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Tag Archives: Use appropriate tools strategically

MP5: The Traveling Point

How do you give students the opportunity to practice “I can use appropriate tools strategically”?
MP5

When we have a new type of problem to think about, I am learning to have students give their best guess of the solution first. I’ve written about The Traveling Point before.

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Students sketched the path of point A. How far does A travel?

Students used paper and polydrons, their hands and string.

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I sent a poll to find out what they were thinking about the distance traveled.

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Students then interacted with dynamic geometry software. Does seeing the figure dynamically move help you better see the path?

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Does seeing the path help you calculate how far A travels?

Traveling Point 2.gif

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And so the journey to make the Math Practices our habitual practice in learning mathematics continues …


And the journey for my own learning continues. Thanks to Howard for correcting me. The second two moves do not travel a distance of 6, but the length of the circumference of the quarter circle.

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One student figured that out by the time the bell rang.

I look forward to redeeming this lesson this year, as the journey continues …

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Posted by on August 23, 2016 in Geometric Measure & Dimension, Geometry

 

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Origami Regular Octagon

We folded a square piece of paper as described in the Illustrative Mathematics task, Origami Regular Octagon. I didn’t want students to know ahead of time that they were creating an octagon, so I changed the wording a bit. We folded (and refolded … luckily, there was not a 1-1 correspondence between paper squares and students). Students worked individually to write down a few observations and then we talked all together.

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It’s an octagon.

There are 8 equal sides.

There are 8 equal angles.

It’s a regular octagon (this is the first year my students have come to me knowing what it means for a polygon to be regular).

How do you know there are 8 equal sides and 8 equal angles?

Because we folded it that way.

How do you know there are 8 equal sides and 8 equal angles?

Because one side is a reflection of its opposite side about the line that we folded.

What is the significance of the lines that you folded?

They are lines of symmetry.

There are 8 of them.

The opposite sides are parallel.

How can you tell?

This took a while. Maybe longer than it needed to.

Another student raised his hand.

I figured out that the sum of the angles in the octagon is 540˚.

(I don’t have the sum of the interior angles of an octagon memorized since I can calculate it, but I did know that 540˚ was too small.)

How did you get that?
I made an octagon and measured the angle. Then I multiplied by 8.

On your handheld?
Yes.

Okay. Let’s see what you have. I made him Live Presenter.

He showed us the angles he measured that were 67.5˚.

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It might help if we can see the sides of your angles. Will you use the segment tool to draw them?

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Other students argued that we needed to double 540 to get the sum of the angles in the octagon, 1080˚.

What else do you notice?

Triangles.

Congruent triangles.

Right triangles.

Students noticed different numbers of triangles.

And they recognized that we knew about congruence because of reflections.

Somehow we asked the question about the value of the angle (x).

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I set up a Quick Poll to collect student responses.

Almost everyone got the correct answer of 22.5˚.

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Can you tell me how you got your answer?

One student used the ¼ square with a 90˚ angle that had been bisected by the folded line to be 45˚ and the bisected again by the folded line to argue that x was 22.5˚.

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Did anyone do something different? Hands went up all around the room.

AC hasn’t talked to the whole class yet today, so I asked what she did.

I saw a circle with 360˚ and divided by 16.

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Then DC’s hand went up. 360/16 is equivalent to 180/8. I saw a line divided into 8 equal parts (or straight angle).

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Then TC showed us the isosceles triangle she used with the 62.5˚ base angles.

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Then someone else showed us the right triangle he used with the complementary acute angles.

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Before we knew it, we had spent almost an hour talking about a regular octagon. And learning math using quite a few Math Practices: construct a viable argument and critique the reasoning of others, look for and make use of structure, use appropriate tools strategically.

I’ve wondered before how much longer we will need to talk about generalizing relationships for interior and exterior angles in polygons. Today I got a glimpse of students being able to figure out those relationships by looking for and making use of structure. The only concern that remains is the length of time it would take to do that on a high stakes standardized test such as the ACT or SAT. And so the journey to do what is best for my students continues …

 
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Posted by on September 21, 2014 in Geometry, Rigid Motions, Tools of Geometry

 

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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]

 
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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.

 
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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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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.

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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.

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What happens to the tangent line at D?

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What happens to the tangent line at C?

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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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Rigid Motions – Which Reflection?

Rigid Motions – Which Reflection?

We added to our introductory lesson on Rigid Motions this year. Sometime last year, I read Which Reflection is Best by Andrew Shauver.

Which_Reflection

I wondered what students would say without measuring and before studying reflections in depth. So I asked.

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And then I asked again, after students had the opportunity to use appropriate tools strategically (most used a ruler).

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Not very many changed their minds from their first glance.

 

Next we talked.

BK offered his argument as to why he chose B over D:

When I put the edges of one end of the ruler on A and A’, the line of reflection didn’t go down the middle of the ruler on D, but it did on B.

 

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Oh…so is there something significant about the line of reflection matching the middle of the ruler?

We made sense of the significance with the yellow-ish angles.

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We developed the vocabulary as it arose, instead of having students write lists of definitions with no context (I cringe to think about how many of my former students suffered through writing geometry vocabulary each unit) – midpoint, bisector, distance from a point to a line, perpendicular bisector.

And we ultimately concluded that the line of reflection will be the perpendicular bisector of the segment that joins a pre-image point with its image.

What a great example of use appropriate tools strategically – BK was using his ruler both to measure distance and to measure for right angles. 

And so as the journey continues, I am thankful for students who are willing to share their thinking and for teachers like Andrew who are willing to share their lessons.

 
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Posted by on August 23, 2014 in Geometry, Rigid Motions

 

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