1. Marian Small Huntsville Math Camp August 2008

2. STARTING OUT SESSION 1 M Small2

3. Goals for Session 1 Recognize your own starting point Consider what differentiating instruction means Learn about some generic strategies Think about how students differ mathematically M Small3

4. Four Corners The best way to differentiate instruction is to: Corner 1: teach to the group, but differentiate consolidation Corner 2: teach different things to different groups Corner 3: provide individual learning packages as much as possible Corner 4: personalize both instruction and assessment M Small4

5. Reflect Have you changed your mind about the best strategies? What new ideas have you heard that you had not thought of before? M Small5

6. Visualization Activity Visualize 4 very different students you will think about as you consider how you will differentiate instruction. Draw and briefly describe these students. You will return to this drawing throughout the week. Clip art licensed from the Clip Art Gallery on DiscoverySchool.com M Small6

7. Current Knowledge Why people care about DI The big guru- Tomlinson (content, process, product) Different sorts of DI Accepted principles: -Focus on key concepts -Choice -Prior assessment M Small7

8. Strategies for differentiating consolidation Menus Tiered lessons (based on any of complexity, resources, product, process, outcome) Tic tac toe (think tac toe) Cubing RAFT Stations M Small8

9. Sample Menu Main Dish: Use transformations to sketch each of these graphs: h(x) = 2(x- 4) 2, g(x) = -0.5(x + 2) 2,…. Side Dishes (choose 2) - Create three quadratic functions that pass through (1,4). Describe two ways to transform each so that they pass through (2,7). - Create a flow chart to guide someone through graphing f(x) = a(x –h) 2 + k…. M Small9

10. Sample Menu Desserts (optional) - Create a pattern of parabolas using a graphing calculator. Write the associated equations and tell what makes it a pattern. -Tell how the graph of f(x) = 3(x +2) 2 would look different without the rules for order of operations…. M Small10

11. Sample Tiers/Lesson on Slope Calculate slopes given simple information about a line (e.g. two points) Create lines with given slopes to fit given conditions (e.g. parallel to … and going through (…)) Describe or develop several real-life problems that require knowledge of slope and apply what you have learned to solve those real-life problems M Small11

12. Sample tic tac toe Complete question # …. on page …. in your text. Choose the pro or con side and make your argument: The best way to add mixed numbers is to make them into equivalent improper fractions. Think of a situation where you would add fractions in your everyday life. Make up a jingle that would help someone remember the steps for subtracting mixed numbers. Someone asks you why you have to get a common denominator when you add and subtract fractions but not when you multiply. What would you say? Create a subtraction of fractions question where the difference is 3/5. Neither denominator you use can be 5. Describe your strategy. Replace the blanks with the digits 1, 2, 3, 4, 5, and 6 and add these fractions: []/[] + []/[] + []/[] Draw a picture to show how to add 3/5 and 4/6. Find or create three fraction “word problems”. Solve them and show your work. M Small12

13. Sample cube: Powers/Exponents Face 1: Describe what a power is. Face 2: Compare using powers to multiplying. How are they alike and how are they different? Face 3: What does using a power remind you of? Why? Face 4: What are the important parts of a power? Why is each part needed? Face 5: When would you ever use powers? Face 6: Why was it a good idea (or a bad idea) to invent powers? M Small13

14. Sample RAFT ROLEAUDIENCEFORMATTOPIC CoefficientVariableEmailWe belong together AlgebraPrincipal of a schoolLetterWhy you need to provide more teaching time for me VariableStudentsInstruction manualHow to isolate me Equivalent fractionsSingle fractionsPersonal adHow to find a life partner M Small14

15. Sample Stations: Surface Area Station 1: Simple “rectangular” or cylinder shape activities Station 2: Prisms of various sorts Station 3: Composite shapes involving only prisms Station 4: Composite shapes involving prisms and cylinders Station 5: More complex shapes requiring invented strategies M Small15

