Differentiating math


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Differentiating math

  1. 1. Differentiating Mathematics at the Middle and High School Levels Raising Student Achievement Conference St. Charles, IL December 4, 2007 "In the end, all learners need your energy, your heart and your mind. They have that in common because they are young humans. How they need you however, differs. Unless we understand and respond to those differences, we fail many learners." * * Tomlinson, C.A. (2001). How to differentiate instruction in mixed ability classrooms (2nd Ed.). Alexandria, VA: ASCD. Nanci Smith Educational Consultant Curriculum and Professional Development Cave Creek, AZ nanci_mathmaster@yahoo.com
  2. 2. Differentiation of Instruction Is a teacher’s response to learner’s needs guided by general principles of differentiationRespectful tasks Flexible grouping Continual assessment Teachers Can Differentiate Through: Content Process Product According to Students’ Readiness Interest Learning Profile
  3. 3. What’s the point of differentiating in these different ways? LearningReadines s Interes t Profile Growth Motivation E fficiency
  4. 4. Key Principles of a Differentiated Classroom• The teacher understands, appreciates, and builds upon student differences. Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD
  5. 5. READINESSWhat does READINESS mean?It is the student’s entry point relative to a particular understanding or skill. C.A.Tomlinson, 1999
  6. 6. A Few Routes to READINESS DIFFERENTIATION Varied texts by reading level Varied supplementary materials Varied scaffolding • reading • writing • research • technology Tiered tasks and procedures Flexible time use Small group instruction Homework options Tiered or scaffolded assemssment Compacting Mentorships Negotiated criteria for quality Varied graphic organizers
  7. 7. Providing support needed for a student to succeed in work slightlyFor example… beyond his/her•Directions that give more structure – or comfort zone. less•Tape recorders to help with reading or writing beyond the student’s grasp•Icons to help interpret print•Reteaching / extending teaching•Modeling•Clear criteria for success•Reading buddies (with appropriate directions)•Double entry journals with appropriate challenge•Teaching through multiple modes•Use of manipulatives when needed•Gearing reading materials to student reading level•Use of study guides•Use of organizers•New American Lecture Tomlinson, 2000
  8. 8. Compacting1. 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.9. Eliminate all drill, practice, review, or preparation for students who have already mastered such things.10. Keep accurate records of students’ compacting activities: document mastery. Strategy: Compacting
  9. 9. Developing a Tiered Activity1 Select the activity organizer 2 •concept Think about your students/use assessments Essential to building •generalization a framework of skills understanding • readiness range reading thinking • interests information • learning profile • talents 3 Create an activity that is • interesting 4 • high level High skill/ • causes students to use Chart the Complexity key skill(s) to understand complexity of a key idea the activity Low skill/ complexity 5 Clone the activity along the ladder as needed to ensure challenge and success for your students, in • materials – basic to advanced 6 • form of expression – from familiar to unfamiliar Match task to student based on • from personal experience to removed from personal experience student profile and task •equalizer requirements
  10. 10. The Equalizer1. Foundational Transformational Information, Ideas, Materials, Applications3. Concrete Abstract Representations, Ideas, Applications, Materials5. Simple Complex Resources, Research, Issues, Problems, Skills, Goals7. Single Facet Multiple Facets Directions, Problems, Application, Solutions, Approaches, Disciplinary Connections9. Small Leap Great Leap Application, Insight, Transfer11. More Structured More Open Solutions, Decisions, Approaches13. Less Independence Greater Independence Planning, Designing, Monitoring15. Slow Pace of Study, Pace of Thought Quick
  11. 11. Adding FractionsGreen Group Use Cuisinaire rods or fraction circles to model simple fraction addition problems. Begin with Blue Group common denominators and work up to denominators with common Manipulatives such as Cuisinaire factors such as 3 and 6. rods and fraction circles will be available as a resource for the group. Students use factor trees Explain the pitfalls and hurrahs of and lists of multiples to find adding fractions by making a common denominators. Using this picture book. approach, pairs and triplets ofRed Group fractions are rewritten using common denominators. End by Use Venn diagrams to model adding several different problems LCMs (least common multiple). of increasing challenge and length. Explain how this process can be used to find common denominators. Use the method on Suzie says that adding fractions is more challenging addition like a game: you just need to know problems. the rules. Write game instructions explaining the rules of adding fractions. Write a manual on how to add fractions. It must include why a common denominator is needed, and at least three ways to find it.
