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					           Learner Profile Card

                     Gender Stripe
Auditory, Visual, Kinesthetic   Analytical, Creative, Practical
Modality                                           Sternberg


Multiple Intelligence Preference
Gardner                                              Inventory
                   Array Interaction Inventory
• 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                     Logical              Spontaneous                        Quiet
                         Sensitive                  Systematic                  creative                   Insightful
                           Caring                   Organized                    Playful                   reflective

              In your normal day-to-day life, you tend to value:

                          Harmony                        Work               Stimulation                    Reflection
                  Relationships are          Time schedules are            Having fun is           Having some time
                         important                   important                important            alone is important

              In most settings, you are usually:

                        Authentic                   Traditional                  Active                    Inventive
                    Compassionate                  Responsible             Opportunistic                  Competent
                      Harmonious                      Parental              Spontaneous                     Seeking

              In most situations, you could be described as:

                      Empathetic                     Practical                Impetuous                 Conceptual
                   Communicative                   Competitive                 Impactful             Knowledgeable
                        Devoted                         Loyal                    Daring                 Composed
   Array Interaction Inventory, cont’d

You approach most tasks in a(n) _________ manner:
        Affectionate               Conventional                  Courageous                     Rational
        Inspirational                  Orderly                  Adventurous                Philosophical
          Vivacious                  Concerned                    Impulsive                    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”               Over-control            “It’s not my fault”                Withdraw
      Make mistakes              Become critical                   Manipulate                 Don’t talk
          Feel badly                Take charge                       Act out          Become indecisive

When you’ve “had a bad day” and you become frustrated, how might you respond?
        Over-please            Be perfectionistic           Become physical                   Disengage
                 Cry             Verbally attack             Be irresponsible                     Delay
      Feel depressed                  Overwork              Demand attention                  Daydream

Add score:

        Harmony                   Production                  Connection                   Status Quo
                        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                  Caring                Logical              Spontaneous                  Quiet
                            Sensitive              Structured              Creative                Imaginative
(Positive Behavior)         Nurturing              Organized                Playful                 Insightful
                          Harmonizing              Systematic            Enthusiastic               Reflective
                         Feeling-oriented       Thinking-oriented       Action-oriented         Inaction-oriented
  RELUCTANT               Overadaptive            Overcritical             Disruptive             Disengaging
                          Overpleasing             Overworks                Blames                Withdrawn
(Negative Behavior)      Makes mistakes           Perfectionist          Irresponsible              Delays
                         Cries or giggles        Verbally attacks      Demands attention          Despondent
                         Self-defeating            Demanding                Defiant               Daydreams
PSYCHOLOGICAL              Friendships           Task completion      Contact with people          Alone time
    NEEDS               Sensory experience        Time schedule         Fun activities              Stability

 WAYS TO MEET           Value their feelings     Value their ideas     Value their activity     Value their privacy
    NEEDS             Comfortable work place        Incentives         Hands-on activities         Alone time
                         Pleasing learning           Rewards           Group interaction      Independent activities
                           environment         Leadership positions          Games              Specific directions
                        Work with a friend          Schedules          Change in routine       Computer activities
                           sharing times            To-do lists                                   Routine tasks
There are two
  keys to
1. Know your kids

2.Know your
      “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
         Howard Gardner
           facts, vocabulary, dates,
These are the
places, names, and examples you want
students to give
                                      Facts 2X3=6,
                                        b  b  4ac
     The know is massively

          forgettable.                  Vocabulary
                                      numerator, slope
           “Teaching facts in isolation is like
           trying to pump water uphill.”
                   -Carol Tomlinson
KNOW (Facts,
 Vocabulary, Definitions)
• Definition of numerator and
• The quadratic formula
• The Cartesian coordinate plane
• The multiplication tables

          • 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
    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,
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
            (Black and Wiliam, 1998))
  Major Concepts and
  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

