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I noticed a pattern in that every time the length increases by 1, the number of blocks needed increases by 4. Therefore:

1, 8
2, 12
3, 16
4, 20
5, 24
6, 28
7, 32
8, 36
9, 40
10, 44

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I originally started with a t-chart showing that the patio stones increase by 4 when the garden increases by 1, but I wanted to do something different so I solved the problem by creating an equation.

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Knowing that younger students will often  require a concrete visual, I took the time to draw the 10 unit long garden and also used a ratio table where I could clearly see that the number of patio stones required increased by 4. I also recognize that I didn't include the extension questions...

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Garden Path - Sonali                                                     At first I counted the number of blocks that were being added to the right side only. I assumed the that since the length was getting bigger, it would only be getting longer, but I realized that the square was growing in size as well. Then I had to count how many blocks they were for the first 3 set of gardens and then found that the size was growing by 4 yellow blocks so I counted up for the rest. 

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Garden Path - Sonali                                                     At first I counted the number of blocks that were being added to the right side only. I assumed the that since the length was getting bigger, it would only be getting longer, but I realized that the square was growing in size as well. Then I had to count how many blocks they were for the first 3 set of gardens and then found that the size was growing by 4 yellow blocks so I counted up for the rest. 

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1- 8
2- 12
3- 16
4- 20
5- 24
6- 28
7-32
8-36
9- 40
10 - 44

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I noticed that there are 4 times the number of patio stones then the garden length, plus 4 corner stones. I therefore, wrote an algebraic expression. 

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Using the diagrams, I looked for a pattern within the first three. Each increased the number of stones by 4 when the garden increased by 1.  Therefore:

1, 8
2, 12
3, 16  meant that a garden length of 4 needed 20 stones.

To solve a garden length of 10 I continued to count up:
5, 24
6, 28
7, 32
8, 36
9, 40
10, 44

While this did not take particularly long to get to a garden length of 10 it would not be effective to problem solve to get the length of 100. I know that I could create an equation that could then make it possible to quickly solve for a garden length of 100.

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When I solved the problem I followed the following steps:

1. identify the pattern between lengths 1-3.

2. apply the pattern rule to determine the length of one side in garden length #4.

3. Multiply by 4 to get the total number of stones needed to surround the garden.