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Differentiating math

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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 learners." * * 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 [email protected]
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Page 1: Differentiating math

Differentiating Mathematics at the Middle and High School LevelsRaising Student Achievement Conference

St. Charles, ILDecember 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 learners." *

* Tomlinson, C.A. (2001). How to differentiate instruction in mixed ability classrooms (2nd Ed.). Alexandria, VA: ASCD.

Nanci SmithEducational ConsultantCurriculum and Professional DevelopmentCave Creek, [email protected]

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Differentiation of Instruction

Is a teacher’s response to learner’s needs

guided by general principles of differentiation

Respectful tasks Flexible grouping Continual assessment

Teachers Can Differentiate Through:

Content Process Product

According to Students’

Readiness Interest Learning Profile

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What’s the point of differentiating in these

different ways?Readiness

Growth

InterestLearning Profile

Motivation Efficiency

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Key Principles of a Differentiated Classroom

Key Principles of a Differentiated Classroom

• The teacher understands, appreciates, and builds upon student differences.

• The teacher understands, appreciates, and builds upon student differences.

Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD

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READINESS

What does READINESS mean?

It is the student’s entry point relative to a particular understanding or skill.

C.A.Tomlinson, 1999

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A Few Routes to READINESS DIFFERENTIATION

Varied texts by reading levelVaried supplementary materialsVaried scaffolding• reading• writing• research• technology

Tiered tasks and procedures Flexible time useSmall group instructionHomework optionsTiered or scaffolded assemssmentCompactingMentorshipsNegotiated criteria for qualityVaried graphic organizers

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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 student’s 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 level•Use of study guides•Use of organizers•New American Lecture

Tomlinson, 2000

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1. Identify the learning objectives or standards ALL students must learn.

2. 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.

3. Plan and offer meaningful curriculum extensions for kids who qualify. **Depth and Complexity

Applications of the skill being taughtLearning Profile tasks based on understanding

the process instead of skill practiceDiffering perspectives, ideas across time,

thinking like a mathematician **Orbitals and Independent studies.

4. Eliminate all drill, practice, review, or preparation for students who have already mastered such things.

5. Keep accurate records of students’ compacting activities: document mastery.

Compacting

Strategy: Compacting

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Developing a Tiered Activity

Select the activity organizer•concept•generalization

Essential to buildinga framework ofunderstanding

Think about your students/use assessments

• readiness range• interests• learning profile• talents

skillsreadingthinkinginformation

Create an activity that is• interesting• high level• causes students to use key skill(s) to understand a key idea

Chart the complexity of the activity

High skill/Complexity

Low skill/complexity

Clone the activity along the ladder as needed to ensure challenge and success for your students, in

• materials – basic to advanced• form of expression – from familiar to

unfamiliar• from personal experience to removed

from personal experience•equalizer

Match task to student based on student profile and task requirements

1

3

5

2

4

6

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Information, Ideas, Materials, Applications

Representations, Ideas, Applications, Materials

Resources, Research, Issues, Problems, Skills, Goals

Directions, Problems, Application, Solutions, Approaches, Disciplinary Connections

Application, Insight, Transfer

Solutions, Decisions, Approaches

Planning, Designing, Monitoring

Pace of Study, Pace of Thought

The Equalizer

1. Foundational Transformational

2. Concrete Abstract

1. Simple Complex

2. Single Facet Multiple Facets

3. Small Leap Great Leap

4. More Structured More Open

5. Less Independence Greater Independence

6. Slow Quick

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Adding FractionsGreen Group

Use Cuisinaire rods or fraction circles to model simple fraction addition problems. Begin with common denominators and work up to denominators with common factors such as 3 and 6.

Explain the pitfalls and hurrahs of adding fractions by making a picture book.

Blue GroupManipulatives such as Cuisinaire rods and fraction circles will be available as a resource for the group. Students use factor trees and lists of multiples to find common denominators. Using this approach, pairs and triplets of fractions are rewritten using common denominators. End by adding several different problems of increasing challenge and length.

Suzie says that adding fractions is like a game: you just need to know the rules. Write game instructions explaining the rules of adding fractions.

Red GroupUse Venn diagrams to model LCMs (least common multiple). Explain how this process can be used to find common denominators. Use the method on more challenging addition problems.

