This can form the basis for the definition of congruence in terms of rigid motions.

Arguments based on superimposing one shape on another and checking that they match up exactly.This can push students toward finding a more general definition. The teacher can ask students how they would apply this approach to figures with curved sides, like those in Set C. 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. This can help establish that looking similar is not sufficient. The same can be asked of circles or ellipses and other non-rectilinear figures. 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. Arguments based on the physical appearance of the shape: they look like they are the same size and same shape.For the first question, students should have access to a variety of tools, including tracing paper or transparencies, scissors, tape, rulers, protractors etc. After 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. Teachers should expect students to approach this in at least three different ways: 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.

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. 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. This is fine as far as it goes, but it isn't very useful for talking about the congruence of figures with curved sides. 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.Ī 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. So by the end of elementary school, students have an idea that the notion of 'sameness' is nuanced in a geometric context.

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). 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). 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 task can also be used to illustrate the importance of crafting shared mathematical definitions (MP 6). 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 congruence in terms of rigid transformations.

Transformations of shapes in geometry