Grade 10

Benchmark 10.3.E Draw and construct representations of two-and three-dimensional geometric objects using a variety of tools, such as straightedge, compass technology. Chapter 6 in the textbook

Benchmark 10.3.F Represent and model transformations in a coordinate plane and describe the results. Chapter 12 in the textbook.

Grade Ten – Mathematics Essential Focus Elements from the SMART Consortium

Need to Know Expected to do

Tips and Skills from Smart Consortium -Have pairs of students do a “describe and draw” where one communicates direction and the other makes the design using GeoBoard or dot paper. -Add excitement by making use of the internet enchantedmind and National Library of Virtual Manipulatives or use tangrams -Use protractors and other tools to explore angles and polygons -Use polar navigation -Use GeoBoards to construct similar and congruent triangles. -Find words, letters and figures that have symmetric patterns; site examples of translation, rotation and reflection in real-life situations. -Use practical problems involving triangles. -Bring out the compass, ruler and protractor for this one.

The Learning Theory The van Hiele Levels of Geometric Understanding Dutch Educators Their theory explains why many students encounter difficulties in their geometry course, especially with formal proofs. Students need many more experiences in thinking at lower levels before learning formal geometric concepts.

Types of spatial difficulties displayed by underachievers 1. Directional Concepts (up, down, right, left) 2. Elementary spatial concepts (inside, above, under) 3. A spatial concept is associated with inappropriate criteria (these are parallel lines, these lines are not parallel) 4. Cannot act mentally on a shape or visualize it being changed or transformed. 5. Concept of angle and the extent of rotation cause difficulty (decide that one angle is larger than another because its arms are longer)

6. Student uses inappropriate perceptual features to categorize shapes. 7. Student has difficulty representing 3-D objects in 2-D.

Why do they form misconceptions 1. Perceptual difficulties—students have difficulty integrating components or parts of a spatial stimulus to form the whole, difficulties discriminating between the main visual information and irrelevant background information. Lack of earlier sensory-motor experiences, such as building, matching, shape manipulation, etc. Difficulties learning visually or tactually, some prefer to learn auditorily. When this is an extreme learning style, difficulty learning visually.

Inadequate teaching. Research by Dickson, Brown and Gibson(1984), many students form spatial concepts are due primarily to inadequate teaching.

Students who have difficulty learning spatial knowledge can be assisted by the following sequence: 1. Students manipulate shapes in free play situations, such as building, solving spatial problems, drawing two and three dimensional objects, explore shapes through physical actions, etc. 2. Students practice describing individual shapes, learn to use distinctive spatial properties to describe individual shapes, learn to classify shapes using spatial labels.

3. Describe differences between shapes, draw shapes and visualize shapes. 4. Students manipulate groups of shapes, describe shapes from different perspectives, fit shapes together. 5. Generate general spatial concepts (i.e. all squares are rectangles, set of polygons) 6. Recognize spatial concepts in different perceptual contexts (i.e. act on one shape to produce the other and discuss the effect of particular transformations)

Techniques and procedures for helping underachievers to learn spatial knowledge GeoBoards Logo Any materials that encourage physical action and visualization From article by Munro, J., Mathematics underachievers learning spatial knowledge

Please focus on the following What levels do you see your students at? What level do you think you teach at? What modifications can you make to your instructional practice to increase student achievement?

Activities Rotate from area to area Geogebra Reflectors for transformations Constructions of polygons You may have to pair up due to lack of computers.