1. Geometry

2. Big Ideas in Geometry
2- and 3-dimensional figures can be classified and distinguished by their properties or attributes
2-dimensional figures are viewed in the rectangular coordinate plane; transformations of 2-dimensional figures within the plane may produce figures that are similar and/or congruent to the original figure

3. (π r2 h) = 3v
Big Ideas in Geometry
Mathematical statements can be used to describe geometric relationships within and between geometric figures
Formulas that describe attributes of geometric figures can be utilized for indirect measurements and problem solving

4. Sorting Geometric Figures
What properties or attributes could be used to sort these figures?

5. Geometric Figures What do you see?
What disadvantages are there for students in learning about attributes of figures when their experiences are in a pictorial mode rather than with models?

6. In the 2-D Rectangular Plane
Grade 6
Transform figures in the coordinate
plane and describe the transformation
Solve problems involving geometric
figures in the coordinate plane
Grade 7
Use scaling and proportional
reasoning to solve problems related to
similar and congruent polygons
Grade 8
Identify, predict, and describe
dilations in the coordinate plane
Note that slides 6 and 7 were designed for a previous adoption. Adjust these to match the current Standards.

7. Objectives into Big Ideas
Decide within your groups which big idea relates to each objective in the Standard Course of Study
Write the grade level and objective on the charts under the appropriate big idea

8. Triangle Thinking On your handout draw 5 different triangles
After the first triangle, each new triangle must be different in some way from those already drawn
Number the triangles and write why you think each is different

9. Triangle Thinking
What are the thought processes you used in completing this activity?
What conversations and types of activities that promote this thinking among out students?

10. Geometric Thinking
With a partner examine the student work and respond to the guiding questions
What do you think helps students be successful in geometry?

11. Habits of Mind Framework
Driscoll’s framework describes
components of geometric thinking
Seeking relationships
Generalizing geometric ideas
Checking transformation effects
Balancing exploration with deduction

12. Seeking Relationships
Actively looking for and applying relationships, within and between geometric figures, in one, two, and three dimensions
How are these figures are alike?
How are these figures different?
What would I have to do to this object to make it like that object?
Have I found all the ones that fit this description?
What if I think about this relationship in a different dimension?

13. Generalizing Geometric Ideas
Wanting to understand and describe the “always” and the “every” related to geometric phenomena
Does this happen in every case?
Why would this happen in every case?
Can I think of examples when this is not true?
Would this apply in other dimensions?

14. Checking Transformation Effects
Noticing and analyzing effects when
transformations act on geometric objects
How did that get from here to there?
What changes? Why?
What stays the same? Why?

15. Balancing Exploration & Deduction
Trying various ways to approach a problem and regularly stepping back to take stock
What happens if I… draw a picture, add to/take apart this picture, work backwards from the ending place, etc.?
What did that action tell me?

16. Circles and Quadrilaterals
Try the Circle Task handout with a partner
Try the Quadrilateral Task handout with a partner
Make lists of all the attributes of
triangles, circles, and quadrilaterals
What habits of mind did you use in
completing the tasks and lists?

17. Shapes & Properties

18. Thinking about Formulas
Formulas that describe attributes of geometric figures can be utilized for indirect measurements and problem solving
Two dimensional figures are viewed in the rectangular coordinate plane; transformations of two dimensional figures within the plane may produce figures that are similar and/or congruent to the original figure

19. Shrinking a Circle
Fold the piece of string in half and place your finger on the 2 loose ends on the mark at the center of the paper
Draw a circle about that fixed point
Fold the string in half again and draw a second circle with the same procedure, around the same fixed point

20. Shrinking a Circle How do the radii of the two circles relate?
How do the circumferences of the two circles relate? Justify your answer with a formula.
How do the areas of the two circles related? Justify your answer with a formula.
Are the circles similar, congruent, both, or neither? How do you know?

21. Enlarging a Drawing
Follow the directions on the handout to enlarge this drawing. 21

22. Enlarging a Drawing Which features of the drawing stayed the same?
Which features of the drawing changed? 22

23. Comparing New to Old
What do you notice about the lengths of the sides?
What do you notice about the angle measures?
How do the areas compare?
What do you notice about the shapes of the figures? 23

24. Similarity and Congruence
Under the original drawing of the figure, write how you know when figures are similar, when they are congruent, and when they are neither 24

25. Similarity and Congruence
Two figures are similar if all of the corresponding angles are congruent and corresponding sides are proportional
Two figures are congruent if the all the corresponding angles are congruent and the ratio of the corresponding sides is 1:1
Van de Walle (2006, p. 199) 25

26. Similar Triangles Activity
Arrange the smaller triangles to form two similar triangles. How do you know they are similar?
Which of the smaller triangles are congruent? How do you know?
Which smaller triangles are similar? How do you know? 26

27. Tools for investigating
What are the types of tools we can use to help students engage in explorations of transformations?
Paper folding, cutting
Miras or mirrors
Patty paper or wax paper
Computer software & applets

28. Similar Triangles Activity
What types of geometric thinking do students need to engage in to solve this task?
How could you modify or extend this task to incorporate more geometric thinking? 28

29. Blue Box Task Dimensions: Height = 12 cm Width = 10 cm Depth = 4 cm
Radius of
cut out
circle = 2 cm
Individually complete the Blue Box worksheet then compare your answers with your table

30. What do you think?
Identify the mathematics that is needed to solve this task
What are the volume and surface area?
What are possible nets?
What additional questions did you write?