  12. 12. Graphing with a Point and a SlopeAll groups:• Given three equations in slope-intercept form, the students will graph the lines using a T-chart. Then they will answer the following questions:• What is the slope of the line?• Where is slope found in the equation?• Where does the line cross the y-axis?• What is the y-value of the point when x=0? (This is the y-intercept.)• Where is the y-value found in the equation?• Why do you think this form of the equation is called the “slope-intercept?”
  13. 13. Graphing with a Point and a SlopeStruggling Learners: Given the points• (-2,-3), (1,1), and (3,5), the students will plot the points and sketch the line. Then they will answer the following questions:• What is the slope of the line?• Where does the line cross the y-axis?• Write the equation of the line. The students working on this particular task should repeat this process given two or three more points and/or a point and a slope. They will then create an explanation for how to graph a line starting with the equation and without finding any points using a T-chart.
  14. 14. Graphing with a Point and a SlopeGrade-Level Learners: Given an equation of a line in slope-intercept form (or several equations), the students in this group will:• Identify the slope in the equation.• Identify the y-intercept in the equation.• Write the y-intercept in coordinate form (0,y) and plot the point on the y-axis.• use slope to find two additional points that will be on the line.• Sketch the line. When the students have completed the above tasks, they will summarize a way to graph a line from an equation without using a T-chart.
  15. 15. Graphing with a Point and a SlopeAdvanced Learners: Given the slope-intercept form of the equation of a line, y=mx+b, the students will answer the following questions:• The slope of the line is represented by which variable?• The y-intercept is the point where the graph crosses the y-axis. What is the x-coordinate of the y-intercept? Why will this always be true?• The y-coordinate of the y-intercept is represented by which variable in the slope-intercept form?Next, the students in this group will complete the following tasks given equations in slope-intercept form:• Identify the slope and the y-intercept.• Plot the y-intercept.• Use the slope to count rise and run in order to find the second and third points.• Graph the line.
  16. 16. BRAIN RESEARCH SHOWS THAT. . . Eric Jensen, Teaching With the Brain in Mind, 1998 Choices vs. Required content, process, product no student voicegroups, 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 EQUALSIncreased intrinsic Increased MOTIVATION APATHY & RESENTMENT
  17. 17. -CHOICE- The Great Motivator!• Requires children to be aware of their own readiness, interests, and learning profiles.• Students have choices provided by the teacher. (YOU are still in charge of crafting challenging opportunities for all kiddos – NO taking the easy way out!)• Use choice across the curriculum: writing topics, content writing prompts, self-selected reading, contract menus, math problems, spelling words, product and assessment options, seating, group arrangement, ETC . . .• GUARANTEES BUY-IN AND ENTHUSIASM FOR LEARNING!• Research currently suggests that CHOICE should be offered 35% of the time!!
  18. 18. AssessmentsThe assessments used in this learning profile section can be downloaded at: www.e2c2.com/fileupload.aspDownload the file entitled “Profile Assessments for Cards.”
  19. 19. How Do You Like to Learn?1. I study best when it is quiet. Yes No2. I am able to ignore the noise of other 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, no matter what. Yes No8. Sometimes I get frustrated with my work and do not finish it. Yes No9. When my teacher gives an assignment, I like to have exact steps on how to complete it. Yes No10. When my teacher gives an assignment, I like to create 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 on an assignment. Yes No14. I like to have a certain amount of time to work on an assignment. Yes No15. I like to learn by moving and doing. Yes No16. I like to learn while sitting at my desk. Yes No
