Today I subbed in a math class at a continuation school -- which means that the students were at many different levels. One thing I noticed near the front of the classroom were textbooks for Integrated Math II and III. These texts were published by Pearson -- and one reason that many people oppose Common Core is that they feel it's less about helping students and more about making money for Pearson.
Unfortunately, I didn't see a copy of their Integrated Math I text anywhere. But let's review the contents of the two texts that I do see.
Pearson Mathematics II
Volume 1
Chapter 1: Reasoning and Proof
Chapter 2: Proving Theorems About Lines and Angles
Chapter 3: Congruent Triangles
Chapter 4: Proving Theorems About Triangles
Chapter 5: Proving Theorems About Quadrilaterals
Chapter 6: Similarity
Chapter 7: Right Triangles and Trigonometry
Chapter 8: Circles
Volume 2
Chapter 9: Surface Area and Volume
Chapter 10: Properties of Exponents With Rational Exponents
Chapter 11: Polynomials and Factoring
Chapter 12: Quadratic Functions
Chapter 13: Probability
Chapter 14: Other Types of Functions
Chapter 15: Sequences and Series
Pearson Mathematics III
Volume 1
Chapter 1: Drawing Conclusions From Data
Chapter 2: Linear Equations, Inequalities, and Functions
Chapter 3: Quadratic Functions and Equations
Chapter 4: Polynomials and Polynomial Functions
Chapter 5: Rational Expressions and Functions
Chapter 6: Radical Expressions and Functions
Volume 2
Chapter 7: Exponential and Logarithmic Functions
Chapter 8: Trigonometric Functions
Chapter 9: Sequences and Series
Chapter 10: Applying Geometric Concepts
Chapter 11: Connecting Algebra and Geometry
Chapter 12: Circles
Since I don't have a copy of the Math I text, I can't make any conclusions. But I've always thought that on the Integrated Pathway, geometry would be split among all three years of math. Yet we notice that almost the entire geometry curriculum appears in Math II, Volume 1, with only surface area and volume appearing in the second volume.
I assume that Pearson intended for the two volumes into which the company divides both Math II and III to correspond to the semesters. So we see that the first semester of Math II is all Geometry, while the second semester is a mixture of Algebra I and II. The first semester of Math III starts the same way that an Algebra II course would, while the second semester ends with a little bit of Geometry (as Chapter 11 implies, it connects the two subjects).
Let's focus on Math II Volume 1, as this text is the most relevant to our geometry blog. Section 1-1 of the text is on basic constructions -- and I already see something that I don't like about the text. We recall that the Prentice-Hall text that David Joyce criticizes on his website also begins with constructions, and Joyce doesn't like that one bit. Joyce writes:
On pages 40 through 42 four constructions are given: 1) to cut a line segment equal to a given line segment, 2) to construct an angle equal to a given angle, 3) to construct a perpendicular bisector of a line segment, and 4) to bisect an angle. Later in the book, these constructions are used to prove theorems, yet they are not proved here, nor are they proved later in the book. There is no indication whether they are to be taken as postulates (they should not, since they can be proved), or as theorems. At the very least, it should be stated that they are theorems which will be proved later. In summary, the constructions should be postponed until they can be justified, and then they should be justified.
Notice that the four constructions in the Pearson text are exactly the same as the four in Prentice-Hall, even numbered in the same order! I checked the list of authors for the Pearson text and saw that there are several, since both algebra and geometry writers are needed. But I notice that one of the geometry authors is Laurie Bass, who is also, according to Joyce, the author for Prentice-Hall -- small wonder, then, that the same four constructions are here. The fact that Bass doesn't prove any of her constructions is especially galling considering that on last Monday's featured PARCC question, students had to justify the angle bisector construction (the last construction of Section 1-1)! Moreover this lesson also requires students to purchase compasses close to the first day of school.
Section 1-2, on Patterns and Inductive Reasoning, is a much better lesson. I've mentioned that many texts have an early lesson on patterns, but the U of Chicago doesn't. If I were teaching a class using the Pearson text, I'd be tempted to skip Section 1-1 entirely and start the course with 1-2. In short, Chapter 1 of the Pearson text corresponds mostly to Chapter 2 of the U of Chicago text, with some material from later chapters. Section 1-5, on Deductive Reasoning, contains some of the logic from U of Chicago's Chapter 13, and Section 1-7, on Proving Angles Congruent, contains some of the material from Chapter 3.
