Answers to General Formulas for |x| + |y| = N:
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
[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?
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.
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!]
Sunday, March 25, 2007