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

Chapter 2. Exploring Infinite Sets

WHAT? Introduction

Usually mathematics courses begin with an exploration of the set of numbers that will be addressed or used in that course. The first lesson in this chapter focuses on defining and describing all of the large sets of numbers commonly used in algebra, followed by reading and writing activities that enhance students' understanding of these sets. Mini-lesson 2.2 discusses the properties of the set of real numbers, the largest set of numbers explored in pre-algebra curricula.

WHY? Objectives

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

· Complete a matching exercise in a magic square format to illustrate the comparison and contrast of certain infinite sets

· Write one-minute summaries for self-assessment

· Work with peers to create graphic organizers to demonstrate the hierarchy or set or subset relationship of infinite sets

· Read, write, and use mathematical symbols in place of words (activity 2.4)

· Create a mathematics glossary using the major terms related to infinite sets of numbers

· Complete a semantic feature analysis (activity 2.7) showing different ways of expressing sets

· Paraphrase text and share their writings with other students

Mini-Lesson 2.1 Infinite Sets

CCSS Standard 6.NS: Number System

Apply and extend previous understandings of numbers to the system of rational numbers.

A set is a collection of elements or objects. An infinite set is a nonterminating list of numbers; it does not end, and it follows a prescribed number pattern. Brackets are used to denote sets.

We will consider the objects to be numbers in this book. For example, in the set of natural numbers |N…“N” can be expressed as a roster or list contained inside the braces:

The ellipses “…” mean the list of numbers continues in that pattern forever.

The set of whole numbers is the set of natural numbers along with the number 0:

The numbers in ||N and ||W are infinite, meaning they go on forever. In fact, for any large natural number, we can find another number by adding 1 to that number.

Another important infinite set is the set of integers (|I):

Note that the integers contain no ending or beginning number. In fact we denote the smallest negative integer with an abstract expression:

and the greatest integer as

Some infinite sets are difficult to express as lists. For example, the set of rational numbers (which is actually the set of fractions or any number that can be written as a fraction) is infinite but cannot be expressed in a roster (which is just a list of numbers). What would the first number after 0 be in this: ? .1?, .01? .001? In fact any of these numbers could be considered in the middle of the set! Try to imagine the smallest fraction or the largest fraction.

Yet the set of fractions can be expressed in what is called set-builder notation. The set of rational numbers is abbreviated as |Q and is defined as this set:

This is read, “The set of all x such that x = a divided by b, where a and b are integers but b cannot be equal to 0.”

We cannot divide any number or expression by 0.

The largest infinite set that we will work with in the pre-algebra course is the set of real numbers, often referred to as the reals, expressed like this: . The real numbers are the numbers that we see and use in the real world, such as counting numbers, integers, fractions, and decimals.

One way to understand and visualize the reals is to think of a horizontal number line that goes on infinitely in the positive direction and infinitely in the negative direction. All the points on the number line have a number value that gives the distance from 0; the sign (positive or negative) determines the direction the value is from 0. Numbers with no sign are considered positive. All values that lie on this number line, including fractions that are not labeled on the number line below, are real numbers:

Real Number Line

Below are some examples of real numbers. Many of them are familiar. Students will see a few of them as they move on in their study of algebra.

Teaching Tip

It is often difficult for students to grasp what the infinity sign () means. It is thought to be a number, but it is not. It is an abstract concept that means the numbers are getting very large () or very small () and are endless.

Activity 2.1: Magic Square

WHAT? Description

The magic square activity combines a matching activity with mathematics and a puzzle of a magic square (Vacca & Vacca, 1999). The format of the matching activity consists of two columns, one for concepts and one for definitions, facts, examples, or descriptions. In solving the matching activity, the student places the numbers in the corresponding lettered squares inside the magic square.

To check their answers, students add the numbers in each row, in each column, and in each diagonal. These sums should be equal. This sum is referred to as the square's “magic number.” Sometimes only the numbers in the rows will sum up to the magic number and one or more diagonal contain values that do not add up to the magic number.

WHY? Objectives

During this activity, pre-algebra students:

· Learn the concept of infinite sets

· See the mix of interest and mystery of the mathematics of infinite sets

· Learn about the concept and properties of magic squares

HOW? Example

The magic squares below are 3-by-3 unit squares. Notice that the cells in these particular magic squares contain all whole numbers from 1 to 9 with no repeats. When you add the numbers in each column, in each row, and in each diagonal, your sum is 15. This number is called the magic number.

