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Tag Archives: CCSS-M Geometry

Triangle Congruence Criteria

CCSS-M G-CO 8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

I wrote last year about spending a class period with my students trying to explain how we could show SSS is true using rigid motions. I was impressed that my students were using the practice make sense of problems and persevere in solving them, however, all in all, the majority of students didn’t successfully come up with a series of rigid motions to show that given three pairs of congruent sides, one triangle was congruent to another.

This lesson was inspired by an Illustrative Mathematics task.

I studied in preparation for this lesson again this year, trying to determine how I could get more students to meet the standard. I am using Usiskin’s Mathematics for High School Teachers – An Advanced Perspective, which I am glad to have from my graduate school days. I only wish I had been able to take the geometry seminar…I studied the first half of the text in a seminar on algebra.

Usiskin starts by proving two segments are congruent if there is a set of rigid motions to map one onto the other. Then proves that two angles are congruent if there is a set of rigid motions to map one onto the other. And then proves that SAS and SSS work.

Reading through his work gives me more confidence in how the standards have been deliberately thought out and sequenced.

CCSS-M 8.G.A.1. Verify experimentally the properties of rotations, reflections, and translations:

a. Lines are taken to lines, and line segments to line segments of the same length.

b. Angles are taken to angles of the same measure.

c. Parallel lines are taken to parallel lines.

We start with lines and segments and angles. And move up to triangles and other figures. It is perfect. Except that my current students didn’t have CCSS-M in grade 8. Eventually we can start with triangles, but I decided that maybe we should start with segments this year instead of jumping straight to SSS.

Well, we actually did start with a conversation about triangles. We have been learning which criteria are sufficient for proving triangles congruent by looking for regularity in repeated reasoning. But I wanted to remind students that we started talking about congruence this year in terms of rigid motions. So I showed a few triangles & asked how we could prove that they were congruent.

It didn’t take students long to recognize that we could show ∆APD ≅ ∆APE by reflecting one triangle about line AP.

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What about these triangles?

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It is easy for students to answer that we can show there is a rotation to map one triangle onto the other. But I wanted them to attend to precision. I wanted them to reason abstractly and quantitatively. What rotation will map one triangle onto the other?

A rotation of ∆APE about A by an angle measure of BAC.

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Or actually by an angle measure of CAB. We learned something about our dynamic geometry software about angle direction and rotation that we didn’t previously know.

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Now to segments. How can we show that segment AB is congruent to segment CD?

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Almost everyone did a translation first. Some translated segment AB by vector BC to get segment CA’.

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Others translated segment AB by vector AC to get segment CB’.

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Everyone had a better understanding of why we can say both that segment AB is congruent to segment CD and that segment AB is congruent to segment DC.

Now that we have translated one of the segments so that one endpoint of one is now mapped to one endpoint of the other, how can we prove that the two segments are congruent?

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A rotation.

What rotation?

At this point in the lesson, TI-Nspire Navigator’s Class Capture feature is invaluable. We have a teacher on maternity leave, and I am currently dealing with 60 students. There is no way I could keep up with what each student is doing. But with Class Capture, I was able to keep taking pictures of what was on each students’ screen to pay attention to interesting approaches. It helps me select which student work to discuss with the whole class. I even used the “Add to Stack” feature to have a record of some of the interesting work in case the student had moved to something else before we had time to discuss it as a class.

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We went next to WA, who had shown that the two segments were congruent. (I could tell because the originally blue-colored segment had a black segment on top of it.) WA shared what she did with us. She rotated segment CA’ about point C by a measure of 41 degrees. How did you know to rotate the segment 41 degrees? I didn’t. I tried a different angle measure first, and that didn’t work, so I edited the angle measure until it worked. Ahh. That is perfect. I couldn’t have chosen a more perfect example for us to learn. So WA has done a great job of reasoning quantitatively. But I want us to move into reasoning abstractly. And here is why. The 41 degrees works now. But it won’t always work.

Another reason for TI-Nspire Navigator. I made WA the Live Presenter. And I asked her to move her original segment. This was the perfect opportunity to begin to understand the difference between the quantitative and the abstract. As soon as WA moved her original segment, the rotated segment no longer mapped to segment CD. 41 degrees was perfect for our original setup. But it doesn’t always work. What angle of rotation can we use to make our mapping always work?

WA

Did everyone use a translation and a rotation to show that the two segments are congruent? BE couldn’t wait to tell us how he used rigid motions differently to show that the two segments were congruent. He used reflections. But I made him wait. I gave everyone a few minutes to work alone to see if they could create a sequence of reflections that would prove the two segments congruent. Then I gave students a few minutes to talk about their solution with a partner. Then BE shared his solution with the whole class.

