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Practicing Formative Assessment: Hopewell Triangles

This year’s Mathematics Assessment Project Hopewell Triangles task on similarity and right triangles played out differently than previous years.

2013 – Right Triangle

2013 – Similarity

2014 – Misconceptions

In general, students didn’t have as many misconceptions as the prior year.

As students were working by themselves, I did see one misconception that I was sure to bring out in our whole class discussion.

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(Look at ∆D above.)

On another trip around the room, I saw this on his handheld, which helped him correct his own mistake for ∆D.

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I particularly enjoy seeing different ways that students explain why the triangles are similar (#3) and why or why not the triangle is a right triangle.

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I sent a TNS document with questions to collect student responses for some of the questions. When I collected it after students worked individually, we had a 70% success rate. (I’ve changed the Student Name Format to Student ID to keep student names concealed.)

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At that point, we changed to Team mode, and students talked with each other about their work and I told them that they could change answers in their TNS document as they discussed their work. Students are making it a practice to not mark an answer unless they have an explanation to go with the answer. When I collected their work the second time, I knew that our whole class discussion needed to start with the third question. (I also knew who needed to come in during zero block for extra support.)

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Also of note is that the first collection of question 3 had these results:

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But the final collected had these results:

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Why is ∆1~∆A?

Why isn’t ∆1 similar to ∆F or ∆E?

We finished the discussion by discussing a misconception that students had last year. Anna thinks that ∆2 is a 30-60-90 triangle. Do you agree? Why or why not?

And so the journey continues, using formative assessment to make instructional adjustments to meet the needs of the students who are currently in my care …

 
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Posted by on February 28, 2015 in Dilations, Geometry, Right Triangles

 

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The Fundamental Theorem of Calculus

How do you provide an opportunity for your students to figure out the relationship between differentiation and antidifferentiation?

We have used the Calculus Nspired activities The First Fundamental Theorem of Calculus and The Second Fundamental Theorem of Calculus for several years now to improve our understanding of the relationship between a function and its accumulation function. I actually do print the student handouts for these activities and give students time during class to make sense of the relationship between differentiation and antidifferentiation.

It was time for our whole class discussion.

We defined the accumulation function using a definite integral. What do we know?

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Students had figured out earlier that the definite integral of f(x) from a to a would be 0 and concluded that F(a)=0.

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Students recognized that the value of the definite integral of f(x) from a to d would be F(d) and that the value of the definite integral of f(x) from a to c would be F(c).

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Suppose we want to calculate the definite integral of f(x) from c to d. They could tell area-wise that was equivalent to finding the definite integral of f(x) from a to d and subtracting the definite integral of f(x) from a to c, which of course gives us F(d)-F(c).

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And what does F(x) have to do with f(x)?

They could tell from the exploration that F(x) is the antiderivative of f(x).

Really? You mean we don’t have to do the limit-sum-infinite-number-of-rectangles every time? Really. You’ve earned the Fundamental Theorem of Calculus.

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We talked for a little while about average value using the Calculus Nspired activity MVT for Integrals, and then checked in on their understanding.

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As we moved into the second part of the Fundamental Theorem of Calculus, I posed a question to see how they would answer. (Remember that at this point, they’ve been using the FTOC for about 20 minutes.)

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I was excited about a few students getting it right. Without discussing the correct responses with the whole group (I showed their answers but had Show Correct Answer deselected, I sent another question, which unearthed their misconception and revealed my initially bad question.

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The students who got the answer correct in the first question had gotten it correct the wrong way, but their mistake wasn’t revealed because sin(-π)=0.

By now we were past the bell, and so we started over the next lesson with the second part of the Fundamental Theorem of Calculus.

It’s always exciting to find both the right questions to ask (the ones that reveal student misconceptions) and the wrong questions to ask (the ones that hide student misconceptions) so that I can continue asking the right ones and discontinue asking the wrong ones. In this lesson, I found at least one of each. And so the journey finding and asking the right questions continues …

 
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Posted by on February 27, 2015 in Calculus

 

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Circles Tasks

We looked at two diagrams as we finished up our unit on Circles. I knew that we didn’t have time for everyone to do both tasks. I wondered whether I could pull off giving students some choice in what they investigated.

