For each pair of figures, decide whether these figures are the same size and same shape. Explain your reasoning.Â
What does it mean for two figures to be the same size and same shape?Â
The purpose of the task is to help students transition from the informal notion of congruence as “same size, same shape” that they learn in elementary school and begin to develop a definition of congruenceÂin terms of rigid transformations. The task can also be used toÂillustrate the importance of crafting shared mathematical definitions (MP 6). Note that the term “congruence” is not used in the task; it should be introduced at the end of the discussion as the word we use to capture a more precise meaning of “same size, same shape.”
The notion of equivalence is a deep one in mathematics, and in first grade, students begin to investigate what it means for two numbers to be equal (1.OA.D.7). But what does it mean for two geometric figures to be “the same”? In first grade, students begin to study what it means for two one-dimensional figures to have the same length (1.MD.A), in third grade students study what it means for two two-dimensional figures to have the same area (3.MD.C), and in fifth grade they study what it means for two three-dimensional objects to have the same volume (5.MD.C). So by the end of elementary school, students have an idea that the notion of “sameness” is nuanced in a geometric context. They also talk about what it means for two figures to have the same shape, and in every-day language, we say that two rectangles have the “same shape.” But what does it mean for two figures to have the “same size and same shape”? If two rectangles have the same area, are they the same size and shape? Without a more precise definition of what we mean by same size, same shape, we can”t say yes or no.
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A common definition for congruence states that two polygons are congruent if there is a correspondence between the vertices and corresponding sides have the same length and corresponding angles have the same measure. This is fine as far as it goes, but it isn”t very useful for talking about the congruence of figures with curved sides. Defining congruence in terms of rigid transformations covers all kinds of figures, and we can show that the traditional definition for polygons follows from it. Since rigid transformations applied to a figure are just a formalization of the idea of picking up that figure and moving it around without stretching or breaking it, this definition also has the advantage of formalizing an intuitive idea of what it means for two figures to be the same size and same shape: if you can move one on top of the other so there are no gaps or overlaps, then they are the same size and same shape.
Used appropriately, this task can initiate a conversation that will lead from the elementary school notion of same size and same shape to the more formal middle school definition of congruence in terms of rigid motions. For the first question, students should have access to a variety of tools, including tracing paper or transparencies, scissors, tape, rulers, protractors etc.ÂAfter students have worked to articulate a definition of “same size, same shape” either alone or in groups, the class can discuss the merits and drawbacks of the different possible definitions.ÂTeachers should expect students to approach this in at least three different ways:
Arguments based on the physical appearance of the shape: they look like they are the same size and same shape. If students suggest this type of approach, the teacher can ask whether that means any two rectangles qualify as in some sense they look the same. The same can be asked of circles or ellipses and other non-rectilinear figures. This can help establish that looking similar is not sufficient. Arguments based on measurements. Students might measure side lengths and angles and if corresponding sides and angles have the same measurements they could conclude that the two shapes have the same size and shape. The teacher can ask students how they would apply this approach to figures with curved sides, like those in Set C. This can push students toward finding a more general definition. Arguments based on superimposing one shape on another and checking that they match up exactly. This can form the basis for the definition of congruence in terms of rigid motions.
Set C raises a very important question about the nature of congruence. If two figures are mirror images, does that “count” as being the same size and shape, or not? (By convention, the mathematical community has agreed that they are, but a different decision could have been made.) Set D raises another question: if a shape has several “”parts,”” can we move them each independently or do we have to move them together? (Again, we agree by convention that the figure must be taken as a whole whenÂdefiningÂcongruence.)
After a discussion of the possible definitions for “same size, same shape,” students who have already been introduced to rigid transformations can then be given the formal definition:
Two figures A and B are congruent if one is the image of the other under a sequence of rigid transformations.
If students have not yet been introduced to rigid transformations, the task can be used to motivate their study. What does it mean to “move one figure on top of another so that they line up exactly?” High school geometry moves in the direction in making these notions ofÂ congruence (“same shape”) much more precise, leading one to the careful study of translations, reflections, and rotations. For example, several major milestones in high school geometry involve using these transformations to determine when two triangles are congruent from information about their sides and angles.Â In this sense, this task is the start of a transition in which students can come to understand that the notion of equivalence of geometric figures is subtle and deep.