Algebra 1 Connect the Dots

Algebra 1 – Connect the Dots! Name______Based on the Discovering Algebra Condensed Lessons

Let’s ease back into this whole “math” thing by starting our first Algebra lesson of the year with a Connect-the- Dots Activity!  This probably seems a little simple for high school math, but what we’re actually studying (under the guise of an early elementary activity) is the algebra inside fractals.

A fractal is a figure that is created by repeating its geometric shape to form a more complex geometric figure. Sierpinski’s Triangle is a specific example of this geometric phenomenon. The investigation below is designed to help you take a look at the algebra involved in the early stages of creating Sierpinski’s Triangle.

Stage 0

Take a look at the original triangle before you connect any dots. It is an equilateral triangle, and let’s say that its total area is 1 square unit.

Area of Sierpinski’s Triangle after Stage 0 –

Stage 1

Connect all first. It’s ok if you have to cross through any of the other symbols along the way!!! Color in any downward-facing triangles. Consider the area of one upward-facing triangle in relation to the total area of the original equilateral triangle. (Assume that each smaller triangle is similar to the original bigger triangle.)

Area of one of the upward-facing triangles after Stage 1 –

Area of all of the upward-facing triangles after Stage 1 –

Stage 2

Then, connect all , but do not cross through any triangle that you have already colored in! Color in any downward-facing triangles. Again, consider the area of one of the smallest upward-facing triangles in relation to the total area of the original equilateral triangle. (Assume that each smaller triangle is still similar to the original bigger triangle.)

Area of all of the upward-facing triangles after Stage 2 – Stage 3

Finally, connect all , but do not cross through any colored triangle! Color in any downward-facing triangles. Consider the area of one of the smallest upward-facing triangles in relation to the total area of the original equilateral triangle. (Assume that each smaller triangle is still similar to the original bigger triangle.)

Area of all of the upward-facing triangles after Stage 3 –

Extending to Algebra

Use your notes from Stages 0-3 to answer the following questions. Be sure to show all work!!!! Each Stage represents a new area amount. For example, a Stage 1 triangle is the area of one of the upward-facing triangles after Stage 1.

1. What is the combined area of one Stage 0 triangle and two Stage 1 triangles?

2. What is the combined area of one Stage 1 triangle and three Stage 3 triangles?

3. What is the combined area of one Stage 1 triangle, three Stage 2 triangles, and two Stage 3 triangles?

4. What is the combined area of one Stage 0 triangle, one Stage 1 triangle, two Stage 2 triangles, and three Stage 3 triangles?

5. What is the combined area of two Stage 0 triangles, three Stage 1 triangles, two Stage 2 triangles, and four Stage 3 triangles? Sierpinski’s Triangle: Connect-the-Dots

Use this triangle for the Connect-the-Dots Activity.