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Carrying a Figure onto Itself

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

By the end of our course, we want students to be able to say, “I can map a figure onto itself using transformations”.

 

In our lesson on Reflections, we asked about reflecting a rectangle onto itself.

What do we need to do a reflection?

An object and a line.

What line do we need to reflect rectangle ABCD onto itself?

 

We talked about the difference between drawing a line and constructing a line.

How many lines will work?

What is the significance of the lines?

 

Students constructed lines different ways. Some students used the midpoints of the opposite sides to create a line.

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Some students constructed the perpendicular bisector of a side to create a line.

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We made Abby the Live Presenter. She changed the size of the rectangle to show us that using the perpendicular bisector always works.

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Several students thought that the diagonals of the rectangle could be the lines of reflection. We made Max the Live Presenter, and he showed us what happens when we reflect a rectangle about its diagonal.

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How many ways can you reflect a regular hexagon onto itself?

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The next day, we explored rotations.

How many ways can you rotate an equilateral triangle onto itself?

Where is the center of rotation?

What is the angle of rotation?

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If we change the size of the equilateral triangle, does the rotation still work?

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One student defined the rotation angle using three points instead of an angle measure. How should you arrange the points of the angle to rotate the triangle onto itself?

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We are now towards the end of the unit. In class yesterday, I asked students to write down any two ways to transform the rectangle onto itself. After a minute, I asked them to look back at what they had written. Have you attended to precision? If you said to reflect, have you described what is the line of reflection? If you said to rotate, have you described what is the center of rotation? Several students rotated the rectangle 360˚ or 720˚ or -360˚ about any point on the rectangle. I guess it’s not completely trivial to recognize that the rotation will work about any point (and not just a vertex), but I asked them to use an angle measure that wasn’t a multiple of 360.

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Students revised their work and then shared with their team. One team member told another that hers was not going to work. They called me over to mediate, which reminded me again how good it is for students to have dynamic action technology in their hands. Try it and see. I don’t have to be the judge … students can use the technology to test their conjectures. MJ wanted to reflect the rectangle first about a diagonal and rotate about the midpoint of the diagonal.

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By the time we made MJ the Live Presenter, she had decided to reflect the rectangle first about a diagonal and then reflect it about the perpendicular bisector of the diagonal.

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As I continued to monitor students working, I saw several who used a sequence of transformations to map the rectangle onto itself. Our standard specifically says to use reflections and rotations, but I asked BB to share her work. She reflected the rectangle about one of its sides and then translated it using a vector equal to the side perpendicular to the first side. Will that always work?

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Another student found the intersection of the perpendicular bisectors of the sides and rotated the rectangle 900˚ about that point. Why does 900 work?

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And so the journey to ease the hurry syndrome continues, often spending 20 minutes on what I had planned to take 5 …

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

 

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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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Visual: SMP-7 Look for and Make Use of Structure #LL2LU

How do our learners determine an equivalent expression to 4(x+3)-2(x+3)?

How would they determine the zeros of y=x2-4?

How might we provide opportunities for them to successfully look for and make use of structure?

 

We want every learner in our care to be able to say

I can make look for and make use of structure.  (CCSS.MATH.PRACTICE.MP7)

But…What if I think I can’t? What if I have no idea what “structure” means in the context of what we are learning?

 

One of the CCSS domains in the Algebra category is Seeing Structure in Expressions. Content-wise, we want learners to

  • “use the structure of an expression to identify ways to rewrite it. For example, see x4–y4 as (x2)2–(y2)2, thus recognizing it as a difference of squares that can be factored as (x2–y2)(x2+y2)”
  • “factor a quadratic expression to reveal the zeros of the function it defines”
  • “complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines”
  • “use the properties of exponents to transform expressions for exponential functions”.

 

How might we offer a pathway for success? What if we provide cues to guide learners and inspire noticing?

 

Level 4
I can integrate geometric and algebraic representations to confirm structure and patterning.

Level 3
I can look for and make use of structure.

Level 2
I can rewrite an expression into an equivalent form, draw an auxiliary line, or identify a pattern to make what isn’t pictured visible.

Level 1
I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole.

SMP7_Number SMP7_Algebra

 

Illustrative Mathematics has several tasks to allow students to look for and make use of structure. We look forward to trying these, along with a leveled learning progression, with our students.

3.OA Patterns in the Multiplication Table

4.OA Multiples of 3, 6, and 7

5.OA Comparing Products

6.G Same Base and Height, Variation 1

A-SSE Seeing Structure in Expressions Tasks

Animal Populations

Delivery Trucks

Seeing Dots

Equivalent Expressions

 

Leveled learning progression posters

[Cross posted on Experiments in Learning by Doing]

 
 

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SMP7: Look For and Make Use of Structure #LL2LU

SMP 7

We want every learner in our care to be able to say

I can look for and make use of structure.
(CCSS.MATH.PRACTICE.MP7)

But…What if I think I can’t? What if I have no idea what “structure” means in the context of what we are learning?

