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Differentiating Mathematics at the Middle and High School Levels Raising Student Achievement Conference St. Charles, IL December 4, 2007


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Title: Differentiating Mathematics at the Middle and High School Levels Raising Student Achievement Conference St. Charles, IL December 4, 2007

Differentiating Mathematics at the Middle and
High School Levels Raising Student Achievement
Conference St. Charles, IL December 4, 2007
  • "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
  • Tomlinson, C.A. (2001). How to differentiate
    instruction in mixed ability classrooms (2nd
    Ed.). Alexandria, VA ASCD.
  • Nanci Smith
  • Educational Consultant
  • Curriculum and Professional Development
  • Cave Creek, AZ
  • nanci_mathmaster_at_yahoo.com

Differentiation of Instruction
Is a teachers response to learners needs guided
by general principles of differentiation
Respectful tasks
Flexible grouping
Continual assessment
Teachers Can Differentiate Through
According to Students
Learning Profile
Whats the point of differentiating in these
different ways?
Learning Profile
Key Principles of a Differentiated Classroom
  • The teacher understands, appreciates, and builds
    upon student differences.

Source Tomlinson, C. (2000). Differentiating
Instruction for Academic Diversity. San Antonio,
  • What does READINESS mean?
  • It is the students entry point relative to a
    particular understanding or skill.
  • C.A.Tomlinson, 1999

  • Varied texts by reading level
  • Varied supplementary materials
  • Varied scaffolding
  • reading
  • writing
  • research
  • technology
  • Tiered tasks and procedures
  • Flexible time use
  • Small group instruction
  • Homework options
  • Tiered or scaffolded assemssment
  • Compacting
  • Mentorships
  • Negotiated criteria for quality
  • Varied graphic organizers

Providing support needed for a student to succeed
in work slightly beyond his/her comfort zone.
  • For example
  • Directions that give more structure or less
  • Tape recorders to help with reading or writing
    beyond the students grasp
  • Icons to help interpret print
  • Reteaching / extending teaching
  • Modeling
  • Clear criteria for success
  • Reading buddies (with appropriate directions)
  • Double entry journals with appropriate challenge
  • Teaching through multiple modes
  • Use of manipulatives when needed
  • Gearing reading materials to student reading
  • Use of study guides
  • Use of organizers
  • New American Lecture
  • Tomlinson, 2000

  • Identify the learning objectives or standards ALL
    students must learn.
  • Offer a pretest opportunity OR plan an alternate
    path through the content for those students who
    can learn the required material in less time than
    their age peers.
  • Plan and offer meaningful curriculum extensions
    for kids who qualify.
  • Depth and Complexity
  • Applications of the skill being taught
  • Learning Profile tasks based on understanding
    the process instead of skill practice
  • Differing perspectives, ideas across time,
    thinking like a mathematician
  • Orbitals and Independent studies.
  • Eliminate all drill, practice, review, or
    preparation for students who have already
    mastered such things.
  • Keep accurate records of students compacting
    activities document mastery.

Strategy Compacting
Developing a Tiered Activity
  • Select the activity organizer
  • concept
  • generalization
  • Think about your students/use assessments
  • readiness range
  • interests
  • learning profile
  • talents

Essential to building a framework of understanding
skills reading thinking information
The Equalizer
  • Foundational Transformational
  • Concrete Abstract
  • Simple Complex
  • Single Facet Multiple Facets
  • Small Leap Great Leap
  • More Structured More Open
  • Less Independence Greater Independence
  • Slow Quick

Information, Ideas, Materials, Applications Rep
resentations, 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
Adding Fractions
  • Green 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

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.
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
  • 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 x0? (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.

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 Slope
  • Advanced Learners Given the slope-intercept form
    of the equation of a line, ymxb, the students
    will answer the following questions
  • The slope of the line is represented by which
  • 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.

Teaching With the Brain in Mind, 1998
  • Choices vs. Required
  • content, process, product no student
  • groups, resources environment restricted
  • Relevant vs.
  • meaningful impersonal
  • connected to learner out of
  • deep understanding only to pass
    a test
  • Engaging vs.
  • emotional, energetic low interaction
  • hands on, learner input lecture
  • Increased intrinsic Increased

-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 . . .
  • Research currently suggests that CHOICE should be
    offered 35 of the time!!

  • The assessments used in this learning profile
    section can be downloaded at
  • www.e2c2.com/fileupload.asp
  • Download the file entitled Profile Assessments
    for Cards.