  20. 20. 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 havelearned to an audience. This survey will help determine the kinds of productsYOU are interested in creating.My Name is: ____________________________________________________Instructions:Read each statement and circle the number that shows to what extent YOU areinterested in creating that type of product. (Do not worry if you are unsure of howto make the product). Not At All Interested Of Little Interest Moderately Interested Interested Very Interested 1. Writing Stories 1 2 3 4 5 2. Discussing what I 1 2 3 4 5 have learned 3. Painting a picture 1 2 3 4 5 4. Designing a 1 2 3 4 5 computer software project 5. Filming & editing a 1 2 3 4 5 video 6. Creating a company 1 2 3 4 5 7. Helping in the 1 2 3 4 5 community 8. Acting in a play 1 2 3 4 5
  21. 21. Not At All Interested Of Little Interest Moderately Interested Interested Very Interested9. Building an 1 2 3 4 5 invention10. Playing musical 1 2 3 4 5instrument11. Writing for a 1 2 3 4 5newspaper12. Discussing ideas 1 2 3 4 513. Drawing pictures 1 2 3 4 5for a book14. Designing an 1 2 3 4 5interactive computerproject15. Filming & editing 1 2 3 4 5a television show16. Operating a 1 2 3 4 5business17. Working to help 1 2 3 4 5others18. Acting out an 1 2 3 4 5event19. Building a project 1 2 3 4 520. Playing in a band 1 2 3 4 521. Writing for a 1 2 3 4 5magazine22. Talking about my 1 2 3 4 5project23. Making a clay 1 2 3 4 5sculpture of acharacter
  22. 22. Not At All Interested Of Little Interest Moderately Interested Interested Very Interested24. Designing 1 2 3 4 5information for thecomputer internet25. Filming & editing 1 2 3 4 5a movie26. Marketing a 1 2 3 4 5product27. Helping others by 1 2 3 4 5supporting a socialcause28. Acting out a story 1 2 3 4 529. Repairing a 1 2 3 4 5machine30. Composing music 1 2 3 4 531. Writing an essay 1 2 3 4 532. Discussing my 1 2 3 4 5research33. Painting a mural 1 2 3 4 534. Designing a 1 2 3 4 5computer35. Recording & 1 2 3 4 5editing a radio show36. Marketing an idea 1 2 3 4 537. Helping others by 1 2 3 4 5fundraising38. Performing a skit 1 2 3 4 5
  23. 23. Not At All Interested Of Little Interest Moderately Interested Interested Very Interested 39. Constructing a 1 2 3 4 5 working model. 40. Performing music 1 2 3 4 5 41. Writing a report 1 2 3 4 5 42. Talking about my 1 2 3 4 5 experiences 43. Making a clay 1 2 3 4 5 sculpture of a scene 44. Designing a multi- 1 2 3 4 5 media computer show 45. Selecting slides 1 2 3 4 5 and music for a slide show 46. Managing 1 2 3 4 5 investments 47. Collecting 1 2 3 4 5 clothing or food to help others 48. Role-playing a 1 2 3 4 5 character 49. Assembling a kit 1 2 3 4 5 50. Playing in an 1 2 3 4 5 orchestra Products TotalInstructions: My Written 1. ___ 11. ___ 21. ___ 31. ___ 41. ___ _____Way …A Profile Oral 2. ___ 12. ___ 22. ___ 32. ___ 42. ___ _____ Artistic 3. ___ 13. ___ 23. ___ 33. ___ 43. ___ _____Write your score Computer 4. ___ 14. ___ 24. ___ 34. ___ 44. ___ _____beside each Audio/Visual 5. ___ 15. ___ 25. ___ 35. ___ 45. ___ _____number. Add each Commercial 6. ___ 16. ___ 26. ___ 36. ___ 46. ___ _____Row to determine Service 7. ___ 77. ___ 27. ___ 37. ___ 47. ___ _____your expression Dramatization 8. ___ 18. ___ 28. ___ 38. ___ 48. ___ _____style profile. Manipulative 9. ___ 19. ___ 29. ___ 39. ___ 49. ___ _____ Musical 10.___ 20. ___ 30 . ___ 40. ___ 50. ___ _____
  24. 24. Learner Profile Card Gender StripeAuditory, Visual, Kinesthetic Analytical, Creative, PracticalModality Sternberg Student’s InterestsMultiple Intelligence Preference ArrayGardner Inventory Nanci Smith,Scottsdale,AZ
  25. 25. Differentiation Using LEARNING PROFILE• Learning profile refers to how an individual learns best - most efficiently and effectively.• Teachers and their students may differ in learning profile preferences.
