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Differentiating Mathematics at the Middle and High School LevelsRaising Student Achievement ConferenceSt. 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 Smith

Educational Consultant

Curriculum and Professional Development

Cave Creek, AZ


Differentiation of Instruction

Is a teacher’s response to learner’s needsguided by general principles of differentiation

Respectful tasks

Flexible grouping

Continual assessment

Teachers Can Differentiate Through:




According to Students’



Learning Profile

what s the point of differentiating in these different ways
What’s the point of differentiating in these different ways?

Learning Profile






key principles of a differentiated classroom
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


What does READINESS mean?

It is the student’s 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





Tiered tasks and procedures

Flexible time use

Small group instruction

Homework options

Tiered or scaffolded assemssment



Negotiated criteria for quality

Varied graphic organizers


Providing support needed for a student to succeed in work slightly beyond his/her comfort zone.


  • For example…
  • Directions that give more structure – or less
  • Tape recorders to help with reading or writing beyond the 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


  • 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


Create an activity that is

  • interesting
  • high level
  • causes students to use
  • key skill(s) to understand
  • a key idea

High skill/


Low skill/


Chart the complexity of the activity

  • 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

Developing a Tiered Activity



  • Select the activity organizer
  • concept
  • generalization
  • Think about your students/use assessments
  • readiness range
  • interests
  • learning profile
  • talents

Essential to building

a framework of











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

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.

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
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?”
graphing with a point and a slope13
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 slope14
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 slope15
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.
brain research shows that eric jensen teaching with the brain in mind 1998
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


Increased intrinsic Increased


choice the great motivator
-CHOICE-The Great Motivator!
  • Requires children to be aware of their own readiness, interests, and learning profiles.
  • Students have choices provided by the teacher. (YOU are still in charge of crafting challenging opportunities for all kiddos – NO taking the easy way out!)
  • Use choice across the curriculum: writing topics, content writing prompts, self-selected reading, contract menus, math problems, spelling words, product and assessment options, seating, group arrangement, ETC . . .
  • Research currently suggests that CHOICE should be offered 35% of the time!!

The assessments used in this learning profile section can be downloaded at:

Download the file entitled “Profile Assessments for Cards.”

how do you like to learn
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

an assignment. Yes No

15. I like to learn by moving and doing. Yes No

16. I like to learn while sitting at my desk. 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: ____________________________________________________


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).


Instructions: My Way …A Profile

Write your score beside each number. Add each Row to determine your expression style profile.

learner profile card
Learner Profile Card

Gender Stripe

Auditory, Visual, Kinesthetic


Analytical, Creative, Practical


Student’s Interests

Multiple Intelligence Preference


Array Inventory

Nanci Smith,Scottsdale,AZ

differentiation using learning profile
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






Group Orientation

independent/self orientation

group/peer orientation

adult orientation





Intelligence Preference













Cognitive Style






People-oriented/task or Object oriented




Easily distracted/long Attention span

Group achievement/personal achievement




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.

  • 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

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 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
Parallel Lines Cut by a Transversal
  • Visual: Make posters showing all the angle relations formed by a pair of parallel lines cut by a transversal. Be sure to color code definitions and angles, and state the relationships between all possible angles.









Smith & Smarr, 2005

parallel lines cut by a transversal30









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 transversal31









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

introduction to change mi
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 mi35
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 mi36
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 mi37
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
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 proofs39
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.
sternberg s three intelligences
Sternberg’s Three Intelligences




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

Thinking About the Sternberg Intelligences


Linear – Schoolhouse Smart - Sequential

Show the parts of _________ and how they work.

Explain why _______ works the way it does.

Diagram how __________ affects __________________.

Identify the key parts of _____________________.

Present a step-by-step approach to _________________.

Streetsmart – Contextual – Focus on Use


Demonstrate how someone uses ________ in their life or work.

Show how we could apply _____ to solve this real life problem ____.

Based on your own experience, explain how _____ can be used.

Here’s a problem at school, ________. Using your knowledge of ______________, develop a plan to address the problem.


Innovator – Outside the Box – What If - Improver

Find a new way to show _____________.

Use unusual materials to explain ________________.

Use humor to show ____________________.

Explain (show) a new and better way to ____________.

Make connections between _____ and _____ to help us understand ____________.

Become a ____ and use your “new” perspectives to help us think about ____________.

triarchic theory of intelligences robert sternberg
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.

  • 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 ___
triarchic theory of intelligences robert sternberg43
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.

  • 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 ___

triarchic theory of intelligences key robert sternberg
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 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 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.

forms of equations of lines
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
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 vocabulary48
Circle Vocabulary


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.


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 vocabulary49
Circle Vocabulary


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.


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 classroom50
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

Journal entry

Short answer test

Open response test

Home learning


Oral response

Portfolio entry


Culminating product

Question writing

Problem solving


Anecdotal records

Observation by checklist

Skills checklist

Class discussion

Small group interaction

Teacher – student conference

Assessment stations

Exit cards

Problem posing

Performance tasks and rubrics

key principles of a differentiated classroom53
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



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.
build a square




Build – A – Square
  • Build-a-square is based on the “Crazy” puzzles where 9 tiles are placed in a 3X3 square arrangement with all edges matching.
  • Create 9 tiles with math problems and answers along the edges.
  • The puzzle is designed so that the correct formation has all questions and answers matched on the edges.
  • Tips: Design the answers for the edges first, then write the specific problems.
  • Use more or less squares to tier.
  • Add distractors to outside edges and

“letter” pieces at the end.

Nanci Smith



The ROLE of writer, speaker,

artist, historian, 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.

angles relationship raft
Angles Relationship RAFT

** 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.

raft planning sheet
RAFT Planning Sheet




How to Differentiate:

  • Tiered? (See Equalizer)
  • Profile? (Differentiate Format)
  • Interest? (Keep options equivalent in learning)
  • Other?
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.



Ideas for Cubing



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.


Think Dots

Nanci Smith



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.


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: Facilitator’s Guide, ASCD, 1997


Personal Agenda

Montgomery County, MD

Personal Agenda for_______________________________________

Starting Date _____________________________________________________

Teacher & student

initials at


Special Instructions


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

proportional reasoning think tac toe
Proportional Reasoning Think-Tac-Toe

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
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 menu74
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 menu75
Similar Figures Menu


  • 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.