In this activity, students calculate total price including sales tax. They repeat the calculation for several items and with two different tax rates, giving them an opportunity to make use of repeated reasoning (MP8). Students also write expressions to represent the amount of the sales tax and the total price including tax for an item with any price, . This builds on their previous work writing equations to represent situations involving percent increase.

Invite students to share their strategies for calculating the sales tax and total price for the laundry soap in each city. Ask students to describe any patterns they see in the table, for example:

• The values in the total column can be found by adding the previous two columns.
• The sales tax is proportional to the price with a constant of proportionality of 0.06 or 0.08.
• The total is proportional to the price with a constant of proportionality of 1.06 or 1.08.

Next, invite students to share their expressions for the last row of each table. Make sure students see the connection between this row and their previous work on percent increase. Point out that sometimes we want to know just the amount of the tax, , and sometimes we want to know the total, which is the price plus the sales tax, .

If time permits, introduce the term “tax rate.” Explain that when there is a certain tax that gets applied to a class of goods, it is called a tax rate. Tax rates are usually described in terms of percentages. Ask students:

• “What is the tax rate in City 1?” (6%)
• “What is the tax rate in City 2?” (8%)

In this activity, students encounter a situation in which rounding error makes it look like the relationship between the price of an item and the sales tax is not quite proportional. Students see that this is due to having a fractional percentage for the tax rate and the custom of rounding dollar amounts to the nearest cent. As students explain which tax rate matches the given amount of sales tax for each item, they attend to precision (MP6).

Some students may say that the relationship is not proportional. Remind them of the activity in a previous unit where they measured the length of the diagonal and the perimeter of several squares and determined that there was really a proportional relationship, even though measurement error made it look like there was not an exact constant of proportionality.

Some students may say that the tax rate is exactly 7%. Prompt them to calculate what the sales tax would have been for the paper towels and the lamp if the tax rate were exactly 7%.

Some students may use 7.25% as the tax rate since that is what comes from the first item (paper towels), but in this case they did not check this number against the tax on the other items provided. Prompt students to use the additional information they have to check their answer before proceeding to solve the row with laundry soap.

The purpose of this discussion is to highlight how a fractional tax rate, along with the custom of rounding dollar amounts to the nearest cent, can explain the apparent inconsistencies in the table.

Consider asking:

• "How did you determine the tax rates for the items in City 1 and City 2 from the previous activity?" (I divided the sales tax by the price.)
• "How was determining the tax rate for City 3 different from Cities 1 and 2?" (Since the tax rates were not the same for each item, I had to determine what tax rate might give each value listed.)
• “Is there a proportional relationship between the price of the item and the amount of sales tax for City 3? How do you know?” (Yes, there is a proportional relationship. The constant of proportionality for each row is approximately 0.073.)

Consider reminding students about when they measured the length of the diagonal and the perimeter of a square. They saw that measurement error made it look like the relationship was not quite a proportional relationship.

In this activity, students practice finding the percentage, the original amount, or the new amount in the context of sales tax and tips. They apply the strategies they learned previously with percent increase, such as representing the percentage as a decimal and writing and solving an equation. As students apply what they learned about percent increase to represent situations involving tax and tip, they are making use of structure (MP7).

Students may attempt to write an equation, but place numbers in the wrong place. Ask them what each piece of their equation means in this situation. In particular, monitor for students who struggle with understanding the second part the first question. Help these students understand by rephrasing the question as, "The total is what percent of the subtotal?" and helping them to see that the answer should be greater than 100% since the total is greater than the subtotal.

Students might need a way to keep track of all the information. Suggest using a table that keeps track of original price and percentage.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how they solved each problem, including which values were given in the situation and which values they were trying to find. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases, such as “sales tax,” “tip,” “subtotal,” and “total including tax.” Ask students to suggest ways to update the display: “Are there any new words or phrases that you would like to add? Is there any language you would like to revise or remove?”

If no student uses the strategy of writing and solving an equation, ask students:

• “How we might use equations to solve the last two problems?” (Set up an equation that shows the percentage times the subtotal equals the new amount, such as or )
• “How can we represent the tax rate 9.5% as a decimal?” (0.095 for just the sales tax; 1.095 for the total including tax)
• “How do you know whether to multiply or divide to solve the problem?” (It depends on whether the number next to the tax rate—in other words, the subtotal or the original amount—is known or unknown.)