Presentation on theme: "Jean J. McGehee University of Central Arkansas"— Presentation transcript:
1Jean J. McGehee email@example.com University of Central Arkansas Building Geometric Thinking with Hands-On Tasks & in Virtual EnvironmentsJean J. McGeheeUniversity of Central Arkansas
2TodayGeometric Habits of Mind—from Paper Folding to Using Sketchpad—in the context of rich problemsTransformations: a Connecting big ideaThe role of good definitions: QuadrilateralsConnecting Sketchpad to the Number and Algebra strands
4Goals of FGT Strengthen understanding of geometry Enhance capacity to recognize and describe geometric thinkingIncrease attention to students’ thinkingEnhance understanding of students’ geometric thinkingPrepare to advance students’ geometric thinking
5The Geometry Curriculum in Arkansas Let’s take a quick look at the frameworks..\..\..\Desktop\frameworks\geometry_06.docEven my student interns at UCA notice how much repetition there is in the curriculum—e.g. The Triangle SumWe do need to revisit ideas—but we need to do it with value added.
6The FGT Project in NE Arkansas Two school districts in grades 5-11I 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.
7All 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.
8It’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!!
9Another 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 werelike what they had done in class "except they didn't give us any paper to fold".
10Structure of FGTThe 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.
11Three content strandsFocus the work on different important areas of geometry & measurement.They are:PropertiesTransformationsMeasurement
12The Structured Exploration Process Stage 1: Doing mathematicsStage 2: Reflecting on the mathematicsStage 3: Collecting student workStage 4: Analyzing student workStage 5: Reflecting on students’ thinking
13FGT G-HOMs Reasoning with Relationships Generalizing Geometric Ideas Looking for InvariantsBalancing Exploration & Reflection
14More about FGT and G-HOMs later First, let’s do exercise our own geometric thinkingFolding, Making Squares, Congruent HalvesPaper-Folding & ConstructionsTangramsDissecting ShapesComparing TrianglesWe will start in detail—but I may have to summarize the latter problems.
15Do Math--Ideally Work problem individually 5-10 minutes Work problem in groups 25 minutesLast 10 minutes groups prepare report either on transparency or chart paperReflect on the problem & Identify G-HOMs minutes
16Let’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 areaGo to Sketchpad
17Review: 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 approachAlso I want to show you this problem in a fun book.
18A Quilter’s Solution—does it work? Go to Sketchpad
19The Number DevilThis little devil beguiles Robert into dreams to give him a glimpse of the beauty & power numbers.In this case, the square root of 2.
20Student intern gave students two squares and asked How many black squares fit into the red square?Show how you knew this.
21Hands-On & SketchpadI have learned both in PD and classes to start with Hands-OnMaking 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.
22Basic Paper FoldingThe 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
23Analyzing 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?
24Student 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?
25Paper Folding related toTangrams You are familiar with the square, but can you make a rectangle that are not squares—2 ways?
26Let’s explore the area problems Tangrams on SketchpadMore shapes with the same area—an understanding based on properties rather than memorized formulas.
27Dissecting ShapesThe ability to dissect and transform shapes is important.Students are also exploring invariance and properties.
29Comparing TrianglesStart 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.
30Comparing TrianglesStart 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?
31In 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.
37The 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
38Development of Definitions Descriptive DefiningConstructive DefiningHierarchical vs. Partition DefiningThe Role of Construction & Measurement
39Quadrilateral 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.
40Geometric Thinking Task Demand Categories Memorization:What is the formula for the area of a triangle? State the SAS congruence postulateProcedures 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.
41Reasoning 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?”
42Generalizing 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?”
43Investigating 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?”
44Sustaining 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?"
45Sketchpad is not limited to Geometry Making figures for any handout-the PentagonThe capabilities of the hide/show buttons and easy text abilities make it ideal for puzzlesIt gives a visual representation of algebra-graphs and algebra tiles.
46This 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.
47ALGEBRA 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.SYMBOLICNUMERICAL/TABLE
48Teaching with the Pentagram CONCRETE/PICTORIALGRAPHVERBALThis 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/TABLESYMBOLIC
50FactoringI have a PowerPoint for you and Sketches that are interactive with the tiles.
51Formulas & Graphing Making sense of geometric formulas A sketch that could be done on NSpire—but it works on the computer, too
52The 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?