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PPT – Differentiating Mathematics at the Middle and High School Levels Raising Student Achievement Conference St. Charles, IL December 4, 2007 PowerPoint presentation | free to download - id: 5c6a87-N2E1M

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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
- 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

Process

Product

Content

According to Students

Readiness

Interest

Learning Profile

Whats the point of differentiating in these

different ways?

Learning Profile

Readiness

Interest

Growth

Motivation

Efficiency

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,

TX ASCD

READINESS

- What does READINESS mean?
- It is the students entry point relative to a

particular understanding or skill. - C.A.Tomlinson, 1999

A Few Routes to READINESS DIFFERENTIATION

- 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.

Scaffolding

- 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

level - Use of study guides
- Use of organizers
- New American Lecture
- Tomlinson, 2000

Compacting

- 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

1

2

- 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

3

4

5

6

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

fractions.

Red Group Use Venn diagrams to model LCMs (least

common multiple). Explain how this process can

be used to find common denominators. Use the

method on more challenging addition

problems. Write a manual on how to add

fractions. It must include why a common

denominator is needed, and at least three ways to

find it.

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 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

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.

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 - EQUALS
- Increased intrinsic Increased
- MOTIVATION APATHY RESENTMENT

-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!!

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.

How Do You Like to Learn?

- 1. I study best when it is quiet. Yes No
- 2. I am able to ignore the noise of
- other people talking while I am working. Yes

No - 3. I like to work at a table or desk. Yes No
- 4. I like to work on the floor. Yes No
- 5. I work hard by myself. Yes No
- 6. I work hard for my parents or teacher. Yes

No - 7. I will work on an assignment until it is

completed, no - matter what. Yes No
- 8. Sometimes I get frustrated with my work
- and do not finish it. Yes No
- 9. When my teacher gives an assignment, I like

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

to - create my own steps on how to complete it. Yes

No - 11. I like to work by myself. Yes No
- 12. I like to work in pairs or in groups. Yes No
- 13. I like to have unlimited amount of time to

work on - an assignment. Yes No

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

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.

- I Prefer reading a story rather than listening to

someone tell it. A D - I would rather watch television than listen to

the radio. A D - I remember faces better than names. A D
- I like classrooms with lots of posters and

pictures around the room. A D - The appearance of my handwriting is important to

me. A D - I think more often in pictures. A D
- I am distracted by visual disorder or movement.

A D - I have difficulty remembering directions that

were told to me. A D - I would rather watch athletic events than

participate in them. A D - I tend to organize my thoughts by writing them

down. A D - My facial expression is a good indicator of my

emotions. A D - I tend to remember names better than faces.

A D - I would enjoy taking part in dramatic events like

plays. A D - I tend to sub vocalize and think in sounds. A

D - I am easily distracted by sounds. A D
- I easily forget what I read unless I talk about

it. A D - I would rather listen to the radio than watch

TV A D - My handwriting is not very good. A D
- 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

D - 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

presented.

Parallel Lines Cut by a Transversal

- Visual Make posters showing all the angle

relations formed by a pair of parallel lines cut

by a transversal. Be sure to color code

definitions and angles, and state the

relationships between all possible angles.

1

2

3

4

5

6

8

7

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

EIGHT STYLES OF LEARNING

TYPE 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 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

EIGHT STYLES OF LEARNING, Contd

TYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BY

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

class.

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

theorem.

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.

Sternbergs 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.

Thinking About the Sternberg Intelligences

ANALYTICAL

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

PRACTICAL

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.

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

____________.

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

activity. - 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

activity. - 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

Sternberg

- 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

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.

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,

TX ASCD

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

THINKING ABOUT ON-GOING ASSESSMENT

- STUDENT DATA SOURCES
- Journal entry
- Short answer test
- Open response test
- Home learning
- Notebook
- Oral response
- Portfolio entry
- Exhibition
- Culminating product
- Question writing
- Problem solving

- TEACHER DATA MECHANISMS
- 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,

TX ASCD

USE OF INSTRUCTIONAL STRATEGIES. The following

findings related to instructional strategies are

supported by the existing research

- Techniques and instructional strategies have

nearly as much influence on student learning as

student aptitude. - Lecturing, a common teaching strategy, is an

effort to quickly cover the material however, it

often overloads and over-whelms students with

data, making it likely that they will confuse the

facts presented - Hands-on learning, especially in science, has a

positive effect on student achievement. - Teachers who use hands-on learning strategies

have students who out-perform their peers on the

National Assessment of Educational progress

(NAEP) in the areas of science and mathematics. - Despite the research supporting hands-on

activity, it is a fairly uncommon instructional

approach. - Students have higher achievement rates when the

focus of instruction is on meaningful

conceptualization, especially when it emphasizes

their own knowledge of the world.

Make Card Games!

Make Card Games!

Build A Square

- Build-a-square is based on the Crazy puzzles

where 9 tiles are placed in a 3X3 square

arrangement with all edges matching. - Create 9 tiles with math problems and answers

along the edges. - The puzzle is designed so that the correct

formation has all questions and answers matched

on the edges. - Tips Design the answers for the edges first,

then write the specific problems. - Use more or less squares to tier.
- Add distractors to outside edges and
- letter pieces at the end.

Nanci Smith

R A F T

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 curriculum content in

greater depth.

RAFT ACTIVITY ON FRACTIONS RAFT ACTIVITY ON FRACTIONS RAFT ACTIVITY ON FRACTIONS 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 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

Cubing

Cubing

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.

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

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?

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

requirements - Students select order of work (except for

required meetings and homework) - 4. The Agreement
- The teacher agrees to let students have freedom

to plan their time - Students agree to use the time responsibly
- Guidelines for working are spelled out
- Consequences for ineffective use of freedom are

delineated - Signatures of the teacher, student and parent (if

appropriate) are placed on the agreement

Differentiating Instruction Facilitators Guide,

ASCD, 1997

Personal Agenda

Montgomery County, MD

Personal Agenda for ______________________________

_________

Starting Date ____________________________________

_________________

Teacher student initials at completion

Special Instructions

Task

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!