## Learning Math: Geometry

# What Is Geometry?

## Explore the basics of geometric thinking using rich visualization problems and mathematical language. Use your intuition and visual tools for geometric construction. Reflect on the basic objects of geometry and their representation.

**In This Session**

**Part A: **Quick Images

**Part B: **Building from Directions

**Part C: **Folding Paper

**Part D: **Basic Objects

**Homework**

In this session, you will use mathematical communication and geometric thinking to solve problems. You will use paper folding as a construction tool because of its visual and kinesthetic properties. (Paper folding is more intuitive and taps into geometric reasoning more easily than traditional straightedge and compass constructions.) The session ends with some reflection on the basic objects of geometry and the difference between an ideal mathematical object and its representation.

For information on required and/or optional materials for this session see the Notes section below.

**Learning Objectives**

In this session, you will learn to do the following:

- Use geometric thinking to solve visualization problems
- Use mathematical language to express your ideas and to understand the ideas of others
- Construct the basic objects of geometry

### Key Terms

**New in This Session:**

**Altitude:** An altitude of a triangle is a line segment connecting a vertex to the line containing the opposite side and perpendicular to that side.

**Angle Bisector: **An angle bisector is a ray that cuts the angle exactly in half, making two equal angles.

**Concurrent: **When three or more lines meet at a single point, they are said to be concurrent. In a triangle, the three medians, three perpendicular bisectors, three angle bisectors, and three altitudes are each concurrent.

**Line: **A line has only one dimension: length. It continues forever in two directions (so it has infinite length), but it has no width at all. A line connects two points via the shortest path, and then continues on in both directions.

**Line Segment: **A line segment is the portion of a line lying strictly between two points. It has a finite length and no width.

**Median: **A median is a segment connecting any vertex of a triangle to the midpoint of the opposite side.

**Midline: **A midline is a segment connecting two consecutive midpoints of a triangle.

**Perpendicular Bisector: **The perpendicular bisector of a line segment is perpendicular to that segment and bisects it; that is, it goes through the midpoint of the segment, creating two equal segments.

**Plane: **A plane is a flat, two-dimensional object. We often represent a plane by a piece of paper, a blackboard, or the top of a desk. In fact, none of these is actually a plane, because a plane must continue infinitely in all directions and have no thickness at all. A plane can be defined by two intersecting lines or by three non-collinear points.

**Point: **A point specifies only location; it has no length, width, or depth. We usually represent a point with a dot on paper, but the dot we make has some dimension, while a true point has dimension 0.

**Ray: **A ray can be thought of as a half a line. It has a point on one end, and it extends infinitely in the other direction.

### Notes

**Note 1**

**The following materials are needed for groups choosing to do hands-on activities:**

- at least 17 pieces of patty paper
- blank white paper (at least 10 pages, 3 of which must be square)
- one straightedge or ruler
- pencils and pens

You can purchase these materials from the following sources:

Key Curriculum Press 1150 65th Street Emeryville, CA 94608 800-995-MATH 800-541-2442 (fax) www.keypress.com |
Delta Education 80 Northwest Boulevard P.O. Box 3000 Nashua, NH 03061-3000 800-442-5444 800-282-9560 (fax) www.delta-education.com |