Not all magic squares have consecutive numbers in its cells. However, all the numbers in each row and in each column must add up to the same (magic) number. There are some magic squares in which the diagonal numbers do not sum up to the same number. Clearly, magic squares must have the same number of rows or columns; thus, they are squares.

The activities in this chapter show some ways to integrate the use of magic squares with algebra. The first three magic squares below contain the numbers 1 through 9, but they in different cells. These 3-by-3 squares are very special and used often to build other different magic squares.

Worksheet 2.1: Building a Magic Square Activity

Name _____ Date _____

Directions: This magic square has the magic number of 39. Note: The diagonal sums do not necessarily yield the same sum as rows and columns in this example.

Use the same numbers shown in the above magic square to construct a different magic square. Does it have the same magic number?

Worksheet 2.2: Magic Square Activity for Infinite Sets

Name _____ Date _____

Directions: Select the best answer for each of the terms on the left from the numbered descriptors or facts on the right. Put the number in the proper lettered cell in the magic square. Add each row and each column. If these sums are the same number, then you have found the magic number and matched the correct terms with their descriptors or rules. You may need to use your text and notes to match the correct sets with their descriptors or rules.

Activity 2.2: 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 writer and reader.

Ask students to free-write (write without worrying about grammar, punctuation, or spelling errors) on a particular topic for one minute. One-minute 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 soon. Free-writing means writing 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 best 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 this activity, pre-algebra students:

· Free-associate and free-write about a particular mathematical topic in a limited amount of time

· Self-assess their comprehension of a concept or lesson without worrying about grammar or spelling and grades

· Demonstrate their comprehension of topics to the teacher

HOW? Examples

Prompts

· “Take out a clean sheet of paper. We are going to free-write for exactly one minute on one topic. To free-write means to write continuously without worrying about grammar rules or correct spelling. Just write everything you think or know about the following topic, without picking up your pencil.”

· “We are going to write one-minute summaries on certain topics. Use the back of your homework and free-write, meaning write everything you know about the set of reals. Do not worry about spelling or grammar. When I say ‘Go,’ you will start, and you will stop when I stay ‘Stop!’”

Sample Topics

· Real numbers

· Infinite sets

· Properties of the reals

· Distributive

Here are some examples of students' one-minute summaries of infinite sets. The summaries are in quotation marks and teacher comments in brackets:

· “Infinite sets go on and on and on forever, like the set of counting numbers 1,2,3,4,… Every time you say a number, I can give one bigger.” [Very good! Is there a smallest counting number? Yes, it is 1.]

· “Infinite is like outer space. You can keep going and going, and then there is still more to go. I guess that's why sets of numbers are infinite ‘cause they keep on going.” [Good. Is there an infinite number of pieces of sand on Earth?]

Mini-Lesson 2.2 Properties of the Real Numbers

CCSS Standard 7.EE: Expressions and Equations

Use properties of operations to generate equivalent expressions.

This lesson involves the set of real numbers and its properties. Understanding these properties gives students more tools to use when problem solving. All of the infinite sets of numbers we've explored in this chapter have certain properties dealing with the operations: addition, subtraction, multiplication, and division. Since the reals are the largest set of numbers used in pre-algebra, we first consider all the real number properties. Each property has a rule, a particular set of numbers, and one or more operations.

The following list sets out all of the properties students will use in pre-algebra and algebra, along with the rule and examples. Let a, b, and c stand for real numbers.

· The commutative property holds for the set of reals with addition and multiplication. Hint: commute means “move.”

In algebra, multiplication is written as “ab” or “2y”; that is, the multiplication symbol is not used. Therefore, the rule is ab = ba or 4(5) = 5(4). The parentheses here are used to denote multiplication.

· The associative property holds for the reals with addition and multiplication. Hint: associate means “pair up.”

When doing several operations, always do what is in the parentheses first.

· The distributive property holds for the reals with multiplication distributed over addition or subtraction. This is one of the most useful properties in algebra! Hint: distribute means “pair off.”

