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foundations
Project M3 : Mentoring
Mathematical Minds is a series
of 12 curriculum units developed
to motivate and challenge
mathematically talented
elementary students.
Standards-Based
3
Project M : 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).
Thinking like Mathematicians
The 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.
Challenging and Engaging
A 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.
Research
Project 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 Winning
A number of modules in the Project M3 series have earned the
National Association for Gifted Children’s Curriculum Studies Award.
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Mathematical Communication
Rich 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.
Differentiation
Understanding 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 Available
Kits are available for most modules and contain hands-on
manipulatives for activities that make it easy for teachers.
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foundations
Meeting the Needs of Mathematically Talented Students
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 Research
The 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.
Results
Extensive 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.
To learn more about the authors or to
read articles about this award-winning
program, visit www.projectm3.org.
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materials:
Teacher and Student
Teacher Guide
The 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
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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materials:
Teacher and Student
Student Mathematician’s Journal
THINK
DEEPLY
Student Mathematician:
Date:
Mathematician’s Journal
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?
The Student Mathematician’s Journal is a unique feature of every unit in
the Project M 3 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.
Copyright © Kendall/Hunt
Publishing Company
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.
Your Thoughts
and Questions
he
Use t
Project M3: At the Mall with
bac
k!
Need more room?
Algebra
9
Chapter 1: Thinking about
Variables and Equations
Lesson 2: Number Tricks
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In each unit of the Project M 3 series, students explore a simulated or reallife 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.
Manipulatives
Manipulatives 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.
Hands-On!
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implementation
strategies
Implementing Project M3
Replacement Units
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.
These 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.
Enrichment Materials
Variety of Learning Settings
There are a variety of ways to implement Project M 3.
With their strong focus on critical and creative mathematical
thinking, Project M 3 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.
The Project M 3 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 impressive
programs’ in 30 years
Project M3 was examined by independent
consultant Susan Carroll, president of Words and
Numbers Research in Torrington, Connecticut.
“I’ve evaluated programs for 30 years, and this is
one of the most impressive I’ve ever seen.”
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RESEARCH
Positive Results with Project M3
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).
For the latest research and articles on examples of successful
implementation, go to www.kendallhunt.com/m3
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Levels 3-4
LEVEL 3
writing and testing game cards, creating rules, and
designing the board.
Unraveling the Mystery of the MoLi Stone:
Place Value and Numeration
What’s the Me in Measurement All About?
In 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.
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.
Awesome Algebra: Looking for Patterns
and Generalizations
Students 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
LEVEL 4
Factors, Multiples, and Leftovers: Linking
Multiplication and
Division
Students 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
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Digging for Data: The
Search within Research
Students 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.
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 with
Variables and Equations
Students 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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Levels 4-5
(LEVEL 4 Continued)
Getting into Shapes
In 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.
LEVEL 5
Treasures from the Attic: Exploring
Fractions
Students 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: Using
Algebra to Analyze Change
Students 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
Analyze This! Representing
and Interpreting Data
In 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.
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 on
Proportional Reasoning
and Similarity
Students 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.
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lesson format
Clear and Organized Lesson Format
Each 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.
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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.
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MATHEMATICAL
COMMUNICATION
Emphasizing the Importance of Communication
Mathematical communication is a key component in the successful use of Project M3
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 Communication
Discussion is very important in nudging student thinking
forward.Students working on Project M 3 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 Communication
The Student Mathematician’s Journal is a unique feature of
every unit in the Project M 3 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 Project
M 3 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.
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NCTM
standards
Teacher Guide–Connecting to NCTM Standards
The introduction outlines how the unit connects to the Principles and Standards for School Mathematics (PSSM).
Sample Pages
The sample pages within this module
were taken from Level 4 Module,“At
the Mall with Algebra: Working with
Variables and Equations.”
Introduction
NOTES
Although 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.
Copyright © Kendall/Hunt Publishing Company
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 problemsolving 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 Content Standards
Lists expectations of students as it
relates to the Principles and Standards
for School Mathematics.
