# Squares – Conceptual Understanding of Square Roots

Last year I made a post on the Squares and Square Roots which examined developing a conceptual understanding of square roots while taking the class outside. I thought the activity was interesting and told a lot about the prior knowledge of the class. I wanted to follow-up on this post based on a similar activity we did this year.

As anyone from Victoria knows the rain comes about this time of year. The last few days have not been ones that anyone would like to take their class outside. I did want to replicate the activity we did last year so it needed to do some modification due to the weather. While I would still have preferred to conduct the task outside the modifications made for a better learning experience.

Students were organized in groups of three and asked to create three squares out of string taping the string to their desks. The squares were to be 1 m^2, 2 m^2 and 3 m^2. To complete the task students first needed to determine the amount of string needed to create the square. Once the length was determined students were to come to me and get their string.

I thought the 1 m^2 would be easy for students to create after last years experience. I was surprised with the amount of time it took some students to complete this square. There were some misconceptions among students about the difference between perimeter and area. These were relatively easily overcome in group discussions and likely in a much more effective manner than I could have done.

I correctly guessed that the 2 m^2 square would be harder to make. Most students tried to use sides 2 metres in length resulting in a 4 m^2 square. Many groups did pick up on their error as the larger square as it was more than double the size of the 1 m^2 square. The key piece of the lesson was to have the two squares overlap so the areas could be compared and have students discover the error on their own. This was an improvement over the prior year and something I plan to continue. As a side note, only one of the groups was able to complete all three squares.

Students enjoyed the activity and even those who pride themselves on being strong computationally had trouble with the task due to some gaps in their conceptual understanding of square roots. To be fair this was an introductory class on square roots so I did not expect everyone to solve the problem easily. However, many had said they understood square roots but I think this understanding was limited to performing the calculations on a calculator. Building the squares lead to a better conceptual understanding of square roots.

Completing this activity underscores my belief that computational fluency while important is no replacement for conceptual understanding. They go hand in hand. To have one without the other is like being blind in one eye. You can see what is in front of you but you can not see the depth of mathematics.