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
  • 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
  • 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
  • have exact steps on how to complete it. Yes No
  • 10. When my teacher gives an assignment, I like
  • create my own steps on how to complete it. Yes
  • 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

My Way An expression Style Inventory K.E.
Kettle J.S. Renzull, M.G. Rizza University of
Connecticut Products provide students and
professionals with a way to express what they
have learned to an audience. This survey will
help determine the kinds of products YOU are
interested in creating. My Name is
Instructions Read each statement and circle the
number that shows to what extent YOU are
interested in creating that type of product. (Do
not worry if you are unsure of how to make the
Not At All Interested Of Little Interest Moderately Interested Interested Very Interested
1. Writing Stories 1 2 3 4 5
2. Discussing what I have learned 1 2 3 4 5
3. Painting a picture 1 2 3 4 5
4. Designing a computer software project 1 2 3 4 5
5. Filming editing a video 1 2 3 4 5
6. Creating a company 1 2 3 4 5
7. Helping in the community 1 2 3 4 5
8. Acting in a play 1 2 3 4 5
Not At All Interested Of Little Interest Moderately Interested Interested Very Interested
9. Building an invention 1 2 3 4 5
10. Playing musical instrument 1 2 3 4 5
11. Writing for a newspaper 1 2 3 4 5
12. Discussing ideas 1 2 3 4 5
13. Drawing pictures for a book 1 2 3 4 5
14. Designing an interactive computer project 1 2 3 4 5
15. Filming editing a television show 1 2 3 4 5
16. Operating a business 1 2 3 4 5
17. Working to help others 1 2 3 4 5
18. Acting out an event 1 2 3 4 5
19. Building a project 1 2 3 4 5
20. Playing in a band 1 2 3 4 5
21. Writing for a magazine 1 2 3 4 5
22. Talking about my project 1 2 3 4 5
23. Making a clay sculpture of a character 1 2 3 4 5
Not At All Interested Of Little Interest Moderately Interested Interested Very Interested
24. Designing information for the computer internet 1 2 3 4 5
25. Filming editing a movie 1 2 3 4 5
26. Marketing a product 1 2 3 4 5
27. Helping others by supporting a social cause 1 2 3 4 5
28. Acting out a story 1 2 3 4 5
29. Repairing a machine 1 2 3 4 5
30. Composing music 1 2 3 4 5
31. Writing an essay 1 2 3 4 5
32. Discussing my research 1 2 3 4 5
33. Painting a mural 1 2 3 4 5
34. Designing a computer 1 2 3 4 5
35. Recording editing a radio show 1 2 3 4 5
36. Marketing an idea 1 2 3 4 5
37. Helping others by fundraising 1 2 3 4 5
38. Performing a skit 1 2 3 4 5
Not At All Interested Of Little Interest Moderately Interested Interested Very Interested
39. Constructing a working model. 1 2 3 4 5
40. Performing music 1 2 3 4 5
41. Writing a report 1 2 3 4 5
42. Talking about my experiences 1 2 3 4 5
43. Making a clay sculpture of a scene 1 2 3 4 5
44. Designing a multi-media computer show 1 2 3 4 5
45. Selecting slides and music for a slide show 1 2 3 4 5
46. Managing investments 1 2 3 4 5
47. Collecting clothing or food to help others 1 2 3 4 5
48. Role-playing a character 1 2 3 4 5
49. Assembling a kit 1 2 3 4 5
50. Playing in an orchestra 1 2 3 4 5
Products Written Oral Artistic Computer Audio/Visual Commercial Service Dramatization Manipulative Musical 1. ___ 2. ___ 3. ___ 4. ___ 5. ___ 6. ___ 7. ___ 8. ___ 9. ___ 10.___ 11. ___ 12. ___ 13. ___ 14. ___ 15. ___ 16. ___ 77. ___ 18. ___ 19. ___ 20. ___ 21. ___ 22. ___ 23. ___ 24. ___ 25. ___ 26. ___ 27. ___ 28. ___ 29. ___ 30 . ___ 31. ___ 32. ___ 33. ___ 34. ___ 35. ___ 36. ___ 37. ___ 38. ___ 39. ___ 40. ___ 41. ___ 42. ___ 43. ___ 44. ___ 45. ___ 46. ___ 47. ___ 48. ___ 49. ___ 50. ___ Total _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
Instructions My Way A Profile Write your score
beside each number. Add each Row to determine
your expression style profile.
Learner Profile Card
Gender Stripe
Auditory, Visual, Kinesthetic Modality
Analytical, Creative, Practical Sternberg
Students Interests
Multiple Intelligence Preference Gardner
Array Inventory
Nanci Smith,Scottsdale,AZ
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.


