The New Approach to High School Geometry Required by Common Core

1. The New Approach to High School Geometry Required by Common Core Fair Warning: The program information does not really describe the content of this talk which is basically about the Mathematics that Teachers Need to Know (very little about book or teaching) Tom Sallee University of California, Davis

2. The Mathematical Bones of CCG The Mathematics that Teachers Need to Know Tom Sallee University of California, Davis

3. BE SURE TO ASK QUESTIONS This talk is for you—not me.

4. Changes being required for geometry courses. Biggest one: rigid transformations are how you determine congruence. Real problem—book is very good now. How do we make necessary changes and preserve all that is good now? The new CCG books will use these ideas to prove some basic results and build on them for the rest of the course.

5. Because I am among friends • I am not going to be precise about the distinctions between a segment and the measure of its length and ditto with angles. I am trying to convey larger ideas and believe you can translate the occasional imprecision into what you need.

6. Most Basic Notions • Reflection Across a Line • Dilation With Respect to a Point

7. Postulate 1 • Every reflection preserves linearity, angles and lengths. (and betweenness)

8. Postulate 2 • Every dilation preserves linearity, angles and ratios of lengths. (and betweenness)

9. Other Basic Motions Translations Rotations about a point We will not be considering slide reflections.

10. Definitions • A rigid motion or rigid transformation (or congruence transformation) is the result of a sequence of reflections, rotations, and translations. • Two geometric figures are congruent if there exists a congruence transformation taking one exactly onto the other.

11. Theorem 1 • Every congruence transformation preserves linearity, angles and lengths. • We are simply saying that since linearity, etc. is preserved at each step, it is preserved from beginning to end.

12. Similarity transformations • What is going to be the difference between a congruence transformation and a similarity transformation? Discuss with someone near you.

13. Definition • A similarity transformationis the result of a sequence of reflections, rotations, translations and dilations. • Two geometric figures are similar if …what??

14. Definitions • A similarity transformationis the result of a sequence of reflections, rotations, translations and dilations. • Two geometric figures are similar if there exists a similarity transformation taking one exactly onto the other.

15. Theorem 2 • Every similarity transformation preserves linearity, angles, and ratios of lengths. Same idea as before.

16. From CCSS-Math First Geometry Standards on Congruence • Experiment with transformations in the plane • Understand congruence in terms of rigid motions

17. Mathematical Agenda Today • Topic 1 Relationships among various congruence and similarity transformations. • Topic 2 Using transformations to prove standard theorems.

18. Congruence Transformations • Experiment with transformations in the plane • Understand congruence in terms of rigid motions

19. What can we do with reflections? • Key point: If you know what happens to a single triangle in an affine transformation (linear map plus translation) you know what happens to the entire space. • For this course, we only care about congruence and similarity transformations.

20. Congruence via reflections • If ABC and A’B’C’ are congruent triangles, can you find a single reflection that takes A to A’? • If ABC and AB’C’ are congruent triangles [note A = A’ here, so the two triangles have a common vertex], can you find a single reflection that keeps A fixed and takes B to B’? • If ABC and ABC’ are congruent triangles [note A = A’ and B = B’ here, so the two triangles have a common edge], and C is not equal to C’, can you find a single reflection that keeps A and B fixed and takes C to C’?

21. Theorem 3 Every congruence transformation is the product of at most 3 reflections.

22. Exploring similarity transforms. • Suppose triangles DEF and D’E’F’ are similar, what sequence of reflections and dilations could show this? • What would be your basic strategy?

23. How many dilations? • If a similarity transform involves two dilations, one that doubles distances and the other than multiplies by 2.5, is there a single dilation that can be used instead?

24. State the theorem • Theorem 4: Every similarity transformation is the product of ….. what?

25. Theorem 4 Every similarity transformation is the product of at most 3 reflections and one dilation.

26. From reflections to other rigid motions • .

27. What do reflections do to the plane? • In particular, are any points fixed by a reflection—i.e. don’t move?

28. Points fixed by reflections Are any points fixed by a reflection across a line? • Answer: points are not moved by a reflection if and only if they are on the line.

