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.

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

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

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.

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).

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

 

 

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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?

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.

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.

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.

Here are the results of the questions that they worked.

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.

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.

How can we graph the circle, limited to functions?

How can we tell which points are correct?

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

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.)

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.

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?

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

 

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.

  

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.

  

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?

   

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?

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

 

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

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.

 

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.

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.

 

Students start with what they know – the Pythagorean Theorem.

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

 

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?

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?

 

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.

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).

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

I sent a Quick Poll to collect their measurements.

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.

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

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?

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.

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]

 

Inscribed Angles

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.
3. (+) 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?

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.

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.

About one-third of the students intuited the relationship.

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?

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.

And then we thought about why.

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

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

 

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

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

 

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).

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

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

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?

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.

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.

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.

 

Midpoint Quadrilaterals

CCSS-M G-CO.C.11

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

While I am not exactly certain what “and conversely” modifies in this standard, I do want my students to think about not only the necessary conditions for naming a figure a parallelogram but also the sufficient conditions.

Our learning goals for the unit on Polygons include the following I can statement:

I can determine sufficient conditions for naming special quadrilaterals.

I’ve sent Quick Polls before asking students to determine whether the given information is sufficient for naming the figure a parallelogram.

Luckily the teachers with whom I work have kindly let me know how pathetic the questions are, and so I no longer send them. So how can we get students to determine the sufficient information for naming a figure a parallelogram without giving them the list from their textbook to use and memorize?

I started this lesson by showing three (pathetically drawn) figures with some given information and sending a poll for them to mark each figure that gives sufficient information for a parallelogram (more than one, if needed). Granted it’s only a bit better than the Yes/No Quick Polls, but it is better, and it did give students more opportunity to construct a viable argument and critique the reasoning of others than the one-at-a-time polls.

For an item like this, I especially like showing students the results without showing the correct answer, as that leaves room for even more conversation about math.

Next I asked them to construct a non-special quadrilateral and then its midpoint quadrilateral.

(Yes, Connor your polygon can be concave.)

What do you notice?

It’s a parallelogram.

How do you know?

I blog to reflect on my practice in the classroom. And so what I know now is that I should have asked students to measure and/or construct auxiliary lines using a sufficient amount of information to show that their midpoint quadrilateral was a parallelogram. Everyone wouldn’t have measured the exact same parts, and I could have used Class Capture to select students to present their information to the class. But I didn’t think of that during the lesson. The students played with their construction, some recognizing that the midpoint quadrilateral is a parallelogram no matter how they arranged their original vertices.

Others recognizing that every successive midpoint quadrilateral would also be a parallelogram.

And none connecting what we had done at the beginning of the lesson with what we were doing now.

And none proving why the figure had to be a parallelogram. I feel like the proof of why should come after we study dilations. But I like students figuring out that the figure is a parallelogram during our unit on polygons.

So maybe, eventually, we will move dilations earlier in the course.

Or maybe we can revisit the why-they-are-parallelograms after or during the dilations unit.

Either way, I’m grateful for a do-over next year as the journey continues …

Or maybe we can revisit the why after or during the dilations unit.

Either way, I’m grateful for a do-over next year as the journey continues …

 

Using Rigid Motions for Parallel Lines Angle Proofs

CCSS-M.G-CO.C.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

After proving that vertical angles are congruent, we turned our attention towards angles formed by parallel lines cut by a transversal.

My students come to high school geometry having experience with angle measure relationships when parallel lines are cut by a transversal. But they haven’t thought about why.

We make sense of Euclid’s 5^th Postulate (wording below from Cut the Knot):

If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

We use dynamic geometry software to explore Parallel Lines and Transversals:

And then traditionally, we have allowed corresponding angles congruent when parallel lines are cut by a transversal as the postulate in our deductive system. It makes sense to students that the corresponding angles are congruent. Then once we’ve allowed those, it’s not too bad to prove that alternate interior angles are congruent when parallel lines are cut by a transversal.

But we wonder whether we have to let corresponding angles in as a postulate. Can we use rigid motions to show that the corresponding angles are congruent?

One student suggested constructing the midpoint, X, of segment BE. Then we created a parallel to lines m and n through X. That didn’t get us very far in showing that the corresponding angles are congruent. (image on the top left)

Another student suggested translating line m using vector BE. So we really translated more than just line m. We really translated the upper half-plan formed by line m. We used took a picture of the top part of the diagram (line m and above) and translated it using vector BE. We can see in the picture on the right, that m maps to n and the transversal maps to itself, and so we conclude (bottom left image) that ∠CBA is congruent to ∠DEB: if two parallel lines are cut by a transversal, the corresponding angles are congruent.

Once corresponding angles are congruent, then proving alternate interior (or exterior) angles congruent or consecutive interior (or exterior) angles supplementary when two parallel lines are cut by a transversal follows using a mix of congruent vertical angles, transitive and/or substitution, Congruent Supplements.

But can we prove that alternate interior angles are congruent when parallel lines are cut by a transversal using rigid motions?

Several students suggested we could do the same translation (translating the “top” parallel line onto the “bottom” parallel line). ∠2≅∠2’ because of the translation (and because they are corresponding), and we can say that ∠2’≅∠3 since we have already proved that vertical angles are congruent. ∠2≅∠3 using the Transitive Property of Congruence. We conclude that when two parallel lines are cut by a transversal, alternate interior angles are congruent.

