1. 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

2. 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

3. What’s the point of differentiating in these different ways?
Learning Profile
Readiness
Interest
Growth
Motivation
Efficiency

4. 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

5. What does READINESS mean?
It is the student’s entry point relative to a particular understanding or skill.
C.A.Tomlinson, 1999

6. 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

7. 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 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

8. 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

9. Developing a Tiered Activity1 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
Create an activity that is
interesting
high level
causes students to use
key skill(s) to understand
a key idea 4
Chart the complexity of the activity
High skill/
Complexity
Low skill/
complexity 5
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 6
Match task to student based on student profile and task requirements

10. 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
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

11. 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.

12. 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?”

13. 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.

14. 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.

15. Graphing with a Point and a Slope
Advanced 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.

16. Relevant vs. Irrelevant
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

17. -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!!

18. Assessments
The assessments used in this learning profile section can be downloaded at:
Download the file entitled “Profile Assessments for Cards.”

19. 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
14. I like to have a certain amount of time to work on
15. I like to learn by moving and doing. Yes No
16. I like to learn while sitting at my desk. Yes No

20. Moderately Interested
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
3. Painting a picture
4. Designing a computer software project
5. Filming & editing a video
6. Creating a company
7. Helping in the community
8. Acting in a play

21. Moderately Interested
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
11. Writing for a newspaper
12. Discussing ideas
13. Drawing pictures for a book
14. Designing an interactive computer project
15. Filming & editing a television show
16. Operating a business
17. Working to help others
18. Acting out an event
19. Building a project
20. Playing in a band
21. Writing for a magazine
22. Talking about my project
23. Making a clay sculpture of a character

22. Moderately Interested
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
26. Marketing a product
27. Helping others by supporting a social cause
28. Acting out a story
29. Repairing a machine
30. Composing music
31. Writing an essay
32. Discussing my research
33. Painting a mural
34. Designing a computer
35. Recording & editing a radio show
36. Marketing an idea
37. Helping others by fundraising
38. Performing a skit

23. Moderately Interested
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
41. Writing a report
42. Talking about my experiences
43. Making a clay sculpture of a scene
44. Designing a multi-media computer show
45. Selecting slides and music for a slide show
46. Managing investments
47. Collecting clothing or food to help others
48. Role-playing a character
49. Assembling a kit
50. Playing in an orchestra
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.

24. Nanci Smith,Scottsdale,AZ
Learner Profile Card
Gender Stripe
Auditory, Visual, Kinesthetic
Modality
Analytical, Creative, Practical
Sternberg
Student’s Interests
Multiple Intelligence Preference
Gardner
Array Inventory
Nanci Smith,Scottsdale,AZ

25. 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.

26. Learning Profile Factors Intelligence Preference
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
creative
verbal/linguistic
logical/mathematical
spatial/visual
bodily/kinesthetic
musical/rhythmic
interpersonal
intrapersonal
naturalist
existential
Cognitive Style
Creative/conforming
Essence/facts
Expressive/controlled
Nonlinear/linear
Inductive/deductive
People-oriented/task or Object oriented
Concrete/abstract
Collaboration/competition
Interpersonal/introspective
Easily distracted/long Attention span
Group achievement/personal achievement
Oral/visual/kinesthetic
Reflective/action-oriented

27. Activity 2. 5 – The Modality Preferences Instrument (HBL, p
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 express my emotions verbally A D
I would rather be in a group discussion than read about a topic. A D

28. Interpreting the Instrument’s Score
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 Instrument’s Score
Total the number of “A” responses in items _____
This is your visual score
Total the number of “A” responses in items _____
This is your auditory score
Total the number of “A” responses in items _____
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.

29. 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

30. 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. 1 2 3 4 5 6 7 8
Smith & Smarr, 2005

31. 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) 1 2 3 4 5 6 7 8
Smith & Smarr, 2005

32. 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 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

33. 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
“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
“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

34. 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.

35. 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.

36. 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.

37. 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.

38. 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.

39. 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.

40. 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.

41. 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 _________________.
PRACTICAL
Streetsmart – Contextual – Focus on Use
Demonstrate how someone uses ________ in their life or work.
Show how we could apply _____ to solve this real life problem ____.
Based on your own experience, explain how _____ can be used.
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 ____________.

42. 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 I’m 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 ___

43. 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 other’s 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 ___

44. 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. ___ ___ ___
4. ___ ___ ___
7. ___ ___ ___
10. ___ ___ ___
13. ___ ___ ___
16. ___ ___ ___
19. ___ ___ ___
22. ___ ___ ___
25. ___ ___ ___
28. ___ ___ ___
Total Number of True:
Analytical ____ Creative _____ Practical _____

45. 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 won’t 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.

46. 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.

47. 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.

48. 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.

49. 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 Twice as nice
Circle Tangent poem You touch me!
Secant Chord voic I extend you.

50. Key Principles of a Differentiated Classroom
Assessment and instruction are inseparable.
Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD

51. 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

52. 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

53. 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

54. 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.

55. Make Card Games!

56. Make Card Games!

57. 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

58. 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.

59. 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
Baker
Directions
To Double the Recipe
Estimated Sum
Fractions/Mixed Numbers
Advice Column
To Become Well Rounded

60. Angles Relationship RAFT
Role
Audience
Format
Topic
One vertical angle
Opposite vertical angle
Poem
It’s like looking in a mirror
Interior (exterior) angle
Alternate interior (exterior) angle
Invitation to a family reunion
My separated twin
Acute angle
Missing angle
Wanted poster
Wanted: My complement
An angle less than 180
Supplementary
angle
Persuasive speech
Together, we’re a straight angle
**Angles
Humans
Video
See, we’re everywhere!
** This last entry would take more time than the previous 4 lines, and assesses a little differently. You could offer it as an option with a later due date, but you would need to specify that they need to explain what the angles are, and anything specific that you want to know such as what is the angle’s complement or is there a vertical angle that corresponds, etc.

61. 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
Humans
Monologue
All that I can be
Algebra students
Instruction manual
How and why to isolate me
Algebra
Public
Passionate plea
Why you really do need me!

62. RAFT Planning Sheet Know Role Audience Format Topic Understand Do
How to Differentiate:
Tiered? (See Equalizer)
Profile? (Differentiate Format)
Interest? (Keep options equivalent in learning)
Other?
Role
Audience
Format
Topic

63. 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.

64. Fraction Think Dots 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.
Fraction
Think Dots
Nanci Smith

65. Fraction
Think Dots
Nanci Smith

66. 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

67. 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?

68. 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?

69. 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.

70. 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: Facilitator’s Guide, ASCD, 1997

71. Personal Agenda Special Instructions Task
Montgomery County, MD
Personal Agenda for _______________________________________
Starting Date _____________________________________________________
Teacher & student
initials at
completion
Special Instructions
Task
Remember to complete your daily planning log; I’ll call on you for conferences & instructions.

72. 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

73. 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.

74. 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.

75. 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.

76. Whatever it Takes!