differentiating instruction in the middle school math...
TRANSCRIPT
8/19/2009
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Differentiating Instruction in the Middle School Math Classroom
Essex Town School District
August 27, 2009
"In the end, all learners need your energy, your heart and your mind. They have that in common because they are young humans. How they need you however, differs. Unless we understand and respond to those differences, we fail many learners." *
* Tomlinson, C.A. (2001). How to differentiate instruction in mixed ability classrooms (2nd Ed.). Alexandria, VA: ASCD.
Nanci Smith, Ph.D.
Educational Consultant
Curriculum and Professional Development
Cave Creek, AZ
Differentiation
Is a teacher’s response to learner’s needs
Shaped by mindset & guided by general principles of differentiation
Continual assessment
Teachers can differentiate through
Content Process Product Affect/Environment
According to students’
Readiness
Through a variety of instructional strategies such as:
RAFTS…Graphic Organizers…Scaffolding Reading…Cubing…Think-Tac-Toe…Learning
Contracts…Tiering… Learning/Interest Centers… Independent Studies….Intelligence
Preferences…Orbitals…Complex Instruction…4MAT…Web Quests & Web Inquiry…ETC.
Respectful tasks Flexible groupingQuality Curriculum Bldg. Community
Interest Learning Profile
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What’s the point of differentiating in these different ways?
Readiness
Growth
InterestLearning Profile
Motivation Efficiency
READINESS
What does READINESS mean?
It is the student’s entry point
relative to a particular
understanding or skill.C.A.Tomlinson, 1999
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A Few Routes to READINESS DIFFERENTIATION
Varied texts by reading levelVaried supplementary materialsVaried scaffolding• reading• writing• research• technology
Tiered tasks and procedures Flexible time useSmall group instructionHomework optionsTiered or scaffolded assemssmentCompactingMentorshipsNegotiated criteria for qualityVaried graphic organizers
Developing a Tiered Activity
Select the activity organizer
•concept
•generalizationEssential to building
a framework of
understanding
Think about your students/use assessments
• readiness range
• interests
• learning profile
• talents
skills
reading
thinking
information
Create an activity that is
• interesting
• high level
• causes students to use
key skill(s) to understand
a key idea
Chart the
complexity of
the activity
High skill/
Complexity
Low skill/
complexity
Clone the activity along the ladder as
needed to ensure challenge and success
for your students, in• materials – basic to advanced• form of expression – from familiar to
unfamiliar• from personal experience to removed
from personal experience• equalizer
Match task to student based on
student profile and task
requirements
1
3
5
2
4
6
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Information, Ideas, Materials, Applications
Representations, Ideas, Applications, Materials
Resources, Research, Issues, Problems, Skills, Goals
Directions, Problems, Application, Solutions, Approaches, Disciplinary Connections
Application, Insight, Transfer
Solutions, Decisions, Approaches
Planning, Designing, Monitoring
Pace of Study, Pace of Thought
The Equalizer
1. Foundational Transformational
2. Concrete Abstract
1. Simple Complex
2. Single Facet Multiple Facets
3. Small Leap Great Leap
4. More Structured More Open
5. Less Independence Greater Independence
6. Slow Quick
Adding FractionsGreen Group
Use Cuisinaire rods or fraction circles to model simple fraction addition problems. Begin with common denominators and work up to denominators with common factors such as 3 and 6.
Explain the pitfalls and hurrahs of adding fractions by making a picture book.
Blue Group
Manipulatives such as Cuisinaire rods and fraction circles will be available as a resource for the group. Students use factor trees and lists of multiples to find common denominators. Using this approach, pairs and triplets of fractions are rewritten using common denominators. End by adding several different problems of increasing challenge and length.
Suzie says that adding fractions is like a game: you just need to know the rules. Write game instructions explaining the rules of adding fractions.
Red Group
Use Venn diagrams to model LCMs (least common multiple). Explain how this process can be used to find common denominators. Use the method on more challenging addition problems.
Write a manual on how to add fractions. It must include why a common denominator is needed, and at least three ways to find it.
