Learner Profile Card
Gender Stripe
Auditory, Visual, Kinesthetic
Analytical, Creative, Practical
Modality
Sternberg
Student’s
Interests
Multiple Intelligence Preference
Gardner
Array
Inventory
Array Interaction Inventory
Directions:
• Rank order the responses in rows below on a scale from 1 to 4 with 1 being “least like me” to 4 being “most like me”.
• After you have ranked each row, add down each column.
• The column(s) with the highest score(s) shows your primary Personal Objective(s) in your personality.
In your normal day-to-day life, you tend to be:
Nurturing
Sensitive
Caring
Logical
Systematic
Organized
Spontaneous
creative
Playful
Quiet
Insightful
reflective
Stimulation
Having fun is
important
Reflection
Having some time
alone is important
Active
Opportunistic
Spontaneous
Inventive
Competent
Seeking
Impetuous
Impactful
Daring
Conceptual
Knowledgeable
Composed
In your normal day-to-day life, you tend to value:
Harmony
Relationships are
important
Work
Time schedules are
important
In most settings, you are usually:
Authentic
Compassionate
Harmonious
Traditional
Responsible
Parental
In most situations, you could be described as:
Empathetic
Communicative
Devoted
Practical
Competitive
Loyal
Array Interaction Inventory, cont’d
You approach most tasks in a(n) _________ manner:
Affectionate
Inspirational
Vivacious
Conventional
Orderly
Concerned
Courageous
Adventurous
Impulsive
Rational
Philosophical
Complex
When things start to “not go your way” and you are tired and worn down, what might your responses be?
Say “I’m sorry”
Make mistakes
Feel badly
Over-control
Become critical
Take charge
“It’s not my fault”
Manipulate
Act out
Withdraw
Don’t talk
Become indecisive
When you’ve “had a bad day” and you become frustrated, how might you respond?
Over-please
Cry
Feel depressed
Be perfectionistic
Verbally attack
Overwork
Become physical
Be irresponsible
Demand attention
Disengage
Delay
Daydream
Production
Connection
Status Quo
Add score:
Harmony
Personal Objectives/Personality Components
Teacher and student personalities are a critical element in the classroom dynamic. The Array Model
(Knaupp, 1995) identifies four personality components; however, one or two components(s) tend to greatly
influence the way a person sees the world and responds to it. A person whose primary Personal Objective of
Production is organized, logical and thinking-oriented. A person whose primary Personal Objective is
Connection is enthusiastic, spontaneous and action-oriented. A person whose primary Personal Objective is
Status Quo is insightful, reflective and observant. Figure 3.1 presents the Array model descriptors and offers
specific Cooperative and Reluctant behaviors from each personal objective.
Personal Objectives/Personality Component
HARMONY
PRODUCTION
CONNECTION
STATUS QUO
COOPERATIVE
(Positive Behavior)
Caring
Sensitive
Nurturing
Harmonizing
Feeling-oriented
Logical
Structured
Organized
Systematic
Thinking-oriented
Spontaneous
Creative
Playful
Enthusiastic
Action-oriented
Quiet
Imaginative
Insightful
Reflective
Inaction-oriented
RELUCTANT
(Negative Behavior)
Overadaptive
Overpleasing
Makes mistakes
Cries or giggles
Self-defeating
Overcritical
Overworks
Perfectionist
Verbally attacks
Demanding
Disruptive
Blames
Irresponsible
Demands attention
Defiant
Disengaging
Withdrawn
Delays
Despondent
Daydreams
PSYCHOLOGICAL
NEEDS
Friendships
Sensory experience
Task completion
Time schedule
Contact with people
Fun activities
Alone time
Stability
WAYS TO MEET
NEEDS
Value their feelings
Comfortable work place
Pleasing learning
environment
Work with a friend
sharing times
Value their ideas
Incentives
Rewards
Leadership positions
Schedules
To-do lists
Value their activity
Hands-on activities
Group interaction
Games
Change in routine
Value their privacy
Alone time
Independent activities
Specific directions
Computer activities
Routine tasks
There are two
keys to
differentiation:
1. Know your kids
2.Know your
content
“In times of change,
the learners inherit the earth
while the learned find
themselves beautifully equipped
to deal with a world that
no longer exists.”
