Private: Learning Math: Geometry

Professional Development > Private: Learning Math: Geometry > 5. Dissections and Proof > 5.1 Part A: Tangrams (15 minutes)

Mathematics

K-2, 3-5, 6-8

Dissections and Proof Part A: Tangrams (15 minutes)

Session 5, Part A

In this part

Exploring Tangrams
Moving Tangrams

Exploring Tangrams

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.

Use a set of tangrams, or print and cut out the set provided here, to work on the following problems.

Think about the angles you need to create in order for a shape to be a square.

Problem A1

Given that the tangram puzzle is made from a square, can you recreate the square using all seven pieces?

Problem A2

Use all seven tangram pieces to make a rectangle that is not a square.

Part A (continued): Moving Tangrams

One set of tangram pieces can make more than one shape. Geometric language helps you give clear descriptions of how to move pieces to change one shape into another. Roll your cursor over each of the three images below to see an animation of that move.

Slide or Translation
Here, a parallelogram made of two small triangles becomes a square as you slide one of the small triangles parallel to the base of the parallelogram. This type of movement is known as a translation. Try this with your tangram pieces.

Flip or Reflection
Here, the parallelogram becomes a triangle when you flip one of its halves (small triangles) at its horizontal midline. This type of movement is called a reflection. Try it.

Rotation
Here, a square becomes a triangle when you rotate one of its two halves (small triangles) 270° around vertex A. This type of movement is called a rotation. Try it.

Problem A3
For each of the pairs of figures below, do the following:

Build the shape on the left with your tangram set.
Turn it into the shape on the right by reflecting, rotating, or translating one of two of the pieces. (This may take more than one step.)
Write a description, telling which piece or pieces you moved and how you moved them.

Solutions

Problem A1

Problem A2

Here is one possible solution:

Problem A3

Translate the leftmost triangle horizontally to the right by twice the length of the side of the square.

Rotate the top triangle 180° about vertex A. Then translate the square to the left by twice its side length.

Imagine a line through the middle of the parallelogram, parallel to its horizontal sides. Reflect the rightmost triangle over it.

Series Directory

Private: Learning Math: Geometry

Credits

Produced by WGBH Educational Foundation. 2003.

ISBN: 1-57680-597-2

Sections

5.1 Part A: Tangrams (15 minutes)

Session 5, Part A

5.2 Part B: Cutting Up (50 minutes)

Session 5, Part B

5.3 Part C: The Midline Theorem (55 minutes)

Session 5, Part C

5.4 Homework

Session 5, Homework

Sessions

Session 2 Triangles and Quadrilaterals

Learn about the classifications of triangles, their different properties, and relationships between them. Examine concepts such as triangle inequality, triangle rigidity, and side–side–side congruence, and look at the conditions that cause them. Compare how these concepts apply to quadrilaterals. Explore properties of triangles and quadrilaterals through practical applications such as building structures.

Session 3 Polygons

Explore the properties of polygons through puzzles and games, then proceed into a more formal classification of polygons. Look at mathematical definitions more formally, and explore how terms can have different but equivalent definitions.

Session 4 Parallel Lines and Circles

Use dynamic geometry software to construct figures with given characteristics, such as segments that are perpendicular, parallel, or of equal length, and to examine the properties of parallel lines and circles. Look past formal definitions and discover the properties and relationships among geometric figures for yourself.

Session 5 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.

Session 6 The Pythagorean Theorem

Continue to examine the idea of mathematical proof. Look at several geometric or algebraic proofs of one of the most famous theorems in mathematics: the Pythagorean theorem. Explore different applications of the Pythagorean theorem, such as the distance formula.

Session 7 Symmetry

Investigate symmetry, one of the most important ideas in mathematics. Explore geometric notions of symmetry by creating designs and examining their properties. Investigate line symmetry and rotation symmetry; then learn about frieze patterns.

Session 8 Similarity

Examine your intuitive notions of what makes a "good copy" and then progress toward a more formal definition of similarity. Explore similar triangles and look into some applications of similar triangles, including trigonometry.

Session 9 Solids

Explore various aspects of solid geometry. Examine platonic solids and why there are a finite number of them. Investigate nets and cross-sections for solids as a way of establishing the relationships between two–dimensional and three–dimensional geometry.

Session 10 Classroom Case Studies, K-2

This is the final session of the Geometry course! In this session, we will examine how geometry as a problem-solving process might look when applied to situations in your own classroom. This session is customized for three grade levels.

Session 11 Classroom Case Studies, 3-5

Session 12 Classroom Case Studies, 6-8

Watch Videos 11 and 12 in the 10th session for grade 6–8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6–8 teachers (former course participants) who have adapted their new knowledge to their classrooms.