Day 2

Professional Learning for Mathematics Leaders and Coaches—Not

just a 3-part series

1

Four Corners activity•Go to the corner designated for the View and Discuss between-session opportunity you took advantage of.

2

Four Corners activity•Share some of your observations and thoughts with those from other boards in your corner.

3

Responses to Sticky Note Pileup

•Did you try this activity?•What do you see as its value?• What concerns do you have?

4

Responses to professional learning protocols sample•For those of you who have been coached or been a coach, which of these protocols is one that drew your attention? Why? 5

Questions from Day 1

•We read your Day 1 feedback and have some answers for you.

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Questions from Day 1

•If you have any other outstanding questions from Day 1, we will try to answer them now.

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More sharing

•Did you try out any goal setting/question writing activities related to our last session? • Can you share what happened? 8

What was our focus in Day 1?We looked at the ideas

that:•when planning a lesson, you design backwards: focus in on a goal and plan your lesson around that goal. 9

•the goal for a lesson is created based on curriculum expectations, but filtered through the lens of big ideas

What was our focus in Day 1?

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•question(s) to consolidate that goal are essential and may inform other parts of the lesson

What was our focus in Day 1?

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• responses to consolidating question(s) are useful assessment for learning information.

What was our focus in Day 1?

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•The consolidation question(s) need to reflect the goal, whether the lesson is a concept-building lesson or a practice lesson.

What was our focus in Day 1?

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•The consolidation question(s) is more than just “do another one”.

What was our focus in Day 1?

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Now let’s change direction a bit•We have been focusing on the math.

•Now let’s focus on the students.

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Thinking about differentiating

•Let’s look at a problem we might plan to use to see how thinking about difficulties students might have could lead you to alter the problem or to differentiating the task.

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Suppose this is the scenario..

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Suppose..• You want your students to solve this problem.Brandon and Alexis counted their money. Together, they had $7.50, but Brandon had $2.90 more than Alexis. 18

Brandon and Alexis counted their money. Together, they had $7.50, but Brandon had $2.90 more than Alexis.

a) How much did each have?b) How do you know there are no other answers? 19

Let’s talk…Let’s make a list of a “top 5” anticipated problems.

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Look at one approach• Have a look at the hand-out showing one approach to dealing with a problem you might anticipate.• Suggest a couple of other approaches in the other boxes.

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Let’s talk

• How did your new questions address the problems?

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This approach might lead to…differentiating

instruction, maybe offering one problem to some students and the other to others.

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DI requires

•a focus on big ideas (to have something big enough to differentiate)

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DI requires

•prior assessment (to know the need to and direction to differentiate)

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DI requires

•choice (to actually differentiate)

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What many of us do now

•The conventional approach to differentiating is to scaffold-- presenting problems in bits. Maybe this is not the only or the best way, to differentiate.

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Our focus…

•Our focus will be on two strategies:- Parallel tasks- Open questions

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You’ve met them

•Remember Day 1’s open question about creating a linear growing pattern beginning at -10 that grew slowly?

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You’ve met them

•Remember Day 1’s task where you matched questions to big ideas but different groups used different questions?

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Anticipated problem

•By anticipating the “problem” some grade 7-9 teachers might have with grade 11 or 12 content, we differentiated the task. 31

In parallel tasks..

•We look at the same instructional goal, with the anticipated student difficulty addressed.

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We could…

•Look at your listing of alternative Brandon/Alexis questions.• You may have already created parallel tasks. 33

All that’s missing…

•is how to handle the class when different students work on different tasks focused on the same goal.• We’ll address that now. 34

An example….

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• You plan to ask: The slope of a line is -2/3. Tell us the coordinates of two points on the line.

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You could anticipate…

• that some students may need more time with positive slopes before they are ready for problems involving negative slopes. 37

Looking at the goal….

•negative slopes are not really required. Students could address the goal using either positive or negative slopes.

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So….

•we create parallel tasks, offering choices accessible to more students.

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Parallel tasks•A line of slope 2/3 goes through (-4,-1). What is the equation?

•A line of slope -2/3 goes through (-4,-1). What is the equation? 40

Common questions

• Do you know which way your line slants? How do you know?

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Common questions•Could (-4, 3) be on your line? How do you know?• Could (-3, 0) be on your line? How do you know?

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Common questions•What do you need to know to write the equation?• How can you get that information?

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Common questions•What is your equation?• How can you be sure you’re right?

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Parallel tasks

•are useful particularly for the main instructional activity in a 3-part lesson.

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The MATCH template

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Where you might use the previous task

•The previous parallel tasks could have been the Action! if slope had just been introduced.

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Where you might use this

•They could have been the Minds On if students already had significant experience with slope.

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Another example

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I changed this from the earlier problem to have something at grade 7

Original Plan•You had planned an activity where students would translate a series of algebraic expressions into words and vice-versa.

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Then…• you realized that some expressions may be much more difficult for students than others.

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So…•You want the more difficult ones addressed, but you need to build success.

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The parallel tasks•Create 3 algebraic expressions using combinations of the given words. Then represent the expressions with models.

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The parallel tasks•Choice 1:

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•Choice 2: Add

1/2Divide by 10

Add the opposite

Multiply by 0.1

Take half

Subtract from 0

Add 1 Divide by 3

Multiply by 4

Multiply by 2

Double Triple

Common questions•What operation signs appeared in your algebraic expression? Why those?•How did your other representation show those operations were happening? 55

Common questions•Did you always end up with two terms (parts) if you used two phrases?• Could you have chosen different words to end up with the same expression? 56

Common questions•What were the advantages and disadvantages of each representation?

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•The particular operation the student uses is irrelevant to the goal.

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So the choices might be:•Use

algebra tiles to model two polynomials that add to 6x2+8x.

