RSS

Category Archives: Polygons

Is This a Rectangle?

Is This a Rectangle?

One of our learning intentions in our Coordinate Geometry unit is for students to be able to say I can use slope, distance, and midpoint along with properties of geometric objects to verify claims about the objects.

G-GPE. Expressing Geometric Properties with Equations

B. Use coordinates to prove simple geometric theorems algebraically

  1. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

We recently used the Illustrative Mathematics Task Is This a Rectangle to provide students the opportunity to practice.

We also used Jill Gough’s and Kato Nims’ visual #ShowYourWork learning progression to frame how to write a solution to the task.

How often do we tell our students Show Your Work only to get papers on which work isn’t shown? How often do we write Show Your Work next to a student answer for which the student thought she had shown her work? How often do our students wonder what we mean when we say Show Your Work?

0 showyourwork-es1.png

The Show Your Work learning progression begins to help students understand what we mean when we say Show Your Work. I have seen it empower students to ask each other for feedback on their work: Can you read this and understand it without asking me any questions? It has been transformative for my AP Calculus students as they write Free Response questions that will be scored by readers who can’t ask them questions and don’t know what math they can do in their heads.

We set the timer for 5 minutes of quiet think time. Most students began by sketching the graph on paper or creating it using their dynamic graphs software. [Some students painfully and slowly drew every tick mark on a grid, making me realize I should have graph paper more readily available for them.]

They began to look for and make use of structure. Some sketched in right triangles to see the slope or length of the sides. Some used slope and distance formulas to calculate the slope or length of the sides.

I saw several who were showing necessary but not sufficient information to verify that the figure is a rectangle. I wondered how I could steer them towards a solution without telling them they weren’t there yet.

I decided to summarize a few of the solutions I was seeing and send them in a Quick Poll, asking students to decide which reasoning was sufficient for verifying that the figure is a rectangle.

10 Screen Shot 2016-03-23 at 9.41.00 AM.png

Students discussed and used what they learned to improve their work.

It occurred to me that it might be helpful for them to determine the Show Your Work level for some sample student work. And so I showed a sample and asked the level.

But I didn’t plan ahead for that, and so I hurriedly selected two pieces of student work from last year to display. I was pleased with the response to the first piece of work. Most students recognized that the solution is correct and that the work could be improved so that the reader knows what the student means.

I wish that I hadn’t chosen the second piece of work. Did students say that this work was at level 3 because there are lots of words in the explanation and plenty of numbers on the diagram? Unfortunately, the logic is lacking: adjacent sides perpendicular is not a result of parallel opposite sides. Learning to pay close enough attention to whether an argument is valid is good, hard work.

Tasks like this often take longer than I expect. I’m not sure whether that is because I am now well practiced at easing the hurry syndrome or whether that is because learning to Show Your Work just takes longer than copying the teacher’s work. And so the journey continues …

 
3 Comments

Posted by on April 11, 2017 in Coordinate Geometry, Geometry, Polygons

 

Tags: , , , , ,

MP6 – Defining Terms

Screenshot 2016-01-27 09.07.23.pngHow do you provide your students the opportunity to attend to precision?

1-screen-shot-2016-10-25-at-1-18-52-pmWriting sound definitions is a good practice for students, making all of us pay close attention to what something is and is not.

I’ve learned from Jessica Murk about Bongard Problems, which give students practice creating sound definitions based on what something is and is not.

1_1 Screen Shot 2016-11-14 at 8.30.14 AM.png

What can you say about every figure on the left of the page that is not true about every figure on the right side of the page? (Bongard Problem #16)

Last year when I asked students to define circle, I found it hard to select and sequence the responses that would best contribute to a whole class discussion without taking too much class time.

2-screen-shot-2016-10-25-at-1-16-23-pm3-screen-shot-2016-10-25-at-1-16-38-pm4-screen-shot-2016-10-25-at-1-16-48-pm

I remember reading Dylan Wiliam’s suggestion in Embedding Formative Assessment (chapter 6, page 147) to have students give feedback to student responses that aren’t from their own class. I think it’s still helpful for students to spend time writing their own definition, and possibly trying to break a partner’s definition, but I wonder whether using some of last year’s responses to drive a whole class discussion this year might be helpful.

