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Presentation on theme: "Differentiating Mathematics at the Middle and High School Levels Raising Student Achievement Conference St. Charles, IL December 4, 2007 "In the end,"— Presentation transcript:


2 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

3 Differentiation of Instruction Is a teacher’s response to learner’s needs guided by general principles of differentiation Respectful tasksFlexible groupingContinual assessment Teachers Can Differentiate Through: Content ProcessProduct According to Students’ ReadinessInterestLearning Profile

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

5 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

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

7 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

8 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

9 1.Identify the learning objectives or standards ALL students must learn. 2.Offer a pretest opportunity OR plan an alternate path through the content for those students who can learn the required material in less time than their age peers. 3.Plan and offer meaningful curriculum extensions for kids who qualify. **Depth and Complexity Applications of the skill being 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. 4.Eliminate all drill, practice, review, or preparation for students who have already mastered such things. 5.Keep accurate records of students’ compacting activities: document mastery. Compacting Strategy: Compacting

10 Developing a Tiered Activity Select the activity organizer concept generalization Essential to building a framework of understanding Think about your students/use assessments readiness range interests learning profile talents skills reading thinking information Create an activity that is interesting high level causes students to use key skill(s) to understand a key idea Chart the complexity of the activity High skill/ Complexity Low skill/ complexity Clone the activity along the ladder as needed to ensure challenge and success for your students, in materials – basic to advanced form of expression – from familiar to unfamiliar from personal experience to removed from personal experience equalizer Match task to student based on student profile and task requirements

11 Information, Ideas, Materials, Applications Representations, Ideas, Applications, Materials Resources, Research, Issues, Problems, Skills, Goals Directions, Problems, Application, Solutions, Approaches, Disciplinary Connections Application, Insight, Transfer Solutions, Decisions, Approaches Planning, Designing, Monitoring Pace of Study, Pace of Thought The Equalizer 1.FoundationalTransformational 2.ConcreteAbstract 1.SimpleComplex 2.Single FacetMultiple Facets 3.Small LeapGreat Leap 4.More StructuredMore Open 5.Less IndependenceGreater Independence 6.SlowQuick

12 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.

13 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?”

14 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.

15 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.

16 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.

17 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, energeticlow interaction hands on, learner input lecture seatwork EQUALS Increased intrinsicIncreased MOTIVATION APATHY & RESENTMENT

18 - 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!!

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

20 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

21 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 InterestedOf Little InterestModerately InterestedInterestedVery Interested 1. Writing Stories Discussing what I have learned Painting a picture Designing a computer software project Filming & editing a video Creating a company Helping in the community Acting in a play12345

22 Not At All InterestedOf Little InterestModerately InterestedInterestedVery Interested 9. Building an invention Playing musical instrument Writing for a newspaper Discussing ideas Drawing pictures for a book Designing an interactive computer project Filming & editing a television show Operating a business Working to help others Acting out an event Building a project Playing in a band Writing for a magazine Talking about my project Making a clay sculpture of a character 12345

23 Not At All InterestedOf Little InterestModerately InterestedInterestedVery Interested 24. Designing information for the computer internet Filming & editing a movie Marketing a product Helping others by supporting a social cause Acting out a story Repairing a machine Composing music Writing an essay Discussing my research Painting a mural Designing a computer Recording & editing a radio show Marketing an idea Helping others by fundraising Performing a skit12345

24 Not At All InterestedOf Little InterestModerately InterestedInterestedVery Interested 39. Constructing a working model Performing music Writing a report Talking about my experiences Making a clay sculpture of a scene Designing a multi- media computer show Selecting slides and music for a slide show Managing investments Collecting clothing or food to help others Role-playing a character Assembling a kit 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.

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

26 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.

