Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
INTERPRETATION OF OBJECTIVE - G.CO.A.3
This objective doesn't need much interpretation - it is all about symmetries of figures. The objective mentions a few specific shapes but that doesn't mean that you can't look at symmetries of other shapes. The symmetries of these particular shapes that are mentioned are not trivial though, knowing the rotational and reflectional symmetries of the quadrilaterals will help us unlock their properties in a later objective. It will also help us solve problems dealing with regular polygons.
(1) The student will be able to identify and describe the different symmetries (line symmetry, rotational symmetry, point symmetry) of a figure.
(2) The student will be able to determine the maximum possible lines of symmetries that exist for a given polygon.
(3) The student will be able to determine the order and angle of a rotational symmetry.
(4) The student will know the symmetries of a parallelogram, rectangle, rhombus, square, trapezoid and regular polygon.
THE BIG IDEA
Symmetry in a figure implies congruence of sides and angles. If you can rotate the shape onto itself then at some point we have an exact match of sides and angles. The same is true about reflectional symmetry; if you can reflect a shape onto itself then it must have some congruent sides and angles. Symmetry is a nice introduction to the general transformations of reflection and rotation. These early notations of symmetry will guide us to some informal arguments about congruence and proof of properties found in figures.
TRAPS & PITFALLS
Not a lot of traps and pits in this area. The only thing I have seen reoccurring over the years is how students struggle to determine if non-vertical or non-horizontal lines are lines of symmetry. For example many students feel that the diagonal of a parallelogram or of a rectangle is a line of symmetry.
The rotational symmetry value of 1 is also very confusing, generally if a shape has no rotational symmetry we would want to say its value is 0 but it is not... its 1... one full revolution about the center placing back in its original location.
Another area I have noticed a weakness is identifying all of the lines of symmetry in an odd sided regular polygon, like the pentagon. Students struggle to find them all.
Students come with some basic understanding of symmetry. In early grades they have discuss shape symmetry with more emphasis on reflectional and less on rotational.
Students will take these early notions of symmetry and apply them as they discuss relationships within figures. Knowing that an isosceles triangle has a single line of reflectional symmetry will helps us discuss that the base angles then must be congruent. Symmetry will help us identify triangle and quadrilateral properties in an informal and easy way.
MY REFLECTIONS (over line l)
I underestimated the importance of this objective – totally!! When I was trying to plan out the unit I decided to move this objective to later in the unit because it seemed to be about quadrilaterals. I thought that I would cover it when I covered the proofs and properties of quadrilaterals in G.CO.11. That was a big mistake - I now see that the intent of this is to help students see relationships within shapes and also to introduce reflections and rotations in familiar way. This objective is there to help students recognize that if shapes have symmetries, then they also have congruence parts within their shape. I missed this whole idea.