  26. 26. Learning Profile Factors Group Orientation Learning Environment Genderindependent/self orientation quiet/noise & warm/cool group/peer orientation adult orientation Culture still/mobile combination flexible/fixed “busy”/”spare” Cognitive Style Intelligence Preference analytic Creative/conforming practical Essence/facts creative Expressive/controlled verbal/linguistic Nonlinear/linear logical/mathematical Inductive/deductive spatial/visual People-oriented/task or Object oriented bodily/kinesthetic Concrete/abstract musical/rhythmic Collaboration/competition interpersonal Interpersonal/introspective intrapersonal Easily distracted/long Attention span naturalistGroup achievement/personal achievement existential Oral/visual/kinesthetic Reflective/action-oriented
  27. 27. 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 D2. I would rather watch television than listen to the radio. A D3. I remember faces better than names. A D4. I like classrooms with lots of posters and pictures around the room. A D5. The appearance of my handwriting is important to me. A D6. I think more often in pictures. A D7. I am distracted by visual disorder or movement. A D8. I have difficulty remembering directions that were told to me. A D9. I would rather watch athletic events than participate in them. A D10. I tend to organize my thoughts by writing them down. A D11. My facial expression is a good indicator of my emotions. A D12. I tend to remember names better than faces. A D13. I would enjoy taking part in dramatic events like plays. A D14. I tend to sub vocalize and think in sounds. A D15. I am easily distracted by sounds. A D16. I easily forget what I read unless I talk about it. A D17. I would rather listen to the radio than watch TV A D18. My handwriting is not very good. A D19. When faced with a problem , I tend to talk it through. A D20. I express my emotions verbally. A D21. I would rather be in a group discussion than read about a topic. A D
  28. 28. 1. I prefer talking on the phone rather than writing a letter to someone. A D2. I would rather participate in athletic events than watch them. A D3. I prefer going to museums where I can touch the exhibits. A D4. My handwriting deteriorates when the space becomes smaller. A D5. My mental pictures are usually accompanied by movement. A D6. I like being outdoors and doing things like biking, camping, swimming, hiking etc. A D7. I remember best what was done rather then what was seen or talked about. A D8. When faced with a problem, I often select the solution involving the greatest activity. A D9. I like to make models or other hand crafted items. A D10. I would rather do experiments rather then read about them. A D11. My body language is a good indicator of my emotions. A D Interpreting the Instrument’s Score12. Total the number of “A” responses in items 1-11 I have difficulty remembering verbal directions if I have not _____ the activity before. done A D 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.
  29. 29. Parallel Lines Cut by a Transversal• Visual: Make posters showing all the angle relations formed by a pair of parallel lines cut by a transversal. Be sure to color code definitions and angles, and state the relationships between all possible angles. 1 2 3 4 5 6 8 7 Smith & Smarr, 2005
  30. 30. Parallel Lines Cut by a Transversal• Auditory: Play “Shout Out!!” Given the diagram below and commands on strips of paper (with correct answers provided), players take turns being the leader to read a command. The first player to shout out a correct answer to the command, receives a point. The next player becomes the next leader. Possible commands: – Name an angle supplementary 1 2 3 supplementary to angle 1. 5 8 4 6 – Name an angle congruent 7 to angle 2. Smith & Smarr, 2005
  31. 31. Parallel Lines Cut by a Transversal• Kinesthetic: Walk It Tape the diagram below on the floor with masking tape. Two players stand in 2 1 3 assigned angles. As a 5 4 team, they have to tell 6 8 what they are called (ie: 7 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
  32. 32. EIGHT STYLES OF LEARNINGTYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BYLINGUISTIC Learns through the Read Memorizing Saying, hearing and manipulation of words. Loves names, places, seeing wordsLEARNER to read and write in order to Write“The Word Tell stories dates and trivia explain themselves. They alsoPlayer” tend to enjoy talkingLOGICAL/ Looks for patterns when Do experiments Math Categorizing solving problems. Creates a set Figure things outMathematical of standards and follows them Reasoning ClassifyingLearner when researching in a Work with numbers Logic Working with abstract“The Questioner” sequential manner. Ask questions Problem solving patterns/relationships Explore patterns and relationshipsSPATIAL Learns through pictures, charts, Draw, build, design Imagining things VisualizingLEARNER graphs, diagrams, and art. and create things Sensing changes Dreaming“The Visualizer” Daydream Mazes/puzzles Using the mind’s eye Look at pictures/slides Reading maps, Working with Watch movies charts colors/pictures Play with machinesMUSICAL Learning is often easier for Sing, hum tunes Picking up sounds RhythmLEARNER these students when set to Remembering music or rhythm Listen to music Melody“The Music melodies Play an instrument MusicLover” Noticing pitches/ Respond to music rhythms Keeping time
  33. 33. EIGHT STYLES OF LEARNING, Cont’dTYPE CHARACTERISTICS LIKES TO IS GOOD AT LEARNS BEST BYBODILY/ Eager to solve problems Move around Physical activities Touching physically. Often doesn’t read (Sports/dance/ MovingKinesthetic directions but just starts on a Touch and talkLearner project Use body acting) Interacting with space“The Mover” language crafts Processing knowledge through bodily sensationsINTERpersonal Likes group work and Have lots of Understanding people Sharing working cooperatively to friends Leading others ComparingLearner solve problems. Has an“The Socializer” interest in their community. Talk to people Organizing Relating Join groups Communicating Cooperating Manipulating interviewing Mediating conflictsINTRApersonal Enjoys the opportunity to Work alone Understanding self Working along reflect and work Focusing inward on Individualized projectsLearner independently. Often quiet Pursue own“The Individual” feelings/dreams Self-paced instruction and would rather work on his/ interests her own than in a group. Pursuing interests/ Having own space goals Being originalNATURALIST Enjoys relating things to their Physically Exploring natural Doing observations“The Nature environment. Have a strong experience nature phenomenon Recording events in NatureLover” connection to nature. Do observations Seeing connections Working in pairs Responds to Seeing patterns Doing long term projects patterning nature Reflective Thinking
  34. 34. Introduction to Change (MI)• Logical/Mathematical Learners: Given a set of data that changes, such as population for your city or town over time, decide on several ways to present the information. Make a chart that shows the various ways you can present the information to the class. Discuss as a group which representation you think is most effective. Why is it most effective? Is the change you are representing constant or variable? Which representation best shows this? Be ready to share your ideas with the class.
