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Tag Archives: Geometry Nspired

Notice and Note: The Equation of a Circle

I wrote in detail last year about how our students practice I can look for and express regularity in repeated reasoning to make sense of the equation of a circle in the coordinate plane.

This year we took the time not only to notice what changes and what stays the same but also to note what changes and what stays the same.

Our ELA colleagues have been using Notice and Note as a strategy for close reading for a while now. How might we encourage our learners to Notice and Note across disciplines?

Students noticed and noted what stays the same and what changes as we moved point P.

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They made a conjecture about the path P follows, and then we traced point P.

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We connected their noticings about the Pythagorean Theorem to come up with the equation of the circle.

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Students moved a circle around in the coordinate plane to notice and note what happens with the location of the circle, size of the circle, and equation of the circle.

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And then most of them told me the equation of a circle with center (h,k) and radius r, along with giving us the opportunity to think about whether square of (x-h) is equivalent to the square of (h-x).

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And so the journey continues … with an emphasis on noticing and noting.

 

 

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Posted by on March 19, 2016 in Circles, Geometry

 

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The Equation of a Circle

Expressing Geometric Properties with Equations

G-GPE.A Translate between the geometric description and the equation for a conic section

  1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.

Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?

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It’s a right triangle.

The hypotenuse is always 5.

The legs change.

What else do you notice? What has to be true for these objects?

The Pythagorean Theorem works.

How?

Leg squared plus leg squared equals five squared.

What do you notice about the legs? How can we represent the legs on the graph?

One leg is always horizontal.

One leg is always vertical.

How can we represent their lengths in the coordinate plane?

x and y?

(I think they thought that the obvious was too easy.)

What do x and y have to do with point P?

Oh! They’re the x- and y-coordinates of point P.

So what can we say is always true?

Is there an equation that is always true?

x²+y²=5²

What path does P travel? (This was preceded by – I’m going to ask a question, but I don’t want you to answer out loud. Let’s give everyone time to think.)

And then we traced point P as we moved it about coordinate plane.

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So P makes a circle, and we have figured out that the equation of that circle is x²+y²=5².

I then let them explore two other pages with their teams, one where they could change the radius of the circle and one where they could change the center of the circle.

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And then they answered a few questions about what they found. I used Class Capture to watch as they practiced look for and express regularity in repeated reasoning.

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Here are the results of the questions that they worked.

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What would you do next?

What I didn’t do at this point was differentiate my instruction. It occurred to me as soon as I got the results that I should have had a plan of what to do with the students who got 1 or 2 questions correct. It turns out that it was a team of students – already sitting together – who needed extra support – but I didn’t figure that out until later. Luckily, my students know that formative assessment isn’t just for me, the teacher – it’s for them, too. They share the responsibility in making a learning adjustment before the next class when they aren’t getting it.

We pressed on together – to make more sense out of the equation of a circle. I used a few questions from the Mathematics Assessment Project formative assessment lesson, Equations of Circles 1, getting at specific points on the circle.

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And then I wondered whether we could begin making a circle. I assigned a different section of the x-y coordinate plane to each team. Send me a point (different from your team member) that lies on the circle x²+y²=64. Quadrant II is a little lacking, but overall, not too bad.

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How can we graph the circle, limited to functions?

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How can we tell which points are correct?

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I asked them to write the equation of a circle given its center and radius, practicing attend to precision.

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54% of the students were successful. The review workspace helps us attend to precision as well, since we can see how others answered.

(At the beginning of the next class, 79% of the students could write the equation, practicing attend to precision.)

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I have evidence from the lesson that students are building procedural fluency from conceptual understanding (one of the NCTM Principles to Actions Mathematics Teaching Practices).

But what I liked best is that by the end of the lesson, most students reached level 4 of look for and express regularity in repeated reasoning: I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

When I asked them the equation of a circle with center (h,k) and radius r, 79% told me the standard form (or general for or center-radius form, depending on which textbook/site you use) instead of me telling them.

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We closed the lesson by looking back at what happens when the circle is translated so that its center is no longer the origin. How does the right triangle change? How can that help us make sense of equation of the circle?

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And so the journey continues, one #AskDontTell learning episode at a time.

 
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Posted by on April 19, 2015 in Circles, Coordinate Geometry, Geometry

 

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Classifying Triangles

We look specifically at 45-45-90 triangles on the first day of our Right Triangles unit. I’ve already written specifically about what the 45-45-90 exploration looked like, but I wanted to note a conversation that we had before that exploration.