How might we offer a pathway for success? What if we provide cues to guide learners and inspire interrogative self-talk?

 

Level 4
I can integrate geometric and algebraic representations to confirm structure and patterning.

Level 3
I can look for and make use of structure.

Level 2
I can rewrite an expression into an equivalent form, draw an auxiliary line to support an argument, or identify a pattern to make what isn’t pictured visible.

Level 1
I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole.

 

Are observing, associating, questioning, and experimenting the foundations of this Standard for Mathematical Practice? It is about seeing things that aren’t readily visible.  It is about remix, composing and decomposing what is visible to understand in different ways.

How might we uncover and investigate patterns and symmetries? What if we teach the art of observation coupled with the art of inquiry?

In The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators, Dryer, Gregersen, and Christensen describe what stops us from asking questions.

So what stops you from asking questions? The two great inhibitors to questions are: (1) not wanting to look stupid, and (2) not willing to be viewed as uncooperative or disagreeable.  The first problem starts when we’re in elementary school; we don’t want to be seen as stupid by our friends or the teacher, and it is far safer to stay quiet.  So we learn not to ask disruptive questions. Unfortunately, for most of us, this pattern follows us into adulthood.

What if we facilitate art of questioning sessions where all questions are considered? In his post, Fear of Bad Ideas, Seth Godin writes:

But many people are petrified of bad ideas. Ideas that make us look stupid or waste time or money or create some sort of backlash. The problem is that you can’t have good ideas unless you’re willing to generate a lot of bad ones.  Painters, musicians, entrepreneurs, writers, chiropractors, accountants–we all fail far more than we succeed.

How might we create safe harbors for ideas, questions, and observations? What if we encourage generating “bad ideas” so that we might uncover good ones? How might we shift perspectives to observe patterns and structure? What if we embrace the tactics for asking disruptive questions found in The Innovator’s DNA?

Tactic #1: Ask “what is” questions

Tactic #2: Ask “what caused” questions

Tactic #3: Ask “why and why not” questions

Tactic #4: Ask “what if” questions

 

What are barriers to finding structure? How else will we help learners look for and make use of structure?

 

[Cross posted on Experiments in Learning by Doing]

 

Dyer, Jeff, Hal B. Gregersen, and Clayton M. Christensen. The Innovator’s DNA: Mastering the Five Skills of Disruptive Innovators. Boston, MA: Harvard Business, 2011. Print.

 
 

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Rigid Motions – Which Reflection?

Rigid Motions – Which Reflection?

We added to our introductory lesson on Rigid Motions this year. Sometime last year, I read Which Reflection is Best by Andrew Shauver.

Which_Reflection

I wondered what students would say without measuring and before studying reflections in depth. So I asked.

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And then I asked again, after students had the opportunity to use appropriate tools strategically (most used a ruler).

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Not very many changed their minds from their first glance.

 

Next we talked.

BK offered his argument as to why he chose B over D:

When I put the edges of one end of the ruler on A and A’, the line of reflection didn’t go down the middle of the ruler on D, but it did on B.

 

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Oh…so is there something significant about the line of reflection matching the middle of the ruler?

We made sense of the significance with the yellow-ish angles.

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We developed the vocabulary as it arose, instead of having students write lists of definitions with no context (I cringe to think about how many of my former students suffered through writing geometry vocabulary each unit) – midpoint, bisector, distance from a point to a line, perpendicular bisector.

And we ultimately concluded that the line of reflection will be the perpendicular bisector of the segment that joins a pre-image point with its image.

What a great example of use appropriate tools strategically – BK was using his ruler both to measure distance and to measure for right angles. 

And so as the journey continues, I am thankful for students who are willing to share their thinking and for teachers like Andrew who are willing to share their lessons.

 
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Posted by on August 23, 2014 in Geometry, Rigid Motions

 

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What’s My Rule

Instead of having a whole lesson of What’s My Rule explorations, we are adding one exploration to each bellringer during our unit on Rigid Motions. From yesterday:

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Students move point Z and observe how W follows. Z is mapped to W according to some rule that the students are trying to determine

I’ve written about this exploration before, so I want to focus on what was different this year.

Students constructed viable arguments and critiqued the reasoning of others. We are learning how to attend to precision, so we were lenient in giving credit to responses for which the oral explanation helped us make sense of the written explanation.

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One team wrote that if you added something to Z and subtracted something from W, then the points would map onto one another. I wouldn’t have worded what they were trying to say like they did. But they were getting at some important mathematics. Ultimately, they were trying to convey that Z and W are the same distance from the origin. We constructed a circle with the origin as the center and Z as one of the points on the circle and noticed that both Z and W always lie on the circle.

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A few other students said that the rule was to reflect Z over the line y=x to get W. Does that always work? We looked back and decided it wasn’t always true. When is it true? When does (x,y)→(-x,-y) also represent a reflection about the line y=x?

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Others thought that the rule was to reflect Z over the line y=-x to get W. Does that always work? We looked back and decided it wasn’t always true. When is it true? When does (x,y)→(-x,-y) also represent a reflection about the line y=-x?

