Breaking Down Barriers
Lesson 10 of 23
Objective: SWBAT create an area model in order to simplify an expression with the distributive property.
This is a reteaching lesson in which we address mistakes being made with the distributive property. With reteaching lessons I like to start by giving my students a chance to first go over their homework with each other. This helps to surface and clarify what they are not understanding. This lesson revisits the definition of multiplication, the connection between mutliplication and addition, and utilizes the area model to support understanding.
On the screen the warmup tells students to check in with their math family group on their homework. They are told to: Compare their answers and discuss any answers that are different! I encourage them to defend their answers as best they can. I ask them to listen to each others reasoning and help their peers find mistakes. I want to send the message that a mistake is a learning opportunity and also that they are resources for each other. This is good practice for developing argumentation skills (mp3) by forcing them to articulate evidence and listen to reasoning.
As I circulate I try to encourage student communication. Rather than answering questions for them or verifying correctness I try to turn inquiries back to the group. I model questions like:
- “why do think that makes sense or doesn't make sense”
- “what makes you say that?”
- "what makes you think that is wrong?"
If I hear students giving a reason that was helpful I will share it with the class.
- “Jose just reminded us that it’s not the number in front that tells us whether terms are alike...”
- “Jessica just came up with a really good way of explaining how to add 3x and x…”
If all the members of a math family group are stuck I may address the class for help. Sometimes their questions provide a need and an opportunity to introduce a new vocabulary term like “coefficient”. Sometimes we discover a knowledge gap that requires more modeling for the whole class. When these 'teachable moments' arise I point out how important it was to ask the question or make the mistake. I emphasize that learning takes place in those moments. This encourages questions and creates a more growth mindset attitude towards asking questions in class. Questions are viewed as learning opportunities not as an indicator of a lack of intelligence.
This is a reteaching lesson that addresses several mistakes with the distributive property:
- combining unlike terms inside parentheses. 4(3x+2) = 4(5x) = 20x
- distributing multiplication only to the first term in the parentheses. 4(3x + 2) = 12x + 2
- combining terms before distributing multiplication. 6x + 2(4+x) = 2(4+7x)
The definition of multiplication helps students think about distributive property problems in a different way. I remind students that multiplication is repeatedly adding equal groupings. For example, 4(3x+2) can be thought of as four groups of (3x+2) or (3x+2)+(3x+2)+(3x+2)+(3x+2), which, when they combine like terms is 12x+8. I ask them to try this strategy to double check some of the homework problems they dissagreed about.
My students have seen an area model used for distributive property but very few have actually drawn an area model themselves, so I don't know how much it makes sense to them. I remind them how we chunk difficult multiplication to make it easier. 5(53)=5(50) + 5(3). I draw an area model to represent the equality of the two expressions and then ask students to draw a model for multiplication chunked like this: 4(2x+1). I ask my students to walk me through the process of calculating the area.
My hope is that one of these models will help students make sense of the math they are doing and give them an alternate strategy with which to check their thinking.
The purpose of this lesson is to teach them how to make an area model to represent the multiplication in a distributive property problem. Up to this point I have started with the model and asked them to find the area or write the expression. This is a way to start with the expression and give them a way to scaffold their own learning process. They are forgetting to distribute to all terms inside the parentheses, but if they can draw the model themselves, this may help them relate the multiplication to the idea of finding area, which forces them to distribute. Teaching them to make the model themselves is a way of giving them access to a model for as long as they need it.
On their white boards I ask them to draw a large rectangle which I do on the overhead as well. I write 3(x + 2) and ask them to label the dimensions of the rectangle if this is how we find its area. One side should be 3 and the other should be x + 2. I tell them that the length across the top is broken into two section x and.... (2) and I draw a line separating the rectangle. I ask them now to find the area of the large rectangle. They should put up 3x + 6. I ask them what the multiplied to get 3x and where that 3 times x is in the original expression 3(x + 2). I do the same for the 6. This helps them relate the physical model to the abstract expression and may help them transfer the skill. We practice on several other similar problems:
5(3x + 2) 4(2x + 5) (2 + x)4 3(2c + x + 4)
If they are able to get the correct solution using the model, I know they will stop making the mistake. If they continue to make the mistake of forgetting to distribute to all the terms I may need to separate the large rectangle into separate rectangles to make it more obvious. I know they won't rely on the area model forever, but I hope this gives them the scaffolding they need for as long as they need. I expect them to naturally stop using it once they have internalized the idea.
They can work on homework distributive with area models for remainder.