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Both tasks are from the Mathematics Assessment Project.

The first can be found as Circles in Triangles in the tasks section and Inscribing and Circumscribing Right Triangles in the formative assessment lessons section.

The second is Temple Geometry from the tasks section.

What mathematical question could we explore?

Some of you saw my tweet.

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Yes. We are doing that “wonder” thing.

Most of the wonderings were about areas of various regions, and most of the questions were about the second diagram. I let each team decide what to explore. Only one team chose the first diagram.

I had copied a handout with questions about both diagrams, but I hesitated to give it to students, as it gave so much away. Instead, I gave the students a copy of the diagram only on which to work.

As has become our practice, students started working by themselves.

They practiced look for and make use of structure.

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Some practiced use appropriate tools strategically to make sense of the problem.

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Then they talked with their teams. Those working the second task almost immediately concluded that the radius of the smallest circle was half the radius of the largest circle. One student had used paper to measure the distance. Another “eyeballed” it correctly. Others constructed the diagram using dynamic geometry software and used the measuring tool to verify their conjecture. But no one was able to prove that the radius of the smallest circle was half the radius of the largest circle.

Time was ticking quickly. I still hadn’t even talked with the team who had chosen to work on the Circles in Triangles. What should I do to move student learning forward?

We moved into whole class mode. I selected a few students to share their thoughts about the length of the radius of the smallest circle. I made CS the Live Presenter so that he could show how his team translated the smallest circle so that its center lay on the point that partitioned the diameter of the medium circle into a 1:3 ratio.

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So if the radius of the smallest circle is one-half the radius of the largest circle, then what is the area of the shaded region? Every team set to work, ensuring that all of members of their team could calculate the area of the shaded region, and they did so successfully. (The evidence is on my computer at school … we are at home today for a “snow” day.)

What would have happened if I had given the teams the handout from the beginning?

From Circles in Triangles:

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From Temple Geometry:

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Would they have learned more mathematics during the lesson? Would they have practiced look for and make use of structure or use appropriate tools strategically during the lesson? Would they have engaged in productive struggle?

What would have happened if I had given the teams the handout that provided the structure for them once they got to a certain point on their own … even though they wouldn’t have had time to complete the investigation together?

I am learning that what works in our classrooms has so much to do with the students we have. Productive struggle isn’t just for students: We can plan great ideas collaboratively, but even so, we must be attentive to meeting the students in our room where they are and moving those students’ learning forward. And so, thankfully, the journey continues …

 
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Posted by on February 26, 2015 in Circles, Geometry

 

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Area between Curves

Our learning goals for the Applications of Definite Integrals unit in calculus are the following:

I can calculate and use the area between two curves.

I can use the disc and washer methods to calculate and use the volume of a solid.

I can use the shell method to calculate and use the volume of a solid.

I can calculate and use volumes of solids created by known cross sections.

During the lesson focusing on the first goal, we used a scenario from a TImath activity The Area Between to start our conversation.

I rarely send the TNS documents as is to my students or give them a copy of the printed student handout (even though I learn from both in my own planning of how the lesson will play out). This activity gave the following information on the first two pages:

Suppose you are building a concrete pathway. It is to be 1/3 foot deep.

To determine the amount of concrete needed, you will need to:

- calculate area (the integral of the top function minus the bottom function

- calculate volume (area multiplied by depth)

The borders for the pathway can be modeled on the interval -2π ≤ x ≤ 2π by

f(x)=sin(0.5x)+3

g(x)=sin(0.5x)

On the next page, graph the functions. Use the Integral tool to calculate the area under f1 and f2. Then, use the Text and Calculate tools to find the volume of the pathway.

Which takes away any opportunity for students to engage in productive struggle.

I shared this instead:

Suppose you are building a concrete pathway that is to be 1/3 foot deep. The borders for the pathway can be modeled on the interval -2π ≤ x ≤ 2π by f(x)=sin(0.5x)+3 and g(x)=sin(0.5x).

