As students enter they begin the warmup on the screen.
Farmer John and Farmer Fred have neighboring fields. Both fields are rectangular and have the same length, but Fred’s field is half the width of John’s. Below is one possibility of what both fields might look like if Fred planted 40 square feet of pumpkins. Draw another possibility. (included in the warmup is a diagram of Fred's field with dimensions of 4 by 10 and a diagram of John's field with dimensions of 4 by 20)
I expect students to come up with field dimensions for John of 8 by 10 as well as several other possibilities for both fields by factoring 40 to find other possible dimensions for Fred's.
Before going over their warmup I want to go over last night's homework, because problem number 2 provides a basis for which further exploration will be done today in class. Problem 2 from their homework is a three part question that gives them dimensions of 10 by 40 for John's field and asks how many square feet of sunflowers John planted, how many square feet of pumpkins Fred planted, and, if they both planted pumpkins the following year, how many square feet of pumpkins they would plant combined. When going over this I want to diagram distributive property model on the board to intro the model for the distributive property. We draw John's field with dimensions of 10ft and 40ft and label it in the center with an area of 400 sq.ft. I ask what the dimension of Fred's field could be. When I draw Fred's field I don't draw it separately, but attached to John's field in whatever orientation matches the dimensions provided by students in their first suggestion. We label the dimensions of Fred's field and label it with the area. Since there are two ways, depending upon which dimension they halved, I also draw a diagram showing this second possibility. When we are asked what their combined area is I ask how they found it. Most if not all will probably have added the areas of the two fields together. If no one suggests multiplying the length and width of the larger combined field I will ask what the dimensions are of the larger rectangular field (the whole pumpkin patch) created by the combined smaller fields? This example works really well for students in my area because we are an agricultural community and kids have experience going to the pumpkin patches which are all over this area- you may want to choose a situation that relates better for your students. If students don't come up with 10 by 60 or 40 by 15 then I would highlight those dimensions on the diagram and ask what the combined width is of both fields together. Then I would ask if there is another way to find the total area of the pumpkin patch besides adding the two smaller fields. After a minute or two of group discussion they should easily come up with the larger multiplication problem. Then I ask them which method they think would be simpler to do in their heads. Adding the two fields separately is easier.
Now I would go over the warm up. I diagram or have students come up and diagram on the overhead the possible diagrams they came up with. If I diagram it I draw the fields connected as I did in the above example from homework. If they diagram it and don't do it this way that's fine, I would leave a little room below to come back to during their exploration.
Using the diagrams we made for the possible dimensions in the warm up I ask students to calculate the total area of the combined pumpkin patch in each case using both methods we looked at in the last example on the board and decide which was an easier method for each case. In the first case shown with areas of 4 by 10 and 4 by 20 or 4 by 10 and 2 by 10, students may find both methods equally easy. 4x10+4x20 and 4x30 as well as 4x10+2x10 and 6x10. They may come to the same conclusion in the second set of diagrams 2 by 20 and 2 by 40 as well as 2 by 20 and 1 by 20. However, in the third set they may find adding the smaller areas easier, especially with 8 by 5 and 8 by 10. They may not like either method for 5 by 8 and 5 by 16. (I would consider bringing this one back in tomorrow's lesson when we get into distributive property and show them how to further break 5x16 into 5x10 and 5x6). In the last example they may find that both methods are equally easy. I really want them to spend some time getting some experience with the distributive property using area models for two reasons. One is that I want them to recognize the connection when I use area models to represent the distributive property, but more importantly, I want them to get a sense of why it is useful and how they can use it to make their job easier, so that it becomes a tool for them. (MP5)
I want to finish up the day by generalizing patterns just as we did in yesterday's lesson (Farmer John and Farmer Fred day 1 of 2). This time I want to take it a step further and give variables for first Fred as we did yesterday and then for John. I start by using the area for the examples we just worked with and enter 40 for the area of Fred's field and then ask them for the other two areas. Then I suggest another area for Fred and ask them for a third suggestion for Fred. After the class has worked out these three examples then I put a variable for Fred's area. Some students will get it right away, but others will need more scaffolding. I circle the three areas we used for Fred and remind them that we did the same thing to every one of these numbers in order to find the area of John's field and I ask them what we did in each case. Some will say 40+40, 30+30, etc. and some will say multiplied it by 2 (if they say we doubled it make sure to ask how we represent "doubling" with numbers). I would then ask them to verbally describe what we would do to any number we choose for Fred's area in order to get John's. "If I use the letter n to represent any number then, what do I do to it?" They may say n+n or n times 2 and I write both in the table for John's. Then we work on the "combined" expression in the same manner, with the same kind of scaffolding quesitons. We may get three different expressions: n+n+n, 2n+n, 3n. I go through the same process with an area for John's field. I enter an area for John's field (20 maybe) and ask what Fred's field would be. Before having students share with the whole group have them first check with their groups, because some may double it instead of halve it. Having them work it out in the group is a quick and easy way to incorporate practice of argumentation (MP3).