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

Chapter 1. The Basic Tools of Algebra

WHAT? Introduction

When middle school students begin a pre-algebra course, they are introduced to a magnificent new world of mathematical symbols and concepts. By this time, they have learned the basic operations of arithmetic: addition, subtraction, multiplication, and division. They still need to continue practicing these skills along with learning higher-order skills. After becoming familiar with the newness of algebra, students are able to incorporate the new with the old to solve more complex problems. This chapter has two mini-lessons: one focusing on the basic symbols and concepts of algebra and the other on solving a simple equation. The first lesson, which introduces pre-algebra students to the basics and general concepts of algebra, lays the foundation for the activities for reading and writing to learn pre-algebra. It may be used as content for any of the activities that follow.

WHY? Objectives

By doing the activities in this chapter, pre-algebra students will:

· Find the major mathematical topics in the lessons that follow and in any pre-algebra text and write them out for use in future activities

· Create a math glossary demonstrating their understanding of the concepts of algebra

· Use semantic word maps to show the relationship between certain algebraic concepts

· Construct and use a concept circle to focus on a large algebraic idea and its components, rules, and examples

· Use algebraic terms creatively, allowing them to learn, understand, and apply these terms in a short story

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

Mini-Lesson 1.1 The Big Ideas of Algebra

CCSS Standard 6.EE: Expressions and Equations

Apply and extend previous understanding of arithmetic expressions.

A toolbox full of all the basic tools of algebra certainly contains a definition of algebra: a generalization of arithmetic in which symbols or letters called variables represent numbers and to which many of the same arithmetic properties and operations apply. This definition encompasses some of the major algebraic concepts and tools. Adding the following definitions to our toolbox yields plenty of tools to help learn about algebra:

· A variable is a symbol that stands for a number or a range of numbers.

· The letters X or Y can represent any number, so they are variables.

· A constant is a fixed number. For example, in 2x + 8, the 8 is the constant.

· A coefficient is the multiplier next to a variable. For example, in 5x or 10a²b, the 5 and 10 are coefficients.

· The arithmetic operations are addition, subtraction, multiplication, and division: +, −, ×, ÷.

· A term is a variable or variables with a coefficient or a constant. For example, in 2, 4x, 6y and −8 are terms.

· An algebraic expression is a mathematical phrase that can contain numbers, operators (add, subtract, multiply, divide), and at least one variable (like x or y.) For example, 4x + 6y - 8 and 24ab - 3a²b are algebraic expressions

· Like terms are terms that have the same variable (raised to the same power) but may have different coefficients. For example, 2x, 5x, and 6x are like terms.

Teaching Tip

When you introduce an exponential expression, such as x², show students the comparison of x with x². For example, if x=3, then x² = 9. This reinforces the idea that x and x² are two different variables and therefore not like terms.

The following example describes many of these concepts:

−2x and 4 are terms. -2x + 4 is an algebraic expression.

These are useful tools for performing arithmetic operations.

Combining Like Terms

Completing an arithmetic operation on an algebraic expression requires collecting and combining like terms. Like terms can be thought of as similar things, like apples. For example, we can add the two like terms 4a and 5a by thinking of adding 5 apples to 4 apples to get 9 apples:

Note that 20ab and 36ab are like terms, but 15a and 23ab are not like terms. The variables a and ab are not the same variables. Similarly, 5x² and 7x² are like terms, but 5x² and 7x³ are not like terms and cannot be combined.

Here are some other examples of combining like terms:

Example 1: . The 5x and 2x can be combined and the 3y and 6y can be combined because they are like terms, but since 3x and 9y are not like terms, the operation is completed when you get 3x + 9y, which is in simplified form.

Example 2: . Here we moved, or commuted, the like terms 14a and 12a so they sat next to each other and then combined them by adding the coefficients. This is because the operation addition has a property called the commutative property. We discuss this property in detail, along with other number properties, in mini-lesson 2.2 in Chapter Two.

Activity 1.1: The Writing Is on the Wall

WHAT? Description

Elementary teachers often use a word wall or bulletin board to help students learn the spelling and vocabulary of math terms. Your wall will be more useful to students if you group terms—for example, the Top Ten, Fabulous Five, or Foremost Four mathematics concepts—from each lesson or chapter. Students should be asked to give the key concepts and explain their choice of concepts.

Students collect significant pre-algebra or algebra concepts and terms that they then use to construct their math glossaries (described in activity 1.3) and add to their mathematical vocabulary.