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Next we moved to SAS. Students had a paper version that they could use to plan their rigid motions.

Paper Copy

They were much better prepared to tackle SAS after our conversation about the segments.

A few examples of student work:

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And the great thing about our dynamic geometry software is that students are able to check their own work. If they can change the original triangle & still have a mapping of the pre-image onto the image, then they have been successful.

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This lesson gives me hope. My students did so much better this year than last. And just think what could happen with students who have been through 8th grade CCSS-M. And so the journey continues …

 
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Posted by on November 10, 2013 in Angles & Triangles, Geometry

 

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Triangle Proofs

CCSS-M Congruence G-CO 10

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

I’m not sure where the idea of inequalities in triangles has fallen with CCSS-M. Maybe it is one of those topics that hasn’t surfaced in our efforts to teach concepts more deeply. Do students need to know that they shortest side is opposite the smallest angle? It is at least helpful to know when solving triangles and evaluating whether solutions are reasonable. So we did a brief exploration with dynamic geometry software and a chart to ensure students know that the smallest angle is opposite the shortest side.

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We didn’t prove the Triangle Inequality Theorem – students have some exposure to it in middle school. But most haven’t thought about why the sum of two sides of the triangle must be larger than the third. So we used dynamic geometry software to show why some side lengths can make a triangle and others can’t and gave student a visual for the “collapsing triangle” that happens when the sum of two side lengths equal the 3rd side.

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And a brief making sense of what values the 3rd side of the triangle can have when two of the side lengths are fixed. (In the diagram, AB=6 cm, and AB=5 cm.)

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My students know that the sum of the measures of the angles of a triangle sum to 180°, but they have not thought about why.

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So how could we prove that the sum of the measures of the angles of a triangle sum to 180°?

What do you know is true in the diagram?

The measures of the angles sum to 180°. Yes – we know that because our teachers told us. But we haven’t proven it yet. If that is what our goal is, how can we get there?

Student struggled for a couple of minutes before a hint. So I am gathering that you are not seeing much in the diagram as it is given that we can use in our proof. Last time, we talked specifically about the practice of look for and make use of structure. You drew auxiliary lines to solve some problems. Are there any auxiliary lines that you can draw for this diagram that might be helpful in proving The Triangle Sum Theorem?

I set the timer again. This time for 3 minutes. And I walked around to look at the auxiliary lines that students were adding to their diagrams. When the timer went off, we shared some of the diagrams with the whole class. Could any of these be helpful in our proof?

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Of course it turns out that drawing in a line parallel to a base of the triangle is helpful. But my students figured that out. I didn’t have to tell them. And even though they all settled on the diagram in the top right for their proof, they didn’t all use exactly the same steps or angles for their argument as to why the sum of the measures of the angles of a triangle is 180°.

During the next few minutes, I heard cries of success from different groups in the class. They had proven The Triangle Sum Theorem using our deductive system. They were proud of themselves. They understand why the sum of the measures of the angles of a triangle is 180° – not just that it is.

We had a short conversation about the implications of our proof. On what does our proof rely? Ultimately, it relies on our Corresponding Angles Postulate, our version of Euclid’s 5th Postulate. Without it, our geometry would be different. Without it, we could be developing spherical geometry or elliptical geometry or hyperbolic geometry. Without it, the sum of the measures of the angles of the triangle could be more than 180° or less than 180°.

—–

We ended the lesson with a Quick Poll.

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Which most students got. How did you get it? I found the sum of the angles and subtracted from 180. Then I subtracted that result from 180. How long did it take you to get that?

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What do you notice in the diagram? Write down your observations.

m∠1 + m∠2 + m∠3 = 180

m∠3 + m∠4 = 180

As we began to list student observations, the students recognized that the exterior angle is equal to the sum of the two remote interior angles. Of course they didn’t use those words exactly. But they concluded that m∠1 + m∠2 = m∠4.

Is this a waste of our time? I don’t think it is. The students are beginning to realize that we have choices in our deductive system, and that those choices affect what is true. The students are beginning to develop arguments as to why one result has to be true and why another result cannot be true.

And so the journey continues …

 
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Posted by on October 28, 2013 in Angles & Triangles, Geometry

 

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Angle Proofs

CCSS-M Congruence G-CO 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.

How do you go about having students prove theorems about lines and angles in your class? We certainly don’t formally prove every theorem in our geometry course that we use, but we prove a lot of them. What we do that I believe is more important than proving theorems is exploring the hypothesis to make sense of possible conclusions. I remember memorizing a lot of theorems out of our geometry textbook when I took high school geometry. I don’t think that is what my students will remember from our class. In fact, I know it is not.