Eric Hoffer
It Begins with Good Instruction
The greatest enemy to
understanding is
coverage.
Howard Gardner
These are the
facts, vocabulary, dates,
places, names, and examples you want
students to give
you.
Facts 2X3=6,
The know is massively
forgettable.
 b  b  4ac
2a
2
Vocabulary
numerator, slope
“Teaching facts in isolation is like
trying to pump water uphill.”
-Carol Tomlinson
KNOW (Facts,
Vocabulary, Definitions)
• Definition of numerator and
denominator
• The quadratic formula
• The Cartesian coordinate plane
• The multiplication tables
Skills
• Basic skills of any discipline
• Thinking skills
• Skills of planning, independent learning, etc.
The skill portion encourages the students to “think”
like the professionals who use the knowledge and skill
daily as a matter of how they do business. This is what
it means to “be like” a mathematician, an analyst, or an
economist.
Research about teaching
suggests that learning by
struggling at first with a concept
enables students to benefit from
an explanation that brings the
ideas together (Schwartz & Bransford,
2000).
BE ABLE TO DO (Skills: Basic Skills,
Skills of the Discipline, Skills of Independence,
Social Skills, Skills of Production)
•
•
•
•
•
•
•
Describe these using verbs or phrases:
Analyze, test for meaning
Solve a problem to find perimeter
Generalize your procedure for any situation
Evaluate work according to specific criteria
Contribute to the success of a group or team
Use graphics to represent data appropriately
Juicy Verbs
compose
influence
adopt
unify
devise
promote
elaborate
designate
detail
substitute
merchandize
limit
deconstruct
prove
formulate
structure
predict
simulate
shadow
illustrate
propose
tailor
inscribe
refresh
eliminate
transform
wonder
transfer
improve
advise
visualize
reflect
expand
emphasize
access
concentrate
minimize
convert
immerse
approximate
connect
ponder
justify
regroup
portray
design
compete
simulate
incorporate
concentrate
disguise
modify
produce
compartmentalize
personify
anchor
energize
integrate
uncover
deviate
Certain methods of teaching,
particularly those that emphasize
memorization as an end in itself tend
to produce knowledge that is
seldom, if ever, used. Students who
learn to solve problems by following
formulas, for example, often are
unable to use their skills in new
situations. (Redish, 1996)
It Begins with Good Instruction
Adding It Up (National Research Council) –
Rule-based instructional approaches that do not
give students opportunities to create meaning for
the rule, or to learn when to use them, can lead to
forgetting, unsystematic errors, reliance on visual
clues, and poor strategic decisions.
Research about teaching suggests
learning may be hindered by
• isolated sets of facts that are not
organized and connected or organizing
principles without sufficient knowledge
to make them meaningful (NRC, 1999)
• Students have become accustomed to
receiving an arbitrary sequence of
exercises with no overarching
rationale.”
(Black and Wiliam, 1998))
Major Concepts and
Subconcepts
These are the written statements of truth, the core to the
meaning(s) of the lesson(s) or unit. These are what connect the
parts of a subject to the student’s life and to other subjects.
It is through the understanding component of instruction that we
teach our students to truly grasp the “point” of the lesson or the
experience.
Understandings are purposeful. They focus on the key ideas
that require students to understand information and
make connections while evaluating the relationships that exist
within the understandings.
UNDERSTAND (Essential Truths That
Give Meaning to the Topic)
Begin with I want students to understand THAT…
– Multiplication can have different meanings in
different contexts, including repeated addition,
groups and creation of area.