Jill and I had recently talked about introducing new learning by drawing on what students already know. I’ve always started 45-45-90 triangles by having students think about what they already know about these triangles (even though many have never called them 45-45-90 triangles before). After hearing about one of Jill’s classes, though, I started by asking students to make a column for triangles, right triangles, and equilateral triangles, noting what they know to always be true for each. This short exercise gave students the opportunity to attend to precision with their vocabulary.

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It occurred to me while we were talking that having students draw a Venn Diagram to organize triangles, right triangles, and equilateral triangles might be an interesting exercise. How would you draw a Venn Diagram to show the relationship between triangles, right triangles, and equilateral triangles?

In my seconds of anticipating student responses, I expected one visual but got something very different.

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What does it mean for an object to be in the intersection of two sets? Or the intersection of three sets? Or in the part of the set that doesn’t intersect with the other sets?

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Then we thought specifically about 45-45-90 triangles. What do you already know? Students practiced look for and make use of structure.

One student suggested that the legs are half the length of the hypotenuse. Instead of saying that wouldn’t work or not writing it on our list, I added it to the list and then later asked what would be the hypotenuse for a triangle with legs that are 5.

10.

I wrote 10 on the hypotenuse and waited.

But that’s not a triangle?

What?

5-5-10 doesn’t make a triangle.

Why not?

It would collapse (students have a visual image for a triangle collapsing from our previous work on the Triangle Inequality Theorem).

Does the Pythagorean Theorem work for 5-5-10?

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Students reflected the triangles about the legs and hypotenuse to compose the 45-45-90 triangle into squares and rectangles.

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And they constructed an altitude to the hypotenuse to decompose the 45-45-90 triangle into more 45-45-90 triangles.

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And then we focused on the relationship between the legs and the hypotenuse using the Math Nspired activity Special Right Triangles.

And so the journey continues … listening to and learning alongside my students.

 
 

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Special Right Triangles: 45-45-90

I gave my students our learning progression for SMP 8 a few weeks ago as we started a unit on Right Triangles and had a lesson specifically on 45-45-90 Special Right Triangles.

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The Geometry Nspired Activity Special Right Triangles contains an Action-Consequence document that focuses students attention on what changes and what stays the same. The big idea is this: students take some kind of action on an object (like grabbing and dragging a point or a graph). Then they pay attention to what happens. What changes? What stays the same? Through reflection and conversation, students make connections between multiple representations of the mathematics to make sense of the mathematics.

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Students start with what they know – the Pythagorean Theorem.

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Looking at the side lengths in a chart helps students notice and note what changes and what stays the same:

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The legs of the triangle are always the same length.

As the legs increase, the hypotenuse increases.

The hypotenuse is always the longest side.

 

Students begin to identify and describe patterns and regularities:

All of the hypotenuses have √2.

The ratio of the hypotenuse to the leg is √2.

 

Students practice look for and express regularity in repeated reasoning as they generalize what is true:

To get from the leg to the hypotenuse, multiply by √2.

To get from the hypotenuse to the leg, divide by √2.

hypotenuse = leg * √2

Teachers and students have to be careful with look for and express regularity in repeated reasoning. Are we providing students an opportunity to work with diagrams and measurements that make us attend to precision as we express the regularity in repeated reasoning that we notice?

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In a Math Practice journal, Kaci writes about “look for regularity in repeated reasoning”. We figured out that half of a square is a 45-45-90 triangle, and students were trying to determine the other two sides of the triangle given one side length of the triangle. She says “To find the length of the hypotenuse, you take the length of a side and multiply by √2. The √2 will always be in the hypotenuse even though it may not be seen like √2. In her examples, the triangle to the left has √2 shown in the hypotenuse, but the triangle to the right has √2 in the answer even though it isn’t shown, since 3√2√2 is not in lowest form. She says, “I looked for regularity in repeated reasoning and found an interesting answer.”

What opportunities can we provide our students this week to look for and express regularity in repeated reasoning and find out something interesting?

 
 

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Trig Ratios

Trig Ratios

How do your students experience learning right triangle trigonometry? How do you introduce sine, cosine, and tangent ratios to them?

NCTM’s Principles to Actions includes build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. In what ways can technology help us help our students build procedural fluency from conceptual understanding?

