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# How Much is in the Bag 2

# Finding Percents

## Overview

Students use informal methods of their own choosing to find percents of randomly generated monetary values.

# Key Concepts

Many approaches work for solving percent problems. This lesson focuses on experimenting with a range of approaches and understanding why and how multiple approaches yield correct results.

# Goals and Learning Objectives

- Find a percent of a given quantity.
- Find a quantity given a part and the percent that part is of the whole.
- Use percents in money calculations.

# Introduction to Percent

# Lesson Guide

- Have students read the text about part-part and part-whole relationships and what percent means.
- Write “%” means “per hundred” on the board.

# Mathematics

The mathematical focus of the Opening is on understanding the meaning of percent from two distinct perspectives: the fact that percent means “per hundred” and the visual representation of a given percent (60%) of a whole amount.

SWD: Write what you are saying so that students can follow along. Put this problem in a table format so that students can clearly identify ways of representing numbers.

For example:

Number in Words | Decimal | Fraction | Percent |

60 hundredths (decimal) 60 over 100 (fraction) 60 out of 100 parts (percent) | 0.60 | $\frac{60}{100}$ | 60% |

## Opening

# Introduction to Percent

Discuss:

- Most of the relationships you have looked at in this unit are part-part relationships.
- For example, 60 students like action movies, and 40 students do not like action movies. The quantity 60 is one part, and the quantity 40 is the other part.

- You can think about this situation in another way.
- You can say that out of 100 students, 60 students like action movies. You can also say that out of 100 students, 40 students don't like action movies.

In this way of expressing the relationships between the quantities, 60 and 40 are parts and 100 is the whole.

- You can say that out of 100 students, 60 students like action movies. You can also say that out of 100 students, 40 students don't like action movies.

# More About Percent

# Lesson Guide

- Write “60%” means “60 per hundred” on the board.
- Have students fill in the grid to represent the students who like action movies.

# Mathematics

Discuss what 60% looks like on the grid. Ask the class to add any useful labels to indicate what represents 60%, what represents the whole, and what each small square represents.

## Opening

# More About Percent

If there are 100 students and 60 of them like action movies:

- You can say that $\frac{60}{100}$ or $\frac{6}{10}$ or $\frac{3}{5}$ of the students like action movies.
- You can say that 0.60 or 0.6 of the students like action movies.
- You can say that 60% (percent) of the students like action movies.

The percent symbol (%), as well as the word *percent* itself, indicates "per hundred" or "out of a hundred." (You can remember that *cent* means 100 because it takes 100 cents to make 1 dollar.) Thus, 60% means "60 per hundred" or "60 out of one hundred."

- Using a 10x10 grid, and fill in 60% of the squares to represent the students who like action movies.
- Write an equation that you could use to find 60% of a number.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will find percents of a given whole and find a whole given a part and a percent.

## Opening

Find percents of a given whole, and find a whole given a part and a percent.

# How Much Is in the Bag?

Have students work in pairs on the problem.

# Mathematical Practices

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

Students are likely to notice patterns in their computation strategies and use the repetition in their reasoning to develop efficient shortcuts for solving the problems.

**Mathematical Practice 5: Use appropriate tools strategically.**

Students are also likely to try different strategies, using different tools, to solve the problems. If any students systematically use one tool throughout the problems, prepare to ask them to share their reasons for their choice.

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Some students may use the familiarity of the context of working with money to support their thinking about whether or not their answers make sense.

# Interventions

**Student does not understand that one whole is 100%.**

- 100% means 100 per 100, or $\frac{100}{100}$ = 1.
- 100% is the whole amount: 100 out of 100.
- If you were to break the whole amount into 100 pieces, all of them together make 100%.

**Student does not understand that percents refer to hundredths.**

- If you were to break the whole amount into 100 equal-sized pieces, what could you call each piece?
- Represent 75% as a decimal, and then read the number aloud to your partner.
- 1% is equivalent to 0.01, which is one hundredth.

**Student uses operations incorrectly or unsystematically with a calculator.**

- Use your reasoning skills instead of the calculator to determine if your answer makes sense.

**Student uses operations incorrectly or unsystematically without a calculator.**

- Check your answer with a calculator.
- Does your answer make sense if you think about the values as amounts of money?

# Answers

## Work Time

# How Much Is in the Bag?

- The label on the bag tells you how much money is in the bag. Take turns finding how much money is indicated by the given percentage.
- Your partner should agree with your answer or challenge it if your explanation is not clear, correct, and complete. When you agree on an answer, enter it in the appropiate blank on the diagram.
- Continue until you have filled in all the blanks.

- What is 10% of the number on the bag?
- Fill that number in the "10% is" box.

# Find Percents and Whole Amounts

# Lesson Guide

Have students work in pairs on the problem.

ELL: This is a good opportunity for students to share ideas with others by working cooperatively. This interaction helps students develop the second language.

# Mathematical Practices

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

Students are likely to notice patterns in their computation strategies and use the repetition in their reasoning to develop efficient shortcuts for solving the problems.

**Mathematical Practice 5: Use appropriate tools strategically.**

Students are also likely to try different strategies, using different tools, to solve the problems. If any students systematically use one tool throughout the problems, prepare to ask them to share their reasons for their choice.

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Some students may use the familiarity of the context of working with money to support their thinking about whether or not their answers make sense.

# Interventions

**Student does not understand that one whole is 100%.**

- 100% means 100 per 100, or $\frac{100}{100}$ = 1.
- 100% is the whole amount: 100 out of 100.
- If you were to break the whole amount into 100 pieces, all of them together make 100%.

