Marian Small Huntsville Math Camp August 2008

STARTING OUT SESSION 1 M Small2

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

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

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

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

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

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

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

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

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

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

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

Sample RAFT ROLEAUDIENCEFORMATTOPIC CoefficientVariable We 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

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

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

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

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

but also the big picture Session 2 M Small19 Thinking about the pieces,

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 Small20

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 Small21

The Pieces M Small22

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 Small23

Integer +/- trajectory M Small24 Multiplication

Big Ideas M Small25

Not…. Demonstrate an understanding of the characteristics of a linear relation or connect various representations of a linear relations (overall expectation) M Small26

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 Small27

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 Small28

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 Small29

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 Small30

Assessment for Learning Session 3 M Small31

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 Small32

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 Its area is ____hectares. There are about _____ residents. A permit fee to build is $___ per $____ estimated value M Small33

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 It was incorporated as a town in Its area is hectares. There are about residents. A permit fee to build is $8 per $1000 estimated value. M Small34

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 Small35

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

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

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

Possible Interview Name three 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

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

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

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

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

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

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

Adapting a Lesson Session 4 M Small46

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 Small47

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 Small48

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 Small49

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 Small50

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 Small51

Or.. Conventional question: What is (-2 3 ) 2 – (-2 2 ) 2 ? vs. Write the number 128 as the difference of powers of negative integers. M Small52

Let’s Practice How could you open up these? 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) Add the first 40 terms of 3, 7, 11, 15, 19,… M Small53

Using Parallel Tasks The idea is to use two similar tasks that meet different students’ needs, but make sense to discuss together. M Small54

Example 1 Task A: 40% of a number is 24. What is the number? Task B: 2/3 of a number is 24. What is the number? M Small55

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 $55. What would each company charge for ½ hour of work? for 2 hours of work? Task B: An electrician charges no automatic fee but an hourly fee of $50. How much would he charge for 40 minutes? M Small56

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 Small57

Let’s Try One M Small58

Let’s Try One M Small59

Or from a text M Small60

An example M Small61 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.

Open up Practice M Small62

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 Small63

Creating an inclusive classroom Session 5 M Small64

Goals for Session 5 Explore aspects of an inclusive climate M Small65

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 Small66

Place Mat Activity Create a place mat like this one. Write for 3 minutes. M Small 67 Open questions vs. parallel tasks- advantages? disadvantages?

Most interesting comments Which comment from your colleagues did you find the most intriguing or thought provoking? M Small68

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 Small69

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 Small70

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 Small71

Reflect How would you complete this??? When a student gives me an unexpected, unusual response, I tend to…. M Small72

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 Small73

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 Small74

Let’s Look at Some Teachers Have a look at how these teachers handle their students. How inclusive do they seem? M Small75

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 Small76