## Insights Into Algebra 1: Teaching for Learning

# Variables and Patterns of Change

## In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.

## Part 1: Translating Words Into Symbols

Translating words into symbols is equivalent to modeling a situation using an equation and **variables**. Similarly, algebraic equations and inequalities can represent the quantitative relationship between two or more objects.

## Part 2: Linear Equations

**Linear equations** can be used to model situations involving one or more unknowns. A linear equation is an algebraic equation whose variable(s) is/are of degree one. For example, 6x + 3 = 27 and a = 2b are linear equations. Solving linear equations is a key component of the algebra curriculum.

### Part 1: Explanation

**Explanation**

Variables have important roles in algebra. Often, they serve as placeholders in equations for which there are unknown quantities. In such cases, finding the specific value of the variable for which an equation is true yields the solution to the problem. Students are likely to be familiar with this from early elementary school, when they filled in the square to make a statement such as 5 + = 13 true.

One example of a problem that uses a variable in this way involves finding the width of a rectangle when the area and length are known. If the area is 36 square units and the length is 9 units, an equation for finding the width (w) of the rectangle is 9w = 36. In this situation, the variable w does not vary; it is a placeholder representing 4 units.

Another use of variables is to represent quantities that truly vary. In the area formula 36 = lw, the value of each variable depends on the value of the other. Consequently, as the value of one variable changes, the values of the other variables change, too. Used in this manner, variables serve several purposes that we will explore in the Role in the Curriculum section of this workshop.

Other examples:

- If a runner jogs one mile in eight minutes, the number of miles the runner covers in t minutes can be represented by the expression . Therefore, the equation could be used to determine how long it took to run miles. More generally, if the distance run is d miles, the linear function can be used.
- To determine the number of 1-foot tiles needed to construct a border around a pool that measures
*l*feet by*w*feet, consider that the two lengths need*l*tiles each, the two widths need*w*tiles each, and four tiles are needed at the corners. The equation could be 2l + 2w + 4. Equivalently, however, the following expressions also represent the number of tiles needed: 2(l + w + 2), (l + 2)(w + 2) – lw and 2(l + 1) + 2(w + 1).

### Part 1: Mathematical Definition

**Variable: **A symbol used to represent an unspecified member of some set. A variable is a “place holder” or a “blank” for the name of some member of the set. Any member of the set is a value of the variable and the set itself is the range of the variable. If the set has only one member, the variable is a constant. The symbols x and y in the expression x^{2} – y^{2} = (x + y)(x – y) are variables that represent unspecified numbers in the sense that the equality is true whatever numbers may be put in the places held by x and y.

(Source: James, Robert C. and Glenn James. *Mathematics Dictionary* [4th edition]. New York: Chapman and Hall, 1976.)

### Part 1: Role in the Curriculum

Variables, expressions, and equations are important parts of the algebra curriculum. The National Council of Teachers of Mathematics (NCTM) states:

Students’ understanding of variable should go far beyond simply recognizing that letters can be used to stand for unknown numbers in equations … The following equations illustrate several uses of variable encountered in [algebra]:

The first equation illustrates the role of variable as “place holder:” x is simply taking the place of a specific number that can be found by solving the equation. The use of variable in denoting a generalized arithmetic pattern is shown in the second equation; it represents an identity when t takes on any real value except 0. The third equation is a formula, with A, L, and W representing the area, length, and width, respectively, of a rectangle. The third and fourth equations offer examples of covariation: in the fourth equation, as x takes on different values, y also varies.

(Principles and Standards for School Mathematics, NCTM, 2000, p. 224)

Students should be able to relate variable expressions and equations to other forms of representations, such as tables, graphs, and verbal descriptions. One way of developing competency in this area is to use a functional approach. Present students with a table of values and have them generate a function to describe the relationship. For instance, consider the following table:

x | y |

1 | 4 |

2 | 7 |

3 | 10 |

4 | 13 |

5 | 16 |

From this table, students should notice that the y value is always equal to one more than three times the x value. Stating this relationship using variables, y = 3x + 1.

In a similar manner, Janel Green used a real-world problem involving the number of tiles needed to form a border around a rectangular pool. This problem provides a context in which students can use variables to represent a situation. NCTM recommends the pool-border problem as a means of developing fluency in using various representations:

Students should become flexible in recognizing equivalent forms of linear equations and expressions. This flexibility can emerge as students gain experience with multiple ways of representing a contextualized problem. For example, consider the following problem, which is adapted from Ferrini-Mundy, Lappan, and Phillips (1997):

A rectangular pool is to be surrounded by a ceramic-tile border. The border will be one tile wide all around. Explain in words, with numbers or tables, visually, and with symbols the number of tiles that will be needed for pools of various lengths and widths.

