Formulas and equations are so abstract to my students. This lesson helps them relate equations and formulas to the physical model of the rectangle. Being able to associate the numbers with the physical objects they represent gives students a way to make sense of the math. (MP1) Taking time to make connections between the different ways of representing the area can help students understand why the formula for area makes sense. On an emotional level my students need permission to struggle and make mistakes. This lesson also points out all the correct thinking involved in making mistakes. It is so important to point out all the things that are correct about the thinking that leads to a mistake the lesson the stigma.
This warm up presents a disagreement between two boys, Angel and Jonah, about how to find the area of a 20 by 30 foot rectangular room. Students are asked to point out what each boy is right and wrong about. This is a good way to scaffold an argument for students as well as engage them in critiquing arguments. (MP3)
Students should point out that Jonah is correct when he says the dimensions must be multiplied and when he uses square feet. His answer of 3,600 square feet is wrong, because he multiplied all four dimensions. Students may not realize this, but instead say the right area is 600 square feet. They should also point out that Angel is right about 3600 being way too much and that only two dimensions are needed to calculate the area. He also used square feet which is right. His mistake was adding the two dimensions to get 50.
Students are not used to pointing out what is correct so if they only point out the mistakes I make sure to tell them that both boys are right about several things and encourage them to look for those. I will keep encouraging them until all the correct things are pointed out. I want students to notice that the things the boys were right about outnumbered the things they were wrong about.
Each group is given two equations that both represent the area or perimeter of a single rectangle(area perimeter sort cards). For example one group might be given A=81 ft^2 and A=9 ft. x 9 ft. or they may get A=5(2x) and A=2x+2x+2x+2x+2x or they may get two equations that represent the perimeter of some rectangle. This activity is easy to differentiate as you can pair more difficult equations with more advanced students and the easier ones with those at a more basic level.
Their first task is to draw the rectangle (The Sisters: Area and Perimeter) They are told that there is another 'sister' group that has equations showing the perimeter of their rectangle and they need to write the equations their 'sister group' might be working with. Next they go find their 'sister' group to see if they correctly predicted the equations they were using. Before letting students find their 'sister' group tell them that they may find they didn't come up with exactly the equations the group has or they may have come up with additional ones. Tell them to double check that the equations they came up with actually work to find the area of their rectangle. Then have them take turns discussing which equations were most helpful in drawing the rectangle.
Have one group at a time come up to the front with the cards they were given and the rectangle they drew to show what they did and explain how they figured out what the rectangle looked like and which equation was more helpful and why. Explaining this forces the students to dissect the equation and relate it to the dimensions of the rectangle. Have the group show how they came up with the expressions their "sister" group might have had.
If you don't want kids roaming around the room you can modify this by not having the two groups find each other. Instead you can have them present and then ask who their sister group is. But then you give up the group discussion examining the predicted equations. The value in this discussion is that they uncover more of the connections between multiple methods.