TRANSFORMATION GEOMETRY? IT S JUST SLIDES, FLIPS AND TURNS, ISN T IT? WHAT S THE BIG DEAL?

Transcription

1 1.1 TRANSFORMATION GEOMETRY? IT S JUST SLIDES, FLIPS AND TURNS, ISN T IT? WHAT S THE BIG DEAL? BROOKHILL INSTITUTE OF MATHEMATICS IHE GEOMETRY SUMMIT APRIL 25, 2016 KEVIN MCLEOD, UWM DEPARTMENT OF MATHEMATICAL SCIENCES MIKE STEELE, UWM DEPARTMENT OF CURRICULUM AND INSTRUCTION

2 1.2 TODAY S AGENDA Introduction Justifying area formulas The new Geometry, 7-8, High School progression document Why the transformational approach to geometry? Rigid motions and congruence (Grade 8) Rigid motions and congruence (High School) Triangle congruence criteria Reflections and next steps

3 INTRODUCTION 1.3

4 William McCallum, The University of Arizona 1.4 K-8 Domains & HS Conceptual Categories K HS Counting & Cardinality Operations & Algebraic Thinking Algebra Expressions and Equations Number & Operations in Base Ten Measurement & Data Number & Operations Fractions The Number System Ratios & Proportional Relationships Number and Quantity Functions Statistics & Probability Modeling Geometry

5 1.5 Standards for Mathematical Practice Standards for Mathematics Content Make sense of problems & persevere in solving them Reason abstractly & quantitatively Construct viable arguments & critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for & make use of structure Look for & express regularity in repeated reasoning K 8 Standards by Grade Level High School Standards by Conceptual Categories Domains Clusters Standards

6 STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. 6. Attend to precision. William McCallum, The University of Arizona 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Reasoning and Explaining Modeling and Using Tools Seeing Structure and Generalizing Which of these do you see as most relevant to geometry?

7 1.7 GEOMETRY VERTICAL ARTICULATION: A VIEW FROM MIKE S RECORDS 20 YEARS AGO 7 th grade Basic area and perimeter Names and types of 3-D shapes Flips, turns, and slides 8 th grade Area and perimeter (again) 2-D shape properties 3-D shapes, surface area and volume Flips, turns, and slides so we can do a tessellation project 10 th grade Area, perimeter, surface area, and volume (again) Definitions and proof Triangles, triangles, triangles Constructions Circles, but we probably won t have time, so we ll get at it in Algebra II

8 JUSTIFYING AREA FORMULAS 1.8

9 1.9 JUSTIFYING AREA FORMULAS Turn and talk: What is area? (Of a 2-dimensional shape?)

10 1.10 JUSTIFYING AREA FORMULAS Find the area of this parallelogram In as many ways as you can. (At least two.)

11 1.11 JUSTIFYING AREA FORMULAS Properties of area: The moving property : the area of a shape is not changed if the shape undergoes a rigid motion The combining property : the total area of two (or more) non-overlapping shapes is the sum of their individual areas. How do these properties help you to justify your strategies for finding the area of the parallelogram?

12 1.12 JUSTIFYING AREA FORMULAS 3.MD.C.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. a. A square with side length 1 unit, called a unit square, is said to have one square unit of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

13 THE NEW GEOMETRY, 7-8 HIGH SCHOOL, GEOMETRY PROGRESSION DOCUMENT 1.13

14 1.14 THE NEW GEOMETRY, 7-8, HIGH SCHOOL PROGRESSION DOCUMENT Overview Geometry has two important streams that begin in elementary grades: understanding properties of geometric figures and the logical connections between them, and developing and using formulas to compute lengths, areas and volumes. A third stream, coordinate geometry, surfaces in Grade 5, gains importance in Grades 6 8, and mingles with algebra to become analytic geometry in high school.

