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foundations

Standards-BasedProject M3 : Mentoring Mathematical Minds incorporates

the National Council of Teachers of Mathematics (NCTM)

Content and Process Standards, Principles and Standards of

School Mathematics (PSSM 2000) and Exemplary Practices in

Gifted Education recommended by the NAGC (National

Association for Gifted Children).

Challenging and EngagingA goal of the authors was to create a challenging and

motivational curriculum, increase math achievement and

improve attitudes toward math in talented and diverse

students.

Project M3 introduces advanced math content focused on

critical and creative problem solving and reasoning. Students

are exposed to standards-based concepts through a variety

of engaging investigations, projects and simulations.

ResearchProject M3 is the result of a five-year collaborative research

effort between a team of national experts in the fields of

mathematics, mathematics education, and gifted education.

Research was funded by the US Department of Education.

Award WinningA number of modules in the Project M3 series have earned the

National Association for Gifted Children’s Curriculum Studies Award.

Thinking like MathematiciansThe Project M3 curriculum is organized to give the student

the opportunity to work and act like real mathematicians.

The “Think Deeply” questions in each unit challenge

students to make sense of mathematics. In each unit,

students explore a simulated real-life problem and use their

Mathematician’s Journals to think, write, and act like

mathematicians to solve the problem.

Mathematical CommunicationRich verbal and written mathematical communication is a key

component in the successful use of Project M3. Mathematical

thinking and learning are deepened and enhanced by the

many activities in which students and teachers talk, write, and

use mathematical language together.

DifferentiationUnderstanding that even students who are gifted

mathematically may come into a unit or topic at different

levels, Project M3 provides differentiation opportunities

through the use of “Hint Cards” and “Think Beyond” cards.

Kits AvailableKits are available for most modules and contain hands-on

manipulatives for activities that make it easy for teachers.

Project M3 : MentoringMathematical Minds is a seriesof 12 curriculum units developedto motivate and challengemathematically talentedelementary students.


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In 1980, the National Council of Teachers of Mathematics

(NCTM) made a bold statement, “The student most neglected in

terms of realizing full potential, is the gifted student of mathematics.”

As test scores indicate, progress since that time has been

slow or nonexistent in this area. It is especially true for

underrepresented students from economically

disadvantaged backgrounds. With funding from the U.S.

Department of Education’s Jacob K. Javits Program,

Project M3 was developed to address this issue.

Project M3 has been a 5-year collaborative research effort of

faculty at the University of Connecticut, Northern Kentucky

University, and Boston University and teachers,

administrators, and students in 11 schools of varying

socioeconomic levels in Connecticut and Kentucky.

A team of national experts in the fields of mathematics,

mathematics education, and gifted education worked

together to create a total of 12 curriculum units of

advanced mathematics.

Development and ResearchThe content units of instruction for Project M3 are based on

the NCTM Content Standards (2000) and include Number,

Algebraic Thinking, Geometry, Measurement, and Data

Analysis and Probability. The NCTM Process Standards (2000)

embedded in these units concentrate on communication,

reasoning, connections, and problem solving. A special focus

emphasizes the development of critical and creative

thinking skills in problem solving.

The curriculum design followed the tenets of The Multiple

Menu Model: A Practical Guide for Developing Differentiated

Curriculum (Renzulli, Leppien, & Hays, 2000) and The Parallel

Curriculum, A Design to Develop High Potential and Challenge

High-Ability Learners (Tomlinson, Kaplan, Renzulli, Purcell,

Leppien & Burns, 2002) published by the National

Association for Gifted Children.

These models adhere to the belief that “most, if not all, learners

should work consistently with concept-focused curriculum, tasks that

call for high level thought, and products that ask students to extend and

use what they learned in meaningful ways” (Tomlinson et al., 2002,

p.13).

Using the Multiple Menu Model and the Curriculum of

Practice from the Parallel Curriculum, the curriculum

encourages students to assume the role of mathematicians

as they develop critical and creative thinking skills in solving

real problems.

Projects are included in the units and used as a way for

students to pursue some of their own interests. Renzulli’s

Enrichment Triad Model (Renzulli, 1977; Gubbins, 1995) is one

of the instructional approaches; students choose a topic to

investigate, receive support and coaching from the teacher,

and produce a product for a real audience.

This combination of the NCTM Content and Process

Standards, an increase in depth and complexity, and

exemplary practices in the field of gifted and talented

curriculum development has created the type of

mathematics that is both truly challenging and enjoyable

for talented math students.

ResultsExtensive field testing went into each Project M3 module

before it was released. This highly acclaimed program has

proven successful in all social economic environments

nationwide and internationally. See page 7 for research

results.

Meeting the Needs of Mathematically Talented Students

To learn more about the authors or toread articles about this award-winning

program, visit www.projectm3.org.

foundations


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materials:Teacher and Student

STAMP

Teacher GuideThe Teacher Guide includes an overview of the unit featuring the

organization of the unit, suggested pacing, materials list, assessments,

and rubrics. It also provides background information on the

mathematics being taught, connections to the National Standards,

the learning environment, mathematical communication, and

differentiating instruction.

Skill assumptions, or what students should know before they begin the

unit, are included in the Teacher Guide.

In effect, the Teacher Guide provides built-in professional development

for the teacher along with easy-to-follow lesson plans.

Hint Cards and Think Beyond Cards Modeled after Japanese pedagogy, these cards are another unique

feature of Project M3 units and encourage students to communicate

about their learning.

There are two types of cards to aid differentiation in the classroom:

Hint Cards to help students move forward in their thinking with a problem

and Think Beyond Cards to challenge students who are ready to move

beyond the lesson concepts.

Project M3 STAMP Another unique feature of Project M3 is the teacher stamp and ink pad.

This stamp is used for teacher feedback in grading student journal work.

It is based on a rubric designed to score students on math concepts,

communication, and vocabulary.


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Student Mathematician’s JournalThe Student Mathematician’s Journal is a unique feature of every unit in

the Project M3 series, encouraging students to communicate in writing. It

includes the student worksheets from each lesson. In these journals, we ask

students to reflect on what they have learned, think deeply about the

mathematics, and write about it.

The focus is on the NCTM Process Standards of Communication and

Reasoning. In effect, they are working and acting like real

mathematicians when they do this.

In each unit of the Project M3 series, students explore a simulated or real-

life problem and use their Mathematician’s Journals to think, write and

act like mathematicians to solve the problem. To this end in each lesson

there are two Think Deeply Questions that focus on the essential

concepts of the lesson and encourage talented students to delve into

more sophisticated mathematics.

ManipulativesManipulatives are a necessary part of creating a successful learning

experience for students. Manipulative kits are available for purchase from

Kendall Hunt for those curriculum units requiring more than common items.

A master list of materials for each unit is included in the Teacher Guide.

Need more room?

Use the back!

THINKDEEPLY

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Mathematician’s Journal

Student Mathematician: Date:

Your Thoughtsand Questions

1. Your best friend missed school but heard that you learned a number trick in math class. You decide to send your friend your set of cards and a note explaining how your number trick works.a. What numbers were on your cards?b. What was your rule?c. Why does your rule work?

Project M3: At the Mall with Algebra

9 Chapter 1: Thinking about Variables and EquationsLesson 2: Number Tricks

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Hands-On!

materials:Teacher and Student


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implementationstrategies

Implementing Project M3

There are a variety of ways to implement Project M3.

Project M3 materials focus on both accelerated and enriched

curriculum and so they can be used as enrichment materials

to enhance instruction as well as replacement units.

Enrichment MaterialsWith their strong focus on critical and creative mathematical

thinking, Project M3 units can be used to enhance the

regular curriculum. Talented students will be taken to new

heights by exploring concepts in depth in unique ways and

with creative twists.