  • a shape with no corners
  • A circle is a shape that is equal distance from the center.
  • a round shape whose angles add up to 360 degrees
  • A circle is a two-dimensional shape, that has an infinite amount of lines of symmetry, and a total of 360 degrees.
  • A 2-d figure where all the points from the center to the circumference are equidistant.

Screenshot 2016-01-27 09.07.23.png

Screenshot 2016-01-27 09.07.17.png

We recently discussed trapezoids.

Based on the diagram, how would you define trapezoid?

5 Trapezoid.png

Does how you define trapezoid depend on how you construct it?

Can you construct a dynamic quadrilateral with exactly one pair of parallel sides?

Trapezoid.gif

And so the #AskDontTell journey continues …

 
Leave a comment

Posted by on November 14, 2016 in Circles, Geometry, Polygons

 

Tags: , , , , , ,

MP7: The Diagonal of an Isosceles Trapezoid

 

SMP7 #LL2LU Gough-Wilson

I’ve written about the diagonals of an isosceles trapezoid before.

When we practice “I can look for and make use of structure”, we practice: “contemplate before you calculate”.

We practice: “look before you leap”.

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

1 Screen Shot 2016-04-10 at 1.33.22 PM.png

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

Or sometimes we see more than is helpful to see all at one time.

2 Screenshot 2015-11-05 09.53.03.png

We persevere.

3 Screenshot 2015-11-05 09.53.24.png

Even with the same auxiliary lines, we don’t always see the same picture.

4 Screen Shot 2016-04-10 at 1.38.32 PM.png 5 Screen Shot 2016-04-10 at 1.38.24 PM.png

We learn from each other.

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

 
Leave a comment

Posted by on August 16, 2016 in Angles & Triangles, Geometry, Polygons

 

Tags: , ,

Midpoint Quadrilaterals

Midpoint Quadrilaterals

CCSS-M G-CO.C.11

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

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

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

I can determine sufficient conditions for naming special quadrilaterals.

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

1 01-05-2015 Image009 2 01-05-2015 Image010

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

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

3 Screen Shot 2014-11-05 at 8.55.19 AM

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

4 Screen Shot 2014-11-05 at 8.57.12 AM

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

6 Screen Shot 2014-11-05 at 9.17.15 AM

(Yes, Connor your polygon can be concave.)

7 Screen Shot 2014-11-05 at 9.19.34 AM

What do you notice?

8 Screen Shot 2014-11-05 at 9.19.51 AM

It’s a parallelogram.

9 Screen Shot 2014-11-05 at 9.20.31 AM

How do you know?

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

10 Screen Shot 2014-11-05 at 9.22.36 AM 11 Screen Shot 2014-11-05 at 9.22.57 AM

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

12 Screen Shot 2014-11-05 at 9.24.07 AM 13 Screen Shot 2014-11-05 at 9.24.34 AM

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

14 Screen Shot 2014-11-05 at 9.27.33 AM

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

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

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

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

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

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

 
1 Comment

Posted by on January 5, 2015 in Geometry, Polygons

 

Tags: , , , , , , , , , ,

Midsegments

Midsegments

CCSS-M-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.

What do you do when the standard for the day gives away what you want students do explore and figure out on their own?

I’ve made a deal with my administrator to post the process standard for the day (Math Practice) instead of the content standard.

In many of our geometry classes, our learning goals include look for and make use of structure and look for and express regularity in repeated reasoning.

1 SMP7_algebra

We defined midsegment:

A midsegment of a triangle is a segment whose endpoints are the midpoints of two sides of the triangle.

The midsegment of a trapezoid is a segment whose endpoints are the midpoints of the non-parallel sides.

Then we constructed the midsegment of a trapezoid. Students observed the trapezoid as I changed the trapezoid.

