Study of Patterns and Variables
Linda V. Migliaccio
Academic Setting
Class Setting
This unit is designed to be used in the second semester of seventh grade both regular and enriched education. My students range from completely unmotivated about math (more importantly, school) to highly engaged. Generally, my enriched students are more motivated about school. It is my belief that students must be comfortable enough in the classroom to take the risk of learning. Due to this my classroom environment is very student-centered. It is my goal to motivate my students to be creative problem solvers and be active participants in their education.
School Setting and Student Population
The students at Garfield Middle School come mainly from low income to medium income families. Garfield serves the North Valley of Albuquerque with an enrollment over 600. Seventy-three percent of our student population qualifies for free or reduced lunch. Eighty-one percent of the students are Hispanic (of these 15% are LEP--Limited English Proficient), approximately 13% are Anglo, 4% American Indian, 1.3% are African American, and 0.3 other ethnicities.
While approximately two-thirds of our students qualify for Title 1 services (a federally funded reading program), we currently service less than twenty-five percent of qualifying students. Our students are reading, on an average, two grades below grade level. The school is currently focusing on improving reading and math. Results from Terra Nova testing shows students scores in English, Math, Social Studies, and Science have dropped from last year.
In addition to being a Title I school, we also receive funding as a bilingual school. Many of our monolingual (languages other than English) and LEP students have a difficult time in academics because their reading level is below that of the text books. About 15% of Garfield's students are enrolled in our Special Education program. Garfield participates in the Reading Renaissance/Accelerated Reader program. This program encourages students to read more books and complete comprehension tests on the computer. Our reading scores on standardized tests have improved thanks to this program.
Unit Objectives
Students will work collaboratively to identify patterns of change in real world situations. Working together will help them develop essential communication skills. Students take verbal communication and turn it into written form as they explore mathematical formulas and the various ways of representing changing situations by using graphs and charts. Through solving problems and discussing various aspects of mathematics, students will become familiar with the symbolic representations used in math as well as the practice of analyzing data within ever-changing systems. By working together and communicating ideas, students will develop a deeper understanding of mathematics and its impact on their everyday lives. This will lead to better grades, test scores, and most importantly, mathematical confidence.
This unit will contain a summary of background information the students must have to be successful. This includes the ability to utilize addition, subtraction, multiplication and division of integers , experience gathering and organizing data, practice identifying patterns, experience with creating graphs including appropriate scales and intervals, and describing the cause and effect relationship of between a constant and a variable.
This unit will be divided into four basic parts. Each part will include daily activities and written explanations, as well as problem solving opportunities for students. Each part will have a number of daily activities followed by leading questions to promote cognitive development mathematical concepts.
The goals of A Study of Patterns and Variables unit are to introduce students to algebra through real-life situations. Students will learn to interpret data in various forms. Students will present data in several forms including tables, graphs, equations, and in written form. Students will make decisions based on information presented in tables, graphs and words. Students will understand and find patterns of change and predict outcomes based on those patterns. Most importantly this unit will be culturally relevant and intriguing, as well as a tool for developing mathematical confidence in seventh grade students.
Part One: Variables, constants and coordinate graphs
Students will become familiar with variables and understand their presence in every day life. Students will also develop their vocabulary in mathematics. Students will conduct an experiment, record the data in a table and utilize this information to create graphs. Students will also experiment with utilizing various scales and intervals for graphing. Students will hypothesize explanations for patterns and changes in the data. Student will develop an understanding of relationship between variables and their representation in graphs.
Part Two: Develops understanding of graphing changes
Students will interpret data presented on a table, graph, and in paragraphs. Students will communicate understanding of data presented in graphs, tables, and written form. Students will translate from one presentation to another (moving from charts to graphs). Students will develop an understanding of the advantages and disadvantages of each.
Part Three: Problem solving by analyzing tables and graphs
Students will change the presentation of the data from graphs to tables or from tables to graphs. Students will compare each representation and determine advantages and disadvantages for each. Students will determine patterns of change within each representation of data. Students will describe change which occurs in predictable ways which will be communicated mathematically as well as verbally.
Part Four: Rules by which patterns are governed
Students will understand the meaning of linear and non-linear relationships. Students will represent linear relationship utilizing tables and graphs. Students will utilize these relationships to compare rates. Students will find patterns of predictable change. Students will learn to express patterns of change in words, and mathematical symbols.