Learning Profile Factors
Learning Environment quiet/noise warm/cool still/
mobile flexible/fixed busy/spare
Group Orientation independent/self
orientation group/peer orientation adult
orientation combination
Gender Culture
Intelligence Preference analytic practical creati
ve verbal/linguistic logical/mathematical spatial/
visual bodily/kinesthetic musical/rhythmic interpe
rsonal intrapersonal naturalist existential
Cognitive Style Creative/conforming Essence/facts
Expressive/controlled Nonlinear/linear Inductive/
deductive People-oriented/task or Object
oriented Concrete/abstract Collaboration/competiti
on Interpersonal/introspective Easily
distracted/long Attention span Group
achievement/personal achievement Oral/visual/kines
thetic Reflective/action-oriented
Activity 2.5 The Modality Preferences
Instrument (HBL, p. 23) Follow the directions
below to get a score that will indicate your own
modality (sense) preference(s). This instrument,
keep in mind that sensory preferences are usually
evident only during prolonged and complex
learning tasks. Identifying Sensory
Preferences Directions For each item, circle A
if you agree that the statement describes you
most of the time. Circle D if you disagree that
the statement describes you most of the time.
  1. I Prefer reading a story rather than listening to
    someone tell it. A D
  2. I would rather watch television than listen to
    the radio. A D
  3. I remember faces better than names. A D
  4. I like classrooms with lots of posters and
    pictures around the room. A D
  5. The appearance of my handwriting is important to
    me. A D
  6. I think more often in pictures. A D
  7. I am distracted by visual disorder or movement.
    A D
  8. I have difficulty remembering directions that
    were told to me. A D
  9. I would rather watch athletic events than
    participate in them. A D
  10. I tend to organize my thoughts by writing them
    down. A D
  11. My facial expression is a good indicator of my
    emotions. A D
  12. I tend to remember names better than faces.
    A D
  13. I would enjoy taking part in dramatic events like
    plays. A D
  14. I tend to sub vocalize and think in sounds. A
  15. I am easily distracted by sounds. A D
  16. I easily forget what I read unless I talk about
    it. A D
  17. I would rather listen to the radio than watch
    TV A D
  18. My handwriting is not very good. A D
  19. When faced with a problem , I tend to talk it
    through. A D

  • I prefer talking on the phone rather than writing
    a letter to someone. A D
  • I would rather participate in athletic events
    than watch them. A D
  • I prefer going to museums where I can touch the
    exhibits. A D
  • My handwriting deteriorates when the space
    becomes smaller. A D
  • My mental pictures are usually accompanied by
    movement. A D
  • I like being outdoors and doing things like
    biking, camping, swimming, hiking etc. A D
  • I remember best what was done rather then what
    was seen or talked about. A
  • When faced with a problem, I often select the
    solution involving the greatest activity.
    A D
  • I like to make models or other hand crafted
    items. A D
  • I would rather do experiments rather then read
    about them. A D
  • My body language is a good indicator of my
    emotions. A D
  • I have difficulty remembering verbal directions
    if I have not done the activity before. A D

Interpreting the Instruments Score Total the
number of A responses in items 1-11
_____ This is your visual score Total the
number of A responses in items
12-22 _____ This is your auditory score Total
the number of A responses in items
23-33 _____ This is you tactile/kinesthetic
score If you scored a lot higher in any one
area This indicates that this modality is very
probably your preference during a protracted and
complex learning situation. If you scored a lot
lower in any one area This indicates that this
modality is not likely to be your preference(s)
in a learning situation. If you got similar
scores in all three areas This indicates that
you can learn things in almost any way they are
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.

Smith Smarr, 2005
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
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
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 minds 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
BODILY/ Kinesthetic Learner The Mover Eager to solve problems physically. Often doesnt 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
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
    change constantly. 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

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 change and 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.

Multiple Intelligence Ideas for Proofs!
  • Logical Mathematical Generate proofs for given
    theorems. Be ready to explain!
  • Verbal Linguistic Write in paragraph form why
    the theorems are true. Explain what we need to
    think about before using the theorem.
  • Visual Spatial Use pictures to explain the

Multiple Intelligence Ideas for Proofs!
  • Musical Create a jingle or rap to sing the
  • Kinesthetic Use Geometer Sketchpad or other
    computer software to discover the theorems.
  • Intrapersonal Write a journal entry for
    yourself explaining why the theorem is true, how
    they make sense, and a tip for remembering them.

Sternbergs Three Intelligences
  • We all have some of each of these intelligences,
    but are usually stronger in one or two areas
    than in others.
  • We should strive to develop as fully each of
    these intelligences in students
  • but also recognize where students strengths
    lie and teach through those intelligences as
    often as possible, particularly when introducing
    new ideas.