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Graphing with a Point and a Slope
All groups:
• Given three equations in slope-intercept form, the students will graph the lines using a T-chart. Then they will answer the following questions:
• What is the slope of the line?
• Where is slope found in the equation?
• Where does the line cross the y-axis?
• What is the y-value of the point when x=0? (This is the y-intercept.)
• Where is the y-value found in the equation?
• Why do you think this form of the equation is called the “slope-intercept?”
Graphing with a Point and a Slope
Struggling Learners: Given the points
• (-2,-3), (1,1), and (3,5), the students will plot the points
and sketch the line. Then they will answer the following
questions:
• What is the slope of the line?
• Where does the line cross the y-axis?
• Write the equation of the line.
The students working on this particular task should repeat this
process given two or three more points and/or a point and a slope.
They will then create an explanation for how to graph a line starting
with the equation and without finding any points using a T-chart.
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Graphing with a Point and a Slope
Grade-Level Learners: Given an equation of a line in slope-intercept form (or several equations), the students in this group will:
• Identify the slope in the equation.
• Identify the y-intercept in the equation.
• Write the y-intercept in coordinate form (0,y) and plot the point on the y-axis.
• use slope to find two additional points that will be on the line.
• Sketch the line.
When the students have completed the above tasks, they will
summarize a way to graph a line from an equation without using a T-
chart.
Graphing with a Point and a SlopeAdvanced Learners: Given the slope-intercept form of the
equation of a line, y=mx+b, the students will answer the following questions:
• The slope of the line is represented by which variable?
• The y-intercept is the point where the graph crosses the y-axis. What is the x-coordinate of the y-intercept? Why will this always be true?
• The y-coordinate of the y-intercept is represented by which variable in the slope-intercept form?
Next, the students in this group will complete the following tasks given equations in slope-intercept form:
• Identify the slope and the y-intercept.
• Plot the y-intercept.
• Use the slope to count rise and run in order to find the second and third points.
• Graph the line.
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Make Card Games!
Assessments
The assessments used in this learning profile section can be downloaded at:
www.e2c2.com/fileupload.asp
Download any file:
“Profile Assessments for Cards”
“Profile Assessment WORD”
“Kid Friendly Learning Profile”
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INTEREST
What does INTEREST mean?
Discovering interest is important;
Creating interest is even
more important.
Inventing Better Schools, Schlechty
BRAIN RESEARCH SHOWS THAT. . .Eric Jensen, Teaching With the Brain in Mind, 1998
Choices vs. Requiredcontent, process, product no student voice
groups, resources environment restricted resources
Relevant vs. Irrelevantmeaningful impersonal
connected to learner out of contextdeep understanding only to pass a test
Engaging vs. Passive
emotional, energetic low interaction
hands on, learner input lecture seatwork
EQUALS
Increased intrinsic Increased
MOTIVATION APATHY & RESENTMENT
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-CHOICE-The Great Motivator!
• Requires children to be aware of their own readiness, interests, and learning profiles.
• Students have choices provided by the teacher. (YOU are still in charge of crafting challenging opportunities for all kiddos – NO taking the easy way out!)
• Use choice across the curriculum: writing topics, content writing prompts, self-selected reading, contract menus, math problems, spelling words, product and assessment options, seating, group arrangement, ETC . . .
• GUARANTEES BUY-IN AND ENTHUSIASM FOR LEARNING!
• Research currently suggests that CHOICE should be offered 35% of the time!!
How Do You Like to Learn?
1. I study best when it is quiet. Yes No
2. I am able to ignore the noise of
other people talking while I am working. Yes No
3. I like to work at a table or desk. Yes No
4. I like to work on the floor. Yes No
5. I work hard by myself. Yes No
6. I work hard for my parents or teacher. Yes No
7. I will work on an assignment until it is completed, no
matter what. Yes No
8. Sometimes I get frustrated with my work
and do not finish it. Yes No
9. When my teacher gives an assignment, I like to
have exact steps on how to complete it. Yes No
10. When my teacher gives an assignment, I like to
create my own steps on how to complete it. Yes No
11. I like to work by myself. Yes No
12. I like to work in pairs or in groups. Yes No
13. I like to have unlimited amount of time to work on
an assignment. Yes No
14. I like to have a certain amount of time to work on
an assignment. Yes No
15. I like to learn by moving and doing. Yes No
16. I like to learn while sitting at my desk. Yes No
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Differentiation Using
LEARNING PROFILE
• Learning profile refers to how an individual learns best - most efficiently and effectively.