Write a manual on how to add fractions. It must include why a common denominator is needed, and at least three ways to find it.

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Graphing with a Point and a Slope

All groups: • Given three equations in slope-intercept form, the

students will graph the lines using a T-chart. Then they will answer the following questions:

• What is the slope of the line?• Where is slope found in the equation?• Where does the line cross the y-axis?• What is the y-value of the point when x=0? (This

is the y-intercept.)• Where is the y-value found in the equation?• Why do you think this form of the equation is

called the “slope-intercept?”

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Graphing with a Point and a Slope

Struggling Learners: Given the points

• (-2,-3), (1,1), and (3,5), the students will plot the points and sketch the line. Then they will answer the following questions:

• What is the slope of the line?

• Where does the line cross the y-axis?

• Write the equation of the line.

The students working on this particular task should repeat this process given two or three more points and/or a point and a slope. They will then create an explanation for how to graph a line starting with the equation and without finding any points using a T-chart.

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Graphing with a Point and a Slope

Grade-Level Learners: Given an equation of a line in slope-intercept form (or several equations), the students in this group will:

• Identify the slope in the equation.• Identify the y-intercept in the equation.• Write the y-intercept in coordinate form (0,y) and plot

the point on the y-axis.• use slope to find two additional points that will be on the

line.• Sketch the line.

When the students have completed the above tasks, they will summarize a way to graph a line from an equation without using a

T-chart.

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Graphing with a Point and a SlopeAdvanced Learners: Given the slope-intercept form of the

equation of a line, y=mx+b, the students will answer the following questions:

• The slope of the line is represented by which variable?• The y-intercept is the point where the graph crosses the y-

axis. What is the x-coordinate of the y-intercept? Why will this always be true?

• The y-coordinate of the y-intercept is represented by which variable in the slope-intercept form?

Next, the students in this group will complete the following tasks given equations in slope-intercept form:

• Identify the slope and the y-intercept.• Plot the y-intercept.• Use the slope to count rise and run in order to find the

second and third points.• Graph the line.

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BRAIN RESEARCH SHOWS THAT. . .Eric Jensen, Teaching With the Brain in Mind, 1998

Choices vs. Required content, process, product no student voice

groups, resources environment restricted resources

Relevant vs. Irrelevant meaningful impersonal

connected to learner out of context deep understanding only to pass a test

Engaging vs. Passive emotional, energetic low interaction

hands on, learner input lecture seatwork

EQUALSIncreased intrinsic Increased MOTIVATION APATHY &

RESENTMENT

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-CHOICE-The Great Motivator!

• Requires children to be aware of their own readiness, interests, and learning profiles.

• Students have choices provided by the teacher. (YOU are still in charge of crafting challenging opportunities for all kiddos – NO taking the easy way out!)

• Use choice across the curriculum: writing topics, content writing prompts, self-selected reading, contract menus, math problems, spelling words, product and assessment options, seating, group arrangement, ETC . . .

• GUARANTEES BUY-IN AND ENTHUSIASM FOR LEARNING!

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

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Assessments

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.”

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How Do You Like to Learn?

1. I study best when it is quiet. Yes No2. I am able to ignore the noise of

other people talking while I am working. Yes No3. I like to work at a table or desk. Yes No4. I like to work on the floor. Yes No5. I work hard by myself. Yes No6. I work hard for my parents or teacher. Yes No7. I will work on an assignment until it is completed, no

matter what. Yes No8. Sometimes I get frustrated with my work

and do not finish it. Yes No9. When my teacher gives an assignment, I like to

have exact steps on how to complete it. Yes No10. When my teacher gives an assignment, I like to

create my own steps on how to complete it. Yes No11. I like to work by myself. Yes No12. I like to work in pairs or in groups. Yes No13. I like to have unlimited amount of time to work on

an assignment. Yes No14. I like to have a certain amount of time to work on

an assignment. Yes No15. I like to learn by moving and doing. Yes No16. I like to learn while sitting at my desk. Yes No

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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 product).

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

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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

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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

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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.

