17119 QC Instructional Unit Covers 2 - Etowah County Schools€¦ · ER.AL1-1.2.1. iii Note QualityCore® Instructional Units illustrate how the rigorous, empirically researched course

Introduction to Algebra I: The Value of a Variable

Algebra IModel Instructional Unit 1

The Research-Driven Solution

to Raise the Quality of High

School Core Courses

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ER.AL1-1.2.1

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Note QualityCore® Instructional Units illustrate how the rigorous, empirically

researched course standards can be incorporated into the classroom. You may use this Instructional Unit as is, as a model to assess the quality of the units in use at your school, or as a soure of ideas to develop new units. For more information about how the Instructional Units fit into the QualityCore program, please see the Educator’s Guide included with the other QualityCore materials.

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C o n t e n t s

Unit 1 Introduction to Algebra I: The Value of a Variable Purpose............................................................................................................ vi Overview ......................................................................................................... vi Time Frame ..................................................................................................... vi Prerequisites ..................................................................................................... 1 Selected Course Standards ............................................................................... 1 Research-Based Strategies ............................................................................... 2 Essential Questions .......................................................................................... 2 Suggestions for Assessment ............................................................................. 2

Preassessment ............................................................................................ 3 Embedded Assessments............................................................................. 3 Unit Assessment ........................................................................................ 4

Unit Description ............................................................................................... 4 Introduction................................................................................................ 4 Suggested Teaching Strategies/Procedures................................................ 6

Enhancing Student Learning .......................................................................... 37 Selected Course Standards....................................................................... 37 Unit Extension ......................................................................................... 37 Reteaching ............................................................................................... 38

Bibliography................................................................................................... 40

Appendix A: Record Keeping ......................................................................A-1 Appendix B: Days 1–3 ................................................................................. B-1 Appendix C: Days 4–5 ................................................................................. C-1 Appendix D: Days 6–8.................................................................................D-1 Appendix E: Days 9–10 ............................................................................... E-1 Appendix F: Day 11 ......................................................................................F-1 Appendix G: Days 12–14.............................................................................G-1 Appendix H: Days 15–16.............................................................................H-1 Appendix I: Days 17–19 ............................................................................... I-1 Appendix J: Enhancing Student Learning..................................................... J-1 Appendix K: Secondary Course Standards ..................................................K-1 Appendix L: Course Standards Measured by Assessments ......................... L-1

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P u r p o s e , O v e r v i e w , a n d T i m e F r a m e

Purpose This unit is an introduction to the use of variables and expressions as the

language of algebra. Students will begin to translate real-world situations into algebraic expressions using patterns in real-world problems. They will use algebraic properties to simplify those expressions. Finally, students will be introduced to the relationship between expressions and other forms of data representation such as tables, charts, and graphs.

Overview According to John Dossey (1998), president of the National Council of

Teachers of Mathematics (NCTM), there are four goals in high school algebra: 1. To help students see and use algebra as a way of representing

quantities and relationships among quantities 2. To predict what happens in quantitative settings 3. To control, where possible, the outcomes to quantitative processes 4. To extend the applications and establish the validity of new

relationships in the structure of algebra To accomplish these goals, students in Algebra I will explore connections between the real world, mathematical expressions, and representations of data. In the process, they will learn the language used to describe mathematical relationships by investigating real-world problems, generalizing from arithmetic relationships, and gaining from those generalizations an abstract understanding of mathematics. Students will gain rich understanding of algebraic expressions by investigating how changes in patterns affect expressions. They will then discover the relevance of algebraic expressions by investigating how problems can be represented with algebraic expressions.

As students discover the connections between algebra and the world around them, they will develop a deep understanding of algebra. Individually and in groups, they will investigate how to think about new information by building on their prior knowledge, explaining their reasoning, and testing their ideas. Through journal writing, responses to reflective questioning, and solving routine and complex problems, they will analyze their thinking and learning processes.

By empowering students to discover algebra’s relevance to the world, this introductory unit is designed to help Algebra I teachers meet rigorous expectations. However, for students to meet the course’s rigorous standards, they must do most of the explaining and the demonstrating. In such a student-centered classroom, the teacher’s role is to develop and expand students’ thinking by posing rich problems and investigations, listening, observing, questioning, and understanding the students—actions that, when emphasized, create a classroom atmosphere of collaboration and achievement.

Time Frame This unit requires approximately nineteen 45–50 minute class periods.

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UNIT 1 INTRODUCTION TO ALGEBRA I: THE VALUE OF A VARIABLE

Prerequisites Add, subtract, multiply, and divide rational numbers Follow the correct order of operations Set up and solve problems with rational numbers

Selected Course Standards

The primary standards, which represent the central focus of this unit, are listed below and highlight skills useful not only in Algebra I, but in other disciplines as well. Secondary standards are listed in Appendix K.

B.1. Mathematical Processes a. Apply problem-solving skills (e.g., identifying irrelevant or

missing information, making conjectures, extracting mathematical meaning, recognizing and performing multiple steps when needed, verifying results in the context of the problem) to the solution of real-world problems

b. Use a variety of strategies (e.g., guess and check, draw a picture) to set up and solve increasingly complex problems

c. Represent data, real-world situations, and solutions in increasingly complex contexts (e.g., expressions, formulas, tables, charts, graphs, relations, functions) and understand the relationships

d. Use the language of mathematics to communicate increasingly complex ideas orally and in writing, using symbols and notations correctly

e. Make appropriate use of estimation and mental mathematics in computations and to determine the reasonableness of solutions to increasingly complex problems

In most scholastic and academic disciplines, what you learn to think about is not as important as how you learn to think. —David Eggenschwiler (2007, January 5)

Instructional programs from prekindergarten through grade 12 should enable all students to understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; [and] analyze change in various contexts. —National Council for Teachers of Mathematics (2000, p. 37)

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f. Make mathematical connections among concepts, across disciplines, and in everyday experiences

g. Demonstrate the appropriate role of technology (e.g., calculators, software programs) in mathematics (e.g., organize data, develop concepts, explore relationships, decrease time spent on computations after a skill has been established)

h. Apply previously learned mathematical concepts in algebraic contexts

C.1. Foundations a. Evaluate and simplify expressions requiring addition, subtraction,

multiplication, and division with and without grouping symbols b. Translate real-world problems into expressions using variables to

represent values c. Apply algebraic properties (e.g., commutative, associative,

distributive, identity, inverse, substitution) to simplify algebraic expressions

D.2. Graphs, Relations, and Functions (in the First Degree)

f. Use the terminology associated with the Cartesian plane in describing points and lines

G.1. Data Relations, Probability, and Statistics b. Interpret data from line, bar, and circle graphs, histograms,

scatterplots, box-and-whisker plots, stem-and-leaf plots, and frequency tables to draw inferences and make predictions

c. Identify arithmetic sequences and patterns in a set of data h. Identify the most efficient way to display data

Research-Based Strategies Group Work (p. 7) Cooperative Learning (p.7) Essential Questions (pp. 11, 12, 14) Reflective Questioning (p. 12) 3-2-1 Assessment (p. 15) Cues and Questions (p. 16) Think-Pair-Share (pp. 19, 21, 23, 27) Misconception Check (p. 20) Whiteboarding (pp. 26, 28, 33)

Essential Questions 1. What is algebra? 2. How can I use algebra in my life? 3. How can algebra be used to model the world around us?

Suggestions for Assessment

Except where otherwise noted, assessments can be given a point value or they can simply be marked off as completed.

Tips for Teachers

The essential questions and the primary course standards for this unit should be prominently displayed in the classroom.

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Preassessment

Introduction Activity—Students demonstrate their facility with arithmetic by creating a math problem. (Day 1)

Worksheet—The Crossnumber Puzzle (p. G-2) preassesses students’ understanding of the properties of real numbers and will serve as a review of addition, subtraction, multiplication, and division of negative integers. (Day 12)

Embedded Assessments Homework—By calculating the sum of a series, students gain additional

practice using patterns to solve problems. This homework also establishes the routine of solving problems every day. (Day 2)

Gallery Walk—After walking around the room viewing posters that introduce the concepts that will be learned in the course, students work in groups to solve the problems and present their solutions to the class. (Day 3)

Homework—The Writing and Arithmetic homework (p. B-3) encourages students to become comfortable writing in class and gives additional preassessment information. (Day 3)

Journal Writing—Assess students’ communication of mathematical ideas by reading students’ journal entries regularly. (Days 4, 5, 8, 9, 11, 14, and 16)

Problem—Walter the Walker is a real-world problem that provides a relevant context for several important algebraic concepts in the unit. (Days 5, 11)

3-2-1 Assessment—Informally assess students’ understanding with a 3-2-1 Assessment. (Day 5)

Homework—The Find the Pattern homework (p. C-9) gives students practice solving problems by finding patterns. (Day 5)

Activity—The activity that accompanies the Types of Graphs and Tables handout (pp. D-2–D-6) introduces students to mathematical models used to describe and display real-world data and events. (Day 6)

Transparency—Introduce students to expressions involving addition by modeling a real-world event with the Interstate Driving transparency (p. D-7). (Days 6–8)

Homework—The Day 8 homework (p. D-8) gives students practice modeling real-world events with expressions involving addition. (Day 8)

Transparency—The Remy’s After-School Job transparency (p. E-2)introduces students to a way of modeling real-world events using expressions involving multiplication. (Day 9)

Worksheet—Introduce students to expressions involving more than one operation with the Matinee Mania Club worksheet (pp. E-3–E-4). (Days 9–10)

Homework—The Compare and Contrast homework (p. E-5) illustrates the relationship between patterns of data and real-world expressions. (Day 10)

Transparency—Remind students of the many ways to represent data and help them connect graphs to their algebraic expressions with the Comparing Graphs of Expressions transparency (p. F-2). (Day 11)

Transparency—Give students additional practice finding expressions to model events with the Population Estimates transparency (p. F-3). (Day 11)

Homework—The Day 11 homework (p. F-4) connects population estimates data to other representations of data and introduces both the concepts

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of adding and subtracting like terms and the distributive property of multiplication over addition. (Day 11)

Activity—The Adding and Subtracting Like Terms activity (p. G-4) uses a hands-on geometric model to illustrate addition and subtraction of like terms. (Days 12–14)

Activity—Several in-class problems illustrate algebraic properties in real-world contexts. (Days 12–16)

Activity—Identify students who are struggling with new concepts with a whiteboarding activity. (Days 14 and 16)

Homework—The Simplifying Expressions homework (pp..G-6–G-7) gives students practice simplifying algebraic expressions. (Day 14)

Homework—The Distributing and More homework gives students practice simplifying algebraic expressions, multiplying monomials, and using the distributive property of multiplication over addition. (p. H-3). (Day 16)

Unit Assessment Project—Students create a Real-World Problem

Poster that displays a real-world problem to represent the concepts presented in the unit (p. I-2). (Days 17–19)

Unit Description

Introduction It is vitally important that students enjoy learning

mathematics, feel confident in their ability to do mathematics, and see the relevance of mathematics in their lives. According to Harry and Rosemary Wong (2004, p. 4), “student achievement at the end of the year is directly related to the degree to which the teacher establishes good control of the classroom procedures in the very first week of the school year.” In other words, preparation for students before they ever enter the classroom is critical—not just for a successful first week, but for a successful school year.

Therefore, before class begins, identify the procedures that you expect in your classroom, and be ready to model them for students. For instance, to build classroom rapport and to demonstrate that their ideas matter, make a point to acknowledge and talk to each student. Begin the first and every class with a warm-up, either written on the board or placed on students’ desks for them to work upon entering the classroom. A warm-up establishes a routine that reinforces the expectation that students are to begin working when they enter the classroom. As Wong and

Wong (2004, p. 95) point out, students perform better when they know what the teacher expects of them. Warm-ups also allow you to take attendance without wasting valuable educational minutes. By piquing students’ interest, focusing their attention, connecting to previous learning, or introducing the topic of the day’s lesson, warm-up activities make the most of the time you have and prepare students for the day’s learning, just as runners prepare for a race.

Before class begins you should also set up the classroom. Display the rules, schedule, essential questions, a calendar for assignments, and procedures

Tips for Teachers

Prior to the first day of school, use the following checklist (Wright, 1989) to identify tasks not yet accomplished or to spark new ways of starting off the new year. Am I energized to be enthusiastic about this class? Is the classroom arranged properly for the day’s

activities? Are my name, course title, and room number on

the chalkboard? Do I have an icebreaker planned? Do I have a way to start learning names? Do I have a way to gather information on student

backgrounds, interests, course expectations, questions, and concerns?

Is the syllabus complete and clear? Have I outlined how students will be evaluated? Do I have announcements of needed information

for the day? Do I have a way of gathering student feedback? When the class is over, will students want to come

back? Will I want to come back?

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around the room to allow students to find needed information conveniently and to encourage them to take responsibility for their learning. Leave empty space on bulletin boards to post student work and projects, thereby emphasizing that their work is important and that you value it and, by extension, them. Finally, ensure that your students will know they have come to the right room by posting your name and room number prominently outside your door.

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Suggested Teaching Strategies/Procedures Days 1–3

Students are introduced to the classroom and the concepts that they will learn.

Materials & Resources

Unit Assignments and Assessments (pp. A-2–A-3) Class syllabus* Introduction (p. B-2) Index cards* Tape* Beach ball* Chalkboard* Standard deck of playing cards* Writing and Arithmetic (p. B-3) Writing and Arithmetic Key (p. B-4)

*Materials or resources not included in the published unit

Day 1

Before class, place a copy of the class syllabus and the Introduction activity (p. B-2) on each desk or in a designated location in the room. To facilitate seating on the first day, assign each desk a unique number, mathematical symbol, or variable. After writing this number, symbol, or variable on two index cards, keep one card to pass out to students, and attach the other card to the desk with tape.

By treating students with respect and courtesy, you will begin to build personal relationships on first impression. Therefore, stand outside your door to greet students as they enter the classroom for the first time. Welcome them to Algebra I, introduce yourself, and hand each of them one of the index cards. Ask them to find the desk with the matching number, symbol, or variable, to read the course syllabus, and to begin working on the Introduction activity.

The Introduction activity asks students to list three interesting things about themselves and to invent a math problem. When students have finished, initiate an icebreaker. First, pair students up. After students introduce themselves to each other, ask them to share two or three interesting things about themselves that they wrote on their index cards with their partner. Listen to the students as they share; learn their names and interests. When they have finished, toss a beach ball to a student at random. Ask him to introduce his partner to the class and to share one or two interesting things about her. When finished sharing, he should then toss the beach ball to another student. (You should be prepared to share, too—icebreakers work best when everyone in the room is involved.) As each student introduces his or her partner, show that you value their thoughts by listening attentively and recording their comments. Your note-taking will also ensure that no one is left out. Later in the course, use your notes about students to make real-world problems that are relevant to both the mathematical concepts being learned and individual students’ lives. After the icebreaker, ask volunteers to write their problems on the board and to challenge the rest of the class to solve them. If the class is large, this activity

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may take the entire class period. As homework, ask students to solve one of the problems shared by a volunteer.

Before students leave, collect the index cards and the Introduction activity. Save the index cards to make a seating chart and to use the numbers, symbols, or variables to group students throughout the course. Provide positive and useful feedback on the math problems students designed and return them the next day. The problems may reveal students who require remediation or additional help, and others who are adept and may be able to help their classmates. Take note of what you learn about them. In addition, some problems might be useful as is or with some modifications to build or reinforce specific skills or concepts such as pattern seeking, simplifying algebraic expressions, or estimating. Throughout the unit, the problems can serve as warm-ups, wrap-ups, or homework problems as appropriate.