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
 – Does it yield optimal depth and breadth of insight into the
 – 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.
• 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
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.    Angel                   37.   Idea                 73.    Punishment
 2.    Animal                  38.   Immortality          74.    Quality
 3.    Aristocracy             39.   Induction            75.    Quantity
 4.    Art                     40.   Infinity             76.    Reasoning
 5.    Astronomy               41.   Judgment             77.    Relation
 6.    Beauty                  42.   Justice              78.    Religion
 7.    Being                   43.   Labor                79.    Revolution
 8.    Cause                   44.   Language             80.    Rhetoric
 9.    Chance                  45.   Law                  81.    Same/Other
 10.   Change                  46.   Liberty              82.    Science
 11.   Citizen                 47.   Life and death       83.    Sense
 12.   Constitution            48.   Logic                84.    Sign/Symbol
 13.   Courage                 49.   Love                 85.    Sin
 14.   Custom and convention   50.   Man                  86.    Slavery
 15.   Definition              51.   Mathematics          87.    Soul
 16.   Democracy               52.   Matter               88.    Space
 17.   Desire                  53.   Mechanics            89.    State
 18.   Dialectic               54.   Medicine             90.    Temperance
 19.   Duty                    55.   Memory/Imagination   91.    Theology
 20.   Education               56.   Metaphysics          92.    Time
 21.   Element                 57.   Mind                 93.    Truth
 22.   Emotion                 58.   Monarchy             94.    Tyranny
 23.   Eternity                59.   Nature               95.    Universe
 24.   Evolution               60.   Necessity            96.    Virtue/Vice
 25.   Experience              61.   Oligarchy            97.    War & Peace
 26.   Family                  62.   One and Many         98.    Wealth
 27.   Fate                    63.   Opinion              99.    Will
 28.   Form                    64.   Opposition           100.   Wisdom
 29.   God                     65.   Philosophy           101.   World
 30.   Good and Evil           66.   Physics
 31.   Government              67.   Pleasure and Pain
 32.   Habit                   68.   Poetry
 33.   Happiness               69.   Principle
 34.   History                 70.   Progress
 35.   Honor                   71.   Prophecy
 36.   hypothesis              72.   Prudence
• Mathematical Concepts
  Number            Error / Uncertainty            Ratio
  Measurement       Proportion                     Behavior
  Symmetry Relationships           Pattern
  ProbabilityFunction                      Truth
  Order             Problem solving                Change
  System            Quantification         Prediction

• Mathematical Understandings
  Our number system maintains order and is rich with patterns.
  Mathematicians quantify data in order to establish real-world
  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
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
• 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         Write problem with
          common denominators

 4 3
        Divide the numerators
 6 6
      3                           2 1 4
                                    
                                  3 2 3
           Dividing Fractions
        Demonstrating the 5 Strands
• Now relate the pattern to the algorithm of invert and
• 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
         • What was a challenge
           for you?
         • What did you learn
           from this activity?
       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
• 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
• Create 9 tiles with math problems and answers along the
• 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
•   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
Improper Fraction         Mixed Numbers               Reconciliation Letter   Were More Alike than
A Simplified Fraction     A Non-Simplified Fraction   Public Service          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   Unlike Denominators in an   Application form        To Become A Like
Additional Problem        Addition Problem                                    Denominator
A Mixed Number that       5th Grade Math Students     Riddle                  What’s My New Name
Needs to be Renamed to
Like Denominators in a    Unlike Denominators in a    Story Board             How to Become a Like
Subtraction Problem       Subtraction Problem                                 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)     Invitation to a family            My separated twin
                                           angle                         reunion

       Acute angle                     Missing angle                 Wanted poster              Wanted: My complement

 An angle less than 180               Supplementary                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

   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!
     ROLE               AUDIENCE                  FORMAT                        TOPIC
Equivalent Fractions         Farmers           Poster Ad for Fertilizer    How do I get bigger
       Event            Mutually exclusive           Love letter               We’ll never be
                              event                                       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     Persuasive Speech             You need me!
                        multi-step problem
   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
Scientific Notation       Large numbers              Health Ad             The benefits of being
      Radius                 Diameter                  Letter              How do I fit into your
       Scale                   Map                      Poem              Why do we need to be
  2 line segments          All segments         Wanted Poster for a        Are you our missing
                                                 complete triangle                link?
     Multiples                Factors                Storyboard               To Infinity and
                RAFT Planning Sheet
How to Differentiate:
• Tiered? (See Equalizer)
• Profile? (Differentiate Format)
• Interest? (Keep options equivalent in
• Other?

