first day, model for the students your expectations. As a class create symbols to represent the numbers 0-12. It could look like:
0= square 1= triangle, 2- circle, 3=heart, 4= star, 5= diamond, 6= smiley face, 7= sad face, 8= flower, 9= rainbow, 10= triangle + square, 11= triangle + triangle, 12= triangle + circle.
Give students various problems to solve keeping in mind that the denominator throughout the problem never changes. The next day the students create their own symbols to represent each number. They create a key and at least ten problems. Collect all the students’ work samples and redistribute the work. Another fun activity to do, would partner up with another class. Each class creates their own number system and posts it. Create class problems with number systems and trade. It will be fun to see the kids solve various problems.
Lesson 16-18 Culminating Activities:
Activity #1: Fraction Fun Class Newspaper
Put the students into equal groups (if possible). Each group will be assigned a task to create a classroom fraction newspaper. I recommend using the Microsoft template for newspapers. It’s important to set the standards for what you are expecting. Each student should be assigned a role in the group in order to prevent conflict. I believe these groups should be created based on interest and talent. The students will create a column on the following:
- Arts and Entertainment- This can be done one of two ways, or both depending on the demand of students interested in this column. The students in this group should be broken into two groups within the group. One group of students should be the artists and the other group should be the reviewers. This column is a variation of “color by number”. If you have students like mine who are very talented artists, they can create their own picture template for coloring by fractions. The students would have to draw a picture break it up into various section sizes. Based on these sizes, the students would have to assign fractions to each section (the sections would have to be about the same size). In fourth grade we work with a lot of “benchmark fractions” (fractions that you can judge/compare other numbers with) which include: 1/4, 1/2, 3/4, 1/3, 2/3 and often 1/10 (because of its importance in decimals). I would limit the fractions the students use to create their picture to these fractions. They are also familiar with them as well. If this idea is too difficult for the students, the students can be given a “color by fraction” picture already designed for them to color. You can take a basic color by number picture, and substitute the basic numbers with fractions. Make sure to include a key in which each fraction is represented by which color. Once the artists have completed this colorful picture, it will be the art reviewers’ job to review the picture. The reviewers must use fraction words in their review. The review must be at least 3 paragraphs long (in fourth grade we make the rule that a paragraph is 5-7 detailed sentences), and have correct grammar and punctuation.
- Sports Page- this assignment would have to be given in advance, and to students who are allowed to watch full football games. The students would have to watch a full football game and keep track of the bloopers they see. For example, they could keep a tally of how many passes a certain quarterback makes, and how many were incomplete. Since they have done an activity like this in a previous lesson, they should understand the concept of it. The students could even do more than one blooper. One student could do incomplete passes, another student could do fraction of fumbles, or watch a certain player and create the fraction of tackles he gets over how many total carries he has. The students must then turn their data into a review. I always tell the students, that bloopers are funny, so your report should be fairly funny as well. The students must use fractions in their report. Since most sports pages include more than one sport, if there are many students who want to complete this task, they could do another sport-like hockey for example.
- Health Section- reporting on a “restaurant” aka- cafeteria food! What student doesn’t like to critique school food! In fourth grade, one of our big science units is Nutrition. The students learn about the five food groups, the key nutrient in each group, and the health benefit of each food group. They also learn recommended serving sizes for each group, as well as how to read the new and old food guide pyramid and portion plate. Over time cafeteria has become a lot healthier, but how much healthier? The recent food guide pyramid says we need to have (per day) 6 servings of the grain group, 5 servings of the vegetable group, 3 servings of the fruit group, 3 servings of the milk group, and 2 servings of the meat group. The students could track the cafeteria lunch options for a week. They create a daily report about which fractional part of their served lunch is from each of the food groups. They draw a conclusion about whether or not school lunches are healthy. Another variation of this article is one group could research and report about school lunches, and another group could interview other fourth grade students about what they eat daily for each meal, and create a review of what fraction of students interviewed are “healthy” (follow the food guide pyramid).
- Cooking Section- the students are given a “Fraction Chips” recipe from the book Eat
Your Math Homework.[viii] The students write up the article including the answers to things such as: how many ways can you serve ¾ of a tortilla? Can you make equal shares with the fraction chips you have? (How else can you make a ¾ serving?) What are the ways you can make a ½ serving size? How about a 2/3 serving size?
- Blogging Fractions- in every newspaper there is an Editorials section. Here people can write their opinions of fractions. Why not have the students write a blog about their opinion of fractions? Students can create a blog on www.blogspot.com and create a week long blog where people can respond to their thoughts. They turn these opinions into an article.
