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
2......4 (1 times 4)
3......16 (4 times 4)
4......64 (16 times 6)
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.
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. Wrap up the class by having a discussion on any patterns the noticed with the number of segments (or shapes) and the total length (or area).
Questions for Students
Refer to the Instructional Plan for Questions.
- Describe your students' level of enthusiasm. What could you change about this lesson to make it more engaging?
- How can you incorporate technology to help students find patterns?
- How can you modify this lesson to help high and low-level achievers?
- Identify patterns in a geometrical figure.
- Build a foundation for the understanding of fractals.
- Make hypotheses and develop experiments to test them.
NCTM Standards and Expectations
- Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes.
- Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.
Common Core State Standards – Mathematics
Grade 4, Algebraic Thinking
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
Classify two-dimensional figures in a hierarchy based on properties.
Common Core State Standards – Practice
Make sense of problems and persevere in solving them.
Model with mathematics.
Use appropriate tools strategically.
Look for and make use of structure.