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, all."— Presentation transcript:
1Differentiating 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 SmithEducational ConsultantCurriculum and Professional DevelopmentCave Creek, AZ
2Differentiation of Instruction Is a teacher’s response to learner’s needs guided by general principles of differentiationRespectful tasksFlexible groupingContinual assessmentTeachers Can Differentiate Through:ContentProcessProductAccording to Students’ReadinessInterestLearning Profile
3What’s the point of differentiating in these different ways? Learning ProfileReadinessInterestGrowthMotivationEfficiency
4Key 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
5What does READINESS mean? It is the student’s entry point relative to a particular understanding or skill.C.A.Tomlinson, 1999
6A Few Routes to READINESS DIFFERENTIATION Varied texts by reading levelVaried supplementary materialsVaried scaffoldingreadingwritingresearchtechnologyTiered tasks and proceduresFlexible time useSmall group instructionHomework optionsTiered or scaffolded assemssmentCompactingMentorshipsNegotiated criteria for qualityVaried graphic organizers
7Providing support needed for a student to succeed in work slightly beyond his/her comfort zone. ScaffoldingFor example…Directions that give more structure – or lessTape recorders to help with reading or writing beyond the student’s graspIcons to help interpret printReteaching / extending teachingModelingClear criteria for successReading buddies (with appropriate directions)Double entry journals with appropriate challengeTeaching through multiple modesUse of manipulatives when neededGearing reading materials to student reading levelUse of study guidesUse of organizersNew American LectureTomlinson, 2000
8CompactingIdentify 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 ComplexityApplications of the skill being taughtLearning Profile tasks based on understanding the process instead of skill practiceDiffering perspectives, ideas across time, thinking like a mathematician**Orbitals and Independent studies.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
9Developing a Tiered Activity 12Select the activity organizerconceptgeneralizationThink about your students/use assessmentsreadiness rangeinterestslearning profiletalentsEssential to buildinga framework ofunderstandingskillsreadingthinkinginformation3Create an activity that isinterestinghigh levelcauses students to usekey skill(s) to understanda key idea4Chart the complexity of the activityHigh skill/ComplexityLow skill/complexity5Clone the activity along the ladder as needed to ensure challenge and success for your students, inmaterials – basic to advancedform of expression – from familiar to unfamiliarfrom personal experience to removed from personal experienceequalizer6Match task to student based on student profile and task requirements
10The Equalizer Foundational Transformational Concrete Abstract Simple ComplexSingle Facet Multiple FacetsSmall Leap Great LeapMore Structured More OpenLess Independence Greater IndependenceSlow QuickInformation, Ideas, Materials, ApplicationsRepresentations, Ideas, Applications, MaterialsResources, Research, Issues, Problems, Skills, GoalsDirections, Problems, Application, Solutions, Approaches, Disciplinary ConnectionsApplication, Insight, TransferSolutions, Decisions, ApproachesPlanning, Designing, MonitoringPace of Study, Pace of Thought
11Adding 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 GroupManipulatives such as Cuisinaire rods and fraction circles will be available as a resource for the group. Students use factor trees and lists of multiples to find common denominators. Using this approach, pairs and triplets of fractions are rewritten using common denominators. End by adding several different problems of increasing challenge and length.Suzie says that adding fractions is like a game: you just need to know the rules. Write game instructions explaining the rules of adding fractions.Red GroupUse Venn diagrams to model LCMs (least common multiple). Explain how this process can be used to find common denominators. Use the method on more challenging addition problems.Write a manual on how to add fractions. It must include why a common denominator is needed, and at least three ways to find it.
12Graphing 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?”
13Graphing 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.
14Graphing 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.
15Graphing 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.