Until I started using TI-Nspire Technology several years ago, right triangle trigonometry is one topic where I felt like I started and ended at procedural fluency. How do you get students to experience trig ratios?

I’ve been using the Geometry Nspired activity Trig Ratios ever since it was published. Over the last year, I also read posts from Mary Bourassa: Calculating Ratios and Jessica Murk: Building Trig Tables about learning experiences for making trigonometric ratios more meaningful for students. Here’s how this year’s lesson played out …

We first established a bit of a need for something called trig (when they finally get to learn about the sin, cos, and tan buttons on their calculator that they’ve not known how to use). I showed a diagram and asked how we could solve it. We reserved “trig” for something they couldn’t yet solve.

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We use TI-Nspire Navigator with our TI-Nspire handhelds, and so I can send Quick Polls to assess where students are. Sometimes Quick Polls aren’t actually so “quick”, but these were, along with letting students think about what we already know and uncovering a few misconceptions along the way (25 isn’t the same thing as 18√2).

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Next I asked each student to construct a right triangle with a 40˚ angle and measure the sides of the triangle.

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I sent a Quick Poll to collect their measurements.

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Then we looked at the TNS document for Trig Ratios. Students can take multiple actions on the diagram. I asked them to start by moving point B. What do you notice? We recorded their statements for our class notes.

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Then I asked them to click on the up and down arrows of the slider. What do you notice?

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What ratio of side lengths is used for the sine of an angle?

You all constructed a right triangle with a 40˚ angle and recorded the measurements. What’s true about all of your triangles?

  • The triangles are all similar because the angles are congruent.
  • The corresponding side lengths are proportional.
  • We know that sin(40˚) is always the same.
  • So the opposite leg over the hypotenuse will be the same?

Will it? We sent their data to a Lists & Spreadsheet page and calculated a fourth column, opp_leg/hyp. What do you notice?

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Of course their ratios aren’t exactly the same, but that’s another good discussion. They are close. And students noticed that one entry has the opposite leg and adjacent leg switched because the leg opposite 40˚ is shorter than the leg opposite 50˚.

We didn’t spend long looking at the TNS pages for tangent and cosine … students were well on their way to understanding a trig ratio conceptually. They just needed to establish which side lengths to use for cosine and which to use for tangent.

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There’s a reason that #AskDontTell has been running through my mind as I have conversations with my students and reflect on them. Jill Gough wrote a post using that hashtag over two years ago: Circle Investigation – #AskDontTell.

What #AskDontTell opportunities can you provide your students this week?

[Cross-posted at T3 Learns]

 
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Posted by on March 12, 2015 in Geometry, Right Triangles

 

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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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30-60-90 Triangles

In our 30-60-90 triangle lesson, we first find out what students already know about 30-60-90 triangles. We deliberately let them practice look for and make use of structure. Some students compose the triangle into an equilateral triangle and note that one side is double the length of the other (we eventually attended to precision to note that the hypotenuse is double the length of the shorter leg).

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Some students rotated the triangle 180˚ about the midpoint of the hypotenuse; others rotated it 180˚ about the midpoint of the longer leg.

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Other students decomposed the triangle by drawing the altitude to the hypotenuse and noted that they formed two additional 30-60-90 triangles.

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Our lesson (partially from the Geometry Nspired activity Special Right Triangles) provided the opportunity for students to look for and express regularity in repeated reasoning using the equilateral triangle. What changes and what stays the same as you grab and move point B?

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Recording side lengths in a table also provided the opportunity for students to look for and express regularity in repeated reasoning. But after our 45-45-90 lesson the day before, I thought it would be okay to skip that step and move to #3.

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As soon as we recorded some of the student responses to #3, I realized I had made a mistake. They weren’t all getting that the longer leg is the shorter leg times √3. I had tried to rush to the result instead of giving students the time to notice when calculations are repeated, to evaluate the reasonableness of intermediate results, and to look for general methods and shortcuts.

We took a step back. They all used the Pythagorean Theorem to determine the altitude for an equilateral triangle with a side length of 10, and then in the little time remaining we used our technology to confirm the result and help us generalize the relationship between the side lengths of a 30-60-90 triangle.

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And so the journey continues … learning more each day that providing students deliberate learning episodes steeped in using the Math Practices is much more effective than having them haphazardly figure out the math.

 
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Posted by on February 7, 2015 in Angles & Triangles, Geometry, Right Triangles

 

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