16. How do students differ? How are they different algebraically? How are they different with respect to proportional reasoning? How are they different spatially? How are they different with respect to problem solving and reasoning behaviours? M Small16

17. What to do Choose one of the four topics (algebra, proportional reasoning, spatial, problem solving and reasoning behaviours). We will form four groups (or more sub-groups) based on your choices. Be ready to articulate to the rest of us what “big picture” differences you are likely to find as a classroom teacher. Problems for you to consider will be provided. M Small17

18. Sharing Thoughts Reflect: How do the differences we discussed relate to your 4 students? M Small18

19. Your questions Let’s take some time to discuss some of the questions you raised and some of the concerns (or disagreements) you have. M Small19

20. but also the big picture Session 2 M Small20 Thinking about the pieces,

21. Thinkermath: Huntsville Huntsville is ____ km to Toronto, ___ km from North Bay via Highway ___, and ___ km to the Ottawa Valley. Captain Hunt arrived there in _____. It was incorporated as a town in 1901. Its area is ____hectares. There are about _____ residents. A permit fee to build is $___ per $____ estimated value. 8 11 130 215 350 1000 1869 1901 18 000 68 716 M Small21

22. Thinkermath: Huntsville Huntsville is 215 km to Toronto, 130 km from North Bay via Highway 11, and 350 km to the Ottawa Valley. Captain Hunt arrived there in 1869. It was incorporated as a town in 1901. Its area is 68 716 hectares. There are about 18 000 residents. A permit fee to build is $8 per $1000 estimated value. M Small22

23. How do you describe what you teach? Think about one of the courses you will teach in September. A parent asks what you will be teaching in the first month or two of the course. What would you say? Share your thoughts with a partner. How common are our descriptions? M Small23

24. Goals for Session 2 Become familiar with the notion of instructional trajectories Become more knowledgeable about big ideas in math and apply that knowledge to consider big ideas in Ontario math courses See the value of big ideas M Small24

25. The Pieces M Small25

26. Planning instructional sequences Instructional trajectories/learning landscapes/knowledge packages A description, usually visual, of the development; helps you see where students come from and where they go to M Small26

27. Integer +/- trajectory M Small27 Multiplication

28. Big Ideas M Small28

29. Not…. Demonstrate an understanding of the characteristics of a linear relation or connect various representations of linear relations (overall expectation) M Small29

30. Not…. Construct tables of values, graphs and equations using a variety of tools to represent linear relations derived from descriptions of realistic situations (specific expectation) or Describe a situation that would explain the events illustrated by a given graph of a relationship between two variables or… M Small30

31. Examples With certain relations, all you need to know are two pieces of data and you can describe the whole relation. With certain kinds of relations, a specific increase in one variable always results in a specific increase in the other. M Small31

32. What are the big ideas you teach? Move to a table with teachers who teach one of the same courses you do. Think/Pair/Share What is a big idea in that course? What is important to teach, but not a big idea? M Small32

33. Closing Session 2 Write down one new idea you learned in Session 1 or 2 that you think might be useful in your teaching. Write down one question you still have on the Question sheet at your table. Write down one thing that you heard that you disagree with or have doubts about. Be ready to challenge that idea in the next couple of days if that belief or concern continues. M Small33

34. Assessment for Learning Session 3 M Small34

35. Goals for Session 3 Become familiar with the importance of and strategies to collect useful assessment for learning data to inform differentiation Practise those strategies M Small35

36. Gathering Information To gather diagnostic information, you might use: -a task, -an interview, -paper and pencil items, -a graffiti exercise, -an anticipation guide,… M Small36

37. Grade 8 Integers The focus in grade 8 in teaching integers is multiplication and division and problems involving all four operations, considering order of operations. M Small37

38. Possible Task First, figure out what you think each of these products might be and why. a) 3 x (-4) b) (-4) x 3 c) (-3) x (-4) d) (-12) ÷ 3 e) (-12) ÷ (-3) f) 12 ÷ (-3) Then choose 4 integers so that the product < quotient < sum < difference M Small38

