Friday, October 10, 2014

Chapter 4 Test (Day 46)

This week is National Metric Week -- so called because it contains the tenth day of the tenth month, and the metric system is based on powers of ten.

The relationship between the Common Core Standards and the Metric System is complex. The following pro-metric site lists the standards relating to the metric system:

Grade 2: “Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes and Estimate lengths using units of inches, feet, centimeters, and meters.”
Grade 3: “Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).”
Grade 4: “Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec.”
Grade 5: “Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.”
So the Common Core Standards mention both US Customary and metric units. There's often a debate regarding whether only US Customary, or only metric, units should be taught in schools. It's likely good, therefore, that the Common Core doesn't favor either, or else the opponents of the favored system would use this as a reason to oppose the standards.
As for myself, I try to remain neutral. Advocates of both systems have their arguments -- the US Customary system has familiarity within this country, while the metric system has compatibility with other nations.
And now here's the Chapter 4 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 could be either positive 90 or negative 90, because it is not specified which of MNP or STU is the preimage and which one 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. There are four pairs of corresponding angles -- 1 and 5, 2 and 6, 3 and 7, 4 and 8. The students only have to name two of the four pairs.
7. Angles 5 and 4 are alternate interior angles.
8. The justification is: the Same-Side Interior Angle Test Theorem.
9. If two lines are cut by a transversal and corresponding angles are equal, then the lines are parallel (this is the Corresponding Angles Postulate of Dr. Franklin Mason and the U of Chicago).
10. I know this one is identical to the review question, but I kept it anyway. It's easy to forget that same-side interior angles need to be supplementary, not equal, in order for the lines to be parallel. A counterexample should be easy to draw -- just make the same-side interior angles both 60, or any other angle (except 90 -- since they'd still be supplementary as well), and the two lines will end up as intersecting lines.
11. Yes, we have a || b (by the Same-Side Interior Angle Test Theorem).
12. Here's a two-fer: yes, we have both a || b and c || d (by the Corresponding Angle Test Theorem).
13. Yes, we have a || b (by the Two Perpendiculars Theorem, which is also a Parallel Test).
14. No, we don't have enough information to conclude lines are parallel. The trick is that a is perpendicular to d while b is perpendicular to c -- in neither case do we have two lines perpendicular to the same line. It would be easy to draw a, b, c, d with a, d are perpendicular, b, c perpendicular, without any of the lines being parallel.
15. Due to formatting issues, I write this as a paragraph proof, but students should still write this as a two-column proof. But this one's easy: we are given that angles 3 and 8 are equal. These are corresponding angles, and therefore by the Corresponding Angles Test Theorem, m || n. QED
16. Notice that this one is basically the same as Question 12, except reflected (or rotated, depending on your perspective). We are given that angles 2 and 4 are equal, and angles 4 and 7 are equal. So, by the Transitive Property of Equality, angles 2 and 7 are equal. These are corresponding angles, and therefore by the Corresponding Angles Test Theorem, lines AB and CD are parallel. QED
17. 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.)
18. The converse is: if the lines are parallel, then same-side interior angles are supplementary. This is another preview, as the converses of the Parallel Tests -- the Parallel Consequences, are coming up.
19. This is our only true U of Chicago Chapter 4 question -- by the Segment Symmetry Theorem, the two lines of symmetry are the perpendicular bisector and the line containing the segment. This is yet another preview, because we'll find out soon that any figure with two intersecting lines of symmetry must have rotational symmetry as well.
20. The justification is: rotations preserve distance. This is basically the same as Question 2, except that the vertices are lettered differently and there is no accompanying pictures. Hopefully students will recognize this and get this question correct!

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