Looking for Imposters – Constructing an Understanding of Similarity

Journey’s end is across the river.    It all starts with a character in a video game.     Specifically, an adorable little guy named Mug Wump.

Mug has a family, all of whom are similar to him.    However, there are also some imposters in this game.   Sneaky little guys who are pretending to be part of the family.  Who knows what nefarious deeds these sly fellows might commit, so it’s important to ferret them out before they can cause too much trouble for the Wumps.

From Ordered Pairs to Coordinate Graphs

In this Connected Math lesson, students are given a set of coordinate pairs for Mug, and a set of rules for creating the other characters.    Students must graph Mug, find the ordered pairs for the other characters using the rules, and then graph the other characters.    Finally, they determine which of the characters are imposters and examine the rules to identify how one might identify similarity using only the rules.

My students love this lesson because Mug is adorable and they like the discovery that goes with creating the other characters in a video game.   I love this lesson because it allows students to build an understanding of similarity visually and then to see it as a scale factor in the diagram and in the rule for the ordered pair.    It also provides practice with graphing and the opportunity to discuss dilation.

Coordinate Graph to Ordered Pairs and Back Again

The next day, students continue to explore how different rules for the coordinate pairs change a figure.    They are given a diagram of a hat for our hero, Mug Wump.    They then use different rules to create different hats.    The activity starts with a graph of the hat.    Students create the list of ordered pairs for the hat and then apply a set of different rules to the ordered pairs to create different hats.    The rules explore how using different coefficients change the hat.    They also explore how adding or subtracting from the ordered pairs change the hat.   Finally, they explore how a coefficient less than one changes the hat.

Students find the idea of making a hat for Mug intriguing.    They also are surprised by how the different rules shift and change the hat.    They sometimes struggle with seeing that different coefficients for x and y don’t create a similar hat.    They seem to see it readily with the Wumps, but then don’t see it readily with the hats.   I think it is because the overall size of the hat is small and the changes are a little more subtle.    This subtlety is something that some kids catch and  others don’t.   It leads to some interesting discussions within table groups as kids decide which hats are similar and which are not.    I always teach this lesson using Kagan’s Numbered Heads Together Cooperative Learning Structure.   I like that this structure gives kids the opportunity to work independently first, a structure for a discussion, and mutual accountability because they never know who is going to be asked to speak for the group.

I like that this lesson deepens students understanding of how rules for coordinate pairs influence similarity.    The discussions are rich and practice with other skills is embedded in the lesson (graphing ordered pairs).    The lesson also gives a nice context to talk a little bit about the idea of dilation and translation of figures on a coordinate plane.    While these ideas aren’t central to the lesson or part of the CCSS standards that I am teaching in this course, it is a nice introduction to vocabulary and concepts that students will see the next year.

I wrap up the lesson with an exit ticket in which I ask students how the rule (2x, 2x+7) will change the hat.   This is a chance to see if students have fully grasped the concept.    Some of them initially say that the shape will be doubled.    Some students are stumped by the combination of a coefficient and an addend in the rule.    Bringing these ideas out as we go over the exit ticket (after I collect them and do a quick scan of them) gives us a chance to solidify everyone’s understanding that the coefficient dilates the figure and the addend translates it.

Exploring Scale Factor and Ratios within Similar Parallelograms & Triangles

With a firm grasp of scale factors in hand, students deepen their understanding of similarity by exploring ratios within similar shapes.    They find the ratio of adjacent sides within a rectangle and then compare that ratio of the corresponding adjacent sides within a similar rectangle.    They repeat this process with parallelograms and triangles.    As they work through this process, I have students use color coding as they identify corresponding sides.    They use the color coding in the diagrams and in the ratios to help them ensure that they are indeed looking at ratios of corresponding sides.   They begin by looking at the angles to find the corresponding sides and then use colors to highlight the pairs of corresponding sides.    This year was the first time that I implemented this color coding.   It was done to support some of my students who are twice exceptional and have visual processing difficulties.    I found that it was really helpful for them.   It was also really helpful for the entire class.    Students grasped the concepts more quickly and with more assurance than in previous years, without me sacrificing the discovery process by “telling” them the answers rather than allowing them to construct their own understanding.

Using the Characteristics of Similar Shapes to Find Missing Side Lengths

Once students have a firm understanding of the characteristics of similar shapes, they use those characteristics to find missing side lengths.    They begin with some similar triangles and parallelograms with missing side lengths.    Then, they proceed on to using these ideas in real-world contexts.    They use the shadow method and the mirror method to create similar triangles and estimate the height of trees, buildings, towers, and basketball hoops.    Finally they usedthe idea of similar triangles to estimate the distance across a river using some trees on one side of the river and some stakes on the other to create similar triangles.   The lessons gives the students the chance to explore how math is used to solve real problems.

Bringing It All Together

Characteristics of similar rectangles parallelograms and triangles

At the conclusion of this series of lessons, I have students complete a foldable summarizing the characteristics of similar figures.    This foldable is designed to help my students who are at the extremes with regard to the conceptual/sequential dimension of cognitive style.    It gives them a picture of the whole and the parts, making explicit connections between the big picture and the details.   This is to help my conceptual learners start with the big picture and then connect the details to that understanding.   It is also to help my sequential learners who tend to compartmentalize information and ideas to see the connections and build the big picture.

I wait until the conclusion of the series of lessons to complete the foldable because I believe students benefit from the opportunity to construct an understanding of these ideas.   The series of lessons are taken from Connected Math.   The premise is that students construct an understanding of concepts and develop a deeper understanding of the math by doing so.   Each lesson includes a summary in which those ideas are drawn out and made explicit after students have had a chance to build their own understanding.    The final foldable is an attempt to summarize the ideas addressed over the course of several lessons.  (Click the link below the photo of the foldable to download it or go to the resources page and click the link below the photo.)