2 01-04-2015 Image008

I sent a Quick Poll: What do you think is true about a midsegment of the trapezoid?

3 Screen Shot 2014-11-03 at 9.32.41 AM

It creates both a triangle and a trapezoid. 1

all the midpoints form a similar triangle    1

mn is parallel to yx   2

((1)/(2))the size of the origin        1

parallel to base of triangle   1

all the midsegments make a similar triangle upside down          1

parallel to the base   1

cre8

Δ

and a trap      1

creates triangle and trapezoid        1

It will form the side of a triangle that is similar to the original.   1

The midsegment is parallel to the side not involved in making the midsegment.        1

it would be a median            1

MN is parallel to YX  1

parallel to base          1

mn parallel to yx       1

MN is parallel to the bottom line     1

XMN=XNM     1

cuts the tri into a trap and tri          1

all the segments will make a similar triangle tothe original         1

all of the midpoints connected make a similar triangle to the original one       1

creates ∆ on top + trap. on bottom 1

Triangle XMN is similar to triangle XYZ.

Line MN is parallel to line YX.          1

2/3 the largest side  1

2/3 o  1

side parallel to the midsegment is a           1

it makes a triangle and a tra            1

mn is ll to yx, mnx is congruent to triangle mon   1

In order to change things up a bit, I quickly printed the students conjectures, cut them up, and distributed a few to each team. Now you decide whether the conjectures you’ve been given are true. And if so, why?

I used Class Capture to monitor while the students talked and worked (and played/explored beyond the given conjectures).

4 Screen Shot 2014-11-03 at 9.39.15 AM 5 Screen Shot 2014-11-03 at 9.39.47 AM 6 Screen Shot 2014-11-03 at 9.43.11 AM

Then I asked what they figured out through a Quick Poll:

seg mn parallel to both seg ab and seg dc 1

parallel to both of the bases            1

it cre8 2 trap 1

mn is parallel to dc and ab  2

Trapezoid ABCD is similar to ABNM.

Lines AB and CD are parallel to the midsegment. 1

MN is parrellel to AB and DC           1

It creates a line that is parallel to the bases and forms two trapezoids. 1

nidsegment is parallel to top and bottom sides    1

((AB+DC)/(2))          2

it would be parallel to the sides above and below it        1

MN is parallel to DC and AB 1

(AB+DC)/(2)=MN     1

its parallel to DC        1

It makes two trapezoids       1

it is // 2 ab and dc    1

it forms two trapozoids        1

It is parallel to the sides above and below it.         1

makes 2 trap. 1

It makes a similar trapezoid.            1

they make 2 trapezoid, ab+dc/2     1

parellel to base of trapezoid            1

ab ll to mn ll to dc. when a parallelogram, creates 2 congruemt trapezoids      1

trapezoid ABNM is similar to trapezoid MNCD      1

all midsegments make a diala         1

And then we talked about how they knew these statements were true.

Jameria had a lot of measurements on her trapezoid. I made her the Live Presenter. What conjectures can we consider using this information?

7 Screen Shot 2014-11-03 at 9.44.22 AM

I made Jared the Live Presenter.

What does Jared’s auxiliary line buy us mathematically?

8 Screen Shot 2014-11-03 at 9.47.16 AM 9 Screen Shot 2014-11-03 at 9.47.33 AM

I made Landon the Live Presenter.

What conjectures can we consider using this information?

10 Screen Shot 2014-11-03 at 9.48.49 AM

I sent a Quick Poll to formatively assess whether students could use the conjecture we made about the length of the midsegment compared to the length of the bases of the trapezoid.

11 Screen Shot 2014-11-03 at 9.53.04 AM

What about the midsegments of a triangle?

12 Screen Shot 2014-11-03 at 9.53.59 AM13 Screen Shot 2014-11-03 at 9.58.40 AM

And why?

We had not yet started our unit on dilations, and so there was more to the why in a later lesson.

