Plenary 4

Summer InstituteThunder Bay

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Consider this relation…

Contrast

Open:

Describe a relation sort of like this one.

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Not Open:

What type of relation is this?

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 the relation the graph represents.

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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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Contrast

What made it open?

What made the first question open?

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Contrast

Open:

Write an equation and solve it.

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Less Open:

Solve. 3x-2 = 8. Describe your strategy.

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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Might tell you that there are giant missing prerequisites that will get in the way of new learning.

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OR assessment for learning

Might tell you that your students already know what you were planning to teach.

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OR assessment for learning

Comparing parallel tasks and open

questions

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Look at this pair of parallel tasks. Choose

one and try it.

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Jennifer drew a graph of y = 2x – 8. She said there was no more work to do to draw a graph of y = . What might she mean?

Jennifer drew a graph of y = 2x2 – 8. She said there was no more work to do to draw a graph of y = . What might she mean?

x+82

x+82

€

±

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• What would the original graph look like? How do you know?

• What would the second graph look like? How do you know?

Common questions

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• What do you think Jennifer meant when she said there was no more work to do?

• Do you think she was right?

Common questions

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• What big idea was implicit in both of the parallel tasks?

What do you think?

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• With a partner, reframe the two parallel tasks into a single open question.

• Once you do, decide if you think the parallel tasks or the open question might be better to use and why.

What do you think?

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• Open questions can work in all three parts of a three-part lesson.

Where to use open questions

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• For example, a minds-on open question might be: Create two linear growing patterns that you think are really similar.

Minds-on

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You could ask:

• What makes them similar?

• How are their pattern rules similar?

Open Minds-on

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• How might such an open question be effective in diagnosing student differences?

• Do you see it more as exposing or evoking thinking?

Purpose Open 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.

More examples – Open Minds-on

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For a lesson on quadratics:

Which two graphs do you think are alike? Why?

Y = 3x2 -2 y = -3x2 -2

Y = 3x2 +2 y = 2x2 + 3

More Open Minds-on

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• How would you open up this question (or is it open now)?

Fill in the missing values:

Your turn – Open Minds-on

1 2 3 6 7 8

8 13 23 28

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Action

How might you use open questions for action?

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Choose two different values for the missing amounts. Make them different kinds of numbers: 3x – = 4 + x

Draw a diagram that would help someone understand how to solve your equation.

Open Action

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• How could this activity help the struggler?

• How could it help the strong student?

• What math is learned?

Open Action

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• What strategy do you think I used to make this question open?

• Is it a strategy that could be used in other situations?

Open Action

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A linear growing pattern has 50 as the 25th term. Create a bunch of possible patterns.

How does the starting value for your pattern relate to the rate at which it grows?

More examples – Open Action

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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.

More Open Action

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Compare the roots of these three equations. What do you notice? Why does it happen?

4x2 – 17x + 4 = 0

6x2 – 37x + 6 = 0

8x2 – 65x + 8 = 0

More Open Action

Add another question that works the same way.

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Make up other sets of quadratic equations that have something in common?

More Open Action

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A trig function goes through the point (π,7). What could it be?

List a bunch of possibilities.

More Open Action

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What combination of $2 coins and $5 bills have a total value of $100?

Your turn – Open Action

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Open questions are also very appropriate for consolidation.

Consolidation questions

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• Imagine that students have learned how to solve linear equations (e.g., 4x – 2 = 9).

Open Consolidation

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Without solving this equation, how do you know that the solution has to be positive?

-4x + 7 = -5x + 30

Open Consolidation

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How are these equations alike and different?

3x – 2 = 6 + 9x

3x – 2 = 6 + 9x – 6x -8

Open Consolidation

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• students have learned the quadratic formula for solving quadratic equations and many other traits of quadratics.

Imagine…

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Besides the solutions to ax2 +bx + c = 0, what else does the quadratic formula tell you?

Open Consolidation

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How is solving a quadratic equation like solving a linear equation?

How is it different?

Open Consolidation

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One parabola is very narrow and one parabola is very wide.

Is there any way you be sure about how their equations differ?

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?

Open Consolidation

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• Create an open consolidation question that you might use having taught students a lesson about how to solve equations of the form

ax +b = cx + d (e.g., 2x – 4 = 3x +1)

Your turn – Open Consolidation

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

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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• Now you try.

• With a partner, choose a lesson.

Your own lesson

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• Open up:

- the minds-on,

- the action,

- the consolidate.

Your own lesson

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• Interact with others at your tables.

• Select one minds-on, one action, and one consolidate question.

• Post them in the right spot.

Share time

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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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• Assessment for learning is your focus.

• An open question should be accessible to all students.

Open Minds-on

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• This is really an unscaffolded investigation, whether large or small.

• Students who need scaffolding receive it, but not everyone needs it.

Open Action

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• I had a lesson goal tied to a big idea. Did I achieve my lesson goal?

• By opening it up, I continue to allow for exposure of student thinking.

Open Consolidate