39. Possible Interview Name two integers between +2 and -8. How would you represent them? Which is greater: their sum or their difference? How do you know? The sum of a positive and negative integer is -4. What could the integers be? What situation might this describe? The difference between two negative integers is +8. What could the integers be? Use a number line or counters to show me why. M Small39

40. Possible paper and pencil items Complete these comparisons: e.g. -2 [ ] -4, -8 [ ] +10, 4 [ ] -1 Complete: -2 + 4 = [ ] -10 – 2 = [ ] [ ] + -4 = 8 etc. Tell why the sum of two negatives has to be negative. OR M Small40

41. Possible paper and pencil items Choose 2 positive and 2 negative integers. Show how to compare them, add them, and subtract them. Which of the tasks was easiest for you to do? Why? M Small41

42. Possible graffiti exercise Questions to which groups respond: When do you ever use integers? How are integers like whole numbers? How are integers different from whole numbers? M Small42

43. Possible anticipation guide Do you agree or disagree? Be ready to explain. You can predict the sign of the product of two integers if you know the sign of the sum. The sign of the quotient of two integers has to be the same as the sign of the product. You can either multiply first or add first when working with integers, e.g. [(-2) x (-3)] + 4 = (- 2) x [(-3) + 4] M Small43

44. Your turn Choose one of these topics: Grade 8 fractions Grade 9 linear relations Grade 10 quadratics Use two approaches to collecting diagnostic information. Prepare tools to collect that information. M Small44

45. Closing Session 3 Write down one new idea you learned in Session 3 that you think might be useful in your teaching. Write down one question you still have on the Question sheet at your table. Write down one thing that you heard that you disagree with or have doubts about. Be ready to challenge that idea in the next couple of days if that belief or concern continues. M Small45

46. Questions re your 4 students Behavioural differences vs cognitive differences (What I’m dealing with and what I’m not.) M Small46

47. Adapting a Lesson Session 4 M Small47

48. Goals for Session 4 Become familiar with and practise the opening up of closed questions Practise adapting both the instructional and consolidation pieces of a lesson to be more inclusive M Small48

49. Differentiating Instruction Let’s focus on differentiating instruction rather than only consolidation. We need tasks that are meaningful for all students, but we want to be able to manage it all. M Small49

50. Your answer is….? A graph goes through the point (1,0). What could it be? What makes this an accessible, or inclusive, sort of question? M Small50

51. Using Open Tasks Conventional question: You saved $6 on a pair of jeans during a 15% off sale. How much did you pay? vs. You saved $6 on a pair of jeans during a sale. What might the percent off have been? How much might you have paid? M Small51

52. Or… You saved some money on a jeans sale. Choose an amount you saved: $5, $7.50 or $8.20. Choose a discount percent. What would you pay? M Small52

53. Or.. Conventional question: What is 5 2 + 6 2 + 3 3 ? vs. Represent 88 as the sum of powers. M Small53

54. Let’s Practice How could you open up these? (Remember your 4 students.) Add: 3/8 + 2/5. A line goes through (2,6) and has a slope of -3. What is the equation? Graph y = 2(3x - 4) 2 + 8. Add the first 40 terms of 3, 7, 11, 15, 19,… M Small54

55. Using Parallel Tasks The idea is to use two similar tasks that meet different students’ needs, but make sense to discuss together. M Small55

56. Example 1 Task A: 1/3 of a number is 24. What is the number? Task B: 2/3 of a number is 24. What is the number? Task C: 40% of a number is 24. What is the number? M Small56 How do you know the number is more than 24? Is the number more than double 24? How did you figure out your number?

57. Example 2 Task A: One electrician charges an automatic fee of $35 and an hourly fee of $45. Another electrician charges no automatic fee but an hourly fee of $85. What would each company charge for a 40 minute service call? Task B: An electrician charges no automatic fee but an hourly fee of $75. How much would she charge for a 40 minute service call? M Small57 How do you know the charge would be more than $40? How did you figure out the fee?