(PSSM, p. 282)

Both examples above give students the opportunity to translate numerical situations into symbolic expressions. However, it is also important for students to translate symbolic expressions into words. For instance, students might be given the expression 6x + 3 and asked to state a situation for which that expression could serve as a representation. A possible response might be, “Harriet has three beads. She can buy more in packets of six beads each. If she buys x packets, she will have 6x + 3 beads.” Exercises of this type could be extended to include more complex expressions and equations. In most classrooms, students typically gain extensive experience translating situations into symbolic expressions, but they generally don’t encounter opportunities in the other direction. Experience with both types of situations – generating expressions for particular situations, and generating situations for particular expressions – is known as “bi-directional practice,” and such practice is crucial for developing in-depth understanding.

Effectively translating words into symbols involves recognizing equivalent forms of the same relationship. Representing the same situation in more than one way provides opportunities for students to understand equivalent algebraic expressions. The pool problem Janel Green used is one such example. NCTM states:

Complex symbolic expressions also can be examined, such as the equivalence of 4 + 2L + 2W and (L + 2)(W + 2) – LW when representing the number of unit tiles to be placed along the border of a rectangular pool with length L units and width W units. (

PSSM, p. 225)

Once students have found more than one expression that describes the number of tiles needed, they should be asked to find as many equivalent expressions as possible, and to discuss reasons why the expressions are equivalent.

In addition to meeting the NCTM Algebra Standard, the ability to interpret and describe situations in various ways helps students attain the goals of the Representation Standard. By the end of high school, students are expected to understand various representations of the same relationship and effectively represent situations using tables, graphs, and symbolic expressions. NCTM explains this further:

By working on problems like the “tiled pool” problem, students gain experience in relating symbolic representations of situations and relationships to other representations, such as tables and graphs. They also see that several apparently different symbolic expressions often can be used to represent the same relationship between quantities or variables in a situation. The latter observation sets the stage for students to understand equivalent symbolic expressions as different symbolic forms that represent the same relationship. In the ’tiled pool’ problem, for example, a class could discuss why the four expressions obtained for the total number of tiles should be equivalent. They could then examine ways to demonstrate the equivalence symbolically. For example, they might observe from their sketches that adding two lengths to two widths (2L + 2W) is actually the same as adding the length and width and then doubling: 2(L + W). They should recognize this pictorial representation for the distributive property of multiplication over addition – a useful tool in rewriting variable expressions and solving equations. In this way, teachers may be able to develop approaches to algebraic symbol manipulation that are meaningful to students. (

PSSM, p. 283)

Students can solve some equations by examination or guess-and-check procedures; other equations may warrant a paper-and-pencil solution or possibly the use of technology and algebra software. A major goal of algebra is for students to acquire fluency with symbols, expressions, and equations, and to be able to represent various situations using algebraic expressions and equations. In this way, they will be able to determine which method of solution is most appropriate for a given problem.

### Part 2: Explanation

A linear equation is a polynomial equation of the first degree, such as x + y = 7. Said another way, a linear equation has no variables raised to a power other than one.

The simplest linear equations involve only one unknown, such as x + 2 = 3, and they are solved by finding the value of the unknown that makes the equation true. For example, the equation above is true when x = 1, because 1 + 2 = 3. More complex linear equations may contain more than one variable, such as x + y = 7 or a – b + c – d + 4 = 14. This workshop will focus on the simplest linear equations and those whose graph is a line.

Linear equations with just one unknown are solved using equivalence transformations, sometimes informally called “inverse operations.” In this process, the operation of multiplication is “undone” using division, because multiplication and division are multiplicative inverses; similarly, addition is “undone” using subtraction, because addition and subtraction are additive inverses. The reason this process works is because the equal sign acts as a balance between the left side and the right side of the equation. As long as the same operation is applied to each side of the equation, the equation remains in balance and the equality is preserved. For example, the equation 2x + 3 = 9 is solved as follows:

- Subtraction is the additive inverse of addition. Therefore, to “undo” the addition of 3, subtract 3 on both sides of the equation: 2x + 3 – 3 = 9 – 3, which yields 2x = 6.
- Because division is the multiplicative inverse of multiplication, to “undo” the multiplication of x by 2, divide both sides by 2: , which yields x = 3. Other examples:
- A traditional example involves people’s ages. For instance, “Becky is 6 years younger than Sally, and Sally is 13 years old. What is Becky’s age?” The equation b + 6 = 13 can represent this situation, and it is true when b = 7, so Becky is 7 years old.
- A person bought 3 cans of soda as well as several six-packs of soda, and she has a total of 27 cans. The linear equation 6x + 3 = 27 determines the number of six-packs she bought, where x is the number of six-packs.
- Linear functions can model the “transmission factor” of two gears. The ratio between the number of revolutions made by a 1 cm gear (b) and the number of revolutions made by a 2 cm gear (a) can be expressed as a = 0.5b and b = 2a.

### Part 2: Mathematical Definition

**Linear Equation or Expression**: An algebraic equation or expression which is of the first degree in its variable (or variables); i.e., its highest degree term in the variable (or variables) is of the first degree. The equations x + 2 = 0 and x + y + 3 = 0 are linear. An equation or expression is linear in a certain variable if it is of the first degree in that variable. The equation x + y^{2} = 0 is linear in x, but not in y.