So bear with me as I share two student reflections about the course:

“I have never really liked math and wasn’t looking forward to this class because of it. Surprisingly, I actually like it much better than I thought I would. Although geometry has confused me sometimes this year, I have actually felt proud when I figure something out that I thought I couldn’t. I think I have been able to get a new mindset and have been able to switch from thinking regularly to analytically when I enter your class. I can now look at a problem, and try to think about it in different ways until I figure out how to solve it.”

“The main memory that I will keep of Geometry class is so complex that everything I say will simply be an understatement.  Although I will attempt to describe it.  The main memory I will keep of this class is the memory of the first math class that I enjoyed.  I liked the fact that in that class I sometimes had the bravery to answer questions.  I was not nervous to have the wrong answer in front of the class or Mrs. Wilson.  I loved the fact that Mrs. Wilson incorporated enjoyable activies throughout her lessons. This was the first math class I really engaged myself in.”

So we make a big deal about accepting postulates and proving theorems in our class deductive system for geometry. We looked at a list of Euclid’s axioms and postulates and noted that Euclid’s 5th postulate took way longer to make sense of than any of the others.

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And then we set out to prove theorem #1.

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Vertical angles are congruent. Most of my students know that vertical angles are congruent. But they have not thought about why vertical angles are congruent.

What do we know to be true in this diagram? Students noted that the sum of angles 2 and 3 is 180°. The sum the sum of angles 3 and 4 is 180°. Students noted that the sum of angles 1, 2, 3, and 4 is 360°. How do we know this? We can observe it using technology. We used the angle measurement tool to verify the measures of other angles in the diagram. But how do we know it will always be true? So we proceeded to formalize that vertical angles are congruent … making use of the Angle Addition Postulate that we had agreed to accept into our deductive system without proof. And making use of Euclid’s axioms – if equals are subtracted from equals, the remainders are equal – that we now call the Subtraction Property of Equality. Eventually we proved that angles 2 and 4 were congruent.

Next I sent students a Quick Poll: ∠3 and ∠2 are both supplementary to ∠1. If m∠3=50°, what is m∠2?

And asked them to make a conjecture: If two angles are supplementary to the same angle, then …

And then we proved that if two angles are supplementary to the same angle, then they are congruent to each other..

Given: ∠2 is supplementary to ∠1. ∠3 is supplementary to ∠1.

Prove: ∠2≅∠3

What about the converse?

Is it true?
It didn’t take students long to decide that it wasn’t true. Their counterexample was to suppose m∠2=m∠3=180.

So we moved next to parallel lines, remembering Euclid’s 5th postulate as we began to explore what happens when parallel lines are cut by a transversal.

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What can we conclude about lines a and b? lines m and n?

We agreed that when corresponding angles are congruent, the lines are parallel. Do we agree enough to let this in to our deductive system as a postulate?

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I heard from another geometry teacher that her students thought of corresponding angles as a translation, which goes right along with figures being congruent if there is a rigid motion that maps one onto the other…and a good reason for us to communicate with each other about what we do and say in the classroom. That idea hasn’t come up in my classroom yet, but I will make sure that it does.

And so from there, we proved that alternate interior angles are congruent when parallel lines are cut by a transversal. We didn’t prove that alternate exterior angles are congruent, since the proof is similar. We proved that consecutive interior angles are supplementary when parallel lines are cut by a transversal. We didn’t prove that consecutive exterior angles are supplementary, since the proof is similar.

And then on the summative assessment, I asked the following, which I believe gave evidence that students are beginning to learn to create arguments and they are not just memorizing proofs to recreate on a test.

Suppose you are structuring your deductive system in such a way that you postulate the following about alternate interior angles:

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

Prove the Corresponding Angles Theorem using the Alternate Interior Angles Postulate and without using results from any parallel line theorems that we proved in class.

Our last part of parallel lines was specifically to think about auxiliary lines. The progression of questions follows.

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I am not convinced that I did them in the right order, although maybe it is significant that all but one student got the 3rd problem correct. Should we have started with the 3rd problem instead of the 1st? Since we did start with the 1st, the effect was subtle for problem #3. Many students drew immediately drew auxiliary lines when they saw the 3rd problem. After the 3rd problem, I drew my students’ attention to the math practice look for and make use of structure, which explicitly states drawing auxiliary lines to support an argument. Student shared their thinking, as their arguments for finding the measure of angle 3 were not all the same.