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;
Project M3: At the Mall with Algebra
Introduction
NOTES
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• Express mathematical relationships using equations; and
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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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NCTM
standards
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.
• 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:
2
Copyright © Kendall/Hunt Publishing Company
By the end of this unit, students should:
Process Standards
Emphasizes the importance of the
PSSM Process Standards and
specifically lists where each area is
addressed within the unit.
Project M3: At the Mall with Algebra
Introduction
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In grades 3-5 all students should:
NOTES
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)
Copyright © Kendall/Hunt Publishing Company
• 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.
Project M3: At the Mall with Algebra
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Unit Overview
Teacher Guide — Planning and Organizing the Unit
The Unit Overview provides the teacher with all the necessary information to plan for the upcoming unit.
Unit Overview
Suggested Unit Pacing
Suggested Unit Pacing
1
—
2 day
Unit Test given as Pretest
Chapter 1: Thinking about Variables and Equations
Lesson 1: I’m Thinking of a Number
Lesson 2: Number Tricks
Lesson 3: Variable Puzzles
Lesson 4: Cover Up!
3 days
3 days
3 days
3 days
Lists the suggested number of
days it should take to complete a
lesson.
Chapter 2: Equations that Go Together
Lesson 1: Pet Parade
Lesson 2: A Penny for Your Thoughts
Lesson 3: Seasonal Symbols
Lesson 4: At the Mall
3 days
3 days
3 —12 days
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
Pacing is based on a 50-minute math class period per day.
Skill Assumptions
Skill Assumptions
Vocabulary
Lists the words students
will be learning and
applying within the lesson.
Teacher
Unit Overview
king about Variables and Equations
Overview
of Lessons
• Experience translating stor y problems to number
sentences
and vice versa.
LESSON/ACTIVITY/CHECK-UP
VOCABULARY
• Facility with fact families for addition, subtraction,
multiplication and division.
Unit Test given as a Pretest
• Experience solving open-number sentences, such as Lesson
5 + Δ = 112.
equation,
– I’m Thinking of a Number
vingwill explore a number game and use variables to
• Experience with making organized lists as a problem-sol
expression,
Students
strategy.
factor,
represent changing quantities. Students are introduced to
• Familiarity with the commutative property of addition
and
inverse operations,
expressions and equations and learn to translate
multiplication.
mathematical relationships from words to symbols. Students prime,
• Familiarity with the properties of equality for addition,
variable
will write equations in which expressions are set equal to
subtraction, multiplication and division, i.e., If a = bspecific
, then values and will use the working-backwards strategy
a + c = b + c.
to undo operations in order to determine the value of the
• Familiarity with the inverse relationship among operations
of various equations.
variable in
whole numbers, i.e., subtraction is the inverse or “undoing” of
addition.
Check-up 1 (Answer Key in Resources section at end of book)
Lists the prior skills
students should
have coming into
the unit.
28
29 days
Copyright © Kendall/Hunt Publishing Company
Total
Project M3: At the MallLesson
with Algebra
2 –Number Tricks
Blackline masters teacher will
need for the lesson.
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variable
# DAYS1
1
—
2
3
3
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.
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Students recognize that
variables can also be used to
represent a specific unknown.
Materials List
At the Mall with Algebra: Working with Variables and Equations
Lesson/Activity/Check-Up
Lesson 3 – Variable Puzzles
• The Teacher Guide includes Hint Cards and Think Beyond Cards as blackline masters in each lesson.
Students will continue to explore the idea that in some
Kendall/Hunt Publishing Company offers a full set of the cards on colored card stock which are
situations a variable has a specific value rather than a
perforated and can be torn apart for use in the classroom.
• Supplies are available from Kendall/Hunt Publishing Company at 1-800-542-6657.
variable
3
Lists the order of lessons, activity or
Check-up within the unit. Provides an
overview of each lesson.
Lesson 1: I’m Thinking of a Number
Students
Teacher
Supplies
14
Variables and Equations
• “Mathematician’s Journal Think • Blackline master “Hint Cards”
(p. 59)
Deeply About…” (Student
Mathematician’s Journal
• Blackline master “ Think
pp. 3–5)
Beyond Cards” (p. 60)
Lesson 2: Number Tricks
Students
Teacher
• “Mathematical Tricker y!”
