Wednesday, January 15, 2014

Cutting Corners -- The Square Transformed Into An Octagon Problem

A regular octagon is formed from a square by making 45° cuts from each corner.

(a) Draw a diagram or construct a model. See figure below.
(b) Since the octagon is "inscribed" in the square, its area is less than that of the square.
Explain using only "Euclidean" methods why the perimeter of the octagon is less than the perimeter of the square.

(c) If the perimeter of the square is 4 show that the perimeter of the octagon is 8(√2 - 1).

Note that the perimeter of the octagon is roughly 20% less than the perimeter of the square. Reasonable?

Saturday, January 4, 2014

Three congruent isosceles right triangles walked into a bar...

Silly title but you might want to try the following problem with your high school geometry students or with middle schoolers doing a unit on right triangles. Furthermore, elementary school children need many hands-on experiences with pattern blocks, tangrams, pentominos and the like to develop their innate spatial sense. They should also be allowed to experiment with two such triangular pieces to make a square, a parallelogram, a larger isosceles triangle, etc. Then have them work with the 3 triangles to make different polygons including the trapezoid. They don't need to consider the area or the 2nd part of the question.

Three congruent isosceles right triangles are joined to form an isosceles trapezoid having an area of 3 sq units.

(a) Draw a possible diagram.
(b) Determine the perimeter of the trapezoid.

Answer: (b) 6+2√2

•How much time would you allow for a discussion of this problem! 10 min? 15? 20? Guess it depends on whether you see this as just an exercise or as an activity.
• How much difficulty do you think most middle and secondary students would have with drawing an appropriate diagram?
•Do you think most will need to draw several figures before arriving at the isosceles trapezoid? Do you think some will come up with a trapezoid which is not isosceles and think they're finished? Can you anticipate that some will miss one of the key words like isosceles (which occurs TWICE!).
• Do you think the spatial "puzzle pieces" part of the problem is more significant than the numerical part or about equal?
• Do you expect some students to hit a wall and express something like "I forgot the formula for the area of a trapezoid!" We should make this a teachable moment -- "WE DON'T NEED TO RECALL THAT FORMULA! WHY!"
•Do you see benefits from students working in pairs here? Would you have them work independently then come together after a few minutes? My view is the stronger spatial student will "see" the correct figure more rapidly and influence the other who may give up and wait for his/her partner to draw it. So I might ask them to draw a few figures on their own for a couple of minutes.
•Do you think any of the older students need manipulatives?
• What is our role here? Catchphrases like"guide on the side" do not tell us what interventions we should actually use? Part of knowing what to do/say comes from our experience and part from instinct but my rule of thumb was "less is more". Allowing them to struggle for awhile is critical or, to put it another way, "without irritation there would never be a pearl!"
• How would you solve this problem? When planning do you feel it's important to think of alternate solutions or let this flow from the students?
•Finally, I think it's important to identify which of the  Mathematical Practice Standards are brought to play in this investigation. All of them? A couple? Guess that depends on you...

I typically get few if any comments from these detailed investigations. That's ok. Just planting seeds I guess...