Other students noticed that we could describe the rule using a rotation of Z 180˚ about the origin.

No one noticed that we could reflect Z about the x-axis and then about the y-axis. So what happens when no one notices something we want them to notice? I could have moved on. It wouldn’t have been detrimental to my students learning of mathematics if they didn’t know that. But I didn’t. Instead I asked whether there was a reflection that we could use to map Z onto W. I gave students just seconds to think alone and then time to talk with their teams. I monitored their team talk. 5 teams said that we could reflect Z about the perpendicular bisector of segment ZW to map Z onto W. Yes. Not what I was expecting … but absolutely true.

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One team said that we could reflect Z about the line y=x and then about the line y=-x to map Z onto W. Oh…we can reflect Z about y=x and then y=-x? How can you show that?

What happens when you reflect (x,y) about y=x? (y,x)

What happens when you reflect (y,x) about y=-x? (-x,-y)

Is there another sequence of reflections that will map Z onto W?

Reflecting about y=-x and then y=x.

Is there another sequence of reflections that will map Z onto W?

Teams worked together – and after another few minutes, they figure out that reflecting about y=0 and then x=0 would work. Or reflecting about x=0 and then y=0.

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And then we were called to the cafeteria for school pictures.

And then a student came up to me in the line for school pictures and asked whether there would be an infinite number of pairs of lines about which we could reflect Z onto W.

Are there an infinite number of pairs of lines that will work? 

What relationship do the pairs of lines have that we found?

y=x and y=-x; y=0 and x=0

What is significant about the pairs of lines?

After a few more questions, the students around us in line for pictures noted that the lines are perpendicular. 

So if we reflected Z about y=2x, then about what other line would we need to reflect Z’ to get W?

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I will be the first to admit both that of course all of this makes sense mathematically, and also that I’ve never thought about it before. And so the journey continues … ever grateful for the students with whom I learn.

 

Thanks to Michael Pershan for sharing Transformation Rules.

 
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Posted by on August 21, 2014 in Coordinate Geometry, Geometry, Rigid Motions

 

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Graphing Stories

How do you and your coworkers effect change in the classroom? Two years ago, I asked our principal whether we could schedule a geometry class during first block with four teachers and about 25 students. It was our first year to implement our new CCSS Geometry standards, and we needed to try it together. I have learned over the years that it doesn’t hurt to ask – he might say no, but he might also said yes. Well, he said yes, and as you can imagine, sharing a class together has been important for us both as learners and as teachers. As one of the teachers reflected recently, “Participating in a class all year with a team of teachers is the best professional development I have ever had.” Last year, our Algebra 2 team had a shared class to implement their new standards, and this year, our Algebra 1 team shares a class. I visit as often as I can.

We are building our course as we go, using all sorts of resources. We are using the framework from EngageNY, and we are using some of the activities and tasks in their lessons. We have started with a unit on Graphing Stories.

 

On the first day, we used Growing Patterns from an NCTM Article Coloring Formulas for Growing Patterns.

 

Lesson Goals (written with Jill Gough this summer):

Level 4: I can represent the number of tiles in a figure in more than one way and show the equivalence between the expressions.

Level 3: I can represent the number of tiles in a figure using an explicit expression or a recursive process.

Level 2: I can apply patterns to predict the number of tiles in a later figure.

Level 1: I can describe the pattern and draw a figure before and after the given figures.

H_Growing Pattern 

How do you see the pattern growing?

How many tiles are in H(5)? H(100)? H(N)?

Some students completed a table of values, and some students drew a graph.

 

On the second day, we used a Mathematics Assessment Project formative assessment lesson, Interpreting Time-Distance Graphs, with a focus on rate of change.

Lesson Goals:

Level 4: I can calculate average rate of change from a graph and a table.

Level 3: I can calculate average rate of change from a graph or a table.

Level 2: I can match a distance-time graph with a story and a table.

Level 1: I can annotate a graph using away from home and towards home, how fast and how slow.

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On the third day, we used Graphing Stories Video 2 to begin the lesson.

Lesson Goals:

Level 4: I can create a believable story for a given graph.

Level 3: I can calculate average rates of change for an elevation graph.

Level 2: I can create an elevation graph from a video, labeling the axes with appropriate units of measure.

Level 1: I can identify time intervals for each piece of an elevation graph.

Level 1: I can calculate the slope of a line.

Whose graph is believable?

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We used additional scenarios from Engage NY Algebra 1, Module 1, Lessons 1-2.

 

There were more videos and scenarios on Day 4 from Engage NY Algebra 1, Module 1, Lessons 3-4.

The lesson from Day 5 comes from David Wees’ webinar at the Global Math Department on Strategic Inquiry.

 

We have read Timothy Kanold’s blogpost on Leaving the Front of the Classroom Behind. And we are trying. And we are most grateful to our administrators for letting us try it together before we have to try it alone.

 
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Posted by on August 18, 2014 in Algebra 1, Graphing Stories

 

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