Chapter 7 Test (Day 46)

Today I subbed in a sixth grade math class. Students were working out of the Glencoe California Common Core math text -- the same text that I mentioned for eighth grade earlier on the blog.

Here is the table of contents for the sixth grade text:

1. Ratios and Rates
2. Fractions, Decimals, and Percents
3. Compute with Multi-Digit Numbers
4. Multiply and Divide Fractions
5. Integers and the Coordinate Plane
6. Expressions
7. Equations
8. Functions and Inequalities
9. Area
10. Volume and Surface Area
11. Statistical Measures
12. Statistical Displays

The students appeared to be finishing the test for Chapter 2 that they started yesterday. We just ended the first quarter, so the class ought to be finishing the Chapter 3 Test today, not the Chapter 2 Test. I am currently scheduled to sub in another middle school math class on Monday and Tuesday.

Returning to the U of Chicago text, here's the Chapter 7 Test. As usual, I provide some rationale for the answers.

1. The magnitude, or angle of rotation, is about 90 degrees. Notice that points M, N, O are collinear, as are O, S, T, in order to encourage students to look at the angle at which these two lines intersect.

2. The justification is: rotations preserve distance. Notice that we have a Reflection Postulate stating that reflections preserve distance, and a rotation is merely the composite of two reflections, each of which preserves distance. Therefore the entire rotation preserves distance as well. Indeed, we see that rotations preserve all four of the ABCD properties, as well as orientation, since the first reflection switches orientation and the second switches it back. Wu proves the properties of rotations independently of the properties of reflections, but in U of Chicago, rotations depend on reflections.

3. Angle POP" measures 180 degrees. The angle of rotation is always double the angle at which the two reflecting lines intersect. Since the reflecting lines are perpendicular, the magnitude is 180.

4. Make sure that the students rotate the triangle counterclockwise, since the angle is positive. (And speaking of positive and negative angles, notice that the answer to Question 1 is positive 90 or negative 90, because it is specified that MNP  is the preimage and STU is the image.) We see that the students are to rotate the equilateral triangle 120 degrees -- and notice that the rotation of an equilateral triangle 120 degrees ends up looking like a translation. This might make the rotation easier to perform, or it could trick the students.

5. Similarly, the rotation of an equilateral triangle 180 degrees ends up looking like a glide reflection.

6. No, since there is no AAA Congruence Theorem.

7. Yes, by SAS.

8. Yes, by HL.

9. Yes, by ASA.

10. a. SAS
      b. Angle G
      c. FG
      d. EG

11. a. SSS
      b. A <-> A, B <-> D, C <-> C

12. a. SAS
      b. N <-> O, O <-> Q, P <-> R

13. 1. FG = IG, EG = HG
      2. Vertical Angles Theorem
      3. SAS Congruence Theorem (steps 1, 2, 1)

14. This one is tricky because it appears that nothing is given!
Proof:
Statements                    Reasons
1. Circle O                    1. Given
2. AO = CO, BO = DO   2. Definition of Circle
3. Angle AOB = COD     3. Vertical Angles Theorem
4. Triangle ABO = CDO 4. SAS Congruence Theorem (steps 2, 3, 2)

15, It's Halloween weekend, so of course there must be a pentagram!
Proof:
Statements                           Reasons
1. Regular Pentagon ABCDE 1. Given
2. AB = BC = CD,                2. Definition of Regular Polygon
    Angle ABC = BCD
3. Triangle ABC = BCD        3. SAS Congruence Theorem (steps 2, 2, 2)
4. AC = BD                          4. CPCTC

16. Here's one of my favorites! We are given that the reflection of A is B, so by the definition of reflection, m is the perpendicular bisector of AB, so m is perpendicular to line AB. Similarly, the reflection of C is D, so by the definition of reflection, m is the perpendicular bisector of CD, so m is perpendicular to line CD. Therefore, by the Two Perpendiculars Theorem, lines AB and CD are in fact parallel. QED (Notice that in a way, these last few questions are previews of Chapter 5 -- we see that ABCD is an isosceles trapezoid.)

Today is a test day, so there should be some discussion about traditionalists. Well, what can I say that hasn't already been posted all this week with the Common Core elementary math problems going viral this week.

A second problem has appeared this week. A student was asked whether 103 - 28 = 75 is reasonable, but the student lost a point for finding the exact answer rather than subtracting.

Recall that in both cases, I agree with the traditionalists. I understand why so many problems like this appear -- in order to help students who can't perform the standard algorithms. They are frustrating when student who can perform them are asked to answer the Common Core way.

Enjoy your Halloween weekend!