•Use algebra tiles to model two polynomials that multiply to 6x2+8x.

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You will notice I changed the expression to stay with grade 9 expectations.

Some common questions•What algebra tiles show

6x2+8x?• Is there any other way to model that polynomial?

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Some common questions•How did you arrange

your tiles?•Is there any other way you could have arranged the tiles?

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Some common questions

• How did you figure out how to start?

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Your turn

• In your group, choose one of the suggested set of parallel tasks. • Develop some common questions. 64

Share

• Go to the one of the four corners for people who did your task. Share your questions with another group in your corner.

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Now…

• Let’s think about open questions.

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Contrast• Open:Describe a relation… sort of like this one.

• Not Open:What type of relation is this?

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The underlying idea

• To which big idea do you think this question might relate?

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Contrast

• Open:A graph passes through the points (2,4) and (3,8). Describe a relation the graph could represent.

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• Not open: A line passes through the points (2,4) and (3,8). Write the equation of the line.

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What made it open?

• What made the first question open?

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Contrast• Open: Write an equation and solve it.

• Less Open: Solve 3x-2 = 8. Describe your strategy.

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Using open questions for assessment for

learning• An open question, as a minds-on activity, provides valuable information about how to proceed with your lesson.

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• It might tell you that there are missing prerequisites that will get in the way.

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• It might tell you that your students already know what you were planning to teach.

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• It might tell you that your lesson is going in the right direction, but a few “tweaks” acknowledging what students have said could help. 77

Comparing open and parallel

• Open questions might be more vague than parallel.• Sometimes an open question can replace two parallel ones.

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For example…

• For the parallel task where we used a positive slope vs a negative one, we could have said:

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For example…

• A line goes through (-4,-1). Choose its slope.• Once you’ve chosen it, tell the equation of the line.

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• Open questions can work in all three parts of a three-part lesson.• Parallel questions suit the action and perhaps Minds On, but not the consolidation where things are pulled together.

Where to use open questions

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• For example, a Minds On open question related to Big Idea 5 might be: Create two linear growing patterns that you think are really similar.

Minds-on

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For these slides, rather than using a lot of space to write goals, I slipped in a possible BI link—see if you agree– we have to acknowledge that sometimes other BIS would be possible.

You could ask:• What makes them similar?• How are their pattern rules similar?

Minds-on

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• For a lesson on inequalities:Create a mathematical statement where any number greater than 10 is a possible solution. (Related to BI 4)

Some more examples

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• For a lesson on quadratics:Which two graphs do you think are most alike? Why?Y = 3x2 -2 y = -3x2 -2Y = 3x2 +2 y = 2x2 + 3

Some more examples

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• How would you open up this question (or is it open now)?Fill in the missing values:

Your turn

1 2 3 6 7 88 13 23 28

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For the action

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• Choose two different values for the missing amounts. Make them different kinds of numbers: 3x – = 4 + xDraw a diagram that would help someone understand how to solve your equation. (BI3)

An Open Action!

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• How could this activity help the struggler?• How could it help the strong student?• What math is learned?

An Open Action!

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• A linear growing pattern has 50 as the 25th term. Create a bunch of possible patterns. • How does where your pattern starts relate to how fast it grows? (BI 6)

More action examples

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•Two lines intersect at (1,3).One is much steeper than the other. What could the pair of lines be? Give several possibilities.(BI6)

More Action! examples

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•Compare the roots of these three equations (BI 5). What do you notice? Why does it happen?4x2 – 17x + 4 = 0 6x2 – 37x + 6 = 08x2 – 65x + 8 = 0

More Action! examples

Add another question that works the same way.

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•Can you make up other sets of quadratic equations that have something in common?

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•A trig function goes through the point (π,7). What could it be? List a bunch of possibilities. (BI 6)

More Action! examples

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• Open questions are also very appropriate for consolidation.• We saw many of these in our last session.

Consolidation questions

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• Imagine that students have learned about the meaning of linear equations, e.g. what 4x – 2 = 9 means.

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• Without solving this equation, how do you know that the solution has to be positive? (BI 6) -4x + 7 = -5x + 30

You might ask…

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• How are these equations alike and different? (BI 5)3x – 2 = 6 + 9x3x – 2 = 6 + 9x – 6x -8

Or you might ask…

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• Imagine that students have learned the quadratic formula for solving quadratic equations.

For example

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• Besides the solutions to ax2 +bx + c = 0, what else does the quadratic formula tell you (BI 6)?OR

You might ask…

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• How is solving a quadratic equation like solving a linear one? How is it different?(BI 5)

You might ask…

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• One parabola is very narrow and one parabola is very wide.• What do you know or what don’t you know about how their equations differ? (BI 4)

Or

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• For a lesson on trigonometric functions:You must graph a trig function. In what situation might you want a y-axis going from -100 to 100? (BI 4)

Some more examples

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• Often you can start from an existing lesson and open parts of it up.

Starting from a source

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• One strategy is to start with an answer and create a question. For example, a growing pattern has 20 as the 9th term. What could the pattern be?

Fail-safe strategies

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• Another is to ask for similarities and differences. For example, how is factoring x2 +5x+6 like and different from factoring 3x2 - 2x – 8.

Fail-safe strategies

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• Another is to let the student choose values. For example, ask students to choose values for and and graph x + y = 8.

Fail-safe strategies

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• From the other examples we have seen, you can see that these are not the only three strategies but they are helpful.

Other strategies

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•Choose one of the following tasks. Make it more open

Your turn

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• You have 14 $5-bills. How many twonies do you need to have $100?• What combination of $2 coins and $5 bills have a total value of $100?

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• Think about why you might use an open question in each part of the lesson. Would your reasons be the same?• Discuss this in your group.

Consolidate

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