A Story of Geometry

1. A Story of Geometry Grade 8 to Grade 10 Coherence

2. Objectives • Articulate and model the instructional approaches to teaching the content. • Examine the coherence of topics and lessons from grade 8 to grade 10.

3. Participant Poll • Classroom teacher • School leader • Principal • District leader • BOCES representative

4. Agenda • Congruence and Rigid Motions • Grade 8: Basic Rigid Motions • Translation, Reflection, Rotation • Grade 10: Basic Rigid Motions • Translation, Reflection, Rotation • Congruence

5. Transformations in Geometry under the CCSS Same Size & Same Shape • Transformations, specifically rigid motions, serve as the foundation of the concept of congruence • Why is congruence defined in terms of rigid motions? • To avoid having to directly measure objects: • Are the opposite sides of a rectangle really equal in length? • Are two angles positioned differently in space really of equal measure? • To develop an intuitive sense of congruence, leading to a definition that can be used with all figures in the plane-not just triangles and polygons.

6. Grade 8

7. Translation • Translation is defined as a motion that “slides” figures along a vector. • A vector is a segment in the plane with a designated starting point and endpoint.

8. Activity • Draw the following on a piece of paper: • A line, • A ray, • A segment, • A point, • An angle, • A curved figure, • A simple drawing of your choice.

9. Properties of Translation • We have experimentally verified that a translation: • Maps lines to lines, rays to rays, segments to segments, and angles to angles. • Preserves lengths of segments. • Preserves angles measures of angles.

10. Translation of Lines • Some properties of translation are highlighted. • Example: What properties can we discuss about translated lines? • There are two possible scenarios: • A line and its translated image coincide (when the vector belongs to the line or is parallel to the line): • A line and its translated image will be parallel (when the vector is not parallel to the line):

11. A Sequence of Translations • Imagine life without an “undo” button on your smart device or computer! • We want to make sure that when we move things around in the plane, we can put them back where they belong, or “undo” the motion. • For that reason, we show students how a translation along a vector can be undone by translating along a vector • This is the beginning of the concept of congruence. It shows that a sequence of two translations can map a figure onto itself.

12. Activity • Take out your paper and transparency. • This time, reflect each of the images you drew by “flipping” your transparency across the line you drew.

13. Properties of Reflection • We have experimentally verified that a reflection: • Maps lines to lines, rays to rays, segments to segments, and angles to angles. • Preserves lengths of segments. • Preserves angles measures of angles. • Additional property that is verified: • When you connect a point and it’s reflected image, the segment is perpendicular to the line of reflection. • Not only is the line of reflection perpendicular to the segment, but it bisects the segment.

14. Activity • Take out your paper and transparency. • This time, rotate each of the images you drew by placing your finger on top of the point you drew and carefully rotate your transparency in one direction and then the other.

15. Properties of Rotation • We have experimentally verified that a rotation: • Maps lines to lines, rays to rays, segments to segments, and angles to angles. • Preserves lengths of segments. • Preserves angles measures of angles.

16. Congruence • With each new rigid motion that is learned, students immediately begin sequencing the motion with a known motion. • For example: • The first sequence is two translations. • Once reflection is learned, students sequence two reflections. Then, students sequence a translation and a reflection. • Once rotation is learned, students sequence two rotations. Then, students sequence a translation and a rotation, or a rotation and a reflection, etc. • Congruence is defined in terms of a sequence of rigid motions, performed using a transparency, that shows the mapping of one figure onto another.

17. Grade 10

18. Rigid Motions in Grade 10 • Students enter Grade 10 with an intuitive sense of congruence and have experimentally verified properties of rigid motions • They know that “same size, same shape” is not a precise way of describing congruence • Defining congruence with the use of rigid motions captures all types of figures • In Grade 10, students formalize the concepts from Grade 8 through language • The visual/experiential understanding of how each rigid motion actually “works” is put into explicit parameters • Students think about the plane and the rigid motions in the plane more abstractly • Constructions are used in the application of rigid motions

19. Grade 10: Reflection • In Grade 10, students clearly define reflection and how to: • Determine the line of reflection by construction • Reflect a figure across a line by construction. • In Grade 8, students have used transparencies to experimentally verify the properties of a reflection AND that the line of reflection is the perpendicular bisector of any segment that joins a pair of corresponding points between the figure and its image

20. Grade 10: Determining the Line of Reflection • Use the construction of a perpendicular bisector to determine the line of reflection for the following figures:

21. Grade 10: Determining the Line of Reflection

22. Grade 10: Mapping over the Line of Reflection

23. Grade 10: Mapping over the Line of Reflection

24. Grade 10: Rotation • In Grade 8, students experimented with a model of a rotation, spinning figures on transparencies to verify that rotations were indeed distance preserving and angle preserving. • In Grade 10, students clearly define rotation and learn to: • Determine the center of rotation • Determine the angle of rotation

25. Grade 10: Determining the Angle of Rotation

26. Grade 10: Determining the Angle of Rotation

27. Grade 10: Determining the Center of Rotation

28. Grade 10: Determining the Center of Rotation

29. Grade 10: Rotation

30. Grade 10: Translation • In Grade 8, students experimented with a model of a translations, sliding figures on transparencies to verify that translations were distance preserving and angle preserving. • In Grade 10, students clearly define translation and learn to: • Apply a translation by constructing parallel lines

31. Grade 10: Applying a Translation • Given the experience students enter Grade 10 with, they can visualize the image of the figure under a translation, provided the vector.

32. Grade 10: Applying a Translation • To apply the translation, we must construct the line parallel to each side in the direction and at a distance equal to the length of the vector.

33. Grade 10: Applying a Translation Follow the instructions to construct the line parallel to AB through P.

34. Grade 10: Applying a Translation Line PQ is parallel to line AB.

35. Grade 10: Translating a Segment • The translation of a segment might look like this:

36. Grade 10: Translating a Triangle • The translation of a triangle might look like this:

37. Grade 10: Translation

38. Congruence Congruent. Two figures in the plane are congruent if there exists a finite composition of basic rigid motions that maps one figure onto the other figure. • Once students are comfortable with rigid motions, they study the link between the concept of rigid motions and congruence • We want students to be able to use the language around congruence in a clear way

39. Congruence Sample Question: Why can’t a triangle be congruent to a quadrilateral?

40. Congruence A triangle cannot be congruent to a quadrilateral because there is no rigid motion that takes a figure with three vertices to a figure with four vertices. Sample Question: Why can’t a triangle be congruent to a quadrilateral? Sample Answer:

41. Coherence • It is important that students leave Grade 8 with a solid, intuitive understanding of the rigid motions • The physical manipulation of and visual understanding of rigid motions in Grade 8 needs be put into careful language in Grade 10 • Properties of rigid motions that make obvious sense need are married with construction, and eventually used in reasoning • The “careful use of language” is mentioned frequently in Grade 10. • Ultimately, we want students to understand that Geometry exists as a axiomatic system- that the establishment of a new fact comes strictly from basic assumptions or existing facts • These assumptions and existing facts appear throughout Module 1, and certainly in the topic of Rigid Motions.

42. Biggest Takeaway • A solid understanding of how rigid motions behave in Grade 8 will lay the groundwork for Grade 10.

43. Key Points • The hands-on, experiential understanding and experimental verification of properties in Grade 8 are formalized through language and construction in Grade 10.