Pearson's Chapter 2 is on Lines and Angles. Section 2-2 introduces a postulate on parallels -- and it happens to be the Same-Side Interior Angles Postulate. This is highly unusual, as most texts choose either Corresponding Angles (U of Chicago) or Alternate Interior Angles as a postulate. Section 2-3 contains the Parallel Tests -- and all of the are listed as theorems. The first test is the Converse of the Corresponding Angles Theorem, and its proof is said to be given in Section 13-5 -- but that chapter is on Probability. I once saw another text that does give the proof -- it's an indirect proof, and so Pearson probably intended to wait until Chapter 4 to give the proof, but it mistakenly omits it. In summary, most of this chapter corresponds to Chapter 3 of the U of Chicago text, except for Section 2-5 on Parallel Lines and Triangles, which contains another Parallel Postulate (Playfair's), which isn't given in the U of Chicago until Chapter 13 (where it's actually proved as a theorem).
Pearson's Chapter 3 is on Congruent Triangles. We notice that in Sections 3-2 and 3-3, SSS, SAS, and ASA are given as postulates. But then in true Common Core fashion, congruence transformations (isometries) are used to verify these postulates in Section 3-8. I notice that as important as the isometries are to Common Core, translations, reflections, and rotations aren't defined anywhere in this section. I suspect that since these isometries form the foundation of Common Core Geometry, Pearson actually defines them in its Math I text. There's no way for me to know for sure since I don't have access to the Math I text. In short this corresponds to the first part of Chapter 7 in U of Chicago, except for isosceles triangles, which appear at the start of Chapter 5.
Pearson's Chapter 4 is on Proving Theorems About Triangles. Here is where the concurrency theorems are shown. These appear in Chapter 10 of Dr. M but nowhere in the U of Chicago. In Section 4-5, indirect proofs appear for the first time, and the rest of the chapter is devoted to proving the inequalities associated with triangles, including Triangle Inequality and Hinge Theorem. These appear in Chapter 5 of Dr. M but are scattered in the U of Chicago, as I discussed here last week.
Pearson's Chapter 5 is on Proving Theorems About Quadrilaterals. Most of this material appears in the same-numbered chapter of the U of Chicago, except some of the theorems on parallelograms don't appear until Chapter 7. Of course I had to check to see that Section 5-7 of Pearson gives the exclusive definition of trapezoid. As I said earlier, the inclusive definition was spotted in a PARCC question, but I have yet to see it. Then the rest of the chapter gets into coordinate geometry -- which I suppose makes sense as many coordinate proofs involve quadrilaterals. The harder concurrency theorems from Chapter 4 are given coordinate proofs here as well. Many of these coordinate proofs require the Distance or Midpoint Formulas, which are never introduced. Once again, I suspect that these formulas are taught in the Math I text and students are expected to remember them.
Pearson's Chapter 6 is on Similarity. This corresponds to Chapter 12 of the U of Chicago. Right triangle similarity appears in Section 6-4, but not until Chapter 14 in the U of Chicago. The chapter ends by defining dilations and similarity transformations -- and just as in Chapter 3, these are used to verify AA, SAS, and SSS Similarity,
Pearson's Chapter 7 is on Right Triangles and Trigonometry. This corresponds to Chapter 14 of the U of Chicago. Angles of elevation/depression, inverse trig, and areas of regular polygons using the apothem all appear in this chapter, but are only mentioned briefly if at all in the U of Chicago.
Pearson's Chapter 8 is on circles. In some ways, this is more like Dr. M's Chapter 9 than the U of Chicago's Chapter 15, since all of the circle material is right here in one chapter. Section 8-2 is on the areas of a circle. Notice that the area of no other basic figure appears in any chapter of this text. In some ways, this is alarming because the text defines the area of a segment of a circle as the area of a sector minus the area of a triangle -- except that the area of a triangle isn't defined in the text. Once again, we can only assume that any gaps must be covered in the Math I text.
As I mentioned earlier, Chapter 9 is the first chapter of Volume 2, and this chapter is on surface area and volume. Whenever I look at the curriculum for any text, I like to see whether I can time it so that the lessons mentioning pi can occur around Pi Day. In this text, Chapters 8 and 9 mention pi. But it would be tough to squeeze in all of Chapters 10-15 between mid-March and the PARCC exam (recall that PARCC does have an Integrated Math test), especially considering that these contain some tough algebra lessons.
For Volume 2, I will only mention the remaining chapters that are relevant to geometry. Chapter 12 is on quadratic equations, including completing the square and the quadratic formula -- but then it ends up with the equation of a circle. This is convenient since we know that equations of a circle that require completing the square appear on the PARCC.
