Equivalent Expressions: Distributive Property

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Objective

SWBAT use the distributive property to demonstrate equivalence of expressions.

Big Idea

This lesson gives students a more in-depth look at the distributive property.

Opening

15 minutes

In this opening activity, students will be playing a version of the four number game.  In this version of the game, the object is to write down as many counting numbers as possible starting with 1, which can be derived using only the numbers 1, 2, 3, and 4 and the operations of addition and multiplication.  Some numbers can be derived in multiple ways while other numbers cannot be derived at all.  Students should work with their partners to create the longest list they can in about 5-10 minutes.  Students are able to use parenthesis where needed to come up with the appropriate number while attending to the order of operations.  This game will help students begin to reason quantitatively in ways that are important during this lesson (MP2).

Example:

1=1

2=2

3=1+2

4=1+3 or 2x2

etc.

Once students have had time to make their list, quickly go around to each pair and make a list of the different ways to write each number starting with 1. After they work on this task for some time, there are several followup questions that I like to ask:

  • What is the first number that cannot be made with the numbers 1, 2, 3, and 4?  (22)
  • What is the largest number that can be made using the numbers 1, 2, 3, and 4? (36)  

The second question motivates an investigation of the mathematical structure of the game (MP7). Pursuing this question is important to my plan for this lesson. So, I plan to ask the following additional questions:

  • If we were playing a three number game, what would the largest number be?  
  • What if we were playing a 5 number game?

After some time for thinking, I plan to point out to students that in each of these cases the form of the largest number is the same:

Three number game: (1+2)(3)

Four number game: (1+2)(3)(4)

Five number game: (1+2)(3)(4)(5)

In each case, the expression combines addition and multiplication to summarize a pattern. From this point, we begin to discuss the distributive property in algebra.

Investigation: Distributive Property

10 minutes

During this section of the lesson, we will be following the Equivalent Expressions Distributive Property presentation. I will lead the students through the slides. For the prompt on Slide 2, I will have students work with a partner.  I expect most of my students to come up with the idea that the area of the first rectangle is "2*x" and the area of the second rectangle is "8".  During this launch I will try to visit any students who struggled with area models yesterday.

Teacher's Note: Slide 3 in the PowerPoint file contains an animation. It is also possible to use the .pdf version without the animation.

I plan to give students a couple of minutes to think about Slide 3.  In this slide my goal is to make the familiar explicit. Students may be applying this reasoning without reflecting on it. Today, it is important to highlight this in order to build new ideas onto existing conceptual understanding.  I encourage students to recognize that breaking up the figure is a process of distributing the quantity 2 as a measure of the dimensions of the rectangle. I find that making this explicit now, helps students to extend this way of thinking to binomials, trinomials, etcetera as they continue to progress in algebra.

In Slide 4, we encounter a common misconception. I hope that students will be motivated to make an area model as they consider the question and construct a response.  I will stay with this question until my students begin to make progress with understanding why these two expressions are not equivalent. The design of this slide targets MP3 as the students work. I hope that by listening, sharing, and critiquing students will construct a deep understanding that will help them to avoid this misconception. 

 

Investigation: Modeling Binomials

10 minutes

Beginning with Slide 5 in Equivalent Expressions Distributive Property, we focus on multiplying binomials using the Distributive Property. My plan is for students to do a Think-Pair-Share as they work on Slide 5. I expect students to come up with different answers to this question, which is great.  Some will say that the trinomial demonstrates the Distributive Property.  I like to show the expression at the bottom because it shows in plain terms how the a is distributed to the entire second term and how the b is distributed to the second term.  This is also represented pictorially in Slide 6. When displaying Slide 6, I will model how distributing "a" results in the first rectangle and distributing "b" produces the second rectangle.

Slide 7 encourages students to begin to extend their thinking to all products of polynomials. Again, the figure is an anchor point for students. The expression is simple enough to explore and I will let students come up with their own explanations, most likely while working with a partner. 

Slide 8 steps back to the work in the earlier part of the class. I will ask my students to record this summary statement in their notebooks. I will specifically point out that the Property applies to all real numbers. In this lesson, we worked with side lengths and therefore did not encounter negative quantities. 

Closure

5 minutes

During today's lesson, students will be working in pairs for a majority of the time. So, today's Ticket_Out_the_Door is a formative assessment. I designed it to get some feedback with respect to student progress with the Distributive Property. I will make sure to ask my students for a written explanation of the answer to Question#3. I will review the explanation for signs of lingering misunderstanding.