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Visual: SMP-5 Use Appropriate Tools Strategically #LL2LU

We want every learner in our care to be able to say

I can use appropriate tools strategically.
(CCSS.MATH.PRACTICE.MP5)

SMP5 #LL2LU

Level 4:
I can communicate details of how the chosen tools added to the solution pathway strategy using descriptive notes, words, pictures, screen shots, etc.

Level 3:
I can use appropriate tools strategically.

Level 2:
I can use tools to make my thinking visible, and I can experiment with enough tools to display  confidence when explaining how I am using the selected tools appropriately and effectively.

Level 1:
I can recognize when a tool such as a protractor, ruler, tiles, patty paper, spreadsheet, computer algebra system, dynamic geometry software, calculator, graph, table, external resources, etc., will be helpful in making sense of a problem.

 

Suppose you are solving an equation.

Are you practicing use appropriate tools strategically if you use the numerical solve command on your graphing calculator?

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Or what about using your calculator to substitute values of x until you find a value that makes a true statement?

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Are you practicing use appropriate tools strategically if you use a computer algebra system to explain your steps?

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Or what if you use the graphing capability of your handheld?

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Consider each of the following learning goals:

I can explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution, and I can construct a viable argument to justify a solution method.  CCSS-M A-REI.A.1.

I can solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. CCSS-M A-REI.B.3.

I can explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. CCSS-M A-REI.D.11.

Does use appropriate tools strategically depend on the learner? Or the learning goal? Or the teacher? Or the availability of tools?

 

[Cross posted on Experiments in Learning by Doing]

 
2 Comments

Posted by on September 15, 2014 in Standards for Mathematical Practice

 

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SMP5: Use Appropriate Tools Strategically #LL2LU

We want every learner in our care to be able to say

I can use appropriate tools strategically.
(CCSS.MATH.PRACTICE.MP5)

SMP5

But…What if I think I can’t? What if I have no idea what are appropriate tools in the context of what we are learning, much less how to use them strategically? How might we offer a pathway for success?

Level 4:
I can communicate details of how the chosen tools added to the solution pathway strategy using descriptive notes, words, pictures, screen shots, etc.

Level 3:
I can use appropriate tools strategically.

Level 2:
I can use tools to make my thinking visible, and I can experiment with enough tools to display confidence when explaining how I am using the selected tools appropriately and effectively.

Level 1:
I can recognize when a tool such as a protractor, ruler, tiles, patty paper, spreadsheet, computer algebra system, dynamic geometry software, calculator, graph, table, external resources, etc., will be helpful in making sense of a problem.

We still might need some conversation about what it means to use appropriate tools strategically. Is it not enough to use appropriate tools? Would it help to find a common definition of strategically to use as we learn? And, is use appropriate tools strategically a personal choice or a predefined one?

Strategic

How might we expand our toolkit and experiment with enough tools to display confidence when explaining why the selected tools are appropriate and effective for the solution pathway used?  What if we practice with enough tools that we make strategic – highly important and essential to the solution pathway – choices?

What if apply we 5 Practices for Orchestrating Productive Mathematics Discussions to learn with and from the learners in our community?

  • Anticipate what learners will do and why strategies chosen will be useful in solving a task
  • Monitor work and discuss a variety of approaches to the task
  • Select students to highlight effective strategies and describe a why behind the choice
  • Sequence presentations to maximize potential to increase learning
  • Connect strategies and ideas in a way that helps improve understanding

What if we extend the idea of interacting with numbers flexibly to interacting with appropriate tools flexibly?  How many ways and with how many tools can we learn and visualize the following essential learning?

I can understand solving equations as a process of reasoning and explain the reasoning.  CCSS.MATH.CONTENT.HSA.REI.A.1

What tools might be used to learn and master the above standard?

  • How might learners use algebra tiles strategically?
  • When might paper and pencil be a good or best choice?
  • What if a learner used graphing as the tool?
  • What might we learn from using a table?
  • When is a computer algebra system (CAS) the go-to strategic choice?

Then, what are the conditions which make the use of each one of these tools appropriate and strategic?

[Cross posted on Experiments in Learning by Doing]

________________________

“The American Heritage Dictionary Entry: Strategically.” American Heritage Dictionary Entry: Strategically. N.p., n.d. Web. 08 Sept. 2014.

 
6 Comments

Posted by on September 14, 2014 in Standards for Mathematical Practice

 

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Locating a Warehouse

We changed the Learning Mode to individual. Where would you place a warehouse that needed to be equidistant from all three roads? (From Illustrative Mathematics.)

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Students started sketching on paper, and I set up a Quick Poll so that we could see everyone’s conjecture at the same time.

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We changed the Learning Mode to whole class. With whom do you agree?

