1. Exploring generations of similar figures. (Each must be the same shape but larger.) For example, how many square pattern tiles will you use to create the next larger square for each generation? How many triangle tiles will you use to create the first generation, second generation, third generation, and so forth? Draw the first through the fifth generations of squares, triangles and trapezoids. Count how many of each shape are needed to create each generation. Explain the pattern that you have observed. If a pattern does hold for each generation, how many tiles would be required at the 20th generation? How do you determine if the generation you are building is similar (by mathematical definition) to the other generations? What happens to the area when you double the dimensions of a given polygon? Triple them? Describe the pattern and give the value needed for the 20th generation. 2.Choose a number between 2 and 10. This is the length of a line segment. If you dilate the line segment by a factor of 3, what is the relationship between the lengths of the line segment and its dilation? How would you describe the relationship of dilation to the original? 3. Choose another number between 2 and 10. This is the length of one side of a cube. If you dilate the cube by a factor of 2, what is the relationship between the volume of the cube and its dilation? What if the cube is a sphere, or a pyramid—does the relationship of the volumes change? 4. Television sets used to come with an aspect ratio of 4:3 (width is 4 units and height is 3 units). Since broadcast channels in the United States are now broadcasting in 16:9, most newer sets come with a 16:9 aspect ratio. Your brother believes that since you can rewrite the ratio 16:9 as 42:32, that means a 4:3 television set is similar to a 42:32 television set. Is this true? What argument would you use to clarify your or your brother’s thinking? What are the dimensions of a television set that would be similar to the 4:3 set?