(Source: James, Robert C. and Glenn James, *Mathematics Dictionary* (4th edition). New York: Chapman and Hall, 1976.)

John McLeish provides a useful explanation of linear: “Linear means an equation of the first power of the unknown; such an equation can be represented by a straight line graph, hence ‘linear.'” (McLeish, John. *Number: The History of Numbers and How They Shape Our Lives*. New York: Fawcett Columbine, 1991.)

### Part 2: Role in the Curriculum

To become comfortable solving and manipulating linear equations, students will need to experience linear relationships in various situations. They will need a significant amount of practice before developing fluency.

Upon the successful completion of an algebra course, students should be able to use symbolic notation to represent and explain mathematical relationships and solve linear equations. The National Council of Teacher of Mathematics (NCTM) states:

Although students will probably acquire facility with equations at different times … students should be able to solve equations like 84 – 2x = 5x + 12 for the unknown number … and to recognize that equations such as y = 3x + 10 represent linear functions that are satisfied by many ordered pairs (x, y).

(Principles and Standards for School Mathematics, NCTM, 2000, p. 226)

Students should also be able to produce two or more equivalent expressions that represent the same situation and to use simple formulas.

Solving linear equations is a key part of attaining a global understanding of linear relationships and involves more than the ability to solve for an unknown. The solution of linear equations is often a necessary step when interpreting a complex situation. Students may describe a linear relationship using a table, graph, or words, and from that description they may generate an expression or equation to represent the situation. For instance, students may describe the cost of a cell phone plan in various ways:

Some students might describe the pattern verbally: “Keep-in-Touch costs $20.00 [monthly] and then $0.10 more per minute [of use.]” Others might write an equation to represent the cost (y) in dollars in terms of the number of minutes (x), such as y = 20.00 + 0.10x.

(PSSM, p. 226)

From the function, students should be able to find the cost for any number of minutes. The cost for 25 minutes occurs when x = 25, and finding the associated cost involves solving the equation y = 20.00 + 0.10(25). Similarly, the number of minutes for which the cost would be $35.00 occurs when y = 35.00, and students can find that value by solving the equation 35.00 = 20.00 + 0.10x.

### Resources

The links below are to pages within stable sites and are current as of the date of publication of this workshop. Due to the ever-changing nature of the Web, it is possible that some links may change. Should you reach a non-working link, we recommend entering a couple words from its description into the site’s search function, or into a Web-based search engine.

**Cooperative Learning:**

Andrini, Beth.* Cooperative Learning & Mathematics (Grades K-8)*. Kagan Cooperative Publishing.

This book offers various cooperative learning activities for elementary and middle school mathematics.

Glosser, Gisele. “Cooperative Learning Techniques.” From *Mrs. Glosser’s Math Goodies*, ©1998-2004.

This article is available online at http://www.mathgoodies.com/articles/coop_learning.html.

Kagan, Spencer. *Cooperative Learning*. Kagan Cooperative Publishing.

This book provides information and tips on forming teams, classroom management, and lesson planning, and provides some of the research and theory that supports cooperative learning.

Kushnir, Dina. *Cooperative Learning & Mathematics High School Activities*(Grades 8-12). Kagan Cooperative Publishing.

This book explores various cooperative learning activities for use in a high school mathematics class.

Slavin, Robert E. “Research on Cooperative Learning: Consensus and Controversy.” *Educational Leadership*, December 1989/January 1990; vol. 47, no. 4: pp. 52-54.

Thirteen Ed Online’s Concept to Classroom Cooperative and Collaborative Learning Workshop

http://www.thirteen.org/edonline/concept2class/month5/index.html

Burns, Marilyn. “7 Musts for Using Manipulatives.” *Scholastic Instructor*, June 1996. *Scholastic Instructor* is available online at http://teacher.scholastic.com/products/instructor/.

Ross, Rita and Ray Kurtz. “Making Manipulatives Work: A Strategy for Success.” *Arithmetic Teacher* [National Council of Teachers of Mathematics], January 1993; issue 40: pp. 254-258.

Ideas of Algebra, K – 12, 1988 Yearbook of the National Council of Teachers of Mathematics. Reston, VA: National Council of Teachers of Mathematics, 1988.

Illuminations

http://illuminations.nctm.org

The interactive math tools and i-math investigations contain links to virtual manipulatives that are aligned with the Standards of the National Council of Teachers of Mathematics.

National Library of Virtual Manipulatives

http://matti.usu.edu/nlvm/nav/

This site contains a collection of applets organized by grade band and standard. The applets in the 6-8 and 9-12 grade bands of the Algebra Standard will be most helpful for this workshop.

ESCOT PoW Applets

http://mathforum.com/escotpow/puzzles/

This collection of applets can be used to solve the Math Forum’s problems of the week.

Shodor Foundation

www.shodor.org

Shodor is a nonprofit organization that creates applets for a variety of concepts and offers them free to the public. Though not always as attractive as other applets, the Shodor applets are educationally sound and highly effective.