## Learning Math: Geometry

# Dissections and Proof

## Review and explore transformations such as translation, reflection, and rotation. Apply these ideas to solve more complex geometric problems. Use your knowledge of properties of figures to reason through, solve, and justify your solutions to problems. Analyze and prove the midline theorem.

### In This Session

**Part A:** Tangrams

**Part B:** Cutting Up

**Part C:** The Midline Theorem

Homework

In this session, you will begin to use the properties of figures to solve geometric problems and to justify their solutions. You will also use visualization and geometric reasoning to solve cutting tasks.

For information on required and/or optional materials for this session, see **Note 1**.

### Learning Objectives

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

- Dissect (cut) geometric figures into pieces you can rearrange to form different geometric figures
- Use geometric language to describe your reasoning, justifications, and solutions to problems
- Prove the midline theorem

### Key Terms

**Previously Introduced
**

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

**Parallelogram: **A parallelogram is a quadrilateral that has two pairs of opposite sides that are parallel.

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

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

**Side-Angle-Side (SAS)Congruence: **Side-angle-side (SAS) congruence states that if any two sides of a triangle are equal in length to two sides of another triangle and the angles bewteen each pair of sides have the same measure, then the two triangles are congruent; that is, they have exactly the same shape and size.

**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**

**Midline Theorem: **The midline theorem states that a midline of a triangle creates a segment that is parallel to the base and half as long.

**Reflection: **Reflection is a rigid motion, meaning an object changes its position but not its size or shape. In a reflection, you create a mirror image of the object. There is a particular line that acts like the mirror. In reflection, the object changes its orientation (top and bottom, left and right). Depending on the location of the mirror line, the object may also change location.

**Rotation: **Rotation is a rigid motion, meaning an object changes its position but not its size or shape. In a rotation, an object is turned about a “center” point, through a particular angle. (Note that the “center” of rotation is not necessarily the “center” of the object or even a point on the object.) In a rotation, the object changes its orientation (top and bottom). Depending on the location of the center of rotation, the object may also change location.

**Tangram: **A tangram is a seven-piece puzzle made from a square. A typical tangram set contains two large isosceles right triangles, one medium isosceles right triangle, two small isosceles right triangles, a square, and a parallelogram.

**Translation: **Translation is a rigid motion, meaning an object changes its position but not its size or shape. In a translation, an object is moved in a given direction for a particular distance. A translation is therefore usually described by a vector, pointing in the direction of movement and with the appropriate length. In translation, the object changes its location, but not its orientation (top and bottom, left and right).

### Notes

**Note 1**

**Materials Needed:**

• |
tangram set, or a large square of stiff construction paper so that you can make your own tangram set (see Homework for instructions) |

• |
blank white paper (at least 12 pages) |

• |
scissors |

• |
ruler |

**Tangram Set**

You can purchase tangram sets from the following source:

ETA/Cuisenaire

500 Greenview Court

Vernon Hills, IL 60061

800-445-5985/847-816-5050

800-875-9643/847-816-5066 (fax)

http://www.etacuisenaire.com