All has been quiet blog-wise on the geometry front the past week or two because I have spent all of my time mulling over the CCSS Geometry scope and sequence. I am using both the PARCC model content framework (geometry is on page 47) and the arrangement into courses by the Bill and Melinda Gates Foundation and Pearson Foundation (geometry starts on page 22).
We have completed two units so far – Unit 0 on Constructions (including medians, altitudes, perpendicular bisectors, and angle bisectors of triangles – I just couldn’t bear to give students a compass, straightedge, and list of directions for making a construction without giving them a reason to do the construction) – and Unit 1 on Rigid Motions (translations, rotations, and reflections). Now we are left with the following learning objectives for Congruence.
G-CO 9: Prove theorems about lines and angles. Theorems include: 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 endpoints.
G-CO 10: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G-CO 8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G-CO 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.
The Illustrative Mathematics site has a task for proving triangles congruent (G-CO 10), but it seems like students will need to prove theorems about lines and angles G-CO 9 before they are going to be able to prove triangles congruent G-CO 8.
This set of four learning objectives basically encompasses six of the units we have taught in our previous geometry course: 1-Basic Terms of Geometry, 2-Reasoning & Proof, 3-Parallel Lines, 4-Triangles, 5-Congruent Triangles, 6-Special Quadrilaterals. So what used to be almost half of our textbook has been whittled down to four learning objectives.
Can we expect students to prove theorems such as “vertical angles are congruent” and “if two parallel lines are cut by a transversal, then alternate interior angles are congruent” without having a conversation about deductive systems, postulates, theorems, and deductive reasoning? Do we expect students to know what the converse of a statement is before they get to this high school geometry course? Do students have any need for knowing that an original statement and its contrapositive are logically equivalent, and likewise, the converse and inverse will be logically equivalent? I have had my geometry students prove this in the past using truth tables and symbolic logic. Am I wasting our time?
I couldn’t totally take the plunge this year. I’m not ready for all of these learning objectives to be covered in one typical unit of study. But I am taking small steps. I have made one unit on Angles for G-CO 9 and 10 where we are talking about pairs of angles and parallel lines. Next, we will do a unit on Polygons for G-CO 8 and 11 where we talk about proving triangles congruent and special quadrilaterals. Maybe once we get high school students who have progressed through CCSS in K-8, we will be less afraid that putting all of these learning objectives together in one small unit will affect our students’ success on college entry exams.
For now, we are trying to anticipate the content of which students need reminding in our bell ringers at the beginning of class. For example, we added a page of terms before asking students to solve problems with complementary, supplementary, and vertical angles. Because we created the diagrams on a Geometry page of TI-Nspire, they are interactive. Students who don’t remember what they should have learned in Middle School can quickly refresh their memories.
In addition, we have students who have no idea how to solve a problem like the following: ∠DFG and ∠JKL are complementary angles. m∠DFG=x+5, and m∠JKL=x-9. Find the measure of each angle. At least at the beginning of the unit, we are trying to scaffold the presentation so that they can begin to learn the process they should go through themselves when solving similar problems:
In another bellringer we gave students questions that reviewed the use of the Reflexive, Symmetric, and Transitive properties. We needed those properties as some of the reasons in our proofs … and some our students couldn’t remember the names of the properties, much less how to use them.
We are focusing our class time on proving the theorems – not just trying to use them. And we are focusing our class time on having students give reasons for going from one step to the next – not just asking for an answer but asking how they got their answer.
On our second day of parallel lines, we focused on problems with auxiliary lines, which in the past is something we might not have gotten to until the fifth or sixth day of the unit. It is something we might not have tried in some of our more challenging classes. But we did it with success. And the students felt accomplished.
In this problem, the students started out slowly, but as they began to talk to each other, several asked if they could resubmit their answer. The second time around with question 2.1, 17 students answered correctly. Note that these results were with no direct instruction from the teacher.
The last question provides students to enter into the Mathematical Practice of “looking for and making use of structure”. Maybe next year we will be brave enough to start with that question.
And so the journey continues….