– Fractions always represent a relationship of parts
and wholes.
– Addition and subtraction show a final count of the
same thing.
– Functions can be represented in many ways
(graphs, words, tables, equations) but all
representations are of the same function.
Some questions for identifying truly
“big ideas”
– Does it have many layers and nuances, not obvious to the
naïve or inexperienced person?
– Do you have to dig deep to really understand its
meanings and implications even if you have a surface
grasp of it?
– Is it (therefore) prone to misunderstanding as well as
disagreement?
– Does it yield optimal depth and breadth of insight into the
subject?
– Does it reflect the core ideas as judged by experts?
Hints for Writing Essential Understandings
Essential understandings synthesize ideas to show an important
relationship, usually by combining two or more concepts.
For example:
People’s perspectives influence their behavior.
Time, location, and events shape cultural beliefs and practices.
Tips:
• When writing essential understandings, verbs should be active and in
the present tense to ensure that the statement is timeless.
• Don’t use personal nouns- they cause essential understanding to
become too specific, and it may become a fact.
• Make certain that an essential understanding reflects a relationship of
two or more concepts.
• Write essential understandings a complete sentences.
• Ask the question: What are the bigger ideas that transfer to other
situations.
Concepts
Some concepts
• span across several subject areas
• represent significant ideas, phenomena,intellectual process,
or persistent problems
• Are timeless
• Can be represented though different examples, with all
examples having the same attributes
• And universal
For example, the concepts of patterns, interdependence,
symmetry, system and power can be examined in a variety of
subjects or even serve as concepts for a unit that integrates
several subjects.
Discipline-based Concepts
•
Art-color, shape, line, form, texture, negative space
Literature-perception, heroes and antiheroes, motivation, interactions, voice
Mathematics-number, ratio, proportion, probability, quantification
Music-pitch, melody, tempo, harmony, timbre
•
Physical Education-movement, rules, play, effort, quality, space, strategy
Science-classification, evolution, cycle, matter, order
Social Science- governance, culture, revolution, conflict, and cooperation
Mortimer Adler’s List of the Most Important Concepts in Western Civilization
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
Angel
Animal
Aristocracy
Art
Astronomy
Beauty
Being
Cause
Chance
Change
Citizen
Constitution
Courage
Custom and convention
Definition
Democracy
Desire
Dialectic
Duty
Education
Element
Emotion
Eternity
Evolution
Experience
Family
Fate
Form
God
Good and Evil
Government
Habit
Happiness
History
Honor
hypothesis
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
Idea
Immortality
Induction
Infinity
Judgment
Justice
Labor
Language
Law
Liberty
Life and death
Logic
Love
Man
Mathematics
Matter
Mechanics
Medicine
Memory/Imagination
Metaphysics
Mind
Monarchy
Nature
Necessity
Oligarchy
One and Many
Opinion
Opposition
Philosophy
Physics
Pleasure and Pain
Poetry
Principle
Progress
Prophecy
Prudence
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
Punishment
Quality
Quantity
Reasoning
Relation
Religion
Revolution
Rhetoric
Same/Other
Science
Sense
Sign/Symbol
Sin
Slavery
Soul
Space
State
Temperance
Theology
Time
Truth
Tyranny
Universe
Virtue/Vice
War & Peace
Wealth
Will
Wisdom
World
• Mathematical Concepts
Number
Error / Uncertainty
Ratio
Measurement
Proportion
Behavior
Symmetry Relationships
Pattern
ProbabilityFunction
Truth
Order
Problem solving
Change
System
Quantification
Prediction
Representation
• Mathematical Understandings
Our number system maintains order and is rich with patterns.
Mathematicians quantify data in order to establish real-world
probabilities.
All measurement involves error and uncertainty.
