Fall 2012 Investigating “Look Alike” Rectangles Objectives: a) Explore how equivalent ratios can be expressed arithmetically, geometrically, and algebraically; b) Compare what happens to the ratio of rectangles’ lengths and widths when adding a number to the dimensions and multiplying the dimensions by the same number. Materials: 10 rectangle patterns, handout “Look-­‐Alike Rectangles,” graph paper, ruler, and scissors Part 1 1. Sort the 10 rectangles into three sets of three rectangles that “look alike” plus one “oddball.” Explain to your table group your rationale for grouping the rectangles the way you did. 2. Measure and record the sides of each rectangle to the nearest half-­‐centimeter. Then, set up a ratio of the side lengths, short-­‐to-­‐long. Record your results on the handout. What do you notice? Do you want to make any adjustments to your groupings based on your results? 3. Draw three different graphs of the first quadrant of a coordinate grid and label the x-­‐
axis “length” and the y-­‐axis “width.” Then separately trace your three groupings of rectangles in a nesting fashion (small to large) onto three different graphs (one grouping per graph). Each rectangle should start at the origin (0, 0). Its shorter side should align with the y-­‐axis and its longer side should align with the x-­‐axis. Next, using a straightedge, draw a line from the origin through the upper right corner of the largest rectangle. Describe what you see. Which rectangles are similar? Which width-­‐to-­‐length ratios are equivalent? How are the width-­‐to-­‐length ratios shown on the coordinate grid? What is the slope of the line you just drew? What do these results mean? Investigating “Look Alike” Rectangles Page 1 of 2 Part 2 Fall 2012 1. Predict what will happen when you compare the ratios of side lengths of a rectangle if you add to the dimensions compared to multiplying the dimensions by a number. Discuss in your table groups. 2. On a separate piece of graph paper, choose one rectangle to trace on a graph. First, multiply the length and width of the rectangle by 2, 3 or 4 and draw the new figure so that it nests with the original rectangle. Then, using a straightedge, draw a line from the lower left vertex through the upper right vertex of the largest rectangle. Next, using the same number, add this number to the length and width of the original rectangle and draw this new figure nesting with the other two rectangles. Record your results. Original rectangle dimensions Multiply by ____ (2, 3, or 4) Add ____ (2, 3, or 4) Width (cm) Length (cm) Describe what you notice. 3. Compare the ratios of the side lengths of a rectangle when increased by addition versus the rectangle increased by multiplication. How do your actual results compare to your prediction? What do these results mean? 4. Is enlarging or shrinking a figure an additive or multiplicative process? Explain how you know. Investigating “Look Alike” Rectangles Page 2 of 2