WHY? Objectives

During this activity, pre-algebra students:

· Choose key concepts from a section in their textbook or that day's lesson

· Collect words to help construct a math glossary over the entire course

HOW? Example

The first two examples that follow come from two different lessons in a pre-algebra textbook. The first list gives what a student might consider the five most important terms from a lesson on linear equations—the “fabulous five.” The second list gives the “top ten” concepts from a lesson on inequalities. The third example consists of what an algebra student might give as the seven most “sensational symbols” from the entire study of algebra.

Fabulous Five Terms from a Lesson on Linear Equations

variable

coefficient

constant

like terms

solution

Top Ten Concepts from a Lesson on Inequalities

Seven Sensational Symbols from Algebra

Worksheet 1.1: The Writing Is on the Wall

Name _____ Date _____

Directions: Choose five concepts that are key to algebra. Look for terms that are related in the same way to these concepts. You may use today's lesson or a chapter from your pre-algebra textbook. Then try finding six symbols that are commonly used in algebra and list them under the Sensational Six Symbols. Be creative!

Fabulous Five Terms from a Lesson on Algebraic Expressions

Six Sensational Symbols from the Basics of Algebra

Activity 1.2: Semantic Word Maps

WHAT? Description

Semantic word maps depict and display the relationships between key concepts and terms. These word maps often resemble flowcharts or webs connecting mathematical terms. Arrows connect related concepts and often display a hierarchy of the terms, where one term is a set and the others are subsets of that set. For example, if algebra is the main set, then equations, variables, and constants are all parts of the set or subsets of the set “algebra.”

After students have completed their semantic word maps, you might pose critical-thinking questions—questions that go beyond the lesson—that encourage students to think of the relationships between the subsets—for example: If equations and variables are both subsets of algebra, is a variable a subset of an equation?

WHY? Objectives

During this activity, pre-algebra students:

· Explore relationships between mathematical concepts and terms

· See the hierarchy of key concepts

· Create study guides displaying key concepts

HOW? Example

Consider the related algebraic concepts in the map shown on . Place the terms in the word map showing their relationship to each other. Add arrows and circles as needed. Hint: Look for the larger set (equations) and decide which of the concepts are parts of this set. There may be more than one set and some concepts (subsets) may be parts of more than one set. Peruse the terms, looking for one word or concept that is made from the other words or parts. Clearly, an equation is formed from the other terms, like variables or constants, expression, term, variable, coefficient, constant, and the equal sign.

Worksheet 1.2: Semantic Word Maps

Consider the following terms related to algebra:

You may want to group some of these terms together before you start.

Place one term in each figure in the diagram, and draw arrows showing how these terms might be related to each other. There is more than one way to fill in the shapes. Be prepared to share and perhaps defend your choices that make up your semantic map.

Activity 1.3: Math Glossary

WHAT? Description

Asking students to collect the 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 math notebooks or using computer word processing or spreadsheet programs, students can input the terms from each chapter so that by the end of this course, they will have created a dictionary or glossary they can use during quizzes or in future courses.

It is important to give students a format for displaying the terms. For example, a student might be asked to give each term's part of speech: a noun, verb, adjective, and so on. Then the students might be asked to give a definition from the reading or class notes and a definition or description of their own. See the suggested format in worksheet 1.3.

WHY? Objectives

In this activity, pre-algebra students:

· Create a math glossary giving the term, part of speech, and definition of a concept they are studying and by following these steps (objectives):

· Read the lesson on algebraic concepts, and take notes on the lesson, highlighting important words or concepts.

· List the significant terms, and give the part of speech for each term.

· Write out the definition from the textbook.

· Write a personal definition or description of each term.

· Write a phrase or statement using the word correctly.

HOW? Example

These are a few terms that could be used in the glossary:

Worksheet 1.3: Math Glossary

Name _____ Date _____

Directions: Reread the first lesson on algebra before you begin a math glossary. Choose at least five concepts or words, and fill in the matrix below. The first one is filled in for you.

Activity 1.4: Concept Circles

WHAT? Description

The creation or use of concept circles is one of the more versatile reading and post-reading activities (Vacca & Vacca, 1999). A concept circle usually focuses on a single concept and its important features. Features, or descriptors of the concept, are placed within sectors of the circle. The circle is generally divided into quarters; however, more sectors may be used if needed. Though the concept circle is traditionally a circle, it could also be diagrammed in a square or any two-dimensional figure.

Concept circles may be teacher created and used to quiz students:

· Given the descriptors contained in the sections of the circle, students identify the concept.

· Given the concept and descriptors, students select which of the descriptors is incorrect.

· Given the concept and a few descriptors, students fill in the rest of the circle.

· Concept circles may also be student created:

· The teacher gives the concept, and students fill in the sections of the circle.