Day 2

As a warm-up on Day 2, ask students to share their solutions to the homework problem. Then, take time to prepare students for the group work they will undertake throughout the course. In rigorous mathematics courses, students are frequently asked to work in cooperative groups. No instructional strategy has been found more effective at improving student achievement than cooperative learning. In rigorous math courses—in which students conjecture, solve problems, and justify their responses using the language of the discipline—the strategy is especially important. Effective cooperative groups have five essential elements: positive interdependence, face-to-face promotive interaction, individual and group accountability, interpersonal and small group skills, and group processing (Johnson, Johnson, & Holubec, 1993).

Positive interdependence means that all members of the group rely on each other. Each member of the group plays a unique role within the group.

Face-to-face interaction requires members to communicate, thereby strengthening their communication skills as well as helping them retain the information they are communicating.

Individual and group accountability means that all students are responsible for their own and their group’s learning. Randomly selecting students within the group to share the group’s responses helps ensure that all students within the group are actively engaged in the tasks presented.

Interpersonal and small group skills are the tools for working effectively with others. These are lifelong skills that many employers find essential.

Group processing means that group members reflect on their performance and adjust it, as necessary, to maintain effectiveness.

Because students will be working in groups and leading class discussions, it is important to establish ground rules for group discussions. Rules encourage appropriate interactions such as active listening, a technique in which a listener repeats in his own words what the speaker just said. This requires listeners to be attentive and enables speakers to know whether they are expressing themselves clearly. Ground rules also discourage students who dominate conversations, show disrespect toward others, or speak out of turn. One way to create ground rules is an acronym method described in the online Teaching Guide for Graduate Student Instructors at the University of

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California at Berkeley (University of California Berkeley Graduate Division, 2007). There are seven steps to the acronym method:

1. Across the top of the board, write Ground Rules. Just below, along the board’s left-hand side, write the word ROPES.

2. Explain that the ROPES will serve as rules for class discussion to which everyone in the classroom agrees (everyone will know the ROPES).

3. For each letter appearing in the word ROPES, ask students to suggest a word beginning with that letter—R for respect, O for open-mindedness, P for patience, etc.—that represents a potential ground rule for group discussion. Each student should explain the importance of the suggested rule.

4. Contribute suggestions of your own. 5. Call for a general consensus as to which rules are best. 6. Copy down the list of ground rules and distribute them in a

handout at the next class for students to insert in their notebooks. 7. Ask the class to revisit the ground rules from time to time, to

determine how well they are being followed. Having established ground rules for discussions, divide the class into

small groups using a deck of cards, as follows: First, limit the number of cards in the deck to match the number of students in the class. For example, for a class of 20 students, select 4 aces, 4 twos, 4 threes, 4 fours, and 4 fives. Have each student pick a card from this limited deck. Students whose cards have the same face value should sit together around one table or pod of student desks. Some students may benefit and work more effectively when assigned a role within the group. Therefore, assign roles according to the suits in the deck: students with hearts can be managers; students with clubs, facilitators; students with diamonds, recorders; and students with spades, checkers. Students should return the cards to the deck, but remember the specific card they drew.

To ensure that all students understand their roles while working in their cooperative groups, write a description of each role on an overhead transparency or on the board when explaining the roles to the class.

Managers pick up any materials needed for the activity and keep track of the time.

Facilitators read the questions to the group, focus the conversation on the current task, and encourage progress towards the next question.

Recorders take note of the group’s responses, from ongoing discussion points through the final consensus. (At the same time, group members should also keep a personal record of the answers on their worksheets or in their class notebooks.)

Checkers will ensure that all group members understand or agree and that each member is ready to present if called upon.

In addition to inviting students to collaborate in their learning, the next activity serves as an informal preassessment of students’ abilities to analyze abstractions. Begin by presenting the following quote from the German mathematician Carl Friedrich Gauss (as cited in Eves, 1990, p. 478): “Mathematics is the queen of the sciences.” Ask groups to interpret Gauss’s words. As students work, remind them of the ground rules for discussion as well as their respective roles in the group. After 5 minutes, choose a card at random from the deck to identify a student to summarize his or her group’s

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discussion. After all groups have shared their thoughts, tell the story of Gauss and his achievements.

Gauss was born in Germany in 1777 to uneducated parents, but was soon revealed to be a child prodigy. Legend has it that when he was ten years old and his teacher asked him to add the integers from 1 to 100, Gauss spotted a pattern almost immediately and gave the correct sum—5,050—on the spot. At the age of 21, Gauss wrote a textbook on number theory, Disquisitiones Arithmeticae, which helped to shape modern number theory into a recognized branch of mathematics. Throughout his life, Gauss contributed to numerous mathematical and scientific fields, including algebra, electricity and magnetism, and astronomy. He is now sometimes called the greatest mathematician since antiquity.

Mathematicians believe that Gauss added the integers from 1 to 100 by “folding” the series in half—that is, by recognizing the series consists of 50 pairs of integers, each pair having a sum of 101. However, do not reveal that pattern to the students yet. Instead, give them the opportunity to discover patterns for themselves by working in groups to find the sum of the integers from 1 to 100. Some may see the same pattern Gauss discovered. Other students may build on the observation that the sum of the integers 1–10 is 55; the sum of the integers 11–20 is 155; the sum of 21–30 is 255, and so forth; in other words, the sum of each set of ten integers is 100 plus the sum of the preceding set of ten integers. Adding these sums produces a total sum of 5,050. Still other students may build on the observation that the sum of the integers 1–9 is 45. Because the value of a sum is unchanged by the addition of 0 (0 is the identity element for addition) their observation may lead to the following strategy:

The sum of the integers from 1–100 is equal to the sum of the integers from 0–100, which in turn is equal to 100 plus the sum of the integers from 0–99. Arranging the integers from 0–99 in a square array (see Figure 1), students can see that, looking down the columns, each of the numbers 0–9 occurs as the ones digit exactly 10 times. Adding together all the ones digits yields a subtotal of 450. Looking across the rows, each of the numbers 0–9 occurs as the tens digit exactly 10 times. Adding together all the tens digits yields a subtotal of 4,500. 450 + 4,500 = 4,950, the sum of the integers 0–99. Adding the final integer, 100, to 4,950 produces the sum of the series, or 5,050.

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0 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

Figure 1

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As homework for Day 2, encourage students to find the sum of a similar series, such as the sum of all odd numbers from 1 through 99, by using either the pattern they discovered or another group’s pattern. Not only will the homework problem give students additional practice using patterns to solve problems, but also it will begin a routine of solving homework problems every day.

To continue to introduce students to the concepts that will be addressed throughout the course, design an “Algebra Gallery” for Day 3. Hang posters around the room, each with a problem that gives an overview of how algebra is used in everyday life. These problems should connect students’ prior knowledge to algebraic concepts. Each problem should be moderately difficult—a real challenge to students—but not so complicated that it blocks students’ efforts to engage it. Each problem should also take approximately the same amount of time to solve. Constructed-response problems fitting these criteria can be found in the QualityCore Algebra I Formative Item Pool, which is available in an online tool called Test Builder. This tool allows teachers to search for, select, and print problems that assess the course standards covered in this unit.

Day 3

Warm up for the Algebra Gallery by having students return to their Day 2 groups and walk around the room to view the problems. After students return to their desks, collect the homework and ask volunteers to share their patterns and solutions. Then, assign each group a problem from the Algebra Gallery. Each group should then work together to solve its problem and represent its work and solution on poster paper. When all groups have finished, randomly select one student from each group to present the problem and its solution to the rest of the class as a wrap-up activity.

Assign the Writing and Arithmetic homework (p. B-3) before class ends. The writing portion of the assignment encourages students to become comfortable writing in class while enabling you to learn more about your students. Many students have had bad experiences with mathematics in the past and dislike it; some experience math anxiety when asked to do math problems. It is important to acknowledge these fears and to give students opportunities—such as this homework assignment—to experience success in mathematics classes. Use the four math problems as an additional preassessment to discover students’ strengths and weaknesses—in this case, the facility to adhere to the correct order of operations with rational numbers.

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Days 4–5

Students investigate the purpose of algebra and work cooperatively to solve a problem that introduces them to algebraic language.


Quotes (p. C-2) The Function of Algebra (pp. C-3–C-4) Card Trick (p. C-5) Your Age (p. C-6) Class notebook* Math journal* Glossary (p. C-7) Class Journal Feedback (p. C-8) Centimeter blocks or graph paper* Clipboard* (one for the teacher) Sticky notes* Find the Pattern (p. C-9) Find the Pattern Key (p. C-10) Deck of cards* Poster paper*


Day 4

As a warm-up, place a copy of the Quotes worksheet (p. C-2) on each desk or in a designated area of the room. Greet students at the door and ask them to have a seat and begin the activity, which asks them to investigate the question “What is mathematics?” based upon what others have said about the subject. The goal of the exercise is to stimulate students’ interest in mathematics as an “open” field. As students work at their desks, collect their homework; when they have finished, direct their attention to the essential questions posted around the room. Essential questions draw attention to the most important concepts of the unit, helping to prevent lessons that present random assortments of facts. According to Heidi Hayes Jacobs (1997, p. 26), “An essential question is the heart of the curriculum. It is the essence of what you believe students should examine and know in the short time they have with you.” Like mathematics, an essential question is not designed to have one right answer, but rather to be explored by students and teachers alike. Explain to students that they will discuss their ideas about algebra and its purpose so that they may begin to see how mathematics relates to the world around them.

After reminding students of the ground rules for discussions established on Day 2, encourage volunteers to share their responses to the Quotes worksheet. Students may make an immediate connection between the quotes and Essential Question 1: “What is algebra?” Ask them to share what they think algebra is and what it involves. Encourage them to ask questions and to come up with a preliminary class definition. Meanwhile, avoid correcting their thinking; instead, make thoughtful observations about

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Tips for Teachers

For many teachers it can be difficult to be patient after asking a question, and yet increasing wait-time beyond 3 seconds is positively correlated to improvements in student achievement and increases in the quality and amount of student contributions to classroom discussion (Rowe, 1986).

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their ideas. Summarize what they say, and encourage their curiosity by observing, “I hadn’t considered that perspective before,” or exclaiming, “That is an interesting observation!”

Encourage further discussion by asking questions: What do you think the author meant by this statement? Do you agree with the author of this quote? Do you think this is more or less true today and why? How do the statements relate to each other?

Because many students may not recognize or acknowledge the wide-ranging uses of algebra in the world around them, they often ask, “When am I ever going to use this stuff?” Sharing your enthusiasm and engaging students in dialogue is critical to helping students to overcome their skepticism and understand that they are as much a part of the process of making meaning of mathematics as René Descartes, Carl Friedrich Gauss, and Pierre de Fermat. Guide them to the realization that math is continually evolving and new applications of mathematics are revealed every day. To support instruction relevant to this point, The Function of Algebra background information (pp. C-3–C-4) is a brief summary of the role of algebra in the field of mathematics. Use it to refamiliarize yourself with the human experience of mathematics and to guide further investigation. It should enable you to facilitate a classroom discussion that can lead your students to begin to appreciate mathematics in ways that are relevant to them. However, avoid simply reading it to students; rather, use it to initiate a discussion that reveals your own personal connections to mathematics. During the discussion, emphasize that even you are still learning about the role of mathematics in the real world. The more sentences that begin with, “I recently learned that math is used in . . .” or “I read a mathematics article that discussed . . .” the more you invite students to share your own curiosity about math. Allow students to share until their thoughts have been exhausted. By the end of the discussion, students should begin to comprehend that algebra is a powerful tool for understanding the world in which we live and begin to address Essential Question 2: “How can I use algebra in my life?”

Next, pique students’ curiosity and fully engage their thinking with a card trick based on algebra. The Card Trick activity (p..C-5) provides instructions for performing an algebra-based card trick. Be aware, however, that it requires practice prior to performing for the students. When you have finished, without revealing the solution, allow the students to try to guess how it was done. Give groups of students a deck of cards and encourage them either to try to replicate the card trick or to look for a pattern that might reveal its solution. Move around the room, observing and listening to students’ conversations. Developing a student-centered classroom requires knowing when to step back and allow students to lead, and when to step in to provide support. Therefore, be encouraging. As you listen, offer compliments such as, “That’s a very good guess,” or “I like your thinking.” Moreover, in order to get more students involved in the discussion, use reflective questioning techniques: ask of group members, “Could someone please restate what _______ said?” or “Can someone else give evidence to support that response?” Later in the unit, students will discover the trick’s solution. (An alternate activity with the same pedagogical end is Your Age [p. C-6].)

Before the end of Day 4, introduce the materials that students will need for the course. Ask students to keep a class notebook in order to organize the various materials they will be responsible for. Let them know that you also

D a y s 4 – 5

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will keep a notebook that students returning from an absence can use to learn what they missed. Use your notebook to illustrate the different sections the notebooks should contain. One method of organizing the notebook is to keep separate sections for class notes, a glossary, homework, and handouts:

Class Notes: Students should keep daily class notes. Specific methods for taking notes—such as annotating—will be offered as the need for notes arises.

Glossary: Define terms and keep vocabulary handouts in the Glossary section. (See the sample Glossary worksheet p. C-7.)

Homework: Keep dated homework assignments in this section. Handouts: Keep handouts and graded assessments in this section.

Some teachers collect notebooks on a periodic basis and award points to students who have kept all necessary materials. Students should also keep a separate math journal (notebook) in which they respond to questions or problems in writing. Journals have not been common in mathematics classrooms, but a substantial amount of research reveals that they are valuable facilitators of students’ learning. According to the NCTM’s Curriculum and Evaluation Standards for School Mathematics (1989, p. 214), “The assessment of students’ ability to communicate mathematics should provide evidence that they can express mathematical ideas by speaking, writing, demonstrating and depicting them visually.” One way to incorporate this standard is through journal writing (Williams & Wynne, 2000). Journal writing helps students to recognize what they do and do not know, to connect prior knowledge with the knowledge they are currently learning, to reflect on and think critically about new ideas, and to keep their thoughts organized (Burchfield, Jorgensen, McDowell, & Rahn, 1993).

Similarly, mathematics journals allow you to gain insight into students’ mathematical thinking, to identify and address student misconceptions, and to assess students’ study habits and attitudes (Rothstein & Rothstein, 2007, p. 22). Although students may at first resist the idea of math journals, they typically come to see their value once they understand that the communication is about math and that you respond to their writing as a math teacher. As students develop the habit of writing, you can assess the effectiveness of daily lessons by collecting their journals. Although you might require students who submit incomplete or poorly conceived entries to resubmit them, grading the journals is unnecessary. More important than issuing a grade is responding to students’ entries with positive, written comments that let students know that you care about their thoughts (McIntosh & Draper, 2001). Therefore, use the Class Journal Feedback rubric (p. C-8), which is adapted from Jim Burke’s Writing Reminders (2003), to provide feedback for their entries. Although students do not need to write in their math journals every day, they should write frequently enough to develop the habit of thinking and communicating about math in writing.

At the end of Day 4, assign journal homework: have students imagine they are part of an advertising firm hired by NCTM to “sell mathematics.” Their job is to create an effective sales pitch for a target audience of bored, disinterested teens. Explain to students that the client will not be satisfied unless their pitch is a positive promotion for learning math. In addition to the journal assignment, assign problems from textbooks or additional resources to allow students to practice routine skills.

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Day 5

Begin Day 5 by having students write their sales pitches on the board, presenting them for the warm-up, and finally voting on the favorite. Next, assign students to groups of three by calling off desk numbers.