16Relevant vs. Irrelevant BRAIN RESEARCH SHOWS THAT. . . Eric Jensen, Teaching With the Brain in Mind, 1998Choices vs. Requiredcontent, process, product no student voicegroups, resources environment restricted resourcesRelevant vs Irrelevantmeaningful impersonalconnected to learner out of contextdeep understanding only to pass a testEngaging vs Passiveemotional, energetic low interactionhands on, learner input lecture seatworkEQUALSIncreased intrinsic IncreasedMOTIVATION 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!!
18AssessmentsThe assessments used in this learning profile section can be downloaded at:Download the file entitled “Profile Assessments for Cards.”
19How Do You Like to Learn? 1. I study best when it is quiet. Yes No 2. I am able to ignore the noise ofother people talking while I am working. Yes No3. I like to work at a table or desk. Yes No4. I like to work on the floor. Yes No5. I work hard by myself. Yes No6. I work hard for my parents or teacher. Yes No7. I will work on an assignment until it is completed, nomatter what. Yes No8. Sometimes I get frustrated with my workand do not finish it. Yes No9. When my teacher gives an assignment, I like tohave exact steps on how to complete it. Yes No10. When my teacher gives an assignment, I like tocreate my own steps on how to complete it. Yes No11. I like to work by myself. Yes No12. I like to work in pairs or in groups. Yes No13. I like to have unlimited amount of time to work onan assignment. Yes No14. I like to have a certain amount of time to work on15. I like to learn by moving and doing. Yes No16. I like to learn while sitting at my desk. Yes No
20Moderately Interested My WayAn expression Style InventoryK.E. Kettle J.S. Renzull, M.G. RizzaUniversity of ConnecticutProducts 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 Interested1. Writing Stories123452. Discussing what I have learned3. Painting a picture4. Designing a computer software project5. Filming & editing a video6. Creating a company7. Helping in the community8. Acting in a play
21Moderately Interested Not At All InterestedOf Little InterestModerately InterestedInterestedVery Interested9. Building an invention1234510. Playing musical instrument11. Writing for a newspaper12. Discussing ideas13. Drawing pictures for a book14. Designing an interactive computer project15. Filming & editing a television show16. Operating a business17. Working to help others18. Acting out an event19. Building a project20. Playing in a band21. Writing for a magazine22. Talking about my project23. Making a clay sculpture of a character
22Moderately Interested Not At All InterestedOf Little InterestModerately InterestedInterestedVery Interested24. Designing information for the computer internet1234525. Filming & editing a movie26. Marketing a product27. Helping others by supporting a social cause28. Acting out a story29. Repairing a machine30. Composing music31. Writing an essay32. Discussing my research33. Painting a mural34. Designing a computer35. Recording & editing a radio show36. Marketing an idea37. Helping others by fundraising38. Performing a skit
23Moderately Interested Not At All InterestedOf Little InterestModerately InterestedInterestedVery Interested39. Constructing a working model.1234540. Performing music41. Writing a report42. Talking about my experiences43. Making a clay sculpture of a scene44. Designing a multi-media computer show45. Selecting slides and music for a slide show46. Managing investments47. Collecting clothing or food to help others48. Role-playing a character49. Assembling a kit50. Playing in an orchestraProductsWrittenOralArtisticComputerAudio/VisualCommercialServiceDramatizationManipulativeMusical1. ___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 ProfileWrite your score beside each number. Add each Row to determine your expression style profile.
25Differentiation Using LEARNING PROFILE Learning profile refers to how an individual learns best - most efficiently and effectively.Teachers and their students maydiffer in learning profile preferences.