• Teachers and their students may
differ in learning profile preferences.
Arithmetic is
answering the
questions,
Mathematics is
questioning the
answers.
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Learning Profile Factors
Group Orientation
independent/self orientation
group/peer orientation
adult orientation
combination
Learning Environment
quiet/noise
warm/cool
still/mobile
flexible/fixed
“busy”/”spare”
Cognitive Style
Creative/conforming
Essence/facts
Expressive/controlled
Nonlinear/linear
Inductive/deductive
People-oriented/task or Object oriented
Concrete/abstract
Collaboration/competition
Interpersonal/introspective
Easily distracted/long Attention span
Group achievement/personal achievement
Oral/visual/kinesthetic
Reflective/action-oriented
Intelligence Preference
analytic
practical
creative
verbal/linguistic
logical/mathematical
spatial/visual
bodily/kinesthetic
musical/rhythmic
interpersonal
intrapersonal
naturalist
existential
Gender
&
Culture
Parallel Lines Cut by a Transversal
• Visual: Make posters showing all the angle
relations formed by a pair of parallel lines
cut by a transversal. Be sure to color code
definitions and angles, and state the
relationships between all possible angles.1
2 34
5
6
7
8
Smith & Smarr, 2005
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Parallel Lines Cut by a Transversal
• Auditory: Play “Shout Out!!” Given the diagram below and commands on strips of paper (with correct answers provided), players take turns being the leader to read a command. The first player to shout out a correct answer to the command, receives a point. The next player becomes the next leader. Possible commands:
– Name an angle supplementary
supplementary to angle 1.
– Name an angle congruent
to angle 2.Smith & Smarr, 2005
12 3
456
78
Parallel Lines Cut by a Transversal
• Kinesthetic: Walk It Tape the diagram below on the floor with masking tape. Two players stand in assigned angles. As a team, they have to tell what they are called (ie: vertical angles) and their relationships (ie: congruent). Use all angle combinations, even if there is not a name or relationship. (ie: 2 and 7)
Smith & Smarr, 2005
12 3
45
6
7
8
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EIGHT STYLES OF LEARNINGTYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BY
LINGUISTIC
LEARNER
“The Word Player”
Learns through the
manipulation of words. Loves
to read and write in order to
explain themselves. They also
tend to enjoy talking
Read
Write
Tell stories
Memorizing
names, places,
dates and trivia
Saying, hearing and
seeing words
LOGICAL/
Mathematical
Learner
“The Questioner”
Looks for patterns when
solving problems. Creates a set
of standards and follows them
when researching in a
sequential manner.
Do experiments
Figure things out
Work with numbers
Ask questions
Explore patterns and
relationships
Math
Reasoning
Logic
Problem solving
Categorizing
Classifying
Working with abstract
patterns/relationships
SPATIAL
LEARNER
“The Visualizer”
Learns through pictures, charts,
graphs, diagrams, and art.Draw, build, design
and create things
Daydream
Look at pictures/slides
Watch movies
Play with machines
Imagining things
Sensing changes
Mazes/puzzles
Reading maps,
charts
Visualizing
Dreaming
Using the mind’s eye
Working with
colors/pictures
MUSICAL
LEARNER
“The Music
Lover”
Learning is often easier for
these students when set to
music or rhythm
Sing, hum tunes
Listen to music
Play an instrument
Respond to music
Picking up sounds
Remembering
melodies
Noticing pitches/
rhythms
Keeping time
Rhythm
Melody
Music
EIGHT STYLES OF LEARNING, Cont’d
TYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BY
BODILY/
Kinesthetic
Learner
“The Mover”
Eager to solve problems
physically. Often doesn’t read
directions but just starts on a
project
Move around
Touch and talk
Use body
language
Physical activities
(Sports/dance/
acting)
crafts
Touching
Moving
Interacting with space
Processing knowledge
through bodily sensations
INTERpersonal
Learner
“The Socializer”
Likes group work and
working cooperatively to
solve problems. Has an
interest in their community.