Context and Background
The Study of Mathematics Rationale
Learning math is learning to think. Many students (and their parents) believe that doing math requires some inherent skill and that they are missing that skill. In my experience this is incorrect. Anyone can learn to do math. Math requires both concrete and abstract thinking skills as well as logic. These are teachable skills (everyone is born with the ability to do them) that must be developed and practiced.
The first step to learning math is developing math confidence. Math confidence is believing you are capable of solving the problem presented. This does not mean you are a math genius and just automatically know how to solve the problem, it means that you are willing to dive in and find a solution. Math confidence is developed through practicing solving problems with increasing difficulty. Math confidence is not reading the problem and saying, "I can't do this." It is saying, "I'm not sure how to do this, but how about we try..." It is essential for students to work in a positive problem-solving environment in order to expand their math confidence and courage.
The second step to learning math is utilizing algorithms to solve concrete problems. This is solving problems with real-life relevance such as: If I have 2 apples and Jamie has 3 apples how many apples do we have all together? Students must build an understanding of number sense, algorithms for adding, subtracting, multiplying, and dividing whole numbers, fractions and decimals. By using concrete objects children can easily grasp physically as well as mentally, they develop a basic understanding of math. Students must be comfortable with concrete thinking in mathematics to build confidence and curiosity about math. The next step to learning math is moving from concrete to abstract thinking. Advanced mathematics is abstract thought. For this step to be successful students must be gradually transitioned from concrete to abstract reasoning. According to Keith Devlin, "...the real challenge--is for the student to come to view the abstract objects of mathematics as real." This can be done with real-life problems and repetitive practice. Having students work with things with which they are familiar gives them something to wrap their brains around while we pull them to abstract reasoning with mathematics. Utilizing puzzles, real life situations, and collaboration the students begin to feel comfortable with abstraction. Students need to practice to develop the thinking skills necessary for abstract thought as well as to build math confidence.
The Importance of Patterns and Variables
There are several reasons to teach algebra, more specifically patterns and variables, in middle school. Some reasons for seventh graders are: 1) to challenge them intellectually, 2) it is an excellence transition from concrete to abstract reasoning, 3) algebra is an important part of the state and district's Standards and Benchmarks for middle school, and 4) it develops logical thought and the problem solving strategies which can be utilized in all subject areas as well as "real life."
Academic Background
Algebra is defined as the branch of mathematics that is the generalization of the ideas of arithmetic (APS, 2001). Basically it is the abstraction of concrete ideas utilized in elementary math. Several algebraic concepts which are necessary for this unit are the ability to "describe, extend, analyze and create a wide variety of patterns. Describe and represent relationships with tables, graphs, and rules. Analyze functional relationships to explain how a change in one quantity results in a change in another. Use patterns and functions to represent and solve problems." (NM Content Standard 12, 2000).
Prerequisites for Teachers
As with any teaching assignment, teachers must have a sense of adventure and a passion for children. This enables them to create an environment in which children will be comfortable taking risks and developing a healthy curiosity about math activities. Teachers should also have working knowledge of basic algebra. Teachers should read the following documents which are fully cited in bibliography: APS District Core Curriculum and Scope and Sequence for Mathematics: Standard 4, "APS K-12 Mathematics Content and Performance Standards," "Building Bridges to Algebraic Thinking" by Day and Jones, "Finding Your Inner Mathematician: As the Abstraction Turns" by Devlin, "Mathematical Problem Solving in Support of the Curriculum?" by Holton and Anderson, Variables and Patterns: Introducing Algebra by Lappan, Fey, Fitszerald, Friel, and Difanis Phillips, and "New Mexico Performance Standards for Mathematics: Standard 12."
Prerequisites for Students
Students will need experience with basic computation of whole numbers, decimals, fractions, and integers. Students with experience writing about their thinking process and explaining their answers in words (yes, math problems can be answered with English) will do even better. The writing process helps increase understanding while incorporating Multiple Intelligences. Utilizing an experience centered curriculum encourages discovery and curiosity about academic concepts presented as well as learning in general. Writing allows students to focus more on the process of problem solving and increases their understanding of the concept. Working collaboratively especially on graphs, removes the mundane but keeps the fundamental skills intact. Additionally, students enjoy working together and their dialogue helps with written communication of ideas.
