Friday, August 15, 2014
Never ASS-U-ME in Geometry: A Triangle Problem to Get Them Thinking!
Not quite back to school for most but the problem above might prove interesting to review some geometric/deductive reasoning.
For new geometry students, replace 'a' by a value, say 40, and ask them to fill in all the missing angles. Most should deduce that angle 5 = 50, but my educated guess is that many will assume b = 40, so
angle 5= angle 6 = 50 and angle 3 = angle 4 = 40. From there to angle 1 = angle 2 = 50, so
angle 2 + angle 3 = 90. QED! Not quite...
Well, the '90' is correct but the reasoning is another story! So this is all about justifying, checking validity of mathematical arguments, sorta' like some of the Eight Mathematical Practices of the Common Core!
In fact, you might ask them to redraw the diagram, keeping the given conditions but making it clear that b does not have to be 40 and that Angles 3&4 also do not have to be 40!
Posted by Dave Marain at 5:37 PM 0 comments
Labels: CCSS, Challenge Problem, Common Core, geometry, Math, mathematical practices, reasoning
Wednesday, December 18, 2013
Two overlapping circles of radius r... - A Common Core Geometry Problem
OVERVIEW
Intersecting circle problems are always interesting and often challenging whether you find them in the text, on SATs or on math contests. The general case involves trig and formulas can be found online.
The objectives of the problem below include:
• Drawing a diagram from verbal description
• Dissecting or subdividing an unknown region into more common parts
• Applying circle theorems and area formulas
• Solving a multistep problem (developing organizational skills, attention to detail)
THE PROBLEM
Two circles of radius r intersect in two points in such a way that the overlap is bounded by two 90° arcs. If the area of the common region is kr^2, determine the value of k.
Answer: (Pi-2)/2
Note: Please verify!
REFLECTIONS FOR MATH TEACHERS
[Note: These are discussion points --- not short answer questions with simple answers!]
• Should the diagram have been given to eliminate confusion?
• Does this problem appear to have any practical application?
• Have you seen a similar problem in your geometry texts? On standardized tests like SATs?
• In similar problems, were the arcs 60° or 90°?
• How would you introduce this problem? Is it worth the time to have students cut out congruent paper or cardboard circular disks, keep one fixed and move the other until it approximates 90° arcs?
Better to use geometry software?
• Assign this for homework? As a group activity in or out of class? As a demo problem with a detailed explanation provided by you?
• How much time would be needed for classroom discussion of this problem?
• Would you plan on providing extensions/generalizations?
• Too ambitious for "regular" classes? Appropriate only for Honors?
• So what makes this a Common Core activity? Are you guided by the Mathematical Practice Standards?
Posted by Dave Marain at 7:10 PM 2 comments
Labels: activities, circle problems, Common Core, geometry, mathematical practices, overlapping circles