27 Learning Profile Factors Group Orientation independent/self orientation group/peer orientation adult orientation combination Learning Environment quiet/noise warm/cool still/mobile flexible/fixed “busy”/”spare” 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 Intelligence Preference analytic practical creative verbal/linguistic logical/mathematical spatial/visual bodily/kinesthetic musical/rhythmic interpersonal intrapersonal naturalist existential Gender & Culture

28 Activity 2.5 – The Modality Preferences Instrument (HBL, p. 23) Follow the directions below to get a score that will indicate your own modality (sense) preference(s). This instrument, keep in mind that sensory preferences are usually evident only during prolonged and complex learning tasks. Identifying Sensory Preferences Directions: For each item, circle “A” if you agree that the statement describes you most of the time. Circle “D” if you disagree that the statement describes you most of the time. 1.I Prefer reading a story rather than listening to someone tell it. A D 2.I would rather watch television than listen to the radio. A D 3.I remember faces better than names.A D 4.I like classrooms with lots of posters and pictures around the room. A D 5.The appearance of my handwriting is important to me. A D 6.I think more often in pictures. A D 7.I am distracted by visual disorder or movement. A D 8.I have difficulty remembering directions that were told to me. A D 9.I would rather watch athletic events than participate in them. A D 10.I tend to organize my thoughts by writing them down. A D 11.My facial expression is a good indicator of my emotions. A D 12.I tend to remember names better than faces. A D 13.I would enjoy taking part in dramatic events like plays. A D 14.I tend to sub vocalize and think in sounds.A D 15.I am easily distracted by sounds. A D 16.I easily forget what I read unless I talk about it. A D 17.I would rather listen to the radio than watch TVA D 18.My handwriting is not very good. A D 19.When faced with a problem, I tend to talk it through.A D 20.I express my emotions verbally. A D 21.I would rather be in a group discussion than read about a topic. A D

29 22.I prefer talking on the phone rather than writing a letter to someone. A D 23.I would rather participate in athletic events than watch them. A D 24.I prefer going to museums where I can touch the exhibits. A D 25.My handwriting deteriorates when the space becomes smaller. A D 26.My mental pictures are usually accompanied by movement. A D 27.I like being outdoors and doing things like biking, camping, swimming, hiking etc. A D 28.I remember best what was done rather then what was seen or talked about. A D 29.When faced with a problem, I often select the solution involving the greatest activity. A D 30.I like to make models or other hand crafted items. A D 31.I would rather do experiments rather then read about them. A D 32.My body language is a good indicator of my emotions. A D 33.I have difficulty remembering verbal directions if I have not done the activity before. A D Interpreting the Instrument’s Score Total the number of “A” responses in items 1-11 _____ This is your visual score Total the number of “A” responses in items 12-22_____ This is your auditory score Total the number of “A” responses in items 23-33_____ This is you tactile/kinesthetic score If you scored a lot higher in any one area: This indicates that this modality is very probably your preference during a protracted and complex learning situation. If you scored a lot lower in any one area: This indicates that this modality is not likely to be your preference(s) in a learning situation. If you got similar scores in all three areas: This indicates that you can learn things in almost any way they are presented.

30 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

31 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,

32 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,

33 EIGHT STYLES OF LEARNING TYPE CHARACTERISTICS LIKES TOIS 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

34 EIGHT STYLES OF LEARNING, Cont’d TYPE CHARACTERISTICS LIKES TOIS GOOD AT LEARNS BEST BY BODILY/ Kinesthetic Learner “The Mover” Eager to solve problems physically. Often doesn’t read directions but just starts on a project Move around Touch and talk Use body language Physical activities (Sports/dance/ acting) crafts Touching Moving Interacting with space Processing knowledge through bodily sensations INTERpersonal Learner “The Socializer” Likes group work and working cooperatively to solve problems. Has an interest in their community. Have lots of friends Talk to people Join groups Understanding people Leading others Organizing Communicating Manipulating Mediating conflicts Sharing Comparing Relating Cooperating interviewing INTRApersonal Learner “The Individual” Enjoys the opportunity to reflect and work independently. Often quiet and would rather work on his/her own than in a group. Work alone Pursue own interests Understanding self Focusing inward on feelings/dreams Pursuing interests/ goals Being original Working along Individualized projects Self-paced instruction Having own space NATURALIST “The Nature Lover” Enjoys relating things to their environment. Have a strong connection to nature. Physically experience nature Do observations Responds to patterning nature Exploring natural phenomenon Seeing connections Seeing patterns Reflective Thinking Doing observations Recording events in Nature Working in pairs Doing long term projects

35 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.

