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SMP7 – The Triangle Sum Theorem

G-CO.C.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.

How do you provide opportunities for your students to look for and make use of structure? I’m finding that deliberate practice in looking for and making use of structure is making the practice a habit for my students.

SMP7 #LL2LU Gough-Wilson

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

We practice: “contemplate before you calculate”.

We practice: “look before you leap”.

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

We persevere.

We learn from each other.

 

Months ago, our goal was to prove the Triangle Sum Theorem.

We thought first about what we already knew … what we had already added to our deductive system.

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Then we practiced “I can look for and make use of structure”.

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

 
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Posted by on July 1, 2016 in Angles & Triangles, Geometry

 

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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 5th 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.

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

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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)

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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.

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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.

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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.

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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.

 
 

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Rigid Motions: Translations

What do you need for a translation?

An object

How far right/left, how far up/down

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In our high school geometry class we can use a directed segment or vector to indicate how far right/left and how far up/down.

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How does vector XY tell us how far right/left and up/down?

Students had the opportunity to look for and make use of structure.

We talked about a right triangle with segment XY as its hypotenuse. The horizontal and vertical legs tell us how far right/left and how far up/down.

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When we translate a triangle using a given segment (or vector), what is congruent?

Students made a list of congruent objects. They shared their list with their teams, and then we discussed with the whole class.

 

The triangles are congruent. Because one is a translation of the other.

The corresponding segments are congruent. Because the triangles are congruent.

The corresponding angles are congruent.

What else is congruent?

The distance from C to C’ is the same as the distance from B to B’.

What else is congruent?

CC’=BB’=AA’

(Yes. I know that I am interchanging equality and congruence. I actually used to spend time specifically teaching notation in geometry. Now students learn notation by observation.)

What else is congruent?

CC’=XY

CX=C’Y

What is CXYC’?

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We ended the lesson with a triangle that had been translated. How can you show that one triangle is a translation of the other?

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One student noted aloud that we could show that the triangles are congruent.

Is showing the triangles are congruent necessary for proving that one triangle is a translation of the other?

Is showing the triangles are congruent sufficient for proving that one triangle is a translation of the other?

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What information is sufficient for proving that one triangle is a translation of the other?

Is it enough to connect A to A’, B to B’, and C to C’?

What must be true about those segments?

 

And so the journey continues …

 
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Posted by on August 29, 2014 in Geometry, Rigid Motions

 

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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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Transformations with Matrices

We finished up our unit on Rigid Motions by taking a brief look at transformations using matrices. At the beginning of class, we made sure each student could transform a given point in the coordinate plane and generalize the transformation.

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Our students do not have any experience multiplying matrices. I didn’t want to get into a huge lesson on multiplying matrices, but I also wanted my students to realize that they had the background they needed to make sense of transforming triangles using matrices.

We used TI-Nspire technology to look for regularity in repeated reasoning. We started by multiplying the matrix that represented the vertices of our triangle (1,4), (4,–3), and (–2,5) by the matrix shown to observe how matrix multiplication works.

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By what matrix should we multiply if we want the output to be the vertices of the triangle after it has been reflected about the x-axis?

It took NR only two tries before she got the correct transformation matrix. It took others more than two tries. But once students had the matrix for a reflection about the x-axis, it didn’t take long to get the matrix for a reflection about the y-axis.

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And then reflections about the lines y=x and y=-x.

 

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And then rotations 90 degrees and -90 degrees about the origin.

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I honestly do not care that students remember the matrix that produces a certain transformation. I am not even sure that I should have spent any class time on this topic (although I have seen questions on some state assessments about this topic). What I care about is that students know that they can make sense of problems and persevere in solving them. I care that they know that they can figure out the matrix that produces a certain transformation instead of me giving them a list to memorize.

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And so the journey to create students who can figure out mathematics instead of being told mathematics continues …

 
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Posted by on September 28, 2013 in Coordinate Geometry, Geometry, Rigid Motions

 

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Constructing a Hexagon

Last week, I gave my students the following task, which I first heard about from an instructor at the University Lab School in Honolulu.

Construct a regular hexagon with segment AB as one of its sides.

-You may not use any Shapes tools.

-You many not use any Measurement tools.

-When you are finished, we can use Measurement tools to justify your construction.

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How would you construct a regular hexagon without knowing the formal steps for the construction of hexagon with a compass and straightedge?

We were finishing up a unit on Rigid Motions, so the hope was that the students would think about the rigid motions that would map one side of a regular hexagon to another side of a regular hexagon.

A few students asked about the measures of angles in the hexagon. Instead of answering directly, I drew two auxiliary objects to see if that would help them determine the measure of an interior angle of a regular hexagon.

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It helped. I didn’t have to tell anyone the angle measure.

While students were working, I monitored their progress using the Class Capture feature of TI-Nspire Navigator. I selected which hexagons we should discuss as a whole class. I sequenced the order in which I called on students.

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Most students started with the rotation of a segment. And by far the majority of the students did the entire construction using rotations. Some students were surprised when they rotated segment AB 120° about point B. Segment BA’ appeared below segment AB. Why did that happen?

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Some students rotated segment AB 120° about point A. Segment AB’ appeared above segment AB. Why did that happen?

 

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Some students realized that they could use the same degree measure for each of they rotations.

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Some students decided to hide the degree measure when they finished to make their construction “look better”.

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The big deal was that we were able to grab and move point A or point B at the end of the construction and preserve the hexagon.

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A few students made use of the symmetry of the hexagon. P.R. rotated segment AB -120° about point B to get BA’. He rotated segment AB 120° about point A to get AB’. Then he connected A’B’ to create one of the lines of symmetry of the hexagon. He reflected all three segments about line A’B’ to get the rest of the hexagon.

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H.J. rotated segment AB -120° about point B to get BA’. Then she found constructed a perpendicular through the midpoint of segment AB. She reflected segment BA’ about the perpendicular bisector of segment AB to get segment AA’’. Then she constructed a line parallel to line AB through A’. She reflected segments AB, BA’ and AA’’ about that parallel line to get the rest of her hexagon.

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I am glad for students who see things differently than most of us. And I am even more glad that they are willing to share what they see with us.

And so the journey continues …. 

 
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Posted by on September 15, 2013 in Geometry, Polygons, Rigid Motions, Tools of Geometry

 

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Mapping an Image Onto Itself

Last year I posted specifically about standard G-CO 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

I wanted to share one of the questions that we put on the performance assessment for this unit.

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Many students created an appropriate polygon for this task. But I want them to learn not just to create – but to also construct. We are using dynamic geometry software on purpose. Can your polygon dynamically map onto itself?

Some could.

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And some could not.

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Which is yet another indication that the journey continues…

 
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Posted by on September 15, 2013 in Geometry, Rigid Motions

 

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I have learned ….

A few years ago, upon Jill Gough’s recommendation, I read How the Brain Learns Mathematics by David Sousa. In 2012, we got to hear Sousa present the opening session at the T^3 International Conference in Chicago. Jill writes briefly about the session here.

I still make sure teachers with whom I work know about the Primacy-Recency Effect, which shows that in a learning episode, we learn best what we learn first. We learn second best what we learn last, and we learn least what is in the middle. Sousa’s research suggests that 20-minute learning episodes offer the most “prime time” learning, with the least amount of down time for students. So even in our 90 minute block periods, we try to focus on 20-minute learning episodes.

Jill writes about a 20-minute experiment in which she participated at The Westminster Schools a few years ago.

I try to provide students some cognitive process every 20 minutes of class so that they can think about what they are learning and what their questions are. I don’t formally ask that question every class period, but as we are coming to the end of our unit on Rigid Motions, I asked students to reflect on what they have learned and what their questions are. The actual prompt was:

Name one thing you have learned in this unit on transformations that you did not know prior to this year. List a question that you still have about transformations that you need to get answered before your summative assessment.

I exported the student responses so that they were easy to organize. I’ve included a few below.

I have learned how to generalize transformations.

My question is is there any easier way to do a rotation.  1

 

I have learned how to rotate a point either clockwise or counterclockwise.

My question is how do you remember which direction y=x and x=y go?          1

 

I have learned why shapes are congruent, not because they look the same but because they can be mapped using rigid motion.

My question is nothing.        1

 

I have learned that a shape does not have only one way of deciding if it is congruent.

My question is…What is the easiest way to map a rotation without memorizing the (x,y) things we learned           1

 

I have learned to rotate a point.

My question is what is orientation. 1

 

I have learned how to rotate the point around the orgin.

My question is           1

 

I have learned . that more things can be congruent than just size and shape

My question is . how do you find rules for rigid motions 1

 

I have learned what vectors are and more information about transformations.

My question is are there different types of vectors.         1

 

I have learned what was orientation.

My question is can a vector be used for anything else?   1

 

I have learned how to use vectors correctly.

My question is without a grid, how would you accurately use the vector? Would you just move it to a relative location?  1

 

I have learned how to reflect over lines y=a, x=b, x=0, and y=0.

My question is-Why do we need a question?        1

 

The responses that I got help me know where to focus on our last class day before the students’ summative assessment.

But I have to admit I am most excited about the first response I saw, highlighted below. This response gives me confidence that our focus on how we are solving problems – our focus on the Standards of Mathematical Practice – is getting us somewhere.

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Make sense of problems and persevere in solving them at its best.

And so the journey continues ….

 

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Reflected Triangles

We used a task from Illustrative Mathematics as part of a performance assessment on Rigid Motions. 

On the next page, △ABC has been reflected across a line into the blue triangle. Construct the line across which the triangle was reflected. Justify your conclusion.

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Students used TI-Nspire to construct the line of reflection, and we asked them to explain their construction on paper. I monitored their work using Class Capture to see what approaches the students were using. This year, almost everyone created segments BB’ and CC’. Some also created segment AA’. Some then constructed the midpoint of the segments then created the line through those midpoints. Some used the perpendicular bisector tool. Our class discussion focused on what were the fewest number of objects we could construct to get the line of reflection.

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And then I asked H.K. if she would share her work. She had the same idea as most of the others, but instead of drawing a segment with a vertex and its image as the endpoints (AA’, BB’, or CC’), she constructed the midpoint of segment BC and the corresponding midpoint of segment B’C’. Then she joined those image and pre-image points with the segment tool (actually vector tool) and constructed the perpendicular bisector of the segment. That’s a big deal. H.K. reminded us of an important property of rigid motions. Most of the segments and angles that we found congruent as we were exploring focused on the vertices of the triangle and their images. But the same is true from every point on the pre-image to its corresponding point on the image – not just from the vertices. Thanks for the reminder, H.K.

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And so the journey continues ….

 

 

 
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Posted by on September 2, 2013 in Geometry, Rigid Motions

 

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Rotations

What do you need for a rotation?

This question is a bit more difficult to answer than for a translation or a reflection.

We need an image. And we need an angle of rotation. But an angle of rotation isn’t enough – we need direction. In our high school geometry course, we begin to let the sign of the angle indicate the direction of the rotation. And that still isn’t enough. We need a center of rotation.

Are the two triangles congruent?

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Yes. The two triangles are congruent because there is a rigid motion that maps one triangle onto the other. A rotation is a rigid motion. A rotation is an isometry.

What does a rotation buy us mathematically? What has to be congruent in the diagram? Of course the triangles are congruent. And their corresponding parts are congruent (angles, segments, etc.). Their perimeters and areas are equal. But what else? OB=OB’, OA=OA’, and OC=OC’. m∠BOB’=m∠COC’=m∠AOA’=angle of rotation.

How does a 90° rotation compare with a –90° rotation? Which one is clockwise? Which one is counter-clockwise? In the past, I would have told students that a positive rotation is clockwise and a negative rotation is counterclockwise. Instead, we let students look for regularity in repeated reasoning as they explored rotations in the coordinate plane.

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They determined which was which and recorded information about different angle rotations in a table so that they could reason abstractly and quantitatively.

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So it’s the next day. Students don’t have out their chart – because while we used the chart to organize our information about rotations and generalize the rotations, we don’t want to rely on having to memorize it.

I sent a Quick Poll to assess student understanding of a rotation. Rotate the point (–4,5) –90° about the origin.

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I monitored the results as they came in – and because only 9 students had it correct, I unchecked “Show Correct Answer” before I showed them the results.

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I asked them to get up, find someone in the room who had answered differently, and convince that person that their answer was correct. I sent the poll again.

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While we are down to two responses, only 10 students had it correct. (Not good. Usually at least a majority has it correct the second time around.) I still didn’t show them the correct answer. We went back to the drawing board (the Graphs page). If (4,5) is the answer, then what has to be true? From our earlier exploration, we knew that AO must equal OA’. Necessary. But not sufficient..because we also knew that m∠AOA’=90°. An angle measure of 77° isn’t good enough.

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What happens when A’ is at (5,4)? AO=AO’ and the angle measure is 90°.

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The students asked for another Quick Poll. This time, they rotated (4,2) 90° about the origin. 73% correct (up from 33% correct in the previous poll). What mistake did those who answered incorrectly make? They rotated clockwise instead of counterclockwise. An easier mistake to correct.

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So it hit me while I was driving home that afternoon that while we had learned some about rotations, we had missed a huge opportunity to look for and make use of structure.

I started class the next day with another rotation question. 84% of the students got it correct.

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We went back to the Graphs page. What has to be true?

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AO=AO’. Yes. What else has to be true?

m∠AOA’=90°. Yes. What else has to be true?

The rotation is clockwise. Yes. What else has to be true?

A’ is in Quadrant II. Yes. What else has to be true?

Someone suggested that the lines containing segments AO and AO’ have to be perpendicular. Really? Yes. Perpendicular lines intersect to form right angles.

What else has to be true? Oh! The slopes of the perpendicular lines have to be negative reciprocals of each other.

Really? How do you know? Because my teacher told me last year. (They didn’t really say that part…I did.)

So this year we will actually prove that the product of the slopes of perpendicular lines is –1. But we are not ready for that yet. For now, we will go on what your Algebra I teacher told you. The slopes of perpendicular lines are negative reciprocals of each other.

How do you calculate slope? Inevitably, someone answered “rise over run”. Someone else answered “the change in y over the change in x”. What is the slope of line OA? 2/3. What is the slope of line OA’? -3/2. The lines are perpendicular.

We looked back at the problem from the day before with which students had difficulty. Can the image of (–4,5) rotated –90° about the origin be (4,5)? No! The line containing the pre-image and the center of rotation has a slope of –5/4. The line containing the other point and the center of rotation has a slope of 5/4. But an image of (5,4) does work. The slope of the line containing (5,4) and (0,0) is 4/5.

And so the journey continues, finding opportunities every day to enter into the Standards of Mathematical Practice with my students….

Note: We used the Transformations – Rotations lesson from Geometry Nspired as a guide for part of our exploration in this lesson.

 
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Posted by on September 2, 2013 in Coordinate Geometry, Geometry, Rigid Motions

 

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