This is an initial draft for the third 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.
Distance is probably the most important concept in geometry, but direction is not far behind. The two concepts complement each other wonderfully. Given two distinct points, A and B, we can think about the distance between the two points as well as the direction which is directed from A to B (which always yields a second opposite from B to A). The direction from A to any point on the ray from A to B is considered to be the same direction as A to B and we assume that translation preserves both distance and direction. Thus in one-dimension we had only two directions: the direction from 0 to 1 defined the positive direction and the direction from 1 to 0 defined its opposite, the negative direction. To extend to higher dimensions we imagine a new point that is not on our number line and thus obtain a new direction. We will still want to be able to use any two points to define a translation in such a way that if the same translation takes A to C and B to D, then the distance and direction from C to D is the same as the distance and direction from A to B and the distance and direction from A to C is the same as the distance and direction from B to D. Central to this discussion will be the notion of parallel lines. This idea of translation now forces us to consider an entire plane of points…our two-dimensional space.
One way of thinking about direction is in this “vector” sense. Two points determine a distance and a direction and define a translation by that distance in that direction. Now let us consider another way that direction will show up throughout high school mathematics. Consider a circle in the plane centered at one point A and passing through another point B. Whereas the ray from A to B considers all the points in the plane that are in the same direction as A to B, but that are at any possible distance. The circle considers all the points the in the plane that are at the same distance from A as B, but that are in any possible direction. So choose a point C on this circle. This defines a direction from A to C, but we can also think about that in terms of a direction relative to the direction from A to B. We do that by defining an angle BAC which is defined as the region between the ray A to B and the ray A to C. We can “measure” that angle in several ways that connect the angle with how much of the circle it intersects.
The notion of direction will not only be key in thinking about translation, rotation, reflection, parallel lines, and angle measurements. It will also be key when we think about the slope of a line. We are thinking of a line already as being connected with a certain direction (and its opposite). So if we establish one horizontal line and positive direction on that line, we can think of a line’s direction as being determined by its “angle of elevation” ie one of the smaller angles formed with the horizontal. Once we establish the notion of similar triangles we can say that this in turn defines a fixed ratio, the slope the line. In a later course on trigonometry we give names to the functions (tangent and arctangent) defined by this correspondence.
(A) Key Ideas: Students should understand the definition of angle, degrees, parallel lines, and perpendicular lines. (Note that we define parallel lines so that a line is also parallel to itself.) This week should also build up the intuitive arguments (which will turn into formal proofs next week) for many of the consequences of parallel lines. In particular it should make sense that opposite angles have the same measure, a transversal intersects parallel lines in angles with the same measure, a triangle’s interior angles sum to 180 degrees, and a polygon’s exterior angles sum to 360 degrees. It is nice to see proofs and one of the themes throughout this course is why we do we want to prove things. That being said, the intuitive arguments are more important because they will be more likely to be retained and applied. Students should have an understanding of translation this week that is starting to build more formally on the intuition. That is they should understand why the notion of parallelograms will be key for formalizing our intuitive idea of a translation moving each point. Again hopefully the desire to extend our thinking to higher dimensions can motivate this move to formalism.
(B) Related standards from the common core:
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
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. (This week we make the intuitive arguments for many of the key theorems that will be followed up with formal proofs in the next chapter when we formally introduce rotations and reflections).
(C) Connections with what students have done: Students have done much of this work intuitively but have not begun to think about them more deeply in a way that can formalize. They’ve often done translations in terms of coordinates, but not without coordinates.
(D) Connections with what students will do: The thinking this week about direction will set us up to formally define rotations and reflections next (chapter 4) and later dilations (chapter 8). It will also be critical as think about lines and slopes (chapter 18). We are also setting up the idea more generally of a vector (chapter 23) and laying the groundwork for a better understanding of the tangent function and its inverse (10th grade) and complex numbers (Precalculus).
(E) Texts and videos to which students and parents could refer to help understand key ideas: In volume 1 of his geometry text (unit 3, pages 48-68), James Tanton uses a “pencil trick” to both explore intuitively why a triangles interior angles should sum to 180 degrees and why the trick is inherently making some hidden assumptions about parallelism. He then returns to the notion of parallelism later (unit 8, pages 156-190). Tanton notes that he is taking a different approach from Euclid, but it is also a somewhat different approach from common core.
I am also hoping to write an essay and video to be used as a reference for these ideas. If you know of any other references or resources that would be particularly good for 9th graders please let know.
(F) Questions and tasks that can generate discussion to explore key ideas in class: I like the debate starter: Is a line parallel to itself? This can lead to the idea that WE make choices when we formally define terms, but those choices are made to try to capture some intuition. In particular it can lead to a discussion of what is the important idea we want to capture with paralellism.
Tanton’s “pencil trick” and its hidden assumptions should lead to a nice discussion. I would love to get any other ideas you may have.
(G) Questions and tasks that can be used to help assess student understanding of key ideas: I would consider asking if a student can explain in his or her own words what the problem with the “pencil trick” is. I am going to try to develop some new tasks as well and any ideas you have would be appreciated.
(H) Interesting and challenging ways to extend and apply the key ideas: In the last chapter when thinking about distance we thought about a geometry where the Pythagorean theorem didn’t hold. This week in thinking about direction, it is nice to consider spherical geometry where our assumptions about parallelism doesn’t hold (and thus where the interior of a triangle doesn’t sum to 180 degrees). Tanton does this naturally as part of the “pencil trick” described above.
(I) Standard textbook problems related to the key ideas: There are many problems concerning finding missing angles in a diagram involving parallel lines, transversal, and triangles. Students should be able to answer any such problem.
(J) Misconceptions and “tricks” to avoid: I don’t know of any.
If you have any questions, comments, and/or resources, please share them. As you can see I’m especially in search of tasks and discussion starters for this week. Thanks for your help.