July 22, 2010

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This lesson will take several days to complete. In it, students learn how to compute percents of change, apply them to their virtual family's data, and learn how percents of change related to taxes and raises.

- Mathematics > General
- Mathematics > Applied Mathematics
- Mathematics > Data Analysis & Probability
- Mathematics > Number Sense & Operations
- Mathematics > Problem Solving
- Mathematics > Statistics

- Grade 6
- Grade 7
- Grade 8

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**Specific materials needed for this lesson:**

1. *Percents and Percent of Change *Worksheet (see Lesson 2 Resources folder).

2. *Example of Percents of Percents *Powerpoint

**Procedure:**

Begin by reviewing the methods for finding percents of change. Two effective ways of identifying percent of change follow. The first (and often initially easier-to-understand) method is the use of a proportion: amt. of change/original amt. = % of change/100. In this method, the student finds the absolute amount of change (positive or negative) and expresses it as a percentage of the original quantity. The second method for finding percent of change is to find what % the new number is of the original quantity, and then to subtract that % from 100%. (The benefit of this method is to always provide an accurate sign for the percent of change as well.)

After ensuring that all students can effectively find percents of change, distribute the ** Percents and Percent of Change** worksheets. Have students break into student pairs. Students should work through question (1) individually, and then when both partners have finished, they should check each other's work. They may have already calculated the percentage of their

Once the entire class (or the majority of the class) has checked their partner's work, have everyone come together to lead a discussion on their findings. The discussion should touch on the following points:

- The difference between the percentages of budget versus income. Why are the figures different? (The students should be able to easily identify the fact that the two figures represent two different comparisons.)

- The difference between individuals' percentage spent on food vs. the US average - do they line up? Why or why not?

- The differences among various individuals' percentages in the class. Why did the students with the lowest incomes have the highest percentage of their income spent on food and housing, and vice versa? (This is an ideal opportunity to reinforce the concept of relations. The teacher can create a quick scatterplot of student responses relating student income (independent) to percentage spent on food (dependent) to show the clear negative trend in the data.)

- The correct answers to question (d) for #1 on the worksheet. After the above discussion, students should be able to quickly draw inferences about the average income size of an Indian family, given that such a high percentage of their income is spent on food. The teacher can also highlight that if such a high percentage is spent on food (and conceivably housing would be high as well), then there is a small percentage of a small income left for what most Americans consider discretionary income (cell phones, video games, televisions, etc. etc. etc.).

After exhausting the talking points for this discussion, have student pairs work on question number (2) on the worksheet. Remember to have partners check each other's work for accuracy. (*This is a good opportunity to reinforce and reteach estimating with percents.* The teacher can ask partners if the answers obtained are reasonable, and why or why not?) The goal with all of these exercises is to generate as much student-to-student and student-to-teacher discourse as possible, as the more that students explain their mathematical thinking, the more likely they are to engage in deeper conceptual learning. Try to create and maintain an atmosphere of respect and open inquiry, where students feel free to make mistakes and to challenge each other's mistakes without derision or superiority. Once all (or a majority) of students have completed question (2), have the class come together to discuss their findings. It is important to highlight the fact that *while the 3% was the same for everyone, everyone's salary was raised by a different absolute amount*. Students in middle school often have trouble with the conceptual difference between percent and absolute increase, so this discussion should serve as an opportunity to solidify that concept.

After adequately discussing question (2), have students work individually on question (3). Circulate through the class to ensure they have understood the concept and are able to perform the required operations. Once students have all completed this question, have them exchange their work with their partner and check each other's work.

After ensuring that every student's answer is reasonable, lead a discussion on the progressive tax system in the United States (that is, the fact that the rate at which people are taxed goes up with their income). You can use some of the following talking points to help guide the discussion:

- Who paid the most taxes (in terms of dollars)?

- Who paid the highest percentage of their income (in terms of dollars)?

- Does that seem fair?

- Why would the government set up the tax system this way?

If students do not understand the motivation of a progressive tax system (regardless of whether or not they agree with it), have them perform the following exercise. Go around the room and have each student report *first* what their tax rate is, and *second* what their net income is *after* taxes. Hopefully, students will be able to quickly realize that although richer students pay a much higher percentage of their income in taxes, they still make sometimes over twice or three times what the poorer students make *after taxes*.

For the next section of the lesson, project the *Example Powerpoint* and ask students to make an estimate of which option (the first or the second) is the best deal. Give them 5 minutes to use paper and pencil, if they'd like. Take a class vote. Then, show the next slide and explain the math behind each option. Show that while the 10% discount in Option B sounds like it should give the customer *twice* as much of a discount as Option A's 5% added discount, it actually provides them with $1 *less* of a discount!

To explain this phenomenon, you can have a student stand facing the doorway of the classroom. Tell them that Option A is like saying, "I was going to let you go ¼ of the way to the door, but I'll add on another ¼ of the way to the door. So, really, you're going ½ of the distance to the door!" In this scenario, I could have added ½ of the distance to another ½ of the distance, and you would make it to the door itself.

Option B, however, is like saying, "Go half of the distance to the door. Now, go half of the distance to the door again (from where you're standing now)!" In this scenario, will you ever get to the door? NO! You'll only get really, really close.

This is because in Option A, you are only finding the percent of one number (i.e. the two percents added together to make a new, bigger percent). In Option B, you're finding the percent of a number *and then doing that again*. Because a fraction of a fraction is always a smaller fraction (but never 0), you'll always get smaller, but you'll never actually get to 0. PLUS, a small fraction (5%) of a bigger number (100) is always bigger than a big fraction (10%) of a smaller number (40).

Now, have students work individually to complete question (4) on their worksheet. When done, they should compare their answers with their partners.

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