(I’m fully aware that giving them even this much information takes away from the modeling process … but there is always give and take, and for this lesson, the learning goal wasn’t whether they could determine functions for modeling the sidewalk.)

They decided to graph the functions.

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And talked about how they could calculate the area between the curves.

They had never used the Integral tool for graphs, much less the Bounded Area tool, so they oohed and aahed gasped in amazement.

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Sydney asked: Is that the only way to get the area between the curves?

(I knew that she was looking for and making use of structure, composing and decomposing the sidewalk into regions with equal area).

I answered: Is it?

We made Sydney the Live Presenter, and she used the Integral tool to calculate the area between f(x)=sin(0.5x)+3 and the x-axis from -2π to 2π.

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So how can we calculate the amount of concrete needed? The integral and bounded area tools are helpful for visualizing what you’re calculating, but you can’t use those tools on the AP Exam.

And so the students decided to calculate the area between the curves and then multiply by 1/3 to get the volume of the pathway.

Because they were able to tell me what to do, I almost didn’t send a Quick Poll to collect a definite integral that would calculate the volume. I wanted to hurry up and get to a card-matching activity similar to Michael Fenton’s that I knew would be helpful, but instead I eased the hurry syndrome and sent the poll.

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What I saw and heard was well worth the time that it took.

Can you spot the students’ misconception?

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Several students were multiplying the definite integral by and by 1/3, to represent height times base times depth, instead of recognize that the definite integral represented height times base (area), and not just height. (They knew this … we had summed the areas of an infinite number of rectangles for a certain base to calculate area under the curve. But they obviously didn’t know this like they needed to.)

When we calculated their integral, we didn’t get (1/3)*37.699, as expected.

 

Next I purposefully choose a region for which the upper and lower boundaries changed.

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We had a nice look for and make use of structure discussion about different ways to write a definite integral for calculating the area of the region.

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Many of you might notice that there is more opportunity to look for and make use of structure for the concrete pathway. I never asked whether you really need calculus to calculate the volume of the pathway. Nevertheless, I feel like I found two good problems/items/tasks to push and probe student thinking. And there’s always next year, as the journey continues …

 
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Posted by on February 25, 2015 in Calculus

 

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Angles & Arcs

Circles: CCSS-M G-C.A Understand and apply theorems about circles

2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

4. (+) Construct a tangent line from a point outside a given circle to the circle.

How intuitive is the relationship between an angle with a vertex inside the circle and the intercepted arcs of the angle and its vertical angle?

We started our lesson with a Quick Poll.

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About one-third of the students intuited the relationship.

Students interacted with the Geometry Nspired activity Secants, Tangents, and Arcs.

What happens as you move point A?

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What’s the least amount of information needed to calculate the angles and arcs? If I give you the measures of two the intercepted arcs, how can you determine the measure of an angle?

I sent the poll again. Do you want to keep your answer or change it based on what you observed with the dynamic geometry software?

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So why is the angle measure half the sum of its intercepted arcs?

We practiced look for and make use of structure. What do you see that isn’t pictured? What do we know so far about circles?

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Students thought about what auxiliary lines might be helpful for proving this relationship.

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And then we looked together at some of their ideas for proving the result. Which of these would be helpful for our proof?

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What about when the vertex of the angle is outside the circle?

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This time only one person got the correct answer. I initially thought that he had intuited the relationship, but after talking with him about how he got 70, I realized that wasn’t true.

We went back to the TNS document. What happens when you move point P? How can you determine the angle measure when given the two intercepted arcs?

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A Class Capture gave me evidence that most students were making observations and testing their conjectures.

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The bell rang as one student shared his conjecture: subtract the arcs, and divide by 2.

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Jill Gough challenges us to provide Ask-Don’t-Tell learning opportunities for our students. What Ask-Don’t-Tell learning opportunities are you already providing for your students? What new Ask-Don’t-Tell learning opportunities can you provide your students this week?

And so the #AskDontTell journey continues … one lesson at a time.

