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# Expressions for Percent Increases and Decreases

Lesson 11 of 15

## 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%.

*40 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...

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#### Independent Problem Solving

*20 min*

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.

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#### Exit Ticket

*5 min*

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.

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- LESSON 1: Multiple Representations of Percents
- LESSON 2: The 10% Benchmark
- LESSON 3: The 1% Benchmark
- LESSON 4: Percent Benchmark Fluency
- LESSON 5: Drawing Bar Models to Represent Percents of Increase and Decrease
- LESSON 6: Solve Problems by Applying Percents of Increase and Decrease
- LESSON 7: Discounts and Sales Tax
- LESSON 8: Finding a Percent of Change
- LESSON 9: Finding an Original Value
- LESSON 10: A Percent Equation
- LESSON 11: Expressions for Percent Increases and Decreases
- LESSON 12: Simple Interest
- LESSON 13: Increasing and Decreasing Quantities by a Percent (Day 1 of 2)
- LESSON 14: Increasing and Decreasing Quantities by a Percent (Day 2 of 2)
- LESSON 15: Percent Assessment