27Activity 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 DI would rather watch television than listen to the radio. A DI remember faces better than names. A DI like classrooms with lots of posters and pictures around the room. A DThe appearance of my handwriting is important to me. A DI think more often in pictures A DI am distracted by visual disorder or movement. A DI have difficulty remembering directions that were told to me. A DI would rather watch athletic events than participate in them. A DI tend to organize my thoughts by writing them down. A DMy facial expression is a good indicator of my emotions. A DI tend to remember names better than faces. A DI would enjoy taking part in dramatic events like plays. A DI tend to sub vocalize and think in sounds. A DI am easily distracted by sounds A DI easily forget what I read unless I talk about it. A DI would rather listen to the radio than watch TV A DMy handwriting is not very good A DWhen faced with a problem , I tend to talk it through. A DI express my emotions verbally A DI would rather be in a group discussion than read about a topic. A D
28Interpreting the Instrument’s Score I prefer talking on the phone rather than writing a letter to someone. A DI would rather participate in athletic events than watch them A DI prefer going to museums where I can touch the exhibits A DMy handwriting deteriorates when the space becomes smaller A DMy mental pictures are usually accompanied by movement A DI like being outdoors and doing things like biking, camping, swimming, hiking etc. A DI remember best what was done rather then what was seen or talked about A DWhen faced with a problem, I often select the solution involving the greatest activity A DI like to make models or other hand crafted items A DI would rather do experiments rather then read about them A DMy body language is a good indicator of my emotions A DI have difficulty remembering verbal directions if I have not done the activity before. A DInterpreting the Instrument’s ScoreTotal the number of “A” responses in items _____This is your visual scoreTotal the number of “A” responses in items _____This is your auditory scoreTotal the number of “A” responses in items _____This is you tactile/kinesthetic scoreIf 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.
29Parallel 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.12345687Smith & Smarr, 2005
30Parallel 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 supplementarysupplementary to angle 1.Name an angle congruentto angle 2.12345678Smith & Smarr, 2005
31Parallel 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)12345678Smith & Smarr, 2005
32EIGHT STYLES OF LEARNING TYPECHARACTERISTICSLIKES TOIS GOOD ATLEARNS BEST BYLINGUISTICLEARNER“The Word Player”Learns through the manipulation of words. Loves to read and write in order to explain themselves. They also tend to enjoy talkingReadWriteTell storiesMemorizing names, places, dates and triviaSaying, hearing and seeing wordsLOGICAL/MathematicalLearner“The Questioner”Looks for patterns when solving problems. Creates a set of standards and follows them when researching in a sequential manner.Do experimentsFigure things outWork with numbersAsk questionsExplore patterns and relationshipsMathReasoningLogicProblem solvingCategorizingClassifyingWorking with abstract patterns/relationshipsSPATIAL LEARNER“The Visualizer”Learns through pictures, charts, graphs, diagrams, and art.Draw, build, design and create thingsDaydreamLook at pictures/slidesWatch moviesPlay with machinesImagining thingsSensing changesMazes/puzzlesReading maps, chartsVisualizingDreamingUsing the mind’s eyeWorking with colors/picturesMUSICAL LEARNER“The Music Lover”Learning is often easier for these students when set to music or rhythmSing, hum tunesListen to musicPlay an instrumentRespond to musicPicking up soundsRemembering melodiesNoticing pitches/ rhythmsKeeping timeRhythmMelodyMusic
33EIGHT STYLES OF LEARNING, Cont’d TYPECHARACTERISTICSLIKES TOIS GOOD ATLEARNS BEST BYBODILY/KinestheticLearner“The Mover”Eager to solve problems physically. Often doesn’t read directions but just starts on a projectMove aroundTouch and talkUse body languagePhysical activities(Sports/dance/acting)craftsTouchingMovingInteracting with spaceProcessing knowledge through bodily sensationsINTERpersonal“The Socializer”Likes group work and working cooperatively to solve problems. Has an interest in their community.Have lots of friendsTalk to peopleJoin groupsUnderstanding peopleLeading othersOrganizingCommunicatingManipulatingMediating conflictsSharingComparingRelatingCooperatinginterviewingINTRApersonal“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 alonePursue owninterestsUnderstanding selfFocusing inward on feelings/dreamsPursuing interests/goalsBeing originalWorking alongIndividualized projectsSelf-paced instructionHaving own spaceNATURALIST“The Nature Lover”Enjoys relating things to their environment. Have a strong connection to nature.Physically experience natureDo observationsResponds to patterning natureExploring natural phenomenonSeeing connectionsSeeing patternsReflective ThinkingDoing observationsRecording events in NatureWorking in pairsDoing long term projects
34Introduction 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.