Replacement UnitsThese units were designed to be one to two grade levels

above the regular curriculum. For students who have

mastered a particular grade-level content area, the units are

an opportunity to explore new mathematics at a higher

level commensurate with their abilities.

Variety of Learning SettingsThe Project M3 units were designed to be used in a variety

of settings:

• pull-out programs for talented students,

• high-ability math groups,

• self-contained gifted math classes,

• cluster groups of talented math students,

• after-school math enrichment clubs, and

• summer math enrichment programs,

• gifted magnet school math programs.

‘One of the most impressiveprograms’ in 30 years

Project M3 was examined by independentconsultant Susan Carroll, president of Words andNumbers Research in Torrington, Connecticut.“I’ve evaluated programs for 30 years, and this isone of the most impressive I’ve ever seen.”


Research was conducted on the implementation of 12 units in

11 different schools, nine in Connecticut and two in Kentucky.

The sample consisted of two intervention groups of students

each comprised of approximately 200 talented students

entering third grade, most of whom remained in the project

through fifth grade. Students in this study demonstrated a

significant increase in understanding across all mathematical

concepts in each unit from pre to post-testing.

There was also a comparison group of like-ability students

identified in each of the schools who did not take part in

the Project M3 classes. The Project M3 students consistently

outperformed these students on standardized testing

(Iowa Tests of Basic Skills) and open-response items from

international and national tests (TIMSS and NAEP assessments).

Positive Results with Project M3

RESEARCH

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For the latest research and articles on examples of successfulimplementation, go to www.kendallhunt.com/m3


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Levels 3-4

LEVEL 3Unraveling the Mystery of the MoLi Stone:Place Value and NumerationIn this unit, students explore our

numeration system in depth.Students

discover a stone with unusual markings.

By the end of the unit, students uncover

the mysteries found on this stone by

applying and extending their new

knowledge of place value and other

numeration systems.

Awesome Algebra: Looking for Patternsand GeneralizationsStudents study patterns and

determine how they change, how

they can be extended or repeated,

and how they grow. They develop

generalizations about mathematical

relationships in the patterns.

Students create an Awesome

Algebra board game, including

writing and testing game cards, creating rules, and

designing the board.

What’s the Me in Measurement All About?In this unit on measurement, students

are actively engaged in the

measurement process and connect it

to their own personal worlds.

Students go “In Search of the Yeti,”

“Build Boxes” about themselves, and

put “The Frosting on the Cake” as they

learn about perimeter, area and

volume.

Digging for Data: TheSearch within ResearchStudents explore the world of the

research scientist and learn how

gathering, representing, and

analyzing data are the essence of

good research. Lessons cover

cafeteria surveys, Olympic

competition data, the effects of

exercise on heart rates, and local weather research.

LEVEL 4Factors, Multiples, and Leftovers: LinkingMultiplication andDivisionStudents develop their number

sense with a focus on a deeper

understanding of multiplication

and division. They encounter a range

of different problem situations and

representations and learn about the

relationship between multiplication and division and the

properties associated with these operations. Students

extend their thinking about multiplication to factors and

multiples, and also look at relationships among prime,

composite, square, odd and even numbers. In the final

lesson of the chapter, they write their own training

manual, Strategies for Becoming a Multiplying Magician, to

highlight their new facility with and understanding of

multiplication.

At the Mall with Algebra: Working withVariables and EquationsStudents discover the mathematics

behind number tricks and variable

puzzles. These experiences and

discussions in the unit provide

a rich context for introducing

students to algebraic thinking,

while strengthening their problem

solving and mathematical

communication skills.


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LEVEL 5Treasures from the Attic: ExploringFractionsStudents are introduced to two

children,Tori and Jordan, who uncover

hidden treasures in their

grandparents’attic from a general

store that their great grandparents

used to own.The focus of the entire

unit is on making sense of fractions

rather than on learning algorithms to

perform computations. Fraction concepts include equivalence,

comparing and ordering, and addition and subtraction.

Record Makers and Breakers: UsingAlgebra to Analyze ChangeStudents learn about algebra as a

set of concepts tied to the

representation of relationships

either by words, tables or graphs.

They extend their notion of variable

from a letter in an equation that

represents a number to a more

broad definition, that of a quantity

that varies or changes. Students study constant and

varying rates of change as an intuitive introduction to

slope. They also learn about algebra as a style of

mathematical thinking for formalizing patterns of change.

Funkytown Fun House: Focusing onProportional Reasoningand SimilarityStudents are introduced to similarity

and congruence. The foundational

mathematics behind these concepts

is proportional reasoning. Students

explore ratio as a comparison of two

quantities.

What Are Your Chances?Students begin their exploration of

probability as a measurement of the

likelihood of events. They move

beyond performing simple probability

experiments to an understanding of

experimental and theoretical

probability and the Law of Large

Numbers. Students also experience

what it means for a game to be fair as they create their

own games for a Carnival of Chance.

(LEVEL 4 Continued)

Getting into ShapesIn Getting Into Shapes, students

explore 2- and 3-dimensional

shapes with a focus on their

properties, relationships among

them and spatial visualization. The

reasoning skills that they build upon

in this unit help them to develop an

understanding of more complex

geometric concepts.They learn new, more specialized

vocabulary and learn how to describe properties of shapes

with this terminology allowing for greater clarity and

precision in their explanations.

Analyze This! Representingand Interpreting DataIn this unit, students develop a

deeper understanding of data

analysis. Specifically, they learn

what categorical data is and how to

represent and analyze categorical

data using new, more sophisticated

ways including Venn diagrams and pie graphs. They also

work with continuous data as they learn to construct and

analyze line graphs.

Levels 4-5


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lesson format

Clear and Organized Lesson FormatEach lesson is written in the following format to allow for easy implementation.

Planning Phase• BIG MATHEMATICAL IDEAS

Here is where teachers are provided an overview of

the essential math understandings that students

should come away with in the lesson.

• OBJECTIVES

These list more specific goals of the lesson. Teachers

can align these objectives with their state objectives.

• MATERIALS

Teachers can see at a glance what materials are

needed for the lesson and plan accordingly.

• MATHEMATICAL LANGUAGE

Important math vocabulary is defined here.

Teachers should encourage use of these terms in

class discussions and student writing.

Teaching Phase• INITIATE

This is an introductory learning experience

designed to engage students.

• INVESTIGATE

Here is where students delve into the activity,

exploring the concepts in interesting hands-on

investigations.

• MATHEMATICAL COMMUNICATION

In this section students reflect upon the

mathematical concepts in the investigation using

the Think Deeply Question. They write about them

in their Mathematician’s Journals and discuss them

with a partner, small group and/or the entire class.

• A QUICK LOOK

A brief summary of the lesson outlining the

sequence of instruction is included to provide a

guide for teachers to use during the

implementation of the lesson.

Assessment Phase• CHECK-UPS

At the end of some lessons a brief assessment is

included that can be used to check students’

understanding of unit concepts and serve as an

ongoing review of regular curriculum content.

• UNIT TEST

At the end of each unit there is an assessment that

can be used to check understanding of the key

math concepts in the unit. Some teachers choose to

pretest students as well using this same test to help

in planning appropriate instruction for the unit.

• CONNECTIONS TO THE STANDARDS AND STANDARDIZED

TESTS

This supplemental practice review is based on

common standardized test formats.

• GLOSSARY

Mathematical definitions of vocabulary used in the

unit is located here for easy reference.

Combining the NCTM Content and Process Standards

(2000), Project M3’s emphasis on depth and complexity

of math concepts using exemplary practices in the field

of gifted and talented curriculum development has

created the type of mathematics that is both

challenging and enjoyable for talented math students.


11

The National Council of Teachers of Mathematics (NCTM)

points out that mathematical thinking and learning are

deepened and enhanced by activities in which students

and teachers talk, write and use mathematical symbols

together, while working hard to convey and clarify their

mathematical ideas.