· The set of reals contains an additive identity: the number 0. This means that there exists a number (0) that, when added to any real number, gives that real number.

· The set of reals contains a multiplicative identity: the number 1. This means that there exists a number (1) that when multiplied by a real number gives that number.

· Each real number has an additive inverse: it is the opposite of that number. The sum of each real number added to its opposite is equal to the additive identity 0.

· Every real number except 0 has a multiplicative inverse, which is called the reciprocal of each real number. The product of a real number and its multiplicative inverse gives the multiplicative identity 1. N: The real number 0 has no multiplicative inverse since does not exist. You cannot divide a nonzero number by zero.

Teaching Tip

Students often ask why it's impossible to divide by 0. One answer might be: “Consider 6 divided by 2. A number, 3, when multiplied by 2, gives 6, which gives our solution, 3. This is one definition of division. However, when 6 is divided by 0, there is no number that when multiplied by 0 gives 6.”

In the following activities, students will consider which properties hold for which sets of numbers.

Worksheet 2.3: Magic Square Activity for Properties of Real Numbers

Name _____ Date _____

Directions: Select the best answer for each of the terms on the left from the numbered descriptors or facts on the right. Put the number in the proper square in the magic square below. Add each row, and then add each column. If these sums are the same number, you have found the magic number and matched the correct terms with their rules.

Activity 2.3: Graphic Organizers

WHAT? Description

Graphic organizers (Barron, 1969) are schematics created to show connections between key concepts, similar to the semantic word maps in Chapter One. A graphic organizer is a geometric figure that draws attention to key concepts. Graphic organizers can be used to display subsets (parts) of larger sets; for example: the set of integers is a subset of the set of real numbers. When completed, graphic organizers may be used as a study guide or as a part of a bulletin board.

Teachers and students can create graphic organizers for use during prereading, reading, or postreading of the lesson. Teachers might present a graphic organizer to the class as a prereading demonstration to elicit students' prior knowledge of the concepts to be studied. Working in groups, students might brainstorm terms related to some larger concept and create their own graphic organizers.

WHY? Objectives

During this activity, pre-algebra students will:

· Activate prior knowledge of concepts

· Make connections between key concepts

· Summarize and organize main ideas from the reading for reviewing purposes

HOW? Example

The graphic at the left is often referred to as a “pyramid,” although it is actually a triangle because pyramids in geometry are three-dimensional solids. This graphic (triangle) can be used to demonstrate the relationships of infinite sets. Three infinite sets are placed on the pyramid, with the largest set, the reals, in the larger part or base of the triangle or pyramid. The rational numbers are part of the reals, and the integers are part of the rational numbers.

Worksheet 2.4: Graphic Organizers

Name _____ Date _____

Directions: Which of the two subsets, rationals or irrationals, of the reals do each of the numbers below belong in? Write each number in the appropriate set:

Activity 2.4: Reading Math Symbols

WHAT? Description

Creating a witty short story using math symbols is a good way to reinforce student recognition of these symbols. Students usually think that this activity is fun, and it helps them remember the meaning of the many symbols they will encounter as they learn algebra. Consider these symbols:

WHY? Objectives

During this activity, pre-algebra students:

· Reinforce their understanding of the meaning of symbols

· Reflect on and retain math symbols

HOW? Example

In the Land of , ∃ a melancholy creature who was not ∈ of any particular set or sect.∀ the creature's desires to be a part of a sect, she felt she was ≠ the other creatures of .↔ she was prettier and smarter, then she could be an ∈ of some set besides .∴she decided that she would… in her humble efforts + remain a unique creature in the Land of .

Worksheet 2.5: Reading Math Symbols

Name _____ Date _____

Directions: Use at least six of the symbols below to help you create a story. Write your story in at least four complete sentences. Your story must display the correct meaning of each symbol you use. You may be witty, dramatic, or serious.

Activity 2.5: 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 own 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 display of the terms. For example, a student might be asked to give the term's part of speech: noun, verb, adjective, or something else. Then students might be asked to give a definition from the reading or their class notes and a definition or description of their own. A suggested format is given in worksheet 2.6.

WHY? Objectives

In this activity, students create a math glossary giving the term, its part of speech, and a definition. These are the directions to give to them:

· Read an assigned chapter highlighting important concepts.