  35. 35. Introduction to Change (MI)• Interpersonal Learners: Brainstorm things that change constantly. Generate a list. Discuss which of the things change quickly and which of them change slowly. What would graphs of your ideas look like? Be ready to share your ideas with the class.
  36. 36. Introduction to Change (MI)• Visual/Spatial Learners: Given a variety of graphs, discuss what changes each one is representing. Are the changes constant or variable? How can you tell? Hypothesize how graphs showing constant and variable changes differ from one another. Be ready to share your ideas with the class.
  37. 37. Introduction to Change (MI)• Verbal/Linguistic Learners: Examine articles from newspapers or magazines about a situation that involves change and discuss what is changing. What is this change occurring in relation to? For example, is this change related to time, money, etc.? What kind of change is it: constant or variable? Write a summary paragraph that discusses the change and share it with the class.
  38. 38. Multiple Intelligence Ideas for Proofs!• Logical Mathematical: Generate proofs for given theorems. Be ready to explain!• Verbal Linguistic: Write in paragraph form why the theorems are true. Explain what we need to think about before using the theorem.• Visual Spatial: Use pictures to explain the theorem.
  39. 39. Multiple Intelligence Ideas for Proofs!• Musical: Create a jingle or rap to sing the theorems!• Kinesthetic: Use Geometer Sketchpad or other computer software to discover the theorems.• Intrapersonal: Write a journal entry for yourself explaining why the theorem is true, how they make sense, and a tip for remembering them.
  40. 40. Sternberg’s Three Intelligences Creative Analytical Practical•We all have some of each of these intelligences, but are usuallystronger in one or two areas than in others.•We should strive to develop as fully each of these intelligencesin students…• …but also recognize where students’ strengths lie and teachthrough those intelligences as often as possible, particularlywhen introducing new ideas.
  41. 41. Thinking About the Sternberg IntelligencesANALYTICAL Linear – Schoolhouse Smart - Sequential Show the parts of _________ and how they work. Explain why _______ works the way it does. Diagram how __________ affects __________________. Identify the key parts of _____________________. Present a step-by-step approach to _________________.PRACTICAL Streetsmart – Contextual – Focus on Use Demonstrate how someone uses ________ in their life or work. Show how we could apply _____ to solve this real life problem ____. Based on your own experience, explain how _____ can be used. Here’s a problem at school, ________. Using your knowledge of ______________, develop a plan to address the problem. CREATIVE Innovator – Outside the Box – What If - Improver Find a new way to show _____________. Use unusual materials to explain ________________. Use humor to show ____________________. Explain (show) a new and better way to ____________. Make connections between _____ and _____ to help us understand ____________. Become a ____ and use your “new” perspectives to help us think about ____________.