Verbal CommunicationDiscussion is very important in nudging student thinking

forward.Students working on Project M3 units will have ample

opportunities to engage in exploratory talk to help them develop

more elaborate ideas.Partner talk gives students a minute or two

to transform their thoughts into words with the help of their

closest neighbor before returning to whole class discussion.

To help foster students’ appreciation of other students’ variety

of thinking and problem solving styles, Project M 3 suggests

strategies known as “Talk Moves.” These include: Revoicing,

Repeat/Rephrase, Agree/Disagree and Why, Adding On, and

Wait Time.

Written CommunicationThe Student Mathematician’s Journal is a unique feature of

every unit in the Project M3 series, encouraging students to

communicate in writing.

Two “Think Deeply”questions are included in each lesson to

challenge students to make sense of the mathematics.These

are the "heart and soul" of each lesson.The Mathematician’s

“Write On!”sheet is provided with guidelines to encourage

thorough responses, and includes making sure students read

and understand the question, brainstorm ideas, write their

preliminary response, read over the response to make sure

they answered the question, and defend their ideas.

A section within the Teacher Guide,“Student Mathematician’s

Journal: Guides for Teaching and Assessment” helps teachers

work with students on developing their written mathematical

communication.

The Learning Environment

The learning environment within ProjectM3 is one of respectful and engaging

mathematical discussions. Students are

expected to be taught to listen carefully

to what their classmates say and to

carefully consider all ideas and opinions,

as good mathematicians would do.

One way to establish a mathematically

vigorous environment that is respectful

of all its participants is to expect students

to uphold particular behaviors. These

behaviors are listed as “Rights” and

“Obligations” and are available to

students on page 1 of the Student

Mathematician’s Journal.

MATHEMATICALCOMMUNICATION

Emphasizing the Importance of CommunicationMathematical communication is a key component in the successful use of Project M3


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NCTMstandards

Introduction 1Project M3: At the Mall with Algebra

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IntroductionAlthough when we hear the word algebra, we generally think

about the high school curriculum, the National Council of

Teachers of Mathematics (2000) has elevated this content area to

one of great importance across the K–12 mathematics curriculum.

In fact, it is one of only five content standards. In the early grades,

the emphasis should be on developing algebraic reasoning, and

that is what this unit emphasizes.

In many respects, algebra is the generalization of arithmetic,

and so we encourage students to approach the study of variables,

expressions and equations using their number sense, logical

reasoning and problem-solving strategies. We introduce students

to these concepts through interesting problem-solving situations.

They are intrigued to figure out the mathematics behind number

tricks and to solve variable puzzles. In discovering the answers to

these problems, students learn about different ways to represent

and solve similar types of problems using variables, expressions

and equations.

As students represent and analyze mathematical situations

using algebraic symbols, they come to understand the basic notions

of equality and equivalent expressions. They learn how variables

are used to represent change in quantities and also to represent a

specific unknown in an equation. The idea that the same variable

represents the same quantity in a given equation or set of equations

is a fundamental algebraic concept that students will use

throughout their mathematical learning. In this unit students’

understanding of these concepts comes out of informal problem-

solving in which they use mathematics to make sense of the

situations posed, just like real mathematicians. We hope that the

experiences and discussions in the unit will provide a rich context

for introducing students to algebraic thinking while strengthening

their problem-solving and mathematical communication skills.

Connections to the Principles and Standards for School Mathematics (NCTM, 2000)

This unit develops concepts in algebraic thinking outlined in the

content standard, Algebra (p. 158), and specifically addresses the

following:

In grades 3-5 students should:

• Represent the idea of a variable as an unknown quantity using

a letter or symbol;

NOTES

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NOTES • Express mathematical relationships using equations; and

• Model problem situations with objects and use

representations such as graphs, tables and equations to draw

conclusions.

The unit also challenges students to think deeply about the

concept of equality as it relates to solving equations. Thus, the

following new expectations are included.

By the end of this unit, students should:

• Distinguish between an expression and an equation;

• Understand the different uses of a variable as a quantity

whose value can change or as a specific unknown in an

equation or situation;

• Solve one- and two-step equations;

• Model and solve contextualized problems that involve two

variables by using guess and test, organized lists and/or

making a diagram; and

• Solve sets of equations that involve two unknowns using

one or more of the following strategies: guess and test, an

organized list and substitution.

In addition, the unit challenges students to create and think deeply

about different kinds of representations and connects data analysis

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Lists expectations of students as itrelates to the Principles and Standardsfor School Mathematics.

Connections to Content Standards

Sample PagesThe sample pages within this module

were taken from Level 4 Module,“At

the Mall with Algebra: Working with

Variables and Equations.”

Teacher Guide–Connecting to NCTM StandardsThe introduction outlines how the unit connects to the Principles and Standards for School Mathematics (PSSM).


13

NOTES

Introduction 3Project M3: At the Mall with Algebra

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In grades 3-5 all students should:

Problem Solving (p. 182)

• Build new mathematical knowledge through problem solving;

• Apply and adapt a variety of appropriate strategies to solve problems;

• Monitor and reflect on the process of mathematical problem solving.

Reasoning and Proof (p. 188)

• Make and investigate mathematical conjectures;

• Develop and evaluate mathematical arguments.

Communication (p. 194)

• Organize and consolidate their mathematical thinking through communication;

• Communicate their mathematical thinking coherently and clearly to peers and teachers;

• Analyze and evaluate the mathematical thinking and strategies of others;

• Use the language of mathematics to express mathematical ideas precisely.

Connections (p. 200)

• Recognize and use connections among mathematical ideas;

• Recognize and apply mathematics to contexts outside of mathematics.

Representation (p. 206)

• Create and use representations to organize, record and communicate mathematical ideas;

• Select, apply and translate among mathematical representations;

• Use representations to model and interpret physical, social and mathematical phenomena.

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2 Project M3: At the Mall with Algebra

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one or more of the following strategies: guess and test, an

organized list and substitution.

In addition, the unit challenges students to create and think deeply

about different kinds of representations and connects data analysis

to other mathematical concepts. Thus the following additional

expectations are included.

By the end of this unit, students should:

• Formulate questions, design studies and collect data about

a characteristic shared by two populations or different

characteristics shared by one population;

• Create and analyze Venn diagrams focusing on the various

attributes of the data;

• Create and analyze pie graphs using knowledge of fractional

relationships;

• Create and analyze line graphs using coordinate geometry to

plot points and draw the graphs.

In addition to focusing on learning new content and conceptual

understanding, this unit also emphasizes all five process standards.

The process standards are an integral part of developing

mathematical thinking and are especially important in nurturing

mathematical talent. In particular, the unit specifically addresses

the following:

Introduction

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Emphasizes the importance of thePSSM Process Standards andspecifically lists where each area isaddressed within the unit.

Process Standards

NCTMstandards


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Unit Overview

Teacher Guide — Planning and Organizing the UnitThe Unit Overview provides the teacher with all the necessary information to plan for the upcoming unit.

28 Project M3: At the Mall with Algebra

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Unit Overview

Suggested Unit PacingUnit Test given as Pretest 1—

2 day

Chapter 1: Thinking about Variables and Equations

Lesson 1: I’m Thinking of a Number 3 days

Lesson 2: Number Tricks 3 days

Lesson 3: Variable Puzzles 3 days

Lesson 4: Cover Up! 3 days

Chapter 2: Equations that Go Together

Lesson 1: Pet Parade 3 days

Lesson 2: A Penny for Your Thoughts 3 days

Lesson 3: Seasonal Symbols 3 1—2 days

Lesson 4: At the Mall 2 days

Written Discourse

(at least two formal lessons to model how to write

Think Deeply questions within the unit) 2 days

Check-ups

(at least one per chapter assigned in class as a

formal assessment and others assigned for

homework) 2 days

Unit Test given as Posttest 1 day

Total 29 days

Pacing is based on a 50-minute math class period per day.

