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**Part A:** Geometry as a Problem-Solving Process

**Part B:** Developing Geometric Reasoning

**Part C:** Activities That Illustrate Geometric Reasoning

**Homework**

In the previous sessions, we explored geometry as a problem-solving process. You put yourself in the position of a mathematics learner, both to analyze your individual approach to solving problems and to get some insights into your own conception of geometric reasoning. It may have been difficult to separate your thinking as a mathematics learner from your thinking as a mathematics teacher. Not surprisingly, this is often the case! In this session, however, we will shift the focus to your own classroom and to the approaches your students might take to mathematical tasks involving geometry.

As in other sessions, you will be prompted to view short video segments throughout the session; you may also choose to watch the full-length video for this session. **Note 1**

In this session, you will do the following:

- Explore the development of geometric reasoning at your grade level, including the van Hiele model of geometric learning
- Review mathematical tasks and their connection to the mathematical themes in the course
- Examine children’s understanding of geometric concepts

**Previously Introduced
**

**Concave Polygon: **A concave polygon is any polygon with an angle measuring more than 180°. Concave polygons look like they are collapsed or have one or more angles dented in.

**Congruent:** Two figures are congruent if all corresponding lengths are the same, and if all corresponding angles have the same measure. Colloquially, we say they “are the same size and shape,” though they may have different orientation. (One might be rotated or flipped compared to the other.)

**Irregular Polygon: **An irregular polygon is any polygon that is not regular.

**Polygon: **A polygon is a two-dimensional geometric figure with these characteristics:

- It is made of straight line segments.
- Each segment touches exactly two other segments, one at each of its endpoints.
- It is closed — it divides the plane into two distinct regions, one inside and the other outside the polygon.

**Quadrilateral: **A quadrilateral is a polygon with exactly four sides.

**Rectangle: **A rectangle is a quadrilateral with four right angles.

**Regular Polygon: **A regular polygon has sides that are all the same length and angles that are all the same size.

**Similar: **Two polygons are similar polygons if corresponding angles have the same measure and corresponding sides are in proportion.

**Square: **A square is a regular quadrilateral.

**Trapezoid: **A trapezoid is a quadrilateral that has one pair of opposite sides that are parallel.

**Vertex: **A vertex is the point where two sides of a polygon meet.

**New in This Session
**

**Van Hiele Levels: **Van Hiele levels make up a theory of five levels of geometric thought developed by Dutch educators Pierre van Hiele and Dina van Hiele-Geldof. The levels are (0) visualization, (1) analysis, (2) informal deduction, (3) deduction, and (4) rigor. The theory is useful for thinking about what activities are appropriate for students, what activities prepare them to move to the next level, and how to design activities for students who may be at different levels.

**Note 1
**

This session uses classroom case studies to examine how children in grades K-2 think about and work with geometry. If possible, work on this session with another teacher or a group of teachers. A group discussion will allow you to use your own classroom and the classrooms of fellow teachers as case studies to make additional observations.