I've been thinking about what I wrote in yesterday's post about the PARCC. When the Common Core standards first came out, no one was very sure how much of the new Common Core transformations would be required. We've seen that there are many theorems which can be proved either directly from the transformations, or through other shortcuts such as triangle congruence. We found out that perhaps transformations aren't required as much as many of us once thought.
I pointed out that many of my lessons and units are set up so that transformations are embedded all throughout them. Indeed, I named the current unit as the "translations unit," even though we won't actually see any translations until next week.
Earlier this week, I mentioned the black box analogy that often appears in computer science. (That day I had subbed in a Computing class -- today I'm subbing one period in a Robotics class!) Let's look back at the proof of the Isosceles Triangle Theorem that I posted yesterday:
Proof:
Statements Reasons
1.
2. Let D be the midpoint of
3.
4.
5. Triangle ADB = CDB 5. SSS Congruence Postulate
6. Angle A = C 6. CPCTC
Step 5 here needs SSS Congruence. But does it really matter to the proof of the Isosceles Triangle Theorem whether SSS was assumed as a postulate or proved as a theorem (say using the proof given here earlier on the blog)? Actually, it doesn't -- all that matters is that SSS Congruence is available to us to use in mid- and higher-level proofs. In other words, SSS is a black box -- we don't need to know what's in it (or how it was proved), but only that it can be used and works properly.
So perhaps a better way to divide my lessons into units is to use a black box. In one unit, we use transformations to develop the major theorems like SSS Triangle Congruence, and then in the next, we use the triangle congruence theorems to develop results such as Isosceles Triangle. That way, a teacher who doesn't want to prove the congruence theorems as much as I have can just skip over to the next unit and focus on the higher proofs.
Of course, many teachers have organized their lessons (including myself) so that they certainly cannot serve as black boxes. We've seen that Dr. Franklin Mason can't use SSS to prove the Isosceles Triangle Theorem because he uses the ITT theorem to prove SSS. So we have to know how SSS is proved (i.e., look inside the box) in order to use it in proofs.
And no lesson of mine is less like a black box than today's lesson. Today I move ahead to Lesson 5-5 on Isosceles Trapezoids, because I can use them to derive the properties of translations next week. I even came up with a special definition of isosceles trapezoid in order to make next week's results so much easier to prove. It would be hard for a teacher to cover my isosceles trapezoid lesson and skip my translation lesson, since they're meant to go together.
At this stage, it's too late to go back and redesign my unit as a black box. It's so much easier just to go on with the lesson as planned, but we see that what I'm working on now -- using isosceles trapezoid properties to define translations, and then using translations to develop the parallel line properties -- might not be necessary for the Common Core tests.
I'm actually wondering what this entire semester might look like if I'd planned it as black boxes. We might try something like this for the units:
1. Tools of Geometry
2. Reasoning and Proof
3. Reflections, Rotations, and Triangle Congruence
4. Using Triangle Congruence
5. Translations, Glide Reflections, and Parallel Lines
6. Using Parallel Lines
And so we see that if we want to use transformations to derive SSS and the other triangle congruence theorems, then we go to Unit 3. But if we want to use SSS to prove the Isosceles Triangle Theorem without worrying how to prove SSS (i.e., treat it as a postulate), then we do so in Unit 4. This means that we'd have to prove all of the triangle congruence theorems -- including the tricky AAS -- in Unit 3. But if we want to use, say, the usual (non-neutral) proof of AAS using Triangle Sum, we'd have to wait all the way until Unit 6 -- meaning that AAS is separated from SSS and the others by three whole units.
It's uncertain where, say, quadrilaterals belong. Some properties of quadrilaterals can be derived using triangle congruence only -- such as the kite properties. But other quadrilateral properties requires parallel lines -- most obviously those of a parallelogram. So the quadrilateral properties can't be given all in the same unit, but instead must appear two units apart -- then again, these are spread out in two different chapters -- 5 and 7 -- in the U of Chicago text.
We'd like to derive the properties in Units 3 and 5 directly from transformations, and then those in Units 4 and 6 from the theorems on triangle congruence and parallel lines. It would still look weird to cover today's lesson in Unit 4, then use it to derive translation properties in Unit 5. But I still suppose that a teacher who skips Units 3 and 5 as black boxes won't notice that we're using isosceles trapezoid properties to derive translations.
Now for today's lesson. Recall that we're defining an isosceles trapezoid to be a quadrilateral with a pair of congruent legs ("isosceles") and a pair of congruent base angles. Using this definition, we'd like to prove the following Isosceles Trapezoid Consequences:
(a) The diagonals are congruent.
(b) The other pair of base angles are congruent.
(c) The bases are parallel (i,e, that an isosceles trapezoid is really a "trapezoid").
