Geometry and Measurement

1. Geometry and Measurement Make sure you’ve taken the geometry test prior to starting!

2. Directions (no audio) • Make sure you are in PowerPoint • Go to View toolbars drawing • You will need the drawing toolbar for some of the activities in this PowerPoint

3. van Hiele Levels • Sequential development • Not age dependent like Piaget • Geometric experience = advancement • Both instruction and language should be developmentally appropriate • Students typically learn algorithms without knowing the concept • BAD IDEA!!!

4. Level 1: Visualization • Sorting and classifying • Examples • Triangles vs. non-triangles • Shapes with all straight sides vs others • Transformations • Tangrams- make a set • Soma cube puzzles • Pattern block puzzles

5. Level 1: Visualization • Constructions with… • Geoboards- Virtual geoboard • Tessellations with pattern blocks • Pentominoes

6. Level 1: Visualization • Shape Hunts • Around the school find as many different two-dimensional shapes as possible • Draw pictures, take photos with a digital camera

7. Level 1: Visualization • Location bingo • Develops students sense of direction • Every placement is dependent on the previous placement • Put a square under the circle. • Put a triangle to the right of the square. • Put a kite above the triangle. • Fill in the rest of your grid with shapes from auto shapes on the drawing tool bar. • What would clues be for the rest of the shapes?

8. Level 2: Analysis • Sorting by characteristics of 2-D or 3-D shapes • Right triangles vs non-right triangles • Quadrilateral sort Squares Rhombus Rectangles Trapezoids Quadrilaterals Kites Parallelogram What similarities do you notice??? What differences?

9. Level 2: Analysis • Sorting by characteristics of 2-D or 3-D shapes • Right triangles vs non-right triangles • Quadrilateral sort 4-sided polygon Quadrilaterals Quad with at least (or only) one pair of parallel sides Trapezoids Kites Quadrilateral with two pairs of adjacent congruent sides Parallelogram Quad with two pairs of parallel sides Rectangles Parallelogram with 4 right angles Other Quadrilaterals Parallelogram with 4 congruent sides Rhombus Quadrilaterals that don’t fit elsewhere Parallelogram with 4 right angles and 4 congruent sides Squares

10. Level 2: Analysis • Sorting by characteristics of 3-D shapes • Faces, vertices, edges Shapes with 6 faces, 8 vertices and 12 edges Shapes with a circular base

11. Level 2: Analysis • Area and perimeter with manipulatives • Use square tiles to determine the area and perimeter of a piece of paper Algorithms for these are appropriate for level two, but students need to see where the algorithms come from by using manipulatives or representations!

12. Level 3: Informal Deduction • MDL’s- “minimal defining lists” • Minimal: if anything is removed the definition is incorrect • Defining- any object with this definition must be that shape • a square= quadrilateral with 4 right angles and 4 congruent sides 1 2 3 triangle with one right angle a right triangle = 1 2 Write an MDL for a rectangle and for a parallelogram.

13. Level 3: Informal Deduction • Shape decomposition (Draw lines on the shapes using the Drawing toolbar) • Start with an isosceles triangle • Make two shapes that have 7 total sides • Make three shapes that have 11 total sides • Start with a regular hexagon • Make two shapes that have 8 total sides • Make two shapes that have 9 total sides • Start with a square • Make three triangles- two of the three need to be congruent

14. Level 3: Informal Deduction • Nets of a cube (no audio on this slide) • Draw six small squares on your paper that all share at least a side with another square. An example is given below. • If you were to cut this shape out and fold it up, it would make a cube. How many other arrangements of six squares can you make that will fold up and make a cube?

15. Level 3: Informal Deduction Answer each question with: always, sometimes or never (no audio) • Triangles have one right angle. • Squares are rectangles. • Quadrilaterals are rectangles. • Parallelograms have a right angle. • Trapezoids have a right angle.

16. van Hiele Levels and Cognitively-appropriate Manipulative Levels

17. Measurement- Length • Concrete • Using non-standard units to measure length • How long is the book (in pencils)? • Using rulers to measure length • Representational • Paper representations with a scale • Symbolic • Conversions • 18 inches = __ feet • 5 inches = ___ cm

18. Measurement- Length (no audio) • Concrete • Measuring with non-standard units before standard units. • Take your pen. Measure the length of three objects in the room. • How long are the objects in terms of pen lengths? • Now use a ruler • How long are the objects in centimeters?

19. Measurement- Length(no audio) • Representational • What is the length of all three sides of the triangle show to the right? • What is its perimeter?

20. Measurement- Length(no audio) • Symbolic • If the perimeter of the triangle were 12 cm, what is the length of the third side? ? 4 cm 3 cm *Note: Figure is not drawn to scale

21. Measurement- Volume • Concrete • Using solids to measure volume • Objects- beans, sand • Liquid (e.g., water) • Example of a task • A box is 3 centimeters long, 4 centimeters wide and 3 centimeters high. Build this with your centimeter blocks. How many blocks did you need?

22. Measurement- Volume(no audio) • Representational • What is the volume of the figure? • Can you draw another rectangular prism with the same volume but with different dimensions?

23. Measurement- Volume(No audio) • Symbolic • Using formulas for volume • Volume = Base area * height (for prisms, cylinders)

24. In summary…. • With the students that you are working with in your diagnostic project, what van Hiele level are they performing on? • What activities from these slides peaked your interest?

25. On the 19th……(no audio) • Measurement Olympics • Gallon Man • Soma Cube Puzzles And some other tasks/projects to help you teach measurement