Strands of Mathematical Proficiency
Adding It Up, 2001
•
•
•
•
•
Conceptual Understanding
Procedural Fluency
Strategic Competence
Adaptive Reasoning
Productive Disposition
Adding It Up: Helping Children Learn Mathematics,
NRC, 2001
Strands of Mathematical Proficiency:
Adding It Up, 2001
• Conceptual Understanding Comprehension of
mathematical concepts,
operations and relations
Strands of Mathematical Proficiency:
Adding It Up, 2001
• Conceptual Understanding “Refers to an integrated and functional grasp of
mathematical ideas. Students with conceptual
understanding know more than isolated facts
and methods. They understand why a
mathematical idea is important and the kinds of
contexts in which it is useful. They have
organized their knowledge into a coherent
whole, which enables them to learn new ideas by
connecting those ideas to what they already
know. Conceptual understanding also supports
retention.” P. 118
Strands of Mathematical Proficiency:
Adding It Up, 2001
• Procedural Fluency Skill in carrying out procedures
flexibly, accurately, efficiently,
and appropriately
Strands of Mathematical Proficiency:
Adding It Up, 2001
• “Understanding makes learning skills
easier, less susceptible to common
errors, and less prone to forgetting.
By the same token, a certain level of
skill is required to learn many
mathematical concepts with
understanding.” Page 122
Strands of Mathematical Proficiency:
Adding It Up, 2001
• Strategic Competence Ability to formulate, represent,
and solve mathematical
problems, especially with
multiple approaches.
Strands of Mathematical Proficiency:
Adding It Up, 2001
• Adaptive Reasoning Capacity for logical thought,
reflection, explanation, and
justification
Strands of Mathematical Proficiency:
Adding It Up, 2001
• Productive Disposition Habitual inclination to see
mathematics as sensible, useful,
and worthwhile, coupled with a
belief in diligence and one’s
own efficacy
Research suggests that
learning is enhanced by
providing opportunities for
•
•
•
•
•
Struggling
Choosing and evaluating strategies
Contrasting cases
Organizing information
Making connections (NRC, 1999)
Dividing Fractions
Demonstrating the 5 Strands
• What does it mean to divide? What
meanings does division have?
– Repeated subtraction
– Partitioning or dividing up into groups
– Measurement (fits into)
• Of these meanings, which one works with
dividing fractions?
Dividing Fractions
Demonstrating the 5 Strands
• Measurement Model
6 Divided by 2
Can you think of examples where you would need
to divide fractions?
Dividing Fractions
Demonstrating the 5 Strands
• Dividing fractions with
fraction strips
4 1

5 5
4 1
 4
5 5
Dividing Fractions
Demonstrating the 5 Strands
• Try dividing some fractions with like
denominators on your own using the
fraction strip model
• Share your findings.
• Do you see a pattern in dividing fractions
with like denominators?
Dividing Fractions
Demonstrating the 5 Strands
• Do you know a rule that can help speed up
the process for dividing fractions without
strips?
• Can you think of a way to use the pattern
discovered with dividing common
denominators to make sense of this rule?
Dividing Fractions
Demonstrating the 5 Strands
2 1
 
3 2
4 3
 
6 6
4
43
3
Write problem with
common denominators
Divide the numerators
2 1 4
 
3 2 3
Dividing Fractions
Demonstrating the 5 Strands
• Now relate the pattern to the algorithm of invert and
multiply…
• Where does the common denominator come from?
2 1 2  2 1 3 4 3 4
 

  
3 2 3 2 2  3 6 6 3
2 1 22 4
Invert and multiply!
 

3 2 3 1 3
Dividing Fractions
Demonstrating the 5 Strands
• How can knowing how to divide fractions help you
in your life?
• Think of as many ideas as you can for the benefits
of knowing how to divide fractions!
Fraction Activity
• What went well for
you?
• What was a challenge
for you?
• What did you learn
from this activity?
USE OF INSTRUCTIONAL
STRATEGIES.
