IM Alg2.1.11 Lesson: Adding Up

He cuts the paper to create four equal pieces, then hands one piece each to the others and keeps one for himself. What fraction of the original piece of paper does each person have? You may use the applet below to draw a diagram.

Tyler then takes his remaining paper and does it again. He cuts the paper to create four equal pieces, then hands one piece each to the others and keeps one for himself. What fraction of the original piece of paper does each person have now?

Tyler then takes his remaining paper and does it again. What fraction of the original piece of paper does each person have now?

What happens after more steps of the same process?

• Step 0 is an equilateral triangle.

• To go from Step 0 to Step 1, take every edge of Step 0 and replace its middle third with an outward-facing equilateral triangle.

• To go from Step 1 to Step 2, take every edge of Step 1 and replace its middle third with an outward-facing equilateral triangle.
• This process can continue to create any step of the design.

Consider a different function , where  is the number of new triangles added when drawing Step . Let . How many new triangles are there in Steps 1, 2, and 3? Explain how you know.

What is the total number of triangles used in building Step 3?

What patterns do you notice?