  42. 42. Triarchic Theory of Intelligences Robert SternbergMark each sentence T if you like to do the activity and F if you do not like to do the activity.3. Analyzing characters when I’m reading or listening to a story ___4. Designing new things ___5. Taking things apart and fixing them ___6. Comparing and contrasting points of view ___7. Coming up with ideas ___8. Learning through hands-on activities ___9. Criticizing my own and other kids’ work ___10. Using my imagination ___11. Putting into practice things I learned ___12. Thinking clearly and analytically ___13. Thinking of alternative solutions ___14. Working with people in teams or groups ___15. Solving logical problems ___16. Noticing things others often ignore ___17. Resolving conflicts ___
  43. 43. Triarchic Theory of Intelligences Robert SternbergMark each sentence T if you like to do the activity and F if you do not like to do the activity.3. Evaluating my own and other’s points of view ___4. Thinking in pictures and images ___5. Advising friends on their problems ___6. Explaining difficult ideas or problems to others ___7. Supposing things were different ___8. Convincing someone to do something ___9. Making inferences and deriving conclusions ___10. Drawing ___11. Learning by interacting with others ___12. Sorting and classifying ___13. Inventing new words, games, approaches ___14. Applying my knowledge ___15. Using graphic organizers or images to organize your thoughts ___16. Composing ___30. Adapting to new situations ___
  44. 44. Triarchic Theory of Intelligences – Key Robert SternbergTransfer your answers from the survey to the key. The column with the most True responses is your dominant intelligence.Analytical Creative Practical1. ___ 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 _____
  45. 45. Understanding Order of OperationsAnalytic Task Make a chart that shows all ways you can think of to use order of operations to equal 18.Practical 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.Creative Task Write a book of riddles that involve order of operations. Show the solution and pictures on the page that follows each riddle.
  46. 46. Forms of Equations of Lines• Analytical Intelligence: Compare and contrast the various forms of equations of lines. Create a flow chart, a table, or any other product to present your ideas to the class. Be sure to consider the advantages and disadvantages of each form.• Practical Intelligence: Decide how and when each form of the equation of a line should be used. When is it best to use which? What are the strengths and weaknesses of each form? Find a way to present your conclusions to the class.• Creative Intelligence: Put each form of the equation of a line on trial. Prosecutors should try to convince the jury that a form is not needed, while the defense should defend its usefulness. Enact your trial with group members playing the various forms of the equations, the prosecuting attorneys, and the defense attorneys. The rest of the class will be the jury, and the teacher will be the judge.
  47. 47. Circle VocabularyAll Students: Students find definitions for a list of vocabulary (center, radius, chord, secant, diameter, tangent point of tangency, congruent circles, concentric circles, inscribed and circumscribed circles). They can use textbooks, internet, dictionaries or any other source to find their definitions.
  48. 48. Circle VocabularyAnalytical Students make a poster to explain the definitions in their own words. Posters should include diagrams, and be easily understood by a student in the fifth grade.Practical Students find examples of each definition in the room, looking out the window, or thinking about where in the world you would see each term. They can make a mural, picture book, travel brochure, or any other idea to show where in the world these terms can be seen.
  49. 49. Circle VocabularyCreative 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 email Twice as niceCircle Tangent poem You touch me!Secant Chord voicemail I extend you.
  50. 50. Key Principles of a Differentiated Classroom• Assessment and instruction are inseparable. Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD
  51. 51. Pre-Assessment• What the student already knows about what is being planned• What standards, objectives, concepts & skills the individual student understands• What further instruction and opportunities for mastery are needed• What requires reteaching or enhancement• What areas of interests and feelings are in the different areas of the study• How to set up flexible groups: Whole, individual, partner, or small group
  52. 52. THINKING ABOUT ON-GOING ASSESSMENTSTUDENT DATA SOURCES TEACHER DATA• Journal entry MECHANISMS• Short answer test 2. Anecdotal records• Open response test 3. Observation by checklist• Home learning 4. Skills checklist• Notebook 5. Class discussion• Oral response 6. Small group interaction• Portfolio entry 7. Teacher – student• Exhibition conference• Culminating product 8. Assessment stations• Question writing 9. Exit cards• Problem solving 10. Problem posing 11. Performance tasks and rubrics
  53. 53. Key Principles of a Differentiated Classroom• The teacher adjusts content, process, and product in response to student readiness, interests, and learning profile. Source: Tomlinson, C. (2000). Differentiating Instruction for Academic Diversity. San Antonio, TX: ASCD
  54. 54. USE OF INSTRUCTIONAL STRATEGIES. The following findings related to instructional strategies are supported by the existing research:• Techniques and instructional strategies have nearly as much influence on studentlearning 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 likelythat they will confuse the facts presented• Hands-on learning, especially in science, has a positive effect on studentachievement.• Teachers who use hands-on learning strategies have students who out-performtheir peers on the National Assessment of Educational progress (NAEP) in theareas of science and mathematics.• Despite the research supporting hands-on activity, it is a fairly uncommoninstructional approach.• Students have higher achievement rates when the focus of instruction is onmeaningful conceptualization, especially when it emphasizes their own knowledgeof the world.