The following findings related to
instructional strategies are supported by
the existing research:
• Techniques and instructional strategies have nearly as much influence on student
learning as student aptitude.
• Lecturing, a common teaching strategy, is an effort to quickly cover the material:
however, it often overloads and over-whelms students with data, making it likely
that they will confuse the facts presented
• Hands-on learning, especially in science, has a positive effect on student
achievement.
• Teachers who use hands-on learning strategies have students who out-perform
their peers on the National Assessment of Educational progress (NAEP) in the
areas of science and mathematics.
• Despite the research supporting hands-on activity, it is a fairly uncommon
instructional approach.
• Students have higher achievement rates when the focus of instruction is on
meaningful conceptualization, especially when it emphasizes their own knowledge
of the world.
Make Card Games!
Make Card Games!
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.
m=3
• Add distractors to outside edges and
b=6
-2/3
“letter” pieces at the end.
Nanci Smith
Flippers!
•
•
•
•
•
•
•
•
You will need 2 sheets of construction paper, of different colors. (You’ll only
use ½ a sheet of the second color though.)
Fold the frame color into fourths horizontally (hamburger folds).
Back-fold the same piece in the opposite directions so that it is well creased
and flexible.
Fold the frame at the center only, and make cuts from the fold up to the next
fold line. 7 cuts for 8 sections is easy to do, but cut as many as you like.
Fold the second color of paper into fourths as well. Cut these apart. You will
only use 2 of the strips.
Basket-weave the two strips into the cut strips of the frame. The two sides
need to be woven in opposite directions.
To use the flipper, write questions on the woven colors. To find the answers,
fold the flipper so that the center is pointed at you, then pull the center apart to
reveal answer spaces.
Flipper works in this way on both sides!
Nanci Smith, 2004
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
Angles Relationship RAFT
Role
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
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!
Keep me in mind
when solving an
equation
ROLE
AUDIENCE
FORMAT
TOPIC
Equivalent Fractions
Farmers
Poster Ad for Fertilizer
How do I get bigger
Event
Mutually exclusive
event
Love letter
We’ll never be
together… sob, sob, sob
3 line segments
Polygons
Application
Do we belong?
Pythagoras
Home Buyers
Floor plan
It’s hip to be Square!
Basic Facts
Students working on a
multi-step problem
Persuasive Speech
You need me!
Denominator
Numerator
Song
You’re a part of me
Equivalent Fractions
TV viewers
Reality TV Show
Biggest Reducer
Divisor
Dividend
Rap Song
Let me Count the ways
3-D shapes
Humans
Photo Journal
Where do you find me?
Area of Circle
Humans
Sales Ad
Get the most pi for your
dollar
Scientific Notation
Large numbers
Health Ad
The benefits of being
small
Radius
Diameter
Letter
How do I fit into your
life?
Scale
Map
Poem
Why do we need to be
together
2 line segments
All segments
Wanted Poster for a
complete triangle
Are you our missing
link?
Multiples
Factors
Storyboard
To Infinity and
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
CUBING
1.
Describe it: Look at the subject closely
(perhaps with your senses as well as
your mind)
2.
Compare it: What is it similar to?
What is it different from?
3.
Associate it: What does it make you
think of? What comes to your mind
when you think of it? Perhaps people?
Places? Things? Feelings? Let your
mind go and see what feelings you have
for the subject.
4.
Analyze it: Tell how it is made? What
are it’s traits and attributes?
5.
Apply it: Tell what you can do with it.
How can it be used?
6.
Argue for it or against it: Take a stand.
Use any kind of reasoning you want –
logical, silly, anywhere in between.
•
•
•
•
•
•
•
•
•
Or you can . . . .
Rearrange it
Illustrate it
Question it
Satirize it
Evaluate it
Connect it
Cartoon it
Change it
Solve it
Cubing
Cubing
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.
Cubing
•
•
•
•
•
•
•
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.