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I didn’t intend for us to talk in detail at this point. I wanted students to be able to test their conjecture using their dynamic geometry software. But we had done that the day before for Placing a Fire Hydrant (post to come), and class was cut short during this lesson because of lock-down and evacuation drills. So we did talk in more detail than I had planned. Is the point outside of the triangle equidistant from the three roads? One student vehemently defended her point: I drew a circle with that point as center that touched all three roads. (We have been talking about the distance from a point to a line.) How do you know that the roads are the same distance from the center? They are all radii of the circle. They are perpendicular to the road from the center.

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Could a point inside the triangle of roads be correct? If so, which? We started drawing distances from the points to the lines. Some points were about the same distance from two of the roads but obviously to close to the third road. What’s significant about the point that will be the same distance from all three sides of a triangle? Several students wondered about drawing perpendicular bisectors. Another student vehemently insisted that the point needed to lie on an angle bisector. Would that always work?

Are you going to let us try it ourselves? Well of course! So with about 12 minutes left, students began to construct.

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With about 3 minutes left, I made a student the Live Presenter who showed us that the angle bisectors are concurrent, and used the length measurement tool to show us that the point is equidistant to the sides of the triangle.

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With about 2 minutes left I made another student the Live Presenter who had made a circle inside the triangle. How did you get that circle? What is significant about the circle? It’s inscribed. The center is the where the angle bisectors intersect. So we call that point the incenter. It’s the center of the inscribed circle of a triangle, and the point of concurrency for the angle bisectors. How is this point different from the circumcenter?

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And with 1 minute left: Do you understand what we mean when we say that every point on an angle bisector is equidistant from the two sides?

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And so while I have some record of what every student did during class through Quick Polls and Class Capture and collecting their TNS document once the bell rang, my efforts at closure are foiled again. Maybe one day I’ll actually send one of the Exit Quick Polls that I have made for every lesson.

 
 

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Visual: SMP-3 Construct Viable Arguments and Critique the Reasoning of Others #LL2LU

We want every learner in our care to be able to say

I can construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP3

SMP3

But…what if I can’t? What if I’m afraid that I will hurt someone’s feelings or ask a “stupid” question? How might we facilitate learning and grow our culture where critique is sought and embraced?

From Step 1: The Art of Questioning in The Falconer: What We Wish We Had Learned in School.

By learning to insert feedback loops into our thought, questioning, and decision-making process, we increase the chance of staying on our desired path. Or, if the path needs to be modified, our midcourse corrections become less dramatic and disruptive. (Lichtman, 49 pag.)

This paragraph connects to a Mr. Sun quote from Step 0: Preparation.

But there are many more subtle barriers to communication as well, and if we cannot, or do not choose to overcome these barriers, we will encounter life decisions and try to solve problems and do a lot of falconing all by ourselves with little, if any, success. Even in the briefest of communications, people develop and share common models that allow them to communicate effectively.  If you don’t share the model, you can’t communicate. If you can’t communicate, you can’t teach, learn, lead, or follow.  (Lichtman, 32 pag.)

How might we offer a pathway for success? What if we provide practice in the art of questioning and the action of seeking feedback? What if we facilitate safe harbors to share thinking, reasoning, and perspective?

 

Level 4:

I can build on the viable arguments of others and take their critique and feedback to improve my understanding of the solutions to a task.

Level 3:

I can construct viable arguments and critique the reasoning of others.

Level 2:

I can communicate my thinking for why a conjecture must be true to others, and I can listen to and read the work of others and offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the work.

Level 1:

I can recognize given information, definitions, and established results that will contribute to a sound argument for a conjecture.

 

SMP3 #LL2LU

[Cross-posted on Experiments in Learning by Doing]

________________________

Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School. New York: IUniverse, 2008. Print.

 

 
 

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SMP3: Construct Viable Arguments and Critique the Reasoning of Others #LL2LU

We want every learner in our care to be able to say

I can construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP3

SMP3

But…what if I can’t? What if I’m afraid that I will hurt someone’s feelings or ask a “stupid” question? How may we create a pathway for students to learn how to construct viable arguments and critique the reasoning of others?

Level 4:

I can build on the viable arguments of others and take their critique and feedback to improve my understanding of the solutions to a task.

 

Level 3:

I can construct viable arguments and critique the reasoning of others.

 

Level 2:

I can communicate my thinking for why a conjecture must be true to others, and I can listen to and read the work of others and offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the work.

 

Level 1:

I can recognize given information, definitions, and established results that will contribute to a sound argument for a conjecture.

 

Our student reflections on using the Math Practices while they are learning show that they recognize the importance of construct viable arguments and critique the reasoning of others.

Jordan says “If you can really understand something you can teach it. Every person relates to and thinks about problems in a different way, so understanding different ways to get to an answer can help to broaden your knowledge of the subject. Arguments are all about having good, logical facts. If you can be confident enough to argue for your reasoning you have learned the material well.”