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Learner Profile Card

Auditory, Visual, Kinesthetic

Modality

Multiple Intelligence Preference

Gardner

Analytical, Creative, Practical

Sternberg

Student’s Interests

Array Inventory

Gender Stripe

Nanci Smith,Scottsdale,AZ

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Differentiation Using LEARNING PROFILE

• Learning profile refers to how an individual learns best - most efficiently and effectively.

• Teachers and their students may differ in learning profile preferences.

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Learning Profile Factors

Group Orientation

independent/self orientationgroup/peer orientation

adult orientationcombination

Learning Environment

quiet/noisewarm/coolstill/mobile

flexible/fixed“busy”/”spare”

Cognitive Style

Creative/conformingEssence/facts

Expressive/controlledNonlinear/linear

Inductive/deductivePeople-oriented/task or Object oriented

Concrete/abstractCollaboration/competitionInterpersonal/introspective

Easily distracted/long Attention spanGroup achievement/personal achievement

Oral/visual/kinestheticReflective/action-oriented

Intelligence Preference

analyticpracticalcreative

verbal/linguisticlogical/mathematical

spatial/visualbodily/kinestheticmusical/rhythmic

interpersonalintrapersonal

naturalistexistential

Gender &Culture

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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 PreferencesDirections: 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 D

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

20. I express my emotions verbally. A D

21. I would rather be in a group discussion than read about a topic. A D

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22. I prefer talking on the phone rather than writing a letter to someone. A D

23. I would rather participate in athletic events than watch them. A D

24. I prefer going to museums where I can touch the exhibits. A D

25. My handwriting deteriorates when the space becomes smaller. A D

26. My mental pictures are usually accompanied by movement. A D

27. I like being outdoors and doing things like biking, camping, swimming, hiking etc. A D

28. I remember best what was done rather then what was seen or talked about. A D

29. When faced with a problem, I often select the solution involving the greatest activity. A D

30. I like to make models or other hand crafted items. A D

31. I would rather do experiments rather then read about them. A D

32. My body language is a good indicator of my emotions. A D

33. I have difficulty remembering verbal directions if I have not done the activity before. A D

Interpreting the Instrument’s 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 presented.

Page 29: Differentiating math

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.

12 3

45

67

8

Smith & Smarr, 2005

Page 30: Differentiating math

Parallel Lines Cut by a Transversal

• Auditory: Play “Shout Out!!” Given the diagram below and commands on strips of paper (with correct answers provided), players take turns being the leader to read a command. The first player to shout out a correct answer to the command, receives a point. The next player becomes the next leader. Possible commands:– Name an angle supplementary supplementary to angle 1.– Name an angle congruent to angle 2.

Smith & Smarr, 2005

12 3

456

78

Page 31: Differentiating math

Parallel Lines Cut by a Transversal

• Kinesthetic: Walk It Tape the diagram below on the floor with masking tape. Two players stand in assigned angles. As a team, they have to tell what they are called (ie: vertical angles) and their relationships (ie: congruent). Use all angle combinations, even if there is not a name or relationship. (ie: 2 and 7)

Smith & Smarr, 2005

12 3

45

67

8

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EIGHT STYLES OF LEARNINGTYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BY

LINGUISTIC

LEARNER“The Word Player”

Learns through the manipulation of words. Loves to read and write in order to explain themselves. They also tend to enjoy talking

Read

Write

Tell stories

Memorizing names, places, dates and trivia

Saying, hearing and seeing words

LOGICAL/

Mathematical

Learner“The Questioner”

Looks for patterns when solving problems. Creates a set of standards and follows them when researching in a sequential manner.

Do experiments

Figure things out

Work with numbers

Ask questions

Explore patterns and relationships

Math

Reasoning

Logic

Problem solving

Categorizing

Classifying

Working with abstract patterns/relationships

SPATIAL LEARNER“The Visualizer”

Learns through pictures, charts, graphs, diagrams, and art.

