Math+Lab+2

Math Lab 2 - Geometry Lesson using Mathematica

van Hiele Model - 5 Stages of Geometric Reasoning Level 0 - visualization: children will recognize shapes based on appearances alone, not their geometric properties Level 1 - analysis stage: children begin to recognize the attributes or properties of a shape but cannot compare/contrast two geometric figures to one another Level 2 - informal deduction: children begin to compare geometric shapes and construct simple proofs Level 3 - deduction: children gain the ability to accept postulates/theorems and write proofs (typical level of high school geometry instruction) Level 4 - rigor: students begin to be able to work in other geometric systems that include abstract, proof-oriented logic (Brahier, 2009)
 * The van Hiele model states that this geometric development is sequential and quality instruction is the key to advancement for students.

__Stage 1 - Desired Results__ Students will know and recognize several polygons, their unique attributes, be able to compare and contrast these polygons based on their attributes, be able to find the area for several polygons, and be able to prove that the area for each polygon is based on the area of a rectangle.
 * Established goals:** California Content Standard Geometry GE10 - Students compute areas of polygons including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms and trapezoids.
 * Understandings**: Validate and understand how the area formula for several polygons are based on the area of a rectangle.
 * Essential Questions**: How are polygons related?

__Stage 2 - Assessment Evidence__
 * Performance Task**: Brahier Geometry lesson example #12 "What's My Area?" pg. 225
 * 1) Start with a grid and draw a rectangle using the squares. See that to determine the area of the rectangle you count how many squares are along the bottom (base) and the side (height); area is then found by multiplying base times height. So Area (Rectangle)=base x height
 * 2) A square is a special type of rectangle where the base and height are the same; therefore students should see that the area is found by taking the square of one side or Area (Square)=side^2.
 * 3) A parallelogram is a rectangle with a triangle piece translated (shifted) horizontally. Since the area has not changed, the area can still be found by multiplying base times height.
 * 4) A triangle can be made by drawing a diagonal through a parallelogram so that is bisected in half thus creating two triangles; therefore a triangle is half of a parallelogram. So the area of a triangle is one half the area of a parallelogram or Area (Triangle)= 1/2bh
 * 5) Since a rhombus is a simple parallelogram, its area is still A=bh
 * 6) A trapezoid is made of a square and two right triangles with equal height, therefore the area of a trapezoid is A= 1/2 (b1+b2)h
 * Other Evidence**: California Content Standards Geometry Released Questions #41-47

__Stage 3 - Learning Plan__ This lesson will allow students to progress through van Hiele's levels of geometric reasoning. First students should view shapes (rectangle, square, triangle, rhombus, trapezoid and parallelogram) and identify what they are based on their shape (Level 0) and then start listing or describing the attributes or properties of each of these shapes (Level 1). Next students should compare the shapes and point out their similarities or differences with regard to each shape's properties (Level 2). Finally, as students work through the lesson and see that the area formula for each shape is based on the area formula for a rectangle, students can prove this using drawings or tangrams and support this by writing out proofs (Level 3).