Introduce Essential Question 3: “How can algebra be used to model the world around us?” The next activity explores problem-solving strategies in a real-world situation, inviting students to realize that algebra is a language used to generalize from patterns. Place the following scenario on an overhead transparency:

To improve his health and fitness, Walter plans to walk every day. During the first week, he will walk around the perimeter of the square block in which he lives. Each subsequent week, he intends to walk one block further west of the original loop (see Figure 2). He will keep track of the distance he walks each day on a calendar, expressing the distance in units, where one unit equals one side of a square block. If Walter continues his walking plan, how many units will Walter walk each day during Week 10 ? How many units will he walk each day during Week 100 ?

Provide each group with centimeter blocks or graph paper. Give them plenty of time to model the situation and discuss strategies for solving the problem. As they work, students will likely suggest and work through many theories of their own. Encourage them to keep track of the process so they do not repeat initial mistakes. Meanwhile, circulate through the room, listening to what they are saying, trying not to interject, and being cautious about giving hints. Ask questions that encourage students to be creative and to refine their thinking. Ask, for instance, “Can you find a shortcut that will help you find the number of units without the need to add them all up?” By allowing students to solve problems at their own levels, they will become independent learners by discovering their own problem-solving strategies.

When groups are close to solving the problem, ask volunteers to share their group’s thinking and the problem-solving strategies they used. While they present, have another volunteer record a class list of problem-solving strategies on poster paper. This list can be added to

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Tips for Teachers

Randy Bomer (1995) recommends taking notes about students on a clipboard while in class. For each student, Bomer makes note of the situations in which he or she seems comfortable or uncomfortable, areas of knowledge he or she might bring to the class, reading or other interests outside of school, anything the student says about school, or anything else that may help him know the student better. Even though these notes are imperfect and incomplete, they nevertheless provide a running history of the students’ class experiences. Bomer explains to students that his notes are a form of valuing what they say. At the end of each week, Bomer places the notes into three-ring binders, one binder for each class.

Tips for Teachers

The website of North Central Regional Educational Laboratory (NCREL) features useful information about collaborative learning and small groups. To support successful group work, they suggest that teachers: Include students of mixed abilities when forming

heterogeneous groups. Provide specific work areas for each group. Clearly communicate rules and assignment

requirements. Encourage all group members to participate. Acknowledge and support positive and effective

work within and between groups.

Figure 2

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and referenced throughout the course. In the course of their discussions, some students may have created a table of values. Others may have modeled the first 10 weeks with the centimeter blocks, counted up the units, and then tried to predict Week 100. Still others may have looked for a pattern in the first several weeks to help them determine the distance Walter walked in the weeks that followed. Instead of providing a correct solution, foster students’ independent learning by encouraging them to explain why they think their solution (or someone else’s solution) is plausible. Allow any volunteers to contribute. Encourage groups to revise their work if, after hearing the presentations, they believe there is an error in their thinking. All students should also record successful problem-solving strategies in their notebooks. Emphasizing problem-solving strategies invites students to understand that algebra is a language used to describe patterns. Different groups see different patterns in data, and the language of algebra allows us to communicate those patterns in ways that are easily understood and interpreted. Encourage the class to brainstorm other strategies they could have used to understand the problem, such as graphing. Add these strategies to the class list. All of the strategies will help students recognize that there are many ways to display patterns and data. Students will revisit Walter the Walker again later in the unit.

Wrap up by asking students to use sticky notes to record three things that they have learned so far, two things that they do not yet fully understand, and one question that they still have. This is called a 3-2-1 Assessment. Divide a large piece of poster paper into three columns, labeling them 3, 2, and 1. Ask students to stick each of their notes in its corresponding space as they leave the classroom. Review the notes after class to identify frequently occurring questions or misconceptions. Use this information to help plan or adjust subsequent lessons.

For homework, assign the Find the Pattern homework (p. C-9). If there are absent students, allow them either to read your notebook when they return, or to find volunteers who will explain what they missed during their absence.

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Days 6–8 Variables are introduced to model a pattern using an

algebraic expression.


Springs* Weights* Types of Graphs and Tables (pp. D-2–D-6) Overhead projector* Class notebooks* Chalkboard* Interstate Driving (p. D-7) Clipboard (one for the teacher)* Math journals* Day 8 (p. D-8) Day 8 Key (p. D-9)


Ask students to share with a partner, as a warm-up, the patterns they identified for Part II of the homework. In addition, ask partners to discuss how they know their real-life event fits a pattern. Then ask volunteers to share their patterns and explanations with the class. This student-led discussion should initiate a rich dialogue that will help students become aware of the representations of mathematical ideas in their everyday lives. They will continually explore and revisit these representations throughout the unit. After the discussion, take time to address questions and misconceptions from the 3-2-1 Assessment. Postpone addressing those questions that will be covered later in the unit. Finally, collect the homework for grading.

The next investigation is intended to help students understand and define independent and dependent variables, which will then be connected to different representations of patterns and data. Building upon students’ past experiences and prior knowledge, ask if they have ever done experiments in science class. Encourage them to share how the experiments were set up and what they were trying to show. As they share, use cues and questions to focus their attention on the controlled parts of the experiments and the observed results. Encourage them to recall the specific vocabulary used and to describe how they remember which variable is which. Then, ask a volunteer to remind the class of the Walter the Walker problem and what they were asked to find. Ask the students to explain how the problem is similar to a science experiment. They may notice that both a science experiment and the problem address a question. In each case, one can draw conclusions by trying different inputs (independent variables) and studying the results. Finally, ask them to determine the independent and dependent variables in the Walter the Walker problem. Because the number of units walked depends on the week, the number of units is the dependent variable; the independent variable is time (as measured in weeks).

To further reinforce the concepts of independent and dependent variables, ask pairs of students to join together in groups of four to design and conduct an investigation using springs and weights, with the goal of answering

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the question, “How is the length of a spring determined by the weight hung on it?” Ask students to

record all data in a table; indicate in the table which variable—length or weight—is

independent and which is dependent; graph the data; and discuss all observations, noting patterns and drawing conclusions.

The investigation should take the rest of Day 6 and part of Day 7. Give the groups time to identify variables and to create visual

representations of the data. The Types of Graphs and Tables handout (pp. D-2–D-6) introduces some of the mathematical models used to represent data. As they work, ask students to discuss the purposes of each type of graph and table. They should use what they learn in this discussion to revise their representations. As students share graphs, they should justify the presentation of the data, explain the variables, and draw whatever conclusions they can. In this unit, students primarily will be graphing in the first quadrant of the coordinate plane.

When students seem reasonably confident with the concepts of independent and dependent variables, explain that in algebra, letters are used to represent numerical variables. In addition, define the term expression: that is, as a combination of symbols (letters and numerals) and operations (such as addition, multiplication, raising to a power) that are executed in a precise order. The problems students created for the Introduction activity on Day 1 are examples of numerical expressions; expressions containing variables are algebraic expressions. Ask students to brainstorm the various symbols that might be found in an algebraic expression. Then, reveal on an overhead selected student problems—both the original and a modified version with variables. Give students different values for the variable and ask them to simplify the resulting algebraic expression. The terms variable and expression, with examples, should be added to the students’ glossaries.

Another activity, the function game, emphasizes how independent and dependent variables can be used in mathematics. Here is how it is played:

1. On the board, draw a box with two openings, one in the upper left labeled Input and the other in the lower right labeled Output. Write the word Function in the middle of the box. Explain that the box is a special function machine. For any number that is put into the function machine, it will put out a corresponding number according to a special rule. The job of the students in the function game is to guess the rule.

2. Select a simple function (e.g., f(x) = 2x + 3), but do not share it with the students.

3. Ask a student to go to the board and write a number to input into the function machine. Explain that you will apply the special rule to obtain the output. Apply the function to the input number and write the output on the opposite side of the box.

4. Repeat Step 3, as needed, with different volunteer-supplied input numbers.

5. When students think they know the rule, they should raise a hand but not say anything. Pick a student with a raised hand to write the next output number. Allow the other students with a raised

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Tips for Teachers

While not all algebraic relations are functions, those used in this unit are functional relations designed to introduce students to the practice of creating a table of values, writing expressions, and identifying the independent and dependent variables in a real-world situation. This will help students understand that the independent variable serves as the input value upon which the dependent variable relies.

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hand to do the same. Do not allow them to say anything other than the output number. If a student supplies an incorrect output, encourage them to keep trying to find the rule and ask another student to correct the mistake.

6. After several students have given correct outputs, ask one to express the rule in words and/or as an algebraic expression.

7. After students have caught on to the game, ask for a volunteer to invent the function rule for the next round.

As students become more confident, introduce proper function notation and terms, such as dependent variable to the game. If students come up with

different correct rules for a given function, take the opportunity to lead a discussion of equivalent forms for expressions. After playing the function game, introduce students to the concept of using an expression to represent a real-world event using independent and dependent variables and patterns. With the Interstate Driving transparency (p. D-7), conduct a class discussion that follows students’ process of discovery. At every opportunity, ask students for input. Based on the example, students may suggest determining

the independent and dependent variables. If so, reply with the query, “What next?” Soon, they will use the same process in small groups.

After students have identified the independent and dependent variables for the Interstate Driving scenario, encourage students to begin with simple input values like 0, 1, 2, and 3, then ask them to find the output values. This process should help them find the pattern: guide the students in graphing, encouraging them to record input values on the horizontal axis and output values on the vertical axis. Next, reverse the process: have students choose output values and allow them to find the appropriate input values. Challenge students to predict the result of a negative input value; in this case, a negative input represents miles driven in a westerly direction. To test their ideas, ask them to find the outputs resulting from several inputs with negative values. Finally, using m to represent an input value, ask students to find an expression that represents the output. When they have found an expression, ask them to determine how to check it. Because they will likely suggest replacing m with a number and performing a calculation to see if the outcome matches the values in the table, encourage them to use proper terminology for this. For example, if they suggest, “take out m and put in a number,” ask for the precise word to describe taking out one thing and replacing it with something equivalent—that is, substitute. Likewise, evaluate refers to the process of finding the value of the expression and a term is a component of an algebraic expression separated from other components by a plus or minus sign. All of these words—substitute, evaluate, and term—should be added to their glossaries.

After students have discovered the expression and evaluated it by substituting values for m, ask them to determine how they could use the expression to answer the question on the transparency, “What mile marker will you be at when you have traveled 84 miles?” Give students at least a full minute to think before asking for a volunteer to respond. Finally, emphasize the relationship between the expression and the pattern represented by the problem by asking students to respond to three prompts:

D a y s 6 – 8

Tips for Teachers

Terminus is a Latin word that means “boundary” or “something bounded.” In an algebraic expression, a term is a mathematical component bounded by a plus or minus sign.

Tips for Teachers

Avoid always using x as the input value and y as the output value. Students often form misconceptions about x and y that are hard to overcome when they begin to learn about inverse functions in later mathematics courses.

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1. Explain how the expression would change if you were to enter the highway at mile marker 15.

2. Use the Interstate scenario to explain what the expression m + 100 could represent.

3. The hair on a human head grows at a rate of about 1 cm each month. In January, Vanessa’s hair measures 24 cm long. Find an expression that models the length of Vanessa’s hair in the months that follow. Explain how the expression models the situation.

Employ the Think-Pair-Share strategy (Lyman, 1981) for this part of the lesson. After students list their ideas in response to the prompts, they should share those ideas with a partner. After sharing, students should report their responses to the class. Think-Pair-Share removes the pressure on a student of being put on the spot to share ideas that are only his or hers. Furthermore, because it provides students time to think about a problem before sharing, it also allows them the opportunity to collaborate to solve problems in their own ways.

To determine whether students understand the relationship between expressions and real-world patterns, and to wrap up Day 7, pose one final scenario before assigning homework:

A friend gave you $15 for your birthday. You decided to save the money and add $1 to the amount every day afterward. How much money did you save after d days?

As students work to solve the problem, they should note their problem-solving processes: identify the independent and dependent variables, make a table of values, sketch a graph, and find an expression to model the scenario. While they work, circulate around the room with your clipboard to assess their work and to provide assistance. To challenge those students who demonstrate full understanding, ask them to find the amount of money they would have saved by their next birthday.

Finally, assign the Day 8 homework (p. D-8). This assignment not only provides more opportunities for students to write expressions that model real-world situations, but also asks them to find real-world situations that model given expressions. In the latter case, students will be required to discover patterns in nature that fit certain constraints, a process that reinforces understanding of what each part of an expression represents. Encourage students to seek help if they have trouble with the homework.

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Tips for Teachers

If students need additional practice, set up several learning stations around the room, each station describing a real-world situation that can be modeled by the expression x + c, where c is a constant. Have the students work in groups to solve each problem.

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Days 9–10 Algebraic expressions are further explored as ways to

represent situations.


Math journals* Remy’s After-School Job (p. E-2) Overhead projector* Class notebooks* Matinee Mania Club (pp. E-3–E-4) Clipboard* (one for the teacher) Chalkboard* Compare and Contrast (p. E-5) Compare and Contrast Scoring Guide (p. E-6)


As you greet students at the door, direct their attention to the board where the scenario in Figure 3 should be written. The scenario serves as both a warm-up for the day and a misconception check. In their journals, students should write a plus sign if the expression accurately represents the amount that Remy earns; if it does not accurately represent his earnings, they should write a minus sign. Students should then work to justify their reasoning.

As students are writing their justifications, check their understanding of the warm-up by scanning the symbols they have written in their journals. This quick check will allow you to determine whether the students are comfortable enough with the material from Days 6–8 to move immediately into a new concept, whether additional time should be spent learning the material, or whether the concepts should be revisited while introducing the new material. Ask a volunteer to explain to the class why the expression does not model the situation. If many students still have trouble understanding algebraic representation, encourage them to share their misunderstandings or questions. Sharing should not only help them clarify what they are struggling with, but also invite others who may have the same or similar problems to speak up. Some students may not feel comfortable enough to share their misunderstandings; however, their journal entries will reveal their thoughts and give you an opportunity to respond to them privately. Therefore, ask students to submit their journals and homework assignments. Respond to journal entries and return the journals as soon as possible.

Use Remy’s After-School Job transparency (p. E-2) to introduce the representation of expressions involving multiplication. Assign students to work in small groups, following a similar process as before: they should

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Remy earns $8 an hour at his after-school job. Does the expression h + 8 represent the amount of money Remy earns for h hours of work? Why or why not?

Figure 3

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identify the independent and dependent variables, fill in a table, create a graph, and then write an expression that represents the solution to the problem. In addition, allow time for groups to write a justification for how they know the expression models the situation. Students will likely suggest “h × 8” for the expression, so write it on the transparency. Then ask students why this representation might be confusing. They should realize that the multiplication sign could be misinterpreted as another variable, x. Explain that multiplication between numbers in algebra is commonly represented by “1 · 8” or “1(8)”. When multiplying a number and a variable, the number is placed first and there is no operation sign at all: 8h. As you discuss notation, take the opportunity to introduce the commutative properties of multiplication and addition and define the terms factor (a part of an expression being multiplied by something else) and coefficient (the numerical factor in a term, or the number in front of the variable). Ask students to explain why “h times 8” means the same as “8 times h” and to decide whether the same is true for addition, subtraction, and division. Repeated exposure to a topic helps students retain the information; therefore, the commutative properties of multiplication and addition will be discussed again. Allow students time to add the terms factor and coefficient to their glossaries with their definitions.

Returning to the transparency, help students connect the expression and the scenario by asking them to think of questions that could be answered using the expression. For example, “If Remy works for 20 hours during a certain week, how much will he earn that week?” Ask volunteers to share their questions. Then ask students to think-pair-share each of the following statements:

1. Explain how the expression would change if Remy earned $10 an hour.

2. Use the after-school job scenario to explain what the expression 15h represents.

3. Suppose you walk at an average speed of 3 mph. Find an expression that represents the distance you can walk over a period of hours at this rate. Explain how the expression models the situation.