Draw, build, design and create things

Daydream

Look at pictures/slides

Watch movies

Play with machines

Imagining things

Sensing changes

Mazes/puzzles

Reading maps, charts

Visualizing

Dreaming

Using the mind’s eye

Working with colors/pictures

MUSICAL LEARNER“The Music Lover”

Learning is often easier for these students when set to music or rhythm

Sing, hum tunes

Listen to music

Play an instrument

Respond to music

Picking up sounds

Remembering melodies

Noticing pitches/ rhythms

Keeping time

Rhythm

Melody

Music

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EIGHT STYLES OF LEARNING, Cont’d

TYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BY

BODILY/

Kinesthetic

Learner“The Mover”

Eager to solve problems physically. Often doesn’t read directions but just starts on a project

Move around

Touch and talk

Use body language

Physical activities

(Sports/dance/

acting)

crafts

Touching

Moving

Interacting with space

Processing knowledge through bodily sensations

INTERpersonal

Learner“The Socializer”

Likes group work and working cooperatively to solve problems. Has an interest in their community.

Have lots of friends

Talk to people

Join groups

Understanding people

Leading others

Organizing

Communicating

Manipulating

Mediating conflicts

Sharing

Comparing

Relating

Cooperating

interviewing

INTRApersonal

Learner“The Individual”

Enjoys the opportunity to reflect and work independently. Often quiet and would rather work on his/her own than in a group.

Work alone

Pursue own

interests

Understanding self

Focusing inward on feelings/dreams

Pursuing interests/

goals

Being original

Working along

Individualized projects

Self-paced instruction

Having own space

NATURALIST“The Nature Lover”

Enjoys relating things to their environment. Have a strong connection to nature.

Physically experience nature

Do observations

Responds to patterning nature

Exploring natural phenomenon

Seeing connections

Seeing patterns

Reflective Thinking

Doing observations

Recording events in Nature

Working in pairs

Doing long term projects

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Introduction to Change(MI)

• Logical/Mathematical Learners: Given a set of data that changes, such as population for your city or town over time, decide on several ways to present the information. Make a chart that shows the various ways you can present the information to the class. Discuss as a group which representation you think is most effective. Why is it most effective? Is the change you are representing constant or variable? Which representation best shows this? Be ready to share your ideas with the class.

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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 class.

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Introduction to Change(MI)

• Visual/Spatial Learners: Given a variety of graphs, discuss what changes each one is representing. Are the changes constant or variable? How can you tell? Hypothesize how graphs showing constant and variable changes differ from one another. Be ready to share your ideas with the class.

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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.

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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 theorem.

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Multiple Intelligence Ideas for Proofs!

• Musical: Create a jingle or rap to sing the theorems!

• 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.

Page 40: Differentiating math

Sternberg’s Three Intelligences

Creative Analytical

Practical

•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.

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Linear – Schoolhouse Smart - SequentialANALYTICALThinking About the Sternberg Intelligences

Show the parts of _________ and how they work.Explain why _______ works the way it does.Diagram how __________ affects __________________.Identify the key parts of _____________________.Present a step-by-step approach to _________________.

Streetsmart – Contextual – Focus on UsePRACTICAL

Demonstrate how someone uses ________ in their life or work.Show how we could apply _____ to solve this real life problem ____.Based on your own experience, explain how _____ can be used.Here’s a problem at school, ________. Using your knowledge of ______________, develop a plan to address the problem.

CREATIVE Innovator – Outside the Box – What If - Improver

Find a new way to show _____________.Use unusual materials to explain ________________.Use humor to show ____________________.Explain (show) a new and better way to ____________.Make connections between _____ and _____ to help us understand ____________.Become a ____ and use your “new” perspectives to help us think about ____________.

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Triarchic Theory of IntelligencesRobert Sternberg

Mark each sentence T if you like to do the activity and F if you do not like to do the activity.

1. Analyzing characters when I’m reading or listening to a story ___2. Designing new things ___

3. Taking things apart and fixing them ___4. Comparing and contrasting points of view ___5. Coming up with ideas ___6. Learning through hands-on activities ___7. Criticizing my own and other kids’ work ___8. Using my imagination ___9. Putting into practice things I learned ___10. Thinking clearly and analytically ___11. Thinking of alternative solutions ___12. Working with people in teams or groups ___13. Solving logical problems ___14. Noticing things others often ignore ___15. Resolving conflicts ___

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Triarchic Theory of IntelligencesRobert Sternberg

Mark each sentence T if you like to do the activity and F if you do not like to do the activity.

