1. Jean J. McGehee jeanm@uca.edu University of Central Arkansas
Building Geometric Thinking with Hands-On Tasks & in Virtual Environments
Jean J. McGehee
University of Central Arkansas

2. Today
Geometric Habits of Mind—from Paper Folding to Using Sketchpad—in the context of rich problems
Transformations: a Connecting big idea
The role of good definitions: Quadrilaterals
Connecting Sketchpad to the Number and Algebra strands

4. Goals of FGT Strengthen understanding of geometry
Enhance capacity to recognize and describe geometric thinking
Increase attention to students’ thinking
Enhance understanding of students’ geometric thinking
Prepare to advance students’ geometric thinking

5. The Geometry Curriculum in Arkansas
Let’s take a quick look at the frameworks..\..\..\Desktop\frameworks\geometry_06.doc
Even my student interns at UCA notice how much repetition there is in the curriculum—e.g. The Triangle Sum
We do need to revisit ideas—but we need to do it with value added.

6. The FGT Project in NE Arkansas
Two school districts in grades 5-11
I wanted the teachers at all grade levels to share their strengths and understand the curriculum vertically.
I wanted them to share their students work and their ideas so that they gained an appreciation for each other.

7. All levels benefited.
The 5th and 6th grade teachers enjoyed the hands on activities, and these very same activities are useful even after the high school geometry course.
With Algebra II and beyond we really downplay geometry—yet these kids have to take the ACT or SAT. With these problems they will also have to explain their thinking.

8. It’s hard to get teachers to focus on content for fun when they deal with state tests.
Cathy (6th grade) and Cindy (Geometry) are teachers who trust rich problems, inquiry investigations, and projects BEFORE the State Tests as a means to prepare for criteria tests---Their scores show it!!

9. Another teacher from Arkansas reported:
One of my 6th grade FGT participants told me that her students' scores on the Arkansas assessment increased from 46% proficient and advanced in 2005 to 68% in That was great news, but it gets better.
She credits the increase to their better understanding of geometry and measurement than in years past. She says that is a direct result of the problems we did (and she did with her students) in FGT. In fact, after the testing her students told her that the problems on the test were
like what they had done in class "except they didn't give us any paper to fold".

10. Structure of FGT
The Structured Exploration Process guides the activities in each part of FGT sessions. There is a cycle of doing math and exploring student thinking.
The Geometry Habits of Mind framework provides a lens to analyze geometric thinking.

11. Three content strands
Focus the work on different important areas of geometry & measurement.
They are:
Properties
Transformations
Measurement

12. The Structured Exploration Process
Stage 1: Doing mathematics
Stage 2: Reflecting on the mathematics
Stage 3: Collecting student work
Stage 4: Analyzing student work
Stage 5: Reflecting on students’ thinking

13. FGT G-HOMs Reasoning with Relationships Generalizing Geometric Ideas
Looking for Invariants
Balancing Exploration & Reflection

14. More about FGT and G-HOMs later
First, let’s do exercise our own geometric thinking
Folding, Making Squares, Congruent Halves
Paper-Folding & Constructions
Tangrams
Dissecting Shapes
Comparing Triangles
We will start in detail—but I may have to summarize the latter problems.

15. Do Math--Ideally Work problem individually 5-10 minutes
Work problem in groups 25 minutes
Last 10 minutes groups prepare report either on transparency or chart paper
Reflect on the problem & Identify G-HOMs minutes

16. Let’s do more with paper folding-Start at b
Construct a triangle with exactly ¼ the area of the original square. Explain how you know it has ¼ the area:
Construct another triangle that also has ¼ the area, which is not congruent to the first one you constructed. Explain how you know is has ¼ the area:
Construct a square with exactly /12 the area of the original square. Explain how you know it has ½ the area:
Construct another square, also with ½ the area which is oriented differently than the one you constructed in (d). Explain how you know it has ½ the area
Go to Sketchpad

17. Review: Investigating Area by Folding
Some comments on the challenge problem.
Recall that it was relatively easy to find a square that is ¼ of the original.
We all found one square that is ½ of the original.
I want to show you a quilter’s approach
Also I want to show you this problem in a fun book.

18. A Quilter’s Solution—does it work?
Go to Sketchpad

19. The Number Devil
This little devil beguiles Robert into dreams to give him a glimpse of the beauty & power numbers.
In this case, the square root of 2.

20. Student intern gave students two squares and asked
How many black squares fit into the red square?
Show how you knew this.

21. Hands-On & Sketchpad
I have learned both in PD and classes to start with Hands-On
Making a gallery of chart paper reports and walking through the gallery is a wonderful way to summarize the problem.
Sketchpad provides a way to solidify conjectures and make a bridge to proof.
Talk about Linda’s and my contrast. Go to Sketches for Comparing triangles.

22. Basic Paper Folding
The perpendicular bisector is the most basic fold. Who can describe this for me? How do you know?
Construct a line that is parallel to your original segment. Describe your method. How do you know your new line is a parallel line to the original segment?
Now start with a fresh segment each time and construct: an isosceles triangle an equilateral triangle a square

23. Analyzing Student Work
What are the important mathematical ideas in the problem?
What strategies do you want to foster and why?
What is the evidence that a student used a strategy? Is it related to a G-HOM?