 
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Posted by on February 24, 2015 in Angles & Triangles, Circles, Dilations, Geometry

 

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Inscribed Angles

Circles: CCSS-M G-C.A Understand and apply theorems about circles

  1. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
  2. (+) Construct a tangent line from a point outside a given circle to the circle.

We started our unit on circles looking at a diagram with a right triangle both inscribed in a circle and circumscribed about a circle. What do you notice? What do you wonder?

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By the end of the unit, we will be able to generalize the relationship between the sides of the right triangle and the radii of the inscribed and circumscribed circles.

My students don’t come to me knowing all of the vocabulary associated with circles, but the longer we teach with our new standards, the more I am convinced students can learn vocabulary through the modeling of using it properly and by practicing using it properly. Geometry vocabulary doesn’t have to be reduced to copying definitions from the glossary of the textbook onto a notecard (an apology those former students who had me before I figured this out).

For example, we started with a brief look at the Geometry Nspired activity Circles – Angles and Arcs.

Before generalize the relationship between a central angle and its intercepted arc, I sent a Quick Poll. The wording of the Quick Poll added “major arc” to students’ vocabulary.

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For an inscribed angle, I started with a poll just to see how intuitive the relationship is between the angle measure and intercepted arc before any kind of learning episode to explore the relationship.

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About one-third of the students intuited the relationship.

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I didn’t show the results. Instead, we looked at another page in the TNS document. What do you notice as you move point A or C?

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I sent the poll again, and we used their results to generalize the relationship between the measure of an inscribed angle and its intercepted arc.

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And then we thought about why.

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We checked again to be sure that everyone was getting what they needed to about central angles, inscribed angles, and intercepted arcs.

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And then we looked at cyclic quadrilaterals. Without me telling them anything, 12 answered correctly before the bell rang.

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And so the next lesson began with the results from this question. Which answer is correct? And why?

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And so the #AskDontTell journey continues … one lesson at a time.

 
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Posted by on February 23, 2015 in Angles & Triangles, Circles, Geometry

 

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My Political Letter

When I asked Google to define political, here’s what I saw:

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I’ve not thought of myself as political before, but apparently, when politicians start making decisions that affect what I’ve given my life to do every day, I will take the time to write a political letter. So here it is – a letter motivated by my beliefs – that will be delivered to the representative and senator for whom I vote – when Mississippi First hosts their Capitol Day today.

 

Dear Representative Campbell and Senator Horhn:

I want to thank you for your continued support of education as shown by your votes – for students, teachers, and parents. I hope that you will vote to keep our current Mississippi College- and Career-Readiness Standards.

I have been teaching high school mathematics for 22 years. My students have always done well on their ACT, SAT, PSAT, and AP Calculus exams. Even so, our new state standards have made a positive impact on the conversations that we have in our classroom as students productively struggle to make sense of mathematics. Our new state standards have made a positive impact on the conversations that I have with other educators across the state as we take a serious look at not only what we have been teaching but also how we have been teaching it.

Our new state standards demand that we graduate students who are proficient at constructing viable arguments and critiquing the reasoning of others. Our new state standards demand that we graduate students who are proficient at making sense of problems and persevering in solving them. Just last week, a student in one of my classes looked up at the Standards for Mathematical Practice posted on the wall in the front of the classroom and remarked, “Those are good practices for life … for any type of learning.”

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Our new state standards have connected me to educators all over the country implementing similar standards. I learn alongside educators not only in Mississippi but also in Washington, Florida, Arizona, and Maryland. We share what works through blogs and tweets. We share what doesn’t work through blogs and tweets. And then we all take back what we learn to our own students and tailor a lesson to fit our learners where they are.

Our new state standards have provided me an opportunity to learn more mathematics than I knew before we had them (and I have an M.S. in mathematics). They demand that we think about multiple ways to solve problems, and so I am constantly learning from my students through their thinking.

I hope that you will give us the opportunity to continue using our new state standards. Give us a chance to show that our students can graduate from high school college- and career-ready.

Sincerely,

Jennifer Wilson

 
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Posted by on February 18, 2015 in Uncategorized

 
 
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