For every pair of figures, decide whether these numbers are the exact same size and also same shape. Describe your reasoning.Â

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What go it average for two numbers to it is in the same size and also same shape?ÂÂ

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## IM Commentary

The function of the job is to help students transition from the informal notion of congruence together "same size, exact same shape" the they find out in elementary school school and also begin to build a definition of congruenceÂin terms of rigid transformations. The task can also be supplied toÂillustrate the importance of crafting common mathematical interpretations (MP 6). Keep in mind that the ax "congruence" is not offered in the task; it need to be introduced at the end of the discussion as words we use to catch a much more precise definition of "same size, same shape."

The concept of equivalence is a deep one in mathematics, and in an initial grade, students begin to inspection what it way for two numbers to be equal (1.OA.D.7). But what go it mean for two geometric numbers to be "the same"? In very first grade, students start to study what it method for two one-dimensional figures to have the same size (1.MD.A), in 3rd grade students research what it method for 2 two-dimensional numbers to have the same area (3.MD.C), and also in 5th grade they examine what it way for two three-dimensional objects to have the same volume (5.MD.C). For this reason by the end of elementary school, students have actually an idea that the id of "sameness" is nuanced in a geometric context. They additionally talk about what it way for two numbers to have the exact same shape, and also in every-day language, us say that 2 rectangles have actually the "same shape." but what go it median for two figures to have the "same size and also same shape"? If two rectangles have actually the same area, room they the same size and shape? without a more precise meaning of what we typical by same size, very same shape, us can"t speak yes or no.

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A common definition for congruence claims that 2 polygons space congruent if there is a correspondence in between the vertices and corresponding sides have the exact same length and also corresponding angles have the same measure. This is good as much as the goes, yet it isn"t very useful because that talking around the congruence of figures with curved sides. Specifying congruence in terms of rigid changes covers all kinds that figures, and we can show that the traditional meaning for polygons follows from it. Due to the fact that rigid transformations applied to a figure are just a formalization of the idea of picking up that figure and also moving it approximately without stretching or breaking it, this meaning also has the benefit of formalizing one intuitive idea the what it means for two figures to be the very same size and also same shape: if you deserve to move one on peak of the other so there room no gaps or overlaps, then they space the very same size and also same shape.

Used appropriately, this task can initiate a conversation that will certainly lead from the elementary school school notion of very same size and same form to the much more formal middle school an interpretation of congruence in terms of rigid motions. Because that the an initial question, students need to have access to a variety of tools, including tracing file or transparencies, scissors, tape, rulers, protractors etc.ÂAfter students have functioned to articulate a definition of "same size, exact same shape" one of two people alone or in groups, the course can comment on the merits and also drawbacks the the different feasible definitions.ÂTeachers should expect student to method this in at the very least three different ways:

disagreements based top top the physical appearance that the shape: lock look favor they are the same size and same shape. If students indicate this type of approach, the teacher deserve to ask even if it is that method any 2 rectangles qualify together in some sense they look the same. The same deserve to be request of circles or ellipses and other non-rectilinear figures. This can assist establish that looking comparable is no sufficient. Arguments based on measurements. Students can measure next lengths and angles and also if equivalent sides and also angles have the same measurements they can conclude that the 2 shapes have the exact same size and shape. The teacher deserve to ask students exactly how they would apply this method to numbers with curved sides, favor those in collection C. This deserve to push students towards finding a more general definition. Arguments based top top superimposing one shape on another and checking that they complement up exactly. This can type the basis because that the an interpretation of congruence in regards to rigid motions.Set C raises a an extremely important question around the nature the congruence. If two figures are winter images, does that "count" as being the very same size and also shape, or not? (By convention, the mathematical community has agreed the they are, yet a various decision could have to be made.) collection D raises one more question: if a shape has several ""parts,"" have the right to we move them each individually or perform we need to move castle together? (Again, we agree through convention that the figure must be taken as a totality whenÂdefiningÂcongruence.)

After a discussion of the feasible definitions because that "same size, exact same shape," students who have already been presented to rigid transformations can then be offered the official definition:

Two numbers A and B room congruent if one is the image of the various other under a succession of rigid transformations.

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If students have actually not yet been introduced to rigid transformations, the task deserve to be supplied to motivate their study. What go it median to "move one figure on optimal of an additional so that they heat up exactly?" High college geometry move in the direction in making these notions ofÂ *congruence *("same shape") much more precise, leading one to the careful study the translations, reflections, and rotations. For example, several significant milestones in high institution geometry involve using these changes to identify when 2 triangles are congruent native information about their sides and also angles.Â In this sense, this job is the start of a change in i beg your pardon students can involved understand that the id of equivalence of geometric numbers is subtle and also deep.