As I mentioned, Integrated Math III contains some geometry. But notice that Chapter 8 of Math II and Chapter 12 of Math III are identical chapters. Every exercise in both chapters is the same, and Section 12-1 of Math III even mentions Postulate 19 and Theorem 72, even though the first 18 postulates and 71 theorems don't appear in the Math III text! I know that much of the material from Algebra I is repeated in Algebra II and then again in Precalculus, but this is the first time that I've seen a textbook publisher try to get away with including identical material in different courses!
As for the other geometry chapters in Math III, Chapter 10 includes some key problems that appear on the PARCC exam. These include some more constructions in Section 10-1 (most notably the inscription of a hexagon in a circle, specifically mention in Common Core), density in Section 10-2 (which we've already seen on the PARCC), areas of similar figures in Sections 10-3 and 10-6.
Chapter 11 contains some area formulas -- including the ones that I thought were missing from the Math II text. I wouldn't be surprised to find out that Chapter 11, like 12, is the repetition of a topic from an earlier text.
I spent much of today's blog discussing the Pearson Integrated Math texts, but notice that none of the students at the continuation school where I was subbing actually studied from any of them. Some of the students were reading the McDougal-Littell Algebra I text -- recall that continuation students are behind in credits, so these students are one of the last cohorts to take an Algebra I course. The other students were reading Mathematics: A Human Endeavor by Harold R. Jacobs. This text appears to teach a stripped-down version of Algebra II, as many Algebra II topics (sequences, quadratic equations, logarithms, conic sections) appear at a basic level. Section 10-2 of this text is on the Seven Bridges of Konigsberg -- my first day of school activity. The students who were reading the Jacobs text were officially enrolled in "Coordinated Math." Most of the students were only on Chapters 2 or 3 of their respective texts. Of course, the students who were classified as juniors still had to take the SBAC -- and they were pulled out of classes today to do so.
I've been thinking about integrated math courses. We've spent time looking at the Pearson Integrated Math texts, and we've already discussed the Saxon and Singapore Integrated texts on the blog. And so, if I'm going to spend an entire post criticizing the Pearson text, I should ask myself, what would my ideal Integrated Math classes look like?
I notice that much of the high school material naturally falls into three dimensions. Think about it -- my Integrated Math I course can begin with geometry, from lines, angles, and transformations all the way up to similarity (Chapters 1 through 7 and 12 of U of Chicago). Then we go from similarity to the coordinate plane and the constancy of a line's slope -- and that's a great point to segue into linear (one-dimensional) equations and algebra. We could probably cover the first semester of an Algebra I course before the end of the first year.
Integrated Math II stands for the second dimension. So on the geometry side, this class should focus on measuring two-dimensional figures -- perimeter and area. Many area questions lend themselves to quadratic equations, and so the algebra side of the class should work on quadratic functions, like the second half of an Algebra I class.
Integrated Math III stands for the third dimension. By now, you've figured out that the geometry side of the class should have surface area and volume. For algebra, we can focus on cubic polynomials, and some non-polynomials functions studied in Algebra II, like exponential functions. So far, this division is incomplete, but if I do it right, I can cover all of the algebra and geometry needed for Common Core and make connections between the topics better than Pearson can, which is what Integrated Math is all about. For example, Chapter 10 in Pearson's Math III ends with the concept of locus -- a set of all points satisfying a certain condition. This would be a great place to segue into conic sections, which are defined as sets of points satisfying certain conditions -- in a word, loci.
Before I leave the classroom where I subbed, I saw that there were a few worksheets left over from an activity that this class did last week. I noticed that this activity was divided into parts, "Act One," "Act Two," "Act Three," "Sequels," and "Title." And of course, as soon as we see three acts, the readers of this blog and I should immediately recognize it as a Dan Meyers Three-Act activity.
The question in Act One is, "Where will the stacks of cups tie?" So I went directly to the dy/dan website and tried to find this Three-Act activity, but I couldn't find it anywhere. The following link was the closest activity that I could find:
http://blog.mrmeyer.com/2008/linear-fun-2-stacking-cups/
How many Styrofoam cups would you have to stack to reach the top of your math teacher’s head?
So I suppose that "tie" here must mean "tied score." If the stack of cups is shorter than the teacher, then the stack "loses," if the stack of cups is taller, then the stack "wins," and so if the stack of cups is the same height as the teacher, then the stack "ties." Yes, this is an entertaining activity that could fit in Geometry, Algebra I, or Integrated Math I (the one-dimensional course).
