Differentiating Mathematics at the Middle and High School Levels Raising Student Achievement Conference St. Charles, IL

1. Differentiating Mathematics at the Middle and High School LevelsRaising Student Achievement ConferenceSt. Charles, ILDecember 4, 2007 "In the end, all learners need your energy, your heart and your mind. They have that in common because they are young humans. How they need you however, differs. Unless we understand and respond to those differences, we fail many learners." * * Tomlinson, C.A. (2001). How to differentiate instruction in mixed ability classrooms (2nd Ed.). Alexandria, VA: ASCD. Nanci Smith Educational Consultant Curriculum and Professional Development Cave Creek, AZ nanci_mathmaster@yahoo.com

2. Differentiation of Instruction Is a teacher’s response to learner’s needsguided by general principles of differentiation Respectful tasks Flexible grouping Continual assessment Teachers Can Differentiate Through: Process Product Content According to Students’ Readiness Interest Learning Profile

3. Create an activity that is • interesting • high level • causes students to use • key skill(s) to understand • a key idea High skill/ Complexity Low skill/ complexity Chart the complexity of the activity • Clone the activity along the ladder as needed to ensure challenge and success for your students, in • materials – basic to advanced • form of expression – from familiar to unfamiliar • from personal experience to removed from personal experience • equalizer Match task to student based on student profile and task requirements Developing a Tiered Activity 1 2 • Select the activity organizer • concept • generalization • Think about your students/use assessments • readiness range • interests • learning profile • talents Essential to building a framework of understanding skills reading thinking information 3 4 5 6

4. Green Group Use Cuisinaire rods or fraction circles to model simple fraction addition problems. Begin with common denominators and work up to denominators with common factors such as 3 and 6. Explain the pitfalls and hurrahs of adding fractions by making a picture book. Blue Group Manipulatives such as Cuisinaire rods and fraction circles will be available as a resource for the group. Students use factor trees and lists of multiples to find common denominators. Using this approach, pairs and triplets of fractions are rewritten using common denominators. End by adding several different problems of increasing challenge and length. Suzie says that adding fractions is like a game: you just need to know the rules. Write game instructions explaining the rules of adding fractions. Adding Fractions Red Group Use Venn diagrams to model LCMs (least common multiple). Explain how this process can be used to find common denominators. Use the method on more challenging addition problems. Write a manual on how to add fractions. It must include why a common denominator is needed, and at least three ways to find it.

5. 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?”

6. 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.

7. 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.

8. 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.

9. 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

10. 1 3 2 4 5 8 6 7 Parallel Lines Cut by a Transversal • Auditory: Play “Shout Out!!” Given the diagram below and commands on strips of paper (with correct answers provided), players take turns being the leader to read a command. The first player to shout out a correct answer to the command, receives a point. The next player becomes the next leader. Possible commands: • Name an angle supplementary supplementary to angle 1. • Name an angle congruent to angle 2. Smith & Smarr, 2005

11. 1 3 2 4 5 8 6 7 Parallel Lines Cut by a Transversal • Kinesthetic: Walk It Tape the diagram below on the floor with masking tape. Two players stand in assigned angles. As a team, they have to tell what they are called (ie: vertical angles) and their relationships (ie: congruent). Use all angle combinations, even if there is not a name or relationship. (ie: 2 and 7) Smith & Smarr, 2005

12. 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.

13. 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.

14. 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.

15. 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.

16. 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.

17. 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.

18. Thinking About the Sternberg Intelligences ANALYTICAL Linear – Schoolhouse Smart - Sequential Show the parts of _________ and how they work. Explain why _______ works the way it does. Diagram how __________ affects __________________. Identify the key parts of _____________________. Present a step-by-step approach to _________________. 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. 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 ____________.

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

20. 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.

21. 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.

22. 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.

23. Circle Vocabulary Creative Find a way to help us remember all this vocabulary! You can create a skit by becoming each term, and talking about who you are and how you relate to each other, draw pictures, make a collage, or any other way of which you can think. OR Role Audience Format Topic Diameter Radius email Twice as nice Circle Tangent poem You touch me! Secant Chord voicemail I extend you.