Skill Assumptions • Experience translating story problems to number sentences

and vice versa.

• Facility with fact families for addition, subtraction,

multiplication and division.

• Experience solving open-number sentences, such as 5 + Δ = 12.

• Experience with making organized lists as a problem-solving

strategy.

• Familiarity with the commutative property of addition and

multiplication.

• Familiarity with the properties of equality for addition,

subtraction, multiplication and division, i.e., If a = b, then

a + c = b + c.• Familiarity with the inverse relationship among operations of

whole numbers, i.e., subtraction is the inverse or “undoing” of

addition.

Unit Overview

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Overview of Lessons

LESSON/ACTIVITY/CHECK-UP VOCABULARY # DAYS1

Unit Test given as a Pretest 1—2

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Lesson 1 – I’m Thinking of a Number

Students will explore a number game and use variables to

represent changing quantities. Students are introduced to

expressions and equations and learn to translate

mathematical relationships from words to symbols. Students

will write equations in which expressions are set equal to

specific values and will use the working-backwards strategy

to undo operations in order to determine the value of the

variable in various equations.

equation,

expression,

factor,

inverse operations,

prime,

variable

3

Check-up 1 (Answer Key in Resources section at end of book)

Lesson 2 –Number Tricks

Students will analyze a number trick and write an equation

that represents a rule in order to generalize the outcome of

the trick. Then students will create their own number trick.

Students recognize that variables can also be used to

represent a specific unknown.

variable 3

Lesson 3 – Variable Puzzles

Students will continue to explore the idea that in some

situations a variable has a specific value rather than a

variable 3

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Materials ListAt the Mall with Algebra: Working with Variables and Equations• The Teacher Guide includes Hint Cards and Think Beyond Cards as blackline masters in each lesson.

Kendall/Hunt Publishing Company offers a full set of the cards on colored card stock which are

perforated and can be torn apart for use in the classroom.

• Supplies are available from Kendall/Hunt Publishing Company at 1-800-542-6657.

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Lesson 1: I’m Thinking of a Number

Students

• “Mathematician’s Journal Think

Deeply About…” (Student


pp. 3–5)

Teacher

• Blackline master “Hint Cards”

(p. 59)

• Blackline master “Think Beyond Cards” (p. 60)

Supplies

Lesson 2: Number Tricks

Students

• “Mathematical Trickery!”

(Student Mathematician’s

Journal p. 7)

• “Mathematician’s Journal Think

Deeply About…” (Student

M h i i ’ J l

Teacher

• Magic Number Trick Cards

(five cards prepared in advance;

see directions in “Initiate”section)

• Blackline master “Hint Cards”( 77)

Supplies

• Calculator

• Blank 3-by-5 index cards (at least 10 per student)

• Colored markers

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Lists the suggested number ofdays it should take to complete alesson.

Suggested Unit Pacing

Lists the prior skillsstudents shouldhave coming intothe unit.

Skill Assumptions

Lists the order of lessons, activity orCheck-up within the unit. Provides anoverview of each lesson.

Lesson/Activity/Check-Up

Blackline masters teacher willneed for the lesson.

Teacher

Lists the words studentswill be learning andapplying within the lesson.

Vocabulary


15

Teacher Guide — Understanding the Chapter Focus

45

Thinking aboutVariables and Equations

1chapter


Teaching NotesThe focus of this chapter is on variables, expressions and

equations. Students learn that the term variable has a number of

meanings. The two meanings explored are: 1) a variable is a

quantity that varies; and 2) a variable is a specific unknown in an

equation. Most adults think variable has only the latter meaning

— that is, that variables are letters that have set values. In the

equation, b + 1 = 5, the variable is the letter b, and the value of b is

4. Namely, when 4 is added to 1, the sentence is true. Yet, this

interpretation of variable is only one definition, and it can be a bit

misleading since nothing is varying or changing, as the word

implies. It is important when working with students to explain

clearly what is meant by the term and to differentiate between

situations when an unknown actually varies in value versus when

the unknown has a particular value. Helping students understand

that the term variable has more than one meaning is key.


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NOTESoperation. In the equation, 3 + N = 10, students know that 3 should

be added to N. What they often do not comprehend is that the

equal sign indicates equality or balance between the expressions on

each side of the equal sign; 3 + N must have the same value as the

other side of the equal sign (10). They also are challenged by the

idea that although an equal sign is used with equations, the equal

sign does not completely define equations. Students learn that

equations are a comparison of two expressions, and the sign

symbolizes this comparison.

In Lesson 1 students play a guessing number game and use

inverse operations to determine starting numbers. Students are

introduced to expressions and equations as they start to use both

to write algebraic rules. In the second lesson students analyze a

card trick, write a rule to use to generalize the outcome of the

trick, and then create their own tricks. In both lessons, students

are using variables to represent changing quantities. They also are

starting to understand the circumstances where a variable

represents a specific unknown.

Lesson 3 asks students to solve variable puzzles as they

become comfortable with the idea of variables representing

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Variables in equations where the degree of the variable is 1

(e.g., 4x1 – 2 = 10, N 1 ÷ 3= 12, x1 + 6 = 23.4) have one value that

makes the equation true. This is in contrast to equations that have

variables of higher degrees (e.g., x2 + 2x = 8, 3x2 = 75); these have

more than one possible solution. This level of differentiation is not

important for students to understand at this age, but one day they

will need to make sense of it.

It is important at this time for students being introduced to

the concept of variable to understand the symbolism. The variable

stands for an actual quantity or quantities and can be represented

by any letter or symbol such as �, � or ��. Some letters have

special meaning in mathematics, such as “i ” for imaginary

numbers, but almost any letter and any symbol can be used as a

variable as long as it is defined. For instance, in the expression

c + 6, c is defined as the number of cards collected.

Students need to understand the process of naming variables.

It is common for mathematicians to use the letters x and y as

variables that vary in relation to each other as x and y are the axes

on the coordinate grid. However, many students at this age are not

familiar with the coordinate plane, so it is essential for them to see

h b fi f i h l h b h

NOTES

M3L4-AMA_TE_045-122.indd 46 1/10/08 1:45:42 PM

Presents the chapter focus andprovides tips to the teacher onteaching the chapter.

Teaching Notes

Points out what students may or maynot know and how to best introducestudents to a new concept.

Teaching Notes

Presents an overview of a particularlesson and the new concept(s)students will be learning.

Teaching Notes

chapteroverview


16

Lesson Overview

Project M3: At the Mall with Algebra 67

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Number Tricks

Big Mathematical IdeasIn this lesson the concepts of variable and equation introduced in

Lesson 1 are revisited. Students practice using variables and

writing equations. First, they analyze a mathematical trick, and

then they represent the solution to the trick algebraically.

Objectives• Students will write equations using variables.

• Students will learn about the sums of even and odd numbers.

• Students will solve equations.

Materials

Students

• “Mathematical Trickery!” (Student Mathematician’s

Journal p. 7)

• “Mathematician’s Journal Think Deeply About…” (Student

Mathematician’s Journal pp. 9–11)



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Lesson2

M3L4-AMA_TE_045-122.indd 67 1/10/08 1:45:47 PM

NOTES Teacher

• Magic Number Trick Cards (five cards prepared in advance;

see directions in “Initiate” section)

• Blackline master “Hint Cards” (p. 77)

• Blackline master “Think Beyond Cards” (p. 78)

Supplies

• Calculator

• Blank 3-by-5 index cards (at least 10 per student)

• Colored markers

Mathematical Language• Variable – a quantity whose value changes or varies.

A variable could also be defined as a specific unknown in an

equation. In algebra, letters often represent variables.

InitiateAsk students if they have ever seen a magic number trick.

