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Conditional Statements & Instructional Adjustments

I wrote recently about A-S-N-T-F.

I also wrote recently about what happens when students don’t recognize that a statement and its converse are different. We explicitly worked on conditional statements after that lesson.

So we defined converse/inverse/contrapositive/biconditional using symbolic logic, and then students decided which was which for a given conditional statement: If a shape has four sides, then the shape is a rectangle.

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What happens when you change which statement is the conditional statement?

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Again, students figured out which was which using the “new” conditional.

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We formatively assessed their progress on conditional statements:

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And then we were ready to start thinking about the truth value of the statements.

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Or maybe we weren’t ready actually ready, but we started thinking about the truth value anyway.

I never click on the student results for a Quick Poll in front of the students for the first time. My projector remote is an extension of my hand (except when I’ve carelessly laid it down and can’t find it), and so while students are working, I freeze the screen to look at the results and decide whether to make an instructional adjustment and if so, what instructional adjustment to make before I show the results to the students. Sometimes (as is the case in the contrapositive QP above), I have time to go talk to students who have answered incorrectly to clear up misconceptions while other students are still working. Sometimes (as is the case with this question), I deselect “Show Correct Answer” before displaying the results to students.

A student usually asks, “So who is correct?”

To which I reply, “So who is correct?”

I asked students to find someone in the room at a different table who answered differently. Convince them you are correct. Then let them convince you they are correct. (Practice construct a viable argument and critique the reasoning of others.) Then send in your answer again.

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Sometimes the responses change to 100% correct. Sometimes they don’t. And so I have to decide my next instructional adjustment. Do we have time to try this again? Or is the clock ticking more quickly?

My decision this time was to have a student come draw her counterexample.

What do we mean when we say two angles are supplementary?

Does that counterexample convince those who say the statement is true?

What if we are evaluating whether this statement is A-S-N?

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And so the journey continues, grateful for technology that gives every student in my classroom a voice – from the quietest to the loudest – so that I can make more informed decisions for when to make instructional adjustments.

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Posted by on November 29, 2014 in Angles & Triangles, Geometry

 

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Angle Bisection and Midpoints of Line Segments

As we finished Unit 2 on Tools of Geometry this year, I looked back at Illustrative Mathematics to see if a new task had been posted that we might use on our “put it all together” day before the summative assessment.

I found Angle Bisection and Midpoints of Line Segments.

I had recently read Jessica Murk’s blog post on an introduction to peer feedback, and so I decided to incorporate the feedback template that she used with the task.

The task:

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What misconceptions do you anticipate that students will have while working on this task?

 

What can you find right about the arguments below? What do you question about the arguments below?

Student A:

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Student B:

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Student C:

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Student D:

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Student E:

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Student F:

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Student G:

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Student H:

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Student I:

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Student J:

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The misconception that stuck out to me the most is that students didn’t recognize the difference between parts (a) and (b). I’ve wondered before whether we should still give students the opportunity to recognize differences and similarities between a conditional statement, its converse, inverse, contrapositive, and biconditional. We decided as a geometry team to continue including some work on building our deductive system using logic, even though our standards don’t explicitly include this work. We know that our standards are the “floor, not the ceiling”. We did this task before our work on conditional statements in Unit 3, and so students didn’t realize that, essentially, one statement was the converse of the other. Which means that what we start with (our given information) in part (a) is what we are trying to prove in part(b). And vice versa.

The feedback that students gave was tainted by this misconception.

Another misconception I noticed more than once is that while every point on an angle bisector is equidistant from the sides of the angle, students carelessly talked about the distance from a point to a line, not requiring the length of the segment perpendicular from the point to the line and instead just noting that that the lengths of two segments from two lines to a point are equal.

It occurred to me mid-lesson that maybe we should look at some student work together to give feedback. (This happened after I saw the “What he said” feedback given by one of the students.)