And so the journey continues, even making deals with my administrators as needed, to create a classroom where students get to make and test and prove their own conjectures instead of being given theorems from our textbook to prove.

 
Leave a comment

Posted by on January 4, 2015 in Angles & Triangles, Geometry, Polygons

 

Tags: , , , , , , , , , ,

Angles in Polygons Problems-Items-Tasks

Angles in Polygons Problems-Items-Tasks

We found several good tasks to use throughout our unit on polygons.

I had heard of the NRICH site before but had not used it. A search on NRICH for polygon angles turned up some out-of-the-ordinary tasks.

A quadrilateral can have four right angles. What is the largest number of right angles an octagon can have?

1 Screen Shot 2015-01-01 at 11.33.48 AM 2 Screen Shot 2015-01-01 at 11.34.04 AM

3 Screen Shot 2015-01-01 at 8.45.17 PM

I was surprised at how quickly my students thought to draw a concave polygon to explore this problem. I went straight to a convex polygon first when I thought about it. I think this has to do with look for and make use of structure. We’ve asked “what do you see that’s not pictured?” We’ve learned geometry by drawing auxiliary lines. I’m convinced that my students think differently than I do because of how they’ve learned geometry.

I can’t remember where I got the next problem. But I like that it asks for both a calculation and a proof of why the triangle must be isosceles. When I saw the student results (27 out of 30 correct), I had to think quickly about whether it was worth the time to go through the proof. I decided that it was. My students had been asking for more practice with proofs, and so I figured this was an opportunity to let them compare their work with the rest of the class. A student led us through her reasoning.

4 Screen Shot 2015-01-01 at 11.39.54 AM

5 Screen Shot 2015-01-01 at 11.36.56 AM

6 Screen Shot 2015-01-01 at 8.44.47 PM

7 Screen Shot 2015-01-01 at 8.56.27 PM

I can’t remember where I got the regular pentagon task, either, and my attempts at googling have failed. I have two favorite responses.

8 Screen Shot 2015-01-01 at 9.02.10 PM

For the trapezoid requested in part (d), a few students listed a parallelogram. I love that my students have embraced our inclusive definition of trapezoids, and I love that they feel confident enough in their understanding to include a parallelogram as the answer to a question about a trapezoid.

9 Screen Shot 2015-01-01 at 9.04.03 PM

For the similar triangles and justification requested in parts (e) and (f), a few students justified the similarity of triangles using transformations. This student suggested that ∆DJC~∆EJB because of a reflection and a dilation. (Of course the response could use more attention to precision. She didn’t say about which line to reflect one of the triangles, but the line is drawn in the diagram. And you wouldn’t believe that I have actually had a conversation with my students about the word being dilate instead of dialate. Dialect prevails in the South.)

10 Screen Shot 2015-01-01 at 9.06.53 PM

We have incorporated Shapedoku into this unit as well after reading Reinforcing Geometric Properties with Shapedoku Puzzles in the October 2013 Mathematics Teacher.

11 Screen Shot 2015-01-01 at 11.40.19 AM

Finally, we tried something different for conversation about angles in quadrilaterals. My students do need some practice with using the properties of quadrilaterals to calculate angles measures … but the regular types of problems can be so boring. So I gave the regular types of problems with a twist.

Given one angle measure, for which figures can you determine all remaining angle measures?

12 Screen Shot 2015-01-01 at 9.15.39 PM

This gave students practice with properties and angle measures, and it provoked conversation that giving problems like this one at a time wouldn’t have.

(I neglected to save the Quick Poll I sent to collect their work, but it went well. I had students work alone for 2-3 minutes before talking with their team.)

12.5 Screen Shot 2015-01-01 at 9.24.33 PM

The next class, I used the same diagrams but gave part of one of the interior angles formed by a diagonal. The question was the same: Given one angle measure, for which figures can you determine all remaining angle measures?

13 Screen Shot 2014-11-03 at 8.38.42 AM

I watched as students worked.