(Student Mathematician’s
Journal p. 7)
• Magic Number Trick Cards
(five cards prepared in advance;
see directions in “Initiate”
section)
• “Mathematician’s Journal Think
• Blackline master “Hint Cards”
Deeply About…” (Student
l
( 77)
M h
i i ’ J
M3L4-AMA_TE_00i-044.indd 29
Supplies
• Calculator
• Blank 3-by-5 index cards
(at least 10 per student)
• Colored markers
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chapter
overview
Teacher Guide — Understanding the Chapter Focus
chapter
1
Copyright © Kendall/Hunt Publishing Company
Thinking about
Variables and
Equations
Teaching Notes
Presents the chapter focus and
provides tips to the teacher on
teaching the chapter.
Teaching Notes
The 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.
Project M3: At the Mall with Algebra
M3L4-AMA_TE_045-122.indd 45
NOTES
Chapter 1: Thinking about Variables and Equations
Teaching Notes
Variables in equations where the degree of the variable is 1
(e.g., 4x1 – 2 = 10, N 1 ÷ 3= 12, x 1 + 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., x 2 + 2x = 8, 3x 2 = 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 n, 2 or I. 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.
45
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
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Points out what students may or mayoperation. In the equation, 3 + N = 10, students know that 3 should
be added to N. What they often do not comprehend is that the
not know and how to best introduceequal
sign indicates equality or balance between the expressions on
each side of the equal sign; 3 + N must have the same value as the
students to a new concept.
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.
NOTES
Teaching Notes
Presents an overview of a particular
lesson and
concept(s)
1 3the 7new
10
students will 6
be learning.
Lesson 3 asks students to solve variable puzzles as they
become comfortable with the idea of variables representing
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Lesson Overview
Teacher Guide–Lesson Planning Phase
Lesson
2
13
7
6 10
Number Tricks
Big Mathematical Ideas
Big Mathematical Ideas
In 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.
Describes the main concept
or idea the lesson will cover.
Copyright © Kendall/Hunt Publishing Company
Objectives
• Students will write equations using variables.
• Students will learn about the sums of even and odd numbers.
Objectives
• Students will solve equations.
Materials
A listing of what students are
expected to learn in the lesson.
Students
• “Mathematical Tricker y!” (Student Mathematician’s
Journal p. 7)
• “Mathematician’s Journal Think Deeply About…” (Student
Mathematician’s Journal pp. 9–11)
Materials
Lists the
Projectmaterials
M : At the Mall with Algebra
needed for the lesson.
3
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
NOTES
M3L4-AMA_TE_045-122.indd 67
67
Teacher
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• 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)
Mathematical Language
• Colored markers
Mathematical Language
Provides the mathematical
definition for vocabulary terms
covered within the lesson.
• 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.
Initiate
Ask students if they have ever seen a magic number trick.
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Lesson Overview
Teacher Guide–Lesson Planning Phase
Initiate
Card A
Card B
Card C
Card D
Card E
Front
1 3 5 7 9
Back
2 4 6 8 10
Initiate
The introductory
learning experience
designed to engage
students.
Copyright © Kendall/Hunt Publishing Company
Ask 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.
NOTES
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
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1:45:47
PMyou
should be in a location so that other students can see
them
but
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.
M3L4-AMA_TE_045-122.indd 68
Investigate
Copyright © Kendall/Hunt Publishing Company
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).
Provides an opportunity
for students to further
explore the concepts
in-depth in a hands-on
investigation.
Project M3: At the Mall with Algebra
Student Mathematician:
Date:
Mathematical Trickery!
Part I: Make the following cards for the math trick.
Card A
Card B
Card C
Card D
Card E
Front
1 3 5 7 9
Back
2 4 6 8 10
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?