**Student does not understand that percents refer to hundredths.**

- If you were to break the whole amount into 100 equal-sized pieces, what could you call each piece?
- Represent 75% as a decimal, and then read the number aloud to your partner.
- 1% is equivalent to 0.01, which is one hundredth.

**Student uses operations incorrectly or unsystematically with a calculator.**

- Use your reasoning skills instead of the calculator to determine if your answer makes sense.

**Student uses operations incorrectly or unsystematically without a calculator.**

- Check your answer with a calculator.
- Does your answer make sense if you think about the values as amounts of money?

# Answers

## Work Time

# Find Percents and Whole Amounts

In this diagram you are given a value that is 75% of the amount of money in the bag.

- Find the amount of money that is in the bag, and enter the value in the 100% box.
- Take turns finding how much money is indicated by the given percentage.
- Your partner should agree with your answer or challenge it. When you agree on an answer, enter it in the appropriate blank on the diagram.
- Continue until you have filled in all the blanks.

Ask yourself:

- Find the value in the bag first.
- Remember the given number is 75% of the value in the bag.

# Jan’s and Martin’s Ideas

# Lesson Guide

Have students work in pairs on the problem.

# Mathematical Practices

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

Students are likely to notice patterns in their computation strategies and use the repetition in their reasoning to develop efficient shortcuts for solving the problems.

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Some students may use the familiarity of the context of working with money to support their thinking about whether or not their answers make sense.

# Answers

- Jan is correct. Possible explanation: The quantity represented by a percent depends on the size of the whole, and many different quantities can be represented by the same percent. For example, 10 and 50 are each 50% of a different whole: 10 is 50% of 20, and 50 is 50% of 100.
- Martin is correct. Possible explanation: The same number can be represented by many different percents, depending on the size of the whole. For example, 20 is 20% of 100, and 20 is 50% of 40.

## Work Time

# Jan's and Martin's Ideas

- Jan said that the same percent can represent different quantities. Is she correct? Explain.
- Martin said that a single quantity can be represented by different percents. Is he correct? Explain.

Ask yourself:

- When considering Jan’s statement, think about 50% of two different “wholes."
- When considering Martin’s statement, think about this situation: you have $100 and your friend has $20, and each of you contributes $10 to your school library fundraiser.

# Prepare a Presentation

# Preparing for Ways of Thinking

Listen and look for the following student thinking to highlight during the Ways of Thinking discussion:

- Students who use a variety of strategies to solve the problems; for example, equations, calculator, mental strategies, converting the percents to decimals, combining operations, or different types of diagrams
- Students who discuss what operations are appropriate and/or efficient for finding the missing values

Presentations will vary.

# Challenge Problem

## Answer

- 10% of 0.1 is 0.01.

## Work Time

# Prepare a Presentation

- Explain the strategy you used to find the percent of a given number. Provide an example.
- Explain the strategy you used to find the whole when given a part and the percent. Provide an example.
- Be prepared to present and justify your explanation about why Jan and Martin are either right or wrong.

# Challenge Problem

- What is 10% of 0.1?

# Make Connections

# Mathematics

Have students who used a variety of strategies present. Have them describe any strategies they used initially but decided to abandon as they worked through the problems and any strategies that worked well in some cases, but not in others.

Examples of different approaches:

- To find 30%, find 10% first and multiply that number by 3.
- To find 5%, find 10% and then divide the result by 2.
- To find 25%, divide by 4.
- To find 12.5%, divide by 4 and then divide the result by 2.

As students present their work, ask them to talk about the following:

- Why they used the operation(s) they did
- Why they used the tool they did (e.g., calculator, diagram, equation)
- How they know their answers make sense (justification can be with a diagram, with reference to the context of money, using an equation, or something else)

As students present their explanations, have the class refine and clarify the presented explanations as needed.

SWD: Find two students that solved the same problem using different strategies. Have both students present, then compare and contrast their solutions.

At this point in the course, students should be comfortable in presenting. Extend student responses by using prompts and targeted questions that are appropriate to the students’ ability.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about the different approaches your classmates used to find percents and whole amounts.

As your classmates present, ask questions such as:

- Do your strategies make use of any patterns? Can you explain the pattern?
- Why did you use that operation to find the missing number?
- Did you check that your answers make sense? If so, how?
- Was it helpful for you to think of the decimal in the Challenge Problem as a monetary value?

# Overview of Percent

# Mathematics

Have pairs quietly discuss the definition of a percent, how to find the percent of a number, and how to find the whole when given a part and the percent. As student pairs work together, make a note to clarify any misunderstandings in the class discussion. After a few minutes, discuss the Summary as a class.

## Formative Assessment

# Summary of the Math: Overview of Percent

**Read and Discuss**

- A ratio of a number to the special denominator 100 is called a percent. Percent means "per hundred." The symbol for percent is
**%**. - For example, if 6 teachers and 44 students went on a field trip, then there were 50 people on the field trip in all. The ratio of teachers to all of the people on the trip is 6:50 or 12:100.

$\frac{\text{teachers}}{\text{people on the field trip}}=\frac{6}{50}=\frac{12}{100}$

Therefore, 12% of the people on the trip were teachers.

Can you:

- Explain what percent means?
- Find the percent of a number?
- Find the whole, given a part and the percent?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out what connections students see between decimals and percents.

ELL: Provide ELLs access to a dictionary and make sure they are given time to discuss with a partner before writing, in order to help them organize their thoughts. Allow ELLs who share the same native language to discuss in their preferred language.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**Some connections that I see between decimals and percents are …**