Thinking About the Sternberg Intelligences
Linear Schoolhouse Smart - Sequential
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 Use
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. Heres a problem at school, ________. Using
your knowledge of ______________, develop a plan
to address the problem.
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
Triarchic Theory of Intelligences Robert Sternberg
  • Mark each sentence T if you like to do the
    activity and F if you do not like to do the
  • Analyzing characters when Im reading or
    listening to a story ___
  • Designing new things ___
  • Taking things apart and fixing them ___
  • Comparing and contrasting points of view ___
  • Coming up with ideas ___
  • Learning through hands-on activities ___
  • Criticizing my own and other kids work ___
  • Using my imagination ___
  • Putting into practice things I learned ___
  • Thinking clearly and analytically ___
  • Thinking of alternative solutions ___
  • Working with people in teams or groups ___
  • Solving logical problems ___
  • Noticing things others often ignore ___
  • Resolving conflicts ___

Triarchic Theory of Intelligences Robert Sternberg
  • Mark each sentence T if you like to do the
    activity and F if you do not like to do the
  • Evaluating my own and others points of
    view ___
  • Thinking in pictures and images ___
  • Advising friends on their problems ___
  • Explaining difficult ideas or problems to
    others ___
  • Supposing things were different ___
  • Convincing someone to do something ___
  • Making inferences and deriving conclusions ___
  • Drawing ___
  • Learning by interacting with others ___
  • Sorting and classifying ___
  • Inventing new words, games, approaches ___
  • Applying my knowledge ___
  • Using graphic organizers or images to organize
    your thoughts ___
  • Composing ___
  • 30. Adapting to new situations ___

Triarchic Theory of Intelligences Key Robert
  • Transfer your answers from the survey to the key.
    The column with the most True responses is your
    dominant intelligence.
  • Analytical Creative Practical
  • 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. ___
  • Total Number of True
  • Analytical ____ Creative _____ Practical _____

Understanding Order of Operations
Make a chart that shows all ways you can think of
to use order of operations to equal 18.
Analytic Task
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
wont necessarily get the correct answers! Give
lots of examples.
Practical Task
Creative Task
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
  • 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.

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.

Circle Vocabulary
  • Creative
  • 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.

Key Principles of a Differentiated Classroom
  • Assessment and instruction are inseparable.

Source Tomlinson, C. (2000). Differentiating
Instruction for Academic Diversity. San Antonio,
  • What the student already knows about what is
    being planned
  • What standards, objectives, concepts skills the
    individual student understands
  • What further instruction and opportunities for
    mastery are needed
  • What requires reteaching or enhancement
  • What areas of interests and feelings are in the
    different areas of the study
  • How to set up flexible groups Whole, individual,
    partner, or small group

  • Journal entry
  • Short answer test
  • Open response test
  • Home learning
  • Notebook
  • Oral response
  • Portfolio entry
  • Exhibition
  • Culminating product
  • Question writing
  • Problem solving
  • Anecdotal records
  • Observation by checklist
  • Skills checklist
  • Class discussion
  • Small group interaction
  • Teacher student conference
  • Assessment stations
  • Exit cards
  • Problem posing
  • Performance tasks and rubrics

Key Principles of a Differentiated Classroom
  • The teacher adjusts content, process, and product
    in response to student readiness, interests, and
    learning profile.

Source Tomlinson, C. (2000). Differentiating
Instruction for Academic Diversity. San Antonio,
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
  • 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.
  • Add distractors to outside edges and
  • letter pieces at the end.

Nanci Smith
The ROLE of writer, speaker, artist, historian,
An AUDIENCE of fellow writers, students,
citizens, characters, etc.
Through a FORMAT that is written, spoken,
drawn, acted, etc.
A TOPIC related to curriculum content in
greater depth.
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 Im 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 Whats 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 Its 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, were a straight angle
Angles Humans Video See, were 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 angles 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!
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
  • 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

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.
Fraction Think Dots
Nanci Smith
Fraction Think Dots
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.
Fraction Think Dots
Nanci Smith
Algebra ThinkDOTS
  • 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?

Algebra ThinkDOTS
  • 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
  • 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?

Algebra ThinkDOTS
  • 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.

Designing a Differentiated Learning Contract
  • A Learning Contract has the following components
  • 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
  • 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
  • A Time Line
  • Teacher sets completion date and check-in
  • 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
  • Signatures of the teacher, student and parent (if
    appropriate) are placed on the agreement

Differentiating Instruction Facilitators Guide,
ASCD, 1997
Personal Agenda
Montgomery County, MD
Personal Agenda for ______________________________
Starting Date ____________________________________
Teacher student initials at completion
Special Instructions
Remember to complete your daily planning log
Ill call on you for conferences instructions.
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
Similar Figures Menu
  • Imperatives (Do all 3)
  • 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.
  • Generate a list of applications for similar
    figures, and similarity in general. Be sure to
    think beyond find a missing side
  • Develop a lesson to teach third grade students
    who are just beginning to think about similarity.

Similar Figures Menu
  • Negotiables (Choose 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.
  • 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
  • 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.
  • Make a photo album showing the use of similar
    figures in the world around us. Use captions to
    explain the similarity in each picture.
  • Write a story about similar figures in a world
    without similarity.
  • Write a song about the beauty and mathematics of
    similar figures.
  • Create a how-to or book about finding and
    creating similar figures.


Whatever it Takes!
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