## What's So Special About Triangles, Anyway?

3-5
1

Students explore ways of building different basic shapes from triangles. They also investigate the basic properties of triangles, as well as relationships among other basic geometric shapes.

Note to Teachers:This lesson includes two activities. Depending upon the ability level of your students, you may accomplish the tasks in one class session, or you may wish to separate them into two lessons.

### Task 1: What Can You Build With Triangles?

Begin the class by reviewing triangles. You may ask questions such as:

• What is a triangle?
• How can we classify triangles based on their sides?
• How can we classify triangles based on their angles?
• Using different quadrilaterals, how can you cut or fold paper to create triangles?

Distribute the Just Two Triangles Activity Sheet to each student, and have them cut out the triangles.

Use the models of the square, triangle, and parallelogram to encourage students to make various shapes. Ask them to try to make these shapes with two triangles. Have the students glue the "new" shapes onto the activity sheet.

In a whole-class discussion, encourage students to share at least one important thing that they noticed about one of the new shapes: likenesses and differences; where it could be seen in the classroom, playground, school, or at home; and so on.

Some possible drawings for the activity sheet include:

### Task 2: How Do You Build Triangles?

Next, display various triangular shapes using pattern blocks, and ask, "How do you know that these shapes are triangular?" The following properties of triangles should emerge from this discussion: three sides, three corners and angles, and straight (rather than curved) sides.

Distribute pattern blocks to each group of two to four students. Have students explore ways to make triangles with the patterning blocks.

Alternatively, you can use the Patch Tool for pattern blocks. This is an applet version of physical pattern blocks.

Have students share their constructions with each other. As a class, share any common and/or unique findings that students may have discovered.

Distribute and follow directions in the How Do You Build Triangles? Activity Sheet.

Have students work in pairs to give or write directions for building one of the triangles, then see if another pair of students can build it by following the directions.

Some possible solutions for the activity sheet include:

Have students compare their drawings with those of several classmates to wrap up the task.

To wrap up the lesson, have a class discussion on what students noticed about using triangles to build other polygons and vice versa.

Assessment Options

1. Use the activity sheets as a form of assessment.
2. Give students a hexagon pattern block and ask them to trace it on a piece of paper several times. Give students a ruler and ask them to break down the hexagon into triangular pieces. Alternatively, students can fold the paper, but this will make it harder for students to undo mistakes. Make it a requirement for students to make at least three different types of triangle.

Extensions

1. In the second half of the lesson, challenge students to make as many different triangles as they can. As they are doing so, ask them to describe the characteristics which make triangles "different."
2. Pose questions such as the following, to correspond to the second half of the lesson:
• Can you make a triangle using only one shape? [4 triangles can make a bigger triangle.]
• Can you make a triangle that uses each of the four shapes? What's the fewest number of pieces you need to make it?
3. Move on to the next lesson, What Does it Take to Construct a Triangle?

Questions for Students

1. What shapes were you able to build with your triangles?

[Students may reply by saying they were able to build squares, parallelograms, and other triangles.]

2. What three-dimensional shapes were you able to build with your triangles?

[Students may reply by saying they were able to build triangular pyramids and triangular prisms.]

3. How many different triangles can be built with two, three, and then four shapes?

4. What happens if all twelve shapes are used to build one "huge" triangle?

[Note: One more small triangle is needed because the pattern for the triangular area is one, four, nine, sixteen, and twenty-five small triangles.]

5. What is the largest triangle that can be built with twelve shapes?

[You may wish to challenge students' responses to this question by asking them how they know they have discovered the largest triangle.]

6. How many different symmetrical designs can be created for the largest triangle?

[It may be helpful to record the various symmetrical designs on chart paper as students discover them.]

Teacher Reflection

• Describe your students' level of enthusiasm with these kinesthetic tasks. How could you structure the class to make it more engaging?
• How is using an online manipulative, such as the Patch Tool, different (or the same as) using physical manipulatives? Which was better for these two tasks?
• What additional questions could you ask students to enrich the class, or peer, conversations?

### Building with Triangles

3-5

Engage students in a study of triangles and their properties.

### What Does it Take to Construct a Triangle?

3-5
Students explore the importance of the side lengths of a triangle and when triangles can or cannot be constructed on the basis of these lengths.

### How Many Triangles Can You Construct?

3-5, 6-8
Students identify patterns in a geometrical figure (based on triangles) and build a foundation for the understanding of fractals.

### Learning Objectives

Students will:

• Explore ways of building different basic shapes from triangles.
• Investigate basic properties of triangles.
• Investigate the relationships among basic geometrical shapes.

### NCTM Standards and Expectations

• Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes.
• Classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids.
• Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes.
• Build and draw geometric objects.