Conditional Statements, Contrapositives, and Indirect Proof

Towards the end of class a few weeks ago, we proved (using truth tables) that the original statement and its contrapositive are logically equivalent.

I sent a Quick Poll to assess student understanding and was fairly pleased with the results. (Students had only learned which was which for conditional/converse/inverse/contrapositive/biconditional statements the day before.)

We began to think about the implications of the original statement and its contrapositive having the same truth value in terms of proof, in particular, indirect proof. And then the bell rang.

When we started class the next day, I sent a Poll specifically pertaining to indirect proof, wondering whether a) what we had learned last class stuck and b) whether they were able to transfer the statement-contrapositive-same-truth-value into how to begin an indirect proof.

Here’s what I got.

What would you have done next?

I didn’t show the results of the poll to the students. Instead I gave them a different question from my stash, with more information.

We talked about the responses, and then they tried an open response question.

We talked about those responses, and then I sent back the first question.

I’ve recently read Embedding Formative Assessment by Dylan Wiliam. Wiliam, along with countless others, suggests planning ahead a sequence of questions for a learning episode along with the instructional moves you’ll make based on the feedback you get from the students. What will you do next if very few of the students get it correct? What will you do next if half of the students get it correct? What will you do next if all of the students get it correct?

If most of my students had gotten the first question about indirect proof correct, I wouldn’t have sent the question with more information about indirect proof. I would have gone straight to the open response question.

I have the luxury of teaching the same classes and planning with the same teachers from year to year. Our stash of questions to ask during the lesson has grown based on responses from students from year to year.

And so the journey continues … teaming together to decide what questions to ask and what instructional adjustments to make, based on the feedback we get from our students.

 

The Circumference of a Circle

Thanks to Andrew Stadel’s CMC-South session, we started our lesson this year with a focus on construct viable arguments and critique the reasoning of others.

Create an argument for comparing the height and circumference of the bottle.

Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility. (Thanks to Andrew for this idea in particular. I’ve had students partner with others with opposing arguments on many occasions; I had not thought about the importance of partnering with others with the same argument to make your argument stronger. In the session I attended, we shared our argument with someone who answered like we did at least twice.)

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

Uh-oh. There were apparently some pretty good convincers from the height < circumference argument. I thought fast about what to do next. I didn’t want to immediately call on someone right or wrong to share her argument with the class – I wasn’t ready for the individual/partner thinking to stop.

So without resolving the first solution, I showed another picture.

And sent another Quick Poll.

Now find someone who answered like you did. Share your arguments. Make yours stronger. Practice applying your math flexibility.

Now find someone who didn’t answer like you did. Share your arguments. Critique each other’s reasoning. Have you been convinced to answer differently?

I could go with a whole class discussion based on these results.

CK reflected on this task in a Math Practices journal: “My first instinct was to say, ‘yes, the height is greater than the circumference’, because just looking at the can gave me the impression that the circumference was not very much. Then I was told to prove my argument, so I drew a diagram. …” (I think it’s interesting that CK chose to reflect on SMP1, make sense of problems and persevere in solving them, for this task, even though I emphasized SMP3, construct viable arguments and critique the reasoning of others, in class. The practices complement each other so well.)

We went on to think a little more about pi, using some data that students had measured at home and submitted via a Google doc and some data through the automatic data capture feature of TI-Nspire.

Based on feedback from students, I think this will be the last year for our What is Pi? lesson in its current form. We are getting students in high school who have learned math with the standard: CCSS-M.7.G.B.4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. And so our students are now coming to us with some understanding of the formulas for the area and circumference of a circle, unlike before.

I’ve recently learned that several of my geometry students wish that we weren’t learning the geometry the way that we are. They like their previous math classes better because they didn’t have to always think about why.

We are trying to change the habits and practice of how students learn mathematics. Focusing on the Standards for Mathematical Practice has required me to think through and plan learning episodes differently than before. Focusing on the Standards for Mathematical Practice requires my students to interact in those learning episodes differently, even though some don’t prefer to. And so the journey continues …

 

The Base Angles of an Isosceles Triangle

Our students come to us knowing that the base angles of an isosceles triangle are congruent. But they don’t know why.

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.

We leave this proof as an exercise on the unit assessment, which is why there is so much setup in the exercise. (Didn’t you always love math textbooks that left proofs as exercises at the end of the section for you to do instead of actually working through the proofs during the section?) I do wonder what would happen if there were no hint. And if this were an exercise in class, of course I wouldn’t give a written hint. But since this is on the test, I am admittedly limiting the amount of productive struggle that I expect from my students.

How would you expect your geometry students to prove the base angles of an isosceles triangle are congruent? What misconceptions might your students have?

This year we got several of the traditional SAS (and SSS) proofs:

And we got a few of the rigid motion – reflection proofs:

I think we still need some work on these proofs … like explicitly stating that A lies on the perpendicular bisector of segment BC because it is the same distance from B as it is from C.