        Role                Audience      Format   Topic
1.   Describe it: Look at the subject closely
     (perhaps with your senses as well as            Or you can . . . .
     your mind)
                                                 •   Rearrange it
2.   Compare it: What is it similar to?          •   Illustrate it
     What is it different from?
                                                 •   Question it
3.   Associate it: What does it make you
     think of? What comes to your mind
                                                 •   Satirize it
     when you think of it? Perhaps people?       •   Evaluate it
     Places? Things? Feelings? Let your
     mind go and see what feelings you have      •   Connect it
     for the subject.
                                                 •   Cartoon it
4.   Analyze it: Tell how it is made? What
     are it’s traits and attributes?
                                                 •   Change it
                                                 •   Solve it
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.
    Ideas for Cubing                                                      Cubing

•   Arrange ________ into a 3-D collage
    to show ________                            Ideas for Cubing in Math
•   Make a body sculpture to show           •   Describe how you would solve ______
    ________                                •   Analyze how this problem helps us use
                                                mathematical thinking and problem solving
•   Create a dance to show
                                            •   Compare and contrast this problem to one
•   Do a mime to help us understand             on page _____.
•   Present an interior monologue with      •   Demonstrate how a professional (or just a
    dramatic movement that ________             regular person) could apply this kink or
                                                problem to their work or life.
•   Build/construct a representation of     •   Change one or more numbers, elements, or
    ________                                    signs in the problem. Give a rule for what that
•   Make a living mobile that shows and         change does.
    balances the elements of ________       •   Create an interesting and challenging word
                                                problem from the number problem. (Show us
•   Create authentic sound effects to           how to solve it too.)
    accompany a reading of _______          •   Diagram or illustrate the solutionj to the
•   Show the principle of ________ with a       problem. Interpret the visual so we
                                                understand it.
    rhythm pattern you create. Explain to
    us how that works.
         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
           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       Explain the difference
                           1 3
              solve             or roll   between adding and
                           5 5
              the die to determine your    multiplying fractions,
              own fractions.

              Compare and contrast         Create a word problem
              these two problems:          that can be solved by
                                                1 2 11
                                                  
                       +                        3 5 15

                      and                  (Or roll the fraction die to
                      1 1
                                          determine your fractions.)
                      3 2

              Describe how people use      Model the problem
              fractions every day.         ___ + ___ .
Nanci Smith
                                           Roll the fraction die to
                                           determine which fractions
                                           to add.
Nanci Smith
              Describe how you would         Explain why you need
                       2 3 1
              solve           or roll      a common denominator
                      13 7 91
              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
              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.

Nanci Smith   do you know you are correct?
                                             Diagram and explain the
                                             solution to ___ + ___ + ___.
                                             Roll the fraction die to
                                             determine your fractions.
     Designing a Differentiated Learning
     Contracthas the following components
A Learning Contract
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
Student Name                                          Date
     What I will do

               What I will use

When I will finish
  How I feel about my project        How my teacher feels about my project

because                           because

Student signature               Teacher’s Signature
                        Learning Contract
                                Chapter: _______
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        Work with a         Make a
                                                          is a foot?   friend to graph     group story
                                                                         the size of at      or one of
                                                                         least 6 things    your own –
                                                                         on the list of      that uses
                                                                          “10 terrific    measuremen
                                                                        things.” Label    t and at least
                                                                       each thing with      one graph.
                                                                       how you know       Turn it into a
                                                                            the size       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    Complete the     Now, make a
                                                              r Who     math madness         math
                                                             Used to    book that goes     madness
                                                             be Rich    with the story    book based
                                                               Last       you read.         on your
                                                            Sunday or                     story about
                                                            Ten Kids,                    kids and pets
                                                             No Pets                       or money
                                                                                          that comes
                                                                                           and goes.
                                                                                           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   Research a kind of    Make a book
                                                    Before        dinosaur or        in which you
                                                   Dark or     airplane. Figure     combine math
                                                    Airport    out how big it is.   and dinosaurs
                                                   Control     Graph its size on     or airplanes,
                                                              graph paper or on     or something
                                                                 the blacktop          else big. It
                                                              outside our room.         can be a
                                                               Label it by name       number fact
                                                                   and size              book, a
                                                                                       book, or a
                                                                                        are at the
                                                                                    author center
                    Proportional Reasoning
□   Create a word problem that   □   Find a word problem from       □    Think of a way that you use
    requires proportional            the text that requires              proportional reasoning in your
    reasoning. Solve the             proportional reasoning.             life. Describe the situation,
    problem and explain why it       Solve the problem and               explain why it is proportional
    requires proportional            explain why it was                  and how you use it.
    reasoning.                       proportional.

