Extending Farmer Frank's Field with the Distributive Property

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SWBAT create an area model and a variable expression to represent the Distributive Property.

Big Idea

Students will distinguish between parameters (variables) and constants in area models and variable expressions.

Intro & Rationale

In today's Warmup my students build on work they have done in previous lessons (see Farmer John homework). They are asked to apply the Distributive Property using numbers, but no variables. They have some experience using the Distributive Property with variables (see Halloween Candy), but not with a continuous model.

Today students will explore equivalent expressions by comparing area models. They will also use Area Models with both constant and variable terms, to begin to think how to use the Distributive Property to model and solve problems. The use of visual models with the Distributive Property helps cement the concept for my students, especially ELL students. 

Since we are synthesizing and applying today, I will encourage active collaboration in Math Family Groups. 



10 minutes

As my students enter they will see the Extending Farmer Frank Warmup on the screen. In the task they are given a rectangular field with dimensions 10 meters by 12 meters. They are told it represents Farmer Frank's existing Pumpkin Patch. Farmer Frank wants to extend the patch by 4 meters in one direction (horizontally) or the other (vertically). Students are asked: Which direction will add more area to his patch? They are also prompted to explain the mathematics they use to answer the question.

I ask my students to collaborate in their Math Family Groups as math brothers and sisters for this Warm Up. When we go over the problem as a class, I will ask students to explain their answers to the class.  

In order to decide which extension will add the most area to the Pumpkin Patch, I expect my students will take one of three approaches.

  1. Some will calculate the entire area of the combined field (12x14 & 10x16)
  2. Others will calculate the area of the original field separately from the areas of the fields to be added and solve it in two parts (12x10+12x4 & 10x12+10x4))
  3. Still, others will not bother to calculate the area of the original field. (10x4 & 12x4)

I plan to model all strategies on the board as students share them. I also plan to lead a whole class discussion to help my students make sense of the following concepts:

  • equivalence
  • constancy
  • variables (or parameters)

I want my students to recognize that Strategies 1 and 2 above are equivalent strategies Seeing Strategy 3 often helps my students distinguish between the portion of the field that remains constant and the portion that is variable.

I have two questions in mind that I will use to advance the discussion: 

  1. How do we know the first two strategies will find the same answer?
  2. How can we use the third strategy to solve the problem without actually calculating the total area? 


40 minutes

During today's Exploration I want my students to practice organizing their data in a table. This step will better enable them to look for and generalize patterns in the data. I want them to be able to write a variable expression describing the patterns that they observe. I expect some students may make the mistake of combining the constant term 120 and the variable 10x, which we will explore in later lessons  using algebra tiles (see Who's Right? and To change or not to change).

To begin I give students another copy of the Farmer Frank problem, this time including a Farmer Frank table.docx to keep track of x (meters extended), the area added, and the total area. I tell students that Farmer Frank is not going to take our advice and extend his field vertically, but horizontally instead. However, he wants to extend it more than 4 meters. I ask them to draw the extended field using x for now to show the number of meters extended.

I ask a student volunteer to choose and share with me some number of meters that Farmer Frank could extend his field and I enter it into the table as an example on the overhead. I will then ask the class, "How much area will this add to the Pumpkin Patch?" When I record their answer I will fill carefully record both the area and an expression representing the calculation that was performed (as explained by the student). I include the expression because it helps students see the structure of the calculation and begin looking for patterns (MP7, MP8).

Next, I will ask students, "What do I have to do to find the resulting total area?" They may say, "add 120 to the area added..." which I will write in the table along with the answer. Or they may just say, "Add 120." If so, I will ask, "to what?" I want to make sure they are thinking in terms of a complete expression of the calculation. 

Now I ask for another number for x, the length to be added. This time I will have a student come up and fill in the table. If the student doesn't include an expression, I plan to ask a different student, "How did __________ calculate the area?" I'll perform the same sequence to record the resulting total area.

After a couple of examples I will ask the students to come up with three more possible areas and include the data in the table on their worksheet. I will definitely encourage working with Math Family Groups. I'll even suggest that they use the same values for x within their groups, if they want. "But," I'll say, "if you use different numbers in your groups and check each other's work you may find the pattern more easily."  

As they work, I will circulate and look for students who have completed their table early. As I find fast workers, I will I ask them to go up to the overhead and record one of their examples in the class table.

Eventually (perhaps after three or four volunteers) I will put a variable, x, in the column for the additional length. As students notice what I have done I will say, "Can anyone tell me what expression would represent the area added for the addition of x meters?"

This is a moment when I may or may not need to scaffold the process of generalization. If no one suggests "10x", I will circle all the numbers that were substituted for x and ask, "How were these numbers used to calculate the area? What operations were performed?" We may spend some minutes describing this in words. Eventually, I will be more direct and ask, "If x were a number what would we do to it? What expression can I write to describe the additional area if I add x meters of length to the plot?"

I want to prepare my students for the moment when we apply the pattern to turn things around and demonstrate the power of the general expression. Here's my approach:

I enter a number (e.g., 50) in the "area added" column. Then, I ask the class to identify the value of x (i.e., how much length was added?) and the new total area.

My goal is for one or more students to explain how they knew what x was in this case. When they explain their thinking, I will write an equal sign between 10x and the number I put in the table for "area added." I'll say, "So, you knew that 50 had to equal 10x?"

We've accomplished a lot at this point. We've worked with patterns and made the connection between modeling with expressions and modeling with equations. To reinforce this point, I will follow the same routine by entering a number for "Total Area" and asking similar questions. It is an unpredictable moment in the lesson, but, I've found that it is a memorable one for my students. 

As time allows, I let them get started on their homework Farmer Frank.docx for just 3 - 5 minutes before I have them pack up. I like to let them get started so that they know they can do it and remember it. I will try my best to give students some time today because many of my students do not have sufficient support at home to overcome obstacles if they forgot what they did in class. So, their Math Family Group may be their best support when it comes to challenging homework.

Flash Card Line Ups

4 minutes

Once students have packed their bags, I like to conclude this lesson with a Flash Card Line Up. This is an activity I do at the end of class when there is just a couple of minutes left. I especially like to use it for to achieve the following goals:

  • We need to review or practice a skill that remains tricky for my students
  • I want to preview previously learned skills or knowledge prior to a future lesson

With bags packed, my students pack up and stand behind their desks. I hold up a flash card (i.e., a recycled manila folder with a problem on it for all to see.) Students do not raise their hands or call out. Instead I Cold Call students and I expect the answer in 1 second. This element of the strategy forces them to focus on the problem and stay quiet. I find that it motivates each student to complete each problem.

When a student answers a question correctly, I have him/her line up at the door. If a student gets one wrong or takes more than one second, I will ask another student the answer. Sometimes I will ask a second student before asking the first student to line up at the door.  This way students don't always know if the first person got it right or not before they answer. If a student does get one wrong, and I see puzzled faces, I will explain it or have a student explain it. (I also do this when I see puzzled faces after a correct answer.)

For today's session we are reviewing two things:

  • exponents 
  • finding the length of missing sides of irregular shapes

Some of my students generally need additional practice multiplying with exponents. Also, students in my district have had trouble in previous years using the fact that opposite sides of a rectangle are equal to calculate the lengths of missing sides, so my flashcards today show pictures of irregular shapes made up of rectangles. Each card has one missing side. Some cards give only the lengths of the sides needed to calculate the missing side and others provide all the other lengths so that they must distinguish the lengths they need.