As students enter class their warmup is on the screen:
Farmer John and Farmer Fred have neighboring fields. Farmer John planted sunflowers and Farmer Fred planted pumpkins. Both fields are rectangular and have the same length, but Farmer Fred's field is half the width of John's. Draw a diagram of what their fields might look like. What could their dimensions be?
These kinds of open ended questions in which there are multiple possible solutions are not what students are used to in math. I expect students to say "but you didn't tell us any of the numbers" or just "I don't understand what to do". Without detailed directions or examples for how to proceed they decide they can't do it and start asking me for help right away. My hope is that with enough experience with open ended questions this will subside. For this problem I go back and read the problem with them and ask them what they know about what the fields look like. I would expect them to say they are rectangular, so I tell them to start with that and then think about what the lengths and widths might be. If students are really resistant then I would do an example on the board with the whole class using the quesitons: "what could the length be for John's?" "then what would the length have to be for Fred's?", etc. Some students will do one example and some will do several. When I see a student who has more than one possible diagram I would definitely highlight that to the class by showing them or saying "Look at what Priscilla has done, she has come up with three different diagrams for how the fields could look!"
After I've circulated and made sure they all have an entry I ask for solutions and write them on the board. I start with two rectangles and ask if they look reasonable. Some students will say yes, because they are rectangles, some will say no because it doesn't look like one of them is half the width of the other. I really want this to emerge because I have seen students make so many mistakes in geometry because they are making an assumption about a dimension because of what the drawing looks like. So, when this comes up I tell them that's okay, because we're going to clarify once we put in some numbers for dimensions. I tell them not to be fooled by a bad drawing.
Now I ask them for some possible dimensions for John's field. I want to avoid asking specifically about length and width here so that all the options can emerge. If I identify too quickly which dimension is the length two things will happen. One is that it limits the possible dimensions of Fred's field and it also distracts them into an unnecessary concern over which way is length and which is width. For example if a student comes up with 10ft by 8ft for John's then Fred's could be either 5 by 8 or 10 by 4, but if we get too concerned with which is which, only one of those possibilities will come up. Now it is possible that when I start to label the dimensions a student might correct me and say "no, I put that number on the bottom" and, in that case, I would just switch it. If students do bring up the question about which is the length and which is the width at this point I would simply say "well, let's see...if the length is 10 and the width is 8 what would the area be?" "If we reverse them, what would the area be?" I also may bring up a rotation, what if we turned the field this way instead, would the dimensions or the area be different? Now when we get a possible solution for one field it should yeild two possible solutions for the other and I hope students would start suggesting that. If they don't I would just ask "what else could Fred's field look like given the dimensions of John's?" until they started bringing up the other possibilities themselves.
I continue asking students for possible dimensions of the two fields that I or they put up on the board. If I call on a student and they say they don't know or that someone already did theirs I tell them we'll come up with one together and ask them the same questions I asked struggling students from the warmup "what could the length be for John's...then what would the length be for Fred's..."
Because this problem mentions John's field first I expect students may start with his field every time and halve the width to get Fred's dimensions. Many of my students have not connected the ideas of halving and doubling, so I want them to notice and use the pattern this way too? I would ask either "who has some dimensions for Fred's field?" or "let's start this time with dimensions for Fred's field". This way they really have to pay attention to the criteria of the problem and not get stuck just in a pattern of calculation. If they get too used to doing the same thing over and over again I'm afraid they will just continue doing the same thing out of habit and I want to reactivate their brain and get them back into the context.
If the discussion of area has already taken place then I would have the class calculate the areas of each of John's and Fred's fields as well as new examples I post on the board. If the areas have not been mentioned then I would bring it up afterwards so that we can use the pattern to help us practice generalizing into variable expressions. This is such an important idea for developing algebraic thinking and modeling with mathematics (MP4) that I try to work it in to as many lessons as possible.
On the screen I have a table to show the areas of John's and Fred's fields separately and combined.
Fred's is listed on top and John's below. I try to show most of my data tables horizontally to reinforce ratios later in that unit.
I ask students for a possible area for Fred. Some may search the board or their papers. We fill it in and then ask what the area of John's field would have to be. Again they may search the board or their papers. Then I ask them to tell what the combined area of their fields would be. We go through this same process three times. I purposely have them going back and forth from searching the board to thinking with their brain, because at some point I want them to stop looking at the board. After 3 examples I stop and ask if they notice a pattern emerging in the table. Give a few quiet moments here. Then I put an area in the table for Fred's field and ask what the area of John's field would be. Some kids may look back at the board, but the ones who have the pattern won't need to. When I see that some kids see it, I have them tell their math family group about the pattern they notice. Then we go back to the table and I put another area (not from the board) for Fred and have them predict John's.
Then I give them an area for John's and ask for the area of Fred's field. Before I have them share out to the class I have them discuss it in their math families, because some will have doubled and not halved and I want them to correct each other. This is a good way to get them used to critiquing the ideas of others and having their own ideas critiqued (MP3). I do another example like the last one.
Then I give them a combined area and ask if anyone can figure out what the areas of John's and Fred's field are. Frequently this conversation starts with the idea of both areas being half of the combined area until someone in the group points out that the areas can't be equal because Fred's should be half of John's. If a group does not get to this point I may also ask them to figure out what the dimensions could be. I don't tell them they are wrong and they often think they are moving ahead because I have given them a new question to work on, but they should notice the problem when they refer back to the criteria for the dimensions. When they have decided on the correct areas for John and Fred I like to go back and bring out some of the "argumentation" that happened in their group. I may ask "who started with the idea of equal fields?" and ask "what did someone say or point out that changed your minds?" and I might say "I like the way you went back to the original criteria in the directions to make sure your numbers fit". I really want to encourage this because it helps them check for reasonableness.
Now I put a variable in the table to represent the area of Fred's field (MP2). Students still may say they can't say the area of John's unless we have a number for Fred's. I will point out that they know enough about the relationship between the two fields to be able to say what to do even without numbers. I may circle all the numbers in the table for Fred's field and ask what we did to every single one in order to get the area for John's. They will say doubled, or multiplied by two. "Then you can tell me what to do with any number I put in for Fred's area to get John's right?" They give the same answer. "Then you can tell me what to do with x right?" By then most students have got it and hopefully fewer will need this much scaffolding the more used to this they get.