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

· Write out the definition from the text.

· Write a personal definition or description of each term.

· Write a phrase or statement using the word correctly.

HOW? Example

These are some sample terms for the glossary:

· Commutative property. Example: “You may add 2 + 3 OR 3 + 2, and both will equal 5.”

· Associative property

· Distributive property

Worksheet 2.6: Math Glossary

Name _____ Date _____

Directions: Refer to your lesson on the properties of real numbers to add to your math glossary. Six properties are listed below. Use the example of the commutative property to help you fill in the rest of the matrix. Consider addition to be the operation for each of the properties.

Activity 2.6: Semantic Feature Analysis

WHAT? Description

Semantic feature analysis (Baldwin, Ford, & Readance, 1981) is a reading strategy that asks students to complete a matrix showing how various terms and concepts are alike or different. The terms or concepts are related or fall under a particular category. The matrix itself consists of several columns. The first column contains a listing of the terms. The remaining columns contain headings that identify features that the terms or concepts might have in common. See the example in the “How?” section.

WHY? Objectives

During this activity, pre-algebra students:

· Explore features of infinite sets

· Learn the properties of the set of real numbers

· Answer questions about the reals

· Compare and contrast features of certain infinite sets

· Summarize the content on infinite sets from the text and from classroom discussions and lecture

· Refer back to the completed matrix when reviewing for exams

HOW? Example

Each of the following properties requires a set of numbers and an operation. Consider the variables a, b, and c, which refer to any real number unless otherwise stated. The properties are arranged in a semantic feature matrix.

The set of real numbers has a multiplicative inverse (reciprocal) for every number except 0. A where A is any real number does not exist. Since only one counterexample is required for a property to fail, the multiplicative inverse property does not hold for the set of reals.

Teaching Tip

Once students have been introduced to the properties, it is important that they see them in use. The real number properties are referred to often in later chapters of a pre-algebra text, as well as in later courses. All higher-level mathematics courses are written with the expectation that students have learned this knowledge and are ready to use it in these courses.

Worksheet 2.7: Semantic Feature Analysis: The Infinite Sets of Algebra

Name _____ Date _____

Directions: Fill in the matrix by writing each of the given infinite sets in each of the three forms. There may or may not be one form that cannot be used for each of the sets.

Worksheet 2.8: Semantic Feature Analysis: Features of Infinite Sets

Name _____ Date _____

Directions: Fill in the matrix by answering each question yes or no as it pertains to the given large set of numbers. Then answer the questions below the matrix referring to your answers.

Solutions for Worksheet 2.8

1. Give the additive identity for the set of reals.

2. The set of irrational numbers has an multiplicative inverse for each irrational number. True or false?

3. Give the multiplicative identity for the set of reals.

4. The set of irrational numbers has an additive inverse for each irrational number. True or false?

5. The set of reals can be split into two parts: the rationals and the irrationals. True or false?

6. Give the multiplicative inverse for .3, a rational number.

7. The reals are commutative with respect to division. True or false?

Activity 2.7: 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

· Critique each other's translations

HOW? Example

Teaching Tip

Negative exponents give fractions; using the expanded form for a number like 5.26 helps student understand this.

Worksheet 2.9: In Your Own Words: A Paraphrasing Activity

Name _____ Date _____

Directions: Read the passage below. Then write out your understanding of scientific notation. Be clear, and use at least two complete sentences. Be prepared to share your paraphrase with your peers. You may use other sources to help develop your definition.

José is giving a report on atoms. He found out that an ordinary penny has about 20,000,000,000,000,000,000,000 atoms. And the average size of an atom is about .00000002906 centimeter across.

José finds that both numbers are hard to read and hard to use in calculations. He searches for a way to abbreviate both numbers and finds scientific notation.

Scientific Notation Steps

1. Pick up all the nonzero digits unless the zero(s) are between the nonzero digits.

Example:

2. Write this number as a number x, where by moving the decimal point.

3. Finish the process by multiplying by 10 raised to some power—positive for a large number and negative for a small number. The value of the exponent is given by the number of spaces from the original place of the decimal point to the new place that you moved it to make it a number between 1 (inclusive) and 10 (noninclusive).

4. Write José’s numbers in scientific notation: