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Unit 3 – Angles

We learned our lesson last year when we tried to cram Angles & Triangles together in a unit that we needed to take a step back & slow down…at least until we have students who have actually experienced the geometry standards in the lower grades as prescribed by CCSS-M. Since this is the first real year of 6-8 implementation by our teachers, we still have a few years before we might be able to make more adjustments. For now, this one unit covers what has traditionally been covered by 3 chapters of our geometry textbook. 3 chapters have been reduced to one CCSS-M standard. The 3 chapters? Points, Lines & Planes; Deductive Reasoning & Logic; Parallel Lines.

Level 1: I can use inductive and deductive reasoning to make conclusions about statements, converses, inverses, and contrapositives.

Level 2: I can use and prove theorems about special pairs of angles. G-CO 9

Level 3: I can solve problems using parallel lines. G-CO 9

Level 4: I can prove theorems about parallel lines. G-CO 9

Congruence G-CO

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.

I know that I have read statements about de-emphasizing the structure and building of a deductive system in CCSS-M. I wish I could remember in which document I read that. I’ve only been emphasizing the structure and building of a deductive system for about 8 years now – ever since I took a graduate course on geometry and realized its importance. So last year, we tried to jump right in to proving theorems without thinking about logic, conditionals, converses, inverses, etc. It was a disaster. Even though conditional statements aren’t explicitly listed in CCSS-M and hence will not be assessed, I believe they are important. I think it is a big deal that we accept the following as a postulate in our system: If two parallel lines are cut by a transversal, then corresponding angles are congruent. That’s our equivalent to Euclid’s 5th Postulate, and our geometry would look differently if we didn’t accept it without proof. The Triangle Sum Theorem relies on this postulate, as do other important results, and I want my students to know that it is our choice of accepting this postulate that brings about results that wouldn’t be true without it.

So we started with a lesson on logic. I asked students to create a word morph – go from WARM to COLD, changing one letter at a time, creating a real word each time. This short exercise emphasizes that we all start with WARM and we all end with COLD, but we don’t all get there the same way.

I read someone’s tweet about having students create a conditional statement for which the statement, converse, inverse, and contrapositive were always true, which we did. I wish I could remember whose it was. Thanks for the idea.

We also gave student groups cards with a conditional statement, its converse, inverse, and contrapositive. The students were to choose a card as the conditional and then decide which of the other cards was the converse, inverse, and contrapositive. Once they have those statements straight, then I choose one of their other cards (like their inverse) and tell them it is now their conditional statement. If it is now the conditional, then which of the other cards are now the converse, inverse, and contrapositive? This short exercise emphasizes that it doesn’t matter how the conditional is written, from that conditional, you can still create the converse, inverse, and contrapositive.

Then we thought about the game of basketball being a deductive system. What would be the undefined terms? Defined terms? Postulates? Theorems? Monopoly and soccer also work for this.

Then we played The Letter Game as an introduction to proofs. I wish I knew the source of The Letter Game. A quick search on the internet did not produce any results (except my Canvas notes from class), so I’ll include it below.

The “Letter Game”

Undefined terms: Letters M, I, and U

Definition: x means any string of I’s and U’s

Postulates:             1. If a string of letters ends in I, you may add U at the end.

2. If you have Mx, then you may add x to get Mxx.

3. If three I’s occur, that is III, then you may substitute U in their place.

4. If UU occurs, you drop it.

Example. Given: MI

Prove: MIIU

1. Given: MIII

Prove: M

2. Given: MIIIUUIIIII

Prove: MIIU

3. Given: MI

Prove: MUI

4. Given: MI

Prove: MIUIU

5. Given: MIIIUII

Prove: MIIUIIU

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I like The Letter Game because it introduces students to the idea that we start with some given information, we have a goal of what to prove, and we use the rules (postulates, theorems, and definitions) in our deductive system to get there. I do not care that students produce a formal two-column proof. I care that students can create a valid, logical argument. I also care that they notice that their valid, logical arguments don’t have to look exactly the same as everyone else’s in the class. There is almost always more than one way to get from the given information to what we are trying to prove. And more than a few students admit they like figuring out a Letter Game proof.

Then we do a few jumbled proofs about segments and angles before we prove our first theorem: Vertical angles are congruent.

And so, right or wrong, the journey continues…

 
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Posted by on October 20, 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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Reflections

What do we need for a reflection?

An object. And a line.

My students have some experience with reflections before they get to my high school geometry course…so it didn’t take them long to articulate what we need for a reflection or to predict where the image of ΔABC would be when it is reflected about line XY.

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Pretty quickly, I let them use their dynamic technology to reflect ΔABC about line XY.

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And I asked them to write down everything they could find that was congruent. This didn’t take as long as it did for the translation, because they got that a reflection is a rigid motion, and so ΔABC is congruent to ΔA’B’C’. And then that means that segment AB is congruent to segment A’B’. And the perimeter of ΔABC is equal to the perimeter of ΔA’B’C’. And the areas of the triangles are equal. And the corresponding angles are congruent. But what else has to be congruent? Someone suggested that the distance from C to line XY will be the same as the distance from C’ to line XY. “Yes!” (I thought to myself as I hid my excitement.) But as we stopped to have a conversation about what one means when they talk about the distance from C to line XY, I realized that my students had no idea what it means mathematically to talk about the distance from a point to a line. I asked if it matters where we draw the distance from C to line XY. All 30 students shook their head no.

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So can any of the red segments drawn from C to line XY represent the distance from C to the line? All 30 students shook their head yes.

Are all of those segments the same length? All 30 students shook their head no. And finally someone talked about “seeing” a right triangle (look for and make use of structure) – and said that segment CF is longer than segment CE because it is the hypotenuse of a right triangle.  At some point we talked about the phrase “as the crow flies” – and the fastest way to get from one location to another, which the students said is a straight line (to which I reply every single time that all lines are straight in our geometry).

It was time to define the distance from a point to a line as the length of the segment that lies on the line that is perpendicular from the point to the line. Other relationships became evident as well. E is the midpoint of segment CC’. Line XY is the perpendicular bisector of segment CC’. Angle CED is congruent to angle CEF, and they are both right angles. And similarly if we talk about segment AA’ or segment BB’ instead.

We all learned something in class today. I know more about student misconceptions for the distance from a point to a line. My students know more about the distance from a point to a line than they did before class. And I hope that my students know more about the distance from a point to a line because of the context and conversation we had than if I had started class with notes and examples about the distance from a point to a line & asked them to work a few problems.

And so the journey continues ….

 

Note: We used the Transformations – Reflections lesson from Geometry Nspired as a guide for part of our exploration in this lesson. http://education.ti.com/en/timathnspired/us/detail?id=A370588B92164483B5C08C3E3833A443&t=E51B8F52AF8A4099A8204BE3B9ED5BD6

 

 

 
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Posted by on August 27, 2013 in Coordinate Geometry, Geometry, Rigid Motions

 

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Rigid Motions

We have started our CCSS geometry course with a unit on Rigid Motions. I’ve included the standards and “I can” statements that we are using at the bottom of the post.

We started last year with a unit called Tools of Geometry, where we did constructions by focusing on special segments in triangles, but we decided to try starting with Transformations this year, since our students have a bit more familiarity with transformations than medians, altitudes, angle bisectors, and perpendicular bisectors. We are hoping to ease them in to “making sense of problems and persevering in solving them” instead of bombarding them with it in the first unit….

We started the lesson with a routine called “Zoom In”, that I read about in Making Thinking Visible. In the routine, students look at a small part of an image. “What do you notice?” “What is your interpretation of what this might be based on what you are seeing?” As the teacher reveals more of the image, students think about the new things they notice & refine their interpretation of what the image might be based on the new information. We used a piece of fabric by Michael Miller.

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The first image was part of the pair of flip flops – and then we zoomed out several times to reveal more of the fabric. Students noticed congruent figures – but they didn’t just say that the figures were congruent because they had the same size and shape. They began to discuss the congruence of the figures in terms of rigid motions. One brown and white cat is congruent to another because there is a translation that will carry one onto the other. One flip flop is congruent to the other in its pair because there is a reflection that will carry one onto the other. One fan is congruent to another because there is a translation that will carry one onto the other. One brown cat is congruent to another brown cat because there is a rotation that will carry one onto the other.

We used the Mathematics Assessment Project formative assessment lesson on Transforming 2D Figures as a guide for the rest of the lesson.

Students had a cutout L shape to transform as requested. We determined whether the figures were congruent based on whether there was a rigid motion (translation, reflection, rotation) that would carry one onto the other.

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Then we asked questions from the MAP lesson such as …

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Where will the L-shape be if it is translated by 1 horizontally and -4 vertically?

Where will the L-shape be if it is reflected over the x-axis?

Where will the L-shape be if it is reflected over the line x=2?

Where will the L-shape be if it is reflected over the line y=x?

Where will the L-shape be if it is rotated through 180° around the origin?

Oh – and just in case you are wondering whether students had remembered the graphs of x=2 and y=x over the summer, that was the topic of the bellringer for the lesson. I think that we often have students memorize that horizontal lines can be written in the form y=# and vertical lines can be written in the form x=# without having the students think about the actual points on the line. The point on the line in the TI-Nspire Graphs page shown below is dynamic, and so students moved the point and began to make sense of what was happening with the coordinates – and ultimately with the equation of the line. So that when we did ask them to reflect their L-shape over the line x=2, there was less discussion about where is the line x=2 and more discussion about the actual reflection.

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The standards that we are using:

G-CO 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G-CO 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G-CO 4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G-CO 5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

G-CO 6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

G-CO 7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

And our “I can” statements:

Unit 1 –Rigid Motions

Level 1: I can identify and define transformations and composite transformations.

Level 2: I can perform transformations and composite transformations.

Level 3: I can determine the congruence of two figures using rigid motions.

Level 4: I can apply transformations and composite transformations to figures in the coordinate plane.

Level 4: I can map a figure onto itself using transformations.

And so the journey continues….

 
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Posted by on August 25, 2013 in Coordinate Geometry, Geometry, Rigid Motions

 

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Standards for Mathematical Practice Student Journals

My goal this year has not just been for me to provide opportunities for my students to enter into the Standards for Mathematical Practice – but for my students to begin to recognize when they are entering into the practices. I gave my students a list of the practices along with a short explanation of each at the beginning of the year. I have that same list posted in the front of my room. This language has become part of how we talk to each other about what we are doing in class – and what our goals are for learning in each unit.

Every quarter this year, I asked my students to reflect on a time when they participated in one of the mathematical practices. When we started, I told them I had no idea what these journals were supposed to look like. I gave them a few reflection questions for each practice, but I didn’t know if they were the right questions to ask. It has been an absolute pleasure to read their responses – especially to get to learn about times when they recognized that they were participating in a mathematical practice and I didn’t observe that happening.

I want to share some of those reflections with you.

1. Hannah writes about the mathematical practice “Make sense of problems and persevere in solving them.” She says, “As I was working in class I didn’t even notice that I was using a mathematical practice. I am going to be honest, I don’t like math in the first place, so when I don’t know how to work something I get very discouraged. I kept working, and even though I was one of the last people to finish, I still was proud of myself. I am learning not to give up and to keep going even though I can’t figure it out.

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The students were given a matrix of the vertices of a triangle. You can see from Hannah’s work that they were trying to figure out by what matrix they could multiply the given matrix in order to produce the vertices of a certain transformed images.

Hannah continues – “Next time I am struggling with a problem I will think of this incident, take a deep breath, and persevere through it.”

2. Franky writes “This semester’s math class has been one of the most challenging classes I have taken. With that being said, it has also been my most beneficial, and it is one of my favorites. No matter how difficult a problem may be, I have learned to continue trying and persevering, for I should be able to figure the problem out. Franky goes on to describe a problem – and then he says to begin solving “of course, draw a triangle”. I have plenty of students who used to never think about drawing a diagram to make sense of a problem – some of them know it will help but just don’t want to take the time to do it – we are having to change the habits and practice of our students.

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Franky continues – “There is no greater feeling than solving a math problem correctly, especially if it is difficult. This class has taught me time and time again to just keep on trying.”

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I think it is significant for our students to know that we are going to give them problems for which they must persevere in solving.

3. Erin talks about the practice model with mathematics. While she was cycling she made a connection between linear distance (which we had studied in trigonometry) and how the sensor on her bike was able to tell her the distance traveled.

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4. While he was at band practice, Sam made a connection between the band’s formation on the field and the transformations that we were studying in geometry. He writes “I am in the center of a diamond shape. We have to translate across the field.” – Note that this was early in the year and he had not yet heard my “a diamond is a gem and not a geometric shape” lecture.

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5. Emilee writes about attending to precision. In her mind, she knows that negative cosecant of an angle is different from inverse cosecant of an angle, but she said negative cosecant, knowing very well she meant inverse cosecant. She writes “My ignorance of precision led to confusion among my table, but I am slowly learning to pay more attention to my words. Saying things, thingys, and whatchamacallit are not acceptable anymore.”

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6. 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 sqrt(2). The sqrt(2) will always be in the hypotenuse even though it may not be seen like sqrt(2). In her examples, the triangle to the left has sqrt(2) shown in the hypotenuse, but the triangle to the right has sqrt(2) in the answer even though it isn’t shown, since 3sqrt(2)sqrt(2) is not in lowest form. She says, “I looked for regularity in repeated reasoning and found an interesting answer.”

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7. Jordan writes about “Construct viable arguments and critique the reasoning of others”. She says, “If you can really understand something you can teach it. Every person relates to and thinks about problems in a different way, so understanding different ways to get to an answer can help to broaden your knowledge of the subject. Arguments are all about having good, logical facts. If you can be confident enough to argue for your reasoning you have learned the material well.”

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8. And Franky says that construct viable arguments and critique the reasoning of others is probably our most used mathematical practice. If someone has a question about a problem, Mrs. Wilson is always looking for a student that understands the problem to explain it. And once he or she is finished, Mrs. Wilson will ask if anyone got the correct answer, but worked it a different way. By seeing multiple ways to work the problem, it is easier for me to fully understand.”

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These student reflections have been very significant in my CCSS-M journey this year. They give me some hope that my students are at least beginning to understand that how we do and learn math is important to me. These student reflections give me hope as our journey continues …

 

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