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I have the Reflector App on my iPad, but between the wireless infrastructure in my room for large files like images and my fumbling around on the iPad, it takes too long to get student work displayed on the board. A document camera would be helpful. But I don’t have one. And I’m not sure how I’d get the work we do through the document camera into the student notes for the day. So I actually did take a picture or too, use Dropbox to get the pictures from my iPad to my computer, and then displayed them on the board using my Promethean ActivInspire flipchart so that we could write on them. And then a few of those were so light because of the pencil (and/or maybe lack of confidence that students had while writing) that the time spent wasn’t helpful for student learning.

Looking back at Jessica’s post, I see that her students partnered to give feedback, since they were just learning to give feedback. That might have helped some, but I’m not sure that would have “fixed” this lesson.

So while I can’t say with confidence that this was a great lesson, I can say with confidence that next year will be better. Next year, I’ll give students time to write their own arguments, and then I’ll show them some of the arguments shown here and ask them to provide feedback together to improve them. Maybe next year, too, I’ll add a question to the opener that gives a true conditional statement and a converse and ask whether the true conditional statement implies that the converse must be true, just so they have some experience with recognizing the difference between conditional statements and converses before we try this task.

And so the journey continues, this time with gratefulness for “do-overs”.

 

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Logic

We spent a day or so on Logic and truth tables. It is a big deal that the conditional statement and the contrapositive are logically equivalent. So big of a deal that indirect proofs are based on that fact. So I want my students to see through truth tables that the conditional and contrapositive produce the same truth values.

We start with this example of the Law of Detachment. (A quick Internet search produced no source, and I have no memory about where I got this.)

Law of Detachment: If p →q is a true conditional and p is true, then q is true.

Example: Jordan knows that if he misses the practice the day before the game, then he will not be a starting player in the game. Jordan misses practice on Tuesday, so he concludes that he will not start in the game on Wednesday.

After an example of the Law of Syllogism (which could otherwise be known as the transitive property for conditional statements), we used a few Venn diagrams with conditional statements. We did one together.

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And then I let students move around the text to match the given conditionals.

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Is the conclusion valid? Or can you find a counterexample?

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Then we moved to a brief introduction of symbolic logic. We created truth tables for and, or, and implies. We talked about the difference between the exclusive or and or. (If you ask my daughter whether she wants chocolate cake OR strawberry pie, she will always answer both.)

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We talked about the truth values for implies. Why is it that if p is true and q is false, we say that p→q is false, but if p is false and q is true, we say that p→q is true? I remember having a hard time grasping this concept in college – and my professor used an example of it raining purple cats and dogs. Which I didn’t get. It’s taken me years to get this, but I think I finally do get it. I used the example from the Law of Detachment. The team has a rule that if you miss practice the day before the game, you don’t get to start. We’re evaluating the validity of that statement.

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The first row is the Law of Detachment. Jordan misses practice. Jordan doesn’t start. The conditional (team rule) is true.

The second row says that Jordan misses practice, but the conclusion is false. Jordan does get to start. If Jordan misses practice and does get to start in the game, then the conditional statement (team rule) is really false.

The third row says that Jordan didn’t miss practice and doesn’t start in the game. Which is fine. And doesn’t contradict the conditional statement (team rule). So we get a true outcome.

The fourth row saw that Jordan didn’t miss practice and does get to start in the game. Which is also fine. And doesn’t contradict the conditional statement (team rule). So we get a true outcome.

After practicing a few truth tables, we created truth tables for an original statement, inverse, converse, and contrapositive.

What do you notice? They each have 3 trues and 1 false. What else do you notice? (Long pauses ensue.) Finally someone notices. The outcomes for the original statement and the contrapositive are the same. The outcomes for the inverse and the converse are the same. Which is a big deal.

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The original statement and the contrapositive are logically equivalent. The inverse and the converse are logically equivalent.

We teach students that to prove a statement indirectly, they should assume the conclusion is false and show that there is a contradiction. I want my students to know why this works.

And then of course it would be nice for them to transfer this information not only to proofs but also to standardized test type questions.

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Important? I think so, but I guess I am not sure. I don’t know how many of these students will go on to major in something where they need to use indirect proofs. I just know that I wish I had some idea about why indirect proofs worked when I was in school. Either way, the journey continues …

 
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Posted by on October 22, 2013 in Angles & Triangles, Geometry

 

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