SMPj1

And Franky says that construct viable arguments and critique the reasoning of others is “probably our most used mathematical practice. If someone has a question about a problem, Mrs. Wilson is always looking for a student that understands the problem to explain it. And once he or she is finished, Mrs. Wilson will ask if anyone got the correct answer, but worked it a different way. By seeing multiple ways to work the problem, it is easier for me to fully understand.”

SMPj2

What if we intentionally teach feedback and critique through the power of positivity? Starting with I like indicates that there is value in what is observed. Using because adds detail to describe/indicate what is valuable. I wonder can be used to indicate an area of growth demonstrated or an area of growth that is needed. Both are positive; taking the time to write what you wonder indicates care, concern, and support. Wrapping up with What if is invitational and builds relationship.

Move the fulcrum so that all the advantage goes to a negative mindset, and we never rise off the ground. Move the fulcrum to a positive mindset, and the lever’s power is magnified— ready to move everything up. (Achor, 65 pag.)

The Mathy Murk has recently written a blog post called “Where do I Put P?” An Introduction to Peer Feedback, sharing a template for offering students a structure for both providing and receiving feedback.

Could Jessica’s template, coupled with this learning progression, give our students a better idea of what we mean when we say construct viable arguments and critique the reasoning of others?

[Cross-posted on Experiments in Learning by Doing]

_________________________

Achor, Shawn (2010-09-14). The Happiness Advantage: The Seven Principles of Positive Psychology That Fuel Success and Performance at Work (Kindle Locations 947-948). Crown Publishing Group. Kindle Edition.

 

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Understanding the Slope of the Tangent Line

My calculus students used a toothpick to explore tangent lines. I heard this idea from Paul Foerster in a workshop at some point along the way. 

We started with the tangent line at A. Everyone didn’t have a horizontal tangent line, but most did. Students quickly determined that A’ would be at the relative minimum of the graph.

E1

Is the slope of the tangent line greater at B or at C?

How do you know? 

I used the TI-Nspire document on the board as we talked about their conjectures.

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Move the tangent line (toothpick) all along the curve. Write down at least two observations (by yourself).

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Then students discussed their observations with their teams. And then each team told the whole class one observation at a time until we had heard them all.

We discussed increasing/decreasing intervals, concavity, relative extrema, and more.

E2

What happens to the tangent line at D?

E3

What happens to the tangent line at C?

E4

Students create their own function with certain requirements for tangent lines at A, B, and C.

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Next we moved to Derivative Trace. The y-coordinate of Point P represents the slope of the tangent line for the each x-coordinate of P. What path does P trace? Students watched a few times. A few said that the path P followed looked like a sine curve; others said that the path P followed looked quadratic.

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We used the Automatic Data capture feature of TI-Nspire to see a scatterplot of the path that P followed.

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We will call that path the derivative.

 

What is true about the derivative when the original function is increasing?

What is true about the derivative when the original function is decreasing?

When is the derivative equal to zero?

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When is the derivative undefined?

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What is the derivative of f(x)=sin(x)?

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What is the derivative of f(x)=ex?

 

So how will we calculate the slope of the tangent line?

What do we need to calculate slope?

Two points

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Then we can calculate the slope of the secant line.

We named the points on the secant line as (x,f(x)), (x+∆x,f(x+∆x)). Students wrote a representation for the slope of the secant line.

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But we don’t really want the slope of the secant line – we want the slope of the tangent line. How can we change the secant line into the tangent line?

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We want ∆x to get as small as possible.

We want ∆x to approach 0.

We want the limit as ∆x approaches 0 of the slope of the secant line.

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And then we calculated the derivative, f’(x) using the definition for one function, f(x)=x2. And we connected what we got for f’(x) when we found f’(4) to the slope of the tangent line.

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And then the bell rang.

 
3 Comments

Posted by on September 3, 2014 in Calculus

 

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Reflecting a Point about a Line

Students had the opportunity to look for and express regularity in repeated reasoning while reflecting a point about the line y=x.

I sent an interactive Quick Poll so that they could move the point, observe its reflection about y=x, and then determine the image of a point not shown on the graph.

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Then I asked students to generalize their results by determining the image of (a,b) reflected about y=x. Which responses would your students accept as correct?

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Students had a more difficult time reflecting a point about the line y=–x. 18 of 26 got it correct for the Quick Poll on which they “played”. (Show Correct Answer has been deselected.)

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But only 12 got it correct for the Quick Poll that simply asked the question.

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Would a graph help?

 

 

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Drawing a representation of the situation is not yet second nature.

 

Then we reflected a point about the horizontal line y=–2 first on the graph:

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And then we reflected a point about the vertical line x=2 on the graph:

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Next, I didn’t give them a graphical representation.

But more of them thought to create their own.

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And then the bell rang while students were submitting.

 

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And so the journey to provide students opportunities to develop mathematical habits of mind continues …

 

 

 
 

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