Project M3: At the Mall with Algebra
7
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
13
7
6 10
Student Mathematician’s Journal p. 7
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
69
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Lesson Overview
Student Mathematician:
Date:
Mathematical Trickery! (continued)
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.
13
7
6 10
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
8
Project M3: At the Mall with Algebra
Student Mathematician’s Journal p. 8
Student Mathematician:
THINK
DEEPLY
Reduced Pages
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 + I ). 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 I , 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.
Reduced pages from
Student
Mathematician’s Journal
are pictured along with
page number.
Mathematical Communication
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.
Students reflect upon the
mathematical concepts in the
investigation using the Think
Deeply Questions, write about them
in their Mathematician’s Journals
and discuss them with a partner,
small group and/or the entire class.
Copyright © Kendall/Hunt Publishing Company
NOTES
Date:
Mathematician’s Journal
Mathematical Communication
Your Thoughts
and Questions
• 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.
1. Your best friend missed school but heard
that you learned a number trick in math
THINK class. You decide to send your friend your
PLY
DEE
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?
k!
bac
the
Use
• 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
Need more room?
Project M3: At the Mall with Algebra
9
13
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
• 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.
7
6 10
Student Mathematician’s Journal p. 9
70
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
M3L4-AMA_TE_045-122.indd 70
NOTES
What to Look for in Responses
After 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.
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
THINK
DEEPLY
Think Deeply
2. a. What is a variable?
b. How did you use a variable to write a rule
for your card trick?
Student Mathematician:
THINK
DEEPLY
Date:
Mathematician’s Journal
1/10/08 1:45:48 PM
2. a. What is a variable?
b. How did you use a variable to write a rule for your card
trick?
Student Responses
Describes to teacher what
they should be looking for in
student responses.
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.
Copyright © Kendall/Hunt Publishing Company
Think Deeply questions from
Student Mathematician’s
Journal in bold in Teacher
Guide.
• 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.
18
the
Use
k!
bac
Need more room?
11
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
13
7
6 10
Student Mathematician’s Journal p. 11
• 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.
Possible Difficulties
• 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.
Possible Difficulties
Potential problems that
might arise as students
attempt to answer questions.
Your Thoughts
and Questions
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Project M3: At the Mall with Algebra
M3L4-AMA_TE_045-122.indd 71
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
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Lesson Overview
NOTES
• 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.)
Sample Discussion
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 variable
means? Scott?
Sample narrative provides teacher with
example of what to expect from class
discussions. These are often written about
common student misconceptions to help
teachers in their class discussions.
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.
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.
Teacher: Ben?
Kyuri: I’m not sure, but I think you have to use
the same letter all the time for the even
numbers.
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.
72
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
M3L4-AMA_TE_045-122.indd 72
Think Beyond
Copyright © Kendall/Hunt Publishing Company
Teacher: Can we use a different letter, maybe a or m
to stand for the number of even numbers
in our number trick rule? Are Sum = 25 + a
and Sum = 25 + m and Sum = 25 + x [writes
these number sentences on the board] the
same rules?
Project M3: At the Mall with Algebra
Suggests ideas to
challenge students
even further. Answers
are provided.
NOTES
1 3
7
6 10
Copyright © Kendall/Hunt Publishing Company
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.
NOTES
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 x
represents 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.
Developing number tricks and writing rules for
the tricks is an enjoyable and challenging activity
for talented students. In the first Think Beyond
THINK
students are asked to figure out how the
BEYOND Card,
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.
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The second Think Beyond Card addresses what is
Project M3: At the Mall with Algebra
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
M3L4-AMA_TE_045-122.indd 73
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).
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
73
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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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Lesson Overview
Student Mathematician:
Date:
Answer Sheet
Mathematical Trickery!
Part I: Make the following cards for the math trick.
Card A
Card B
Card C
Card D
Card E
1 3 5 7 9
Front
2 4 6 8 10
Back
1. If you used only Card A and Card B to play the game, what is the minimum sum
possible?
4
Which numbers give the minimum sum?
6
What is the maximum possible sum?
1 and 3
2. If you used only Card A, Card B and Card C to play the game, what is the
minimum sum possible?