14 Screen Shot 2014-11-03 at 8.55.32 AM

15 Screen Shot 2014-11-03 at 8.56.05 AM

16 Screen Shot 2014-11-03 at 9.04.04 AM

And eventually had them talk with each other.

I can show student responses separated (as above) or grouped together (as below) to help decide which students to ask for explanations.

17 Screen Shot 2015-01-01 at 11.20.12 AM

Again, we had good conversation that wouldn’t have happened had I asked the problems one at a time.

18 Screen Shot 2015-01-01 at 9.18.11 PM

Incorporating new tasks each year keeps the class interesting for us as teachers as much as it does for our students. When is the last time you looked for a new task at NRICH or Illustrative Mathematics or the Mathematics Assessment Project? Or when is the last time you used a task you read about on someone’s blog or found through someone’s Tweet?

And so the journey continues … with thanks to all of you for your suggestions along the way.

And so the journey continues … with thanks to all of you for your suggestions along the way.

 
Leave a comment

Posted by on January 1, 2015 in Angles & Triangles, Polygons

 

Tags: , , , , , , ,

Sorting Teams of Students

We have had students change teams on the first day of each unit this year. Which is something, as I’ve always hesitated to change up teams that are working well together in the past. But since we are now on Unit 5, it’s the expectation.

For the Logic/Angles/Triangles unit, we gave students different conditional statements, along with their converses, inverses, contrapositives, and biconditionals. So, for example, the following five cards made a team:

If the sum of two angles is 90°, then the angles are complements of each other.

If two angles are complements of each other, then their sum is 90°.

If the sum of two angles is not 90°, then the angles are not complements of each other.

If two angles are not complements of each other, then their sum is not 90°.

The sum of two angles is 90˚ if and only if the angles are complements of each other.

Students learned during the first lesson about converse, inverse, etc., and each team had their own card set for discussion.

 

For the Polygons unit, we used different non-special rectangles, trapezoids, parallelograms, kites, rhombi, and squares. Which wasn’t totally easy, as students were unfamiliar with kites and don’t always distinguish rhombi from squares.

 

For the Dilations unit, we gave students cards such as the following:

3-4-5

32-60-68

18-80-82

9-40-41

12-16-20

Chaos ensued.

2014-11-13 08.24.19

I overheard several comments:

11-60-61: November 60, 1961

Where is she?

We sorted a team by 2 digits – 2 digits – 3 digits

Is it a right triangle?

 

Is it a right triangle? Can you tell whether you have a right triangle?

Then they realized that they all had right triangles.

But they still didn’t know how to sort themselves into a team.

More chaos ensued.

2014-11-13 08.23.28

And at some point, I realized that I would either have to intervene, or we would spend our entire first day of the new unit on Dilations sorting into new teams.

Does anyone know what our new unit is?

Dilations.

Oh. Dilations.

Can someone give me a card?

3-4-5.

How did I get so lucky?

5-12-13.

22-120-122.

11-60-61.

We wrote several of them on the board.

And then they found their Pythagorean families with the primitive patriarchs.

Screen Shot 2014-11-13 at 8.34.39 AM

Another class sorted themselves by columns instead of rows (see card image below).

And another class sorted by rows correctly, thinking about scale factor, but had no idea that they were side lengths of right triangles.

Screen Shot 2014-12-01 at 5.59.44 PM

Card sets are available at this link if you’re interested. Share them back with us if you improve them!

And let us know if you have any ideas for our remaining units, as we have a lot to live up to after the Dilations team sort.

6-Right Triangles

7-Circles

8-Coordinate Geometry

9-Geometric Measure & Dimension

10-Modeling with Geometry

 
2 Comments

Posted by on December 3, 2014 in Angles & Triangles, Dilations, Geometry, Polygons

 

Tags: , ,

The Diagonals of a Rectangle

CCSS-M.G-CO.C.11.Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

How can you use rigid motions to show that the diagonals of a rectangle are congruent?

I had a few students come in during zero block to work on proofs. (If you read my last post on the diagonals of an isosceles trapezoid, you’ll know why.)