Have lots of
friends
Talk to people
Join groups
Understanding people
Leading others
Organizing
Communicating
Manipulating
Mediating conflicts
Sharing
Comparing
Relating
Cooperating
interviewing
INTRApersonal
Learner
“The Individual”
Enjoys the opportunity to
reflect and work
independently. Often quiet
and would rather work on
his/her own than in a group.
Work alone
Pursue own
interests
Understanding self
Focusing inward on
feelings/dreams
Pursuing interests/
goals
Being original
Working along
Individualized projects
Self-paced instruction
Having own space
NATURALIST
“The Nature
Lover”
Enjoys relating things to their
environment. Have a strong
connection to nature.
Physically
experience nature
Do observations
Responds to
patterning nature
Exploring natural
phenomenon
Seeing connections
Seeing patterns
Reflective Thinking
Doing observations
Recording events in Nature
Working in pairs
Doing long term projects
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Introduction to Change
(MI)
• Logical/Mathematical Learners: Given a set of data that
changes, such as population for your city or town over
time, decide on several ways to present the information.
Make a chart that shows the various ways you can present
the information to the class. Discuss as a group which
representation you think is most effective. Why is it most
effective? Is the change you are representing constant or
variable? Which representation best shows this? Be ready
to share your ideas with the class.
Introduction to Change
(MI)
• Interpersonal Learners: Brainstorm things that changeconstantly. Generate a list. Discuss which of the things change quickly and which of them change slowly. What would graphs of your ideas look like? Be ready to share your ideas with the class.
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Introduction to Change
(MI)
• Visual/Spatial Learners: Given a variety of graphs, discuss
what changes each one is
representing. Are the changes
constant or variable? How can you
tell? Hypothesize how graphs
showing constant and variable
changes differ from one another.
Be ready to share your ideas with
the class.
Introduction to Change
(MI)
• Verbal/Linguistic Learners: Examine articles from newspapers or magazines about a situation that involves changeand discuss what is changing. What is this change occurring in relation to? For example, is this change related to time, money, etc.? What kind of change is it: constant or variable? Write a summary paragraph that discusses the change and share it with the class.
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Learner Profile Card
Auditory, Visual, Kinesthetic
Modality
Multiple Intelligence Preference
Gardner
Analytical, Creative, Practical
Sternberg
Student’s
Interests
Array
Inventory
Gender Stripe
Nanci Smith,Scottsdale,AZ
Linear – Schoolhouse Smart - SequentialANALYTICAL
Thinking About the Sternberg Intelligences
Show the parts of _________ and how they work.
Explain why _______ works the way it does.
Diagram how __________ affects __________________.
Identify the key parts of _____________________.
Present a step-by-step approach to _________________.
Streetsmart – Contextual – Focus on UsePRACTICAL
Demonstrate how someone uses ________ in their life or work.
Show how we could apply _____ to solve this real life problem ____.
Based on your own experience, explain how _____ can be used.
Here’s a problem at school, ________. Using your knowledge of
______________, develop a plan to address the problem.
CREATIVE Innovator – Outside the Box – What If - Improver
Find a new way to show _____________.
Use unusual materials to explain ________________.
Use humor to show ____________________.
Explain (show) a new and better way to ____________.
Make connections between _____ and _____ to help us understand ____________.
Become a ____ and use your “new” perspectives to help us think about
____________.
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Understanding Order of Operations
Analytic Task
Practical Task
Creative Task
Make a chart that shows all ways you can think of to use order of operations to equal 18.
A friend is convinced that order of operations do not matter in math. Think of as many ways to convince your friend that without using them, you won’t necessarily get the correct answers! Give lots of examples.
Write a book of riddles that involve order of operations. Show the solution and pictures on the page that follows each riddle.
Forms of Equations of Lines
• Analytical Intelligence: Compare and contrast the various forms of equations of lines. Create a flow chart, a table, or any other product to present your ideas to the class. Be sure to consider the advantages and disadvantages of each form.