 

We got a long paragraph proof with the misconception that the two smaller triangles formed by the altitude will be 45˚-45˚-90˚, but then ending with an argument for reflecting the triangle about its altitude to show why the base angles are congruent.

We got another argument for constructing the altitude/angle bisector/perpendicular bisector/median from the vertex and using HL to show that the decomposed triangles are congruent.

And an argument I haven’t seen before using inequalities in triangles.

What we didn’t get were blank responses. Our students are learning to make sense of problems and persevere in solving them. Our students are learning to look for and make use of structure. Our students are learning to construct viable arguments and critique the reasoning of others. As the journey continues, our students are becoming the mathematically proficient students that we want them to become.

 

Vertical Angles Are Congruent

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.

How do we know that vertical angles are congruent, other than “my teacher told me”, or “the dynamic geometry software convinces me”. (Even though we did let our dynamic geometry software convince us, as most students had not before seen measured vertical angles move.)

Students worked individually first. I monitored their work.

How many times have you heard a student say that they don’t know where to start when writing a proof?

Can the leveled learning progression that Jill Gough (@jgough) and I have written for construct a viable argument and critique the reasoning of others help?

What information is given (or implied) in the diagram?

One student marked given information on the diagram so that she could understand it.

Another student is on her way to establishing given information and is working on communicating why her conjecture must be true.

Another student uses his given information and can get to m∠2=m∠4 but should probably show that m∠2+m∠3=m∠3+m∠4 more directly.

Without realizing it, another student is on her way to establishing the Congruent Supplements Theorem. We can see from her work that she used some angle measures to make sense of why vertical angles have to be congruent.

And another student with a “congruent supplements” argument but not written exactly the same way.

So 1 of the 31 students suggested that vertical angles are congruent because of a reflection.

What information do we need to know to define a reflection?

An object and a line.

So about what line are you reflecting ∠2 or ∠4 to show that the figures are congruent?

By the time I had made it around the room again, TL had decided that the angles should be reflected about the angle bisector of ∠3 and ∠1.

When we were ready for the whole class discussion, we started with the progression of traditional Euclidean proofs – letting each student I called on adding a bit more to the argument. Then we considered TL’s proof with rigid motions.

His argument makes sense to the class – and in fact if we test the conjecture using technology we can see that it is true:

But I wonder how we can prove the angle bisector of ∠1 is collinear with the angle bisector of ∠3 without technology. Maybe an indirect proof would work?

So is there another rigid motion that would let us show the congruence of vertical angles?

A rotation?

A rotation of what object about what point using how many degrees?

And so, together, we came up with the following argument to show that vertical angles are congruent using a rotation.

And so the journey continues … learning more about transformational geometry every day from my students, who see geometry unfold differently than I, because their study of geometry started with rigid motions.

 

Origami Regular Octagon

We folded a square piece of paper as described in the Illustrative Mathematics task, Origami Regular Octagon. I didn’t want students to know ahead of time that they were creating an octagon, so I changed the wording a bit. We folded (and refolded … luckily, there was not a 1-1 correspondence between paper squares and students). Students worked individually to write down a few observations and then we talked all together.

It’s an octagon.

There are 8 equal sides.

There are 8 equal angles.

It’s a regular octagon (this is the first year my students have come to me knowing what it means for a polygon to be regular).

How do you know there are 8 equal sides and 8 equal angles?

Because we folded it that way.

How do you know there are 8 equal sides and 8 equal angles?

Because one side is a reflection of its opposite side about the line that we folded.

What is the significance of the lines that you folded?

They are lines of symmetry.

There are 8 of them.

The opposite sides are parallel.

How can you tell?

This took a while. Maybe longer than it needed to.

Another student raised his hand.

I figured out that the sum of the angles in the octagon is 540˚.

(I don’t have the sum of the interior angles of an octagon memorized since I can calculate it, but I did know that 540˚ was too small.)

How did you get that?
I made an octagon and measured the angle. Then I multiplied by 8.

On your handheld?
Yes.

Okay. Let’s see what you have. I made him Live Presenter.

He showed us the angles he measured that were 67.5˚.

It might help if we can see the sides of your angles. Will you use the segment tool to draw them?

Other students argued that we needed to double 540 to get the sum of the angles in the octagon, 1080˚.

What else do you notice?

Triangles.

Congruent triangles.

Right triangles.

Students noticed different numbers of triangles.

And they recognized that we knew about congruence because of reflections.

Somehow we asked the question about the value of the angle (x).

I set up a Quick Poll to collect student responses.

Almost everyone got the correct answer of 22.5˚.

Can you tell me how you got your answer?

One student used the ¼ square with a 90˚ angle that had been bisected by the folded line to be 45˚ and the bisected again by the folded line to argue that x was 22.5˚.