1, 3, 5
9
Which numbers give the minimum sum?
What is the maximum sum possible?
numbers give the maximum sum?
Copyright © Kendall/Hunt Publishing Company
Answers
Which numbers give the
2 and 4
maximum sum?
12
Answers to Student
Mathematician’s
Journal questions.
Which
2,Student
4, 6 Mathematician:
Mathematical Trickery! (continued)
3. What is special about the numbers used to get the minimum and maximum sums?
Date:
Answer Sheet
The minimum sum is the sum when all6.odd
numbers
arenumber
chosen,trick
andworks.
the
Explain
how the
maximum sum is the sum when all even numbers
chosen.
First you are
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
4. What number patterns do you see on a) the fronts?
b) the
backs?
c) the
front
to back?
a) all odd numbers
b) all even numbers
number to the minimum, and you get the sum.
c) the back is the next consecutive number
7. Write a rule using variables for how to find the sum of the numbers on the cards.
Sum
= 25
+ x,even
where x stands for the number of even
5. The person performing the number trick alwaysThe
asksrule
to beis:told
how
many
numbers are used. Why do you think you have to
know
how
many
even
numbers
numbered cards showing.
are used in order to predict the sum?
Make
own set
of number
Since the even numbers are 1 morePart
thanII:
the
oddyour
numbers,
you
need to trick cards.
up a setsum.
of cards to use for the number trick. You may use any number of
know how much extra you will add to8.theMake
minimum
cards. Indicate here what numbers are on the fronts and backs of your cards.
Answers will vary.75
Chapter 1: Thinking about Variables and Equations
Front: Lesson 2: Number Tricks
Project M3: At the Mall with Algebra
Back: Answers will vary.
M3L4-AMA_TE_045-122.indd 75
9. Write the directions for your number
trick.
1/10/08 1:45:50
PM
10. What is the minimum sum possible using your set of cards? Answers will vary.
11. What is the maximum sum possible using your set of cards? Answers will vary.
12. What question will you ask that will give you more information about the sum?
Students should ask a question that enables them to add or
subtract an amount to either the minimum or maximum sum.
13. Write a rule with variables that can be used to determine the sum of your numbers.
Copyright © Kendall/Hunt Publishing Company
Answers will vary. Look for clarity.
Answers will vary depending on the question and the numbers on
the cards.
20
76
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
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Lesson Overview
Think Beyond Cards
Blackline Masters of Think Beyond
Cards. Supports differentiation in the
classroom by challenging students.
Also available separately on
perforated cardstock.
Hint Cards
Blackline Masters of Hint Cards.
Supports differentiation in the
classroom by helping students
move forward with a problem.
Also available separately on
perforated cardstock.
21
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Lesson Overview
A Quick Look
A QUICK
LOOK
Chapter 1: Lesson 2
Number Tricks
A Quick Look
NOTES
Objectives:
Summarizes steps for teaching
the lesson and includes the
breakdown of the phases
within the lesson and the
suggested time.
• Students will write equations using variables.
• Students will learn about the sums of even and odd numbers.
• Students will solve equations.
1
Initiate: ( —
day)
2
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:
Copyright © Kendall/Hunt Publishing Company
Card A
Card B
Card C
Card D
Card E
Front
1 3 5 7 9
Back
2 4 6 8 10
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.
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Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
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Lesson Overview
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!
NOTES
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.)
1
Investigate: (1 —
days)
2
1. Refer students to “Mathematical Trickery!” in the Student
Mathematician’s Journal and have them work individually on
Part I.
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.
NOTES
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.
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.
7. When finished, have students complete Part II of
“Mathematical Trickery!” and write a rule for finding the sum
for their 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:
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.
Sum = 25 + x
Assessment: (1 day)
S = 25 + x
80
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
Copyright © Kendall/Hunt Publishing Company
• The second Think Deeply question asks students to define
a variable and describe how a variable is used to write a M3L4-AMA_TE_045-122.indd
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.
Project M3: At the Mall with Algebra
Sum = 25 + I
Some students might also suggest that the sum varies each
time; they’ll want to use a letter to represent that quantity.
