Most of my students needed a retest for the area and perimeter. There were several different mistakes being made. They were getting the two mixed up together and were finding the perimeter when asked for the area and visa versa. They were adding only two dimensions or multiplying all four, and they were getting the units wrong. I think much of the problem stemmed from not understanding what area and perimeter referred. I also think students were probably given the formula too soon and did not have a chance to figure out where it came from or discover it for themselves. It's hard to backfill this type of experience when they already know there is an algorithm. They kept trying to blindly use an algorithm without thinking about it or making sense of it. I am hoping our focus on multiple methods improves their understanding.
The warm up gives students a rectangular playground with given dimensions. They are asked to figure out how much grass is needed to cover it and are then asked whether they found the area or perimeter to figure it out. I was careful not to give any clues in the questions like "how many square yards" or "how much to cover the whole area". The second question asks how long a pathway would be to surround the playground and asks whether they needed the area or perimeter. As we go over the work I ask them how they know whether area or perimeter were needed. As they describe "covering" and "surrounding" I ask them to come up and show us on the figure. For example I have them show me where the grass would go, shade the entire inside and do the same for a pathway. I also ask how they know the grass would be measured in square units and the pathway would be measured in linear units.
We also go over possible answers to their homework. I give them time to look at each others arguments for or against my statements and then ask them to choose the student with the strongest argument for number 2 from their math family. I tell them to make sure they have given evidence to support their statement and can prove that they are right. We go around each group and see if they can show us or add anything to that argument to make it stronger. Having students come up to the board and point to what they are talking about or to the distinguishing features really helps ELL students separate the concepts. I want them to go deeper than simply saying "you switched area and perimeter". I want them to tell exactly what is correct and incorrect. I want to leave them with the distinguishing characteristics of area and perimeter before they take their test.
There homework has a pretty low entry point for the first section. The second section has multiple solutions, which I model for them with the first example. I am hoping this first example will be enough for some of the students that still struggle with prior knowledge, but some of my more advanced kids will enjoy listing as many possible dimensions as they can. I expect everyone to finish the test with a couple of minutes left. If not, I will stop and collect the remaining tests (they can have additional time at the beginning of another class during the warm up) and we will discuss the two possibilities I have listed for the 15 square foot carpet. I would tell them that one of my suggestions is really not reasonable (15x1) and ask them to discuss why. I would like them to notice that no one could have a hallway that is 1 foot wide. I ask them if it should even be suggested given the context of the situation (no) and we cross it out. I want them to remember to consider the context and only make reasonable suggestions. I will be interested to hear what they say about the 2500 sq. ft. hallway.