December 3, 2012

Geometry Unit 2: Lines and Angles

Most geometry courses start with the study of "point, line, plane" but I was tired of that discussion.  Kids have been plotting points and graphing lines on the plane for years and I need to start assuming they learned in past courses (and fill in gaps if I see them) rather than planning to review everything.  I want more depth in my classes, so that means trusting students have enough background that when I say "point" they can conceptualize what I'm referring to.  Given all of that, we jumped right into lines and angles.

To introduce each unit I have some open question(s) for students to explore.  For this unit we looked at: "What can you make with 2 lines? 3 lines?"  Before we started I asked them to write down everything they knew about lines.  Students recalled words like parallel, perpendicular, intersecting and slope.  Then they started drawing lines, observing angles and previewing ideas that would come up (all the way through when we draw 2 parallel lines and a transversal).  I used some more hint cards which asked questions about max and min numbers of angles, number of intersections, types of angles etc.  We ended up with open questions such as "are obtuse angles always opposite each other?" which we picked right up on the next class.

Our first 'lesson' (focused exploration) I had everyone draw a pair of intersecting (but not perpendicular) lines and make observations about that particular diagram.  They discovered equal angles (which I shared we call vertical angles) and supplementary angles next to each other (which we call adjacent).  The journal question at the end of class asked "What are two ways we used the word vertical today?"  (i.e. yup, I know it's weird but let's just get it out there now- there are vertical lines and vertical angles and they're different things.)

Next day: Now that we've been talking all about angles, we should probably decide what an angle really is.  So, we break them apart and realize they're made of a point (vertex) and parts of lines (rays).  Good to know.  Lines and rays are both infinite, but somehow lines seems longer.  That's an interesting concept!  Journal: Compare and contrast lines and rays.

Continuing on we mix in some algebra, learn to bisect and define linear pair.

To many people this unit seems backwards.  Rays are one of the last things we define and the word segment has yet to appear, we talk all about angles for two blocks without defining them!  I'm striving to have vocabulary come after students have already studied something and they are struggling to describe it, or are tired of saying an entire phrase (we could say: "the angles opposite from each other when lines intersect" but mathematicians are lazy and so they say: "vertical angles").  I also recognize that (especially at the beginning) high school geometry is refining knowledge that students already have.  They can draw an angle without knowing that it's made of a vertex and two rays, they also have probably heard the words vertex and ray before, but we don't need that vocabulary to discuss when angles are equal.  I want to introduce words when we need them.  Otherwise, words are things to memorize.  Geometry is still a lot of vocabulary that they need to learn, but at least if students have some experience with a concept they can attach the word to the memory rather than trying to understand the concept while learning the word.

Next up is parallel lines and transversals, and I put together a GeoGebra experiment for them to explore these angles.  Apparently I haven't blogged this yet.  I'll add it to the 'to write' list.  [12/9/12 edited to add link and strikethrough sentences about not having a link.] Again though, students explore, make observations, then get the names for the angle pairs.

To apply all that they have learned about angles in an interesting way, students fold a piece of origami paper in a specific way that creates lots of nice angle pairs.  Ever so appropriately I call this task Origami Angles. This activity is also our introduction to proof!

In the past I've had some fun with writing crazy sentences and having kids write the inverse, converse and contrapositive.  However, none of those terms are in the common core and we never used inverse or contrapositive again after this one section on writing sentences, so I dropped it down to just introducing the term converse this year.  In particular, we wrote the converse of all the parallel line and transversal statements- once students were convinced that congruent angles meant parallel lines we knew another way to prove lines parallel.

We ended this unit with some more formal proofs using proof cards.

Common Core State Standards:

G.CO.9: prove theorems about lines and angles (vertical angles, alternate interior, corresponding)
G.GPE.5: prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.
G.CO.1know the precise definitions of: angle, parallel, perpendicular 

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