15 1.15 THE NEW GEOMETRY, 7-8, HIGH SCHOOL PROGRESSION DOCUMENT Overview: jigsaw reading Count off by threes at your table. Read one of the three paragraphs on the three streams of geometry. Look for references to transformations in your paragraph, and think about how a transformational approach might be useful in teaching and learning your stream. Also consider: what else do you see that is new or under-addressed? Compare with others at your table.

16 1.16 THE NEW GEOMETRY, 7-8, HIGH SCHOOL PROGRESSION DOCUMENT Grade 8 Read the section on Rigid Motions and Congruence on page 9 of the progressions document. Summarize the expectations for student understandings of rigid motions and congruence in Grade 8. Consider: What aspects of this section does your school/university address particularly well? What are the areas for change?

17 1.17 THE NEW GEOMETRY, 7-8, HIGH SCHOOL PROGRESSION DOCUMENT Why transformation geometry? There is at present an almost total disconnect in the standard curriculum between the geometry of middle school and that of high school. On the one hand, congruence and similarity are defined in middle school as same size and same shape and same shape but not necessarily same size, respectively, and transformations (rotations, reflections, and translations) are taught seemingly only for the purpose of art appreciation,... On the other hand, the concepts of congruence and similarity are defined anew in a high school geometry course, but only for polygons, and at the end of the year, transformations are sometimes brought up as enhancement of the concept of polygon congruence. H. Wu, Teaching Geometry According to the Common Core (2012)

18 1.18 THE NEW GEOMETRY, 7-8, HIGH SCHOOL PROGRESSION DOCUMENT Why transformation geometry? One cannot overstate the fact that the CCSS do not pursue transformational geometry per se. Transformations are merely a means to an end: they are used in a strictly utilitarian way to streamline the existing school geometry curriculum. H. Wu, Teaching Geometry According to the Common Core (2012)

19 RIGID MOTIONS AND CONGRUENCE (GRADE 8) 1.19

20 1.20 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Given two segments AB and CD, which could be very far apart, how can we tell if they have the same length without measuring them individually? Do you think they have the same length? How can you check? 2014 Common Core, Inc.

21 1.21 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Turn and talk: What is a translation? A reflection? A rotation? CCSSM vocabulary: translations, reflections, and rotations are collectively known as basic rigid motions.

22 1.22 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Describe, intuitively, what kind of transformation will be required to move the figure on the left to each of the figures (1)-(3) on the right. To help with this exercise, use a transparency to copy the figure on the left. Note: begin by moving the left figure to each of the positions (1), (2) and (3) Common Core, Inc.

23 1.23 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Definitions A congruence transformation (or just congruence, for short) is a sequence of (one or more) basic rigid motions. Two geometric figures are congruent if there is a congruence which takes one of them onto the other. 8.G.A.2 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; given two congruent figures, describe a sequence that exhibits the congruence between them.

24 1.24 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Properties of congruence transformations Explain why any congruence transformation has the same three properties as the basic rigid motions: 1. It takes lines to lines, segments to segments, rays to rays, and angles to angles; 2. It preserves distances between points (and so also lengths of segments); 3. It preserves the degree measure of angles.

25 1.25 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Prove it! Let L and L be parallel lines, and let M be another line which crosses L and L. Explain why the degree measures of angles 1, 2, and 3 are equal L L M

26 1.26 RIGID MOTIONS AND CONGRUENCE (GRADE 8) 8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines.

27 1.27 REFLECTING ON GRADES 7-8 From your perspective (IHE content, IHE pedagogy, school district): What aspects of the geometry progression in grades 7-8 are strengths? What aspects of the geometry progression in grades 7-8 are new or underaddressed? What changes would support stronger teacher and student learning in grades 7-8 geometry?

28 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) 1.28

29 1.29 THE NEW GEOMETRY, 7-8, HIGH SCHOOL PROGRESSION DOCUMENT High School Read the high school introduction and the Congruence section. (You may also want to scan the Analytic Geometry section.) In what ways are rigid motions connected to notions of congruence in high school? Consider: What aspects of this section does your school/university address particularly well? What are the areas for change?

