Open and Shut
Lesson 4 of 11
Objective: SWBAT explain that polynomials form a system that is closed for addition, subtraction and multiplication.
Set the Stage
I begin class by making a series of statements and asking my students if they agree. I say, "All dogs are mammals so if Fluffy is a dog, he is also a mammal" and "I have never seen a dog that had six legs so dogs don't have six legs." Some students will immediately agree with both statements while others will cite possible exceptions. (MP3) I listen to all the comments then explain that my first statement is an example of deductive reasoning and my second is an example of inductive reasoning. Few if any of my students have heard these phrases before even if they have an idea about what deduction and induction are, so I give the broad definition that inductive reasoning moves from specific examples to a generalization and deductive reasoning starts with a generalization and uses it to explain a specific example. I go on to say that both types of reasoning are used in mathematics and today we will be using inductive reasoning to explore patterns and make generalizations about polynomial arithmetic. I explain understanding inductive reasoning supports "proving" closure in my Open and Shut video.
Put it into Action
I have my students work individually (if you have a large class of varied ability levels you may want to assign students to teams to give more support to struggling students). to solve an assortment of Polynomial Arithmetic examples of addition, subtraction, multiplication, and division so that they can build their own "experience" file of examples. (MP1, MP2) While they are working, I walk around offering encouragement and redirection as needed. Some of my students become easily frustrated with new material and become distracted and/or distract others as a way to avoid the frustration. I try to catch these students before they get too far off track and redirect with leading questions based on their areas of confusion. When everyone is done (or after about 10 minutes) I tell my students that during the next part of this activity they will work with their back partner to compare their answers and reconcile any discrepancies. (MP3) I suggest that one way to check their work is to substitute integers for the variables (avoiding "1" and "2" as problematic).
Now I ask the teams to group the problems by operation (addition, subtraction, multiplication, division) then look for patterns. (MP8) I tell them that they will be sharing their observations during a class discussion. I remind them that there are an assortment of graphic organizers if they want to use one for this activity. Some teams really struggle to find any patterns because they focus on just looking at the answers within each group, so I hint that looking at the groups might be more productive.
As everyone finishes their grouping I ask if any teams have any observations to share. I'm hoping that someone will notice that the answers to the division problems are not always polynomials, but if they don't, I ask leading questions like, "Is there any pattern to the answers between groups?" or "What kinds of expressions are the answers in each group?" I continue guiding the discussion until my students reach a consensus that all the examples show that polynomials can be added, subtracted and multiplied to give polynomial results, but can't always be divided. I remind them that the mathematical words to describe this (I call it "mathlish") are to say that polynomials are "closed" for the operations of addition, subtraction, and multiplication just like integers.
Wrap it Up
To close this lesson I ask my students to add a definition and explanation of closure for polynomials to their notes. (MP6) I tell them my expectations are at least four good sentences using appropriate vocabulary and grammar. I set the minimum at four because I think they can have a sentence each operation at minimum. They need to recognize that polynomials, like integers, are closed for addition, subtraction and multiplication. I've included some explanation examples for your reference.