So let's get back to PARCC now. Question 28 of the PARCC Practice Test is still another transformations question. There are some tricky reflections and rotations mentioned in this one:
Section 1-2, on Patterns and Inductive Reasoning, is a much better lesson. I've mentioned that many texts have an early lesson on patterns, but the U of Chicago doesn't. If I were teaching a class using the Pearson text, I'd be tempted to skip Section 1-1 entirely and start the course with 1-2. In short, Chapter 1 of the Pearson text corresponds mostly to Chapter 2 of the U of Chicago text, with some material from later chapters. Section 1-5, on Deductive Reasoning, contains some of the logic from U of Chicago's Chapter 13, and Section 1-7, on Proving Angles Congruent, contains some of the material from Chapter 3.
Pearson's Chapter 2 is on Lines and Angles. Section 2-2 introduces a postulate on parallels -- and it happens to be the Same-Side Interior Angles Postulate. This is highly unusual, as most texts choose either Corresponding Angles (U of Chicago) or Alternate Interior Angles as a postulate. Section 2-3 contains the Parallel Tests -- and all of the are listed as theorems. The first test is the Converse of the Corresponding Angles Theorem, and its proof is said to be given in Section 13-5 -- but that chapter is on Probability. I once saw another text that does give the proof -- it's an indirect proof, and so Pearson probably intended to wait until Chapter 4 to give the proof, but it mistakenly omits it. In summary, most of this chapter corresponds to Chapter 3 of the U of Chicago text, except for Section 2-5 on Parallel Lines and Triangles, which contains another Parallel Postulate (Playfair's), which isn't given in the U of Chicago until Chapter 13 (where it's actually proved as a theorem).
Pearson's Chapter 3 is on Congruent Triangles. We notice that in Sections 3-2 and 3-3, SSS, SAS, and ASA are given as postulates. But then in true Common Core fashion, congruence transformations (isometries) are used to verify these postulates in Section 3-8. I notice that as important as the isometries are to Common Core, translations, reflections, and rotations aren't defined anywhere in this section. I suspect that since these isometries form the foundation of Common Core Geometry, Pearson actually defines them in its Math I text. There's no way for me to know for sure since I don't have access to the Math I text. In short this corresponds to the first part of Chapter 7 in U of Chicago, except for isosceles triangles, which appear at the start of Chapter 5.
Pearson's Chapter 4 is on Proving Theorems About Triangles. Here is where the concurrency theorems are shown. These appear in Chapter 10 of Dr. M but nowhere in the U of Chicago. In Section 4-5, indirect proofs appear for the first time, and the rest of the chapter is devoted to proving the inequalities associated with triangles, including Triangle Inequality and Hinge Theorem. These appear in Chapter 5 of Dr. M but are scattered in the U of Chicago, as I discussed here last week.
Pearson's Chapter 5 is on Proving Theorems About Quadrilaterals. Most of this material appears in the same-numbered chapter of the U of Chicago, except some of the theorems on parallelograms don't appear until Chapter 7. Of course I had to check to see that Section 5-7 of Pearson gives the exclusive definition of trapezoid. As I said earlier, the inclusive definition was spotted in a PARCC question, but I have yet to see it. Then the rest of the chapter gets into coordinate geometry -- which I suppose makes sense as many coordinate proofs involve quadrilaterals. The harder concurrency theorems from Chapter 4 are given coordinate proofs here as well. Many of these coordinate proofs require the Distance or Midpoint Formulas, which are never introduced. Once again, I suspect that these formulas are taught in the Math I text and students are expected to remember them.
Pearson's Chapter 6 is on Similarity. This corresponds to Chapter 12 of the U of Chicago. Right triangle similarity appears in Section 6-4, but not until Chapter 14 in the U of Chicago. The chapter ends by defining dilations and similarity transformations -- and just as in Chapter 3, these are used to verify AA, SAS, and SSS Similarity,
Pearson's Chapter 7 is on Right Triangles and Trigonometry. This corresponds to Chapter 14 of the U of Chicago. Angles of elevation/depression, inverse trig, and areas of regular polygons using the apothem all appear in this chapter, but are only mentioned briefly if at all in the U of Chicago.
Pearson's Chapter 8 is on circles. In some ways, this is more like Dr. M's Chapter 9 than the U of Chicago's Chapter 15, since all of the circle material is right here in one chapter. Section 8-2 is on the areas of a circle. Notice that the area of no other basic figure appears in any chapter of this text. In some ways, this is alarming because the text defines the area of a segment of a circle as the area of a sector minus the area of a triangle -- except that the area of a triangle isn't defined in the text. Once again, we can only assume that any gaps must be covered in the Math I text.