2014-11-11 08.22.35

We were trying to show that AC=BD. Students used look for and make use of structure to compose the rectangle into two triangles. Which two triangles should we show congruent if we want to show that the diagonals are congruent?

∆ACD and ∆BDC.

One student showed the two triangles congruent by SAS. (Opposite sides of a rectangle are congruent, all angles in a rectangle are right and thus congruent, and CD=CD by reflexive.)

But a pair of girls wanted to use a rigid motion to show that the triangles were congruent.

First up: rotating.

Rotating which triangle?

∆ACD

About what point?

The center of the rectangle. Where the diagonals meet.

How many degrees?

90˚

(I’m not sure whether they really thought we should rotate by 90˚ or they chose 90˚ because we seem to rotate by 90˚ and 180˚ more than any other angle measure.)

We need paper. And scissors.

What is the image of ∆ACD when we rotate it 90˚ about the intersection of the diagonals?

Not ∆BDC.

What is the image of ∆ACD when we rotate it 180˚ about the intersection of the diagonals?

∆ABC

2014-11-11 08.23.50

How can we show that ∆ACD and ∆BDC are congruent?

A reflection?

About what line?

About the perpendicular bisector of segments AB and CD.

Are you sure?

Yes.

Once the triangles are congruent, then the corresponding parts are congruent, and so we can conclude that the diagonals are congruent.

And so the journey continues … with an apology to my former students for not using scissors more often.

 
Leave a comment

Posted by on December 2, 2014 in Geometry, Polygons, Rigid Motions

 

Tags: , , ,

The Diagonals of an Isosceles Trapezoid

It was the day before the test on Polygons, and so I thought that writing a proof and then giving feedback on another team’s proof might be helpful.

1a Screen Shot 2014-11-30 at 5.05.51 PM

Students worked alone for a few minutes, thinking about what was given and what could be implied. Then they worked with their team to talk about their ideas and to begin to plan a proof.

Some were off to a good start.

1b Screen Shot 2014-11-30 at 4.25.41 PM

2 Screen Shot 2014-11-30 at 4.28.16 PM

Some were obviously practicing look for and make use of structure.

3 Screen Shot 2014-11-30 at 4.27.20 PM

Some were stuck.

3b Screen Shot 2014-11-30 at 5.13.27 PM

I talked to several groups, listening to their plan, asking a few questions to get them unstuck.

And then I got out colored paper on which to write the team proof.

 

The clock was ticking, but I thought that surely they would be able to trade proofs with another team for feedback within a few minutes.

 

I talked to another group. They were reflecting ∆ABC about line AC.

What will be the image of ∆ABC about line AC?

The answer? ∆ACD.

Of course that is wrong. It seems so obvious that ∆ABC is not congruent to ∆ACD. And I’m also wondering how that helps us prove that AC=BD, since BD isn’t in either of those triangles. But that’s where this team of students is. I now have the opportunity to support their productive struggle, or I can stop productive struggle in its tracks by giving them my explanation.

 

My choice? Scissors. And Paper. And more time.

What happens if you reflect ∆ABD about line AC?

4 Screen Shot 2014-11-30 at 4.31.22 PM

 

Oh! The triangles aren’t congruent.

5 Screen Shot 2014-11-30 at 4.31.34 PM

 

So are there triangles that are congruent that can get us to the diagonals?

∆ABC is congruent to ∆BAD.

6 Screen Shot 2014-11-30 at 4.31.49 PM

 

How do you know?

A reflection.

About what?

This pencil!

7 Screen Shot 2014-11-30 at 4.32.03 PM

 

So what is significant about the line that the pencil is making?

It’s a line of symmetry for the trapezoid.

It goes through the midpoints.

8 Screen Shot 2014-11-30 at 4.32.18 PM 9 Screen Shot 2014-11-30 at 4.32.27 PM

 

(One of the team members was using dynamic geometry software to reflect ∆ABC in the midst of our conversation, but I don’t have pictures of her work.)