24. Make Card Games!

25. Make Card Games!

27. Angles Relationship RAFT ** This last entry would take more time than the previous 4 lines, and assesses a little differently. You could offer it as an option with a later due date, but you would need to specify that they need to explain what the angles are, and anything specific that you want to know such as what is the angle’s complement or is there a vertical angle that corresponds, etc.

28. Algebra RAFT

29. 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 Cubing Ideas for Cubing Cubing

30. Describe how you would Explain the difference • solve or roll between adding and • the die to determine your multiplying fractions, • own fractions. • Compare and contrast Create a word problem • these two problems: that can be solved by • + • and (Or roll the fraction die to • determine your fractions.) • Describe how people use Model the problem • fractions every day. ___ + ___ . • Roll the fraction die to • determine which fractions • to add. Fraction Think Dots Nanci Smith

31. Fraction Think Dots Nanci Smith

32. Describe how you would Explain why you need solve or roll a common denominator the die to determine your when adding fractions, own fractions. But not when multiplying. Can common denominators Compare and contrast ever be used when dividing these two problems: fractions? Create an interesting and challenging word problem A carpet-layer has 2 yards that can be solved by of carpet. He needs 4 feet ___ + ____ - ____. of carpet. What fraction of Roll the fraction die to his carpet will he use? How determine your fractions. do you know you are correct? Diagram and explain the solution to ___ + ___ + ___. Roll the fraction die to determine your fractions. Fraction Think Dots Nanci Smith

33. Algebra ThinkDOTS Level 1: 1. a, b, c and d each represent a different value. If a = 2, find b, c, and d. a + b = c a – c = d a + b = 5 2. Explain the mathematical reasoning involved in solving card 1. 3. Explain in words what the equation 2x + 4 = 10 means. Solve the problem. 4. Create an interesting word problem that is modeled by 8x – 2 = 7x. 5. Diagram how to solve 2x = 8. 6. Explain what changing the “3” in 3x = 9 to a “2” does to the value of x. Why is this true?

34. Algebra ThinkDOTS Level 2: 1. a, b, c and d each represent a different value. If a = -1, find b, c, and d. a + b = c b + b = d c – a = -a 2. Explain the mathematical reasoning involved in solving card 1. 3. Explain how a variable is used to solve word problems. 4. Create an interesting word problem that is modeled by 2x + 4 = 4x – 10. Solve the problem. 5. Diagram how to solve 3x + 1 = 10. 6. Explain why x = 4 in 2x = 8, but x = 16 in ½ x = 8. Why does this make sense?

35. Algebra ThinkDOTS Level 3: 1. a, b, c and d each represent a different value. If a = 4, find b, c, and d. a + c = b b - a = c cd = -d d + d = a 2. Explain the mathematical reasoning involved in solving card 1. 3. Explain the role of a variable in mathematics. Give examples. 4. Create an interesting word problem that is modeled by . Solve the problem. 5. Diagram how to solve 3x + 4 = x + 12. 6. Given ax = 15, explain how x is changed if a is large or a is small in value.

36. Designing a Differentiated Learning Contract • A Learning Contract has the following components • A Skills Component • Focus is on skills-based tasks • Assignments are based on pre-assessment of students’ readiness • Students work at their own level and pace • A content component • Focus is on applying, extending, or enriching key content (ideas, understandings) • Requires sense making and production • Assignment is based on readiness or interest • A Time Line • Teacher sets completion date and check-in requirements • Students select order of work (except for required meetings and homework) • 4. The Agreement • The teacher agrees to let students have freedom to plan their time • Students agree to use the time responsibly • Guidelines for working are spelled out • Consequences for ineffective use of freedom are delineated • Signatures of the teacher, student and parent (if appropriate) are placed on the agreement Differentiating Instruction: Facilitator’s Guide, ASCD, 1997

37. Proportional Reasoning Think-Tac-Toe Directions: Choose one option in each row to complete. Check the box of the choice you make, and turn this page in with your finished selections. Nanci Smith, 2004

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

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

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