As I mentioned earlier, Chapter 9 is the first chapter of Volume 2, and this chapter is on surface area and volume. Whenever I look at the curriculum for any text, I like to see whether I can time it so that the lessons mentioning pi can occur around Pi Day. In this text, Chapters 8 and 9 mention pi. But it would be tough to squeeze in all of Chapters 10-15 between mid-March and the PARCC exam (recall that PARCC does have an Integrated Math test), especially considering that these contain some tough algebra lessons.
For Volume 2, I will only mention the remaining chapters that are relevant to geometry. Chapter 12 is on quadratic equations, including completing the square and the quadratic formula -- but then it ends up with the equation of a circle. This is convenient since we know that equations of a circle that require completing the square appear on the PARCC.
As I mentioned, Integrated Math III contains some geometry. But notice that Chapter 8 of Math II and Chapter 12 of Math III are identical chapters. Every exercise in both chapters is the same, and Section 12-1 of Math III even mentions Postulate 19 and Theorem 72, even though the first 18 postulates and 71 theorems don't appear in the Math III text! I know that much of the material from Algebra I is repeated in Algebra II and then again in Precalculus, but this is the first time that I've seen a textbook publisher try to get away with including identical material in different courses!
As for the other geometry chapters in Math III, Chapter 10 includes some key problems that appear on the PARCC exam. These include some more constructions in Section 10-1 (most notably the inscription of a hexagon in a circle, specifically mention in Common Core), density in Section 10-2 (which we've already seen on the PARCC), areas of similar figures in Sections 10-3 and 10-6.
Chapter 11 contains some area formulas -- including the ones that I thought were missing from the Math II text. I wouldn't be surprised to find out that Chapter 11, like 12, is the repetition of a topic from an earlier text.
I spent much of today's blog discussing the Pearson Integrated Math texts, but notice that none of the students at the continuation school where I was subbing actually studied from any of them. Some of the students were reading the McDougal-Littell Algebra I text -- recall that continuation students are behind in credits, so these students are one of the last cohorts to take an Algebra I course. The other students were reading Mathematics: A Human Endeavor by Harold R. Jacobs. This text appears to teach a stripped-down version of Algebra II, as many Algebra II topics (sequences, quadratic equations, logarithms, conic sections) appear at a basic level. Section 10-2 of this text is on the Seven Bridges of Konigsberg -- my first day of school activity. The students who were reading the Jacobs text were officially enrolled in "Coordinated Math." Most of the students were only on Chapters 2 or 3 of their respective texts. Of course, the students who were classified as juniors still had to take the SBAC -- and they were pulled out of classes today to do so.
I've been thinking about integrated math courses. We've spent time looking at the Pearson Integrated Math texts, and we've already discussed the Saxon and Singapore Integrated texts on the blog. And so, if I'm going to spend an entire post criticizing the Pearson text, I should ask myself, what would my ideal Integrated Math classes look like?
I notice that much of the high school material naturally falls into three dimensions. Think about it -- my Integrated Math I course can begin with geometry, from lines, angles, and transformations all the way up to similarity (Chapters 1 through 7 and 12 of U of Chicago). Then we go from similarity to the coordinate plane and the constancy of a line's slope -- and that's a great point to segue into linear (one-dimensional) equations and algebra. We could probably cover the first semester of an Algebra I course before the end of the first year.
Integrated Math II stands for the second dimension. So on the geometry side, this class should focus on measuring two-dimensional figures -- perimeter and area. Many area questions lend themselves to quadratic equations, and so the algebra side of the class should work on quadratic functions, like the second half of an Algebra I class.
Integrated Math III stands for the third dimension. By now, you've figured out that the geometry side of the class should have surface area and volume. For algebra, we can focus on cubic polynomials, and some non-polynomials functions studied in Algebra II, like exponential functions. So far, this division is incomplete, but if I do it right, I can cover all of the algebra and geometry needed for Common Core and make connections between the topics better than Pearson can, which is what Integrated Math is all about. For example, Chapter 10 in Pearson's Math III ends with the concept of locus -- a set of all points satisfying a certain condition. This would be a great place to segue into conic sections, which are defined as sets of points satisfying certain conditions -- in a word, loci.
Before I leave the classroom where I subbed, I saw that there were a few worksheets left over from an activity that this class did last week. I noticed that this activity was divided into parts, "Act One," "Act Two," "Act Three," "Sequels," and "Title." And of course, as soon as we see three acts, the readers of this blog and I should immediately recognize it as a Dan Meyers Three-Act activity.
