# Expressions for Percent Increases and Decreases

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## Objective

SWBAT write equivalent expressions for percent increases and decreases by combining like terms

#### Big Idea

An increase of 15% is the same as multiplying by 115%. A decrease of 20% is the same as multiplying by 80%.

## Introduction and Guided Problem Solving

15 minutes

I will begin with the first essential question:  How can you write equivalent expressions for a percent of increase?

This lesson especially exercises MP2.

Then we'll work through the first example.  A bar model is given to provide a visual and to tie into work we've already done with bar models.  The only difference now is that the cost is represented by a variable.  When labeling the bar model the variable, m is the original or regular cost so it goes above 100%.  I'll draw in an additional unit to represent the 10% tax.  This shows that the total with tax can be written as m + 10% of m or 110% of m.

Next I will use the two steps given to show how to write this as two equivalent algebraic expressions.  First we write the sum:  m + 0.1m.  Then we combine like terms to get 1.1m.  So we see that m + 0.1m = 1.1m.  Both expressions represent the total.

For those students who still don't see the connection (perhaps because of the variables) it may be helpful to substitute a value in for m and evaluate it using each expression.  They will see that the result is the same.

Next students work through 4 guided problem solving problems.  Students may need to be reminded that percent values will be represented as decimals in all expressions.

Now I will present the second essential question:  How can you write equivalent expressions for a percent of decrease?

I will go through the same process as above for the increases.  So on the bar model students will see that a sale price 25% discount can be seen as c - 25% of c  or 75% of C.  Following the two steps in the notes will lead to c - 0.25c and 0.75c as equivalent expressions.  Again, some students may find it helpful to prove these are equivalent by substituting values in for c and evaluating each expression.

Common errors

Students often forget to include the variable on their sums and differences.  So instead of writing c - 0.25c they may write c - 0.25.  If this occurs, it is worth discussing what this means.  Perhaps even by substituting a value for c.  If c = \$80, how much is the discount?  How much is the sale price?  What does 80 - 0.25 equal?  Does this represent the sale price?  Etc...

## Independent Problem Solving

20 minutes

Students work on this set of problems independently.  The first 8 problems are structurally identical to the problems solved in the introduction.

Problems 9 is designed to prepare students for a multiple answer multiple choice item types that may occur on upcoming assessments.  Notice that students should explain their thinking.  For example, a student would eliminate 9i and state that this only represents the discount, not the sale price.

Problem 10 is similar to 9 yet students may have to apply the distributive property to recognize all of the equivalent values.

## Exit Ticket

5 minutes

Before beginning the exit ticket, we will discuss the two steps to writing equivalent expressions.  Write a sum for a markup and simplify; wirte a difference for a markdown and simplify.

Problems 1 and 2 are each worth 2 points for each equivalent expression.  Note:  I am expecting a student to write x - 0.4x and 0.6x for problem 1, however I will accept any two equivalent expressions.  So it will be okay if a student writes x(1-0.4).  This applies two problem 2 as well.

Problem 3 requires a brief explanation of how to use substitution.  I will expect student to provide an example.

Earning at least 4 of 5 points will be the mark of a successful exit ticket.