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I solved this problem by first drawing a picture of the next image using what I noticed about the green area of the garden.  Then I did a t-chart of the garden (green) and the patio stones (yellow) .  I drew one extra just to make sure I was seeing that pattern correctly and I like to just find one more than what the question provides to double check.
Jenn

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I solved the problem using a t-chart as well. However, using a t-chart to solve for a garden length of 100 would be rather time consuming so I have also suggested a formula that could be used.
- Jessica

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I did a T-chart. I thought of drawing it, but felt it would take more time. I like the t-chart because it helps me organize the numbers so I am able to see the pattern in the numbers. Once I had the numbers in the t-chart organizing the length and yellow stones, I started to consider if there was a pattern rule that could apply horizontally as well as vertically.
-Alaa

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I also used a t-chart. This was the easiest way for me to notice patterns in the increasing patio stones. From there, I created the equation to determine how many stones I would need for a garden length of 100. I think if I had manipulatives to utilize, I would have used these as well!
- Taylor

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Solution one - draw and count

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Solution two - t-chart pattern (+4)

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Solution Three - algebraic equation (P=Lx4+4)

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I know this solution was already used, but I also did a t-chart to solve this problem. This is what came to mind right away and I like to write out all the information to have in front of me. I also like the t-chart for this problem because I can double check my work when I look at it all written down. For example, I wrote down the first 3 garden lengths and the patio stones that corresponded, then I went all the way up to 10 like Kezia also did. This way I could make sure that the rule for the pattern I found (side length x 4 + 4), was actually correct. Then after confirming my pattern rule I could then use this to find the answer for a garden with a side length of 100.

Christie

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I solved mine very different! I needed to make a picture to solve this problem. I am a very visual learner. I find the equation posts very overwhelming! When I began my picture, I ran into the problem of needing to know how much green I had to add each time. I figured out that the green rule is: increase the number you add by 2 each time.  I drew the garden until garden 5, then I was sure that this was the rule for the green. After that I did not need to draw the gardens to get to the 10th. Relationship: Yes, there is a relationship as your garden gets bigger you do not need as many stones! I would not be able to use my strategy for 100 because I was starting to have to add larger numbers.  Below is a picture of my thinking.
Amanda

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I also did the t-chart.  But another way this can be solved would be to subtract the squares from each other to find the difference. First, square the perimeter (patio stones) and subtract the square of the garden length.  For example, to solve for the g.l. of 4, you could  square the perimeter 6x6 and subtract the square of the garden length 4x4 giving you an answer of 20.  (6x6) - (4x4) = 20. To solve for a g.l. of 10, you would square the perimeter 12x12 and subtract the garden length 10x10 giving you 44.
(12x12) - (10 x 10) = 44. The algebraic equation that I used was:
x² - (x-2)2
Natalie

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I realized that the corners of the garden were the constant (4) and that having 4 sides to the square meant I would need to multiply the garden length by 4. My equation was P= L x4 + 4. So a garden length of 4 would be P= 4 x4 + 4=20 and a garden length of 10 would be P= 10 x4 + 4= 44.  For a garden length of 100 P-100 x4 + 4=404. There is a relationship between the length of the garden and the patio stones. As you go up in length you increase by 4 patio stones each time remembering that you also need to enclose your garden with 4 corner patio stones that never change.
-Marc

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One of the strategies I used was to make a T-chart to organize my data from the visuals for garden lengths 1-3 and my drawing for number 4. I identified a growing pattern that helped me solve for 10 garden lengths. Kezia :)

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-Steffanni

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I solved the problem by taking the given length number, multiplying it by 4 to represent the 4 sides of the garden, and adding 4 more to represent the stones that would be on each corner. For example, with a length of 10, I did 10x4=40 and added 4 more for the corner stones, for an answer of 44. For a garden length of 100, I did 100x4=400 and added 4 more for the corners, so the answer was 404.
- Kelly