4. Think of a real-world quantity that could be modeled by the expression 8h. Explain how the expression fits your situation.

When students have demonstrated that they are comfortable with the concepts, assign them to groups of three by asking them to identify their favorite movie genre (comedy, drama, horror, etc.). Not only does this grouping strategy give you additional insight into your students’ lives, it also introduces the next activity. Distribute Part I of the Matinee Mania Club worksheet (pp. E-3–E-4). As students work in their groups to outline a process for solving the problem, they should also test their ideas by finding the independent and dependent variables, completing the table, creating a graph, and finding an expression that models the situation. As students work, listen to their conversations, with clipboard in hand, and gauge their understanding of algebraic expressions. When groups have finished, ask for volunteers to share their answers. (Make sure that students are aware that not all groups will have the same input and output values in the chart.) Ask the volunteers how they know their answers are correct. If some groups have 8 + 8m for their expression and others have 8m + 8, take the opportunity to review the commutative properties of addition and multiplication. Lead a discussion of the importance of the correct order of operations. Building upon the

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discussion, ask students to suggest ways to remember which operation to perform first when simplifying an expression such as 6 + 3(4 – 1)2. Write their ideas on the board and help them clarify their thinking.

Preview the concept of combining like terms by asking students if the expression 8m + 8 is equivalent to the expression 16m. Allow students to work in their groups for several minutes to test conjectures and formulate a strategy to justify their responses. Then have the groups reveal their justifications. Combining like terms will be explored further in the unit, but connecting it to the real-world context of the Matinee Mania Club worksheet will help students understand that 8m + 8 and 16m are not equivalent.

Groups should next complete Part II of the worksheet. After groups have had time to respond to each statement, facilitate a student-centered class discussion as you call on students to share their answers. After a student shares a response, call on the other students to rephrase, to agree or disagree, and to explain. In this way, rather than relying on you to supply the correct answers, students practice reasoning to come up with a correct response and to justify that response. This approach may take more time than a traditional lecture, but students will have greater understanding and retention of the material.

As a wrap-up activity, have each group share the real-world problem they came up with for Statement 5, and ask the rest of the class to provide the expression that models it. The group responsible for creating the real-world problem will act as the facilitator, calling on volunteers to share their expressions and explanations.

For homework, ask students to complete the writing assignment described on the Compare and Contrast homework (p. E-5). The assignment is a constructed-response prompt like those students will see on standardized exams. It further illustrates the relationship between patterns of data and algebraic expressions. Help students understand how they will be scored by asking them to identify from the directions and the rubric what is important (e.g., correctly using new terminology). An accurate response requires a description of the rate of increase of the output as it relates to the expression as well as a description of the “starting value” of the output (the value of the output with an input of zero) as it relates to the expression. The Compare and Contrast Scoring Guide (p. E-6) is for your use when reviewing students’ responses.

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Tips for Teachers

Operations are ordered in a logical manner based on level of complexity. Because addition and subtraction are inverses, they occupy the same place in the hierarchy. Consider the natural numbers: since multiplication is repeated addition, it is higher on the hierarchy. It occupies the same place as its inverse, division. Because powers are repeated multiplication, they occupy the next level of the hierarchy, along with their inverses, roots. Partitioning symbols—such as parentheses, brackets, braces, and the vinculum (the horizontal line, most frequently used with the radical sign)—are used if there is a need to interrupt the natural order.

While a mnemonic such as “Please Excuse My Dear Aunt Sally” (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction) may be used to remember the order of operations, it is important that students also understand the logical reasoning behind the order of operations so that they can make sense of their work and avoid misconceptions the mnemonic may cause. For instance, students may mistakenly assume that addition comes before subtraction because it is first in the mnemonic, when in fact they are of the same order. Another good reason to know which operations come first are the limitations of technology: not all calculators adhere to the correct order.

Tips for Teachers

At this point in the course, students are not yet making careful distinctions between the various number systems that constitute the real number system (natural numbers, integers, rational and irrational numbers). Number systems will be discussed in detail in Unit 2. Bear in mind, however, that even in the real number system, certain operations are undefined: division by zero, square roots of negative numbers, and so forth. The only set of numbers that is closed under all the operations in the hierarchy (with the understanding that division by zero is not allowed) is the complex number system, which is studied in Algebra II.

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Day 11 Additional representations of expressions are introduced.


Comparing Graphs of Expressions (p. F-2) Overhead projector* Clipboard* (one for the teacher) Class notebooks* Population Estimates (p. F-3) Day 11 (p. F-4) Day 11 Key (p. F-5) Math journals*


Warm up by asking students to match the graphs to the expressions on the Comparing Graphs of Expressions transparency (p. F-2), which are from the Compare and Contrast homework. When they have finished, ask volunteers to share their responses. Encourage students to take the lead in the discussion by asking questions that invite wider participation: “What do the rest of you think?” “Would anyone else like to add to that response?” Finally, have students submit their homework for grading.

Next, in order to reinforce the concept that expressions model real-world situations, return to Walter the Walker. Ask students to find an expression that represents the number of units Walter the Walker walks each day during any week after he began walking. While students think-pair-share their ideas, circulate around the room with your clipboard and informally assess their problem-solving strategies, their responses, and their strategies for testing ideas and checking responses. When students share their expressions with the class, facilitate a student-centered discussion of their strategies and solutions. If students come up with an expression other than 2x + 2 to represent the number of units walked each day during week x, help them see the relationship between the expression and the perimeter of the rectangular path that Walter walks by labeling a diagram with the length (1) and width (x) of the rectangle. The perimeter consists of two xs and two 1s, or 2x + 2. Students should also begin to understand that algebraic expressions can be written in various equivalent forms: for example, 2x + 2 is equivalent to 2(x + 1).

The Population Estimates transparency (p. F-3) will further reinforce students’ understanding of the use of expressions to model real-world data. Have students discuss in small groups how to represent the population data with an expression. As groups work, they should also explain their reasoning in writing, taking note of any questions about the data or problems with the process. While they work, ask questions to prompt their thinking: “How is this data similar to other data that we have seen before?” “How is it different?” “What are we assuming?” Some students may notice that the amount of population increase is inconsistent over time. Encourage students to use the strategies they have been practicing throughout the unit to find an expression that best fits the data. Challenge groups who find an accurate expression by asking them to think of questions that could be answered using the expression.

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When all groups have finished, have volunteers share their expressions, observations, and questions with the class. The data itself may raise questions about population growth. Estimates of population growth are theories based upon assumptions about birth and death rates that are not altogether agreed upon. The data in the table suggests an almost linear growth, but it is an extremely small snapshot of population growth in its entirety. Population growth can be much faster than linear (exponential, hyperbolic, hyperexponential), but can be slowed by disease epidemics (such as AIDS in developing countries) or declining birthrates (as in much of Europe today). Encourage interested students to research more about population growth models and projections. All students should come to realize that, in the Population Estimates scenario, an expression can be used to approximate the increase from year to year, but to use the expression to estimate population in future years requires us to assume that the pattern remains consistent over time, which is not a safe assumption. While this discussion may stray from a strictly algebraic focus, it will again remind students of the application of mathematics to the real world and emphasize mathematics as a field open to new developments.

Finally, assign the Day 11 homework (p. F-4). The population data prompt should have reinforced students’ knowledge of graphs and charts and should have helped them further understand the connections between tables, graphs, and expressions. The homework invites students to see algebraic expressions as generalized arithmetic.

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Days 12–14 The properties of real numbers are used to introduce students

to simplifying algebraic expressions.


Crossnumber Puzzle (p. G-2) Crossnumber Puzzle Key (p. G-3) Clipboard* (one for the teacher) Class notebooks* Nickels, dimes, and pennies* Chalkboard* Adding and Subtracting Like Terms (p. G-4) Adding and Subtracting Like Terms Key (p. G-5) Graph paper* Uncooked spaghetti* Whiteboards* Markers* Simplifying Expressions (pp. G-6–G-7) Simplifying Expressions Key (p. G-8) Math journal*


After collecting the Day 11 homework, divide the class into groups of 3 or 4. They will explore the properties of real numbers. Each group will need one pencil and a Crossnumber Puzzle worksheet (p. G-2). Explain to students that each numerical expression can be evaluated mentally and without a calculator by simplifying it first. In order to encourage everyone to participate, students should work together to discuss how to simplify each expression to make it easier to evaluate. Then, each recorder should jot down notes on the back of the sheet to help the group remember how the expressions were simplified. While students work, circulate throughout the room, informally assessing their understanding. Because the crossnumber puzzle also serves as a review of addition, subtraction, multiplication, and division of both positive and negative integers, be sure to offer assistance and determine students’ proficiency with the calculations. On your clipboard, take note of the concepts students are struggling with and review those concepts before continuing.

When groups finish the puzzle, ask them to share the different strategies they used to simplify the expressions. At first, rephrase students’ answers to connect each strategy to its respective property of real numbers (i.e., commutative properties of addition and multiplication, associative properties of addition and multiplication, additive and multiplicative inverses, and identity properties of addition and multiplication). After a while, prompt students to connect the correct mathematical language to the strategies they used. At the end of the discussion, explain that these properties are also used to simplify algebraic expressions.

Using a concrete illustration, reintroduce students to the concept of addition and subtraction of like terms. On a table or desk, place several groups of coins:

D a y s 1 2 – 1 4

Tips for Teachers

Consistently using the names of the properties in their entirety will help students understand each property and its appropriate use. For instance, naming the commutative property in conjunction with its appropriate operation connects the commutative property to addition and multiplication only and emphasizes that there is a distinction between the two. Direct students to write these properties in their glossaries; at the same time, post them in the room.

26

2 nickels 3 dimes 4 pennies 3 nickels 4 dimes

Ask each student to summarize the collection as simply as possible—that is, 5 nickels, 7 dimes, and 4 pennies. Instruct them to write their summaries on individual whiteboards or on paper. As they finish, ask students to share their summaries with a partner before initiating a class discussion. As soon as you are certain students grasp the concept of like terms, write the expression 2n + 3d + 4p + 3n + 4d on the board and ask each student to simplify it. Quickly, have students share their ideas and then discuss the results. The concept of like terms can be further examined by asking students to determine the total value of the coins. The entire exercise works because it moves from concrete examples to abstract representations, a strategy that grounds the educational process in the real world.

To provide another concrete exercise that addresses adding and subtracting like terms, distribute the Adding and Subtracting Like Terms activity (p. G-4). Divide the class into pairs of students. Each pair will need graph paper and one piece of uncooked spaghetti. Have students break the pasta into three pieces of unequal lengths. Then, define the shortest piece as 1 unit, the medium-length piece as a unit of length x, and the longest piece as a unit of length y. Students will use the pasta pieces to complete the activity. When pairs have finished the activity, ask them to share their answers with the class and explain their reasoning.

Finally, give students practice simplifying algebraic expressions in real-world scenarios. Students should be given the opportunity to explore for themselves how a change in the scenario changes the algebraic expression. This will allow them to own their understanding, as opposed to being introduced to a collection of facts to be memorized.

Write the following scenario on the board to introduce combining like terms in a real-world context:

One day during his regular walk Walter decides to visit a friend. Walter departs his planned loop and walks 4 blocks to the north. After leaving his friend’s, Walter walks back to the point of departure to complete his planned loop. If this walk occurs during week w, how many units will Walter write on his calendar?

After asking students to think and work on the problem independently, allow them to pair with a partner to discuss the scenario. If they need additional help, allow pairs to join with other pairs to get additional ideas or help. Circulate around the room with clipboard in hand and listen for problem-solving strategies. Encourage students to make and test conjectures. Some may discover the expression 2w + 10; others may find (2w + 2) + 8. Ask them to reconcile their responses. Advocates for each expression should explain why they think it is correct. As usual, students should take the lead in the discussion. Intervene when they are stuck, but do no more than prompt them to identify the next step as opposed to telling them what to do. Students should realize quickly that 2w + 10 is equivalent to (2w + 2) + 8. If they neglect to verify the expressions’ equivalence by substituting a known value for w in both expressions and evaluating them, remind them to do so. (For instance, ask

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27

them to figure out how many units Walter would walk if he visits his friend during week 4.) If students wonder why 8 is added only to the constant term and not the coefficient of 2w, encourage them to conjecture by evaluating different similar expressions. The original pattern had an increase of 2 units per week, expressed as 2x. Because the scenario only adds 8 units to the total number of units walked and because it does not change the rate of increase, the 8 can only be added to the constant in the expression. Revisit combining like terms by asking a volunteer to remind the class of the definition of term and asking students which terms in the expression (2w + 2) + 8 they would consider “like.” Before moving on, have them identify and explain the real number property used in the scenario (the associative property of addition).

Provide another example to illustrate the concept of combining like terms in real-world problems. In the plumber problem from Part II of the Matinee Mania Club worksheet, a plumber charges $30 for a service call and $50 per hour for each hour that he works. Suppose you have a coupon for $10 off. Have students test conjectures for an expression that models how much you would pay to hire the plumber for h hours and explain their reasoning. Again, listen and take note of students’ strategies and solutions. After they have tried the problem and shared their responses, call on a student with a correct answer to explain why the expression is 50h – 10 and not (50 – 10)h. That is, if you were to subtract 10 from the 50, it would be equivalent to subtracting $10 per hour for every hour that he was there, not $10 total.

For students who still struggle to understand the concept of adding like terms, write one more problem on the board:

On Sundays the same plumber charges an additional $25 for each hour worked (time-and-a-half pay). Write an expression to model the total amount of money he charges for working on a Sunday.

Again, students should complete a Think-Pair-Share. In this example, the plumber is charging the original rate plus an additional $25 per hour, which can be expressed (50h + 30) + 25h, which is equivalent to $75 per hour plus the $30 service charge, or 75h + 30. The problem can also be solved by multiplying the hourly rate by 1.5, 1.5(50h) + 30. In order to reinforce the relationship between the problem and the change in the expression, encourage students to describe the plumber’s rate verbally and to supply the algebraic expression. Ask students to point out and explain the real number properties being used (the associative and commutative properties of addition). In addition, ask them to explain how combining like terms helped to simplify this algebraic expression.

Expand on the previous scenario by asking students to write an expression to model how much the same plumber would charge on holidays, when he charges double time. In the previous scenario, many students may have discovered the correct expression by adding an additional $25 per hour to the original expression; in this scenario, most students will likely multiply the hourly rate by 2. This simple twist on the same problem emphasizes the relationship between combining like terms and multiplying an algebraic term by a number. Encourage students to explain in words that this relationship exists because multiplication by a natural number is merely repeated addition of the same quantity.

Discussing these scenarios requires students to shift from concrete real-world examples to the abstract nature of algebra. Students should be ready to

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28

make the connection between repeated addition of a variable and multiplication by a natural number (e.g., x + x + x = 3x). Draw a 3-column, 9-row table on the board. Leaving three blank rows at the top of the table, place an x in the left column, an equal sign in the center column, and 1x in the right column of the fourth row. Remind students of the identity property of multiplication and ask them to explain how it applies to the x in the table. In the third row, just above the x in the table, write x + x in the first column and = in the second column. Ask students to create their own table in their notes and to fill in the far right column of the third row with 2x. As students fill in the remaining rows using the same pattern, assist as you are needed. Next, direct students’ attention to the blank row below x; ask them to look for a pattern to fill in the row. As before, remind students of the inverse property of addition (x – x = 0x = 0) and ask them to explain its application to the remaining entries in the table. Allow students to continue filling in the table according to the same pattern. (See Table 1.) When students have finished, facilitate a class discussion that checks their understanding of the concept of like terms and of the properties of real numbers.