 

So the plan was for team to write their proofs on the colored paper and then trade with other teams for feedback. Great idea, right? So how do you proceed with 15 minutes left? Proceed as planned and let them give feedback with no whole class discussion? Or have a whole class discussion to connect student work? Because as it turned out, no two teams proved the diagonals congruent the same way. I chose the latter.

 

I asked the first team to share their work.

 

Their proof needs work. But they have a good idea.

10 Screen Shot 2014-12-01 at 9.07.05 AM

They proved ∆AMD≅∆BMC, which makes the corresponding sides congruent, so with substitution and Segment Addition Postulate, we can show that the diagonals are congruent.

11 Screen Shot 2014-11-30 at 4.37.50 PM

 

Next I asked the team to share who proved ∆ABC≅∆BAD using a reflection about the line that contains the midpoints of the bases. Their written proof needs work, too. But they had a good idea.

12 Screen Shot 2014-11-30 at 4.37.34 PM

 

Another team proved ∆ACD≅∆BDC.

 

Another team constructed the perpendicular bisectors of the bases. Since the bases are parallel, a line perpendicular to one will be perpendicular to the other. I’m not sure they got to a reason that the perpendicular bisectors have to be concurrent. They could have used ∆AZD≅∆BZC to show that. Instead, they used a point Z on both of the perpendicular bisectors (they know that any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment) to reason that ∆AZB and ∆DZC are isosceles & then used Segment Addition Postulate and substitution to show that the diagonals are congruent. Not perfect. But a good start.

13 Screen Shot 2014-11-30 at 4.36.55 PM

 

NCTM’s Principles to Actions discussion on support productive struggle in learning mathematics says, “Teachers sometimes perceive student frustration or lack of immediate success as indicators that they have somehow failed their students. As a result, they jump in to ‘rescue’ students by breaking down the task and guiding students step by step through the difficulties. Although well intentioned, such ‘rescuing’ undermines the efforts of students, lowers the cognitive demand of the task, and deprives students of opportunities to engage fully in making sense of the mathematics.”

 

So while I didn’t rescue my students, we also never made it to an exemplary proof that the diagonals of an isosceles trapezoid are congruent. Did they learn something about make sense of problems and persevere in solve them? Sure. Is that enough?

 

Would it be helpful to lead off next year’s lesson with this student work? Or does that take away the productive struggle?

 

Is it just that we have to find a balance of productive struggle and what exemplary work looks like, which is easier in some lessons than others? If so, I failed at that balance during this lesson. Even so, the journey continues …

 
5 Comments

Posted by on December 1, 2014 in Geometry, Polygons, Rigid Motions

 

Tags: , , , ,

A-S-N-T-F

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

A trapezoid is ___ a parallelogram.

A parallelogram is ___ a trapezoid.

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

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

A trapezoid is a parallelogram.

A parallelogram is a trapezoid.

2014-10-30 09.06.49

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

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

Screen Shot 2014-10-30 at 9.00.46 AM

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

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

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

Screen Shot 2014-10-30 at 9.00.31 AM

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

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

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

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

Graphic Organizer

Here’s what I got back.

Screen Shot 2014-10-30 at 9.01.55 AM

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

So what next?

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

Screen Shot 2014-11-04 at 2.15.06 PM Screen Shot 2014-11-04 at 2.15.26 PM

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

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

Screen Shot 2014-10-30 at 9.16.02 AM

And decent results.

Screen Shot 2014-10-30 at 9.45.35 AM

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

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

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

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

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

Screen Shot 2014-10-30 at 9.52.42 AM Screen Shot 2014-10-30 at 9.51.43 AM

How did ten students get 144˚?

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

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

Screen Shot 2014-10-30 at 9.43.07 AM Screen Shot 2014-10-30 at 9.40.35 AM

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

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

11-05-2014 Image001 11-05-2014 Image002

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

 
2 Comments

Posted by on November 5, 2014 in Geometry, Polygons, Rigid Motions

 

Tags: , , , , , , ,