Grids can be used to embody exactness of mathematical artifacts without having the precision of pencil and ruler skills–concepts of precise lines without reality of line widths or sloppiness getting in the way.
Here are some examples of polygons where only the square is regular.
| concept | description | details |
|---|---|---|
| precision | Using grid points to create polygons. These examples contain angled lines with small integer (1 or 2) x & y components. A focus could be on developing descriptions of non-horizontal and non-vertical lines as. “so many up (down) so many right (left).” | |
| midway points | To make better use of grids, midway points can be used to either simply make smaller polygons or debatably improve upon the regularness of polygons | |
| simple spirals | Use grid polygons to make spirals. The angled lines help to ensure lines of the spirals' lines remain parallel. Each of the straight lines lengths is a multiple of the original polygon's side. There is a pattern to the multipliers of the originals. For the triangle, the multipliers follow this sequence: 1, 2, 3, 4, 5, 6,… The multipliers 2 and 5 (bold in the sequence) are two and five times the original horizontal side not to be confused with the lengths of the sides 4 and 10. Similarly, the diagonal sides are some multiple of root 2 but since they have no additional grid points, it is easier to see the size of the multipliers. For the square, there is no such confusion; the multipliers follow this sequence: 1, 1, 2, 2, 3, 3, 4, 4, 5,… | |
| tricky spirals | What are the sequences of multipliers for these spirals? | |
One tool to understand patterns is to construct a table that breaks up the information. Below I have constructed a table with the length of each segment starting from the center of the spiral. As well, a running total length is calculated.
| length per side | total length |
|---|---|
| 1 | 1 |
| 1 | 2 |
| 2 | 4 |
| 2 | 6 |
| 3 | 9 |
| 3 | 12 |
| 4 | 16 |
| 4 | 20 |
| 5 | 25 |
| 5 | 30 |
| 6 | 36 |
| 6 | 42 |
| 7 | 49 |
| 7 | 56 |
| 8 | 64 |
| 8 | 72 |
| 9 | 81 |
A pattern in the total length can be seen. There are a number of ways to think about why this arises. For instance, each unit line segment in the spiral can be associated with a single dot in the grid which can in turn be associated with the square up and to its left. Total length becomes equivalent to area of a rectangle (which half the time is a square).
In a related way, the sequence of total lengths can be shown zig-zagging down the times table. Since the sequence of square numbers are so useful, it's probably a good idea to highlight them. In this sequence, you may recognize that each square can be seen as a sum of consecutive odd numbers.
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 |
The triangular spiral to the right needs a little twist to better understand its pattern. Previously, the idea of multipliers was discussed as a means of talking about the lengths of line segments. Here, I would like to replace the term multiplier by using the idea of steps. To describe the original triangle we would say, 3 steps: one DR (Down & Right), one L (Left), one UR (Up&Right). Even though the steps are different lengths, we will consider them the same unit. This imprecision is to overcome the problem of not using a triangular grid.
| # steps per side | total # steps |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 6 |
| 4 | 10 |
| 5 | 15 |
| 6 | 21 |
| 7 | 28 |
| 8 | 36 |
| 9 | 45 |
Here the path the sequence of total # steps has a bit of a jumping zigzag.
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 |