M3L4-AMA_TE_045-122.indd 68 1/10/08 1:45:47 PM

Describes the main conceptor idea the lesson will cover.

Big Mathematical Ideas

Lists the materialsneeded for the lesson.

Materials

A listing of what students areexpected to learn in the lesson.

Objectives

Provides the mathematicaldefinition for vocabulary termscovered within the lesson.

Mathematical Language

Teacher Guide–Lesson Planning Phase


Lesson Overview

17

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InitiateAsk students if they have ever seen a magic number trick.

Encourage anyone who knows a trick to share it with the class.

Tell them that today they are going to learn a trick that they can

use to impress their friends and family! Not only will they learn

how to do the trick, but they also will learn why the trick works.

Prior to having students analyze the trick, you’ll show it to them.

In advance, prepare five cards as shown below. (Blank index cards

can be used as cards.) Note that the front and back of each card is

different from its back.

1 3 5 7 9

2 4 6 8 10

Card A

Front

Back

Card B Card C Card D Card E

M3L4-AMA_TE_045-122.indd 68 1/10/08 1:45:47 PM

NOTES


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Ask a student to select 5 of the 10 numbers on the cards and

place the cards so the chosen numbers are face up. The cards

should be in a location so that other students can see them but you

cannot, perhaps at the back of the room. Tell the class that without

having seen the cards, you will be able to predict the sum of the

numbers after you ask just one question! Have students determine

the sum but not tell you what it is. Then ask the question, “How

many of the numbers showing are even?” When you are told how

many of the numbers are even, add this amount to 25, and that

will be the sum. Your students will be amazed and will want to

know how you did it! Perform the trick two more times.

You might be interested in knowing why this trick works.

(Do not reveal this information to students yet!) Notice that each

card consists of a consecutive odd and even pair of numbers, one

on each side. The smallest possible sum occurs when all the

numbers are odd: 1 + 3 + 5 + 7 + 9 = 25. If students say there are

no even numbers showing on the cards, then you know that all

numbers are odd and that their sum is 25. Because the cards

consist of consecutive pairs of odd/even numbers, the number of

even numbers chosen increases the sum by that amount (e.g., two

evens raise the sum by 2 since now there are two numbers that are

each 1 more than their odd partner). This card trick also works if

you ask the question, “How many of the numbers showing are

odd?” and use this information to determine the number of cards

that are even. Students are challenged to uncover the mathematics

behind this trick in the first Think Beyond Card.

Investigate

Distribute five blank 3-by-5 cards to each of the students and tell

them they are going to make a set of number trick cards with a

partner. Refer students to “Mathematical Trickery!” in the Student

Mathematician’s Journal and ask them to work individually on the

questions in Part I. Then split the class into pairs or groups of

three and ask them to discuss their answers. The focus of the first

part of this lesson is on figuring out why the number trick works.

Encourage students to look for patterns that will help them

analyze the trick. Resist the temptation to tell them how the trick

works, but do reinforce the idea that it isn’t really a trick, just a

nice use of mathematics. Instead, ask leading questions, such as

“How are odd and even numbers the same, and how are they

different?” or “Why do you think it is important that consecutive

odd/even pairs are on each card?” Hold a class discussion in order

for students to share their ideas with each other and further make

sense of the mathematics behind the trick (see the sample dialogue

for the second Think Deeply question below).


Mathematical Trickery!

Project M3: At the Mall with Algebra 7 Chapter 1: Thinking about Variables and Equations


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Part I: Make the following cards for the math trick.

1 3 5 7 9

2 4 6 8 10

Card A

Front

Back


1. If you used only Card A and Card B to play the game, what is the minimum sum

possible? Which numbers give the minimum sum?

What is the maximum possible sum? Which numbers give the

maximum sum?

2. If you used only Card A, Card B and Card C to play the game, what is the

minimum sum possible? Which numbers give the minimum sum?

What is the maximum sum possible? Which

numbers give the maximum sum?

3. What is special about the numbers used to get the minimum and maximum sums?

4. What number patterns do you see on a) the fronts? b) the backs? c) the front to back?

5. The person performing the number trick always asks to be told how many even numbers are used. Why do you think you have to know how many even numbers are used in order to predict the sum?

Student Mathematician’s Journal p. 7



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The introductorylearning experiencedesigned to engagestudents.

Initiate

Provides an opportunityfor students to furtherexplore the conceptsin-depth in a hands-oninvestigation.

Investigate

Teacher Guide–Lesson Planning Phase


18

Lesson Overview

NOTES

Project M3: At the Mall with Algebra70

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After discussing the questions on Part I of “Mathematical

Trickery!” see if students are able to generalize a rule for finding

the sum in this trick (Sum = 25 + number of even numbers).

Reintroduce the concept of variable to the class as a quantity that

varies. Ask students what varies in the number trick each time (the

number of even numbers) and tell them we can represent this

quantity using a letter (e.g., Sum = 25 + x) or another symbol (e.g.,

Sum = 25 + ��). You might want to record the possibilities on the

board so that students can see what parts of the equations are

changing. Demonstrate how different values can be substituted for

x or ��, depending on the number of even cards. Some students

might also mention that the sum varies each time and would like

to use a letter to represent that quantity. They could use S to

represent the sum, and thus the algebraic rule for the first equation

would be S = 25 + x, where S represents the sum and x represents

the number of even numbers.

Part II of “Mathematical Trickery!” asks students to create a

similar number trick using different cards. It is recommended that

students use consecutive odd/even numbers on the fronts and backs

of their cards. However, the number of cards used and the number

pairs chosen for the cards can vary dramatically. Finally, ask students

to write a rule for finding the sum for their trick. For example, a

student might make up six different cards that have the following

numbers on the front/back: 11/12, 13/14, 15/16, 17/18, 19/20 and

21/22. The minimum sum of these numbers is 96 (all odds). The

rule would be Sum = 96 + n, where n is the number of even

numbers. Afterwards, each student can practice his/her number

trick with a classmate and then present it to the whole class.

Mathematical CommunicationAfter students have worked on understanding the number

trick and have developed their own trick, you can discuss these

Think Deeply questions as a whole class.

THINKDEEPLY

1. Your best friend missed school but heard that you learned a number trick in math class. You decide to send your friend your set of cards and a note explaining how your number trick works.

a. What numbers were on your cards? b. What was your rule? c. Why does your rule work?



6. Explain how the number trick works.

7. Write a rule using variables for how to find the sum of the numbers on the cards.

Part II: Make your own set of number trick cards.

8. Make up a set of cards to use for the number trick. You may use any number of cards. Indicate here what numbers are on the fronts and backs of your cards.

Front:

Back:

9. Write the directions for your number trick.

10. What is the minimum sum possible using your set of cards?

11. What is the maximum sum possible using your set of cards?

12. What question will you ask that will give you more information about the sum?

13. Write a rule with variables that can be used to determine the sum of your numbers.

Mathematical Trickery! (continued)


Chapter 1: Thinking about Variables and Equations 8 Project M3: At the Mall with Algebra


36

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Need more room?

Use the back!

THINKDEEPLY







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NOTES


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What to Look for in Responses

• Students should clearly explain their number trick, including

the numbers they used on the cards and the minimum and

maximum sums. They might draw pictures of their cards or

describe the number combinations in words.

• Students then need to explain what question they will ask and

how they will interpret the answers. An advanced response

includes an equation using a variable that represents the sum

of the numbers on the card.

Possible Difficulties

• Students might be confused by how to create their own number

tricks and how to analyze them so as to use them convincingly.

Suggest they make a game that is very similar to the one

discussed in class. You may need to help them determine the

minimum sum and the question they will ask their friends.

THINKDEEPLY

2. a. What is a variable? b. How did you use a variable to write a rule for your card trick?

What to Look for in Responses

• Students should define a variable as a quantity that varies.