□   Create a story about a       □   How do you recognize a         □    Make a list of all the
    proportion in the world.         proportional situation?             proportional situations in the
    You can write it, act it,        Find a way to think about           world today.
    video tape it, or another        and explain proportionality.
    story form.

□   Create a pict-o-gram, poem   □   Write a list of steps for      □    Write a list of questions to ask
    or anagram of how to solve       solving any proportional            yourself, from encountering a
    proportional problems            problem.                            problem that may be
                                                                         proportional through solving

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 ____)

                                          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
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
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                                    High-Prep Differentiation
Choices of books                                            Tiered activities and labs
Homework options                                            Tiered products
Use of reading buddies                                      Independent studies
Varied journal Prompts                                      Multiple texts
                                                            Alternative assessments
                                                            Learning contracts
Varied pacing with anchor options                           4-MAT
Student-teaching goal setting                               Multiple-intelligence options
Work alone / together                                       Compacting
Whole-to-part and part-to-whole explorations                Spelling by readiness
Flexible seating                                            Entry Points
Varied computer programs                                    Varying organizers
Design-A-Day                                                Lectures coupled with graphic organizers
Varied Supplementary materials                              Community mentorships
Options for varied modes of expression                      Interest groups
                                                            Tiered centers
Varying scaffolding on same organizer
                                                            Interest centers
Let’s Make a Deal projects                                  Personal agendas
Computer mentors                                            Literature Circles
Think-Pair-Share by readiness, interest, learning profile   Stations
Use of collaboration, independence, and cooperation         Complex Instruction
Open-ended activities                                       Group Investigation
Mini-workshops to reteach or extend skills                  Tape-recorded materials
Jigsaw                                                      Teams, Games, and Tournaments
Negotiated Criteria                                         Choice Boards
Explorations by interests                                   Think-Tac-Toe
Games to practice mastery of information
                                                            Problem-Based Learning
Multiple levels of questions                                Graduated Rubrics
                                                            Flexible reading formats
                                                            Student-centered writing formats
     To Differentiate                              To Differentiate                                To Differentiate
      Instruction By                                Instruction By                                  Instruction by
        Readiness                                      Interest                                    Learning Profile

‫ ٭‬equalizer adjustments (complexity,       ‫ ٭‬encourage application of broad            ‫ ٭‬create an environment with flexible
open-endedness, etc.                       concepts & principles to student interest   learning spaces and options
‫ ٭‬add or remove scaffolding                areas                                       ‫ ٭‬allow working alone or working with
‫ ٭‬vary difficulty level of text &          ‫ ٭‬give choice of mode of expressing         peers
supplementary materials                    learning                                    ‫ ٭‬use part-to-whole and whole-to-part
‫ ٭‬adjust task familiarity                  ‫ ٭‬use interest-based mentoring of adults    approaches
‫ ٭‬vary direct instruction by small group   or more expert-like peers                   ‫٭‬Vary teacher mode of presentation
‫ ٭‬adjust proximity of ideas to student     ‫ ٭‬give choice of tasks and products         (visual, auditory, kinesthetic, concrete,
experience                                 (including student designed options)        abstract)
                                           ‫ ٭‬give broad access to varied materials &   ‫ ٭‬adjust for gender, culture, language
                                           technologies                                differences.

       useful instructional strategies:           useful instructional strategies:            useful instructional strategies:
- tiered activities                        - interest centers                          - multi-ability cooperative tasks
- Tiered products                          - interest groups                           - MI options
- compacting                               - enrichment clusters                       - Triarchic options
- learning contracts                       - group investigation                       - 4-MAT
- tiered tasks/alternative forms of        - choice boards
assessment                                 - MI options
                                           - internet mentors                                            CA Tomlinson, UVa „97
Whatever it Takes!

Description: Fold-Up Invitation Template document sample