Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
INTERPRETATION OF OBJECTIVE - G.CO.D.13
Duh!! No clarification needed here. I guess all I would say is that these can be done early on in the year following the basic constructions in G.CO.D.12 but they can also be when you work with circles because all of these are inscribed in a circle.
One question that we might ask, why only inscribed? All regular polygons can be inscribed in a circle. In this form, we learn the terms of radius and apothem, as well as we see as the regular polygon increases in the number of sides we gain a closer approximation of the circle.
DIRECTLY IMPLIED SKILLS
(1) The student will be able to construct the following:
(a) an equilateral inscribed in a circle.
(b) a square inscribed in a circle.
(c) a regular hexagon inscribed in a circle
INDIRECTLY IMPLIED SKILLS
(2) The student will be able to construct the following:
(a) an equilateral
(b) a square
(c) a regular hexagon
THE BIG IDEA
The regular polygons hold many important properties and to be able to construct them in different ways reveals key relationships about them. These constructions lead us to the special right triangles of 30-60-90 and 45-45-90. They also connect the language of radius of a square or apothem of an equilateral.
Actually when you think about it these shape hold a nice summary of many of the topics found in this unit.
TRAPS & PITFALLS
When teaching these constructions don't just walk them through these... let them discover them. The basic constructions and the triangle and quadrilateral properties should hold enough previous knowledge that they should be able to piece together these constructions on their own or in a group. You will find the alternate way, different from the classic construction will be discovered and those techniques will be a basis for great conversation.
These three regular polygons are a beautiful summary of much of what is covered in this unit - symmetry, congruence, triangle/quadrilateral properties, transformational relationships within the shapes, etc.... To be able to construct these students will use this previous knowledge.
These three shapes link us to some of the powerful similarity concepts. They hold the special right triangles and form the basis for exact answers for 30, 45 and 60 degree angles. The area of these shapes also becomes important when finding volumes.
MY REFLECTIONS (over line l)
When I introduced these I did them after we had done the basic constructions to act as a summary of their construction skills. That worked well. I re-constructed them when we came to similar triangles and the special right triangles and we discussed why those specific angles were formed and what the side relationships were. I re-constructed them again, when we were in the circle unit and we looked at inscribed angles, congruent arcs, inscribed angles on a diameter, and many other nice relationships. The regular shapes hold the maximum number of properties for their polygon classification - they are beautiful!!