They should give an example of an expression or equation

that includes a variable.

• Students should mention that a variable is represented in

algebra using a variety of symbols, such as letters and shapes.

• Other characteristics of a variable that students might

mention are that different letters can be used to represent

the variable, and that they used the variable to represent the

number of even numbers in the game.

• Students also need to give their number trick rule and explain

how, in their case, the value of the variable changes each time

the trick is performed. Be sure the number trick equation or

expression accurately represents their trick.


• Students may not understand the idea that any letter can

be used to represent quantities. Using the problem in the

dialogue below, the rules of S = 25 + a, S = 25 + m and

S = 25 + x are the same, and any letter can be used to stand

for the number of even numbers.



Need more room?

Use the back!

THINKDEEPLY




2. a. What is a variable?b. How did you use a variable to write a rule for your card

trick?



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M3L4-AMA_TE_045-122.indd 71 1/10/08 1:45:49 PM

Students reflect upon themathematical concepts in theinvestigation using the ThinkDeeply Questions, write about themin their Mathematician’s Journalsand discuss them with a partner,small group and/or the entire class.

Mathematical Communication

Reduced pages fromStudentMathematician’s Journalare pictured along withpage number.

Reduced Pages

Think Deeply questions fromStudent Mathematician’sJournal in bold in TeacherGuide.

Think Deeply

Potential problems thatmight arise as studentsattempt to answer questions.


Describes to teacher whatthey should be looking for instudent responses.

Student Responses


Lesson Overview

NOTES


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• Students often think that a variable is a label (e.g., b stands

for “boys” rather than “the number of boys”).

• In this situation, the variable represents a quantity that varies.

(Students sometimes think that if during the first occurrence

of the trick there are three even numbers, from then on, the

variable will represent 3.)

A discussion centered on the second Think Deeply question

about using a variable to make a rule might go something like this.

Teacher: Who can tell me what the term variablemeans? Scott?

Scott: It’s the letter in math — the x or the y.

That’s the variable.

Teacher: Yes, a letter often stands for a variable, but

what does it mean? Kinne, what can you

add to Scott’s response?

Kinne: Scott said that a variable is a letter, and I

agree but I also think we can use a little

square or something like that to be the

variable, too.

Teacher: Ben?

Ben: I also agree with Kinne and Scott, but isn’t

a variable like what you don’t know? Like

in our trick, we don’t know the number of

even numbers at first.

Kinne: Yes, we don’t know the number of even

numbers, and the number of even numbers

changes, and that’s why it is a variable. A

variable can change.

Teacher: Puja, would you please summarize the

points that Scott, Kinne and Ben have

made?

Puja: Well, a variable is what you don’t know in

a problem. You always use x to stand for

the even numbers.

Teacher: Can we use a different letter, maybe a or mto stand for the number of even numbers

in our number trick rule? Are Sum = 25 + aand Sum = 25 + m and Sum = 25 + x [writes

these number sentences on the board] the

same rules?



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Puja: I don’t think so. I think x is what stands for

the even numbers.

Teacher: What do other people think? Kyuri? Then

Rudy.

Kyuri: I’m not sure, but I think you have to use

the same letter all the time for the even

numbers.

Teacher: Actually, both sentences are the same

rule. We can use any letter we want to

represent the even numbers as long as we

indicate what the letter stands for.

So, if we say that a represents the number

of even numbers, we can use the rule

Sum = 25 + a, and if we say that xrepresents the number of even numbers,

we can use the rule Sum = 25 + x.

Rudy: Does that mean we can decide what letter

to use?

Teacher: Yes, as long as you tell everyone else what

quantity the letter is going to represent.

Teacher: So, a variable represents a quantity that

changes, such as the number of even

numbers in our trick. Sometimes there are

zero even numbers, and other times there

are five even numbers. The amount varies.

We can use any letter or symbol we want

to stand for the number of even numbers

as long as we tell others, which is called

“defining the variable.” Let’s write some

of these key ideas on the board and then

have everyone remember the ideas by

writing in their Mathematician’s Journal.

THINK BEYOND

Developing number tricks and writing rules for

the tricks is an enjoyable and challenging activity

for talented students. In the first Think Beyond

Card, students are asked to figure out how the

number trick works when the question posed is

“How many odd numbers are showing?” In this case, students

need to know that the maximum sum is 30 when all even numbers

are showing. They can then subtract 1 for every odd number

showing. For example, if two odd numbers are showing, the

student would subtract 2 from 30, and the sum would be 28.



NOTES

M3L4-AMA_TE_045-122.indd 73 1/10/08 1:45:50 PM

NOTES The second Think Beyond Card addresses what is

fundamental to this trick. That fact is that each card has one odd

and one even number on it, and the even number is a predictable

amount greater than the odd number. If all the numbers on both

sides of the cards were odd (or even), it would be impossible to

predict the sum. But if the odd/even pairs are consecutive, then

the even numbers must be exactly one more than each of the odd

numbers. If the even number in each odd/even pair is a set

amount greater than the odd number, the trick will also work. For

example, the third Think Beyond Card suggests using even

numbers that are all three more than the odd number on each card

(10, 8, 6, 4). The sum of the odd numbers (7, 5, 3, 1) is 16. So to

answer the question “How many even numbers are there?” the

number of even numbers must be multiplied by 3 and added to

the sum of the odd numbers (16 + 3x).

In summary, there must be a predictable pattern between the

numbers on the sides of the cards. However, it doesn’t matter how

many cards are used for the trick or what numbers are on the cards

as long as the minimum sum is calculated prior to trying it out.

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M3L4-AMA_TE_045-122.indd 74 1/10/08 1:45:50 PM

Suggests ideas tochallenge studentseven further. Answersare provided.

Think Beyond

Sample narrative provides teacher withexample of what to expect from classdiscussions. These are often written aboutcommon student misconceptions to helpteachers in their class discussions.

Sample Discussion

19


20

Lesson OverviewC

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Answer Sheet

Project M3: At the Mall with AlgebraChapter 1: Thinking about Variables and Equations




1 3 5 7 9

2 4 6 8 10

Card A

Front

Back





maximum sum?








4

6

1 and 3

2 and 4

9

1, 3, 5 12

2, 4, 6

The minimum sum is the sum when all odd numbers are chosen, and the

maximum sum is the sum when all even numbers are chosen.

a) all odd numbers b) all even numbers

c) the back is the next consecutive number

Since the even numbers are 1 more than the odd numbers, you need to

know how much extra you will add to the minimum sum.

M3L4-AMA_TE_045-122.indd 75 1/10/08 1:45:50 PM

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Answer SheetMathematical Trickery! (continued)







Front:

Back:






First you figure out the sum of the odd numbers, which is the minimum

sum. When you know how many even numbers there are, you add that

number to the minimum, and you get the sum.

The rule is: Sum = 25 + x, where x stands for the number of even

numbered cards showing.

Answers will vary.Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary. Look for clarity.

Students should ask a question that enables them to add or

subtract an amount to either the minimum or maximum sum.

Answers will vary depending on the question and the numbers on

the cards.

M3L4-AMA_TE_045-122.indd 76 1/10/08 1:45:50 PM

Answers to StudentMathematician’sJournal questions.

Answers


21

Blackline Masters of Think BeyondCards. Supports differentiation in theclassroom by challenging students.Also available separately onperforated cardstock.

Think Beyond Cards

Lesson Overview

Blackline Masters of Hint Cards.Supports differentiation in theclassroom by helping studentsmove forward with a problem.Also available separately onperforated cardstock.

Hint Cards


22

Lesson Overview

NOTES


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A Quick Look

Chapter 1: Lesson 2Number Tricks

Objectives:

• Students will write equations using variables.

• Students will learn about the sums of even and odd numbers.

• Students will solve equations.

Initiate: ( 1—2 day)

1. Ask students if they have ever seen a magic number trick.

Encourage them to share some tricks with the class.