As a wrap-up activity, provide each student with a whiteboard and a marker to practice simplifying algebraic expressions. (If whiteboards are not

available, have students use paper and pencil instead.) The expressions that students simplify should become increasingly difficult at a rate that corresponds to students’ progress so far. Simpler problems require combining like terms with addition and subtraction of positive and negative integers, such as –6d + 7 + 8d – 12. Multiplication of a monomial and an integer, such as –3c(–4), is more difficult, and problems adhering to the correct order of operations, such as 4e + 2(6), are more difficult still. At this point, the most difficult problems combine all of the above skills, as in the expression 3a + 6(–2a) – a. Give students ownership of the activity by adding variables to the problems students created on Day 1. Students should write each simplified expression on the whiteboards and, when asked to do so, reveal them in unison. Because students need experiences in which they all help one another and are accountable for their own learning and understanding (Johnson, Johnson, & Holubec, 1993), students themselves should reconcile discrepancies by volunteering justifications for their responses while other students confirm or refute them. This practice will allow you to identify students who still struggle with simple computations and those who still

D a y s 1 2 – 1 4

x + x + x + x = 4x

x + x + x = 3x

x + x = 2x

x = 1x

0x = 0

–x = –1x

(–x) + (–x) = –2x

(–x) + (–x) + (–x) = –3x

(–x) + (–x) + (–x) + (–x) = –4x

Table 1

29

struggle with the new concepts. These students may need additional help outside of class.

Finally, assign the Simplifying Expressions homework (pp. G-6–G-7). Demonstrate the amount of work students should show on assignments by solving a problem similar to those on the homework. Question 8 deals with the card trick you performed on Day 4. Students should begin to understand how the trick works.

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30

Days 15–16 The distributive property of multiplication over addition is introduced

using a real-world example. Students also learn the symbolic manipulation associated with multiplying variables.


Math journals* Uncooked spaghetti* Graph paper* Walter’s New Plan (p. H-2) Overhead projector* Whiteboards* Markers* Distributing and More (p. H-3) Distributing and More Key (p. H-4)


Day 15

Collect students’ homework for grading. As a warm-up, review combining like terms and introduce the distributive property of multiplication over addition by discussing the following scenario:

After finishing his walk one day, Walter decides that he would like to repeat it. Write an expression that represents the number of units Walter will record on his calendar.

Ask students to work independently. In their journals they should write the problem’s solution and a short explanation of the process they used in developing it. When finished, ask students to compare their answers in groups of three. Groups should then share their responses with the class. The process many students will take begins with (2w + 2) + (2w + 2). By combining like terms, the expression simplifies to 4w + 4. Other students may share (verbally or symbolically) that the expression 2w + 2 must be multiplied by two. Therefore, ask a volunteer to write it on the board, using parentheses to indicate that the entire quantity is multiplied by two: 2(2w + 2). Encourage students to substitute several variables into all three expressions to test their equivalency. Because students already know that (2w + 2) + (2w + 2) simplifies to 4w + 4, ask them to explain how to simplify 2(2w + 2) to yield 4w + 4. In the process, they are explaining the distributive property of multiplication over addition in their own words.

Provide another scenario to illustrate the distributive property:

Upon returning from a business trip to Japan, Maria gives her nephews Japanese coins according to their ages. The coins are worth c cents each. If Robert receives 15 coins and Ivan receives 12 coins, what is the total value of the money they receive?

Have students explain verbally the process they would use to solve the problem. One way to solve it is to add up the total number of coins and

D a y s 1 5 – 1 6

31

multiply that number by c to determine the total value. Pointing out the use of parentheses to indicate that addition is performed first, encourage students to write an expression for this process that adheres to the correct order of operations: (15 + 12)c or c(15 + 12). Another way to solve the problem is to multiply the number of coins each nephew receives by c to determine the value of their coins and then to add the two values together: (15c + 12c). By emphasizing both processes as legitimate ways to solve the problem, students should begin to see how the distributive property of multiplication over addition relates to the order of operations. In both expressions, like terms combine to form the expression 27c.

The distributive property of multiplication over addition can also be represented geometrically. Give each student a piece of uncooked spaghetti and ask them to break the pasta into three pieces of unequal lengths, defining, as before, the shortest piece as 1 unit, the medium-length piece as x units, and the longest piece as y units. Each student will use the lengths of spaghetti to draw a rectangle on graph paper with side lengths 2 and (4x + 3). The rectangle should be divided according to its measures, with four xs and three 1s along the top and bottom, and two 1s on the sides (see Figure 4). Adding together the areas of each individual rectangle produces the expression 8x + 6; calculating the sums of the areas of each individual rectangle in each row separately and combining produces 2(4x + 3). The equivalence of the two expressions illustrates the distributive property.

To introduce multiplication of variables, show the transparency Walter’s New Plan (p. H-2) and ask students to work in groups of three to solve the problem. As you have done before, while students work, listen to and take note of their problem-solving strategies and their solutions. After students have finished, invite them to share those strategies and solutions. Meanwhile, add new strategies to the class list begun on Day 5. Because the number of blocks Walter records on his calendar is equal to the area of the square with side lengths equal to the week, during week 10 the area will be 102 or 100, and during week 100 the area will be 1002 or 10,000. Likewise, during week x, the area will be x². Encourage the class to create a table of values along with a graph of the scenario. Ask students to contrast the table, the graph, and the geometric model to others they have already encountered. They should recognize that the output values of the table do not increase the same amount for each consecutive input; the graph is not linear; and the geometric model is a square of equal sides (x) instead of a rectangle with sides of lengths x and 1.

At this point, shift the discussion from concrete, real-world problems to abstract mathematics. Assign a few problems for students to solve that address the relationship between division and multiplication by the reciprocal. For

D a y s 1 5 – 1 6

Tips for Teachers

The Walter the Walker problems may raise questions about whether someone could in fact walk the distances implied by the problem. If the students stray from a strictly algebraic discussion, encourage them to ground their problem-solving in the real world by determining the reasonableness of their solutions. For example, invite students to assign Walter a walking speed that they think is reasonable, and to estimate the size of the blocks to determine how many weeks Walter could continue his route before he could no longer complete a walk in a reasonable time. Allowing students to pose questions and discuss options fosters their curiosity and maintains their interest in the real-world applications of mathematics.

1

1

x x x x 1 1 1

Figure 4

32

instance, ask students to arrange the answers to the following problems in order, from least to greatest:

The number of pieces of pizza each of 4 persons get when they divide 16 pieces of pizza equally

The number of nickels needed to make 80 cents

8(0.05)

The number of dimes needed to make 40 cents

Ask volunteers to share their responses and justifications for the problems that produce the same values. By analyzing those problems with the same answers,

students will gain insight into the relationship between division and multiplication by the reciprocal.

Create another equivalency table (See Table 1), this time for powers of variables and reciprocals. Draw a 3-column, 7-row table on the board. (See Table 2.) Leaving two blank rows at the top of the table, place an x in the left column, an equal sign in the center column, and x1 in the right column of the third row. Encourage students to copy the table in their notes and to independently fill in the rows

above x = x1. As usual, take notes as you circulate through the room, and offer assistance as needed. Stop to check for understanding of those rows before moving on. The problems from the previous exercise should help students to see in the left column that

x ¸ x = x · = 1.

Then, take it a step further: demonstrate that

1 ¸ x = 1 · .

1 164

1805

805

1 xx x

1 1x x

D a y s 1 5 – 1 6

x · x · x = x3

x · x = x2

x = x1

1 = x0

1x = x–1

1x x = x–2

1x x x = x–3

Table 2

Tips for Teachers

If necessary, scaffold instruction by starting with a table raising numbers to various powers before creating a table using variables.

33

D a y s 1 5 – 1 6

While students will not be working with negative exponents at this point in the school year, introducing the concepts now will lay a foundation for that future work. Take time to compare and contrast addition and multiplication of variables so that students can clearly see the difference in the symbolic notation of each operation. For example, ask students to simplify both x² + x² and x² · x². Especially when exponents are involved, students often get confused between the two representations.

Once students understand multiplication of terms with one variable, illustrate multiplication of terms with more than one variable by building on the Adding and Subtracting Like Terms activity from Days 12–14. Introduce another type of geometric model. Again, students will need spaghetti with lengths of 1 unit, x units, and y units. Have students create 3-space rectangular solids with the following volumes: x3, y3, x2y, and xy2. Geometric representations should illustrate for students the concept of unlike terms involving 2 variables (see Figure 5). In order for two terms to be like terms, each term must contain the same variable(s) and corresponding variables must be raised to the same power.

Then have students evaluate each of the four expressions (x3, y3, x2y, and xy2) for several values of x and y, including negative values, to show again how they are not like terms. Once students understand why these are not like terms, again compare and contrast addition and multiplication of terms with more than one variable. For example, ask students to simplify two different expressions such as 2ab2 + 3a2b and 2ab2 · 3a2b. Because the terms are not like terms, the first expression cannot be simplified. Students should apply the rules of multiplication from Table 2 to simplify the second expression.

Return to the whiteboards to assess, as on Day 14, every student’s level of understanding of the concepts introduced in the lesson. Ask students to simplify expressions that can be simplified by the distributive property of multiplication over addition, such as 6(3 – 2a). Quiz them on how to multiply monomials such as 3cd(–2d 2). As students become more comfortable, combine the two concepts, as in the expression –7g2(–3fg + 9). If students’ answers differ, again have volunteers justify their answers for the rest of the class to confirm or refute. As before, supply increasingly difficult problems or ask students to create their own. Not only will students gain practice simplifying expressions, but also they will learn from each other and review all of the concepts studied in the unit. Meanwhile, the process allows you to identify students who are struggling and may need additional assistance.

Figure 5

x x

x

x

y

x

x

y

y

y

y

y

34

Finally, assign the Distributing and More homework (p. H-3) to give additional practice with multiplication of variables and the distributive property of multiplication over addition. Question 3 returns to Your Age trick from Day 4. Students may begin to see how algebra reveals the trick.

D a y s 1 5 – 1 6

35

Days 17–19 Students are introduced to the unit assessment, a poster of a real-world

problem, created by students, that represents the concepts presented in the unit.


Graphing calculator or computer* Overhead projector* Real-World Problem Poster (p. I-2) Poster board*


Day 17

Collect homework and take time to address students’ questions about it. For Part II, if a graphing calculator (or a computer) with projection capabilities is available, project the graphs (from Prompts 2 and 3 of the Day 11 homework) on an overhead screen to help students recognize the multiple representations of expressions.

As a unit assessment, students will work in groups of 4 to create their own real-world problems that can be represented with algebraic expressions. Students will write a prompt that describes the problem and seeks to answer a specific question related to the problem. Students will then represent the problem on poster board along with several mathematical models including an expression, a table of values, and a graph. They should provide a response to the question and a verbal justification for the response. Finally, students should demonstrate an understanding of the properties of algebraic expressions presented in the unit (e.g., the distributive property of multiplication over addition, adding like terms) either through the problem itself or through variations of the problem. Share the Real-World Problem Poster prompt (p. I-2) with students so they are aware of the necessary elements of the poster.

The first day of the project should be devoted to writing the problem. If students have trouble coming up with a problem, encourage them to refer to problems presented in the unit, the real-life event that suggests a pattern from the Find the Pattern homework (Day 5), or the problems that they created for homework on Day 8. All students in the group should be actively engaged in this process, discussing ideas for problems that can be represented with an algebraic expression, language that makes the problem clear and comprehensible, and variations of the problem that illustrate the properties of algebraic expressions. As students discuss possibilities, circulate around the room, listen to ideas, and offer assistance if needed. Make sure that every group has written a problem by the end of the first day. Collect the problems to review students’ progress and provide feedback.

Day 18

On the second day, students should spend their time developing each of the mathematical models. Each student in the group should be responsible for a portion of the design, although students must collaborate to be sure that the parts fit together. For example, one student could be responsible for the problem and its variations, another for a table of values, another for creating a

D a y s 1 7 – 1 9

36

graph, and the fourth student could be responsible for the response and justification for the answer. As students create each of their portions, they should check their work with that of the other members of the group. Each student should record his or her own work.

Day 19

By the third day, after all of the parts of the problem have been created and students are sure that the parts fit together, they should create a poster to illustrate and display each part. At the end of the day, students should submit their poster and all of their written work for grading.

D a y s 1 7 – 1 9

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ENHANCING STUDENT LEARNING

Selected Course Standards

C.1. Foundations a. Evaluate and simplify expressions requiring addition, subtraction,

multiplication, and division with and without grouping symbols b. Translate real-world problems into expressions using variables to

represent values

Unit Extension

Suggested Teaching Strategies/Procedures


Scratch paper* Markers* Transparency sheets* Wood blocks*


Extend activities done on Day 2 with a more challenging problem: For a regional mathematics competition, a team of students was asked to design a series of awards stands for future winners. The group’s designs were simple. They had learned that in mathematics, as in life, simple solutions are often the best. The award stands were built by connecting two-foot cubes in a stairs pattern, as shown in Figure 6.

A stand for 1 winner consisted of one cube and 5 painted faces; a stand for 2 winners consisted of 3 cubes and 12 painted faces; a stand for 3 winners consisted of 6 cubes, and so on. They began to wonder whether there was a pattern in the number of painted faces. They also challenged each other to be the first to model the number of cubes needed for any number of place winners.

For this investigation, materials managers should pick up scratch paper, markers, transparency sheets, and wood blocks.

E n h a n c i n g S t u d e n t L e a r n i n g

One winner Two winners Three winners

Figure 6

38

Reteaching

Suggested Teaching Strategies/Procedures


Markers* Poker chips* Chip Toss (p. J-2) Standard deck of playing cards*


Real-life examples can help students having trouble understanding the rules for adding, subtracting, multiplying, and dividing positive and negative integers. Addition and subtraction of positive and negative numbers can be modeled through football (yards gained and yards lost), temperature, or checkbook registers. Multiplication can be explained using automatic debits from the bank. For instance, suppose a person has a $125 car payment automatically debited from her bank account on the first day of each month. To find the change in her bank balance due to car payments after 6 months, she could multiply 6 · (–125) to obtain –750 dollars. However, suppose she is unable to work for 3 months and the bank must suspend deductions. This suspension is represented by (–3) · (–125), which equals $375, or the amount that remains in her account during the 3-month break in payments.

Another activity that reinforces the concept of addition and subtraction of positive and negative integers is the chip toss game. With a permanent marker, mark white poker chips with a plus sign on one side and a minus sign on the other. The key to the lesson is to demonstrate to the students that (+1) + (–1) = 0. The game can be completed in one class session.

In the chip toss game, students are put into pairs. One student tosses his or her chips while the other records the score on the Chip Toss worksheet (p. J-2). The score is obtained by pairing the opposites to make zeros and counting what is left. For example, if the student tossed 6 negative chips and 4 positive chips, then the score for the team would be –2 because there were two more negative chips than positive in the toss, or (–6) + (4) = –2. Students then switch roles and repeat the toss. Each toss is considered one round, and a game is complete after 15 rounds. When all games are complete, discuss and record several rounds on the board. Emphasize the process of adding the numbers together. Meanwhile, encourage students to look for patterns to form a rule that always predicts the outcome correctly. For example, a round’s final score can be found by subtracting the absolute value of the numbers and assigning to the result the sign of the number having greater absolute value.

Extend the chip toss game by subtracting, multiplying, or dividing using the chips. Treat multiplication the same as addition. The number of positive chips should be multiplied by the number of negative chips to find the score. For the noncommutative operations, subtraction and division, students should calculate both values and choose the greater of the two, taking care not to divide by zero. (In the case of subtraction, this produces the absolute value of the difference between the two numbers. In the case of division, since one value is negative and the other is positive, the result will always be a negative value with the smaller digit in the numerator.) In subtraction, emphasize both that a pair of opposites yields zero and that subtraction means “take away.”


