Lesson 14

Expand Expressions Using the Distributive Property

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Objective

SWBAT expand and evaluate expressions using the distributive property

Big Idea

Students extend their 6th grade knowledge of the distributive property in order to expand expressions involving integers

Introduction

10 minutes

I will begin with the essential question:  How can you use the distributive property to evaluate an expression?

Then as a quick hook I'll present a problem like:  "Marchand earns $8 per hour on a part time job.  She worked 4 hours on Friday and 6 hours on Saturday.  How much did she earn in these two days?"

Students will quickly answer the question but then I'll ask them to explain their process.  Inevitably some will say that she earned $8 * 4 hours = $32 on Friday and $8 * 6 hours = $48 on Saturday for a total of $32 + $48 = $80.  Others will say she worked a total of 6 + 4 hours = 10 hours so she earned $8 * 10 hours = $80.   As students respond I will record their responses on the SmartBoard.  It will look something like this:  8 * 4 + 8 * 6 = 8 (4 + 6) = 8 (10) = 80

I will ask students to discuss in a quick turn-and-talk what each part of the above expressions represents. (MP2).  I'll then say this is an example of the distributive property.

Next I will present the area model.  This will be a reminder of some of the work that they did in 6th grade (at least for the ones who were at my school)!

The area model presents a nice visual model of the distributive property.  This part is to help students start to focus in on the appearance of equivalent expressions generated by the distributive property (MP7).  In other words, it creates expressions that can be seen as a product of two factors, 120 * 50.  A product of two factors, where one factor is a sum, 120 * (30 + 20).  Or it can be a sum of two products.  The focus today will be to go from the 2nd form to the 3rd form:  120 * (30 + 20) = 120 * 30 + 120 * 20.

 

Note:  It may be helpful to model an simpler problem using color tiles to reach students who may need a concrete model.  A problem like 3 * 5 could be modeled as 3 * 2 + 3 * 3 using six color tiles of one color and 9 of another color, all grouped together to make one rectangle.

 

So by the last question on the page, I will focus students' attention to the mechanics of the distributive property - specifically expanding an expression - by asking students to study the expression.  I may ask:  How do the second and third expressions relate?  I want to lead students to discover or re-learn that the factor outside of the parenthesis can be multiplied by each addend.

If needed, I will make up another similar problem using the area model.   It would be great if students can already come to this conclusion, but if not, the next section allows them to explore this even more.

Problem Solving

15 minutes

In this section students explore expanding expressions even further.  And now the lessons starts to become more of a 7th grade lesson as negative integers begin to appear.  Students will solve 10 problems by evaluating expressions with the order of operations.  The set has 5 pairs of equivalent expressions.  

The questions are then set up so that they match the expressions and complete equations using the two expressions.  In question 3, students examine the matched expression and work to explain how the distributive property was used to expand the expression.  I will look for precise language (MP6) in these discussions.  So words such as factor, addend, difference, product should be part of the conversation.  Using one problem as an example:  -3 * (4+8).  A quality student answer will be something like this:  "Multiply the factor outside of the parenthesis by each addend in the parenthesis.  Then take the sum of those two products."  Drawing a visual with arrows may be helpful.

Finally, students put their conclusions to work by expanding and evaluating 4 more expressions. 

 

As students are working on these, I will be watching to make sure values are being expanded correctly.  Sometimes students will only make a product using the first addend and forget to "distribute" to the second addend.

Independent Practice & Extension

20 minutes

The independent practice is all about the fluency of evaluating expressions with the distributive property.  Some students will want to simply solve expressions using the order of operations or mentally.  It's fine if they want to do this, but I will also insist that they show how each expression can be expanded.  If they are discouraged by my request, I will complement them on their abilities to evaluate expressions while reminding them that the main focus is to generate an equivalent expression with the distributive property.  I'll tell them that this is an important skills for evaluating expressions with variables.

Problems 6 and 8 may be especially tricky because they have a different where the subtrahend is a negative.  It is okay if students want to use the additive inverse and re-write these as sums.  When we go over solutions, I will look for students who initially re-wrote the problem as a sum and for those who did not.  This way we can see that either method is fine.

The last two problems involve 3 addends in parenthesis.  I hope that students can apply what they have learned to multiple addends.  If they are stuck, I'll suggest they evaluate the expression first using order of operations and then generate an expanded expression and check for the same value.

The next page has some equations and algebraic expressions to solve.  This is a preview of the work to come to unit from now!  The first 4 equations can be solved with or without the distributive property.  I hope students solve a problem like 11, using mental math.  I hope they see 3(x+4) as two factors whose product must each 27, so x must equal 5.  

 

 

 

 

Exit Ticket

5 minutes

The exit ticket requires students to expand four different expressions and then evaluate.  Two of the problems have sums the other two have differences.  I have included a variety of combinations of positive and negative integers in the problems.  

 

Before students take the exit ticket, I will remind them that for each problem they must have an expanded expression AND an evaluated value.  Many students will be in a rush just to get the answer, but again that is not the point of the lesson.

 

A student who answers at least 3 of the 4 problems will be considered successful.