High School: Geometry
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
You are cuddled up with a cup of joe at your favorite coffee shop, Joe's Joe, to do some grading. You decide to take break when you overhear a couple of high school students debating over one of life's toughest questions to answer: which types of cheese and bread make the best grilled cheese sandwich?
The answer is obvious to you. It's pumpernickel, mozzarella, and cheddar. But these students are making arguments for choices that go beyond personal preference. You are impressed with their critical thinking when they agree on a different answer: American and Swiss on rye. And in fact, they've managed convinced you of it, too.
The critical thinking skills used by these students are much like those that students must learn when they prove mathematical theorems. They should learn how to link what they know in a logical string to prove or disprove an argument. They should have already done this with lines and triangles, so the next logical step would be to head right into the world of parallelograms.
In order to prove theorems about parallelograms, students might want to know what a parallelogram is. (Spoiler alert: it's a quadrilateral with opposite sides that are parallel.) From there, students will want to come up with an argument to prove, whether it's that opposite sides of a parallelogram are congruent, opposite angles are congruent, or that the diagonals bisect each other.
You wouldn't order a grilled cheese just to sit there and stare at it. (Marveling at the beauty of the perfect grilled cheese is excusable and even understandable, but not tasting its delicious cheesy goodness is near blasphemous.) Just the same, students didn't learn about parallel lines, transversals, congruent triangles, and complementary angles for nothing. They should use the knowledge they already have and apply it to parallelograms.
If students are struggling, tell them that pictures are always an excellent way to start proofs. That way, they should at least be able to get off the ground to start with. Refreshing their memory about the theorems and definitions they'll be using might be helpful as well. Also, using the two-column proof format might help students organize their thoughts better, at least in the beginning. Those paragraph proofs can get messy to the point of uselessness.
Of course, that's not always the case for everything. When it comes to grilled cheese sandwiches, for example, messiness has got nothing to do with it.