Intent

In this first activity of The Shape of It, which is devoted to developing basic ideas of similarity, students use concrete situations to develop an intuitive idea of the meaning of “same shape.”

Mathematics

In this introductory activity, which sets the stage for eventually defining what it means for two plane figures to be similar, students are encouraged to reflect on what it means for two figures to be the same shape. Does orientation matter? Are two figures the same shape if they are simply the same type of figure (for example, two rectangles)? The house in the activity has a set of length and angle measurements. Students investigate which measurements change as they enlarge the house.

Progression

Students work on the activity individually and then discuss their work as a class.

Approximate Time

35 minutes

Classroom Organization

Individuals, followed by whole-class discussion

Materials

Grid paper

Protractors

Doing the Activity

Be sure students recall how to use a protractor. They may also need a review of angles and of measuring polygon angles.

Discussing and Debriefing the Activity

Question 1, in which students draw a simple picture on grid paper and then draw one exactly like it, only larger, probably doesn’t need discussion.

For Question 2, ask two or three students to draw their figures, labeling the measurements for each side and angle, and let the class discuss whether the figures are the same shape as Renata’s house. At present, you can leave the question unresolved, perhaps explaining that, for now, students might disagree about what “same shape” means, but that they will be working toward the formal definition that mathematicians use.

Similarly, for Question 3, there will probably be some differences of opinion. Some students may say that “same shape” includes “facing the same way,” so they will not consider the pair of rectangles in Question 3a or the pair of triangles in Question 3d to be the same shape. On the other hand, some may consider that all rectangular figures, for example, are “the same shape” (namely, they are all rectangles).

For Question 3c, some students may consider the triangles to be the same shape even though the height-to-width ratios are different. Some may even consider the pentagon and hexagon of Question 3b to be the same shape (because they are both polygons), even though the number of sides differs.

No conclusions need to come out of this discussion, but help students to define the issues that they agree and disagree about. You might post the following, for example, as questions to be resolved: Is the mirror image of something considered the same shape?Does changing the size of something change its shape?

Highlight comments that hint at the idea of ratio or scale, as this is one of the foundations of the formal definition of similar. For example, a student might say about the triangles in Question 3c, “The triangle on the right is taller so it should also be wider, but it isn’t, so it isn’t the same shape,” which suggests that an increase in one dimension ought to be matched by an increase in another dimension.

Key Questions

Is the mirror image of something considered the same shape?

Does changing the size of something change its shape?

Supplemental Activity

Instruct the Pro (reinforcement) offers students additional practice with basic ideas about angle measurement and the use of protractors.