But I did finally tutor my geometry student today, and once again, I showed him some worksheets from earlier. Today, his school covered Section 1-6 of a text published by Glencoe, a section called "Two-Dimensional Figures." This section covers the perimeter and area of rectangles, circles, and triangles, as well as the Distance Formula and coordinate geometry.
Recall that as David Joyce points out with the Prentice Hall text, giving the Distance Formula, well before its proof using the Pythagorean Theorem, is a terrible idea. I find it fascinating that so many texts would give the Distance Formula in Chapter 1. And I've mentioned many times that I don't want to scare students just coming off of algebra with so many heavy-duty formulas in the first chapter -- so I'm glad that the U of Chicago waits until the second half of the book before teaching area and the distance formula (Chapters 8 and 11).
I showed my student my worksheets from Sections 2-1 and 2-7 of the U of Chicago text -- about convexity and polygons. My student did well on them -- to him, they were a breath of fresh air after seeing so many hard triangle perimeter problems requiring the Distance Formula.
And today, we now see why my plan to post the Fifth Postulate Lesson on Day 55 -- all those fives coming together -- backfired. Aligning my schedule to the school calendar at my first district meant that I needed to finish Chapter 5 by Day 56 in order to finish Chapters 6 and 7 by the end of the semester, so today, Day 55, needs to be test review, not a new lesson.
There are many important concepts covered on this test -- isosceles triangles, quadrilaterals, and parallel lines. Here are the types of questions that appear on this test -- all of these coming from the SPUR section of Chapter 5 of the U of Chicago text.
- The first two questions direct the student to draw an example of the figure using ruler, compass, or protractor. I don't expect most classrooms to have a compass -- and if the students don't have a ruler and protractor, close enough is good enough. For students with a straightedge and compass, the necessary constructions are perpendicular lines (for the scalene right triangle) and congruent line segments (for the isosceles acute triangle -- a single arc is sufficient).
- Question 3 requires use of the Trapezoid Angle Theorem and algebra. Once again, this test will require the most algebra of any first semester chapter.
- Questions 4-5 require use of the Parallel Consequences and algebra. The first question is the Alternate Interior Angles Consequence while technically, the second one is actually Same-Side Exterior Angles. Savvy students should be able to figure it out by using, for example, the Linear Pair Theorem followed by Corresponding Angles. One way I remember is that any two angles formed by two parallel lines and a transversal are either equal or supplementary -- and one can eyeball it to see whether the angles are acute or obtuse to tell which ones are which.
- Questions 6-7 require use of the Isosceles Triangle Theorem -- neither one is straightforward, though, as either the Linear Pair or Vertical Angles Theorems are necessary. Once again, the students can look for acute and obtuse angles to find which ones are equal or supplementary.
- Questions 8-9 require use of the Triangle-Sum Theorem. The first one only needs arithmetic, while the second needs algebra.
- Question 10 requires use of the Quadrilateral-Sum Theorem. Once again, linear pairs, vertical angles, and algebra are needed.
- Question 11 requires use of the Quadrilateral Hierarchy Theorem. Students must arrange the various shapes from most general to most specific.
- In Questions 12-13, students must identify the quadrilaterals. The second one is a rhombus, not necessarily a square, since nowhere is it stated that the angles are right angles. Some might note that here I appear to contradict myself -- didn't I just say that in Question 5, students are supposed to assume that angle 1 is obtuse and angle 7 is acute in order to determine that they are supplementary, but here in Question 13 students should not assume that any of the angles are right angles? The difference is that in Question 5, identifying acute and obtuse angles is used as a mnemonic to remember actual theorems such as Linear Pair or Corresponding Angles Consequence, but in Question 13 nothing warrants knowing that there are any right angles.
- Question 14 is a true-or-false question about the Quadrilateral Hierarchy. If the statement is false, the students should draw a square that isn't a parallelogram. But they shouldn't -- because the statement happens to be true.
- In Question 15, students must determine whether the quadrilateral has any symmetry lines. As it turns out, the figure is a kite, so by the Kite Symmetry Theorem, it has a symmetry line.
- Question 16 is another true-or-false question about the Quadrilateral Hierarchy, but this time students are not directed to draw a counterexample. As it turns out, the statement is true -- that the bases have a common bisector is mentioned in the Isosceles Trapezoid Symmetry Theorem.
- In Question 17, students are to evaluate a conjecture. The U of Chicago text asks students to rank on an A-E scale whether they believe the conjecture to be true or false -- but I don't do this on a test, or otherwise every student would just choose C ("I'm not sure!"). So the students' only choices are true and false. As it turns out, the conjecture is true, and it can proved simply by knowing that half of 180 is 90. But students don't have to prove it -- proofs can wait until...
- ...Questions 18-19. The keys to the first question are the Alternate Interior Angles Test and the definition of trapezoid. The keys to the second question are two definitions -- those of circle and kite. Notice that the first proof underscores the preference for inclusive rather than exclusive definitions. Under the exclusive definition, EFHG could be a parallelogram -- and the way it is drawn, it almost looks like a parallelogram. We can't prove that it's a parallelogram -- but we can't prove that it's not a parallelogram either, which means that under the exclusive definition, we can't prove that it's a trapezoid either! The second question isn't a problem for the exclusive definition -- OQPR is clearly nonconvex while all parallelograms are convex. But even if point O had been drawn outside of circle P, we could still prove that it's an inclusive kite, but it could be a parallelogram (hence a rhombus) and not an exclusive kite.
- Question 20 has the students draw part of the Quadrilateral Hierarchy. It goes without saying that inclusive definitions should be used, so a rectangle is an isosceles trapezoid.