· The student selects the concept from the reading and fills in the sections of the circle.

WHY? Objectives

During this activity, pre-algebra students:

· Categorize information from the reading

· Review features and descriptors of a certain concept

· Self-assess their reading comprehension

HOW? Example

Example of a Concept Circle for an Algebraic Expression

In this example, the circle is divided into four sectors. However, you may divide the concept circle into as many sectors needed. Each sector gives information that helps form a word picture of what an algebraic expression is, and the sectors may include any of the following: definition, description, examples, nonexamples, uses, facts, or features. A few of these categories are explained below.

· Nonexamples are examples that are related to the concept but are not examples of it. The nonexample for an algebraic expression in the concept circle here is the equation , which is actually two algebraic expressions divided by the equals sign, giving an equation.

· Uses may include where the concept is used in mathematics, other courses, or in the physical world.

· Facts might be rules or theorems that make up or apply to the concept.

· Features are parts or attributes of the given concept. Other descriptors or components of the concept may be used. Examples include numerical expressions, pictures, symbols, or words that depict the concept.

Worksheet 1.4: Concept Circle Activity: Algebra

Name _____ Date _____

Directions: Divide the circle into four sectors using definition, example, nonexample, and uses for each sector. Fill in each sector with the correct explanation according to its category. We will share our word pictures of algebra when we are all finished.

Worksheet 1.5: Concept Circle Activity: Combining Like Terms

Name _____ Date _____

Directions: Decide on how many categories/sectors to divide the circle into and give their titles. Then fill in each sector using the correct explanation according to its category. We will share our word pictures of combining like terms when we are all finished.

Mini-Lesson 1.2 Solving Simple Equations

CCSS Standard 6.EE: Expressions and Equations

Reason about and solve one-variable equations and inequalities.

This mini-lesson introduces problem-solving techniques to pre-algebra students. Algebra is all about problem solving, and a vital tool for mathematical problem solving is the equation. An equation consists of an algebraic expression set equal to another algebraic expression, one of which may be a constant.

An equation is a statement asserting the equality of two algebraic expressions that are separated into a left side and a right side joined by an equals sign. Some examples are:

This mini-lesson focuses on simple equations that contain only one variable. The goal is to solve each equation for the variable. Remember that variables stand for numbers or a range of numbers. For example, the equation gives the solution . By replacing x with the number 4, that is, checking work, we see that . Therefore is the correct solution. can be written as 2(4); the parentheses here mean multiplication.

Many equations can be solved visually or by mental trial and error. In fact, these methods are good ways to strengthen and sharpen problem-solving skills. However, there are many equations whose solutions are much more difficult to see.

There is a method that works for solving all simple equations (equations with one variable and that variable is not raised to a power other than 1): using opposite operations. Opposite operation is used to undo every operation that is being applied to the variable. The goal is to get x by itself on one side (often the left side) of the equality sign and a constant on the right. There is one major rule to remember, and I call it the Golden Rule for Solving Equations: do unto one side of the equation as you would do to the other side.

Doing the same thing to both sides of the equation yields an equivalent equation, an equation with the same solution. For example,

Always check your answer!

Teaching Tip

When demonstrating how to solve an equation with more than one or two steps, show all of the work. Students get frustrated when they do not understand how to get from one equation to the next equivalent equation. Only after much practice will they understand when two opposite operations can be completed in one step when solving an equation.

Activity 1.5: Math Story

WHAT? Description

For this activity, give students a list of terms or concepts specific to algebra and ask them to use all of the terms correctly to write a short story. This activity may be used during or after the lesson containing the specified terms. Asking students to use this language requires that they learn the meaning of these terms.

Encourage the student writers to be creative, but also to pay close attention to the meanings of the terms they use. The stories may be fiction or nonfiction. They may be witty, silly, sad, or dramatic. However, give some guidelines for them to follow when creating their stories (see the assignment criteria on worksheet 1.6). Being explicit about what constitutes an A paper is equally important to the student writer and the teacher reader/grader.

WHY? Objectives

During this activity, pre-algebra students will:

· Write a short story using the given algebraic terms

· Use a dictionary or algebra text to find definitions of unknown words

· Practice using and writing about math terms correctly

HOW? Example

For this example, the terms are solve, equal, terms, equation, and problem.

This is a silly short story about problem solving:

Ex had a problem he needed to solve involving dividing a solution into two equal parts. So he approached Wy for help. Together they composed an equation, setting the term 2x equal to 10 cups. Ex said, “We must state that x = amount of cups of the solution when divided into 2 parts.” Solving the equation seemed very simple: find the amount of one of the parts. Ex and Wy brainstormed and found 5 was the solution. Ex and Wy decided they work well together.

