Monday, March 26, 2007
Lattice Points Problem Part 2: Circles, Gauss' Circle Problem and Pick's Theorem
Eric Jablow inspired me to develop the following extended enrichment actvity/project for Geometry students. You can research the general solution of Gauss' Circle Problem in MathWorld but for this application it isn't necessary to use that formula.
Consider the circle of radius 5 centered at the origin.
(a) Determine the coordinates of the 12 lattice points on this circle. Recall that lattice points are points whose coordinates are both integers.
Note: Is it reasonable from symmetry arguments that the number of lattice points is divisible by 4?
(b) Using graph paper, show there are 69 lattice points INSIDE this circle. Describe your counting method.
The total number of lattice points inside or on our circle is 69+12 = 81. This agrees with the result from Gauss' formula but we will now 'approximate' this result using Pick's Theorem which gives the relationship among interior, boundary points and the area of a polygon whose vertices are lattice points.
(c) Consider the inscribed dodecagon formed by connecting the 12 lattice points from part (a). Determine the lengths of the sides of this polygon.
(d) By dividing the polygon into 12 triangles show that the area of this polygon is 74. No trigonometry, just Pythagorean and basics!
Hint: This is not a regular polygon but you can still divide it into 12 isosceles triangles.
Comment: Do you find it surprising that the area is rational (in fact, integral), considering that the sides are irrational?
(e) Pick's Theorem states that the area of a polygon whose vertices are lattice points is given by the formula A = I + B/2 - 1 where I = the number of interior lattice points and B = the number of boundary lattice points, that is, points on the polygon.
Show that Pick's Theorem leads to I = 69.
(f) For our problem the number of lattice points inside the circle matched the number of points inside the inscribed polygon. A coincidence? Whether it's true or not, explain why this result seems to make sense.
(g) To further investigate this 'coincidence', change the radius to 10.
(i) Show, by counting, that there are 317 lattice points inside or on this circle.
(ii) Show that there are still 12 lattice points on this larger circle.
(iii) Show that there are 303 lattice points inside or on the resulting dodecagon.
[As before, find the area w/o trig and use Pick's Thm. Note: To find the area of the polygon use (d) and ratios!]
(iv) Show that the point (4,9) is inside this circle but outside the dodecagon! This suggests why Pick's theorem fails in this case! Why?
Good luck! This is an extended challenge that I will leave up for several days and invite comment. Whether you implement it in a classroom or not, enjoy!
[By the way, some of the numerical results (like 303) have not been independently verified. If you find an error, let me know!]
Posted by Dave Marain at 6:55 PM 1 comments
Labels: area, circles, dodecagon, Gauss Circle Problem, geometry, lattice points, Pick's Theorem
Sunday, March 25, 2007
In-Depth Investigation of Patterns: Lattice Points, Algebra,...
Update!
Answers to General Formulas for |x| + |y| = N:
ON: 4N
INSIDE or ON: N2 + (N+1)2 = 2N(N+1) + 1 = 1+4+8+...+4N
OUTSIDE the 'diamond' but INSIDE or ON surrounding square:
2N(N+1) = 4+8+...+4N
TOTAL: (2N+1)2
[Pls indicate any discrepancies you find!]
The following starts with a fairly simple pattern but watch out -- students from Middle School through Algebra 2 and beyond can take this as far as their imagination and skill can carry them. The questions below review absolute values, inequalities, graphs, geometry, etc., but that is the tip of the iceberg. Prealgebra students should start with simpler values to get the idea. Have them make a table of their findings as suggested at the bottom.
One could begin with a variation on an SAT-I, SAT-II type or math contest question:
How many ordered pairs (x,y), where x and y are both integers,
satisfy |x| +|y| ≤ 6?
Note: If this were to appear on the 'new' SAT, the inequality would be replaced by an equality. or the '6' would be replaced by a smaller number.
Here's a restatement using the terminology of lattice points:
In the coordinate plane, a point P(x,y) is said to be lattice point if both coordinates are integers.
How many lattice points are inside or on the graph of |x| + |y| = 6?
Answer: 85
Sorry for giving it away, but the objective is to discover ways to derive this result, not the result itself.
When I see questions like this, my inclination (as a mathematician) is to generalize, that is, try to understand a general relationship if the '6' were replaced by N.
Here is the more advanced generalization (one could go further!):
We are investigating the number of lattice points inside or on the graph, G, of
|x| + |y| = N, where N is a positive integer. We will also consider the smallest square, S, whose sides are parallel to the coordinate axes in which the graph of
|x| + |y| = N is inscribed. [This really needs to be drawn but I'll leave it to the reader's imagination. The top side of this square is part of the line y = N.]
(a) Derive a formula for the number of lattice points inside or on G.
(b) Derive a formula for the number of lattice points inside or on S
(c) Derive a formula for the number of lattice points outside G but inside or on S.
Notes, Comments:
For the younger students, or for any group that isn't quite ready for the above, I strongly recommend building the pattern one step at a time and to record their results in a table:
|x| + |y| = 0 (o is not a positive integer but it makes sense to start here) to |x| + |y| = 1 to
|x| + |y| = 2,...
Also, there are alternate formulations (or a variety of patterns) for the general number of lattice points. Don't be surprised if your students find them! Of course, they should be encouraged to show they are all equivalent. Questions about the geometry of the figures could also be raised (if time permits). For example, some students do not fully appreciate that the diamond-shaped graph of |x| +|y| = N is in fact a square, Now, what would its area and perimeter be...
This investigation can be started in class then assigned to be finished outside of class, preferably with a partner. I plan on starting this with my 9th graders on Mon or Tue but I will need to review absolute values, graphs, etc., and I will develop it incrementally. I will be more than happy if, in one 40 minute period, they can complete a table showing the correct number of lattice points for N = 0,1,2,3!
[Additional but critical point: I am not suggesting that these kinds of extensive explorations should ever replace the need to deliver content and develop skills. These are intended to be enrichment activities, no more, no less!]
Posted by Dave Marain at 6:11 AM 8 comments
Labels: advanced algebra, algebra 2, geometry, investigations, lattice points, open-ended, patterns, prealgebra