Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

 Exploring Communication
 Are These Shapes Congruent? | Try It Yourself: Congruent Shapes | Problem Reflection | Your Journal

In this section, we take a look at your, rather than your students', approach to a geometric task and your own internal communication about the task at hand. As you work through the activity, pay attention to the methods and vocabulary you use to think about the problem, as well as the questions and any new geometric insights that come to mind. Also, notice the role of vocabulary as you use language to explain and justify your reasoning.

Before you move on to exploring the Interactive Activity, answer the questions below involving two-dimensional shapes. Imagine that you can move the shapes by sliding, rotating, or flipping (reflecting) them.

 Question: Are these two shapes the same? Explain. Show Answer
 Our Answer: They are the same shape because you can rotate the first one and put it exactly on top of the second one. The figures both have four sides, and they fit perfectly on top of each other.
 Question: Are these two shapes congruent? Explain why or why not. Note: "Congruent" means that two shapes exactly match. It's more specific than "same shape." Congruent figures are the same shape and size, even though their orientations may differ. Their sides all match because they are the same lengths, and their angles match because they are the same measures. Sometimes you have to turn a shape to see that it's the same as another. Sometimes you have to flip one over. Pay attention to how and why you use "same shape" and "congruent" in your own thinking. Show Answer
 Our Answer: The shape on the left has five sides, and the shape on the right has six. This is sufficient to show that the two shapes are not congruent, but there are other ways to demonstrate this as well. Examining the shapes' angles shows you that the angles don't match up either (the angle in the middle is a different size in each figure; the shape on the left has five angles and the shape on the right has six; etc.) Also, if you rotate the first shape and put it on top of the second one, the shapes don't match up. Note that placing shapes on top of each other to check congruence is not a proof but simply a visual tool to demonstrate that the shapes are or are not congruent.
 Question: Are these two shapes congruent? How do you know? What vocabulary could you use to describe the figures more clearly? Show Answer
 Our Answer: They are the same shape. Both are triangles, and in fact both are right triangles. However, they are not congruent. Their shape is the same, but their size is not. If you slide one on top of the other, you'll see that they do have the same angles but the sides are not the same lengths. The second figure is an enlarged version of the first. They are similar triangles because their angles are the same, even though the sides of the second figure are magnified or enlarged. Sometimes teaching and using a precise vocabulary word, such as "congruent," can prevent misunderstandings and facilitate communication with students. Less precise phrases, such as "same shape," may sound familiar and friendly, but they are often attached to a different meaning in everyday language, which can lead to misunderstandings. In addition, notice how the term "similar triangle" would need to be pointed out and defined to emphasize the difference between the everyday use of "similar" and the geometric term (i.e., two figures are similar if their corresponding angles have the same measure and their corresponding sides are in proportion –– in our example, the sides of the larger triangle are twice as large as the corresponding sides of the smaller one), and the angles, as noted, are the same.

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