CCSS in Secondary Mathematics: Changing Expectations

1. CCSS in Secondary Mathematics:Changing Expectations Patrick Callahan Co-Director California Mathematics Project

2. Plan for this morning • Changing expectations for Algebra • Do some algebra! • Changing expectations for Geometry • Do some geometry!

3. The course titles may be the same, but the course content is not! Common Core Algebra and Geometry are quite different than previous CA Algebra and Geometry courses!

4. Conrad Wolfram’s TED Talk: What is math? Posing the right questions Real world  math formulation Computation Math formulation  real world, verification

5. Conrad Wolfram’s TED Talk: What is math? Posing the right questions Real world  math formulation Computation Math formulation  real world, verification Humans are vastly better than computers at three of these.

6. Conrad Wolfram’s TED Talk: What is math? Posing the right questions Real world  math formulation Computation Math formulation  real world, verification Yet, we spend 80% or more of math instruction on the one that computers can do better than humans

7. Conrad Wolfram’s TED Talk: What is math? Posing the right questions Real world  math formulation Computation Math formulation  real world, verification Note: The CCSS would indetify Wolfram’s description of math to be Mathematical Modeling, one of the Mathematical Practices that should be emphasized K-12.

8. Sample Algebra Worksheet This should look familiar. What do you notice? What is the mathematical goal? What is the expectation of the student?

10. A sample Algebra Exam

11. A sample Algebra Exam I typed #16 into Mathematica

14. Look at the circled answers. What do you notice?

15. Algebra ≠ Bag of Tricks To avoid the common experience of algebra of a “bag of tricks and procedures” we adopted a cycle of algebra structure based on a family of functions approach.

16. HS Algebra Families of Function Cycle CONTEXTS ABSTRACTION (structure, precision) FUNCTIONS (modeling) EQUATIONS (solving, manipulations • Families of Functions: • Linear (one variable) • Linear (two variables) • Quadratic • Polynomial and Rational • Exponential • Trigonometric

17. Context From Dan Meyer’s blog

18. Model with functions

19. Equations You can’t “solve” a function. But functions can be analyzed and lead to equations, which can be solved. What was the maximum height of the ball? How close did the ball get to the hoop? Symbolizing, manipulating, Equivalence…

20. Abstracting (structure, generalization) Examples: The maximum or minimum occurs at the midpoint of the roots. The sign of the a coefficient determines whether the parabola is up or down (convexity) The c coefficient is the sum of the roots. The roots can be determined in multiple ways: quadratic formula, factoring, completing the square, etc.

21. HS Algebra Families of Function Cycle CONTEXTS ABSTRACTION (structure, precision) FUNCTIONS (modeling) EQUATIONS (solving, manipulations • Families of Functions: • Linear (one variable) • Linear (two variables) • Quadratic • Polynomial and Rational • Exponential • Trigonometric

22. Algebra Area A = Perimeter P = W L

23. Algebra Area A = LW Perimeter P = 2(L+W) W L

24. Can you find a rectangle such that the perimeter and area are the same? ? The name “Golden Rectangle” was taken, So let’s call such a rectangle a “Silver Rectangle”

25. “Silver Rectangles” Area = 16 Perimeter = 16 4 4

26. Silver square symbolic solution 4 4

27. Other silver rectangles 2k k

28. Algebra outside the box Volume V = ? Surface Area S = ? Edge length E = ? H W L

29. Algebra outside the box Volume V = LWH Surface Area S = 2(LW+HW+LH) Edge length E = 4(L+W+H) H W L

30. BONUS QUESTION:Can you find a “Silver Rectangular Prism” (aka Box)? Volume V = LWH Surface Area S = 2(LW+HW+LH) Edge length E = 4(L+W+H) H W L Can you find a box with V=S=E?

31. Geometry

32. Why geometric transformations?

33. NAEP item examples… The 2007 8th grade NAEP item below was classified as “Use similarity of right triangles to solve the problem.”

34. Why is this so difficult? The 2007 8th grade NAEP item below was classified as “Use similarity of right triangles to solve the problem.” Only 1% of students answered this item correctly.

35. The 1992 12th grade NAEP item below was classified as “Find the side length given similar triangles.”

36. The 1992 12th grade NAEP item below was classified as “Find the side length given similar triangles.” Only 24% of high school seniors answered this item correctly.

37. Why are these items so challenging?

38. Are these “the same”?

39. Are these “the same”?

40. Are these “the same”?

41. Are these “the same”?

42. Precision of meaning (or lack thereof) Much of mathematics involves making ideas precise. The example at hand is the challenge of making precise the concept of Geometric Equivalence. There is some common sense notion of “shape” and “size”. Same “shape” and same “size” (“CONGRUENT”) Same “shape” and different “size” (“SIMILAR”) In a survey of 48 middle school teachers, 85% gave these definitions

43. Are these “congruent”? Well, they seem to have the same shape and same size. But one is …”upside down”… “pointing a different way”… “they are the same but different” If we think these are geometrically equivalent/congruent, then we are implicitly ignoring where and how they are positioned in space. We are allowed to “move things around”

44. More precision needed… The main problem with the definition “same shape, same size” is “shape” and “size” are not precise mathematical terms

45. Congruence and Similarity Typical (High School) textbook definitions: Pg 233: Figures are congruent if all pairs of corresponding sides angles are congruent and all pairs of corresponding sides are congruent. Pg 30: segments that have the same length are called congruent. Pg 36: two angles are congruent if they have the same measure. Pg 365: Two polygons are similar polygons if corresponding angles are congruent and corresponding side lengths are proportional.

46. Another implicit problem… Typical textbook definitions: Pg 233: Figures are congruent if all pairs of corresponding sides angles are congruent and all pairs of corresponding sides are congruent. Pg 30: segments that have the same length are called congruent. Pg 36: two angles are congruent if they have the same measure. Pg 365: Two polygons are similar polygons if corresponding angles are congruent and corresponding side lengths are proportional. What does “corresponding” mean?

47. The 1992 12th grade NAEP item below was classified as “Find the side length given similar triangles.” “Correspondence” causing problems?

48. Geometric Transformations An alternate approach to congruence and similarity is using geometric transformations (1 to 1 mappings of the plane). An isometry is a transformation that preserves lengths. Definition: Two figures are congruent if there is an isometry mapping one to the other. The Common Core State Standards puts it this way: CCSS 8G2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations CCSS 8G4: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations

49. A rotation and a dilation show the corresponding sides of the similar triangles. The 1992 12th grade NAEP item below was classified as “Find the side length given similar triangles.” Recall, only 24% of high school seniors answered this item correctly. I conjecture that the students didn’t see the correspondence, hence set up the problem incorrectly, e.g. 6/8 = 5/x .

50. Simple example: Vertical Angle Theorem