The moral to the story is this: if Ex doesn't give you the answer, try Wy!

Worksheet 1.6: Math Story Activity

Name _____ Date _____

Directions: Use terms from the following list to write a short story:

· Variable

· Constant

· Coefficient

· Algebraic expression

· Term

· Like terms

· Combining like terms

· Equation

· Solution

· Problem solving

· Opposite operation

· Equivalent

Assignment Criteria

1. Use at least 9 of the 12 listed terms in your story.

2. Use each term correctly. Use the text or dictionary to check on definitions and correct use of terms. You may be creative with each term; but in at least one place, use the term correctly or in a manner that clearly demonstrates you understand what the term means.

3. Your story may be fiction or nonfiction.

4. Your story should contain an introduction and a conclusion and follow a logical story line.

5. Your story should be at least one page but no more than two pages. You may begin your story on the line below.

Activity 1.6: Algorithm Writing

WHAT? Description

An algorithm is a step-by-step, computational problem-solving procedure with a finite number of steps. During the algorithm writing activity, students are asked to take a computational procedure or equation and write out the problem-solving process in words. This activity helps demystify algebraic problem solving by allowing students to explain each step of the process integrating their own language with the appropriate algebraic terms and concepts. The math glossary is useful for this activity.

This activity is also an excellent way to address students' use or misuse of math terms. Although algorithms are the “how to do it” part of problem solving, this activity helps students to see why each step works. Asking a student to explain a process to another student who does not know how to do it or is struggling to understand should encourage the writer/explainer to explain in more detail or to consider questions students might have.

WHY? Objectives

During this activity, pre-algebra students will:

· Focus on the steps used to solve an algebraic equation

· Practice writing and communicating mathematics to a certain audience

· Learn how to use and write mathematical terms correctly

HOW? Example

Here's an example of a pre-algebra student's algorithm writing and how it could be improved.

Solve the following algebraic equation and write out each step as you complete it using both the appropriate algebraic terms and your own words. Imagine that you are explaining the problem-solving process to a student who has not yet mastered the process.

Although this student offered adequate explanations, I would encourage her to write “I distributed” instead of “I first multiplied,” and “I simplified” rather than “I did the adding.” The asterisk indicates that the teacher asked the student to explain what she meant by “because of opposite operations.” However, that phrase shows that the student writer understood why she must add 12 to both sides of the equation.

Worksheet 1.7: Algorithm Writing

Name _____ Date _____

Directions: Solve the following algebraic equation, and write out each step as you complete it using both the appropriate algebraic terms and your own words. Imagine that you are explaining the problem-solving process to a student who has not yet mastered the process.

Write each step in the left column. Then for each step, explain in detail what you did and why you did it in the corresponding cell in the right column. Be careful not to leave any steps out!

Problem:

Activity 1.7: One-Minute Summary

WHAT? Description

The one-minute summary or one-minute essay is a familiar assignment to many teachers in other disciplines. This brief writing activity yields a wealth of information to both writer and reader.

Ask students to free-write (write without worrying about grammar, punctuation, or spelling errors) on a particular topic for one minute. These summaries may be used at the beginning, middle, or end of the course or unit or more than once during a lesson or over the semester, providing students with a quick assessment of their grasp of a topic. One-minute summaries can also be handed in to assess how well each student knows the topic and see how much they have learned. They need not be graded.

Begin this activity by saying to the students: “Take out a sheet of paper. We are going to free-write for exactly one minute on one topic, which I will give you. Free-writing means to write down everything that comes to mind about the topic, whether it be definition, shapes, examples, words associated with our word, or any other related terms that come to mind, without worrying about spelling or grammar rules. Just write everything you know or remember about the topic without stopping as you write. In exactly one minute, I will say, ‘Stop,’ and you must stop writing. When you are finished, we will share our writings with each other. Then you will hand these in, and at the end of the unit we will repeat this activity. I will hand back your original one-minute summary and you will see how much you have learned.”

One-minute summary works better if you choose terms with a narrower scope. For example, pre-algebra is too broad; however, integers would be a very good concept for a one-minute summary.

WHY? Objectives

During the one-minute summary, pre-algebra students:

· Free-associate or free-write about an algebraic topic in a limited amount of time

· Self-assess their comprehension of a lesson or a concept

· Demonstrate their comprehension of concepts to their teachers

HOW? Example

Here are some sample algebra topics:

Example of a one-minute summary: “A linear equation has a variable like x, and it has an equal sign and some other numbers called constants and coefficients. It has only one answer all the time unless it has two variables x and y. Then it is a line with all the points on the line as solutions for x and y.”