The student has just finished Section 3-2 of the Glencoe text, "Angles and Parallel Lines." Even though he's finished this lesson, he asked for further clarification on a certain proof, which turned out to be the Same-Side Interior Angles Consequence -- actually, the Glencoe text uses the term "consecutive interior angles" instead of "same-side interior angles."
Unfortunately, this proof is not included in my worksheets for this blog. This is because it is not emphasized in the U of Chicago text on which my worksheets are based. The only reference to the same-side interior angles is in Section 5-5, in the context of trapezoids:
Trapezoid Angle Theorem:
In a trapezoid, consecutive angles between a pair of parallel sides are supplementary.
Notice that technically speaking, this text also uses the phrase "consecutive angles," and that Dr. Franklin Mason also uses the term "consecutive interior angles," so maybe this term is becoming more popular than the term to which I'm accustomed, "same-side interior angles."
Here is the two column proof that the U of Chicago provides for the Trapezoid Angle Theorem -- as usual, with the headings changed to "Statements" and "Reasons" and an extra step for "Given."
Prove: Angles 1 and D are supplementary
(Note: In the diagram, angle 1 refers to BAD, and angle 2 refers to BAE, where E is a point on the transversal line AD, chosen so that angles 2 and D are corresponding angles.)
2. angle 1 + angle 2 = 180 2. Linear Pair Theorem
3. angle 2 = angle D 3. Corresponding Angles Consequence
4. angle 1 + angle D = 360 4. Substitution (step 2 into step 1)
5. Angles 1 and D are supplementary 5. Definition of supplementary
I told my student about this proof, or something similar. Of course, if we follow the Dr. Hung-Hsi Wu plan, our proof would use the Alternate Interior Angles Consequence instead of Corresponding Angles, since AIA is the basic consequence that is used to prove the others.
Recall that much of my planning includes how I would revise my course for next year, based on how much my lessons were able to help an actual student. So far, I see one glaring omission -- namely a proof of this Same-Side/Consecutive Interior Angle Consequence. I will include it next year, and maybe I will use the name that is currently en vogue, "consecutive interior angles."
Here is how my course will begin next year, based on my tutoring sessions so far this year:
Days 1-3: Opening Activity (This is now three days because my first day of school is Wednesday.)
Day 4: Section 1-6
Day 5: Section 1-7
Day 6: Section 1-8 ("One-Dimensional Figures")
Day 7: Two-Dimensional Figures and Review (new)
Day 8: Quiz
Day 9: Section 2-1
Day 10: Section 2-2
Day 11: Section 2-3
Day 12: Review
Day 13: Chapter 1 Test
Day 14: Activity: Identifying Patterns (new)
Day 15: Section 2-4
Day 16: Section 13-1 ("The Logic of Making Conclusions," moved up)
Day 17: Activity
Day 18: Section 2-5
Day 19: Section 2-6
Day 20: Section 2-7
Day 21: Section 3-1
Day 22: Activity (possibly another quiz instead)
Day 23: Section 3-2
Day 24: Section 3-3
Day 25: Section 4-4 ("The First Theorem in Euclid's Elements," moved up)
Day 26: Review
Day 27: Chapter 2 Test
The new sections that I mentioned throughout the semester as being helpful to my student have been added in. A new test for Chapter 1 has been added in -- otherwise one will go the entire first quaver without a test. Of course, making Chapters 1-2 longer means making something else shorter.
My student moved on to Section 3-3 of the Glencoe text -- "Slopes of Lines." This is covered in Sections 3-4 and 3-5 of the U of Chicago text -- but I am saving it for 2nd semester.
And this brings me to my 2nd semester preview. So far, I have stuck mostly to the U of Chicago order -- almost to a fault. This explains why I need to make so many changes to my 1st semester plan next year. But I will jumble things up for 2nd semester, in order to give lessons in an order that makes more sense for Common Core:
Chapter 12. Similarity
Chapter 11. Coordinate Geometry
Chapter 13. Logic and Indirect Reasoning
Chapter 14. Trigonometry and Vectors
Chapter 8. Measurement Formulas
Chapter 15. Further Work with Circles
Chapter 9. Three-Dimensional Figures
Chapter 10. Surface Area and Volumes
This also reflects the order in many other texts -- similarity first, volume last. In Common Core, similarity must come before coordinates, so we begin with Section 12-2, "Size Changes (Dilations) Without Coordinates." Much of the similarity lessons will be based on Wu.
Merry Christmas! Stay tuned for a winter break post on Common Core Debate, Grades 8-12.