36 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.

37 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.

38 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.

39 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.

40 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.

41 Sternberg’s Three Intelligences CreativeAnalytical 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.

42 Linear – Schoolhouse Smart - Sequential ANALYTICAL Thinking About the Sternberg Intelligences Show the parts of _________ and how they work. Explain why _______ works the way it does. Diagram how __________ affects __________________. Identify the key parts of _____________________. Present a step-by-step approach to _________________. Streetsmart – Contextual – Focus on 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. Here’s a problem at school, ________. Using your knowledge of ______________, develop a plan to address the problem. CREATIVEInnovator – 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 ____________.

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. 1.Analyzing characters when I’m reading or listening to a story ___ 2.Designing new things___ 3.Taking things apart and fixing them___ 4.Comparing and contrasting points of view___ 5.Coming up with ideas___ 6.Learning through hands-on activities___ 7.Criticizing my own and other kids’ work___ 8.Using my imagination___ 9.Putting into practice things I learned___ 10.Thinking clearly and analytically___ 11.Thinking of alternative solutions___ 12.Working with people in teams or groups___ 13.Solving logical problems ___ 14.Noticing things others often ignore ___ 15.Resolving conflicts ___

44 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. 16.Evaluating my own and other’s points of view___ 17.Thinking in pictures and images___ 18.Advising friends on their problems___ 19.Explaining difficult ideas or problems to others___ 20.Supposing things were different___ 21.Convincing someone to do something___ 22.Making inferences and deriving conclusions___ 23.Drawing___ 24.Learning by interacting with others___ 25.Sorting and classifying___ 26.Inventing new words, games, approaches___ 27.Applying my knowledge___ 28.Using graphic organizers or images to organize your thoughts___ 29.Composing___ 30.Adapting to new situations___

45 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. AnalyticalCreativePractical 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 _____

46 Understanding Order of Operations Analytic Task Practical Task Creative Task Make a chart that shows all ways you can think of to use order of operations to equal 18. A friend is convinced that order of operations do not matter in math. Think of as many ways to convince your friend that without using them, you won’t necessarily get the correct answers! Give lots of examples. Write a book of riddles that involve order of operations. Show the solution and pictures on the page that follows each riddle.

47 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.

48 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.

49 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.

50 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 RoleAudience Format Topic DiameterRadius Twice as nice CircleTangent poemYou touch me! SecantChordvoic I extend you.

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

52 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

53 THINKING ABOUT ON-GOING ASSESSMENT STUDENT DATA SOURCES 1.Journal entry 2.Short answer test 3.Open response test 4.Home learning 5.Notebook 6.Oral response 7.Portfolio entry 8.Exhibition 9.Culminating product 10.Question writing 11.Problem solving TEACHER DATA MECHANISMS 1.Anecdotal records 2.Observation by checklist 3.Skills checklist 4.Class discussion 5.Small group interaction 6.Teacher – student conference 7.Assessment stations 8.Exit cards 9.Problem posing 10.Performance tasks and rubrics

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

55 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.