16. Evaluating my own and other’s points of view ___17. Thinking in pictures and images ___18. Advising friends on their problems ___19. Explaining difficult ideas or problems to others ___20. Supposing things were different ___21. Convincing someone to do something ___22. Making inferences and deriving conclusions ___23. Drawing ___24. Learning by interacting with others ___25. Sorting and classifying ___26. Inventing new words, games, approaches ___27. Applying my knowledge ___28. Using graphic organizers or images to organize your thoughts ___29. Composing ___30. Adapting to new situations ___

Page 44: Differentiating math

Triarchic Theory of Intelligences – KeyRobert Sternberg

Transfer your answers from the survey to the key. The column with the most True responses is your dominant intelligence.

Analytical Creative Practical1. ___ 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 _____

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Understanding Order of Operations

Analytic Task

Practical Task

Creative Task

Make a chart that shows all ways you can think of to use order of operations to equal 18.

A friend is convinced that order of operations do not matter in math. Think of as many ways to convince your friend that without using them, you won’t necessarily get the correct answers! Give lots of examples.Write a book of riddles that involve order of operations. Show the solution and pictures on the page that follows each riddle.

Page 46: Differentiating math

Forms of Equations of Lines• Analytical Intelligence: Compare and contrast the various

forms of equations of lines. Create a flow chart, a table, or any other product to present your ideas to the class. Be sure to consider the advantages and disadvantages of each form.

• Practical Intelligence: Decide how and when each form of the equation of a line should be used. When is it best to use which? What are the strengths and weaknesses of each form? Find a way to present your conclusions to the class.

• Creative Intelligence: Put each form of the equation of a line on trial. Prosecutors should try to convince the jury that a form is not needed, while the defense should defend its usefulness. Enact your trial with group members playing the various forms of the equations, the prosecuting attorneys, and the defense attorneys. The rest of the class will be the jury, and the teacher will be the judge.

Page 47: Differentiating math

Circle VocabularyAll 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.

Page 48: Differentiating math

Circle Vocabulary

AnalyticalStudents 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.

PracticalStudents 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.

Page 49: Differentiating math

Circle VocabularyCreative

Find a way to help us remember all this vocabulary! You can create a skit by becoming each term, and talking about who you are and how you relate to each other, draw pictures, make a collage, or any other way of which you can think.

ORRole Audience Format Topic Diameter Radius email Twice as niceCircle Tangent poem You touch me!Secant Chord voicemail I extend you.

Page 50: Differentiating math

Key Principles of a Differentiated Classroom

Key Principles of a Differentiated Classroom

• AssessmentAssessment and and instructioninstruction are are inseparableinseparable..

• AssessmentAssessment and and instructioninstruction are are inseparableinseparable..

Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD

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Pre-Assessment• 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

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THINKING ABOUT ON-GOING ASSESSMENT

STUDENT DATA SOURCES1. Journal entry2. Short answer test3. Open response test4. Home learning5. Notebook6. Oral response7. Portfolio entry8. Exhibition9. Culminating product10. Question writing11. Problem solving

TEACHER DATA MECHANISMS

1. Anecdotal records2. Observation by checklist3. Skills checklist4. Class discussion5. Small group interaction6. Teacher – student

conference7. Assessment stations8. Exit cards9. Problem posing10. Performance tasks and

rubrics

Page 53: Differentiating math

Key Principles of a Differentiated Classroom

Key Principles of a Differentiated Classroom

• The teacher adjusts The teacher adjusts content, content, process, and productprocess, and product in response to in response to student student readiness, interestsreadiness, interests, and , and learning profilelearning profile..

• The teacher adjusts The teacher adjusts content, content, process, and productprocess, and product in response to in response to student student readiness, interestsreadiness, interests, and , and learning profilelearning profile..

Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD

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USE OF INSTRUCTIONAL STRATEGIES.

The following findings related to instructional strategies are supported by

the existing research:• Techniques and instructional strategies have nearly as much influence on student learning as student aptitude.

• Lecturing, a common teaching strategy, is an effort to quickly cover the material: however, it often overloads and over-whelms students with data, making it likely that they will confuse the facts presented

• Hands-on learning, especially in science, has a positive effect on student achievement.

• Teachers who use hands-on learning strategies have students who out-perform their peers on the National Assessment of Educational progress (NAEP) in the areas of science and mathematics.

• Despite the research supporting hands-on activity, it is a fairly uncommon instructional approach.