39

When subtracting a positive from a negative number (or a negative from a positive number), show students that zeros can be added in the form of pairs before taking away. For example, to evaluate (–6) – (+2) students begin with 6 negative chips. In order to take away 2 positive chips, you must first add 2 zeros (a zero is a pair of positive and negative chips) before +2 can be taken away.

Another activity that can be used to teach operations between positive and negative integers requires a regular deck of cards. Deal a hand of 4 cards to each student or pair of students. Red cards are negative and black cards are positive. Set the rules regarding the numerical value of the face cards before you begin. The game is for students to determine the numerical value of their hand. Depending on the rules you set, either the highest or the lowest hand wins.


Reflecting on Classroom Practice Were students encouraged and invited to collaborate, share

ideas, and learn from each other? Did students show evidence of growth?

40

Bibliography

References

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B i b l i o g r a p h y

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McIntosh, M. E., & Draper, R. J. (2001). Using learning logs in mathematics: Writing to learn. Mathematics Teacher, 94(7), 554-557.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: The National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: The National Council of Teachers of Mathematics.

North Central Regional Educational Laboratory. (n.d.). Effective teaching strategies. Retrieved from http://www.ncrel.org/sdrs/areas/issues/envrnmnt/drugfree/sa3effec.htm

O’Connor, J. J. & Robertson, E. F. (2006). The MacTutor history of mathematics archive. Available from http://www-groups.dcs.st-and.ac.uk/~history/

OnLineMathLearning.com. (n.d.). Algebra math quotes. Retrieved from http://www.onlinemathlearning.com/algebra-math-quotes.html

Rothstein, A., & Rothstein, E. (2007). Writing and mathematics: An exponential combination. Principal Leadership, 7 (5), 21–25.

Rowe, M. (1986). Wait time: Slowing down may be a way of speeding up. Journal of Teacher Education, 37(1), 43-50.

University of California Berkeley Graduate Division. (2007). Acronym Method. Retrieved from http://gsi.berkeley.edu/resources/discussion/ropes.html

U.S. Census Bureau. (2007). Total midyear population for the world: 1950–2050. Available from U.S. Census Bureau website: http://www.census.gov/ipc/www/idb/worldpop.html

Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1). Retrieved from http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Williams, N. B., & Wynne, B. D. (2000). Journal writing in the mathematics classroom: A beginner’s approach. Mathematics Teacher, 93, 132–135.

Wong, H. K, & Wong, R. T. (2004). The first days of school: How to be an effective teacher. Mountain View, CA: Harry K. Wong Publications.

Wright, D. L. (1989). The most important day: Starting well. Teaching at UNL, 11(1), 1–3.

B i b l i o g r a p h y

Contents Unit Assignments and Assessments ................................................................................................................A-2

Example

Unit Assignments and Assessments ................................................................................................................A-3 Record Keeping

A p p e n d i x A : R e c o r d K e e p i n g

Unit Assignments and Assessments Example

Name: _____________________________ Period: Unit 1: The Value of a Variable

Directions to the teacher: Prior to starting the unit, complete the log on the next page according to the example below and distribute it to students as an organizational tool.

Day Assigned Assignment/Assessment

In Class

Home-work

Date Due

Feedback (Completed/

Points)

Days 1–3 Introduction X Sum of a Series X Algebra Gallery X Writing and Arithmetic X

Days 4–5 Journal entry X

Walter the Walker problem X

Find the Pattern X Days 6–8 Types of Graphs and Tables X Interstate Driving X Day 8 homework X Days 9–10 Journal entry X Remy’s After-School Job X Matinee Mania Club X Compare and Contrast X

Day 11 Comparing Graphs of Expressions X

Walter the Walker expression X Population Estimates X Day 11 homework X Days 12–14 Crossnumber Puzzle X

Adding and Subtracting Like Terms X

In-class problems X

Whiteboarding X Simplifying Expressions X

Days 15–16 In-class problems X

Whiteboarding X

Distributing and More X Days 17–19 Assessment Project X X

3-2-1-Assessement

E x a m p l e A - 2

Unit Assignments and Assessments

Name: _____________________________ Period: Unit 1: The Value of a Variable

R e c o r d K e e p i n g A - 3

Day Assigned Assignment/Assessment

In Class

Home-work

Date Due

Feedback (Completed/

Points)

Contents Introduction......................................................................................................................................................B-2

Activity

Writing and Arithmetic....................................................................................................................................B-3 Homework

Writing and Arithmetic Key ............................................................................................................................B-4 Key

A p p e n d i x B : D a y s 1 – 3 B - 1

Introduction

Name: ______________________________ Period: ______________________ Date: ________________

Directions: First, fold the index card in half lengthwise to form a table tent. Write your first and last name on the outside of the tent so it can be seen. On the inside of the tent, provide contact information for your parent or guardian (e.g., home or work phone number, e-mail address) and list three interesting facts about yourself.

Second, in the space below, create a math problem that, when solved, yields a number that is significant to you such as your lucky number, the number of your favorite athlete, or the number of songs in your MP3 player. Be creative and try to make it complicated! Use several operations (e.g., addition, division, use of exponents) and grouping symbols (e.g., parentheses, brackets, fraction bars).

A c t i v i t y B - 2

Writing and Arithmetic

Name: ______________________________ Period: ______________________ Date: ________________

Part I

Directions: On the back, respond to one of the following questions.

1. What is your earliest math memory, and how has it shaped you?

2. What has been your greatest math success, and why?

3. What is your least favorite math topic, and why?

Part II

Directions: Showing all of your work, simplify the following expressions.

1.

2.

3. 10[12 – (0.2 + 11.8)]

4. A baseball player’s batting average is calculated by dividing the number of hits he makes by his number of times at bat. In his first year at Florida State University, Ryne Malone had a .330 batting average with 67 hits. How many times did he appear at bat? Explain how you know your answer is correct.

( )24 2 8

6 2 4 2⋅ −

− + ⋅

( )219 3 2 25

− +

H o m e w o r k B - 3

Writing and Arithmetic Key

Part I 1–3. Answers will vary.

Part II 1. –2

2. 6

3. 0

4. 203 times. Explanations will vary.

K e y B - 4

Contents Quotes ..............................................................................................................................................................C-2

Worksheet

The Function of Algebra..................................................................................................................................C-3 Background Information

Card Trick........................................................................................................................................................C-5 Activity

Your Age .........................................................................................................................................................C-6 Activity

Glossary ...........................................................................................................................................................C-7 Worksheet

Class Journal Feedback....................................................................................................................................C-8 Rubric

Find the Pattern................................................................................................................................................C-9 Homework

Find the Pattern Key ......................................................................................................................................C-10 Key

A p p e n d i x C : D a y s 4 – 5 C - 1

Quotes

Directions: As you read each of the quotes below, use the annotation symbols to help with your reading. At the bottom of the page, write your thoughts about what you have read. Include questions you have or observations you make. No one else will see what you write, but you are welcome to share during class discussion.

“Number rules the universe.” —Pythagoras

“One person’s constant is another person’s variable.” —Susan Gerhart

“I was x years old in the year x2.” —Augustus De Morgan (when asked about his age)

“Men are liars. We’ll lie about lying if we have to. I’m an algebra liar. I figure two good lies make a positive.” —Tim Allen

“We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra.”

—William Kingdon Clifford

“The most powerful single idea in mathematics is the notion of a variable.” —A. K. Dewdney

“There is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.”

—Nikolai Lobachevsky

Underline any words that you do not understand.

Circle any phrases or sentences that are unclear to you.

Place stars next to anything that interests you.

Annotation Symbols:

W o r k s h e e t C - 2

Quotations compiled from the MacTutor History of Mathematics Archive and OnlineMathLearning.com.

The Function of Algebra

Algebra is the branch of mathematics in which symbols, usually letters, are used to represent numbers. It is a language used to study the structure of mathematical systems and to express relationships and generalize from patterns. Across the span of its 4,000-year history, algebra has had important applications in the fields of art, science, engineering, and economics. Such application of mathematics to solve problems in other fields is called applied mathematics. Algebra is also used for reasons other than application, such as to advance theoretical understanding. This is the motivation underlying pure mathematics. Some pure mathematicians study algebra for its intrinsic beauty as a mathematical “art.”

It is often easier for students to understand the immediate usefulness of mathematics as it relates to other fields. Mathematical Moments, a website developed by the American Mathematical Society (2007), provides numerous examples of the application of math to other fields. However, it is essential to understand that applied and pure mathematics are not separate fields. The theoretical premises derived from pure mathematics often inform real-world applications, sometimes years or even centuries later, that culminate in inventions and new ways of thinking and behaving. In this manner, math functions as a driving force of culture.

Many who engage in and study pure mathematics do so because they perceive an inherent beauty in mathematics; they are motivated by the intellectual challenge that the study of math presents. This is aptly expressed by G. H. Hardy, a prominent pure mathematician who, in A Mathematician’s Apology (1992, p. 150), said, “I have never done anything ‘useful.’ No discovery of mine has made, or is likely to make, directly or indirectly, for good or

ill, the least difference to the amenity of the world.”

Hardy’s statement illustrates that the meaning of pure mathematics relative to application is not often apparent at the time of its conception. Much of Hardy’s work has ultimately found practical application in other branches of science. For example, his work in number theory has been useful in cryptology, the scientific study of codes and ciphers. Another example of the later application of a work of pure mathematics can be found on the labels of merchandise in most retail stores. The last digit in a universal product code (UPC) is called a “check digit,” the function of which is to detect and prevent data entry errors. By computing a mathematical algorithm on other digits in the sequence and comparing it to the check digit, a user can verify that the combination of digits is valid. These algorithms are based on principles of number theory, once considered to be the purest of the pure mathematics. Different check-digit schemes have found use in the publishing industry through the use of ISBN numbers; the auto industry utilizes check digits in Vehicle Identification Numbers; and the financial industry employs check digits in credit card numbers. The first check-digit scheme to detect all single digit errors and transpositions was devised as recently as 1969, although the underlying concepts have been studied for centuries.

Eugene Wigner, in “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” (1960, ¶4), addresses the profundity of applications that have derived from the discoveries of pure mathematics. Wigner says “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.” His remark could be considered an expression of what has come to be known

B a c k g r o u n d I n f o r m a t i o n C - 3

as the “mathematical realism” perspective, which holds that mathematical entities are real and exist independently of the human mind. According to this conception, mathematics exists independently and therefore must be discovered, rather than invented. An alternate perspective, “constructivism,” maintains that mathematical entities do not exist and must be constructed to be real. In this view, humans create—but do not discover—mathematics. Thus, constructivists would argue that the usefulness of mathematics in the application to other fields is such because humans created mathematics to explain the world around them.

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, by George Lakoff and Rafael E. Nunez (2000), is an important recent work written in the constructivist perspective. The authors address the cognitive structure of mathematical ideas and tasks from a linguistic premise that maintains that the language of mathematics facilitates and enables understanding and meaning of phenomena.

The human experience with mathematics is perpetually unfolding and is c h a r a c t e r i z e d b y a d y n a m i c interrelationship between theory and application across time. As new applications for theory are conceived, new problems for exploration through the lens of pure mathematics are generated. Theory feeds application which, in turn, feeds theory. There are currently seven “Millenium Prize Problems” for which the Clay Mathematics Institute has offered $1 million for the first correct proof. These problems are considered “among the most important unsolved problems in branches of pure mathematics” (Clay Mathematics Instititute, 2007). Clearly, those who would claim that all mathematics has already been generated are incorrect.

References Clay Mathematics Institute. (2007).

Millennium problems. Retrieved from http://www. claymath. org/millennium

Hardy, G. H. (1992). A mathematician’s apology. Cambridge, England: Cambridge University Press. (Original work published 1940.)

Lakoff, G., & Nunez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic Books.

American Mathematical Society. (2007). Mathematical moments. Retrieved from http:// w w w . a m s . o r g /mathmoments

Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1). Retrieved August 17, 2007, from http://www.dartmouth.edu/~matc/ MathDrama/reading/Wigner.html

T h e F u n c t i o n o f A l g e b r a

B a c k g r o u n d I n f o r m a t i o n C - 4

Card Trick

Directions: Perform a card trick to interest students in applications of mathematics by following the instructions below.

1. Remove the jokers from a full deck of cards. Explain that this card trick is based on algebra. Allow students to choose a volunteer to shuffle the deck and choose a secret number between 1 and 20. Ask the volunteer to share the number with the other members of the class, either by writing it down on paper or by using hand signals, and to remove that many cards from the deck. Meanwhile, turn your back so the students know you cannot see the secret number.

2. When every student knows the number, explain that you are going to show students the first 20 cards of the remainder of the deck. Students should memorize the card in the position of the secret number. (Warn them, however, to avoid giving it away through facial expressions!) This card is the “magic” card that you will attempt to guess by “reading” the cards. Start with the top card in the deck, and lay each of the first 20 cards face up in a pile on a desk.

3. Next, ask the class to call out a random number that is smaller than the number of cards left in your hand. Choose one of the numbers that students call out and count out that many cards; placing them face up on top of the pile of 20. For effect, claim that it is important that you do not know how many cards are in the deck because that would give away the secret number. Meanwhile, pick up the pile of face up cards, flip them over, and place them beneath the remaining cards in your hand.

4. Now, begin passing cards face down from the top of the deck from one hand to the other, pretending that you are “reading” them without looking at the faces. Don’t let the students see the faces, either. In reality, you are counting the cards, starting with the number 20 + the random number + 1 as the first card. For instance, if the random number were 16, you would start counting the first card as 37. The card that you count as 52 is the magic card that students were to memorize. When you get to that card, distract the students by stating that the trick is not working and you are unable to get a “feel” for the magic card. While saying this, bring the bottom remainder of the deck up and place it on top of the counted cards, so that the magic card is now on the bottom of the deck. You might be inclined to give away the magic card at this point, but if you do, students may too easily figure out the trick.

5. Explain that since you are unable to get a feel for the cards, you will instead try to read the students minds. Meanwhile, tap the deck on a desk to square them off and peek at the bottom card, making sure that students don’t see you peek. Encourage students to think hard about the magic card. Meanwhile, cut the deck and shuffle it. The remainder of the trick will be a show to demonstrate your mind-reading abilities. State that you sense a particular student is thinking of the color of the card; while another student is sending signals as to the value. Finally, reveal the identity of the magic card.

How it works: There are 52 cards in the deck. The secret number of cards, n, are removed from the deck. This leaves 52 – n cards in the deck. Then another 20 cards are removed and set aside, leaving 52 – n – 20 cards, or 32 – n cards. Finally, let m represent the random number students chose. Those cards are also removed from the remainder of the deck and placed under the 20 cards set aside. This leaves 32 – n – m cards that are placed on top of the cards that were set aside. Thus, the magic card is in the position (32 – n – m) + n, or 32 – m. Since you know m, you know the position of the given card. Since there are 52 cards, you begin counting at 52 – (32 – m) + 1, or 20 + m + 1.

A c t i v i t y C - 5

Your Age

Directions: Perform the following trick for students to get them interested in applications of mathematics. This trick will work as written in 2007. For each subsequent year, add one to the numbers in Step 5.

1. Pick the number of times you would like to go out to eat a week (more than one but less than 10).

2. Multiply this number by 2 (just to be bold).

3. Add 5.

4. Multiply it by 50.

5. If you have already had your birthday this year add 1757; if you haven’t, add 1756.

6. Now subtract the 4-digit year that you were born.

You should have a 3-digit number. The first digit is how many times you want to go out to restaurants in a week. The next 2 numbers are your age!

