Pre-Algebra Out Loud: Learning Mathematics Through Reading and Writing Activities (2013)

Chapter 6. Equations and Inequalities

WHAT? Introduction

In 2000, the National Council of Teachers of Mathematics recommended that the mathematics curriculum include explorations of algebraic concepts and processes so that students can develop confidence in solving linear equations and investigating inequalities. Today one of the ten goals of the sixth-grade Common Core State Standards requires students to reason about and solve one-variable equations and inequalities. Pre-algebra students learn the basics of algebra early in the course and practice higher-order arithmetic skills so they can learn to solve multistep equations and related applications.

This chapter contains two mini-lessons: the first on solving multistep, one-variable equations and the second on solving one-variable inequalities. The reading and writing strategies in this chapter involve multistep equations, inequalities, and applications using linear equations.

WHY? Objectives

Using the activities in this chapter, pre-algebra students:

· Learn to paraphrase math content

· Examine, write out, and explain each step in the process of an algebraic algorithm or method of operation

· Translate algebra into words

· Create a written word problem that involves a linear equation in one variable

· Collect words and concepts to build a math glossary

· Write a biography on a notable mathematician

Mini-Lesson 6.1 Solving Multistep Equations

CCSS Standard 7.EE: Expressions and Equations

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

An equation is a mathematical sentence in which the left side is equal to the right side. Think of an equation as a balance scale, where the quantities on the left and right sides of the equal sign balance or are equal in value. Moreover, if we perform an operation on one side, we must perform the same operation on the other side so that the equation stays in balance.

Example 1

Example 2

The great thing about solving equations is that we can check our solutions. If placing 4 into the original equation gives a true statement, then it is the correct solution.

Check

, which gives

, which gives

, which gives a true statement.

Linear equations generally have one solution. Linear equations in one variable are equations with a single variable, like x. The variable may have coefficients and the equation may contain constants.

Some equations have no solutions, and some have an infinite number—for example:

Therefore, there is no value for x that will make this equation true, and there is no solution.

Here's another example:

This equation is an identity because the left side is identical to the right side when simplified. Although both sides of the original identity do not look the same, any number substituted for the variable will make the equation true and eventually give an identical equation, like above. The solution set for an identity is infinite and consists of all the real numbers.

When solving any algebraic equation, use a series of equivalent equations—equations that have the same solutions, often referred to as solution sets.

Teaching Tip

After students learn to solve equations, they may make this mistake: when asked to simplify the expression , they add an equal sign and 0 to it. Then they solve it. Algebraic expressions get simplified and equations get solved.

Simplified:

Activity 6.1: In Your Own Words: A Paraphrasing Activity

WHAT? Description

One of the most common excuses that students give for not reading material in their textbooks is that they do not understand the language. This activity helps students target and interpret key concepts. By rewriting portions of mathematical text, students demystify and make personal meaning of mathematical content.

Assign students small portions of the pre-algebra text to read and rewrite in their own words. This activity works equally well with concept definitions, theorems, and examples. Having students read their own versions to each other allows student writers to consider different interpretations and pinpoint misconceptions. If the writing is handed in, you can assess your students' understanding of the material.

WHY? Objectives

During this activity, pre-algebra students:

· Read the content from assigned readings

· Paraphrase the content in their own words

· Share their ideas with peers from the completed paraphrase activity

· Discuss and critique each other's translations

HOW? Examples

Example 1

Textbook: The solution set for an equation that is unsolvable is the empty set.

Student paraphrase: Some equations have no solutions so their solution set is empty and it looks like {}.

Example 2

Textbook: Linear equations always have x raised to the first power. These equations have one, none, or an infinite number of solutions and are said to be consistent, inconsistent, or dependent (respectively).

Student paraphrase: Linear equations never have x squared or cubed; x is always by itself. A consistent equation has one solution. An inconsistent equation has no solutions. And the dependent equation has an infinite number of solutions.

Worksheet 6.2: In Your Own Words: A Paraphrasing Activity

Name _____ Date _____

Directions: Review mini-lesson 6.1 on solving multistep equations. Then use the prompts below to help you paraphrase the material. Be clear, and use complete sentences. Be prepared to share your translation with your peers. You may use other sources to help develop your answers.

My definition of an algebraic equation is:

My understanding of solving multistep equations is:

Show how to solve the following equation: :

Activity 6.2: Method of Operation

WHAT? Description

Most mathematical content consists of processes. Students need to know the “how-tos,” or methods for arriving at or simplifying solutions. Asking students to write out methods of operation (MO) reinforces their understanding of how to solve problems and helps them consider the fine points of exceptions to the rule.

You might ask students to write out the MO for finding the solutions for a linear inequality. We want students to understand the algorithm for solving inequalities, as well as why the algorithm works. Students should be encouraged to write out the MO as if they were talking to another student who is just learning this process. This motivates clear and concise work and helps each student consider useful algebraic vocabulary.

WHY? Objectives

During this activity, pre-algebra students:

· Complete an MO for an algebraic process

· Use a completed MO to conduct an algebraic process

· Share their MOs with their peers (optional)

HOW? Example

Ask students to write out the MO for finding the solution for a multistep linear equation in one variable: ax + b = c.

1. Add or subtract any constants from both sides of the equation.

2. If there are terms on both sides of the equation that contain the variable, add or subtract to get only one term containing the variable.

3. If the variable has a coefficient, divide both sides of the equation by that number.

4. You should have the variable equal a number that is the solution.

Worksheet 6.2: Method of Operation

1. Write out the method of operation for the following inequality:

, where A is a positive number

2. Give the method of operation for finding the solution of any inequalities of the forms or in interval notation.

3. Sandy is at least 21 years old. Her brother Sam is 2 years younger than she is. Using x as the variable that represents Sam's age, write and solve the inequality that represents Sam's age. Then write the method of operation for solving this inequality and give the solution in interval notation.

Mini-Lesson 6.2 Solving Linear Inequalities

CCSS Standard 6.EE: Expressions and Equations

Reason about and solve one-variable equations and inequalities.

Solving a linear inequality is very similar to solving a linear equation; the difference is that the solution for an inequality includes a range of numbers. To get started, consider the definition of a linear inequality in one variable. A linear inequality in one variable can be expressed in the form: ax + b < 0, where a and b are real numbers but a .

The inequality sign will be one of the following: <, >, ≤, or ≥.

A number line can be used to visualize solutions to inequalities in one variable—for example, :

The inequality can also be expressed in what is called interval notation, a form of writing a solution that contains an interval of infinite numbers that are solutions to the inequality. Parentheses or brackets are used to enclose the solutions between two numbers or a number and positive or negative infinity. The symbol for positive infinity is and for negative infinity (). Infinity in mathematics is an abstract symbol for “goes on forever.” Positive infinity means the solutions or numbers get large, in fact, very large. Negative infinity is an abstraction for a very, very small number, a negative number.

The parentheses in interval notation tell us that the numbers given inside are not part of the solution, and brackets [], as shown in the examples below, show the numbers are included in the solution set. A parenthesis is always placed next to the positive or negative infinity symbol −∞−:

x ≤ −2 is expressed as (−∞, −2]; −2 is included in the solution

x < − 2 is expressed as (− ∞, −2); −2 is not part of the solution

x ≥ −2 is expressed as (− 2, +∞); −2 is included in the solution

x > − 2 is expressed as (−2, +∞); −2 is not part of the solution

The same opposite operations used for solving equations are used to solve inequalities. However, there is one more rule: when multiplying or dividing by a negative number during opposite operations, change the direction of the inequality sign—for example:

Dividing by −2 gives .

In interval notation, this is .

Checking a value, −4, in this range gives , which is clearly greater than 6, as the original inequality states.

Teaching Tip

Students often ask why the direction of the inequality sign switches for multiplication and division of negative coefficients (multipliers).

You can explain it in this way: “Consider 3 < 4. Now multiply each side by the positive number 2, which gives 6 < 8, a true statement. However, multiplying by −2 gives , which is not true unless we switch the signs.”

Let's do a couple of examples.

Solve the inequalities for x giving the solution as an inequality and in interval notation.

Example 1

Example 2

(remember to change the direction of the inequality sign)

A compound inequality is an inequality that has two inequality signs, usually less than (<) signs (for example, ), and is expressed in interval notation as . is read “x is greater than negative 2 but less than positive 4,” or “ x is a number between negative 2 and 4.”

Here is an example for solving a compound inequality:

(subtract 2 from all three sides)

or, in interval notation,

Activity 6.3: Translating Words into Algebra

WHAT? Description

This lesson asks students to translate English phrases or sentences into algebraic expressions or equations. This activity is best used as a precursor to activity 2.4, in which students create their own word problems. To be successful at this activity, pre-algebra students must work with their knowledge of basic ideas and concepts of algebra or use their textbooks or concept circles to find definitions and descriptions of math and algebra concepts.

WHY? Objectives

During this activity, pre-algebra students:

· Practice translating algebraic symbols into words

· Learn more about how linear applications are constructed and solved

· Practice deciphering mathematical word problems

· Practice using and translating algebra concepts into real-world problem solving

· Write out the definitions of the terms expression, equation, and inequality and identify the differences among them

HOW? Examples

Here are three examples of translations:

· Five times a number plus + 6 is an expression.

· One-half of 10 times a number and 3 is twice that is an equation.

· A number is at most 13 times . The (x) is an inequality.

Worksheet 6.3: Translating Words Into Algebra

Name _____ Date _____

Directions: Translate each of the following phrases or sentences into algebraic terms. Label each problem as an expression, equation, or inequality by circling the correct term. There is no need to solve any of the equations or inequalities. Use as many numbers and symbols as possible in your translation. If you feel there is more than one way to translate your phrase, choose one way and write it out in numbers and symbols.

1. The product of 13 and a number, all divided by 2 = _____

Circle one: Expression Equation Inequality

2. The sum of five times a number and 2 = _____

Circle one: Expression Equation Inequality

3. The opposite of a negative 3 minus 2 times a number = _____

Circle one: Expression Equation Inequality

4. Seven times the quantity of 2 and a number is at least 4 = _____

Circle one: Expression Equation Inequality

5. A number is at most the quotient of 40 and negative 4 = _____

Circle one: Expression Equation Inequality

6. Write an example like the ones above of your own and translate it into algebra. Label it an expression, equation, or inequality.

Activity 6.4: Writing Word Problems

WHAT? Description

Often students find solving word problems difficult and confusing. This activity asks students to create their own applications, which allows them to look at solving written problems from a different angle. Students do best with this activity when they can look at examples of a “good” problem and a “bad” one. It should be clear how many variables are needed and whether the problem should be linear or quadratic, for example.

Students work in pairs or small groups of three or four to create the problem. Problems can be written on the board, on overhead transparencies, or typed on a word processor that can be projected onto a screen. Each group may then exchange problems and work out solutions on the board. While doing this activity, students learn:

· They must work backward to get the numbers to work out.

· They must choose the words they use carefully so others can understand the question.

· They must understand why the problem works and not just how.

WHY? Objectives

During this activity, pre-algebra students:

· Work together and research word problems

· Create a word problem that can be solved

· Exchange problems with their classmates and try to solve them

· See how problems are constructed

HOW? Example

The following is a student-created problem and a good example of what types of words should or should not be used for your assignment:

A swimming pool in the shape of a rectangle is 35 feet by 15 feet. A deck around the pool is the same width all around. The swimming pool and the deck form a large rectangle with an area of 800 square feet. Find the width of the deck

Worksheet 6.4: Creating a Word Problem

Name _____ Date _____

Directions: One way to create a word problem is to start with an equation and construct a situation or story using the equation. It will be simpler to write a story or word problem about events or things that are important or interesting to you.

For example, think of a situation for which might be appropriate. First, decide what you want the variable x to stand for; for example, of dollars earned. Then you notice that this equation has only one operation in it, addition. Now you can add something from your own experience—for example:

x represents Ashley's weekly allowance that she gets for vacuuming the house. She recently got a raise of $5 and now receives a weekly allowance of $23. Find her original allowance.

Use the following equations to create word problems. You will share these with your peers.

Worksheet 6.5: Creating a Word Problem

One way to create a word problem is to ask a question that requires a numerical solution. Then fill in the facts needed to answer that question. Here is an example:

Question: What are the width and length of a rectangle with a perimeter of 146 feet?

Word problem: A rectangular-shaped room has a length of 16 feet and a perimeter of 146 feet. Find the width of the room.

Solution: Using the perimeter formula for a rectangle, P = 2l + 2w, gives:

Assignment: Create word problems for the following questions.

1. How tall is the tallest boy in our class?

2. How many magic tricks can Adrian perform in 2 hours?

Activity 6.5: Math Glossary

WHAT? Description

Asking students to collect important words or concepts from a chapter or section of text is one of the first steps in building mathematical literacy. Whether by handwriting in their own math notebooks or using computer word processing or spreadsheet programs, students can input the terms from each chapter. By the end of this course, they will have created a pre-algebra dictionary or glossary that they can use during quizzes or in future courses.

It is important to give students a format for the display of the term. For example, a student might be asked to give the term's part of speech (noun, verb, adjective, or something else). Then the student might be asked to give a definition from the reading or class notes and a definition or description of his or her own.

WHY? Objectives

During this activity, pre-algebra students:

· Create a math glossary giving the term, its part of speech, and definition

· Read an assigned chapter highlighting important concepts

· List the significant terms, and consider each term's part of speech

· Write out the word's definition

· Brainstorm, create, and write a personal definition or description of each term

· Write a phrase or statement using the word correctly

HOW? Example

This example gives one term regarding equations and inequalities:

Term: solution

Part of speech: noun

Text definition: answer to a problem or question

My definition: the number you get when solving an equation

Sentence: The solution to the equation is 3.

Worksheet 6.6: Math Glossary

Name _____ Date _____

Directions: Refer to our lesson on algebraic equations to create part of a math glossary. Choose at least five concepts or words and fill in the matrix. One is done for you.

Activity 6.6: Biographies of Algebraists

WHAT? Description

A good way to understand algebra is to explore how the field of algebra came to be. Several mathematicians contributed to the development of the field of mathematics that came to be algebra.

Researching a mathematician's life and contributions helps students appreciate the step-by-step development of the various algebraic theorems and processes. Several of the mathematicians and their contributions to the field of algebra are listed below. Assign students a particular mathematician or have them choose one of their liking. Ask students to share their biographies with the class to broaden the learning experience.

WHY? Objectives

During this activity, pre-algebra students:

· Research a mathematician of their choosing and take notes

· Construct a write-up of a mathematician's life story

· Explore and prepare a brief presentation on their mathematician's math inventions

HOW? Example

The assignment of a biography of a mathematician should include a format for the write-up. For example, each biography might consist of the following parts:

· Early life

· Education and career

· Family or social life

· Algebraic contributions

· Later life

The rubric on the next page may be used to grade student work.

Grading Rubric

Content score (10 Possible) = _____

Contains accurate information on mathematician? 1 2 3 4 5

Correctly explains the mathematics introduced? 1 2 3 4 5

COMMENTS:

Mechanics score (10 Possible) = _____

Grammar, spelling, clarity, transitions, introduction, conclusion?

1 2 3 4 5

Contains required parts and follows guidelines? 1 2 3 4 5

COMMENTS:

Resources score (5 Possible) = _____

Format? Amount of sources, library source? 1 2 3 4 5

COMMENTS:

Final grade = _____

Worksheet 6.7: Math Biographies

Name _____ Date _____

Directions: Choose one of the following mathematicians and write a biography to share in class.

Guidelines

1. Each biography must contain the following parts:

· Early life

· Education and career

· Family or social life

· Mathematical contributions

· Later life

· Resource list

2. Use at least three different sources, with at least one from the school's library.

3. Your biography should be a minimum of four typed double-spaced pages and a maximum of six typed double-spaced pages in length.

4. The last page should be a bibliography in a consistent format.

5. Identify in the biography your mathematician's influence on the field of algebra.