The Investigating_Properties opening activity has some interesting elements and really makes student think and understand the problem before beginning to come up with a solution (MP1). Students could take a variety of approaches depending on how they view the exterior quadrilaterals of the figure. The area of the square in the middle is clearly x^2, but the students now need to account for the area around that square without counting any areas twice. I encourage students to write their answer in simplest form. There are some elements of understanding repeated reasoning (MP8), due to the fact that all the rectangles have and area of "ax" and all of the squares on the corners have an area of "a^2". However, I allow students to look at the picture from their own perspective as there are many different ways to approach this problem, all of which can eventually arrive at the same conclusion.
I allow students to work on Investigating_Properties_Launch without any prior instruction. I want to see what my students are able to come up with. The understanding portion will come with the synthesis in the next phase of the lesson. My idea is to have students "discover" the Commutative and Associate properties of addition and multiplication by seeing how different perspectives on an expression can actually lead to equivalent expressions. Namely, if you and I are both looking at the same figure from different perspectives but, we can both account for the number of blocks (length or area) then we are naming equivalent expressions.
When debriefing this activity, I go through one question at a time and after a group shares their ideas, I have other groups build off of that idea before injecting their own idea into the conversation. I will probably use Think-Pair-Share or turn and talk to slow down the class and allow students to process an idea that might not seem particularly profound at the start of the conversation. I do my best to keep in mind where the lesson is going, and to avoid tangents. At the same time, I want to allow students to voice their thoughts and think out loud.
During this section of the lesson, I will lead the class through the slides in the investigating_properties_direct PowerPoint.
I project this slide and I ask students to make a connection between the structure seen in the properties and the exercises that they just completed (MP7). I tell students that each of these properties is reflected in one of the exercises we have completed today. I want them to see if they can work with a partner to match them up.
Next, I plan to have students read slide three and do a Think-Pair-Share to allow them to discuss the idea more in depth. Some students may need to see the "note" at the bottom of the slide emphasized so they can visualize what the associative property actually allows them to do.
Now I explain to students that in middle school, some work that they have done with expressions has taken the Commutative and Associative properties for granted. I say, "In Algebra, we want to be more specific about what the 'rules' of mathematics allow us to do." I direct students to look at the left hand column of this table and have them notice the changes that are being made from line to line. I let students attempt to come up with the properties that should be filled in in the right hand column and then go through the solution as a class.
Before moving on, I always explain to students that there is more than one way to arrive at the solution. Then, I encourage them to work with their partner to construct another proof that will allow arrive at the same expression.
As a closure for this activity, I allow students to practice this process of proving two expressions are equivalent by using the same properties for multiplication. This is a fairly quick task for most of my students, so I like to have students who finish a little more quickly come up to the board and post solutions (preferably more than one student) so that the rest of the class can see other ways to go about the process.
For this section, I will use Investigating Properties.pptx Slide 6. My goal is to give students another take on the process of proving why two expressions are equivalent. I will typically use this as a Ticket out the Door so that I can see where each student is individually. However, if the class is still in "processing" mode, I may allow the students to work with their partners again so that they can talk through the ideas within this proof.
I like this activity because the premise may not be immediately obvious to all students (the equivalence of the two expressions). In this way, students can see how useful the properties can be in proving that the assumption really is true. For more detail on closing out this lesson, see my Closing Reflection.