In this warmup Reducing Farmer Fred students are given a diagram of a rectangular field with outer dimensions of 10 and 20 meters. They are told the farmer needs to reduce the width of his field in order to create a gap between his field and his neighbor's field in order to prevent cross-polination of their crops. They are asked what the area would be if it is reduced by 2 meters, then by 5 meters.
When I go over this warmup I set up a table warmup Reducing Farmer Fred complete with table at the bottom and first ask what the area would be if it wasn't reduced at all and I fill in an area for x=0. Students volunteer to share and explain their solutions to the other two questions (x=2 & x=5). I expect them to have solved in one of two ways:
Then I ask them to calculate the area if x = 10 and and I include that in the table and ask them to explain how they did the math. Next I put a variable in the table for "x" and ask how they would represent the area (how they would show what to do with whatever value x is). I expect they will come up with 10(20-x) or 200-10x. Finally, I ask them how they can both be true. If they can't explain I would ask how they would show someone that 10(20-x) was equal to 200-10x, which should prompt them to use the distributive property.
This is an open ended problem with multiple possible solutions. I give students a diagram warmup Reducing Farmer Fred and John of two neighboring farmer's fields with dimensions. I tell them they need a 10 meter gap between their fields and ask what the areas of the two fields could be. I expect some student push back on this type of problem, because students are used to following directions and finding the one right answer. It often throws them when they have a little more autonomy in the decision making process and in this case they need to see that it depends on how much each farmer reducing the length of his field.
Prompting questions may help them get started:
As I talk and listen to each group I may share some highlights with the class like:
I may also highlight some of the ways they are working together.
I draw a diagram of two fields, one 20 by 20 and one 5 by 10 and ask if it would be fair if both farmers reduced their fields by 5 meters if these were the original sizes.
Most of them realize intuitively that it would be unfair, but I tell them they should try to use some mathematical evidence to try to convince us.
I am hoping students will calculate the original areas of both fields (400 sq. m. and 50 sq. m.) and also the resulting areas after reducing each one by 5 meters. I will look for a student who has started and ask them to share this idea if not everyone has. This way the idea is coming from them and not me.
Most students will be able to articulate in some way that the farmer with the smaller field loses a larger part of his field. When I ask what they mean by a larger part I am hoping someone may point out that he lost half of his field while the other farmer lost much less than half or a quarter. Requiring them to give mathematical evidence is part of developing the practice of argumentation while at the same time making connections to related math.
If there is time, I may also ask them what they think would be a fair way to reduce these fields. This is a good way to get them to articulate and support an argument and to critique the argument of another. It also helps to put them in touch with their sense of ratio & proportion.