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 formally
assess the lesson.
or
Copyright © Kendall/Hunt Publishing Company
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.
80
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
or
S = 25 + I
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Lesson Overview
Sample Pages
Pages 7, 8, 9, and 11 are from the unit
“At the Mall with Algebra” Student
Mathematical Journal.
Student Mathematician:
Date:
Mathematical Trickery!
Part I: Make the following cards for the math trick.
Card A
Card B
Card C
Card D
Card E
1 3 5 7 9
Front
2 4 6 8 10
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
Copyright © Kendall/Hunt Publishing Company
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?
Project M3: At the Mall with Algebra
24
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7
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
13
7
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Lesson Overview
Student Mathematician:
Date:
Mathematical Trickery! (continued)
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:
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.
13
7
6 10
M3L4-AMA_SE_03-25.indd 8
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
8
Copyright © Kendall/Hunt Publishing Company
9. Write the directions for your number trick.
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Lesson Overview
THINK
DEEPLY
Student Mathematician:
Date:
Mathematician’s Journal
Copyright © Kendall/Hunt Publishing Company
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?
Your Thoughts
and Questions
t he
Use
bac
k!
Need more room?
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26
M3L4-AMA_SE_03-25.indd 9
9
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
13
7
6 10
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Lesson Overview
THINK
DEEPLY
Student Mathematician:
Date:
Mathematician’s Journal
Copyright © Kendall/Hunt Publishing Company
2. a. What is a variable?
b. How did you use a variable to write a rule for your card
trick?
Your Thoughts
and Questions
b
t he
Use
!
ack
Need more room?
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M3L4-AMA_SE_03-25.indd 11
11
Chapter 1: Thinking about Variables and Equations
Lesson 2: Number Tricks
13
7
6 10
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Assessment
Multiple Assessment Options
Project M 3 provides the teacher with a variety of assessment tools.
Rubric
Student Mathematician’s Journal:
Guides for Teaching and Assessing
Rubric for Student Mathematician’s Journal
Assists the teacher with
teaching and assessing the
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.
An assessment stamp and ink pad is available
to expedite the process of grading students’
The
Overall, student demonstrates a partial understanding ofjournal
concepts and,work
if applicable,
uses grading technique is based
appropriate strategy but may have major errors. The student may not have answered all questions.
on the rubric that scores students on math
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. communication, and vocabulary.
concepts,
This
stamp can also be used in other classes.
MATHEMATICAL COMMUNICATION
1
Copyright © Kendall/Hunt Publishing Company
0
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.
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.
MATHEMATICAL VOCABULARY
0
Student does not use any mathematical vocabulary.
3
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M3L4-AMA_TE_00i-044.indd 19
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Student Mathematician’s Journal: Guides for Teaching and Assessing
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Assessment
THINK
DEEPLY
Student Mathematician:
Date:
Mathematician’s Journal
Think Deeply
Example of actual answer as it would
appear on a Think Deeply Student
Mathematician’s Journal. Each Teacher
Guide includes three different
scenarios of student answers.
2. Here is an equation: 7 + 2x = 9.
a. Explain two different ways to determine the value of x.
b. W hat is the value of x?
c. W hich method do you think is better? W hy?
Rubric Explanation
Copyright © Kendall/Hunt Publishing Company
Ties in student answers with
expectations of rubric.
Your Thoughts
and Questions
the
Use
Project M3: At the Mall with Algebra
Student Mathematician’s Journal: Guides for Teaching and Assessing
Concepts
Understands 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.
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.
k!
bac
Need more room?
20
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?
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Rubric Score
Lists score as tabulated from
student answers and how
well they met rubric criteria.
Vocabulary
Describes 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?”
Copyright © Kendall/Hunt Publishing Company
M3L4-AMA_TE_00i-044.indd 20
Teacher Feedback:
E
SCOR
3
2
epts –
Conc nication –
u
Comm lary – 1
bu
Voca
Great! I see how you used both the cover-up and working-backwards
methods to solve the equations. You say that you prefer the workingbackwards 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.
