Framing Rectangles On grid paper use this framing method to make concentric rectangles. Start with a rectangle that is 3 units by 6 units in the center of your paper. “Frame” this rectangle in another, being sure to keep the distance between the two shapes constant. Note the dimensions of your new rectangle. Draw a few more (larger and smaller). Keep track of your rectangle’s dimensions in a table!

Framing Rectangles Compare your data. What is happening to your rectangles? Are some or all or none of them similar? Make an argument to support your answer using the dimensions recorded in your table. The rectangles are NOT similar! The smaller the length of the sides, the “thinner” the rectangle looks. As the length of sides increases, the rectangle comes closer to resembling a square.

When are Two Rectangles Similar? Two figures are similar if the ratios of all pairs of corresponding lengths are equivalent. For instance, these rectangles are similar with a scale factor of 2! Maria: Using between ratios, equivalent ratios divide to the scale factor of 2. We can also write it as a scale ratio of 2:1. Update: This is where we began on Day 9, Thursday.

Revisiting Additive and Multiplicative Thinking Framing Rectangles illustrates a central idea related to proportional reasoning. Proportion is a multiplicative concept. In order for two quantities to be proportional, there must be a constant multiplicative relationship between them. Using the framing method, we created figures additively because the difference around the shape is constant—we are adding the same amount of space at all times. It does NOT preserve the proportion of the shapes.

Teaching Note about Scaling and Equivalent Ratios Scaling a rectangle one time might not be enough to help students question additive reasoning, but scaling it multiple times can make it more obvious. From M2RAP Maria: Using absolute thinking, additive strategy of adding 2, not using relative thinking. May want to have the teachers use both the aspect and between ratios to ‘test’ whether there are similar rectangles. Also, focus on the 1st and last rectangles connected with the arrow – some students would say they are similar because you can multiply 2 times 4 = 8 and 3 times 3 = 9, not realizing they have used different scale factors and treated the quantities separately, not as a ratio. I THINK WE NEED TO THINK ABOUT THE WORDING HERE…SHOULD WE CALL THIS Scaling? Since scaling is multiplying by a factor? More noticeably NOT the same ratio!

Some of our Framing Rectangle Work