  55. 55. Make Card Games!
  56. 56. Make Card Games!
  57. 57. Build – A – Square• Build-a-square is based on the “Crazy” puzzles where 9 tiles are placed in a 3X3 square arrangement with all edges matching.• Create 9 tiles with math problems and answers along the edges.• The puzzle is designed so that the correct formation has all questions and answers matched on the edges.• Tips: Design the answers for the edges first, then write the specific problems.• Use more or less squares to tier. m=3• Add distractors to outside edges and b=6 -2/3 “letter” pieces at the end. Nanci Smith
  58. 58. The ROLE of writer, speaker, R A F Tartist, historian, etc. An AUDIENCE of fellow writers, students, citizens, characters, etc. Through a FORMAT that is written, spoken, drawn, acted, etc. e e ron l ct ne r ut on p on rot A TOPIC related to curriculum content in greater depth.
  59. 59. RAFT ACTIVITY ON FRACTIONS Role Audience Format TopicFraction Whole Number Petitions To be considered Part of the FamilyImproper Fraction Mixed Numbers Reconciliation Letter Were More Alike than DifferentA Simplified Fraction A Non-Simplified Fraction Public Service A Case for Simplicity AnnouncementGreatest Common Factor Common Factor Nursery Rhyme I’m the Greatest!Equivalent Fractions Non Equivalent Personal Ad How to Find Your Soul MateLeast Common Factor Multiple Sets of Numbers Recipe The Smaller the BetterLike Denominators in an Unlike Denominators in an Application form To Become A LikeAdditional Problem Addition Problem DenominatorA Mixed Number that 5th Grade Math Students Riddle What’s My New NameNeeds to be Renamed toSubtractLike Denominators in a Unlike Denominators in a Story Board How to Become a LikeSubtraction Problem Subtraction Problem DenominatorFraction Baker Directions To Double the RecipeEstimated Sum Fractions/Mixed Numbers Advice Column To Become Well Rounded
  60. 60. Angles Relationship RAFT Role Audience Format Topic One vertical angle Opposite vertical angle Poem It’s like looking in a mirror Interior (exterior) angle Alternate interior (exterior) Invitation to a family My separated twin angle reunion Acute angle Missing angle Wanted poster Wanted: My complement An angle less than 180 Supplementary Persuasive speech Together, we’re a straight angle angle **Angles Humans Video See, we’re everywhere! ** This last entry would take more time than the previous 4 lines, and assesses a little differently. You could offer it asan option with a later due date, but you would need to specify that they need to explain what the angles are, and anything specific that you want to know such as what is the angle’s complement or is there a vertical angle that corresponds, etc.
  61. 61. Algebra RAFT Role Audience Format Topic Coefficient Variable Email We belong togetherScale / Balance Students Advice column Keep me in mind when solving an equation Variable Humans Monologue All that I can be Variable Algebra students Instruction manual How and why to isolate me Algebra Public Passionate plea Why you really do need me!
  62. 62. RAFT Planning Sheet Know Understand DoHow to Differentiate:• Tiered? (See Equalizer)• Profile? (Differentiate Format)• Interest? (Keep options equivalent in learning)• Other? Role Audience Format Topic
  63. 63. Cubing Cubing Ideas for Cubing Cubing• Arrange ________ into a 3-D collage to show ________ Ideas for Cubing in Math• Make a body sculpture to show • Describe how you would solve ______ ________ • Analyze how this problem helps us use mathematical thinking and problem solving• Create a dance to show • Compare and contrast this problem to one• Do a mime to help us understand on page _____.• Present an interior monologue with • Demonstrate how a professional (or just a dramatic movement that ________ regular person) could apply this kink or problem to their work or life.• Build/construct a representation of • Change one or more numbers, elements, or ________ signs in the problem. Give a rule for what• Make a living mobile that shows and that change does. balances the elements of ________ • Create an interesting and challenging word problem from the number problem. (Show us• Create authentic sound effects to how to solve it too.) accompany a reading of _______ • Diagram or illustrate the solutionj to the• Show the principle of ________ with a problem. Interpret the visual so we understand it. rhythm pattern you create. Explain to us how that works.