• Students have higher achievement rates when the focus of instruction is on meaningful conceptualization, especially when it emphasizes their own knowledge of the world.

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Make Card Games!

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Make Card Games!

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Build – A – Square• Build-a-square is based on the “Crazy” puzzles where 9

tiles are placed in a 3X3 square arrangement with all edges matching.

• Create 9 tiles with math problems and answers along the edges.

• The puzzle is designed so that the correct formation has all questions and answers matched on the edges.

• Tips: Design the answers for the edges first, then write the specific problems.

• Use more or less squares to tier.• Add distractors to outside edges and

“letter” pieces at the end.

m=3

b=6 -2/3

Nanci Smith

Page 58: Differentiating math

The ROLE of writer, speaker,artist, historian, etc.

An AUDIENCE of fellow writers,students, citizens, characters, etc.

Through a FORMAT that is written, spoken, drawn, acted, etc.

A TOPIC related to curriculumcontent in greater depth.

electron

neutron

proton

R A F T

Page 59: Differentiating math

RAFT ACTIVITY ON FRACTIONS

Role Audience Format Topic

Fraction Whole Number Petitions To be considered Part of the Family

Improper Fraction Mixed Numbers Reconciliation Letter Were More Alike than Different

A Simplified Fraction A Non-Simplified Fraction Public Service Announcement

A Case for Simplicity

Greatest Common Factor Common Factor Nursery Rhyme I’m the Greatest!

Equivalent Fractions Non Equivalent Personal Ad How to Find Your Soul Mate

Least Common Factor Multiple Sets of Numbers Recipe The Smaller the Better

Like Denominators in an Additional Problem

Unlike Denominators in an Addition Problem

Application form To Become A Like Denominator

A Mixed Number that Needs to be Renamed to Subtract

5th Grade Math Students Riddle What’s My New Name

Like Denominators in a Subtraction Problem

Unlike Denominators in a Subtraction Problem

Story Board How to Become a Like Denominator

Fraction Baker Directions To Double the Recipe

Estimated Sum Fractions/Mixed Numbers Advice Column To Become Well Rounded

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Angles Relationship RAFTRole Audience Format Topic

One vertical angle Opposite vertical angle Poem It’s like looking in a mirror

Interior (exterior) angle Alternate interior (exterior) angle

Invitation to a family reunion

My separated twin

Acute angle Missing angle Wanted poster Wanted: My complement

An angle less than 180 Supplementaryangle

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.

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Algebra RAFT

Role Audience Format Topic

Coefficient Variable Email We belong together

Scale / Balance Students Advice column Keep me in mind when solving an

equation

Variable Humans Monologue All that I can be

Variable Algebra students Instruction manual How and why to isolate me

Algebra Public Passionate plea Why you really do need me!

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RAFT Planning Sheet

Know

Understand

Do

How to Differentiate:

• Tiered? (See Equalizer)

• Profile? (Differentiate Format)

• Interest? (Keep options equivalent in learning)

• Other?

Role Audience Format Topic

Page 63: Differentiating math

Ideas for Cubing

• Arrange ________ into a 3-D collage to show ________

• Make a body sculpture to show ________

• Create a dance to show • Do a mime to help us understand• Present an interior monologue with

dramatic movement that ________• Build/construct a representation of

________• Make a living mobile that shows and

balances the elements of ________• Create authentic sound effects to

accompany a reading of _______• Show the principle of ________ with a

rhythm pattern you create. Explain to us how that works.

Ideas for Cubing in Math• Describe how you would solve ______• Analyze how this problem helps us use

mathematical thinking and problem solving• Compare and contrast this problem to one

on page _____.• Demonstrate how a professional (or just a

regular person) could apply this kink or problem to their work or life.

• Change one or more numbers, elements, or signs in the problem. Give a rule for what that change does.

• Create an interesting and challenging word problem from the number problem. (Show us how to solve it too.)

• Diagram or illustrate the solutionj to the problem. Interpret the visual so we understand it.

CubingCubing

Cubing

Page 64: Differentiating math

Nanci Smith

Describe how you would Explain the difference

solve or roll between adding and

the die to determine your multiplying fractions,

own fractions.