56 Make Card Games!


58 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

59 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. R A F T

60 RAFT ACTIVITY ON FRACTIONS RoleAudienceFormatTopic FractionWhole NumberPetitionsTo be considered Part of the Family Improper FractionMixed NumbersReconciliation LetterWere More Alike than Different A Simplified FractionA Non-Simplified FractionPublic Service Announcement A Case for Simplicity Greatest Common FactorCommon FactorNursery RhymeI’m the Greatest! Equivalent FractionsNon EquivalentPersonal AdHow to Find Your Soul Mate Least Common FactorMultiple Sets of NumbersRecipeThe Smaller the Better Like Denominators in an Additional Problem Unlike Denominators in an Addition Problem Application formTo Become A Like Denominator A Mixed Number that Needs to be Renamed to Subtract 5 th Grade Math StudentsRiddleWhat’s My New Name Like Denominators in a Subtraction Problem Unlike Denominators in a Subtraction Problem Story BoardHow to Become a Like Denominator FractionBakerDirectionsTo Double the Recipe Estimated SumFractions/Mixed NumbersAdvice ColumnTo Become Well Rounded

61 Angles Relationship RAFT RoleAudienceFormatTopic One vertical angleOpposite vertical anglePoemIt’s like looking in a mirror Interior (exterior) angleAlternate interior (exterior) angle Invitation to a family reunion My separated twin Acute angleMissing angleWanted posterWanted: My complement An angle less than 180Supplementary angle Persuasive speechTogether, we’re a straight angle **AnglesHumansVideoSee, 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.

62 Algebra RAFT RoleAudienceFormatTopic CoefficientVariable We belong together Scale / BalanceStudentsAdvice columnKeep me in mind when solving an equation VariableHumansMonologueAll that I can be VariableAlgebra studentsInstruction manualHow and why to isolate me AlgebraPublicPassionate pleaWhy you really do need me!

63 RAFT Planning Sheet Know Understand Do How to Differentiate: Tiered? (See Equalizer) Profile? (Differentiate Format) Interest? (Keep options equivalent in learning) Other? RoleAudienceFormatTopic

64 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. Cubing

65 Nanci Smith Describe how you wouldExplain the difference solve or rollbetween adding and the die to determine yourmultiplying 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 useModel the problem fractions every day.___ + ___. Roll the fraction die to determine which fractions to add.

66 Nanci Smith

67 Describe how you wouldExplain why you need solve or rolla common denominator the die to determine yourwhen 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 yardsthat can be solved by of carpet. He needs 4 feet___ + ____ - ____. of carpet. What fraction ofRoll the fraction die to his carpet will he use? Howdetermine your fractions. do you know you are correct? Diagram and explain the solution to ___ + ___ + ___. Roll the fraction die to determine your fractions.

68 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?

69 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 = Explain why x = 4 in 2x = 8, but x = 16 in ½ x = 8. Why does this make sense?

70 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 Given ax = 15, explain how x is changed if a is large or a is small in value.

71 Designing a Differentiated Learning Contract A Learning Contract has the following components 1.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 2.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 3.A Time Line Teacher sets completion date and check-in requirements Students select order of work (except for required meetings and homework) The Agreement 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

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

73 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

74 Similar Figures Menu Imperatives (Do all 3): 1.Write a mathematical definition of “Similar Figures.” It must include all pertinent vocabulary, address all concepts and be written so that a fifth grade student would be able to understand it. Diagrams can be used to illustrate your definition. 2.Generate a list of applications for similar figures, and similarity in general. Be sure to think beyond “find a missing side…” 3.Develop a lesson to teach third grade students who are just beginning to think about similarity.

75 Similar Figures Menu Negotiables (Choose 1): 1.Create a book of similar figure applications and problems. This must include at least 10 problems. They can be problems you have made up or found in books, but at least 3 must be application problems. Solver each of the problems and include an explanation as to why your solution is correct. 2.Show at least 5 different application of similar figures in the real world, and make them into math problems. Solve each of the problems and explain the role of similarity. Justify why the solutions are correct.

76 Similar Figures Menu Optionals: 1.Create an art project based on similarity. Write a cover sheet describing the use of similarity and how it affects the quality of the art. 2.Make a photo album showing the use of similar figures in the world around us. Use captions to explain the similarity in each picture. 3.Write a story about similar figures in a world without similarity. 4.Write a song about the beauty and mathematics of similar figures. 5.Create a “how-to” or book about finding and creating similar figures.

77 Whatever it Takes!

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