- Comics- through researching about fractions in pop culture, I have found a lot of funny comic strips about fractions- for example:
Activity #2: Prezi Presentation
Students are expected to learn “21^{st} Century Skills”. This incorporates technology and their use of technology into their daily education. Since worksheets for many students aren’t the most valuable learning tools for students, why not have the students use technology to create a presentation about what they learned. Using www.prezi.com, the students can create a study guide to show to the class about what they have learned about fractions.
Lesson 19-20 Fraction Review
Again, using modern technology in the form of our SMARTboard, I have created (using the Jeopardy template on SMARTtools), Jeopardy game that reviews all concepts of fractions taught. Since our math time is only 90 minutes per day, this game is played over two days. The students are split into two heterogeneous teams. Students “buzz” in when they know the answer. The game keeps all students engaged. If you do not have a SMARTboard in your classroom, putting questions on index cards works just as well.
Lesson 21- Fraction test
This test is not included in this unit, as it is a common assessment created by the entire fourth grade team.
Resources
Chartier, Tim . Interview by author. Email interview. Email, November 8, 2011.
” nrich.maths.org :: Mathematics Enrichment :: History of Fractions .” nrich.maths.org :: Mathematics Enrichment :: November 2011 Front Page . http://nrich.maths.org/2515 (accessed November 26, 2011).
“Delta Scape: Fractions.” Delta Scape. http://deltascape.blogspot.com/search/label/Fractions (accessed November 26, 2011).
“Fraction Chips.” In Eat Your Math Homework. Charlesbridge: Charlesbridge Pub Inc, 2011. 12-17.
Geometry and Fractions with Tangrams. Lincolnshire, Ill.: Learning Resources, inc., 1995.
Chicago does not offer any guidelines for citing a photograph in a bibliography.
“iss.schoolwires.” iss.schoolwires. http://iss.schoolwires.com/cms/lib4/NC01000579/Centricity/Domain/2862/Peanuts_Fractions_Cartoon.jpg (accessed November 26, 2011).
Chicago formatting by BibMe.org.
Appendix: Implementing District Standards
4.N.1 Understand the value of whole numbers and decimal representations from 0.01 to 100,000
4.N.1.1 Represent whole numbers and decimals using models, words and numbers (symbolic).
4.N.3 Understand the concept of equivalence with models as it applies to fractions, improper fractions, mixed numbers and decimals.
4.N.3.1 Identify equivalent fractions (using halves, fourths, eights; thirds, sixths, twelfths and fifths, tenths, hundredths)
4.N.3.2 Compare fractions, decimals and mixed numbers using models, benchmarks and reasoning
4.N.3.3 Represent mixed numbers as improper fractions and improper fractions as mixed numbers.
4.N.4 Use models to represent addition and subtraction of fractions and decimals.
4.N.4.1 Illustrate addition and subtraction of fractions, with like denominators, using area and length models.
Synopsis
This unit is designed to help make this abstract concept of fractions a little more clearly. This unit can be used for grade levels 3-5 as this is where fractions is most heavily taught in elementary school. This unit begins with a basic review of what is a fraction, how do you create fractions. It then moves into different aspects of fractions such as: fractional parts of a whole, improper fractions and mixed numbers, and adding and subtracting fractions. The unit culminates with a project that brings together all areas of fractions.
This unit is aligned with the North Carolina Standard Course of Study for Mathematics in fourth grade.
Endnotes
“nrich.maths.org :: Mathematics Enrichment :: History of Fractions .” nrich.maths.org :: Mathematics Enrichment :: November 2011 Front Page . http://nrich.maths.org/2515 (accessed November 26, 2011).
“nrich.maths.org :: Mathematics Enrichment :: History of Fractions .” nrich.maths.org :: Mathematics Enrichment :: November 2011 Front Page . http://nrich.maths.org/2515 (accessed November 26, 2011).
nrich.maths.org :: Mathematics Enrichment :: History of Fractions .” nrich.maths.org :: Mathematics Enrichment :: November 2011 Front Page . http://nrich.maths.org/2515 (accessed November 26, 2011).
[iv] Chartier, Tim . Interview by author. Email interview. Email, November 8, 2011.
[v] Geometry and Fractions with Tangrams. Lincolnshire, Ill.: Learning Resources, inc., 1995.
[vi] Geometry and Fractions with Tangrams. Lincolnshire, Ill.: Learning Resources, inc., 1995.
“Delta Scape: Fractions.” Delta Scape. http://deltascape.blogspot.com/search/label/Fractions (accessed November 26, 2011).
[viii] “Fraction Chips.” In Eat Your Math Homework. Charlesbridge: Charlesbridge Pub Inc, 2011. 12-17.
“iss.schoolwires.” iss.schoolwires. http://iss.schoolwires.com/cms/lib4/NC01000579/Centricity/Domain/2862/Peanuts_Fractions_Cartoon.jpg (accessed November 26, 2011).