1-2. constructions (or drawings). Notice that a construction for #2 is halfway to constructing a square inscribed in a circle for Common Core.

3. 95 degrees.

4. 152.5 degrees.

5. 87 degrees.

6. 86 degrees.

7.

*x*degrees. This is almost like part of Euclid's proof of the Isosceles Triangle Theorem (except I think that his proof focused on the linear pairs, not the vertical angles).

8. 27 degrees.

9a.

*x*= 60

9b. 61, 62, 58 degrees.

10. 25, 69, 128, 138 degrees.

11. polygon, quadrilateral, parallelogram, rectangle, square.

12. kite.

13. rectangle.

14. false. A counterexample is found easily.

15. Yes, the perpendicular bisector of the bases.

16. 46. Although I mentioned it briefly this year during Chapter 2,

*perimeter*is a concept that could be developed more in these early lessons. My question actually defines

*perimeter*since my lessons haven't stressed the concept yet. This is the simplest possible perimeter problem that I could have covered, where only the definition of kite is needed to find the two missing lengths. I could have given an isosceles trapezoid instead, where the Isosceles Trapezoid Theorem is needed to find a missing side length. Or since I squeezed in the Properties of a Parallelogram Theorem in our Lesson 5-6 (as part of proving that every rhombus is a parallelogram), I could have even put a parallelogram here with only two consecutive side lengths given.

17. The conjecture is true, and is a key part of the proof of Centroid Concurrency Theorem.

18. Statements Reasons

1. angle

*G*= angle

*FHI*1. Given

2.

*EG*| |

*FH*2. Corresponding Angles Test

3.

*EFHG*is a trapezoid 3. Definition of trapezoid (inclusive def. -- it could be a parallelogram)

19. Statements Reasons

1.

*O*and

*P*are circles 1. Given

2.

*OQ*=

*OR*,

*PQ*=

*PR*2. Definition of circle (meaning)

3.

*OQPR*is a kite 3. Definition of kite (inclusive def. -- it could be a rhombus)

20. Figure is at the top, then below it is quadrilateral. Branching out from it are kite, trapezoid. Then below trapezoid is parallelogram. Kite and parallelogram rejoin to have rhombus below. (Once again, these are inclusive definitions!).

As today is a test day, it ought to be time for another discussion about traditionalists. At first I wasn't going to have another traditionalism topic so soon, but I decided that I will anyway. This is because there is still so much going on with -- you guessed it -- the

*Atlantic*article written by traditionalists Drs. Katharine Beals and Barry Garelick.

When I first started discussing the article, it had drawn 250 comments. Since then, the number of comments have nearly doubled. Rather than pointing out specific comments, I will link to and quote Dr. Beals, who divides the criticisms of her article into seven categories:

http://oilf.blogspot.com/2015/11/explaining-your-math-highly.html

--

*What the Common Core actually says:*Some people stated that there’s nothing in the Common Core itself that requires students to explain answers and thought processes verbally and diagrammatically. (We noted that there are parts of the Common Core that nonetheless can be, and are being, interpreted in this way).

--

*Our specific examples:*Some claimed that examples we used to illustrate work showing aren’t representative of what’s generally going on, and that good teachers would be more flexible and reasonable about what constitutes adequate explanations. (That would be nice, if so.)

--

*Who is actually affected*: Some claimed that our objections apply only to very small subsets of kids. (We pointed out that these practices are problematic for all students, and in particular for second language learners and children with language impairments).

--

*The virtues of showing your work:*Some people conflated showing work (which we agree is reasonable wherever there’s work to show) with explaining answers and thought processes verbally and diagrammatically.

--

*The virtues of doing math proofs*: Some people conflated doing math proofs (which we agree there should be more of in high school math) with explaining answers and thought processes verbally and diagrammatically. (We pointed out that there is actually relatively little emphasis on mathematic proofs in both the various Common Core-inspired curricula and tests).

--

*Communication skills necessary for math-related professions:*Some people believe that having students provide the sorts of verbal and diagrammatic explanations we critique in our article will help prepare future engineers and scientists for the communicative demands of their jobs. (The question then is whether engineers and scientists who learned math in pre-answer-explaining times are deficient in their communication skills compared with their more contemporary counterparts.)

--

*Counter-exemplary anecdotes:*Some people described how well explaining answers and diagramming thought processes work for their students or kids.

--

*Faith*: Some people are sure that meta-cognitive processes are the best way to develop conceptual understanding. (We would say that a better way is to emulate the countries that outcompete us in math, giving kids more direct instruction and individualized practice in conceptually challenging math problems).

Let me comment on some of these categories. We see that the first category is "What the Common Core actually says." Actually, here I'd agree somewhat with the authors of the article. If many teachers, schools, and districts making significant changes to their curriculum in the name of Common Core, then those changes are

*de facto*Common Core. This includes those changes that lean towards progressive math that we've seen in the various elementary school horror stories -- such changes were driven by the transition to Common Core math.

Indeed, the individual Common Core Standards might not require students to explain their answers, but we

*know*that the PARCC and SBAC (especially the performance task) do. Certainly, any curriculum change inspired by the PARCC and SBAC exams is part of Common Core.