2. Inform students that they will learn a trick that they can use

to impress their friends and family. Moreover, they will learn

why the trick works.

3. Using blank index cards, prepare in advance the following five

cards:

1 3 5 7 9

2 4 6 8 10

Card A

Front

Back


4. Ask students to select 5 of the 10 numbers on the cards and

place the cards so the chosen numbers are face up. (Note:The cards should be in a location so that other students can see them but you cannot.)

5. Tell students that you will predict the sum of the numbers

after you ask just one question. Have students determine the

sum of the numbers but not tell you what it is.

A QUICKLOOK



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Summarizes steps for teachingthe lesson and includes thebreakdown of the phaseswithin the lesson and thesuggested time.

A Quick Look



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6. Ask, “How many of the numbers showing are even?” When you are told this amount, add this amount to 25, and the result is the sum of the numbers. Tell students the sum and they will be amazed!

7. Perform the trick two more times. Do not reveal how the trick works at this time! (Note: The smallest sum occurs when all

the numbers selected are odd: 1 + 3 + 5 + 7 + 9 = 25. Because

the cards contain consecutive even/odd pairs of numbers,

when given the number of even cards selected, you simply

add that amount to 25 to get the overall sum. This trick also

works if you ask how many odd numbers are showing. You

simply subtract the number of odd cards from 5 and add the

resulting difference to 25.)

Investigate: (1 1—2 days)

1. Refer students to “Mathematical Trickery!” in the Student Mathematician’s Journal and have them work individually on Part I.

2. When they are finished, split the class into pairs or groups of three and have them discuss their responses. Have students focus their group discussion on looking for patterns that will help them analyze the trick and figure out why it works.Resist the temptation to tell students how the trick works. Help them realize the trick is a fun way to use mathematics.

3. Ask leading questions to guide their discussion, such as “How are odd and even numbers the same and how are they different?” or “Why do you think it is important that consecutive odd/even pairs are on each card?” You may also use Hint Cards for those students who need additional support.

4. Hold a class discussion to encourage students to share their ideas with each other to deepen their understanding of the mathematics behind the trick.

5. Have students generalize a rule for finding the sum in this trick: Sum = 25 + number of even numbers. Reintroduce the idea of a variable and ask students what varies in this trick each time (the number of even numbers). Show students how to represent the variable with a letter or symbol:

Sum = 25 + x or Sum = 25 + ��

Some students might also suggest that the sum varies each time; they’ll want to use a letter to represent that quantity.

S = 25 + x or S = 25 + ��



NOTES

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6. Distribute at least five blank index cards to each student.Have students work individually to create a similar number trick using their own cards. Have students use consecutive odd/even numbers on the fronts and backs of their cards.However, the number of cards used and the number pairs chosen can vary.

7. When finished, have students complete Part II of “Mathematical Trickery!” and write a rule for finding the sum for their trick.

8. Afterwards, have each student practice his/her number trick with a classmate and then present the trick to the whole class.

9. Encourage students to present their magic number trick to friends and family at home.

Assessment: (1 day)

Assign and discuss the Think Deeply questions.

• Have students work individually on the first Think Deeply question. Students are asked to explain how their number trick works to a friend who was absent from math class.Students are asked to describe the numbers they chose for their cards, give the rule for the trick, and explain why the rule works. This question is a good one to use to formallyassess the lesson.

• The second Think Deeply question asks students to define a variable and describe how a variable is used to write a rule for a magic number trick. If class time is limited, this question may be assigned for homework. When discussing this question with the class, be sure to emphasize the fact that a variable is a quantity that varies, and that in algebra,a variable can be represented using a variety of symbols such as letters and shapes.



NOTES

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Lesson Overview

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Project M3: At the Mall with Algebra 7 Chapter 1: Thinking about Variables and EquationsLesson 2: Number Tricks

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1 3 5 7 9

2 4 6 8 10

Card A

Front

Back





maximum sum?








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Sample PagesPages 7, 8, 9, and 11 are from the unit

“At the Mall with Algebra” Student

Mathematical Journal.


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Front:

Back:






Mathematical Trickery! (continued)


Chapter 1: Thinking about Variables and Equations 8 Project M3: At the Mall with AlgebraLesson 2: Number Tricks

36

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Lesson Overview


26

Lesson Overview

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27

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2. a. What is a variable?b. How did you use a variable to write a rule for your card

trick?


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Lesson Overview


28

Assessment

Multiple Assessment OptionsProject M3 provides the teacher with a variety of assessment tools.

19Project M3: At the Mall with Algebra

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Student Mathematician’s Journal: Guides for Teaching and Assessing

Rubric for Student Mathematician’s Journal

MATHEMATICAL CONCEPTS

3 Overall, student demonstrates a strong understanding of concepts and, if applicable, uses appropriate and efficient strategy to solve problem correctly. The student answers all parts of the question/prompt.

2 Overall, student demonstrates a good understanding of concepts and, if applicable, uses appropriate and efficient strategy but with minor errors or incomplete understanding. The student answers all parts of the question.

1 Overall, student demonstrates a partial understanding of concepts and, if applicable, uses appropriate strategy but may have major errors. The student may not have answered all questions.

0 Overall, student demonstrates a lack of understanding of concepts and, if applicable, does not use appropriate strategy. The student may not have answered all questions.

MATHEMATICAL COMMUNICATION

3 Student states ideas/generalizations that are well developed and reasoning is supported with clear details, perhaps using a variety of representations such as examples, charts, graphs, models and words.

2 Student states adequately developed ideas/generalizations and reasoning is supported with some details. When appropriate, representations may be limited.

1 Student states partially developed ideas/generalizations and reasoning is incomplete.

0 Student does not state ideas/generalizations correctly and reasoning is unclear with little or no support.

MATHEMATICAL VOCABULARY

3 Student uses all mathematical vocabulary appropriately, including mathematical vocabulary related to the major math concept(s) from the unit.

2 Student uses most mathematical vocabulary appropriately or may have minor misunderstanding.Student may have misused or omitted an appropriate vocabulary term.

1 Student uses some mathematical vocabulary or may have a major misunderstanding. Student may have misused or omitted several appropriate vocabulary terms or a key vocabulary term related to the major math concept(s) from the unit.

0 Student does not use any mathematical vocabulary.


M3L4-AMA_TE_00i-044.indd 19 1/10/08 1:47:19 PM

An assessment stamp and ink pad is available

to expedite the process of grading students’

journal work The grading technique is based

on the rubric that scores students on math

concepts, communication, and vocabulary.

This stamp can also be used in other classes.

Assists the teacher withteaching and assessing theStudent Mathematician’sJournal.

Rubric


29

20

Need more room?

Use the back!Your Thoughtsand Questions


Mathematician’s JournalTHINKDEEPLY

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Project M3: At the Mall with AlgebraStudent Mathematician’s Journal: Guides for Teaching and Assessing

2. Here is an equation: 7 + 2x = 9.a. Explain two different ways to determine the value of x. b. What is the value of x?c. Which method do you think is better? Why?

M3L4-AMA_TE_00i-044.indd 20 1/10/08 1:47:19 PM

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ConceptsUnderstands how to use the cover-up and working-backwards methods to

solve an equation, and correctly determines the value of x to be 1 using the cover-up method. Does not completely show how to determine the value of x using

the working-backwards method.

VocabularyDescribes inverse operations using “changes the signs” rather than the

appropriate term. Also could have used more precise mathematical language in some places, such as “what is the value

of x” rather than “what would work for x” or “What number plus 7 = 9?” rather

than “What plus 7 = 9?”


1. First you cover up the 2 and x. Then you think: What plus 7 = 9? The anser is 2. Then you think what would work for x. 2 x 1 = 2. 2 + 7 = 9. So x = 1. There’s your answer!