24. Student work on Paper Folding
What do you think students typically do?
How do you think students use geometric language?
Go Back to the Demand of the Task.
Are we actually requiring students to write and speak the language of geometry?
Or do we practice Multiple Choice items and work problems in which the language task is low?

25. Paper Folding related toTangrams
You are familiar with the square, but can you make a rectangle that are not squares—2 ways?

26. Let’s explore the area problems
Tangrams on Sketchpad
More shapes with the same area—an understanding based on properties rather than memorized formulas.

27. Dissecting Shapes
The ability to dissect and transform shapes is important.
Students are also exploring invariance and properties.

28. Dissecting Shapes--Conclusions

29. Comparing Triangles
Start with a piece of paper (you can also use different size rectangular paper). Fold your paper so that point A is directly on top of point C. Some triangles appear.
In the picture below—you should see 3 triangles.

30. Comparing Triangles
Start with another piece of paper. This time fold A onto any point between D and C. Again there are 3 triangles which are all right triangles. What else do you notice about the triangles?

31. In your report, consider
Describe your construction method in pictures and words.
Before you tried your method, why did you think it would work?
Were there methods you tried that didn’t work? What were they?
What are the properties of the constructed shapes? How do you know your shape has these properties.

32. Sorting by Symmetry & more Advanced Properties

33. Transformations

34. Mira—a transition to the Computer
Rotation
Translation
Finding Centers of Rotation

35. Coordinate work
Wumps

36. Dilations

37. The Role of Definitions
To me it appears a radically vicious method, certainly in geometry, . . .to supply a child with ready made definitions, to be memorized after being more or less carefully explained.. . .The evolving of a workable definition by the child’s own activity stimulated by appropriate questions, is both interesting and highly educational.
Bechara, Blandford, 1908

38. Development of Definitions
Descriptive Defining
Constructive Defining
Hierarchical vs. Partition Defining
The Role of Construction & Measurement

39. Quadrilateral Activities
Geometric Structures: If we had time, we would go through these activities—you may think they are repetitive, but students need all of these experiences to deal more flexibly with properties and definitions.
Let’s do Sketchpad activity from Restructuring Proof –think about this activity from High and Low levels.

40. Geometric Thinking Task Demand Categories
Memorization:
What is the formula for the area of a triangle? State the SAS congruence postulate
Procedures without Connections:
Given this drawing, find the area of the triangle?
Given these marked triangles, are they congruent?
Procedures with Connections:
Draw a rectangle around the triangle and find the area.
Fold the paper and identify the relationship between the triangles.
Doing mathematics:
If we don’t want to count the squares that cover the triangle, how can we find the area?
Verify by measurement; Reason through your conjecture about the triangles.

41. Reasoning with relationships
Actively looking for and applying geometric relationships, within and between geometric figures. Internal questions include:
“How are these figures alike?”
“In How many ways are they alike?”
“How are these figures different?”
“What would I have to do to this object to make it like that object?”

42. Generalizing geometric ideas
Wanting to understand and describe the "always" and the "every" related to geometric phenomena. Internal questions include:
“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?”

43. Investigating invariants
An invariant is something about a situation that stays the same, even as parts of the situation vary. This habit shows up, e.g., in analyzing which attributes of a figure remain the same when the figure is transformed in some way. Internal questions include:
“How did that get from here to there?”
“What changes? Why?”
“What stays the same? Why?”

44. Sustaining reasoned exploration
Trying various ways to approach a problem and regularly stepping back to take stock. Internal questions include:
"What happens if I (draw a picture, add to/take apart this figure, work backwards from the ending place, etc.….)?"
"What did that action tell me?"

45. Sketchpad is not limited to Geometry
Making figures for any handout-the Pentagon
The capabilities of the hide/show buttons and easy text abilities make it ideal for puzzles
It gives a visual representation of algebra-graphs and algebra tiles.

46. This simple model—shows both teacher actions and student responses
This simple model—shows both teacher actions and student responses. It also suggests a balanced cycle. I got this idea from 4-MAT and adapted.

47. ALGEBRA CONCRETE/ PICTORIAL GRAPH VERBAL SYMBOLIC NUMERICAL/ TABLE
In addition to balancing learning styles and strategies-we balance representation—which also connects to learning styles (visual, audio, etc.) This model was at the heart of the day—I kept coming back to it.
SYMBOLIC
NUMERICAL/
TABLE

48. Teaching with the Pentagram
CONCRETE/
PICTORIAL
GRAPH
VERBAL
This slide explains the pentagram in more detail. The diagonals show that you can connect any two types and you can always get from one type to another.
NUMERICAL/
TABLE
SYMBOLIC

49. Liz’s Pattern

50. Factoring
I have a PowerPoint for you and Sketches that are interactive with the tiles.

51. Formulas & Graphing Making sense of geometric formulas
A sketch that could be done on NSpire—but it works on the computer, too

52. The Possibilities are Endless!
I have a CD for you with many sketches—start playing with them—imagine how you can use them—even change them.
When you think of a concept for a lesson—visualize the geometric representation of it, then either play with GSP or me.
Any questions? Comments?