The question in Act One is, "Where will the stacks of cups tie?" So I went directly to the dy/dan website and tried to find this Three-Act activity, but I couldn't find it anywhere. The following link was the closest activity that I could find:
http://blog.mrmeyer.com/2008/linear-fun-2-stacking-cups/
How many Styrofoam cups would you have to stack to reach the top of your math teacher’s head?
So I suppose that "tie" here must mean "tied score." If the stack of cups is shorter than the teacher, then the stack "loses," if the stack of cups is taller, then the stack "wins," and so if the stack of cups is the same height as the teacher, then the stack "ties." Yes, this is an entertaining activity that could fit in Geometry, Algebra I, or Integrated Math I (the one-dimensional course).
So let's get back to PARCC now. Question 28 of the PARCC Practice Test is still another transformations question. There are some tricky reflections and rotations mentioned in this one:
Quadrilaterals ABCD and EFGH are shown in the coordinate plane.
(Here are the vertices: A(-4, 4), B(-3, 7), C(1, 6), D(-2, 2), E(4, -4), F(3, -7), G(-1, -6), H(2, -2).)
Part A:
Quadrilateral EFGH is the image of ABCD after a transformation or sequence of transformations.
Which could be the transformation or sequence of transformations?
Select all that apply.
(A) a translation of 3 units to the right, followed by a reflection across the x-axis
(B) a rotation of 180 degrees about the origin
(C) a translation of 12 units downward, followed by a reflection across the y-axis
(D) a reflection across the y-axis, followed by a reflection across the x-axis
(E) a reflection across the line with equation y = x
Part B:
Quadrilateral ABCD will be reflected across the x-axis and then rotated 90 degrees clockwise about the origin to create quadrilateral A'B'C'D'. What will be the y-coordinate of B'?
The answer to Part A may be obvious just by looking at the coordinates above, but I only post them because there's no other way to convey to the readers of this blog in ASCII what the quadrilaterals look like. But the student will see only the quadrilaterals, not the list of coordinates. So let's approach this from a student's perspective.
The easiest thing for the student to check is not the coordinates of the vertices, but the orientation. So we notice that both ABCD and EFGH have a clockwise orientation. This means that we can eliminate three of the choices -- the reflection and the two glide reflections. As for the two remaining choices, one might recall that the composite of a y-axis reflection and an x-axis reflection actually is a rotation of 180 degrees about the origin. After a quick check of corresponding coordinates -- for example, we count off the graph to see the coordinates B(-3, 7) and F(3, -7) -- we see that the this transformation really is a point inversion about the origin. So both of the remaining choices, (B) and (D), are correct.
For Part B, we start with point B(-3, 7). The x-axis reflection switches the sign of the y-coordinate only, so that gives us (-3, -7). The 90-degree rotation clockwise about the origin rotates the point from the third back into the second quadrant, to B'(-7, 3). So the y-coordinate of B is 3.
Common mistakes for Part A include choosing transformations that describe some of the points, but not all of them. For example, if we reflect A(-4, 4) about the line y = x, we do get E(4, -4), and likewise the reflection of D(-2, 2) is H(2, -2). But the mirror images of B and C are not F and G, so choice (E) is wrong. In a similar manner, performing the glide reflection in choice (C) maps C to F, but none of the other points are mapped correctly. Only (B) and (D) map all of the points on the graph to their proper images.
There are a few more comments to make about this problem. First of all, we notice that the transformations in choices (A) and (C) really are glide reflections. The phrase "glide reflection" doesn't appear on the PARCC, but they are listed by the information that matters most for a glide reflection, namely the translation and the reflection. We see that the PARCC will perform the translation first -- as opposed to the U of Chicago which always performs the reflection first. It doesn't really matter as the translation and reflection will always commute in this case, but it's good to know how glide reflections will be written on the PARCC.
I wrote earlier that there are some transformations that are easy enough to appear on the PARCC, and some that shouldn't appear at all. All translations are easy, but only certain reflections and rotations are easy enough for the students. (By this measure, a glide reflection is easy provided the reflection part of it is.) The easiest reflections and rotations are the ones that appear in choices (B) and (D) -- reflections across the axes, rotation 180 about the origin. The next best reflections and rotations are the ones that appear in choice (E) and Part B: reflection across the line y = x (or y = -x), and rotation of 90 degrees in either direction about the origin.