2. The backwards method changes the signs so + is – and x is ÷. You would do 9 – 7 = 2. So x = 1 again!

I prefer the backward method because it is easier for me.

CommunicationDescribes most of the steps taken

for each method. Needs to elaborate more on how the value of x was determined using the working-

backwards method.


2. Here is an equation: 7 + 2x = 9.a. Explain two different ways to determine the value of x. b. What is the value of x? c. Which method do you think is better? Why?

Teacher Feedback:

Great! I see how you used both the cover-up and working-backwards

methods to solve the equations. You say that you prefer the working-

backwards method because it is easier for you. Tell me more about

this. I wonder why you decided to change from addition to subtraction

to solve this problem. What mathematical words can you use to

describe what you mean? I also would like to hear more about how

you determined the value of x using the working-backwards strategy.

Concepts – 3

Communication – 2

Vocabulary – 1

SCORE

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Example of actual answer as it wouldappear on a Think Deeply StudentMathematician’s Journal. Each TeacherGuide includes three differentscenarios of student answers.

Think Deeply

Ties in student answers withexpectations of rubric.

Rubric Explanation

Lists score as tabulated fromstudent answers and howwell they met rubric criteria.

Rubric Score

Suggestions on providing feedback tothe student. Includes questions toencourage and assess deeperunderstanding.

Teacher Feedback

Assessment


30

Assessment

At the end of some lessons abrief assessment is includedthat can be used to checkstudents’ understanding ofunit concepts and serve asan ongoing review of regularcurriculum content. AnAnswer Key for the Check-ups is included in theResources section at the endof the book.

Ongoing AssessmentName Date

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Project M3: At the Mall with Algebra 189Chapter 2: Equations that Go Together

Check-up

1. Label the following as equations or expressions.

a. 2P + 5

b. (N × 3) ÷ 2 = 6

c. 65 = 30 + 35

d. (T – 3) + 12

2. These numbers follow a pattern.

5, 10, 16, 23,

What number is next in this pattern? Explain why you think that number is the

next number.

3. Snack boxes of raisins cost 25¢ each. Michelle bought 12 boxes of raisins for her

class. What could Michelle do to find out the total cost of the raisins?

a. add 25 and 12

b. subtract 12 from 25

c. multiply 25 by 12

d. divide 25 by 12

M3L4-AMA_TE_123-206.indd 189 1/10/08 1:48:46 PM

Found in the Resourcessection of the Teacher Guide,the Connections to theStandards and StandardizedTests is a supplementalpractice review based oncommon standardized testformats.

Test Practice

Resources 219Project M3: At the Mall with Algebra

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Connections to the NCTM Standards () and Standardized Tests

1. If 67 + �� = 98, which of the following equations is also true?

a. 67 + 98 = ��

b. 98 – 67 = ��

c. 98 + �� = 67

d. �� – 98 = 67

2. Fill in the missing numbers below to make the addition fact true.

3 �� 6+ �� 4 7

6 8 3

3. What is the missing number?

• It is between 80 × 2 and 4 × 60.

• Its hundreds digit is greater than its ones digit.

a. 185

b. 193

c. 211

d. 251

4. If there are 4 circles and a 4-pound block on the left side of a balance and 2 circles and a 10-pound block on the right side, how much would each circle have to weigh in order for both sides of the balance to be equal in value?

4 10

5. What is the missing factor?

72 ÷ = 9

M3L4-AMA_TE_207-228.indd 219 1/10/08 1:46:38 PM


31

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Name Date

Unit TestAt the Mall with Algebra:

Working with Equations and Variables

1. Rewrite the following expressions or equations using variables and

mathematical symbols.

a. Seven plus a number

b. Number times 3, plus 5, results in an answer of 29

c. Two times a number divided by 3

d. Number multiplied by 4, minus 2, results in an answer of 18

e. Circle the letter(s) of the statements above (a, b, c, d) that represent

equations. Explain why they are equations. Include your own definition of an

equation as part of your explanation.

Unit Overview

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Unit Test Rubric: At the Mall with Algebra Question 1

Problem Mathematical Focus

Points Expected Student Response

1a–1d

(max 2)

Focus: translating mathematical expressions and equations using variables and mathematical

symbols

0.5 Accurately translates a statement using a variable and mathematical symbols.

A. 7 + N or N + 7

B. 3N + 5 = 29 or (N × 3) + 5 = 29 or N × 3 + 5 = 29

C. 2N ÷ 3 or N × 2 ÷ 3 or (N × 2) ÷ 3

D. 4N – 2 = 18 or (N × 4) – 2 = 18 or N × 4 – 2 = 18

(Note: Students may use any letter or any appropriate symbol to represent the variable

for “number.” Student responses may include other equivalent expressions. No credit

should be given if the answer instead of a variable is included or an equal sign is used

with an expression.)

1e

(max 3)

Focus: differentiating between expressions and equations

1 Correctly circles the equations they wrote for 1a–1d. (Note: Students may get

full credit for this response if they incorrectly translated the written statements to

mathematical symbols in 1a–1d. The letters that are circled must include a correct

equation, or two expressions that are being compared with an equal sign. The

equation may or may not be solvable.)

Focus: understanding the meaning of an equation and justifying a solution

2

1

Clearly explains that an equation is a statement about equality comparing

expressions that have the same value. Student may or may not note that

equations use an equal sign. For example, states, “b and d are equations because

the expressions on either side of the equal sign have the same value.”

–OR–

Attempts to justify choices by explaining what an equation is, but response is

not clear or fully developed beyond stating that equations have an equal sign

or that both sides are “balanced.” For example, states, “An equation has an

equal sign, so b and d are equations,” or, “The expressions in b and d are in

balance.”

–or–

Attempts to justify choices by explaining that b and d are more than just

expressions because they have an equal sign. For example, states, “b and d are

more than just expressions because they have operations, numbers, letters and

an equal sign.”

(Note: Equations do not have to involve operations, such as in 3 = x or 15 = 15.

Also, no credit should be given for a statement like “Equations have an answer,”

because this is not a defining characteristic of an equation. Students who incorrectly

answered 1a–1d or the first part of 1e can still receive full credit — 2 points —

for this section.)

TOTAL POINTS: 5

Unit Overview

M3L4-AMA_TE_00i-044.indd 39 1/10/08 1:47:33 PM

Assesses studentunderstanding of unitsubject matter.

Unit Test

Suggests points to beawarded based onstudent answers toUnit Test.

Unit Test Rubric

Assessment


LEVEL 3 LEVEL 4 LEVEL 5

Unraveling the Mystery ofthe MoLi Stone: Place Value

and Numeration

Factors, Multiples andLeftovers: Linking

Multiplication and Division

Treasures from the Attic:Exploring Fractions

Awesome Algebra:Looking for Patternsand Generalizations

At the Mall with Algebra:Working with Variables

and Equations

Record Makers and Breakers:Using Algebra to Analyze

Change

What’s the Me inMeasurement All About?

Getting into ShapesFunkytown Fun House:

Focusing on ProportionalReasoning and Similarity

Digging for Data: The Searchwithin Research

Analyze This! Representingand Interpreting Data

What Are Your Chances?

Numberand

Operations

Algebra

Geometry andMeasurement

DataAnalysis

andProbability

N3088602

Advanced math content focused on critical andcreative problem solving and reasoning

Engaging investigations, projects and simulations

Rich verbal and written mathematicalcommunication

Students as practicing mathematicians

Alignment with NCTM content and processstandards

Project M3: MentoringMathematical Minds is a series of12 curriculum units developed to

motivate and challengemathematically talented studentsat the elementary level. Highlights

of this curriculum include:

Project M3 units have

been recognized by

the National

Association for Gifted

Children (NAGC)

with their Curriculum

Division Award.

To order call 1-800-542-6657 or visit www.kendallhunt.com/m3

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