Teacher Feedback
Suggestions on providing feedback to
the student. Includes questions to
encourage and assess deeper
understanding.
Communication
Describes most of the steps taken
for each method. Needs to elaborate
more on how the value of x was
determined using the workingbackwards method.
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M3L4-AMA_TE_00i-044.indd 21
Student Mathematician’s Journal: Guides for Teaching and Assessing
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Assessment
Name
Date
Ongoing Assessment
Check-up 
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. An
Answer Key for the Checkups is included in the
Resources section at the end
of the book.
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,
Copyright © Kendall/Hunt Publishing Company
What number is next in this pattern? Explain why you think that number is the
next number.
3.
Connections to the
NCTM Standards ()
and Standardized Tests
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?
1. If 67 + I = 98, which of the following equations is also true?
a. 67 + 98 = I
a. add 25 and 12
b. 98 – 67 = I
b. subtract 12 from 25
c. 98 + I = 67
c. multiply 25 by 12
d. I – 98 = 67
d. divide 25 by 12
2. Fill in the missing numbers below to make the addition fact true.
3I6
+I47
683
Project M3: At the Mall with Algebra
Chapter 2: Equations that Go Together
189
3. What is the missing number?
• It is between 80 × 2 and 4 × 60.
M3L4-AMA_TE_123-206.indd 189
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• Its hundreds digit is greater than its ones digit.
a. 185
Found in the Resources
section of the Teacher Guide,
the Connections to the
Standards and Standardized
Tests is a supplemental
practice review based on
common standardized test
formats.
Copyright © Kendall/Hunt Publishing Company
b. 193
Test Practice
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
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30
Resources
219
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Assessment
Name
Date
Unit Test
Unit Test
Assesses student
understanding of unit
subject matter.
At 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
Unit Test Rubric: At the Mall with Algebra
Question 1
d. Number multiplied by 4, minus 2, results in an answer of 18
Mathematical Focus
Points
Expected Student Response
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
1a–1d
(max 2)
Copyright © Kendall/Hunt Publishing Company
Problem
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.
(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.)
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.)
Project M3: At the Mall with Algebra
Unit Overview
35
Focus: understanding the meaning of an equation and justifying a solution
Copyright © Kendall/Hunt Publishing Company
2
1e
(max 3)
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
M3L4-AMA_TE_00i-044.indd
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.)
35
1/10/08 1:47:32 PM
Unit Test Rubric
Suggests points to be
awarded based on
student answers to
Unit Test.
TOTAL POINTS: 5
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M3L4-AMA_TE_00i-044.indd 39
Unit Overview
39
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Page 32
Project M3: Mentoring
Mathematical Minds is a series of
12 curriculum units developed to
motivate and challenge
mathematically talented students
at the elementary level. Highlights
of this curriculum include:
math content focused on critical and
Advanced
creative problem solving and reasoning
Project M3 units have
been recognized by
the National
Association for Gifted
Children (NAGC)
with their Curriculum
Division Award.
Engaging investigations, projects and simulations
Rich verbal and written mathematical
communication
Students as practicing mathematicians
with NCTM content and process
Alignment
standards
LEVEL 3
LEVEL 4
LEVEL 5
Number
and
Operations
Unraveling the Mystery of
the MoLi Stone: Place Value
and Numeration
Factors, Multiples and
Leftovers: Linking
Multiplication and Division
Treasures from the Attic:
Exploring Fractions
Algebra
Awesome Algebra:
Looking for Patterns
and Generalizations
At the Mall with Algebra:
Working with Variables
and Equations
Record Makers and Breakers:
Using Algebra to Analyze
Change
Geometry and
Measurement
What’s the Me in
Measurement All About?
Getting into Shapes
Funkytown Fun House:
Focusing on Proportional
Reasoning and Similarity
Data
Analysis
and
Probability
Digging for Data: The Search Analyze This! Representing
within Research
and Interpreting Data
What Are Your Chances?
To order call 1-800-542-6657 or visit www.kendallhunt.com/m3
N3088602