As students enter class today they will begin the warm up. It looks similar to the one they did in yesterday's lesson. They are given a rectangular field with two sections and told that Farmer Frank can't remember the dimensions of his pumpkin patch, but he knows the area of the patch in two sections because he just spread bags of fertilizer (12 sq. m and 18 sq. m).
After yesterday's lesson, we know Farmer Frank is forgetful. He still can’t remember the dimensions of his pumpkin patch, but now he remembers that the height (dotted line) was the greatest number he could possibly make it. Can you help him figure out the dimensions by making this side as long as possible?
I circulate to make sure students are not labeling opposite sides with different dimensions, which is the most common mistake that my students make. I often ask, "If the left side is 6 meters in length what must the right side be?" I am also looking to see if students are working with common factors that are not the greatest common factor. If they have other common factors like 2 or 3, I ask if that is the longest it could be.
During this time I am also checking a couple of problems that I've picked out from last night's homework, factoring distributive with area models. For example, I am looking at Number 3 because the two areas being factored are 5x and 10. I suspect several of my students to mistakenly choose 5x as a common factor. If they have, I ask them if x is a factor of 5x and 10. For students who struggled with this problem, I ask that they check their work on Number 4 as well. I am also looking to make sure they have not factored out a 1 and I tell them to use a greater factor because they haven't actually found a relationship between the two products.
When we go over the Warmup I will tell a little story about how the greatest common factor is not just any little old factor but instead is like the superhero of common factors, because he is the biggest. We call him Common Factor The Great or The Greatest Common Factor in the World, or at least in Farmer Frank's Pumpkin patch!
For this segment of the lesson my students will work on one problem at a time. I will write the problems on the board and they will raise up their final answers on the count of three.
I adjust the challenge level and pace to match their proficiency and energy level. I start with some simple problems without variables (20 & 12, 18 & 24) to make sure they are not still labeling opposite sides differently and to make sure they are finding the greatest common factor and not just any factor. I also recreate the rectangle from their warm up and giving the first pair of numbers as the areas of the two sections. If they are factoring correctly and most of them are using the GCF, then I will point out another way to represent our solution without using an Area Model. For example 20 + 12 is one way to represent the area of the entire pumpkin patch using the dimensions 4(5+3).
When I feel that my students are ready, I move on to expressions with variables, then to expressions with three terms:
I expect many of my students will find a common factor, but not always the GCF. This gives me the opportunity to show my students how to test whether or not they have correctly identified the GCF: by checking if the dimensions at the top (or inside the parentheses) still have a common factor.
If lesson is going really well, I will eventually give them something like:
Here, I am probing to see how far they can go on their own with the concepts that we are learning. If most of them are correctly factoring out the GCF, I will ask them to create their own problem to challenge their Math Family Group. I'll say, "Write your own problem on your board and then trade boards with members of your Math Family." As I circulate, I look to see if anyone has created a problem without a common factor. If so, I will ask them to try again since this indicates that they may not understand as well as I want them to. I tell them it is sometimes hard to come up with a problem that works. If other members of the Math Family have written an appropriate expression, I will ask them to explain how they wrote an expression so that it would have a GCF.
Flash Cards is a closing I use to either wrap up what we are doing or to refresh ideas we haven't looked at for a while. Today I have random numbers on Flash Cards, which are just recycled manila folders cut in half.
First, I have my students pack up and stand behind their desks. I hold up two cards and I ask students to find the greatest common factor. No one raises their hand or shouts out the answer. Everybody should be getting ready to answer, because they don't know who I'm going to call on and they need to answer in 1 second after I call on them. If a student gets it right they sit down. If a student gets it wrong I move on to another student. If a student who got the answer wrong looks really confused, I may ask the student who eventually came up with the right answer to explain their thinking.
We keep going until the period ends. If everyone sits down we go for another round.