Marian SmallHuntsville Math Camp

August 2008

STARTING OUTSESSION 1

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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

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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 groupsCorner 3: provide individual learning packages as

much as possibleCorner 4: personalize both instruction and

assessment

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Reflect

• Have you changed your mind about the best strategies?

• What new ideas have you heard that you had not thought of before?

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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

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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

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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

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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….

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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….

<could include text or other exercises as part of the menu>

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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

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Sample tic tac toeComplete 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.

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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?

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Sample RAFTROLE AUDIENCE FORMAT TOPIC

Coefficient Variable Email We belong together

Algebra Principal of a school Letter Why you need to provide more teaching time for me

Variable Students Instruction manual How to isolate me

Equivalent fractions Single fractions Personal ad How to find a life partner

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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

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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?

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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.

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Sharing Thoughts

Reflect: How do the differences we discussed

relate to your 4 students?

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but also the big pictureSession 2

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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?

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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

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The Pieces

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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

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Integer +/- trajectory

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Multiplication

Big Ideas

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Not….

• Demonstrate an understanding of the characteristics of a linear relation or connect various representations of a linear relations (overall expectation)

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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…

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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.

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What are the big ideas you teach?

• Move to a table with teachers who teach one of the same courses you do.

• Think/Pair/ShareWhat is a big idea in that course?What is important to teach, but not a big idea?

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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.

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Assessment for LearningSession 3

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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

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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 10001869 1901 18 000 68 716

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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.

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Your questions

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

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Gathering Information

• To gather diagnostic information, you might use:

- a task, - an interview, - paper and pencil items, - a graffiti exercise, - an anticipation guide,…

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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.

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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

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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.

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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

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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?

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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?

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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]

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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.

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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.

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Adapting a LessonSession 4

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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

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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.

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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?

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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?

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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?

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Or..

• Conventional question: What is (-23)2 – (-22)2?vs.• Write the number 128 as the difference of

powers of negative integers.

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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)2 + 8.• Add the first 40 terms of 3, 7, 11, 15, 19,…

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Using Parallel Tasks

• The idea is to use two similar tasks that meet different students’ needs, but make sense to discuss together.

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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?

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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?

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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.

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Let’s Try One

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Let’s Try One

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Or from a text

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An example

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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

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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.

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Creating an inclusive classroomSession 5

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Goals for Session 5

• Explore aspects of an inclusive climate

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Your questions

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

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Place Mat Activity

• Create a place mat like this one. Write for 3 minutes.

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Open questions vs. parallel tasks- advantages? disadvantages?

Most interesting comments

• Which comment from your colleagues did you find the most intriguing or thought provoking?

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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.

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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?

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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?

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Reflect

How would you complete this??? • When a student gives me an unexpected,

unusual response, I tend to….

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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?

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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?

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Let’s Look at Some Teachers

• Have a look at how these teachers handle their students. How inclusive do they seem?

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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.

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