Glossary of Terms
Algebra the branch of mathematics that is the generalization of the ideas of arithmetic.
Equation a mathematical sentence stating that two quantities are equal.
Expression a combination of numbers, variables, and operation signs.
Function a relationship (a set of ordered pairs) in which for every x value there can be one and only one y value. Hint: a particular x value can appear only once in a set of ordered pairs.
Integers the set of numbers consisting of the counting numbers (i.e., positive whole numbers, negative whole numbers, and zero).
Linear function (equation) a function (equation) that has a constant rate of change and can be demonstrated graphically as a line.
Variable a quantity that may assume any one of a set of values. Also, a letter or symbol that represents an unknown value (e.g., In the equation 2x + y = 9, x and y are variables). (APS, 2001)Strategies for Implementation
As stated earlier, students are better able to understand abstract concepts which relate to concrete objects with which they have experience. For this reason, this unit is written as an experiential approach to teaching (and learning) mathematics. It incorporates problem solving in real-life situations. These problems promote "thinking out of the box," higher-order thinking skills, and mathematical confidence. Due to the written basis of this unit it will also enhance reading comprehension and writing skills. Student will be given the opportunity to practice communicating with their peers, parents, and other adults in a mathematical way.
Implementation
Unit Design
This unit, which will last approximately three weeks, deals with five eighth grade students who decide to work together to raise money for their trip to New York City and Washington DC. John, Georgia, Linda, Paul, and Rigo brainstorm several ways to earn money. Their favorite idea is to hold a dance at the community center.
This unit has four sections. Each section has a varying number of lessons. It is recommended to allot one day for each lesson. For assessment purposes, keep a portfolio including at least some experiments/activities, analysis and Reflective Reasoning journal writing. Each section is followed by "Reflective Reasoning" which are journal questions to stimulate metacognition and review concepts learned in that section. Also after each section there is a list of ideas for "Continued Practice and Assessment."
Each lesson includes all or some of the following: "Story Time" which is the set up for that days lesson. (I generally tell it to the students, however it can be made into a handout); "What Do You Think?" which is a chance for students to reflect on their previous experience or practice estimation; "Experiment" or "Activity" which is usually set up for the student to complete in a cooperative learning environment; and "Analysis" which includes follow-up questions to wrap up the lesson or be given as homework.
Materials necessary for this unit are a clock or watch with a second hand, grid paper, and color pencils for student use. Teachers will need transparencies for tables and graphs.
Albuquerque Public Schools Standards and Benchmarks
Objectives: APS K-12 Mathematics Content and Performance Standards, Draft 2001.
Strand V: Patterns, Functions, and Algebraic Concepts
Content Standard: The student demonstrates an understanding of algebraic skills and concepts through experiences with meaningful mathematical problems that focus on discovering, describing, modeling, and generalizing patterns and functions, representing and analyzing relationships and finding and supporting solutions.
6-8 Benchmark: The student uses tables, graphs, and symbolic representations of patterns. The student understands and uses variable and linear equations in algebraic problem solving.
Grade 7 Performance Standards:
Variable Expressions
- Identifies
and uses variable expressions and formulas to solve a variety of real-life situations (e.g. Simple interest: I = prt) (13F).- Represents, describes,
and analyzes numerical patterns and linear relationships using tables, graphs, words, and standard algebraic notation (13B).Functional Relationships
- Develops
and tests strategies for solving two-step equations (1C, 13D, 13F).- Translates
hypotheses into formal methods of solving algebraic equations (13F).- Recognizes
and applies the properties of equality (13C) (APS, 2001).
Section One
Lesson One
Story Time: (Read to or copy and distribute this problem set up for each student)Five eighth grade students decide to work together to raise money for their trip to New York City and Washington DC. John, Georgia, Linda, Paul, and Rigo brainstorm several ways to earn money. Their favorite idea is to hold a dance at the community center. The community center costs $115 (which includes the sound system). The students get to bring their CDs and play their favorite songs. Each person attending the dance gets 2 slices of pizza and a soda. It costs $7.50 per person at the door. To encourage their friends to attend they decide to include a dance contest. They will have a Dance-A-Thon with the winner and three friends getting a limousine ride home from the dance.
The students need to figure out how long to schedule the community center and when to have the limousine arrive to take the winners home. To figure this out, they did some dancing.