30 1.30 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Work with a partner to complete the task. You are allowed a protractor, compass, and straightedge. Partner A: Without showing the card to your partner, describe how to draw the transformation indicated on the card. When you have finished, compare your partner s drawing with the transformed image on your card. Did you describe the motion correctly? Partner B: Your partner is going to describe a transformation to be performed on the figure on your card. Follow your partner s instructions, and then compare the image of your transformation to the image on your partner s card.

31 1.31 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) What did you discover as you worked on the previous task? Would your students reactions to the task be essentially the same as yours? Why or why not?

32 1.32 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Definitions of the basic rigid motions Work at your table to provide precise definitions of each of the following: 1. A translation; 2. A rotation; 3. A reflection. What information is required in each case to specify the transformation completely?

33 1.33 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) G-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

34 1.34 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Provide a transformational proof of the Side-Angle-Side (SAS) triangle congruence criterion Theorem 1 (SAS triangle congruence criterion) Let ΔABC and ΔA B C be two triangles, with AB = A B, AC = A C, and. Then ΔABC and ΔA B C are congruent. Hint. What must you do in order to show that two figures are congruent?

35 1.35 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

36 1.36 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Provide a transformational proof of this theorem Theorem 2 (Isosceles triangle symmetry) Let ΔABC be isosceles, with AB = AC. Then the angle bisector at A is the perpendicular bisector of segment BC.

37 1.37 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) G-CO.C.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.

38 VERTICAL ARTICULATION, 2016 VERSION 1.38

39 VERTICAL ARTICULATION, 2016 VERSION 1.39

40 1.40 REFLECTING ON HIGH SCHOOL From your perspective (IHE content, IHE pedagogy, school district): What aspects of the geometry progression in high school are strengths? What aspects of the geometry progression in high school are new or underaddressed? What changes would support stronger teacher and student learning in high school geometry?

41 REFLECTIONS AND NEXT STEPS 1.41

42 1.42 REFLECTIONS AND NEXT STEPS What will you take away from this session? What will your next steps be at your institution (K-12, IHE, K-16) to align geometry instruction to CCSSM expectations?

43 1.43 THANK YOU! Questions? Kevin: Mike:

44 1.44 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Vocabulary A transformation of the plane is a rule that assigns to each point of the plane exactly one (unique) point. If we denote the transformation by F, we use F(P) to denote the unique point assigned to P by F. (We will sometimes write P instead of F(P).) The point F(P) will be called the image of P by F. We also say F maps P to F(P). If given any two points P and Q, the distance between the images F(P) and F(P) is the same as the distance between P and Q, then the transformation F preserves distance, or is distance-preserving. A distance-preserving transformation is called a rigid motion (or an isometry); the name suggests that it moves the points of the plane around in a rigid fashion.

45 1.45 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Let s play! Use transparencies to model: A translation (along a given vector ) A reflection (across a given line) A rotation (of a given angle about a given point) Note: translations, reflections, and rotations, are collectively known as basic rigid motions.

46 1.46 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Vocabulary A vector is a line segment with a specified direction If the segment is AB, and the specified direction is from A to B, we denote the corresponding vector by. Vector Vector

47 1.47 RIGID MOTIONS AND CONGRUENCE (GRADE 8) Prove it! Let P be the image of P by a reflection in line L. Use the properties of basic rigid motions to explain why L must be the perpendicular bisector of the segment PP. P. L. P

48 1.48 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Provide a transformational proof of the Angle-Side-Angle (ASA) triangle congruence criterion Theorem 3 (ASA triangle congruence criterion) Let ΔABC and ΔA B C be two triangles, with, AB = A B, and. Then ΔABC and ΔA B C are congruent.

49 1.49 RIGID MOTIONS AND CONGRUENCE (HIGH SCHOOL) Provide a transformational proof of the Side-Side-Side (SSS) angle congruence criterion Theorem 4 (SSS triangle congruence criterion) Let ΔABC and ΔA B C be two triangles, with AB = A B, AC = A C, and BC = B C. Then ΔABC and ΔA B C are congruent.