29. Reflections across intersecting lines • Are any points fixed by two reflections (in sequence) across intersecting lines? Discuss

30. Reflections across intersecting lines • Answer: a point is fixed by a sequence of two reflections if and only if it is at the intersection of the lines. [One way is obvious; other way is also true but less obvious.]

31. Only fixed point is intersection. • Set up coordinate system so that first line of reflection is the x-axis. Reflect first across it preserves x-value. Reflecting then across any other line will change x-value. Thus point will not have been fixed.

32. So we get??? • A point is fixed by a sequence of two reflections if and only if it is at the intersection of the lines. What kinds of rigid motions keep exactly one point fixed?

33. Theorem 5 The product of two reflections across two lines that intersect at C is a rotation about the center C.

34. Theorem 5 The product of two reflections across two lines that intersect at C is a rotation about the center C. Is the converse true?

35. Converses • Recall that the converse of H C is CH. • I think it is worth making a big deal about this distinction in class because so many otherwise well-educated adults often can’t seem to tell them apart. Knowing the difference is essential to clear thinking—certainly in math.

36. Theorem 5 The product of two reflections across two lines that intersect at C is a rotation about the center C. What is the converse in this case?

37. Theorem 5 The product of two reflections across two lines that intersect at C is a rotation about the center C. The converse is true and we will sketch a proof in a minute.

38. Reflections across parallel lines • What do you think happens if the lines are parallel?

39. Theorem 6 The result of two reflections across parallel lines is a translation.

40. Theorem 6 The result of two reflections across parallel lines is a translation. Let’s see if the converse is true by constructing those lines.

41. Can you find those parallel lines? For a particular translation, we want to see if we can find two parallel lines such that the product of reflections across these lines is the given translation. If this is going to work, what do we know about these lines? Specifically, if a translation takes (x,y) (x+2,y), what could the lines be? You have 5 minutes. Work together.

42. Hint • You don’t want the y value to change for either reflection.

43. General result • If I give you an arbitrary translation, can you find two parallel lines whose reflections generate the translation? • Draw an arbitrary translation on your paper and have a neighbor find the lines. • Find the “simplest one.”

44. Theorem 6 (improved) • Every translation is equivalent to two reflections across parallel lines that are perpendicular to the translation segment. The distance between these lines is half the translation distance. • The easiest pair to describe probably is where the first line is the perpendicular bisector of the segment and the second one is through the “proper” end of the segment.

45. Theorem 5 (improved) • Every rotation is the product of two reflections across lines that intersect at the center of rotation. The angle between these lines is half the rotation angle. Essentially the same argument works. Again the easiest pair to describe is probably the one that takes the x-axis to where it will end up after the rotation and then to reflect around that line.

46. Congruences • Experimentwith transformations in the plane • Understand congruence in terms of rigid motions.

47. New View of Congruence A standard triangle congruence theorem takes three (possibly separate) congruence transforms and replaces them by a single one that implies three other congruences.

48. SAS Congruence • So SAS on the triangles ABC and DEF says that if there is one congruence transform that maps AB to DE, another congruence transform that maps angle B to angle E, and a third congruence transform that maps BC to EF, then there exists a singlecongruence transform that simultaneously takes A to D, B to E and C to F thereby showing the congruence of all corresponding sides and corresponding angles.

49. Proof of SAS • You would think we had done it earlier in proof of Thm. 3, but we didn’t. There we made the assumption that we had the congruence of the conclusion, but that is what we need to find.

50. Proof sketch of SAS Idea is similar. Take the congruence transformation that takes the rays that describe angle ABC to angle DEF. Call the image A’B’C’. Then B’ = E, and A’ lies along ray ED with B’A’ (that is, EA’) being congruent to BA and BA is given congruent to ED. Hence, EA’ is congruent to ED and thus, A’ = D. Similarly, C’ = F, so the triangle A’B’C’ is the triangle DEFand so ABC and DEF are congruent.