Wednesday, April 15, 2015

PARCC Practice Test Question 2 (Day 147)

Last night I tutored my geometry student. He is now in Chapter 8 of the Glencoe text, which is on Right Triangles and Trigonometry. It therefore corresponds to Chapter 14 of the U of Chicago. Here is an overview of this chapter:

Section 8-1: Geometric Mean
Section 8-2: The Pythagorean Theorem and Its Converse
Section 8-3: Special Right Triangles
Section 8-4: Trigonometry
Section 8-5: Angles of Elevation and Depression
Section 8-6: The Law of Sines and Law of Cosines
Section 8-7: Vectors

The first section corresponds to Section 14-2 of the U of Chicago, since this is where geometric means are introduced. We notice that Glencoe's Section 8-2, on the Pythagorean Theorem, actually fits the Common Core Standards better than U of Chicago's Section 8-7, since the standards state that similarity, not area, should be used to prove Pythagoras. Section 8-3 in Glencoe is clearly the same as 14-1 in the U of Chicago. Glencoe's 8-4 introduces all three major trig ratios, while U of Chicago takes two sections to cover them, 14-3 and 14-4. On the other hand, the U of Chicago doesn't have a separate section for angles of elevation like Glencoe's 8-5 -- instead they are included with tangents in Section 14-3 -- and it doesn't teach the Laws of Sines and Cosines as in Glencoe's 8-6 at all. These laws are mentioned in the Common Core Standards:

(+) Prove the Laws of Sines and Cosines and use them to solve problems.
(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

but once again, this is an issue of how one divides the standards into courses. The U of Chicago delays the laws until Precalculus -- of course, we'll find out soon enough when we cover the PARCC questions each day whether the Laws of Sines and Cosines appear on the Geometry test. Finally, Section 8-7 of Glencoe is on vectors, and information covered in all of Sections 14-5 through 14-7 of U of Chicago appear in this section.

For Glencoe's Section 8-1, I gave my student the U of Chicago Section 14-2 worksheet. He was able to figure out what geometric means are quickly. To demonstrate how a right triangle can be divided into two similar triangles, I showed him the following activity: take a rectangular sheet of paper and cut it along its diagonal to obtain two congruent right triangles. One of these triangles we keep intact, and the other we cut along its altitude to obtain two smaller right triangles. Then we show that each triangle has two of its angles congruent to those of the other triangles, so that by AA Similarity all three triangles are similar. Notice that I chose not to include this activity on the blog -- mainly because the day that we covered Section 14-2 was nowhere near an activity day. But it's still a great activity to cover at this point if one has the chance.

Once again, we discovered another mistake on my worksheet. Question 5 gives the altitude and one segment of the hypotenuse, but then asks for a leg instead of the other segment of the hypotenuse. As asked, the question could be solved using the Pythagorean Theorem rather than geometric means. I plan on changing it to the other segment of the hypotenuse, especially since this leads into Question 6 from the worksheet.

At the end I showed my student Question 7 from my worksheet -- the derivation of the Pythagorean Theorem that he will encounter in Glencoe's Section 8-2. Because of this, and the fact that I'd already shown him an activity, I didn't show him my favorite Pythagorean Theorem activity based on area and puzzle pieces -- the activity that I did post here on the blog. I may or may not show him that activity, depending on how he is doing in this chapter.

Question 2 from the PARCC Practice EOY exam is on transformations:

2. The figure shows two perpendicular lines s and r intersecting at point P in the interior of a trapezoid. Line r is parallel to the bases and bisects both legs of the trapezoid. Line s bisects both bases of the trapezoid.

Which transformation will always carry the figure onto itself? Select all that apply.
(A) a reflection across line r
(B) a reflection across line s
(C) a rotation of 90 degrees clockwise about point P
(D) a rotation of 180 degrees clockwise about point P
(E) a rotation of 270 degrees clockwise about point P

Now unlike yesterday's question, today's is the type of question that will disturb traditionalists. It requires students to be familiar with the label "trapezoid," and it's mentions the transformations that are strongly associated with Common Core.

We notice that this trapezoid is not a parallelogram, so we don't have to worry about the inclusive definition of trapezoid. Indeed, this figure is an isosceles trapezoid. To answer this question, we go back to the Isosceles Trapezoid Symmetry Theorem from Section 5-5 of U of Chicago. This theorem tells us that an isosceles trapezoid has one line of symmetry -- the perpendicular bisector of both bases of the trapezoid. This is line s. Recall that a symmetry line of a figure, as defined in Section 4-7 of U of Chicago, is a line such that the reflection image of the figure in that mirror is itself. Therefore there is only one correct answer, (B) a reflection across line s.

One way to understand this question is to notice what the correct answer would have been if we replaced "isosceles trapezoid" with another figure from the Quadrilateral Hierarchy:

kite: a reflection across line r (the line containing the diagonal bisecting the end angles)
parallelogram: a rotation of 180 degrees clockwise about point P
rectangle: a reflection across lines r and s (both perpendicular bisectors), a rotation of 180 degrees
rhombus: a reflection across lines r and s (both angle bisectors), a rotation of 180 degrees
square: all 5 answers

The U of Chicago text gives Symmetry Theorems to give us the reflecting lines. As for the rotations, recall that a rotation is defined as the composite of reflections in intersecting lines. So if a figure has two intersecting lines of symmetry, then it also has rotational symmetry -- which is why the rectangle and rhombus have rotational symmetry. But the converse is false -- a figure can have rotational symmetry without having any lines of symmetry, for example the parallelogram.

Notice that the only quadrilateral that has 90-degree rotational symmetry is the square -- and it has all five types of symmetry. On the other hand, I made no mention of a trapezoid that isn't isosceles -- that's because the general trapezoid has no symmetry. Recall that this was British mathematician John Conway's lament about the non-isosceles trapezoid.

As far as students taking the computerized tests go, one concern is that students may not realize that they can choose more than one answer. Fortunately, the isosceles trapezoid has only one symmetry line so this issue is irrelevant, but suppose the actual test uses a rectangle, or even a square, which require the student to choose multiple answers.

Because this lesson deals with the figures on the Quadrilateral Hierarchy, the lesson follows naturally from my most popular lesson that I mentioned on Monday, Section 5-2. In particular, the symmetry theorems appear in the next three sections of the U of Chicago.

PARCC Practice Test Question 2
U of Chicago Correspondence: Section 5-5, Properties of Trapezoids
Key Theorem: Isosceles Trapezoid Symmetry Theorem

The perpendicular bisector of one base of an isosceles trapezoid is the perpendicular bisector of the other base and a symmetry line for the trapezoid.

Common Core Standard:
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Commentary: There are no questions in the text exactly like this PARCC Question. There are questions asking the students to draw the symmetry lines, and the students have to know how many lines to draw. This includes Question 6, an isosceles trapezoid with a single line of symmetry, and Question 11, a rectangle with two such lines. The text does discuss the line symmetries of various quadrilaterals in addition to isosceles trapezoids. Unfortunately, the text does not cover rotational symmetry adequately. This worksheet includes questions about these other quadrilaterals.

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