Multiplication Think Dots
• Struggling to Basic Level
It’s easy to remember how to multiply by 0 or 1! Tell how to remember.
Jamie says that multiplying by 10 just adds a 0 to the number. Bryan
doesn’t understand this, because any number plus 0 is the same number.
Explain what Jamie means, and why her trick can work.
Explain how multiplying by 2 can help with multiplying by 4 and 8. Give at
least 3 examples.
We never studied the 7 multiplication facts. Explain why we didn’t need to.
Jorge and his ____ friends each have _____ trading cards. How many
trading cards do they have all together? Show the answer to your problem
by drawing an array or another picture. Roll a number cube to determine
the numbers for each blank.
What is _____ X _____? Find as many ways to show your answer as
possible.
Multiplication Think Dots
• Middle to High Level
There are many ways to remember multiplication facts. Start with 0 and go through 10 and tell
how to remember how to multiply by each number. For example, how do you remember how
to multiply by 0? By 1? By 2? Etc.
There are many patterns in the multiplication chart. One of the patterns deals with pairs of
numbers, for example, multiplying by 3 and multiplying by 6 or multiplying by 5 and
multiplying by 10. What other pairs of numbers have this same pattern? What is the pattern?
Russell says that 7 X 6 is 42. Kadi says that he can’t know that because we didn’t study the 7
multiplication facts. Russell says he didn’t need to, and he is right. How might Russell know
his answer is correct?
Max says that he can find the answer to a number times 16 simply by knowing the answer to
the same number times 2. Explain how Max can figure it out, and give at least two examples.
Alicia and her ____ friends each have _____ necklaces. How many necklaces do they have all
together? Show the answer to your problem by drawing an array or another picture. Roll a
number cube to determine the numbers for each blank.
What is _____ X _____? Find as many ways to show your
answer as possible.
Describe how you would
1 3

5 5
solve
or roll
the die to determine your
Explain the difference
between adding and
multiplying fractions,
own fractions.
Compare and contrast
Create a word problem
these two problems:
that can be solved by
+
Nanci Smith
1 2 11
 
3 5 15
and
(Or roll the fraction die to
1 1

3 2
determine your fractions.)
Describe how people use
Model the problem
fractions every day.
___ + ___ .
Roll the fraction die to
determine which fractions
to add.
Nanci Smith
Describe how you would
solve
2 3 1
 
13 7 91
or roll
Explain why you need
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?
1 1
3 1
 and 
3 2
7 7
Create an interesting and
challenging word problem
Nanci Smith
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.
Designing a Differentiated Learning
Contract
A Learning Contract has the following components
1. A Skills Component
Focus is on skills-based tasks
Assignments are based on pre-assessment of students’ readiness
Students work at their own level and pace
2. A content component
Focus is on applying, extending, or enriching key content (ideas, understandings)
Requires sense making and production
Assignment is based on readiness or interest
3. A Time Line
Teacher sets completion date and check-in requirements
Students select order of work (except for required meetings and homework)
4. The Agreement
The teacher agrees to let students have freedom to plan their time
Students agree to use the time responsibly
Guidelines for working are spelled out
Consequences for ineffective use of freedom are delineated
Signatures of the teacher, student and parent (if appropriate) are placed on the agreement
Differentiating Instruction: Facilitator’s Guide, ASCD, 1997
MY CONTRACT
Date
Student Name
What I will do
What I will use
When I will finish
How I feel about my project
because
Student signature
How my teacher feels about my project
because
Teacher’s Signature
Learning Contract
Chapter: _______
Name:______________________
Ck Page/Concept
Ck Page/Concept
Ck Page/Concept
___ ___________
___ ___________
___ ___________
___ ___________
___ ___________
___ ___________
___ ___________
___ ___________
___ ___________
___ ___________
___ ___________
___ ___________
Enrichment Options: ______________________________________________
Special Instructor
______________________________ _____ _____ _____ _____ _____ _____ _____
______________________________ _____ _____ _____ _____ _____ _____ _____
______________________________ _____ _____ _____ _____ _____ _____ _____
Your Idea:
______________________________ _____ _____ _____ _____ _____ _____ _____
Working Conditions
________________________________________________________________________
________________________________________________________________________
_________________________________
___________________________________
Teacher’s signature
Student’s signature
Work Log
Date Goal
Actual
The Red Contract
Key Skills: Graphing and Measuring
Key Concepts: Relative Sizes
Note to User: This is a Grade 3 math contract for students below grade level in these skills
Read
Apply
Extend
How big
is a foot?
