<p>Algebra 1 – Connect the Dots! Name______Based on the Discovering Algebra Condensed Lessons</p><p>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. </p><p>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.</p><p>Stage 0</p><p>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.</p><p>Area of Sierpinski’s Triangle after Stage 0 – </p><p>Stage 1</p><p>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.)</p><p>Area of one of the upward-facing triangles after Stage 1 –</p><p>Area of all of the upward-facing triangles after Stage 1 –</p><p>Stage 2</p><p>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.)</p><p>Area of one of the upward-facing triangles after Stage 2 –</p><p>Area of all of the upward-facing triangles after Stage 2 – Stage 3</p><p>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.)</p><p>Area of one of the upward-facing triangles after Stage 3 –</p><p>Area of all of the upward-facing triangles after Stage 3 –</p><p>Extending to Algebra</p><p>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.</p><p>1. What is the combined area of one Stage 0 triangle and two Stage 1 triangles?</p><p>2. What is the combined area of one Stage 1 triangle and three Stage 3 triangles?</p><p>3. What is the combined area of one Stage 1 triangle, three Stage 2 triangles, and two Stage 3 triangles?</p><p>4. What is the combined area of one Stage 0 triangle, one Stage 1 triangle, two Stage 2 triangles, and three Stage 3 triangles?</p><p>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</p><p>Use this triangle for the Connect-the-Dots Activity.</p>