• Practical Intelligence: Decide how and when each form of the equation of a line should be used. When is it best to use which? What are the strengths and weaknesses of each form? Find a way to present your conclusions to the class.
• Creative Intelligence: Put each form of the equation of a line on trial. Prosecutors should try to convince the jury that a form is not needed, while the defense should defend its usefulness. Enact your trial with group members playing the various forms of the equations, the prosecuting attorneys, and the defense attorneys. The rest of the class will be the jury, and the teacher will be the judge.
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Circle Vocabulary
All Students:
Students find definitions for a list of
vocabulary (center, radius, chord, secant,
diameter, tangent point of tangency, congruent
circles, concentric circles, inscribed and
circumscribed circles). They can use
textbooks, internet, dictionaries or any other
source to find their definitions.
Circle Vocabulary
Analytical
Students make a poster to explain the definitions in their own words. Posters should include diagrams, and be easily understood by a student in the fifth grade.
Practical
Students find examples of each definition in the room, looking out the window, or thinking about where in the world you would see each term. They can make a mural, picture book, travel brochure, or any other idea to show where in the world these terms can be seen.
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Circle VocabularyCreative
Find a way to help us remember all this vocabulary! You can create a skit by becoming each term, and talking about who you are and how you relate to each other, draw pictures, make a collage, or any other way of which you can think.
OR
Role Audience Format Topic
Diameter Radius email Twice as nice
Circle Tangent poem You touch me!
Secant Chord voicemail I extend you.
Build – A – Square• Build-a-square is based on the “Crazy” puzzles where 9
tiles are placed in a 3X3 square arrangement with all edges matching.
• Create 9 tiles with math problems and answers along the edges.
• The puzzle is designed so that the correct formation has all questions and answers matched on the edges.
• Tips: Design the answers for the edges first, then write the specific problems.
• Use more or less squares to tier.
• Add distractors to outside edges and
“letter” pieces at the end.
m=3
b=6 -2/3
Nanci Smith
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The ROLE of writer, speaker,artist, historian, etc.
An AUDIENCE of fellow writers,students, citizens, characters, etc.
Through a FORMAT that is written, spoken, drawn, acted, etc.
A TOPIC related to curriculumcontent in greater depth.
electron
neutron
proton
R A F T
RAFT ACTIVITY ON FRACTIONS
Role Audience Format Topic
Fraction Whole Number Petitions To be considered Part of the
Family
Improper Fraction Mixed Numbers Reconciliation Letter Were More Alike than
Different
A Simplified Fraction A Non-Simplified Fraction Public Service
Announcement
A Case for Simplicity
Greatest Common Factor Common Factor Nursery Rhyme I’m the Greatest!
Equivalent Fractions Non Equivalent Personal Ad How to Find Your Soul Mate
Least Common Factor Multiple Sets of Numbers Recipe The Smaller the Better
Like Denominators in an
Additional Problem
Unlike Denominators in an
Addition Problem
Application form To Become A Like
Denominator
A Mixed Number that
Needs to be Renamed to
Subtract
5th Grade Math Students Riddle What’s My New Name
Like Denominators in a
Subtraction Problem
Unlike Denominators in a
Subtraction Problem
Story Board How to Become a Like
Denominator
Fraction Baker Directions To Double the Recipe
Estimated Sum Fractions/Mixed Numbers Advice Column To Become Well Rounded
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Angles Relationship RAFTRole Audience Format Topic
One vertical angle Opposite vertical angle Poem It’s like looking in a mirror
Interior (exterior) angle Alternate interior (exterior)
angle
Invitation to a family
reunion
My separated twin
Acute angle Missing angle Wanted poster Wanted: My complement
An angle less than 180 Supplementary
angle
Persuasive speech Together, we’re a straight angle
**Angles Humans Video See, we’re everywhere!
** This last entry would take more time than the previous 4 lines, and assesses a little differently. You could offer it as
an option with a later due date, but you would need to specify that they need to explain what the angles are, and anything
specific that you want to know such as what is the angle’s complement or is there a vertical angle that corresponds, etc.