Work with a
friend to graph
the size of at
least 6 things
on the list of
“10 terrific
things.” Label
each thing with
how you know
the size
Make a
group story
or one of
your own –
that uses
measuremen
t and at least
one graph.
Turn it into a
book at the
author center
The Green Contract
Key Skills: Graphing and Measuring
Key Concepts: Relative Sizes
Note to User: This is a Grade 3 math contract for students at or near grade level in these skills
Read
Apply
Extend
Alexande
r Who
Used to
be Rich
Last
Sunday or
Ten Kids,
No Pets
Complete the
math madness
book that goes
with the story
you read.
Now, make a
math
madness
book based
on your
story about
kids and pets
or money
that comes
and goes.
Directions
are at the
author center
The Blue Contract
Key Skills: Graphing and Measuring
Key Concepts: Relative Sizes
Note to User: This is a Grade 3 math contract for students advanced in these skills
Read
Apply
Extend
Dinosaur
Before
Dark or
Airport
Control
Research a kind of
dinosaur or
airplane. Figure
out how big it is.
Graph its size on
graph paper or on
the blacktop
outside our room.
Label it by name
and size
Make a book
in which you
combine math
and dinosaurs
or airplanes,
or something
else big. It
can be a
number fact
book, a
counting
book, or a
problem
book.
Instructions
are at the
author center
Proportional Reasoning
Think-Tac-Toe
□
Create a word problem that
requires proportional
reasoning. Solve the
problem and explain why it
requires proportional
reasoning.
□
Find a word problem from
the text that requires
proportional reasoning.
Solve the problem and
explain why it was
proportional.
□
Think of a way that you use
proportional reasoning in your
life. Describe the situation,
explain why it is proportional
and how you use it.
□
Create a story about a
proportion in the world.
You can write it, act it,
video tape it, or another
story form.
□
How do you recognize a
proportional situation?
Find a way to think about
and explain proportionality.
□
Make a list of all the
proportional situations in the
world today.
□
Create a pict-o-gram, poem
or anagram of how to solve
proportional problems
□
Write a list of steps for
solving any proportional
problem.
□
Write a list of questions to ask
yourself, from encountering a
problem that may be
proportional through solving
it.
Directions: Choose one option in each row to complete. Check the box of the choice you make, and turn
this page in with your finished selections.
Nanci Smith, 2004
Menu Planner
Use this template to help you plan a menu for your classroom
Menu: ____________________
Due: All items in the main dish and the specified number of side dishes must be
completed by the due date. You may select among the side dishes and you may decide to
do some of the dessert items as well.
.........................................................
Main Dish (complete all)



...........................................................
Side Dish (select ____)


..........................................................
Dessert


Winning Strategies for Classroom Management
Similar Figures Menu
Imperatives (Do all 3):
1. Write a mathematical definition of “Similar Figures.” It
must include all pertinent vocabulary, address all
concepts and be written so that a fifth grade student
would be able to understand it. Diagrams can be used to
illustrate your definition.
2. Generate a list of applications for similar figures, and
similarity in general. Be sure to think beyond “find a
missing side…”
3. Develop a lesson to teach third grade students who are
just beginning to think about similarity.
