## Tangram Puzzles

- Lesson

In this activity, students choose a picture and use all seven tangram pieces to fill in the outline. They describe and visualize what figures look like when they are transformed through rotations, reflections, or translations, and are put together or taken apart in different ways. This activity helps students to develop their spatial memory and spatial visualization skills, to recognize and apply transformations, and to become more precise in their use of vocabulary about space. Students use an internet-based tool to explore tangram puzzles.

This mathematics excursion is a brief mathematics activity. To maximize student learning, certain prerequisites are necessary to use this activity. Thus, it would be appropriate to include this activity as part of a more fully developed Standards-based lesson, but it should not be used as a complete stand-alone lesson.

*Initiating the Excursion*

Young students' experiences with puzzles provide a background for undertaking this activity. Because similar puzzles are available for use with plastic or paper tangrams, students could move back and forth between concrete materials and the computer environment. To facilitate this, you could have each student cut out the tangram pieces from the second page of the Slide, Flip, Turn Activity Sheet.

Slide, Flip, Turn Activity Sheet

Then, before going online, place a triangle, square, and parallelogram on an overhead projector. Ask student volunteers to demonstrate the three transformations that will be used in this activity (flips, turns, and slides) as preparation for solving the tangram puzzles. Engage students in a brief discussion about each of the three transformations. Discuss the effect that each of these transformations has on the shapes.

*Developing the Excursion*

Group students into teams of two or three. Give each group a copy of the Slide, Flip, and Turn Activity Sheet.

Students should go to the Tangram Puzzles E‑Example to complete the following activity.

Tangram Puzzles E-Example (Java)

a newer, Flash version is available here:

Tangram Puzzles E-Example (Flash)

Have partners work together to solve four different tangram puzzles. For example, one can give directions as the other "drives the mouse." Encourage the students to think about how they need to manipulate the tangram pieces before they begin moving the pieces. Encourage them to make a plan about how to move the pieces, rather than using trial‑and‑error as their strategy. Discuss with students the plans they develop. Ask them to defend their reasoning. This will elicit use of accurate mathematical vocabulary and associate it with the appropriate transformation.

Teams should record their solutions on the activity sheet. Partners should switch roles until all four puzzles are completed.

*Closing*

The closing should be structured so that students can review and pull together what they have learned. Include questions or tasks that encourage students to reflect on their work, particularly as related to the specific stated Learning Objectives. For example, after students complete the activity, you could have them consider questions similar to the guiding questions below. In so doing, they will consolidate what they have learned. Furthermore, this will provide an opportunity for you and your students to assess what they have learned and what they still want or need to understand. This will help you plan for further instruction.

When the four puzzles are completed, ask students to review their solutions. Help them do this by asking questions like the following:

- Show me where you used a flip, a turn, and a slide in solving a puzzle.
- Look at this figure [put a tangram piece or other geometric shape on the overhead], draw what it will look like when it is flipped [or rotated or slid].
- Look around the classroom for examples of transformed shapes. Describe the transformation.
- Were any of the puzzles more challenging than the others? Why?
- Go back and review the solution of each puzzle. How were the solutions alike? How were the solutions different?
- Compare your solutions with some other students' solutions. How were the solutions alike? How were the solutions different?

- Slide, Flip, Turn Activity Sheet (which includes a template for Tangram pieces)
- Computers with internet access

**Extensions**

It will be useful for students to do some related activities using physical manipulatives. For example, teams of students could use the paper tangrams to create different puzzles which they challenge other teams to solve.

**Questions for Students**

- What happens if that parallelogram is rotated? Is the result different than if you flipped it?
- Make a plan before you start moving the pieces. Explain why your plan will work.
- Try giving your directions to your partner without pointing or showing, just by talking.
- Try to fill in this shape in another way.
- How many different ways are there to fill in this shape?
- I see that you used the Hint to solve the puzzle. That's fine. Now, can you do it again without using the Hint?

### Learning Objectives

Students will:

- Recognize and apply flips (reflections), slides (translations), and turns (rotations).
- Use spatial memory and spatial visualization skills.

### NCTM Standards and Expectations

- Recognize, name, build, draw, compare, and sort two- and three-dimensional shapes.

- Describe attributes and parts of two- and three-dimensional shapes.

### Common Core State Standards – Mathematics

-Kindergarten, Geometry

- CCSS.Math.Content.K.G.A.1

Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

-Kindergarten, Geometry

- CCSS.Math.Content.K.G.B.5

Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

-Kindergarten, Geometry

- CCSS.Math.Content.K.G.B.6

Compose simple shapes to form larger shapes. For example, ''Can you join these two triangles with full sides touching to make a rectangle?''

Grade 1, Geometry

- CCSS.Math.Content.1.G.A.1

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size) ; build and draw shapes to possess defining attributes.

Grade 1, Geometry

- CCSS.Math.Content.1.G.A.2

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Grade 2, Geometry

- CCSS.Math.Content.2.G.A.1

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP7

Look for and make use of structure.