How it works: Let x be the number in Step 1. The steps below correlate with Steps 2 through 6 above.

2. 2x

3. 2x + 5

4. 50(2x + 5)

5. 50(2x + 5) + 1757 (or 1756)

6. 50(2x + 5) + 1757 (or 1756) – birth year

Distributing 50 in the expression in Step 6 yields 100x + 250 + 1757 (or 1756) – birth age. Ignoring 100x for now and combining the like terms 250 and 1757 (or 1756) yields 2007 (or 2006) – birth year. Therefore, when the birth year is subtracted from the CURRENT year (for those who have already had a birthday) or from LAST year (for those who haven’t), the age is revealed. Since the original number is multiplied by 100 in the expression 100x, that digit appears in the hundreds place in the final answer.

A c t i v i t y C - 6

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W o r k s h e e t C - 7

Class Journal Feedback

Name: ______________________________ Period: ______________________ Date: ________________

Directions: Use this rubric to give feedback to students on their journal entries.

Effort Completion: Your journal includes all assigned work.

Legibility: Your journal is readable, presentable, and coherent.

Use: Your journal is used to think, learn, practice, understand.

Improvement: Your journal shows overall improvements since last time.

Writing Fluency: You write with ease about a range of subjects.

Development: Your writing includes examples, details, or quotations when appropriate.

Understanding Thoroughness: Your entries show you are trying to fully understand or communicate an idea through writing or

illustrations.

Insight: Your writing demonstrates deep understanding of ideas and goes beyond the obvious.

Requirements Format: All entries clearly list in the margin the date and title of the entry (e.g., Day 1 homework).

Organization: Entries appear in chronological sequence or as otherwise assigned.

Notes:

Adapted from Jim Burke, Writing Reminders. ©2003 by Jim Burke.

R u b r i c C - 8

Find the Pattern

Name: ______________________________ Period: ______________________ Date: ________________

Part I

Directions: Look for a pattern to answer the problems. Explain the problem-solving skills you used to find the pattern.

1. Between 1904 and 1934, hexagonal city planning caught the attention of various planners, engineers, and architects. They saw it as a replacement for more traditional rectangular street grids (Ben-Joseph & Gordon, 2000). Suppose Walter lives in a city with a hexagonal layout and begins walking around the perimeter of the hexagonal block in which he lives. Each consecutive week, he adds one more hexagon (in a row) to his route (as pictured below). How many units will Walter walk in the 10th week? The 100th week? (Each unit represents one side of a hexagon.)

2. Suppose you are painting the surfaces of a cube. How many surfaces will you paint? Now, glue 2 cubes together (sharing a face). If none of the faces are painted, how many square faces will you paint? Glue 3 cubes together in a row. How many square surfaces will you paint? Figure out how many surfaces you would paint if 10 cubes were glued together. Then figure out how many surfaces you would paint if 100 cubes were glued together.

Part II

Directions: Respond to the following prompt in your journal.

Find an event in life that suggests a numerical pattern, such as Walter the Walker’s walking plan. You may be able to find ideas in the newspaper, on the Internet, or in your everyday tasks. Describe how you know that the event fits a pattern.

References Ben-Joseph, E., & Gordon, D. (2000). Hexagonal planning in theory and practice. Journal of Urban Design, 5(3),

235–265. Retrieved from http://web.mit.edu/ebj/www/Hexagonal.pdf

Start

H o m e w o r k C - 9

Find the Pattern Key

Part I 1. In the 10th week, he will walk 42 units; the 100th week, he will walk 402 units.

2. One cube has 6 surfaces; 10 cubes have 42 surfaces; and 100 cubes have 402 surfaces.

Part II Responses will vary.

K e y C - 1 0

Contents Types of Graphs and Tables ............................................................................................................................D-2

Handout

Interstate Driving .............................................................................................................................................D-7 Transparency

Day 8................................................................................................................................................................D-8 Homework

Day 8 Key ........................................................................................................................................................D-9 Key

A p p e n d i x D : D a y s 6 – 8 D - 1

Types of Graphs and Tables

Mathematical data can be visually represented in various ways depending on which aspects of the data one wishes to display or compare. The following pages contain several different types of graphs, all of which portray some part of the data found in Table 1.

Table 1 organizes information about a Social Studies class in which pairs of students were assigned to research and write a report about a world nation.

T y p e s o f G r a p h s a n d T a b l e s

H a n d o u t D - 2

Student Team Country Continent

Area (sq. miles) Population

Day Turned In

Doreen & Connor Argentina South America 1,073,500 40,134,425 Monday Monique & Izzy Chile South America 292,260 16,928,970 Wednesday David & Peter Ethiopia Africa 426,371 73,918,505 Monday Sydney & David France Europe 212,900 62,793,432 Thursday Susan & Jo Germany Europe 137,847 82,689,210 Thursday Russell & Dave Iceland Europe 40,000 319,246 Friday Randall & Elizabeth Japan Asia 145,898 127,380,000 Wednesday Lori & Florence Kazakhstan Asia 1,052,100 15,776,492 Friday Susan & Joan Madagascar Africa 226,658 20,653,556 Tuesday Earl & Rodney Mali Africa 478,841 14,517,176 Friday Diego & Dan Mexico North America 756,066 107,550,697 Friday Teague & James Monaco Europe 1 33,000 Thursday Holly-Lynn & Lee Pakistan Asia 310,400 169,393,000 Tuesday Robin & Lisa Singapore Asia 274 4,987,600 Monday Susan & Will Slovakia Europe 18,932 5,424,057 Thursday Henry & Roxy Switzerland Europe 15,940 7,761,800 Monday Jane & Marie Togo Africa 21,925 6,619,000 Thursday J.J. & Mike Turkey Asia 302,535 72,561,312 Tuesday Elsa & Beth New Zealand Australia 103,483 4,369,439 Thursday

Table 1: Fifth Period Nation Reports

Bar Graphs Bar graphs are used to

compare two or more variables. The length of the bar indicates the value of the variable.

Circle Graphs A circle graph (also

called a pie graph) is used to compare parts of a whole. Circle graphs can represent some data in a way that is easy to interpret. However, because it can be difficult to compare sections of relatively the same size, care should be taken when using circle graphs.


H a n d o u t D - 3

0

1

2

3

4

5

6

7

Af rica Ant arct ica Asia Aust ralia Europe Nort h America Sout h America

Number of Nations by Continent

Africa

Antarctica

Asia

Australia

Europe

North America

South America

Nations by Continent


H a n d o u t D - 4

Histograms Histograms are bar graphs where each bar represents a subrange of an ordered variable, instead of a single value. The range of the ordered variable is divided into equal, continuous, nonoverlapping segments, and the histogram indicates how many data fall into each subrange.

0

20

40

60

80

100

120

140

160

180

Argenti

naChil

e

Ethiopia

France

German

y

Icelan

dJa

pan

Kazak

hstan

Madaga

scar Mali

Mexico

Monaco

Pakist

an

Singap

ore

Slovakia

Switzerl

and

Togo

Turkey

New Zea

land

(in m

illio

ns)

Bar Graph: Population per Nation

0

1

2

3

4

5

6

7

8

9

0-14.9 15-29.9 30-44.9 45-59.9 60-74.9 75-89.9 90-104.9 105-119.9 120-134.9 135-149.9 150-164.9 165-179.9

P op. ( i n mi l l i ons)

Histogram: Population Ranges

Num

ber o

f Nat

ions


H a n d o u t D - 5

Line Graphs Line graphs are used to

compare ordered variables. In standard convention, the independent variable is labeled along the x-axis (horizontal) and the dependent variable along the y-axis (vertical). Frequently, line graphs are used to show changes over a period of time.

0

1

2

3

4

5

6

Monday Tuesday Wednesday Thursday Friday

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f rep

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sub

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Bar Graph: Submission Date

0

1

2

3

4

5

6

Monday Tuesday Wednesday Thur sday Fr iday

Num

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s S

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Line Graph—Submission Date of Reports

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5

10

15

20

25

M onday Tuesday Wednesday Thursday Friday

Total

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Sub

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Number of Submissions by Date


H a n d o u t D - 6

Scatterplot A scatterplot illustrates the sort of relationship that exist between two dependent, quantitiative aspects of the data. While scatterplots might reveal a correlation between the aspects (large values of one variable might occur in the same data that demonstrate large values in another variable), this correlation does not imply causation.

Area vs. Population

0

20,000,000

40,000,000

60,000,000

80,000,000

100,000,000

120,000,000

140,000,000

160,000,000

180,000,000

0 200,000 400,000 600,000 800,000 1,000,000 1,200,000

Area (sq. miles)

Popu

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Interstate Driving

You are traveling east along an interstate highway. When you enter the highway, you are at mile marker 8. Mile marker signs always start numbering at the western border of a state for east/west interstate highways. What mile marker will you be at when you have traveled 84 miles?

Independent variable:

Dependent variable:

Expression:

T r a n s p a r e n c y D - 7

Input Output

Input

Out

put

Day 8

Name: ______________________________ Period: ______________________ Date: ________________

Part I

Directions: Look for a pattern to answer the problems. Explain the problem-solving skills you used to find the pattern.

1. The Mobile Transporter is a rail line approximately 300 ft long located outside of the International Space Station. It is designed to transport heavy space equipment at the rate of 1 inch per second. Suppose the Mobile Transporter begins transporting a piece of equipment 30 ft from the north end of the track toward the south end. Write an expression that models the distance (in inches) from the north end of the track after s seconds. (Hint: you will need to convert feet to inches.)

2. Suppose you have a credit card that offers airline travel rewards each time you make purchases on the card. The credit card company will reward you with one travel point for every dollar spent. These travel points can then be cashed in for airline tickets. When you signed up for the card, they rewarded you with 300 points automatically. Write an expression that models the number of points you will have after spending d dollars.

Part II

Directions: Respond to the following prompt in your journal.

Find your own real-world situation that can be modeled by the expression m + 8. Explain how the expression fits your situation.

H o m e w o r k D - 8

Day 8 Key

Part I 1. s + 30

2. d + 300

Part II Responses will vary.

K e y D - 9

Contents Remy’s After-School Job................................................................................................................................. E-2

Transparency

Matinee Mania Club ........................................................................................................................................ E-3 Worksheet

Compare and Contrast ..................................................................................................................................... E-5 Homework

Compare and Contrast Scoring Guide ............................................................................................................. E-6 Scoring Guide

A p p e n d i x E : D a y s 9 – 1 0 E - 1

Remy’s After-School Job

Remy earns $8 an hour at his after-school job as a cashier. How much money does Remy earn for h hours of work?

Independent variable:

Dependent variable:

Expression:

T r a n s p a r e n c y E - 2

Input Output

Input

Out

put

Matinee Mania Club

Names: _______________________________________________________________________________

Period: _______________________ Date: _________________

Part I

Directions: Determine how to answer the question below. Find the independent and the dependent variables. Then, choose values for the independent variable and fill in the table and the graph. Find an expression to represent the cost of m matinees with a membership. Write a justification for how you know your expression models the situation. Finally, think of a different question that could be answered using the expression.

Your family enjoys going to the matinee every Saturday afternoon. The movie theater has a new promotion called the Matinee Mania Club. If you buy a one-time membership to the club for $8, your entire family can go to the matinee for only $8 each Saturday. With a club membership, how much will it cost altogether to go to m matinees?

Independent variable: ___________________

Dependent variable: ____________________

Expression: ___________________________

Justification:

Question:

W o r k s h e e t E - 3

Input Output

Input

Out

put

Part II

Directions: Discuss the statements below. Choose one person to record your explanations. Each member of your group should be ready to share what you have discussed.

1. Explain how the expression would change if the one-time cost of membership were $10.

2. Explain how the original expression would change if the cost of attending each matinee were $6.

3. Use the matinee example to explain what the expression 9m + 7 could represent.

4. Suppose you hire a plumber to fix a leaky pipe in your basement. The plumber charges $30 for the service call and $50 per hour for each hour that he works. Write an expression to represent the cost of hiring the plumber for a number of hours, and explain how the expression represents the situation.

5. Think of a real-world situation that can be modeled with an expression that has two different arithmetic operations. Other students in the class will be asked to write an expression based on your situation, so be prepared to explain your response.

6. Scenario: Raj is hired as a tutor over the summer. He is paid $50 up front and $10 for each tutoring session. Does the expression 50v + 10 model the scenario? Why or why not?

7. In your own words, write the steps necessary to solve a problem like question 4 above.

M a t i n e e M a n i a C l u b

W o r k s h e e t E - 4

Compare and Contrast

Name: ______________________________ Period: ______________________ Date: ________________

Directions: Below are input/output tables for three expressions. Write a paragraph that compares and contrasts the patterns you identify. Describe what each number in the expression (coefficient or constant) tells you about each pattern. Your work will be scored according to the rubric.

Score Description

3 A response at this level includes an accurate description of how both parts of the expression (coefficient and constant) relate to the patterns in the tables. A student with writing at this level accurately compares and contrasts the patterns and expressions given but does not describe how the parts of similar expressions relate to similar patterns. The student correctly uses some of the new terminology associated with the unit.

2 A response at this level includes an accurate description either of how the coefficient relates to the patterns in the tables or of how the constant relates to the patterns in the tables. A student with writing at this level accurately compares and contrasts some aspects of the patterns with respect to the expressions given but fails to interpret others. The student may or may not correctly use the new terminology associated with the unit.

1 A response at this level fails to include an accurate description of how either part of the expression (coefficient or constant) relates to the patterns in the table. A student with writing at this level does not accurately compare or contrast any aspects of the patterns with respect to the expressions given. The student may or may not correctly use the new terminology associated with the unit.

0 A response at this level is not scorable. The response is off-topic, hostile, blank, or otherwise not scorable.

4 A response at this level includes an accurate description of how both parts of the expression (coefficient and constant) relate to the patterns in the tables. A student with writing at this level not only accurately compares and contrasts the patterns and expressions given, but also is able to describe how the parts of similar expressions relate to similar patterns. The student correctly uses most or all of the new terminology associated with the unit.

Rubric

H o m e w o r k E - 5

Table 1 Table 2 Input x Output x + 8

0 8

1 9

2 10

3 11

4 12

5 13

Input x Output 8x

0 0

1 8

2 16

3 24

4 32

5 40

Input x Output 8x + 8

0 8

1 16

2 24

3 32

4 40

5 48

Table 3

Compare and Contrast Scoring Guide

Scoring Criteria A correct response should include the following points:

Evidence of understanding that the coefficient 8 in the expressions 8x and 8x + 8 relates to an increase in 8 output units for every one increase in input units

Evidence of understanding that the constant 8 in x + 8 and 8x + 8 relates to a starting output value of 8 at input value zero

Evidence of a general understanding that the coefficient of the variable determines the rate of output and that the constant term determines the starting value of pattern at input zero

Evidence of the proper use of terminology associated with the unit (e.g., variable, expression, coefficient)

S c o r i n g G u i d e E - 6

Contents Comparing Graphs of Expressions .................................................................................................................. F-2

Transparency

Population Estimates ....................................................................................................................................... F-3 Transparency

Day 11.............................................................................................................................................................. F-4 Homework

Day 11 Key ...................................................................................................................................................... F-5 Key

A p p e n d i x F : D a y 1 1 F - 1

Comparing Graphs of Expressions

Determine which graph represents each expression from the Compare and Contrast writing assignment (x + 8, 8x, or 8x + 8). Be prepared to explain your reasoning.

T r a n s p a r e n c y F - 2

Input

Out

put

1.

0 5 10 15

5

10

15

Input

Out

put

2.

0 5 10 15

5

10

15

Input

Out

put

3.