Finally, Garelick points out that the need to explain answers is mentioned by the Core after all:

Regarding the issue that CC doesn't require explanations, etc. We quote something that appears on the CC website, that calls for students to justify their answers. Website is not the standards, true, but still... Then, the content standards themselves contain the phrases "students shall explain.." and "student shall understand that..." . The intro to CC standards states that whenever "understand" appears in the content standards, to link that with the Standards for Mathematical Practice, the first of which is "Make Sense of Problems and Perservere in Solving Them". That SMP states in part:

"Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches."

Such language has served as gasoline thrown on the fire of math reform that has been burning for many years, and which has fetishized conceptual understanding.

"Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches."

Such language has served as gasoline thrown on the fire of math reform that has been burning for many years, and which has fetishized conceptual understanding.

*math*problems asked in the name of Common Core are apolitical (and to me, there aren't enough of the few that are for me to say that Common Core Math is political).

And I've already mentioned that I agree with Beals's second point, "Our specific examples" (which leads to her fourth point, "The virtues of showing your work"). I don't dispute the fact that many Common Core questions require the students to explain their answers more thoroughly than necessary to demonstrate understanding.

Of course, the point most relevant to Geometry is her fifth, "The virtues of doing math proofs." Beals points out that writing proofs is

*not*the same as explaining answers, and that she wants to see more of the former, but less of the latter. In fact, she says in her next post:

http://oilf.blogspot.com/2015/11/explaining-your-math-highly_17.html

And there were those who conflated explaining answers and thought processes verbally and diagrammatically with showing work. Others seemed to think that the kind of work displays Barry and I were endorsing was work consisting only of mathematical symbols. But there are plenty of words that go into work-showing (and proofs), including reasons (“given"; “side-angle-side"; “without loss of generality”) and units (“miles per hour”; "liters of water loss per minute"; “pounds of salt per pounds of total mixture”).

I can only imagine what a traditionalist would say if he or she were looking at the test that I'm putting up today. A traditionalist would be glad that this test doesn't require long explanations of answers, and that there are two proofs on the test. But a traditionalist would probably find these two particular proofs too simple, and would also criticize the test for containing too many "label" questions -- especially those referring to the Quadrilateral Hierarchy. He or she might appreciate Question 16 -- even though it mentions a quadrilateral label (kite), it is also a concept question as it requires the students to use knowledge of the definition to solve a

*perimeter*problem.

Let's look at Beals's final point, "Faith." She writes that pro-Common Core reformers take it as a matter of faith that metacognition is the key to to developing understanding, and argues that it would be better to follow strong math nations and give students "more direct instruction and individualized practice in conceptually challenging math problems." Beals repeats this in her following post:

In short, my answer is more teacher-led discussion of underlying concepts, with teachers calling on students, as appropriate, to develop concepts and explanations (for which Japanese classrooms as described in this discussion [linked from her website] are a good model); and more individualized practice with conceptually challenging math problems.

Here is the link that Beals gives on her blog:

http://www.cs.nyu.edu/faculty/siegel/ST11.pdf

Naturally, the Japanese geometry lesson mentioned in the PDF caught my eye:

*Then a student presents his solution. The construction is clearly correct, and he starts out with a correct explanation. However, when the time comes to find the solution, he gets lost and cannot see how to apply the area preserving*

**transformation**that solves the problem. The teacher then tells him to use “the red triangle” as the target destination.[emphasis mine]

And we notice something strange here --

*transformation*? But I thought that traditionalists like Beals hate transformations in the Common Core, yet here she appears to be endorsing a Japanese geometry lesson that mentions transformations! Let's keep on reading:

*Next, the teacher explains how to solve the problem. There are two equivalent answers that correspond to moving vertex C, in the context of Figure 1, to the left or to the right. Both directions solve the problem, and he shows this. Such duality should not be surprising, since the word problem is not described in a way that, in the context of Figures 1, 6, and 7, can distinguish left from right. For completeness, we show the two ways that the triangle*

**transformation**technique can be used to solve the problem. In order to make the connection between the review material and the follow-up Eda-Azusa exercise absolutely clear, the solution with its two versions have been rotated to present the same perspective as in Figure 1, which introduced this triangle**transformation**technique.[emphasis mine]

The transformations being referred to in the Japanese lesson aren't reflections, rotations, translations, or dilations, but something called

*transvections*. (I mentioned transvections on the blog last month, when I was discussing Mandelbrot fractals.) Just as a translation is often called a "slide" and a rotation is often called a "turn," a transvection is often called a "shear."

Transvections have the following properties:

-- Transvections preserve betweenness, collinearity, and orientation.

-- Transvections, like reflections, have a fixed line (that is, not only is the line invariant, but also every point on it is a fixed point).

-- Every line parallel to the fixed line is invariant.

-- Along each invariant line, the distance between a point and its image is proportional to the distance between the invariant line and the fixed line. The constant of proportionality is called

*shear factor.*

-- Points on opposite sides of the fixed line move in opposite directions.

-- Parallel lines are mapped to parallel lines.

-- Distance is

*not*preserved (unless both points are on the same invariant line), nor are angle measures (except 180), but...

-- Area

*is*preserved.

This last property is the most important one. A figure and its transvection image always have the same area. The Japanese geometry lesson shows that the area of a triangle depends only on its base and height -- not by using an area formula, but by using transvections.

Let's conclude this post by pointing out regarding "Faith" -- it's

*traditionalists*who show faith. They have faith that simply by giving "individual practice with conceptually challenging math problems," the students will actually

*do*the math problems. Again, students learn more by

*doing*a progressive lesson than by

*throwing away*a traditionalist problem set on the way out the door.

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