Of these, reflection in y = x is the easiest -- the image of (x, y) is just (y, x). Reflection in the other line y = -x maps (x, y) to (-y, -x), so it switches the coordinates along with their signs. Rotations of 90 degrees are important because they appear in the proof of the perpendicular slope formula. Rotating (x, y) 90 degrees clockwise gives (y, -x), and rotating it 90 degrees counterclockwise gives (-y, x). In either case, a rotation of 90 degrees maps the line passing through the origin and (x, y) (notice that the slope is y/x) to the line passing through the origin and (y, -x) (notice that the slope is -x/y). This constitutes a coordinate proof of the slopes of perpendicular lines as opposite reciprocals. QED
All four of the transformations switch the coordinates of their preimages, but they each have a different effect on the signs of the coordinates. Since tomorrow is Memorial Day weekend, I can make today an activity day. The following is a short activity -- one of my favorites that I like to show during tutoring -- that helps students remember how to perform these four transformations.
We take a small coordinate plane and label its axes x and y -- actually, let's mark the signs as well, so we have all four directions +x (east), +y (north), -x (west), and -y (south). We should label these on the front and back (or just use marker, so the directions bleed through the paper). We cut it out and place it on a larger coordinate plane, also with the four directions marked.
Now we can perform all of the various transformations. Let's say we want to do the transformations listed in Part B. To reflect over the x-axis, we fold and ultimately flip the smaller plane. Now the +y of the larger plane lines up with the -y of the larger plane. This tells us that to reflect over the x-axis, we must switch +y with -y (i.e., change the sign of y). Restoring the smaller plane to its original position, we now fold and flip over the diagonal line y = x. Now the +y of the larger plane lines up with the +x of the smaller plane, indicating that we switch x and y. If we want to try a rotation, let's put the plane in its original position and try a 90-degree rotation clockwise. Now the +x of the larger plane lines up with +y of the smaller and +y of the larger plane lines up with -x of the smaller. Thus (x, y) of the larger (original) plane maps to (y, -x) of the smaller (rotated) plane.
To perform a composite, we perform one transformation right after the other. First let's reflect across the x-axis, so +y original is now -y transformed, but +x didn't change. Now we rotate 90 degrees in the clockwise direction. So now we have +x original lined up with -y transformed and +y original lines up with -x transformed, so (x, y) now maps to (-y, -x). Thus B(-3, 7) maps to B'(-7, 3) and so the answer to Part B is 3.
Notice that (x, y) maps to (-y, -x) in Part B. This is the same as a reflection across y = -x. The set of all eight rotations and reflections mentioned today form a group -- the symmetry group of a square centered at the origin and sides parallel to the axes.
Oh, and here's the answer to yesterday's question -- find some (discontinous) transformation T such that T o T is reflection in the x-axis:
Divide the plane into horizontal strips. Each strip covers a different section of the y-axis. One strip is for y = 0 (i.e., the x-axis itself). The other strips cover the interval (n - 1, n] on the y-axis for each natural number n. Then T acts on each horizontal strip as follows:
n = 0: T maps the x-axis to itself
n > 0 and odd: T translates one unit up
n > 0 and even: T reflects across the line y = 1/2
n < 0 and odd: T translates one unit down
n < 0 and even: T reflects across the line y = -1/2
I was going to make this challenge part of the activity today, but I realized that this will be too difficult even for the brightest high school students. So I threw it out and replaced it with Dan Meyer's cup activity instead.
Thus concludes this post. My next post will be Tuesday, May 26th. Happy Memorial Day!
PARCC Practice EOY Exam Question 28
Common mistakes for Part A include choosing transformations that describe some of the points, but not all of them. For example, if we reflect A(-4, 4) about the line y = x, we do get E(4, -4), and likewise the reflection of D(-2, 2) is H(2, -2). But the mirror images of B and C are not F and G, so choice (E) is wrong. In a similar manner, performing the glide reflection in choice (C) maps C to F, but none of the other points are mapped correctly. Only (B) and (D) map all of the points on the graph to their proper images.
There are a few more comments to make about this problem. First of all, we notice that the transformations in choices (A) and (C) really are glide reflections. The phrase "glide reflection" doesn't appear on the PARCC, but they are listed by the information that matters most for a glide reflection, namely the translation and the reflection. We see that the PARCC will perform the translation first -- as opposed to the U of Chicago which always performs the reflection first. It doesn't really matter as the translation and reflection will always commute in this case, but it's good to know how glide reflections will be written on the PARCC.
I wrote earlier that there are some transformations that are easy enough to appear on the PARCC, and some that shouldn't appear at all. All translations are easy, but only certain reflections and rotations are easy enough for the students. (By this measure, a glide reflection is easy provided the reflection part of it is.) The easiest reflections and rotations are the ones that appear in choices (B) and (D) -- reflections across the axes, rotation 180 about the origin. The next best reflections and rotations are the ones that appear in choice (E) and Part B: reflection across the line y = x (or y = -x), and rotation of 90 degrees in either direction about the origin.
