This is an initial draft for the fourth chapter of a course guide for a year-long 9th grade Algebra & Geometry course. I am requesting your assistance. For more information on the course and this project, please see this post.
One of the biggest challenges in a high school geometry course is to motivate the formalism. Why do we need precise definitions? Why must we prove things that seem obvious? Even things that don’t seem obvious, like the Pythagorean Theorem, why not just accept them as facts because we’ve been “taught” that they’re true? Why are we often asked to prove things that seem irrelevant? One approach is to give up on the formalism all together. Have students measure the angles of a bunch of triangles with a protractor and “observe” that they all seem to add up to 180 degrees. Another approach is to just follow Euclid (mistakes and all) and painstakingly work ones way up carefully until eventually something interesting is proven. The common core seeks to strike a balance by starting with a few assumptions and definitions that seem reasonable, but which also allow one to quickly prove key geometric facts such as those needed to define the concepts of slope and coordinates.
Even with that compromise, though, there still remains the issue of motivation. In this course I’ve been trying my best to build up some motivation for the formalism instead of starting off on Day 1 with a bunch of definitions and axioms. In chapter one we hopefully came to realize that formalism would be necessary to abstract to study something like higher dimensions where we cannot rely on intuition and that even in lower dimensions patterns don’t always generalize as we might expect them to. In chapter two we got a glimpse of a situation where the Pythagorean Theorem wouldn’t hold and in chapter three a glimpse of a situation in which a triangle’s angles didn’t sum to 180 degrees. These explorations, even if quite brief, will hopefully challenge us to be explicit about our assumptions. Finally we saw when looking at translations in chapter three that some understanding of parallelograms is going to be essential.
So that brings us to this week where we formally define the notion of rotation and reflection, and fill in the necessary details from the previous chapter concerning translation and parallelism. I expect to make great use of Geogebra this week, and perhaps a final act of motivation for formalism can be thinking about how does the computer determine precisely where to put the image of a point under these transformations.
(A) Key Ideas: Students should understand the formal definitions of rotation, reflection, and translation. Parentheses notation, however, can wait until the next chapter when there is more motivation for it. More than just knowing the definitions of these rigid transformations, though, students should also understand how those formal definitions allow us to prove statements.
(B) Related standards from the common core:
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (This week will just focus on translations and to formally define them we’ll need some facts about parallelograms which will come out of rotations. So the formal definitions of rotations, reflections, and translations will come next week, but the groundwork is laid in this week.)
Prove theorems about lines and angles, triangles, and parallelograms. Theorems we’re likely to prove are: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoint; measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; opposite sides and angles of a parallelogram are congruent.
(C) Connections with what students have done: Students have often worked with reflecting over an axis using coordinates, but have not thought about how one would formally define reflection over an arbitrary line. In general this week is about providing some formalism to many ideas they have already explored intuitively.
(D) Connections with what students will do: The formal definitions of these rigid transformations will allow us to define congruence and explore symmetry in the next chapter. This is then in turn key to defining similarity (chapter 8) and congruent triangles (chapter 9). The definition of translation and parallelograms is central to our system of coordinates (chapter 18).
(E) Texts and videos to which students and parents could refer to help understand key ideas: I have yet to see a common core geometry text. So for now, the best reference I’ve found is courtesy of Hung-Hsi Wu, but designed for teachers of geometry. His Teaching Geometry in Grade 8 and High School According to the Common Core Standards can serve as a reference. In this week the focus is on defining the rigid motions (pages 96-110), perpendicular bisectors (pages 118-119) and parallelism (pages 127-130). In Wu’s notes the last two sections come after defining congruence, but the proofs depend only on our definitions and assumptions concerning rigid motions.
(F) Questions and tasks that can generate discussion to explore key ideas in class: Much of the discussion driven intuitive development has already been done in past weeks. One idea I have for the move to formalism is to ask students to try to consider (1) what information we need to provide to Geogebra in order to define a rotation or reflection and (2) how does Geogebra use that information to determine where a point should go.
(G) Questions and tasks that can be used to help assess student understanding of key ideas: One task I like is to give students a triangle with vertices labelled and ask them to find the image of given points under given transformations. NO COORDINATES ARE USED. Even though the placement of image points will need be somewhat imprecise, where students place them can reveal misunderstandings and whether students are actually using definitions. For example, students often place the reflection of a point C across the line AB on the line AC even if the angle CAB is not right. Students often neglect to realize that rotating point C in an angle CAB (oriented appropriately) around point A should place it on the ray AB.
In terms of proofs, I like a task where I ask students to take a point A, reflect it over a line PQ (not passing through A), to obtain an image B. They are then asked to translate that point B by the vector PQ to obtain an image C. Finally the image of C by reflection across the line PQ gives a point D. Students are then asked to prove that ABCD is a rectangle. A proper proof requires knowledge of the precise definitions of rectangle, reflection, and translation. Attempts to prove it without those definitions can bring to light some student misunderstandings.
(H) Interesting and challenging ways to extend and apply the key ideas: Our definition of reflection makes use of the concept of perpendicular bisector which we are then able to show is the set of all points equidistant from two points. A circle is the set of all points equidistant from a single point. It can be interesting to explore this notion of equidistance further. What would happen if we wanted a point (or points?) equidistant from three points or four points. How would any of these figures change if we used taxicab geometry?
(I) Standard textbook problems related to the key ideas: As I noted above I have yet to see a common core geometry text, so I know of no standard textbook problems dealing with these precise definitions.
(J) Misconceptions and “tricks” to avoid: In my experience the main thing to watch out for is students not realizing a need for definitions and thinking it is okay to just rely on an intuitive sense of what is meant by these transformations.
If you have any questions, comments, and/or resources, please share them. I would especially love reference resources or suggestions for how to help lead students from intuition to formalism. Thanks for your help.