Another team suggested constructing the midpoint M of segment XY (top image). They rotated the given lines and transversal 180˚ about M (bottom image). ∠2 has been carried onto ∠3 and ∠3 has been carried onto ∠2. We conclude that when two parallel lines are cut by a transversal, alternate interior angles are congruent.

Another team constructed the same midpoint as above with a line parallel to the given lines through that midpoint. They reflected the entire diagram about that line, which created the line in red. They used the base angles of an isosceles triangle to show that alternate interior angles are congruent.

Note 1: We are still postulating that through a point not on a line there is exactly one line parallel to the given line. This is what textbooks I’ve used in the past have called the parallel postulate. And we are postulating that the distance between parallel lines is constant.

Note 2: We haven’t actually proven that the base angles of an isosceles triangle are congruent. But students definitely know it to be true from their work in middle school. The proof is coming soon.

Note 3: Many of these same ideas will show that consecutive (or same-side) interior angles are supplementary. We can use rigid motions to make the images of two consecutive interior angles form a linear pair.

After the lesson, a colleague suggested an Illustrative Mathematics task on Congruent angles made by parallel lines and a transverse, which helped me think through the validity of the arguments that my students made. As the journey continues, I find the tasks, commentary, and solutions on IM to be my own textbook – a dynamic resource for learners young and old.

 

A-S-N-T-F

In geometry, we often use Always, Sometimes, or Never:

A trapezoid is ___ a parallelogram.

A parallelogram is ___ a trapezoid.

(Be careful how you answer those if you are using the inclusive definition of trapezoid.)

And in geometry, we often use True (A) or False (S or N):

A trapezoid is a parallelogram.

A parallelogram is a trapezoid.

(Apparently we were talking about squares and trapezoids, not parallelograms and trapezoids, when we were figuring out which had to be true.)

And in geometry, we often use implied True (A):

So when a few students asked about this question, we asked whether you could draw any parallelogram that doesn’t have four right angles. Since you can, we don’t say that the statement is (A) true.

[Note: the green marks indicate the number of students who answered both and only rectangle and square.]

In geometry, we are still learning the implications of the inclusive definition of trapezoid. Another of our questions was the following.

And thank goodness, in geometry, I have students who question me, even those with a voice so quiet we have to lean in to hear. “But I can draw a trapezoid that doesn’t have exactly one pair of parallel sides. I don’t think the trapezoids should be marked correct.”

And of course, he is right. In our deductive system, we don’t name any quadrilaterals with exactly one pair of parallel sides.

So how could I recover our lesson and be sure that my students understood both what we mean by (A) true and our inclusive definition of trapezoid?

I asked students to look ahead to a graphic organizer (borrowed from Mr. Chase, who borrowed from mathisfun.com), review it, and answer the new question. I took the question from the opener and changed “exactly one” to “at least one”. I asked students to work alone.

Here’s what I got back.

Without showing them the results, I asked students to talk with their teams and answer the poll one last time. All 31 students answered correctly.

So what next?

I’ve been determined over the past three years to stay away from the quadrilateral checklist. You remember the one, right? This is mine from the first 18 years of teaching geometry. I didn’t complete the list for them – each team had a different figure, and they measured (with rulers and protractors before we had the technology with measurement tools) and figured out which properties were always true. But still – how effective is it to complete a checklist, even when you and your classmates are figuring it out?

We wanted to get at how the quadrilaterals are the same and how they are different in a way that was more engaging than just showing a few figures and asking students to calculate a missing measurement.

So another Quick Poll to get the conversation going. Students immediately began talking with their teams. For which figure(s) will the one angle measure be enough for us to determine the remaining interior angle measures?

And decent results.

We went to figure B. Why isn’t one angle measure enough?

And then figure C. Why isn’t one angle measure enough?

And then figure D. Why is one angle measure enough?

Student justifications included words like “rotation”, “reflection”, “decompose into triangles”, “isosceles triangle”. We talked about how we knew the triangles in the kite pictured were not congruent, and in fact not similar either, when decomposed by the horizontal diagonal. Informal justifications … but justifications, nonetheless … and hopefully ammunition for students to realize they can make sense out of these exercises transformationally without having a list of properties for each figure memorized.

We spent a little more time on rhombi using a Math Nspired document for exploration, after which I sent another Quick Poll:

How did ten students get 144˚?

The students figured out the error: a misreading of which angle is 36˚ … not a misunderstanding of angle measure relationships in rhombi.

And then more about kites using the same Math Nspired activity, during which time a student asked to be made the Live Presenter so that he could show his concave kite to the class. What properties do concave and convex kites share? (More than I expected. I’m not the what-can-I-do-to-break-the-rule kind of person. But I am surrounded by students and daughters who are.) And I am still amazed that SC asked to be the Live Presenter since that was usually the time that he excuses himself to go to the restroom.

So what information is enough angle-wise in the kite for you to determine the rest?

We ended with a bit of closure with two final Quick Polls & results that provide evidence of student learning.

And so the journey continues … always rethinking and revising lessons and questions to get the most out of our time and conversations together.