I put (a) the diagonals before (b) the other pair of angles because (a) is easier to prove -- indeed, we end up proving (a) en route to (b). (Oh, and even though we had Saccheri quadrilaterals in mind when we came up with our definition of isosceles trapezoid, we won't use the usual terms "summit" and "summit angles" when referring to the other base or base angles. Just as with the term "Saccheri quadrilateral" itself, use of the word "summit" in a high school Geometry class is bad pedagogy.)
But (c) -- which is the property we need for translations -- is in fact the hardest to prove. Here is the proof that I had in mind:
Given: ABCD with AD = BC, Angle ADC = BCD
Prove: ABCD is a trapezoid
Proof:
Statements Reasons
1. AD = BC, Angle ADC = BCD 1. Given
2. Let M be the midpoint of
3. DM = CM 3. Definition of midpoint
4. Triangle AMD = BMC 4. SAS Congruence (steps 1, 1, 3)
5. AM = BM, Angle AMD = BMC 5. CPCTC
6. Drop a perpendicular from M 6. Uniqueness of Perpendiculars Theorem
to
7. MN = MN 7. Reflexive Property of Congruence
8. Triangle AMN = BMN 8. HL Congruence (steps 5, 7)
9. Angle AMN = BMN 9. CPCTC
10. Angle DMN = CMN, 10. Angle Addition Postulate
Angle DMN + CMN = 180
11.
12.
13. ABCD is a trapezoid 13. Definition of trapezoid (sufficient)
We see that this proof is quite long. In fact, this is one of those (rare) proofs that would be much shorter if we could use reflections instead of triangle congruence! This proof is similar to the that of the Isosceles Trapezoid Symmetry Theorem from the U of Chicago text, except using our definition of isosceles trapezoid instead of the text's:
Given: ABCD with AD = BC, Angle ADC = BCD
Prove: ABCD is a trapezoid
Proof:
Statements Reasons
1. AD = BC, Angle ADC = BCD 1. Given
2. Let m perpendicular bisector
3. D reflected over m is C 3. Definition of reflection (sufficient)
4. Ray DA reflected over m is CB 4. Reflections preserve angle measure.
5. A reflected over m is B 5. Reflections preserve distance.
6. m is perpendicular bisector
7.
8. ABCD is a trapezoid 8. Definition of trapezoid (sufficient)
Perhaps in a black box situation, we could just use the U of Chicago's definition of isosceles trapezoid and prove the necessary theorem for translations without using trapezoids. We will discuss this in more detail next week.
This is an activity day. I'd much rather post activities on Friday -- if not Friday, then Monday -- since these are when the students' minds are least alert for note taking. But we already used up an activity day on Wednesday with the Conjectures activity, and we need to cover this lesson in preparation for next week.
There's one more thing that I want to mention in this post. Again, today is not a test day, so I'm not supposed to discuss the traditionalists. But somehow, my traditionalism schedule has changed from "post every time there's a test" to "post every time either some Common Core math problem goes viral, or Dr. Katherine Beals posts her Problem of the Week on Geometry." In today's case, it's the latter -- Beals is posting this as a third grade problem, but not only is it a geometry question, but it's relevant to this week's lessons:
http://oilf.blogspot.com/2015/11/math-problems-of-week-3rd-grade-common.html
Beals asks her usual "extra credit question" to which her readers already know the intended answer:
2. For the purposes of preparing children for 21st century colleges and careers, are labels more important than concepts?
And the intended answer, of course, is "To prepare students for colleges and careers (the stated goal of Common Core), labels are very unimportant and concepts are very important." Therefore, as the traditionalist thinking goes, there should be very few label questions and very many concept questions on a test that all third graders should take.
Of course, we already know that for third graders, the emphasis should be on multiplication. A compromise can be made between simply identifying figures and doing calculations such as perimeters -- give perimeter questions where only some of the sides are given, and the students need to know the labels in order to find the missing sides. I included such problems on the worksheet.
END
There's one more thing that I want to mention in this post. Again, today is not a test day, so I'm not supposed to discuss the traditionalists. But somehow, my traditionalism schedule has changed from "post every time there's a test" to "post every time either some Common Core math problem goes viral, or Dr. Katherine Beals posts her Problem of the Week on Geometry." In today's case, it's the latter -- Beals is posting this as a third grade problem, but not only is it a geometry question, but it's relevant to this week's lessons:
http://oilf.blogspot.com/2015/11/math-problems-of-week-3rd-grade-common.html
Beals asks her usual "extra credit question" to which her readers already know the intended answer:
2. For the purposes of preparing children for 21st century colleges and careers, are labels more important than concepts?
And the intended answer, of course, is "To prepare students for colleges and careers (the stated goal of Common Core), labels are very unimportant and concepts are very important." Therefore, as the traditionalist thinking goes, there should be very few label questions and very many concept questions on a test that all third graders should take.
Of course, we already know that for third graders, the emphasis should be on multiplication. A compromise can be made between simply identifying figures and doing calculations such as perimeters -- give perimeter questions where only some of the sides are given, and the students need to know the labels in order to find the missing sides. I included such problems on the worksheet.
END
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