58. Let’s Practice You are teaching students to solve a system of two equations in two unknowns (or division of fractions or….). You know some of your students are just not ready. What are parallel tasks you might set up? How did you make sure they were parallel? What are the benefits? M Small58

59. Systems of linear equations Task 1: Task 2 M Small59 Solve: (2x + y) ÷ 4 = 3 and 2(x – y) = -36 How did you use the first piece of information? The second piece? How did you know the numbers could not both be negative? How did you know that their difference was -18?

60. Division of Fractions Task 1: You need 1 1/4 cups of sugar in a recipe. You only have a 1/2 cup measure. How many times do you have to use it to measure all the sugar? Task 2: You divide two fractions. The quotient is 2 ½. What fractions did you divide? OR….. M Small60

61. Division of Fractions Task 1: You divide two numbers. The quotient is 2 ½. What numbers did you divide? Task 2: You divide two fractions. The quotient is 2 ½. What fractions did you divide? M Small61

62. Starting with a Provided Lesson Use a TIPS lesson or a text lesson as a beginning point. Develop a strategy to make the main teaching activity and the consolidation more inclusive. Use the notion of open tasks or parallel tasks. M Small62

63. M Small63

64. Let’s Try One M Small64

65. Let’s Try One M Small65

66. Possible changes Don’t assign the experiment; let them choose. Find a less intimidating way to ask then to examine first and second differences or to “make a scatter plot and a line of best fit.” M Small66

67. Or from a text M Small67

68. And… M Small68

69. And… M Small69

70. An example M Small70 It would be easy to open this up by asking how many hours he could work at each job instead of the fewest hours OR change it to one job OR let the kids pick the goal or the salary.

71. Open up Practice M Small71

72. You try Join a group of other teachers who want to work at the same level as you do. Your job is to work together to take one of the provided TIPS or text lessons or one that you happen to have with you that you teach and make it accessible to as many of the groups of YOUR 4 students as you can. Include one suggestion for differentiating assessment as well. Be ready to describe what you did and how you considered your 4 students. M Small72

73. Creating an inclusive classroom Session 5 M Small73

74. Goals for Session 5 Explore aspects of an inclusive climate M Small74

75. Your questions Let’s take some time to discuss some of the questions you raised and some of the concerns (or disagreements) you have. M Small75

76. Place Mat Activity Create a place mat like this one. Write for 3 minutes. M Small 76 Open questions vs. parallel tasks- advantages? disadvantages?

77. Most interesting comments Which comment from your colleagues did you find the most intriguing or thought provoking? M Small77

78. One more task Go back to the lesson you differentiated. Consider the overall topic and try one other differentiating strategy. You can use menus, tiering, tic tac toe, cubing, RAFTS, or stations. M Small78

79. Sharing Highlights First, let’s consider the work you did in differentiating your lesson. What was the hardest thing for you to deal with? How did you consider your 4 students? How much did it help to do it with colleagues? M Small79

80. Sharing Highlights Now, let’s consider the second differentiating task you did. Did you consider what you did as differentiating instruction or consolidation? How do you think your students would respond? How frequently do you think you could realistically use that strategy? M Small80

81. Reflect How would you complete this??? When a student gives me an unexpected, unusual response, I tend to…. M Small81

82. Developing a Climate of Inclusion What does your body and face convey? Do you welcome unusual response? Do you talk a lot less than your students do? When your students respond to you, do you pick up on what they say and always use their response in some way? Do you provide opportunities for students who are shy and those who are not? M Small82

83. Developing a Climate of Inclusion What does your body and face convey? Do you provide opportunities for students who are weak to not feel weak? Do you provide opportunities for students who are strong to go farther? M Small83

84. Conclusion You will find some references you may want to read later on about differentiated instruction. You may want to think about working with a colleague or in a small PLC to work on adapting lessons or parts of lessons. You may want to start by working on how you respond to students. Think about those 4 students again. Think about how good you will feel meeting their needs. M Small84