Algebra RAFT
Role Audience Format Topic
Coefficient Variable Email We belong together
Scale / Balance Students Advice column Keep me in mind
when solving an
equation
Variable Humans Monologue All that I can be
Variable Algebra students Instruction manual How and why to
isolate me
Algebra Public Passionate plea Why you really do
need me!
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RAFT Planning Sheet
Know
Understand
Do
How to Differentiate:
• Tiered? (See Equalizer)
• Profile? (Differentiate Format)
• Interest? (Keep options equivalent in
learning)
• Other?
Role Audience Format Topic
Ideas for Cubing
• Arrange ________ into a 3-D collage to show ________
• Make a body sculpture to show ________
• Create a dance to show
• Do a mime to help us understand
• Present an interior monologue with dramatic movement that ________
• Build/construct a representation of ________
• Make a living mobile that shows and balances the elements of ________
• Create authentic sound effects to accompany a reading of _______
• Show the principle of ________ with a rhythm pattern you create. Explain to us how that works.
Ideas for Cubing in Math
• Describe how you would solve ______
• Analyze how this problem helps us use mathematical thinking and problem solving
• Compare and contrast this problem to one on page _____.
• Demonstrate how a professional (or just a regular person) could apply this kink or problem to their work or life.
• Change one or more numbers, elements, or signs in the problem. Give a rule for what that change does.
• Create an interesting and challenging word problem from the number problem. (Show us how to solve it too.)
• Diagram or illustrate the solutionj to the problem. Interpret the visual so we understand it.
CubingCubing
Cubing
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Nanci Smith
Describe how you would Explain the difference
solve or roll between adding and
the die to determine your multiplying fractions,
own fractions.
Compare and contrast Create a word problem
these two problems: that can be solved by
+
and (Or roll the fraction die to
determine your fractions.)
Describe how people use Model the problem
fractions every day. ___ + ___ .
Roll the fraction die to
determine which fractions
to add.
5
3
5
1
2
1
3
1
15
11
5
2
3
1
Nanci Smith
8/19/2009
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Nanci Smith
Describe how you would Explain why you need
solve or roll a common denominator
the die to determine your when adding fractions,
own fractions. But not when multiplying.
Can common denominators
Compare and contrast ever be used when dividing
these two problems: fractions?
Create an interesting and
challenging word problem
A carpet-layer has 2 yards that can be solved by
of carpet. He needs 4 feet ___ + ____ - ____.
of carpet. What fraction of Roll the fraction die to
his carpet will he use? How determine your fractions.
do you know you are correct?
Diagram and explain the
solution to ___ + ___ + ___.
Roll the fraction die to
determine your fractions.
91
1
7
3
13
2
7
1
7
3 and
2
1
3
1
Level 1:
1. a, b, c and d each represent a different value. If a = 2, find b, c, and d.
a + b = c
a – c = d
a + b = 5
2. Explain the mathematical reasoning involved in solving card 1.
3. Explain in words what the equation 2x + 4 = 10 means. Solve the problem.
4. Create an interesting word problem that is modeled by
8x – 2 = 7x.
5. Diagram how to solve 2x = 8.
6. Explain what changing the “3” in 3x = 9 to a “2” does to the value of x. Why is this true?
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Level 2:
1. a, b, c and d each represent a different value. If a = -1, find b, c, and d.
a + b = c
b + b = d
c – a = -a
2. Explain the mathematical reasoning involved in solving card 1.
3. Explain how a variable is used to solve word problems.
4. Create an interesting word problem that is modeled by
2x + 4 = 4x – 10. Solve the problem.
5. Diagram how to solve 3x + 1 = 10.
6. Explain why x = 4 in 2x = 8, but x = 16 in ½ x = 8. Why does this make sense?
Level 3:
1. a, b, c and d each represent a different value. If a = 4, find b, c, and d.
a + c = b
b - a = c
cd = -d
d + d = a
2. Explain the mathematical reasoning involved in solving card 1.
3. Explain the role of a variable in mathematics. Give examples.
4. Create an interesting word problem that is modeled by
. Solve the problem.
5. Diagram how to solve 3x + 4 = x + 12.
6. Given ax = 15, explain how x is changed if a is large or a is small in value.
7513 xx