Students in Ms. Mason's and Ms. Wine's Class completed an activity from the National Council of Teachers of Mathematics book Navigating Through Algebra. In this activity, students create a garden made with brown tiles surrounded by white tiles. The soil (the brown tiles) must form a square, and the white tiles surround the edges of the brown tiles. The first garden consisted of one brown tile surrounded by 8 white tiles. The second garden consisted of four brown tiles surrounded by 12 white tiles. Below, you can see gardens 3-6.

The brown tiles can be described as 3 x 3 or as 3 squared (reviewing exponents). The total number of tiles can be described as 5 x 5 or 5 squared.

When the students record the number of brown, white and total tiles, they start to recognize different patterns.

Garden Number	Brown Tiles	White Tiles	Total Tiles
1	1	8	9
2	4	12	16
3	9	16	25
4	16	20	36
5	25	24	49
6	36	28	64

Students start to notice that the brown tiles are increasing by squaring the garden number. So, garden 1 is 1 squared, or 1. Garden 2 is 2 squared, or 4, etc. The white tiles, however, are increasing in an arithmetic pattern which starts at 8 and increases by 4. The total number of tiles are also square numbers. Students can continue filling in the chart without using the tiles, reasoning that the 7th garden will have 7 squared, or 49 tiles. There will be 28 + 4 or 32 white tiles, and 81 tiles total.

This student writes a rule for the finding the number of brown tiles. By giving the students a context, variables can be introduced in a meaningful way. The student has explained that the patio (or garden) number times itself gives the number of brown tiles. It is then easy to suggest to the students to use a letter to represent the variable (in this case the patio or garden number). Here, the student creates a function table to figure out a rule for the number of white tiles. She has made notations on the right that she must multiply by 4, and then add by 4. For patio number 3, she sees that (3 x 4) + 4 = 16 white tiles.

By comparing the growth of the brown tiles to the growth of the white tiles, students can see the difference between arithmetic patterns (the white tiles that increase by 4) and geometric patterns (the brown tiles).

Finally, students can graph the values of the brown tiles and the white tiles (using the garden number as the x-coordinate) to appreciate how the geometric pattern increases to far greater values more quickly than the arithmetic pattern.

Indicators:
1.5.1.2 analyze patterns and generalize rules illustrated in patterns.
1.5.4.1 represent relationships using graphs and tables.
1.5.1.1 recognize, describe, and extend numerical and geometric patterns and functional relationships.