Of these, reflection in y = x is the easiest -- the image of (x, y) is just (y, x). Reflection in the other line y = -x maps (x, y) to (-y, -x), so it switches the coordinates along with their signs. Rotations of 90 degrees are important because they appear in the proof of the perpendicular slope formula. Rotating (x, y) 90 degrees clockwise gives (y, -x), and rotating it 90 degrees counterclockwise gives (-y, x). In either case, a rotation of 90 degrees maps the line passing through the origin and (x, y) (notice that the slope is y/x) to the line passing through the origin and (y, -x) (notice that the slope is -x/y). This constitutes a coordinate proof of the slopes of perpendicular lines as opposite reciprocals. QED
All four of the transformations switch the coordinates of their preimages, but they each have a different effect on the signs of the coordinates. Since tomorrow is Memorial Day weekend, I can make today an activity day. The following is a short activity -- one of my favorites that I like to show during tutoring -- that helps students remember how to perform these four transformations.
We take a small coordinate plane and label its axes x and y -- actually, let's mark the signs as well, so we have all four directions +x (east), +y (north), -x (west), and -y (south). We should label these on the front and back (or just use marker, so the directions bleed through the paper). We cut it out and place it on a larger coordinate plane, also with the four directions marked.
Now we can perform all of the various transformations. Let's say we want to do the transformations listed in Part B. To reflect over the x-axis, we fold and ultimately flip the smaller plane. Now the +y of the larger plane lines up with the -y of the larger plane. This tells us that to reflect over the x-axis, we must switch +y with -y (i.e., change the sign of y). Restoring the smaller plane to its original position, we now fold and flip over the diagonal line y = x. Now the +y of the larger plane lines up with the +x of the smaller plane, indicating that we switch x and y. If we want to try a rotation, let's put the plane in its original position and try a 90-degree rotation clockwise. Now the +x of the larger plane lines up with +y of the smaller and +y of the larger plane lines up with -x of the smaller. Thus (x, y) of the larger (original) plane maps to (y, -x) of the smaller (rotated) plane.
To perform a composite, we perform one transformation right after the other. First let's reflect across the x-axis, so +y original is now -y transformed, but +x didn't change. Now we rotate 90 degrees in the clockwise direction. So now we have +x original lined up with -y transformed and +y original lines up with -x transformed, so (x, y) now maps to (-y, -x). Thus B(-3, 7) maps to B'(-7, 3) and so the answer to Part B is 3.
Notice that (x, y) maps to (-y, -x) in Part B. This is the same as a reflection across y = -x. The set of all eight rotations and reflections mentioned today form a group -- the symmetry group of a square centered at the origin and sides parallel to the axes.
Oh, and here's the answer to yesterday's question -- find some (discontinous) transformation T such that T o T is reflection in the x-axis:
Divide the plane into horizontal strips. Each strip covers a different section of the y-axis. One strip is for y = 0 (i.e., the x-axis itself). The other strips cover the interval (n - 1, n] on the y-axis for each natural number n. Then T acts on each horizontal strip as follows:
n = 0: T maps the x-axis to itself
n > 0 and odd: T translates one unit up
n > 0 and even: T reflects across the line y = 1/2
n < 0 and odd: T translates one unit down
n < 0 and even: T reflects across the line y = -1/2
I was going to make this challenge part of the activity today, but I realized that this will be too difficult even for the brightest high school students. So I threw it out and replaced it with Dan Meyer's cup activity instead.
Thus concludes this post. My next post will be Tuesday, May 26th. Happy Memorial Day!
PARCC Practice EOY Exam Question 28
U of Chicago Correspondence: Section 6-6, Isometries
Key Theorem: Definition of glide reflection
Let r_m be a reflection and T a translation with nonzero magnitude and direction parallel to m. Then G = T o r_m is a glide reflection.
Let r_m be a reflection and T a translation with nonzero magnitude and direction parallel to m. Then G = T o r_m is a glide reflection.
Common Core Standard:
CCSS.MATH.CONTENT.HSG.CO.A.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Commentary: The U of Chicago defines glide reflections as T o r_m, but on the PARCC, what appears is r_m o T. Reflections over y = x and 90-degree rotations don't appear at all.
CCSS.MATH.CONTENT.HSG.CO.A.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Commentary: The U of Chicago defines glide reflections as T o r_m, but on the PARCC, what appears is r_m o T. Reflections over y = x and 90-degree rotations don't appear at all.
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