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Are we wasting our time proving a few of these theorems in class? We don’t write formal two-column proofs for every theorem that we prove, but we do come up with a logical argument as to why the theorems are true. What we do is reason abstractly and quantitatively. What we do is construct a viable argument and critique the reasoning of others. My students recognize that we are figuring out geometry instead of being told geometry, whether they appreciate it yet or not.

And so the journey continues …

 
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Posted by on October 28, 2013 in Angles & Triangles, Geometry

 

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Exploring the Equation of a Circle

CCSS-M G-GPE-1

Expressing Geometric Properties with Equations G-GPE

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.

CCSS-M 8.G.8

Understand and apply the Pythagorean Theorem.

8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

We used the Geometry Nspired activity Exploring the Equation of a Circle to begin our exploration of circles.

We also used some ideas from the Mathematics Assessment Project formative assessment lesson Equations of Circles 1.

The TNS document begins by having students observe what they know about the given triangle.

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It is a right triangle.

As students move point P, what happens?

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The triangle is still right.

The hypotenuse stays 5.

The legs change length depending on the location of P.

Some students might say that a2+b2=52, if we let a and b represent the legs of the right triangle.

Then we do a geometry trace of P as we move P.

What path does P follow?

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If we let x represent the length of the horizontal length of the leg and y represent the vertical length of the leg, then we can say that x2+y2=52 for this circle. Alternatively, if we let (x,y) represent the coordinates of point P, then we can say that x2+y2=52. Then we explored what happens as we make the radius of the circle shorter and longer.

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After exploring the equation of a circle centered at the origin, we translate the center in the coordinate plane. Now what can we say about the right triangle that is pictured?

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After the exploration, we used the sorting activity in the Mathematics Assessment Project’s formative assessment lesson.

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And then TI-Nspire Navigator provided a good opportunity for formative assessment – and for students to attend to precision.

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My students left class not only with an understanding of how the Pythagorean Theorem is related to the Distance Formula and the Equation of a Circle, but they also got some good practice attending to precision through the formative Quick Poll that I sent and by categorizing circle equations. This was a much better lesson than I have had in previous years of teaching the equation of a circle.

And hopefully next year will be even better as the journey continues …

 
 

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Hopewell Geometry – Right Triangle

A while back we gave our students the Mathematics Assessment Project assessment task called Hopewell Geometry.

I have just finished reading Smith and Stein’s book 5 Practices for Orchestrating Productive Mathematics Discussions, and so I have been thinking a lot about sequencing.

Students are given a set of Hopewell Triangles (along with a historical explanation, which you can see at the link above).

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And they are given a diagram with the layout of some Hopewell earthworks.

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The second question is for students to explain whether or not the shaded triangle is a right triangle.

With which student explanation would you start in a class discussion?

How would you sequence the student explanations? Are there any you would be sure to include? Some you would leave out?

Student A

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Student B

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Student C

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Student D

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Student E

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Student F

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Student G

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Student H

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Student I

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Student J

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Student K

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Student L

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And so the journey continues …

 
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Posted by on June 18, 2013 in Geometry, Right Triangles

 

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Hopewell Geometry – Similarity

A while back we gave our students the Mathematics Assessment Project assessment task called Hopewell Geometry.

I have just finished reading Smith and Stein’s book 5 Practices for Orchestrating Productive Mathematics Discussions, and so I have been thinking a lot about sequencing.

Students are given a set of Hopewell Triangles (along with a historical explanation, which you can see at the link above).

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And they are given a diagram with the layout of some Hopewell earthworks.

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The first question is for students to explain which triangle is similar to Triangle 1. Some student responses are below. With which student explanation would you start in a class discussion? How would you sequence the student explanations? Are there any you would be sure to include? Some you would leave out?

Student A

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Student B

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Student C

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Student D

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Student E

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Student F

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Student G

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Student H

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Student I

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Student J

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Student K

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Student L

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Student M

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Student N

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Student O

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Student P

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I have to say that the most thrilling responses are those that justify the similarity of the triangles through dilations and scale factor. Teaching CCSS-M Geometry this year has forced me to change how we talk about similarity and congruence. And while we still discuss similarity postulates such as AA~, SSS~, SAS~ and congruence postulates such as SSS, SAS, ASA, our focus has been on talking about congruence of figures through rigid motions – and similarity of figures through a dilation and if needed, rigid motions. I will be even more comfortable having the transformational geometry-congruence-similarity discussions next year than I was this year. And I will be even more comfortable with sequencing the student work in our classroom discussion so that students can make connections between the different ways to justify similarity of figures.

And so I look forward in great anticipation as the journey continues …

 
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Posted by on June 18, 2013 in Dilations, Geometry, Right Triangles

 

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