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.
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...
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.
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.