Similar Figures Menu
Negotiables (Choose 1):
1. Create a book of similar figure applications and
problems. This must include at least 10 problems. They
can be problems you have made up or found in books,
but at least 3 must be application problems. Solver each
of the problems and include an explanation as to why
your solution is correct.
2. Show at least 5 different application of similar figures in
the real world, and make them into math problems.
Solve each of the problems and explain the role of
similarity. Justify why the solutions are correct.
Similar Figures Menu
Optionals:
1. Create an art project based on similarity. Write a cover
sheet describing the use of similarity and how it affects
the quality of the art.
2. Make a photo album showing the use of similar figures
in the world around us. Use captions to explain the
similarity in each picture.
3. Write a story about similar figures in a world without
similarity.
4. Write a song about the beauty and mathematics of
similar figures.
5. Create a “how-to” or book about finding and creating
similar figures.
Begin Slowly – Just Begin!
Low-Prep Differentiation
Choices of books
Homework options
Use of reading buddies
Varied journal Prompts
Orbitals
Varied pacing with anchor options
Student-teaching goal setting
Work alone / together
Whole-to-part and part-to-whole explorations
Flexible seating
Varied computer programs
Design-A-Day
Varied Supplementary materials
Options for varied modes of expression
Varying scaffolding on same organizer
Let’s Make a Deal projects
Computer mentors
Think-Pair-Share by readiness, interest, learning profile
Use of collaboration, independence, and cooperation
Open-ended activities
Mini-workshops to reteach or extend skills
Jigsaw
Negotiated Criteria
Explorations by interests
Games to practice mastery of information
Multiple levels of questions
High-Prep Differentiation
Tiered activities and labs
Tiered products
Independent studies
Multiple texts
Alternative assessments
Learning contracts
4-MAT
Multiple-intelligence options
Compacting
Spelling by readiness
Entry Points
Varying organizers
Lectures coupled with graphic organizers
Community mentorships
Interest groups
Tiered centers
Interest centers
Personal agendas
Literature Circles
Stations
Complex Instruction
Group Investigation
Tape-recorded materials
Teams, Games, and Tournaments
Choice Boards
Think-Tac-Toe
Simulations
Problem-Based Learning
Graduated Rubrics
Flexible reading formats
Student-centered writing formats
OPTIONS FOR DIFFERENTIATION OF INSTRUCTION
To Differentiate
Instruction By
Readiness
To Differentiate
Instruction By
Interest
To Differentiate
Instruction by
Learning Profile
‫ ٭‬equalizer adjustments (complexity,
open-endedness, etc.
‫ ٭‬add or remove scaffolding
‫ ٭‬vary difficulty level of text &
supplementary materials
‫ ٭‬adjust task familiarity
‫ ٭‬vary direct instruction by small group
‫ ٭‬adjust proximity of ideas to student
experience
‫ ٭‬encourage application of broad
concepts & principles to student interest
areas
‫ ٭‬give choice of mode of expressing
learning
‫ ٭‬use interest-based mentoring of adults
or more expert-like peers
‫ ٭‬give choice of tasks and products
(including student designed options)
‫ ٭‬give broad access to varied materials &
technologies
‫ ٭‬create an environment with flexible
learning spaces and options
‫ ٭‬allow working alone or working with
peers
‫ ٭‬use part-to-whole and whole-to-part
approaches
‫٭‬Vary teacher mode of presentation
(visual, auditory, kinesthetic, concrete,
abstract)
‫ ٭‬adjust for gender, culture, language
differences.
useful instructional strategies:
- tiered activities
- Tiered products
- compacting
- learning contracts
- tiered tasks/alternative forms of
assessment
useful instructional strategies:
- interest centers
- interest groups
- enrichment clusters
- group investigation
- choice boards
- MI options
- internet mentors
useful instructional strategies:
- multi-ability cooperative tasks
- MI options
- Triarchic options
- 4-MAT
CA Tomlinson, UVa ‘97
Whatever it Takes!