0 5 10 15

5

10

15

Total Midyear Population Estimates for the World

2007 6,602,274,812 2008 6,679,532,264 2009 6,757,062,760 2010 6,834,934,808 2011 6,913,263,561

Year 1: Year 2: Year 3: Year 4: Year 5: Year 6: 2012 6,991,796,293 Year 7: 2013 7,070,248,599 Year 8: 2014 7,148,361,369 Year 9: 2015 7,225,918,656 Year 10: 2016 7,302,863,077

From

U.S

. Cen

sus B

urea

u, “

Tota

l Mid

year

Pop

ulat

ion

for t

he w

orld

: 195

0–20

50.”

200

7.

Population Estimates

The table below shows world population estimates for the years 2007 through 2016. Round each estimate to the nearest 10 million in order to approximate a pattern. Using the pattern, write an expression that models the population estimate in Year Y. Write a paragraph that explains your reasoning.

T r a n s p a r e n c y F - 3

Day 11

Name: ______________________________ Period: ______________________ Date: ________________

Directions: In your journal, respond to the following prompt.

1. In class, you were presented with a table of population estimates for the years 2007 through 2016. You then wrote an expression to model the data. Both the table and the expression are mathematical models of the real world. In what other ways could the data be modeled? Think of as many different representations as you can and explain what each representation can be used to show.

2. Order the algebraic expressions below from least to greatest for all POSITIVE values of x. Substitute 2 or 3 values for x to test your order.

3. Order the following algebraic expressions from least to greatest for ALL values of x. Be sure to substitute some negative values of x when you test the expressions.

x – 5

5x

4x – 1

5x – x

4x + x

x – 4

4x – 4

4x

4x – 4

4(x – 1)

4x + 4

4x

4(x + 1)

4(x – 4)

4(x + 4)

H o m e w o r k F - 4

Day 11 Key

1. Answers will vary.

2.

3. 4(x – 4)

4(x – 1) = 4x – 4

4x

4x + 4 = 4(x + 1)

4(x + 4)

x – 5

x – 4

4x – 4

4x – 1

5x – x = 4x

4x + x = 5x

K e y F - 5

Contents Crossnumber Puzzle ........................................................................................................................................G-2

Worksheet

Crossnumber Puzzle Key.................................................................................................................................G-3 Key

Adding and Subtracting Like Terms................................................................................................................G-4 Activity

Adding and Subtracting Like Terms Key ........................................................................................................G-5 Key

Simplifying Expressions ..................................................................................................................................G-6 Homework

Simplifying Expressions Key ..........................................................................................................................G-8 Key

A p p e n d i x G : D a y s 1 2 – 1 4 G - 1

1. 1 2

3

4 5

6

7

8

9

10

Crossnumber Puzzle

Names: _______________________________________________________________________________

Period: _______________________ Date: _________________

Part I

Directions: Rotate the role of recorder in your group. The recorder is the only member of the group who should have a pen or pencil and his or her job is only to record the answers, not to do any calculations on paper. All calculations must be done in your head. Look for a way to simplify each problem to make the calculations easier. Discuss ways to simplify them with your group and have the recorder jot down notes on the back of the worksheet. These notes will be used during the class discussion to present and compare the strategies used.

Across

1. 853 7719 147+ +

4. 4119 4119 406, 217− + +

6.

1 34 4609+ +

7. 32 590 268− + −

8.

150 0

6 1 0 3 4+ +

9.

( )307 118 82+ +

10.

( ) ( )4 5 30, 268 4 5− × + + ×

Down 2.

( )764 2 500× ×

3.

1 917 202 × ×

5.

( ) ( )25 612 4− × × −

8.

14 3333 4× ×

9.

( )32 630 1232 0 512− + − × +

W o r k s h e e t G - 2

Crossnumber Puzzle Key

K e y G - 3

Across 1. 8719

4. 406,217

6. 610

7. 290

8. 30

9. 507

10. 30,268

Down 2. 764,000

3. 9170

5. 61,200

8. 3333

9. 512

Adding and Subtracting Like Terms

Names: _______________________________________________________________________________

Period: _______________________ Date: _________________

Part I

Directions: Each expression in the left column is equal to one expression in the right column. On graph paper, trace the length of each piece of spaghetti needed to find the total length of each expression and determine which lengths match. In the case of subtraction, take away the length being subtracted from the total length. Then, on the worksheet, draw a line to connect the equivalent expressions.

Part II

Directions: Write an expression in the right column that is equal to the expression in the left column. In the space provided, explain why the lengths are equal.

1. 3 a. 3x – x

2. 2x b. 2x + y + x

3. x + 2 c. y + 2 – 1

4. x + y + 1 d. y – x + y

5. y + 1 e. x + 1 + x + x

6. 2y – x f. x + 2y + 1 – y

7. 3x + 1 g. 1 + 1 + 1

8. 3x + y h. 1 + 1 + x

9. 2x + y + 3 a.

A c t i v i t y G - 4

Adding and Subtracting Like Terms Key

1. g

2. a

3. h

4. f

5. c

6. d

7. e

8. b

9. Answers will vary. One example is x + y + 4 + x – 1.

K e y G - 5

Simplifying Expressions

Name: ______________________________ Period: ______________________ Date: ________________

Directions: Justify your responses using full sentences and correct mathematical language.

1. You are working together on an algebra assignment with a friend. A problem asks you to simplify the expression 7x – 6 – 9x + 4. Your friend gets 2x – 10 and you get –2x – 2. Your friend assures you that his is the correct answer because he evaluated both expressions for x = 2 and got the same answer on both sides (see his work in Figure 1). On the back of this sheet, explain to your friend why your answer is the only correct answer.

Directions: Simplify the following expressions. Evaluate both forms of the expression for at least three different values of the variable to show that they are equivalent. Show all of your work.

Directions: Simplify the following expressions.

4. 4w – 3 – 5w

5. 7 – 8b + 16

6. –12a + 3(5a) + 6c – 10

7. d + d – 2d

2. 5x + 7 – 3x 3. 2(5y) + 2y – 18

7x – 6 – 9x + 4 2x – 10

14 – 6 – 18 + 4 4 – 10

8 – 18 + 4 –6

–10 + 4

–6

Figure 1

7 · 2 – 6 – 9 · 2 + 4 2 · 2 – 10

H o m e w o r k G - 6

8. Suppose a card trick that requires a deck of 52 cards begins when a random number (n) between 1 and 20 is chosen and n cards are removed from the deck. Second, from the remainder of the deck, 20 cards are then counted off the top and also removed. Finally, another random number (m) is chosen, and m cards are also removed. Write an expression that tells how many cards remain in the deck after the steps are complete and then simplify the expression.

9. Review your journal responses for Prompts 2 and 3 from the Day 11 homework. Given what you have learned since, as a separate entry revise or add to your previous responses.

S i m p l i f y i n g E x p r e s s i o n s

H o m e w o r k G - 7

Simplifying Expressions Key

1. Correct responses include a) evaluating the friend’s response for a different value to show that they are not equivalent, b) pointing out the correct use of the number properties to prove the correct answer, and c) evaluating one’s own response for several values to demonstrate that it is correct.

2. 2x + 7

3. 12y – 18

4. –w – 3

5. 23 – 8b

6. 3a + 6c – 10

7. 0

8. 52 – n – 20 – m 32 – n – m

K e y G - 8

Contents Walter’s New Plan ...........................................................................................................................................H-2

Transparency

Distributing and More......................................................................................................................................H-3 Homework

Distributing and More Key ..............................................................................................................................H-4 Key

A p p e n d i x H : D a y s 1 5 – 1 6 H - 1

Walter’s New Plan

Walter the Walker, bored with his old plan, decided to begin a new one. In the new plan, Walter will again begin the first week by walking clockwise around the square block on which he lives. Each subsequent week, Walter will walk the length of one additional block in each direction before he returns home. Each week his route encloses consecutively larger square paths (see diagram). On his calendar, Walter will record the number of square blocks within the perimeter of his path. How many square blocks will Walter record on his calendar during Week 10 ? How many will he record during week 100? During week x?

T r a n s p a r e n c y H - 2

Wee

k 1

Wee

k 2

Wee

k 3

Distributing and More

Name: ______________________________ Period: ______________________ Date: ________________

Part I

Directions: Writing in complete sentences, explain your reasoning. Then write and simplify an algebraic expression using appropriate symbols.

1. Gabrielle is designing a bulletin board for the math club. The board is h feet high and l feet long. She would like to find the total length of border trim she will need. Describe 2 methods using multiplication that she could use to find the total length of border trim. Write an expression for each method.

2. Paula visits the $10 Outlet Store to buy school clothes. Every piece of clothing costs $10. Paula buys s shirts, p pairs of pants, and k skirts. Describe 2 methods Paula could use to determine the total amount of money she will spend before taxes. Write an expression for each method.

3. Suppose a number trick that reveals your age begins with a secret number x. This number is then multiplied by 2. Next, 5 is added. The entire expression is multiplied by 50, and finally the number 1757 is added at the end. Write these steps as an algebraic expression and then simplify.

Directions: Simplify each of the following expressions.

4. n(6np)

5. –12g(–2h)

6. 2(2n – 7)

7. –4(–a + 10b)

8. w(7 – w)

9. c(2cd – c2)

10. –6uv(u – v) + 8uv2

Part II Review your journal responses for Prompts 2 and 3 from the Day 11 homework as well as the revisions that you made as a separate entry. Again, as another separate entry, revise or add to your previous responses. This time, write a rationale based on your current thinking for why you ordered each expression as you did.

H o m e w o r k H - 3

Distributing and More Key

1. She could add the height and the length and multiply the sum by two: 2(h + l). She could multiply the height and length of the board by 2 and add them together: 2h + 2l.

2. She could add the number of items together and then multiply the sum by 10: 10(s + p + k). She could multiply the number of each item by 10 and add them together: 10s + 10p + 10k.

3. (2x + 5)50 + 1757 100x + 2007

4. 6n2p

5. 24gh

6. 4n – 14

7. 4a – 40b

8. 7w – w2

9. 2c2d – c3

10. –6u2v + 14uv2

K e y H - 4

Contents Real-World Problem Poster .............................................................................................................................. I-2

Prompt

A p p e n d i x I : D a y s 1 7 – 1 9 I - 1

Real-World Problem Poster

Name: ______________________________ Period: ______________________ Date: ________________

Directions: Create a poster of your real-world problem with the features given below.

Criteria Explanation of Elements Score Real-world problem

The prompt describes the problem in an easily-understandable way and asks a specific related question. The problem and its variations (if any) demonstrate an understanding of the concepts presented in the unit (e.g., adding like terms, the distributive property of multiplication over addition).

Expression(s) The expression includes a clearly-defined independent variable and correctly represents the real-world problem. Additional expressions are included, as necessary, to represent variations of the problem, along with the steps used to simplify the expressions.

Table of values The table displays the values of the dependent variable (outputs) for several different values of the independent variable (inputs).

Graph The graph is correctly labeled and displays all the information in the table of values.

Response to question and justification

TOTAL SCORE:

The response uses the different mathematical models (expression, table, and graph) to answer the specific question and provides a written justification for the response based on the different models.

P r o m p t I - 2

Contents Chip Toss .......................................................................................................................................................... J-2

Worksheet

A p p e n d i x J : E n h a n c i n g S t u d e n t L e a r n i n g J - 1

Chip Toss

Name: ______________________________ Period: ______________________ Date: ________________

Directions: Record the results of each toss in the correct rows below. Use this worksheet to look for patterns that will allow you to form a rule for adding integers with like and unlike signs.

Based on the information recorded above, develop two rules for adding integers, one for when the signs are the same and another for when the signs are different. State the rules in your own words:

When adding two numbers with the same sign:

When adding two numbers with opposite signs:

Toss Number Number of

Positive Chips Number of

Negative Chips Number of

Opposite Pairs Total Value 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

W o r k s h e e t J - 2

Secondary Course Standards A primary course standard

is the central focus of the unit and

is explicitly assessed in an embedded assessment and/or in the summative assessment.

A secondary course standard

is less important to the focus of the unit, but is one that students need to know and use when completing activities for this unit and

may or may not be explicitly assessed by the summative assessment or an embedded assessment.

Course standards considered primary for this unit are listed on pages 1–2. Below is a list of secondary course standards associated with this unit.

Selected Secondary Course Standards

A.1. Skills Acquired by Students in a Previous Course and Refined in This Course a. Set up and solve problems following the correct order of operations

(including proportions, percent, and absolute value) with rational numbers (integers, fractions, decimals)

c. Use rational numbers to demonstrate knowledge of additive and multiplicative inverses

d. Simplify ratios

f. Add, subtract, multiply, and divide rational numbers, including integers, fractions, and decimals, without calculators

A p p e n d i x K : S e c o n d a r y C o u r s e S t a n d a r d s K - 1

Course Standards Measured by Assessments This table represents at a glance how the course standards are employed throughout the entire unit. It identifies

those standards that are explicitly measured by the embedded and unit assessments. The first column lists course standards by a three-character code (e.g., A.1.a.); columns 2–11 (this page), 2–10 (p. L-2), and 2–5 (p. L-3) list the assessments.

Embedded Assessments

Introduction Activity

Sum of a Series

Algebra Gallery

Writing and Arithmetic

Journal Entries

Walter the

Walker Find the Pattern

Types of Graphs

Interstate Driving Day 7

A.1.a. X X

A.1.c. X

A.1.d. X

A.1.f. X X

B.1.a. X X X X X X X

B.1.b. X X X X X X

B.1.c. X X X X X X X

B.1.d. X X X X X X

B.1.e. X X X X

B.1.f. X X X X X X X X

B.1.g.

B.1.h. X X X X X

C.1.a. X X X

C.1.b. X X X X

C.1.c. X

D.2.f. X

G.1.b. X

G.1.c. X X X X X

G.1.h. X X X

Coded Course

Standards

A p p e n d i x L : C o u r s e S t a n d a r d s M e a s u r e d b y A s s e s s m e n t s L - 1

Embedded Assessments Remy’s After-

School Job

Matinee Mania Club

Compare and

Contrast

Comparing Graphs of

Expressions Population Estimates Day 10

Cross-number Puzzle

Adding and Subtracting Like Terms

In-Class Problems

A.1.a. X X

A.1.c. X X

A.1.d. X X

A.1.f. X X

B.1.a. X X X X

B.1.b. X X X X X

B.1.c. X X X X X X X X

B.1.d. X X X X X X

B.1.e. X X

B.1.f. X X X X X X X

B.1.g. X

B.1.h. X

C.1.a. X X X X

C.1.b. X X X X X

C.1.c. X X

D.2.f. X

G.1.b. X

G.1.c. X X X X X X X

G.1.h. X X

Coded Course

Standards

C o u r s e S t a n d a r d s M e a s u r e d b y A s s e s s m e n t s L - 2

Coded Course

Standards

Embedded Assessments Unit

Assessment

Whiteboarding Simplifying

Expressions Distributing and More Assessment Project

A.1.a. X

A.1.c. X

A.1.d. X

A.1.f. X

B.1.a. X X X

B.1.b. X X

B.1.c. X X X

B.1.d. X X X

B.1.e. X X X

B.1.f. X X X

B.1.g. X X

B.1.h. X X

C.1.a. X X X X

C.1.b. X X X X

C.1.c. X X X X

D.2.f. X

G.1.b. X

G.1.c. X

G.1.h. X

C o u r s e S t a n d a r d s M e a s u r e d b y A s s e s s m e n t s L - 3

17119 QC Instructional Unit Covers 2 - Etowah County Schools€¦ · ER.AL1-1.2.1. iii Note QualityCore® Instructional Units illustrate how the rigorous, empirically researched course

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