  64. 64. Describe how you would Explain the difference 1 3 solve + or roll between adding and 5 5 the die to determine your multiplying fractions, own fractions. Compare and contrast Create a word problem these two problems: that can be solved by 1 2 11 + = + 3 5 15 and (Or roll the fraction die to 1 1 + determine your fractions.) 3 2 Describe how people use Model the problem fractions every day. ___ + ___ .Nanci Smith Roll the fraction die to determine which fractions to add.
  65. 65. Nanci Smith
  66. 66. Describe how you would Explain why you need 2 3 1 solve + + or roll a common denominator 13 7 91 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? 1 1 3 1 + and + 3 2 7 7 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.Nanci Smith do you know you are correct? Diagram and explain the solution to ___ + ___ + ___. Roll the fraction die to determine your fractions.
  67. 67. Level 1: 1. a, b, c and d each represent a different value. If a = 2, find b, c, and d. a+b=c a–c=d a+b=5 2. Explain the mathematical reasoning involved in solving card 1. 3. Explain in words what the equation 2x + 4 = 10 means. Solve the problem. 4. Create an interesting word problem that is modeled by 8x – 2 = 7x. 5. Diagram how to solve 2x = 8. 6. Explain what changing the “3” in 3x = 9 to a “2” does to the value of x. Why is this true?
  68. 68. Level 2: 1. a, b, c and d each represent a different value. If a = -1, find b, c, and d. a+b=c b+b=d c – a = -a 2. Explain the mathematical reasoning involved in solving card 1. 3. Explain how a variable is used to solve word problems. 4. Create an interesting word problem that is modeled by 2x + 4 = 4x – 10. Solve the problem. 5. Diagram how to solve 3x + 1 = 10. 6. Explain why x = 4 in 2x = 8, but x = 16 in ½ x = 8. Why does this make sense?
  69. 69. 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 3x − 1 ≤ 5 x + 7. Solve the problem. 5. Diagram how to solve 3x + 4 = x + 12. 6. Given ax = 15, explain how x is changed if a is large or a is small in value.
  70. 70. Designing a Differentiated Learning Contracthas the following componentsA Learning Contract2. 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 pace3. 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 interest4. A Time Line Teacher sets completion date and check-in requirements Students select order of work (except for required meetings and homework)4. The Agreement The teacher agrees to let students have freedom to plan their time Students agree to use the time responsibly Guidelines for working are spelled out Consequences for ineffective use of freedom are delineated Signatures of the teacher, student and parent (if appropriate) are placed on the agreement Differentiating Instruction: Facilitator’s Guide, ASCD, 1997
  71. 71. Montgomery Personal Agenda County, MD Personal Agenda for _______________________________________Starting Date _____________________________________________________ Teacher & student initials at Special Task completion InstructionsRemember to complete your daily planning log; I’ll call on you forconferences & instructions.
  72. 72. Proportional Reasoning Think-Tac-Toe□ Create a word problem that □ Find a word problem from □ Think of a way that you use requires proportional the text that requires proportional reasoning in your reasoning. Solve the proportional reasoning. life. Describe the situation, problem and explain why it Solve the problem and explain why it is proportional requires proportional explain why it was and how you use it. reasoning. proportional.□ Create a story about a □ How do you recognize a □ Make a list of all the proportion in the world. proportional situation? proportional situations in the You can write it, act it, Find a way to think about world today. video tape it, or another and explain proportionality. story form.□ Create a pict-o-gram, poem □ Write a list of steps for □ Write a list of questions to ask or anagram of how to solve solving any proportional yourself, from encountering a proportional problems problem. problem that may be proportional through solving it.Directions: Choose one option in each row to complete. Check the box of the choice you make, and turn this page in with your finished selections. Nanci Smith, 2004
  73. 73. Similar Figures MenuImperatives (Do all 3):2. 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.3. Generate a list of applications for similar figures, and similarity in general. Be sure to think beyond “find a missing side…”4. Develop a lesson to teach third grade students who are just beginning to think about similarity.
  74. 74. Similar Figures MenuNegotiables (Choose 1):2. 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.3. Show at least 5 different application of similar figures in the real world, and make them into math problems. Solve each of the problems and explain the role of similarity. Justify why the solutions are correct.
  75. 75. Similar Figures MenuOptionals:2. 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.3. Make a photo album showing the use of similar figures in the world around us. Use captions to explain the similarity in each picture.4. Write a story about similar figures in a world without similarity.5. Write a song about the beauty and mathematics of similar figures.6. Create a “how-to” or book about finding and creating similar figures.
  76. 76. Whatever it Takes!