What Do You Think? (Students need to complete these estimates before doing the experiment):
- How long do you think you could dance without stopping?
- How do you think the speed of your dancing would change during the course of the afternoon?
- What conditions would affect the speed and length of time you could dance?
Experiment: Divide students into lab groups of four:
- a dancer (dances for 120 seconds)
- a timer (watches the clock and says "time" every 10 seconds)
- a counter (watches the dancer and counts complete number of twists)
- a recorder (writes the count every 10 seconds in the table)
If necessary, a group of three may be used where the counter is also the recorder. Another possibility is to have the teacher be the timer for the entire class, thus keeping all groups together and moving the experiment along. (I do not suggest dancing to music. This lesson requires a trend of having a decreasing number of Twists as the time increases. In my experiments dancing to the beat of music does not allow students to see the signs of fatigue as well in the 120 second allotment.) Each lab group decides who will start at which role. Prepare four tables starting at 0 seconds with intervals of 10 seconds ending at 120 seconds.
Figure 1-1.1
Time (in Seconds) 0
10
20
30
40
50
60
70
80
90
100
110
120
Twists (Total #)
For counting purposes we will define one twist as starting at a stand. The student will raise arms and turn them one direction (ex: to the left) while bending at the knees and turning feet in the other direction (ex: to the right). This is one-half a twist. When the student reverses and twists in the other direction they have completed 1 twist. Have entire class practice twisting to get the idea of how to dance and count this dance move. Warning: some students may be uncomfortable with dancing in front of their peers or physically unable, please be prepared for this to avoid embarrassment.
Have fun. The students will laugh, compare themselves, some even stop dancing due to fatigue or uncontrollable laughter. Repeat 4 times so everyone has a chance to be in each role.
Analysis:1. How did your twisting rate (number of twists per 10 second interval) change as time passed?
- How is this (Analysis #1) shown in your table?
- What does this pattern suggest about the change of your twist speed during a Dance-A-Thon?
Example tables:
Figure 1-1.2
Beth's
Time (in Seconds) 0
10
20
30
40
50
60
70
80
90
100
110
120
Twists (Total #)
0
11
22
32
40
51
62
74
86
94
102
103
108
Figure 1-1.3
Linda's
Time (in Seconds) 0
10
20
30
40
50
60
70
80
90
100
110
120
Twists (Total #)
0
7
20
31
43
55
66
78
85
93
102
114
124
Figure 1-1.4
John's
Time (in Seconds) 0
10
20
30
40
50
60
70
80
90
100
110
120
Twists (Total #)
0
9
20
29
38
46
55
64
76
86
96
105
115
Story Time: In the Twist experiment, the number of twists and the time are variables. These variables are in a relationship (one affects another). A variable is a quantity which changes (or varies). Variables can be categorized as independent or dependent. An independent variable can stand alone, it does not rely on the other variable. Time is often the independent variable. A dependent variable is determined by or depends on the other variable. A fun way to display the variables is in a coordinate graph. A coordinate graph is a visual way to show the relationship between variables.
What Do You Think? Which variable do you think is the independent variable? Which variable do you think is the dependent variable? Why?
Activity: Create a coordinate graph:
- Select two variables. We will use our Dance-A-Thon data from yesterday, so our variables will be time and twists.
- Select an axis for each variable. Generally, the independent variable is put on the x-axis and the dependent variable goes on the y-axis.
- Select a scale for each axis and label each interval. Determine the largest and smallest value you want to show on each axis of your graph. Since the time in our Dance-A-Thon experiment starts at 0 seconds and ends with 120 seconds we will want our graph to begin with 0 and end with 120 seconds. On the table we used 10 second intervals so we will use the same on our graph. In the Dance-A-Thon experiment the dependent variable is Twists. Look at your table from Lesson one to determine the scale. When I did this experiment, I complete 124 twists. So my y-axis scale will start with 0 and end at 125. I will label my intervals every 5 twists.
- Plot the points. In my example, at 50 seconds I did 55 twists. Starting on the x-axis, I go over to 50 seconds. Follow this line up to 55 on the y-axis. Put a dot on the intersection of 50 seconds and 55 twists. Continue plotting for each of data point. Remind students to "go over to the time first then up to the twists."
Example grids:
Figure 1-2.1
Twists
Figure 2-2.2 (graph of 2-2.1 data)