Compare and contrast Create a word problem

these two problems: that can be solved by

+

and (Or roll the fraction die to

determine your fractions.)

Describe how people use Model the problem

fractions every day. ___ + ___ .

Roll the fraction die to

determine which fractions

to add.

5

3

5

1

2

1

3

1

15

11

5

2

3

1

Page 65: Differentiating math

Nanci Smith

Page 66: Differentiating math

Nanci Smith

Describe how you would Explain why you need

solve or roll a common denominator

the die to determine your when adding fractions,

own fractions. But not when multiplying.

Can common denominators

Compare and contrast ever be used when dividing

these two problems: fractions?

Create an interesting and challenging word problem

A carpet-layer has 2 yards that can be solved by

of carpet. He needs 4 feet ___ + ____ - ____.

of carpet. What fraction of Roll the fraction die to

his carpet will he use? How determine your fractions.

do you know you are correct?

Diagram and explain the solution to ___ + ___ + ___.

Roll the fraction die to

determine your fractions.

91

1

7

3

13

2

7

1

7

3 and

2

1

3

1

Page 67: Differentiating math

Level 1:1. a, b, c and d each represent a different value. If a = 2, find b, c, and d.

a + b = ca – c = da + b = 5

2. Explain the mathematical reasoning involved in solving card 1.

3. Explain in words what the equation 2x + 4 = 10 means. Solve the problem.

4. Create an interesting word problem that is modeled by 8x – 2 = 7x.

5. Diagram how to solve 2x = 8.6. Explain what changing the “3” in 3x = 9 to a “2” does to the value of x. Why is this true?

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Level 2:1. a, b, c and d each represent a different value. If a = -1, find b, c, and d.

a + b = cb + b = dc – a = -a

2. Explain the mathematical reasoning involved in solving card 1.

3. Explain how a variable is used to solve word problems.4. Create an interesting word problem that is modeled by

2x + 4 = 4x – 10. Solve the problem.5. Diagram how to solve 3x + 1 = 10.6. Explain why x = 4 in 2x = 8, but x = 16 in ½ x = 8. Why does this make sense?

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Level 3:1. a, b, c and d each represent a different value. If a = 4, find

b, c, and d.a + c = bb - a = ccd = -dd + d = a

2. Explain the mathematical reasoning involved in solving card 1.

3. Explain the role of a variable in mathematics. Give examples.4. Create an interesting word problem that is modeled by

. Solve the problem.5. Diagram how to solve 3x + 4 = x + 12.6. Given ax = 15, explain how x is changed if a is large or a is

small in value.

7513 xx

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Designing a Differentiated Learning Designing a Differentiated Learning ContractContract

A Learning Contract has the following components1.1. A Skills ComponentA Skills Component

Focus is on skills-based tasksAssignments are based on pre-assessment of students’ readinessStudents work at their own level and pace

2.2. A content componentA content componentFocus is on applying, extending, or enriching key content (ideas, understandings)Requires sense making and productionAssignment is based on readiness or interest

3.3. A Time LineA Time LineTeacher sets completion date and check-in requirementsStudents select order of work (except for required meetings and homework)

4. The AgreementThe AgreementThe teacher agrees to let students have freedom to plan their timeStudents agree to use the time responsiblyGuidelines for working are spelled outConsequences for ineffective use of freedom are delineatedSignatures of the teacher, student and parent (if appropriate) are placed on the agreement

Differentiating Instruction: Facilitator’s Guide, ASCD, 1997

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Personal AgendaPersonal Agenda for _______________________________________

Starting Date _____________________________________________________

Teacher & studentinitials at completion

TaskSpecial Instructions

Remember to complete your daily planning log; I’ll call on you for conferences & instructions.

Montgomery County, MD

Page 72: Differentiating math

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

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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.

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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.

Page 75: Differentiating math

Similar Figures Menu

Optionals:1. Create an art project based on similarity. Write a cover

sheet describing the use of similarity and how it affects the quality of the art.

2. Make a photo album showing the use of similar figures in the world around us. Use captions to explain the similarity in each picture.

3. Write a story about similar figures in a world without similarity.

4. Write a song about the beauty and mathematics of similar figures.

5. Create a “how-to” or book about finding and creating similar figures.

Page 76: Differentiating math

Whatever it Takes!


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