35Introduction 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.
36Introduction 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.
37Introduction 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.
38Multiple 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.
39Multiple 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.
40Sternberg’s Three Intelligences CreativeAnalyticalPracticalWe 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.
41Thinking About the Sternberg Intelligences ANALYTICALLinear – Schoolhouse Smart - SequentialShow 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 _________________.PRACTICALStreetsmart – Contextual – Focus on UseDemonstrate 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 - ImproverFind 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 ____________.
42Triarchic 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 ___
43Triarchic 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 ___
44Triarchic 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 Practical1. ___ ___ ___4. ___ ___ ___7. ___ ___ ___10. ___ ___ ___13. ___ ___ ___16. ___ ___ ___19. ___ ___ ___22. ___ ___ ___25. ___ ___ ___28. ___ ___ ___Total Number of True:Analytical ____ Creative _____ Practical _____
45Understanding Order of Operations Make a chart that shows all ways you can think of to use order of operations to equal 18.Analytic TaskA 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 TaskCreative TaskWrite a book of riddles that involve order of operations. Show the solution and pictures on the page that follows each riddle.
46Forms 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.
47Circle 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.
48Circle 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.PracticalStudents find examples of each definition in the room, looking out the window, or thinking about where in the world you would see each term. They can make a mural, picture book, travel brochure, or any other idea to show where in the world these terms can be seen.
49Circle 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.ORRole Audience Format TopicDiameter Radius Twice as niceCircle Tangent poem You touch me!Secant Chord voic I extend you.
50Key Principles of a Differentiated Classroom Assessment and instruction are inseparable.Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD
51Pre-AssessmentWhat the student already knows about what is being plannedWhat standards, objectives, concepts & skills the individual student understandsWhat further instruction and opportunities for mastery are neededWhat requires reteaching or enhancementWhat areas of interests and feelings are in the different areas of the studyHow to set up flexible groups: Whole, individual, partner, or small group
52THINKING ABOUT ON-GOING ASSESSMENT STUDENT DATA SOURCESJournal entryShort answer testOpen response testHome learningNotebookOral responsePortfolio entryExhibitionCulminating productQuestion writingProblem solvingTEACHER DATA MECHANISMSAnecdotal recordsObservation by checklistSkills checklistClass discussionSmall group interactionTeacher – student conferenceAssessment stationsExit cardsProblem posingPerformance tasks and rubrics
53Key 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
54USE 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 presentedHands-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.
57Build – A – SquareBuild-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=3b=6-2/3Nanci Smith
58R A F T The ROLE of writer, speaker, artist, historian, etc. An AUDIENCE of fellow writers,students, citizens, characters, etc.Through a FORMAT that iswritten, spoken, drawn, acted, etc.A TOPIC related to curriculumcontent in greater depth.
59RAFT ACTIVITY ON FRACTIONS RoleAudienceFormatTopicFractionWhole NumberPetitionsTo be considered Part of the FamilyImproper FractionMixed NumbersReconciliation LetterWere More Alike than DifferentA Simplified FractionA Non-Simplified FractionPublic Service AnnouncementA Case for SimplicityGreatest Common FactorCommon FactorNursery RhymeI’m the Greatest!Equivalent FractionsNon EquivalentPersonal AdHow to Find Your Soul MateLeast Common FactorMultiple Sets of NumbersRecipeThe Smaller the BetterLike Denominators in an Additional ProblemUnlike Denominators in an Addition ProblemApplication formTo Become A Like DenominatorA Mixed Number that Needs to be Renamed to Subtract5th Grade Math StudentsRiddleWhat’s My New NameLike Denominators in a Subtraction ProblemUnlike Denominators in a Subtraction ProblemStory BoardHow to Become a Like DenominatorBakerDirectionsTo Double the RecipeEstimated SumFractions/Mixed NumbersAdvice ColumnTo Become Well Rounded
60Angles Relationship RAFT RoleAudienceFormatTopicOne vertical angleOpposite vertical anglePoemIt’s like looking in a mirrorInterior (exterior) angleAlternate interior (exterior) angleInvitation to a family reunionMy separated twinAcute angleMissing angleWanted posterWanted: My complementAn angle less than 180SupplementaryanglePersuasive 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.
