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Seeing Limits

Big Idea 1 from the 2016-2017 AP Calculus Curriculum Framework is Limits.

Enduring Understanding 1.1: The concept of a limit can be used to understand the behavior of a function.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

Students can connect concepts to their visual representation with and without technology.

Mathematical Practice for AP Calculus (MPAC) 4: Multiple Representations

Students can associate tables, graphs, and symbolic representations of functions.

Students can develop concepts using graphical, symbolical, verbal, or numerical representations with and without technology.

 

We begin calculus with a discussion of limits. I throw a dart at the dart board and ask what just happened.

You hit the bullseye.

(I really did this year – on the second try – after missing the board completely on the first try.) But how did the dart make it to the board? It had to go half the distance to the board. And then half the distance again. And again. And again. How did the dart make it to the board?

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The other demonstration is at the recommendation of my grandfather, now 101, who taught high school mathematics for over 50 years. I take a measured piece of yarn (1 meter), cut it in half, and begin to make a pile of the “halves”. How long does the yarn measure in the pile if we put it end to end? It gets closer and closer to measuring 1 meter, but does it ever make it to 1 meter?

How do you provide opportunities for students to “see” limits?

I used to [ineffectively] wave my hands. Now we use technology. Students won’t use this type of visualization while they’re taking a test, but we find it invaluable while they’re learning.

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Posted by on August 24, 2016 in Calculus, Limits & Continuity

 

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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?

Traveling Point 1.gif

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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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Assessing the Centroid of a Triangle

The centroid of a triangle is often called the balancing point of the triangle. It is the point at which the medians of the triangle intersect.

Students used technology to explore the relationship between the vertices of a triangle in the coordinate plane and the vertices of the centroid.

If your students knew the relationship between the vertices of a triangle and the vertices of the centroid, how would you expect them to answer the following question? (I included this question on an end of unit assessment.)

The vertices of a triangle are (a,b–c), (b,c–a), and (c,a–b). Prove that its centroid lies on the x-axis.

A few of my student responses are below.

What learning opportunities could I have provided in class to better prepare my students for this question without just giving them a similar problem?

And so the journey to provide meaningful learning episodes that prepare students to answer questions they haven’t seen before continues …

 
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Posted by on August 22, 2016 in Angles & Triangles, Geometry

 

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When Your Team Is Working Well Together

Have you seen Jill Gough’s blog post Strategic Teaming: leadership, voice, our hopes and dreams? Jill reminds us that strong teams both set norms for their work together and then self assess to ensure that they are functioning within their norms.

How do you provide your students the opportunity to set norms for the work that we have to do together?

I asked my students what it looks like when your team is working well together.

Here’s a wordle of their responses.

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I see communicating, cooperating, talking, participating, strategies, but what strikes me most from their suggestions is everyone.

Some lengthier responses from the students:

We are all talking about our strategies. Everyone considers all possibilities presented by the team. Everyone is contributing and listening to what each other has to say, respecting each other. We communicate reasons the answers may be correct or wrong. We will work together to figure out multiple solutions, or the one correct solution, or if there is no solution.

We’ve agreed to these norms.

Everyone …

Respects

Contributes

Listens

Questions

Collaborates

Communicates

Since I want to be transparent about formative assessment being for students as well as teachers, I showed them Popham’s levels of formative assessment.

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We are working well together when the whole class is using formative assessment (and not just the teacher). We want all students in our class to meet the learning goals. Not just the “smartest”; not just the fastest. This isn’t survival of the fittest where some can adapt and others will grow extinct. Everyone can learn. Everyone will learn.

The start of another school year has come and gone as the journey continues …


Popham, W. James. Transformative Assessment. Alexandria, VA: Association for Supervision and Curriculum Development, 2008. Print.

 
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Posted by on August 18, 2016 in Geometry, Student Reflection

 

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MP8: The Centroid of a Triangle

We had been working on a unit on Coordinate Geometry.

How do you give students the opportunity to practice “I can look for and express regularity in repeated reasoning”?

SMP8 #LL2LU Gough-Wilson

When we have a new type of problem to think about, I am learning to have students estimate the answer first.

I asked them to “drop a point” at the centroid of the triangle. We looked at the responses on the graph first and then as a list of ordered pairs.

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What is significant about the coordinates of the centroid?

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Students then interacted with dynamic geometry software.

Centroid_1.gif

What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between the coordinates of the vertices of a triangle and the coordinates of its centroid?

Some students needed to interact on a different grid setup to see a relationship.

Centroid_2.gif

After a few minutes, I sent another poll to find out what they figured out.

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And then we confirmed student conjectures as a whole class.

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

 

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MP7: The Diagonal of an Isosceles Trapezoid

 

SMP7 #LL2LU Gough-Wilson

I’ve written about the diagonals of an isosceles trapezoid before.

When we practice “I can look for and make use of structure”, we practice: “contemplate before you calculate”.

We practice: “look before you leap”.

We ask: “what you can you make visible that isn’t yet pictured?”

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We make mistakes; the first auxiliary line we draw isn’t always helpful.

Or sometimes we see more than is helpful to see all at one time.

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We persevere.

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Even with the same auxiliary lines, we don’t always see the same picture.

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We learn from each other.

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

 
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Posted by on August 16, 2016 in Angles & Triangles, Geometry, Polygons

 

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MP8: The Medians of a Triangle

How do you give students the opportunity to practice “I can look for and express regularity in repeated reasoning”?

SMP8 #LL2LU Gough-Wilson

When we have a new type of problem to think about, I am learning to have students estimate the answer first.

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I asked for their estimate in two slightly different problems because I wanted them to pay attention to what was given and what was asked for.

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Students then interacted with dynamic geometry software.

Centroid_1.gif

What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between a median of a triangle and its segments partitioned by the centroid?

As students moved the vertices of the triangle, the automatic data capture feature of TI-Nspire collected the measurements in a spreadsheet.

Centroid_2.gif

I sent another poll.

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And then we confirmed student conjectures on the spreadsheet.

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

 
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Posted by on August 15, 2016 in Angles & Triangles, Geometry

 

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