Did anyone do something different? Hands went up all around the room.

AC hasn’t talked to the whole class yet today, so I asked what she did.

I saw a circle with 360˚ and divided by 16.

Then DC’s hand went up. 360/16 is equivalent to 180/8. I saw a line divided into 8 equal parts (or straight angle).

Then TC showed us the isosceles triangle she used with the 62.5˚ base angles.

Then someone else showed us the right triangle he used with the complementary acute angles.

Before we knew it, we had spent almost an hour talking about a regular octagon. And learning math using quite a few Math Practices: construct a viable argument and critique the reasoning of others, look for and make use of structure, use appropriate tools strategically.

I’ve wondered before how much longer we will need to talk about generalizing relationships for interior and exterior angles in polygons. Today I got a glimpse of students being able to figure out those relationships by looking for and making use of structure. The only concern that remains is the length of time it would take to do that on a high stakes standardized test such as the ACT or SAT. And so the journey to do what is best for my students continues …

 

Visual: SMP-3 Construct Viable Arguments and Critique the Reasoning of Others #LL2LU

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

I can construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP3

But…what if I can’t? What if I’m afraid that I will hurt someone’s feelings or ask a “stupid” question? How might we facilitate learning and grow our culture where critique is sought and embraced?

From Step 1: The Art of Questioning in The Falconer: What We Wish We Had Learned in School.

By learning to insert feedback loops into our thought, questioning, and decision-making process, we increase the chance of staying on our desired path. Or, if the path needs to be modified, our midcourse corrections become less dramatic and disruptive. (Lichtman, 49 pag.)

This paragraph connects to a Mr. Sun quote from Step 0: Preparation.

But there are many more subtle barriers to communication as well, and if we cannot, or do not choose to overcome these barriers, we will encounter life decisions and try to solve problems and do a lot of falconing all by ourselves with little, if any, success. Even in the briefest of communications, people develop and share common models that allow them to communicate effectively.  If you don’t share the model, you can’t communicate. If you can’t communicate, you can’t teach, learn, lead, or follow.  (Lichtman, 32 pag.)

How might we offer a pathway for success? What if we provide practice in the art of questioning and the action of seeking feedback? What if we facilitate safe harbors to share thinking, reasoning, and perspective?

 

Level 4:

I can build on the viable arguments of others and take their critique and feedback to improve my understanding of the solutions to a task.

Level 3:

I can construct viable arguments and critique the reasoning of others.

Level 2:

I can communicate my thinking for why a conjecture must be true to others, and I can listen to and read the work of others and offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the work.

Level 1:

I can recognize given information, definitions, and established results that will contribute to a sound argument for a conjecture.

 

[Cross-posted on Experiments in Learning by Doing]

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Lichtman, Grant, and Sunzi. The Falconer: What We Wish We Had Learned in School. New York: IUniverse, 2008. Print.

 

 

SMP3: Construct Viable Arguments and Critique the Reasoning of Others #LL2LU

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

I can construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP3

But…what if I can’t? What if I’m afraid that I will hurt someone’s feelings or ask a “stupid” question? How may we create a pathway for students to learn how to construct viable arguments and critique the reasoning of others?

Level 4:

I can build on the viable arguments of others and take their critique and feedback to improve my understanding of the solutions to a task.

 

Level 3:

I can construct viable arguments and critique the reasoning of others.

 

Level 2:

I can communicate my thinking for why a conjecture must be true to others, and I can listen to and read the work of others and offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the work.

 

Level 1:

I can recognize given information, definitions, and established results that will contribute to a sound argument for a conjecture.

 

Our student reflections on using the Math Practices while they are learning show that they recognize the importance of construct viable arguments and critique the reasoning of others.

Jordan 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.”

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.”

What if we intentionally teach feedback and critique through the power of positivity? Starting with I like indicates that there is value in what is observed. Using because adds detail to describe/indicate what is valuable. I wonder can be used to indicate an area of growth demonstrated or an area of growth that is needed. Both are positive; taking the time to write what you wonder indicates care, concern, and support. Wrapping up with What if is invitational and builds relationship.

Move the fulcrum so that all the advantage goes to a negative mindset, and we never rise off the ground. Move the fulcrum to a positive mindset, and the lever’s power is magnified— ready to move everything up. (Achor, 65 pag.)

The Mathy Murk has recently written a blog post called “Where do I Put P?” An Introduction to Peer Feedback, sharing a template for offering students a structure for both providing and receiving feedback.

Could Jessica’s template, coupled with this learning progression, give our students a better idea of what we mean when we say construct viable arguments and critique the reasoning of others?

[Cross-posted on Experiments in Learning by Doing]

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Achor, Shawn (2010-09-14). The Happiness Advantage: The Seven Principles of Positive Psychology That Fuel Success and Performance at Work (Kindle Locations 947-948). Crown Publishing Group. Kindle Edition.