61Algebra RAFT Role Audience Format Topic Coefficient Variable Email We belong togetherScale / BalanceStudentsAdvice columnKeep me in mind when solving an equationHumansMonologueAll that I can beAlgebra studentsInstruction manualHow and why to isolate meAlgebraPublicPassionate pleaWhy you really do need me!
62RAFT 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?RoleAudienceFormatTopic
63Ideas for Cubing Arrange ________ into a 3-D collage to show ________ Make a body sculpture to show ________Create a dance to showDo a mime to help us understandPresent 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 MathDescribe how you would solve ______Analyze how this problem helps us use mathematical thinking and problem solvingCompare 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.
64Fraction Think Dots Nanci Smith Describe how you would Explain the differencesolve or roll between adding andthe die to determine your multiplying fractions,own fractions.Compare and contrast Create a word problemthese two problems: that can be solved by+and (Or roll the fraction die todetermine your fractions.)Describe how people use Model the problemfractions every day. ___ + ___ .Roll the fraction die todetermine which fractionsto add.FractionThink DotsNanci Smith
66Fraction Think Dots Nanci Smith Describe how you would Explain why you needsolve or roll a common denominatorthe die to determine your when adding fractions,own fractions. But not when multiplying.Can common denominatorsCompare and contrast ever be used when dividingthese two problems: fractions?Create an interesting and challenging word problemA carpet-layer has 2 yards that can be solved byof carpet. He needs 4 feet ___ + ____ - ____.of carpet. What fraction of Roll the fraction die tohis carpet will he use? How determine your fractions.do you know you are correct?Diagram and explain the solution to ___ + ___ + ___.Roll the fraction die todetermine your fractions.FractionThink DotsNanci Smith
67Algebra ThinkDOTS Level 1: 1. a, b, c and d each represent a different value. If a = 2, find b, c, and d.a + b = ca – c = da + b = 52. 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 by8x – 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?
68Algebra ThinkDOTS Level 2: 1. a, b, c and d each represent a different value. If a = -1, find b, c, and d.a + b = cb + b = dc – a = -a2. 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 by2x + 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?
69Algebra ThinkDOTS Level 3: 1. a, b, c and d each represent a different value. If a = 4, find b, c, and d.a + c = bb - a = ccd = -dd + d = a2. 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.
70Designing a Differentiated Learning Contract A Learning Contract has the following componentsA Skills ComponentFocus is on skills-based tasksAssignments are based on pre-assessment of students’ readinessStudents work at their own level and paceA content componentFocus is on applying, extending, or enriching key content (ideas, understandings)Requires sense making and productionAssignment is based on readiness or interestA Time LineTeacher sets completion date and check-in requirementsStudents select order of work (except for required meetings and homework)4. The AgreementThe teacher agrees to let students have freedom to plan their timeStudents agree to use the time responsiblyGuidelines for working are spelled outConsequences for ineffective use of freedom are delineatedSignatures of the teacher, student and parent (if appropriate) are placed on the agreementDifferentiating Instruction: Facilitator’s Guide, ASCD, 1997
71Personal Agenda Special Instructions Task Montgomery County, MDPersonal Agenda for _______________________________________Starting Date _____________________________________________________Teacher & studentinitials atcompletionSpecial InstructionsTaskRemember to complete your daily planning log; I’ll call on you for conferences & instructions.
72Proportional 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 problemsWrite 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
73Similar 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.
74Similar 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.
75Similar 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.