31. Habits of Mind
How might giving students tasks like this promote “habits of mind”?
List everything you know that could help you determine the volume and surface area of the box and draw its net
Is there a relationship between success in geometry and habits of mind?

32. Transformations Match
Use the cards to match the term and its definition
Write the words for your matches on the recording sheet
As the terms are discussed add personal notes to make clearer the vocabulary 32

33. What are transformations?
Transformations are about the movement of figures
If all the points of a geometric figure are moved according to the same rules, the figure is transformed and a new figure is created
Transformations are 1-to-1 relationships from points in a pre-image to points in the image 33

34. Images and Pre-images Pre-image: the points of the original figure
Image: the points of the new figure
One-to-one implies that for every point in the image there is exactly one point in the pre-image 34

35. Rigid Transformations
Translation: points in a figure are slid across a translation vector the same distance along parallel paths
Rotation: turns a set of points about the center of rotation an identical number of degrees
Reflection: points in a figure are reflected across a line (mirror image) 35

36. Non-rigid Transformation
Dilation
Every point of a figure is scaled using a scale factor from the center
The scale factor reduces or enlarges a figure
The shape is preserved, but not the size 36

37. Type of Transformations
What types of transformations were involved in…
The Shrinking Circle?
Enlarging a Drawing?
Similar Triangles Task?

38. Movin’ On Reflect the shape over the x-axis
What will the new coordinates be? 38 38

39. Movin’ On Now reflect the new shape over the y-axis
What will the coordinates be? 39 39

40. Movin’ On C C’
C’’
How far away from the original point C is the final location for C?
Explain your reasoning 40 40

41. Movin’ On Rotate the pre- image (original shape) 180 degreesC
Rotate the pre- image (original shape) 180 degrees
Dilate the final shape by a factor of 2 41 41

42. Movin’ On
What is the perimeter of the new shape?
the area? 42 42

43. Levels of Geometric Thought
Pierre and Dina van Hiele were secondary mathematics teachers concerned with their students’ struggle to reach high levels of thinking in geometry
Read through the handout describing van Hiele Levels of Geometric Thought

44. van Hiele Levels of Geometric Thought*
Level 0: Visualization/Recognition
Level 1: Descriptive/Analytic
Level 2: Abstract/Relational
Level 3: Formal Deduction
Level 4: Rigor
*Sequential Levels—These are not dependent on age; rather they are dependent on experiences

45. Sorting Geometric Figures
Think about attributes, defining and sorting figures through the lens of van Hiele

46. Reflection
What are instructional implications for helping students acquire knowledge to move them up the levels?
What experiences do students need to understand properties and definitions in geometry?
What are the roles of language and experience versus memorization of definitions?

47. Reflecting on MS Geometry
What ideas from this professional development can help you better understand your students’ geometric thinking?
In what ways do the objectives in the geometry strand for your grade relate to objectives in the measurement strand?

48. DPI Mathematics Staff Everly Broadway, Leanne Barefoot Robin Barbour
Carmella Fair
Chief Consultant
Donna Thomas
Mary H. Russell
Johannah Maynor
Partners for Mathematics Learning is a Mathematics-Science Partnership Project funded by the NC Department of Public Instruction. Permission is granted for the use of these materials in professional development in North Carolina Partner school districts.
Partners
for Mathematics Learning

49. PML Consultants Amanda Baucom Julia Cazin Anna Corbett Gail Cotton
Ryan Dougherty
Tery Gunter
Kathy Harris
Joyce Hodges
Karen McCain
Vicki Moss
Kayonna Pitchford
Ron Powell
Susan Riddle
Judith Rucker
Shana Runge
Kitty Rutherford
Penny Shockley
Pat Sickles
Nancy Teague
Bob Vorbroker
Jan Wessell
Carol Williams
Stacy Wozny

50. PML Writers Ana Floyd Jeane Joyner Rendy King Katherine Mawhinney
Gemma Mojica
Elizabeth Murray
Wendy Rich
Catherine Stein
Please give appropriate credit to the Partners for Mathematics Learning project when using these materials. Permission is granted for their use in professional development in North Carolina Partner school districts.
Jeane Joyner, Project Director

51. Geometry