## How Many Triangles Can You Construct?

- Lesson

Students identify patterns in a geometrical figure (based on triangles) and build a foundation for the understanding of fractals.

Distribute and follow the directions on the How Many Triangles? activity sheet.

How Many Triangles? Activity Sheet |

Initially, students should attempt the activity sheet individually. You may wish for students to work together after they have had a chance to work independently.

Ask the following questions to stimulate a whole class discussion:

- How did your triangle change?
- How did you find out the number of triangles that were possible?
- What did you notice about the number patterns?

*Solutions to the Activity Sheet:*

Students should see the following pattern emerge for Triangle 1:

Stage...Number of Triangles

1......1

2......4

3......16

4......64

Students should see the following pattern emerge for Triangle 2:

Stage...Number of Shaded Triangles(and Reason)

1......3(3 to the power of 1)

2......9(3 to the power of 2)

3......27(3 to the power of 3)

4......81(3 to the power of 4)

Ask students if they have heard the term **fractal** previously.
Students who are familiar with the term will know that a fractal is a
geometric shape that can be split into parts, where the parts are
smaller versions of the original geometric shape. Introduce the Fractal Tool, which allows students to explore and create their own fractals.

Fractal Tool |

- How Many Triangles? Activity Sheet
- Ruler, pencils, or fine-line markers
- Writing paper

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

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

### Learning Objectives

- Identify patterns in a geometrical figure
- Build a foundation for the understanding of fractals
- Make hypotheses and then develop experiments to test them

### Common Core State Standards – Mathematics

Grade 4, Algebraic Thinking

- CCSS.Math.Content.4.OA.C.5

Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule ''Add 3'' and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

Grade 5, Geometry

- CCSS.Math.Content.5.G.B.4

Classify two-dimensional figures in a hierarchy based on properties.

### 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.