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# Ratio Table Tool

# Percents Greater than 100%

## Overview

Students use percents greater than 100% to solve problems about rainfall, revenue, snowfall, and school attendance.

# Key Concepts

Percents greater than 100% are useful in making comparisons between the values of a single quantity at two points in time. When a later value is more than 100% of an earlier value, it means the quantity has increased over time. This percent comparison can be used to find unknown values, whether the earlier or later value is unknown.

# Goals and Learning Objectives

- Understand the meaning of a percent greater than 100% in real-world situations.
- Use percents greater than 100% to interpret situations and solve problems.

# Too Much Rain

# Lesson Guide

Have students read the prompt. Ask students to talk with a partner about whether the percent of normal rainfall is greater than or less than 100%. (Answer: *It is greater than 100%.* ) One possible way to find the answer is to divide 25 by 20.

SWD: Help students identify key information from the problems. One way is to have them underline key words and numbers that will help them solve the problem. Allow time for students to master this skill.

ELL: As with other oral instructions, ensure that the pace of your speech is appropriate for ELLs. Pause frequently to allow students to pose questions. Alternatively, ask questions as your explanation unfolds to monitor understanding.

# Mathematics

The start of the lesson introduces the idea of a percent greater than 100% in the context of comparing rainfall between a normal year and the current year. There is a conceptual shift in this situation that students may need to make: instead of a percent representing a comparison between part of a quantity and the whole quantity, as in the previous lessons, the percent represents a comparison between an initial value, or “base,” and another value.

## Opening

# Too Much Rain

The city of Valley View normally gets 20 inches of rain per year.

Last year the city got 25 inches of rain.

Think of using percents to compare 25 inches of rain as a percent of the normal rainfall.

- Will this be greater than or less than 100%?
- What strategies can be used to solve this problem?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will solve problems that involve percents greater than 100%.

## Opening

Solve problems that involve percents greater than 100%.

# Rain

# Lesson Guide

Have students work in pairs on the problem.

SWD: Investigative work is intended to be a time for students to grapple with the mathematics, and it is crucial that you only help students understand the tasks, and not solve the problem.

# Mathematical Practices

Students can draw on all of the resources they have developed so far in this unit (and before) to make sense of today’s problems. Some students may consider analogous problems as a way of getting started, some may make conjectures about values that would make sense before doing any calculations, and many will show perseverance in solving the problems—especially the problem about the company’s earnings (Task 4), which is less routine than the others.

# Interventions

**Student has difficulty setting up computations.**

- Make a diagram to represent the situation.
- How much greater than 100% is the given value? (100% plus what?)
- Represent the percent in decimal form and write an equation to solve the problem. Use a letter to represent the unknown quantity.

# Answers

- Last year’s rainfall is 125% of normal.

## Work Time

# Rain

The city of Valley View normally gets 20 inches of rain per year.

Last year the city got 25 inches of rain.

- Calculate last year’s rainfall as a percent of normal.

To calculate the percent, would you divide 20 by 25 or divide 25 by 20?

# Company Earnings

# Lesson Guide

Have students work in pairs on the problem.

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

Students can draw on all of the resources they have developed so far in this unit (and before) to make sense of today’s problems. Some students may consider analogous problems as a way of getting started, some may make conjectures about values that would make sense before doing any calculations, and many will show perseverance in solving the problems.

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

This lesson also presents opportunities for students to shift back and forth between the computations needed to solve the problems and the meanings of the values as given by the context of each problem. Listen for students who are doing this explicitly in their work, whether it’s in verbal exchange with their partner, in labels for diagrams, in setting up equations, or otherwise.

# Interventions

**Student has difficulty setting up computations.**

- Make a diagram to represent the situation.
- How much greater than 100% is the given value? (100% plus what?)
- Represent the percent in decimal form and write an equation to solve the problem. Use a letter to represent the unknown quantity.

**Student has trouble interpreting the situation about the company’s earnings.**

- Will your answer be greater or less than $9,000?
- Write an equation to solve the problem. Use a letter to represent the unknown quantity.
- In all of today’s problems, there is a base, which is often the “normal” amount, or the initial value. Do you know the base in this situation?
- You know that $9,000 is 120% of normal for July. This means that $9,000 is 100% of normal, plus 20% of normal. How can you represent 100% and 20% in decimal form?

# Answers

- The company’s normal earnings for the month of July are $7,500.

## Work Time

# Company Earnings

A company earned $9,000 in July.

That amount is 120% of the company’s normal earnings for the month of July.

- What are the company’s normal earnings for the month of July?

Will the answer to this problem be greater than $9,000 or less than $9,000?

# True Statement?

# Lesson Guide

Have students work in pairs on the problem.

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

Students can draw on all of the resources they have developed so far in this unit (and before) to make sense of today’s problems. Some students may consider analogous problems as a way of getting started, some may make conjectures about values that would make sense before doing any calculations, and many will show perseverance in solving the problems.

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

This lesson also presents opportunities for students to shift back and forth between the computations needed to solve the problems and the meanings of the values as given by the context of each problem. Listen for students who are doing this explicitly in their work, whether it’s in verbal exchange with their partner, in labels for diagrams, in setting up equations, or otherwise.

# Answers

- The statement cannot be true. Possible explanation: The number of students absent cannot be greater than all of the students, and if all of the students were absent, 100% would be absent.
- The statement could be true. Possible explanation: If normal annual snowfall is 12 inches and last year’s snowfall was 18 inches, the snowfall last year was 150% of normal because 18 is 150% of 12.

## Work Time

# True Statement?

Look at each statement and decide whether or not it could be true.

If the statement could be true, provide details about the situation in the statement that explain how the statement could be true.

If the statement could not be true, explain why.

- 125% of the students are absent today.
- The snowfall last year was 150% of normal.

Think about the situation. Does the statement make sense for that situation?

# 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 approaches to solving the problems, including diagrams, equations, and expressions
- Students who discuss what the base is (its value) and who also use the meaning of the base, given by the context of each problem, to better understand a particular computation

# Challenge Problem

## Answers

- 150% of 0.5 is 0.75
- Solutions will vary. Possible solution:

150% of $\frac{1}{2}$ = 100% of $\frac{1}{2}$ + 50% of $\frac{1}{2}$

100% of $\frac{1}{2}$ = $\frac{1}{2}$

50% of $\frac{1}{2}$ = $\frac{1}{4}$

$\frac{1}{2}$ + $\frac{1}{4}$ = $\frac{2}{4}$ + $\frac{1}{4}$ = $\frac{3}{4}$, or 0.75

## Work Time

# Prepare a Presentation

Explain your solution to one of the Work Time tasks. Support your explanation with diagrams and equations.

# Challenge Problem

- What is 150% of 0.5?
- Use fractions or sketch a diagram to represent the problem and show your solution strategy.

# Make Connections

# Mathematics

Have students share a variety of approaches to solving the problems, especially those who created diagrams and wrote equations. Ask students to connect the parts of a diagram with the parts of an equation, clarifying or developing each first, as needed. Then ask students for different ways of checking that the solutions are correct.

ELL: Use every opportunity for students to work on their speaking skills and to practice reading numbers and equations.

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

Highlight work from students who showed perseverance. Point out students who made conjectures before jumping into the computations and who considered analogous problems if they got stuck.

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

Identify students who used the given context to set up and/or make sense of the computations. Ask them to talk about how the context helped them understand the quantities involved. Also ask how the context might help someone confirm that an answer is correct.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about the approaches your classmates used for solving problems with percents greater than 100%.

As your classmates present, ask questions such as:

- Can you make a diagram to illustrate what you are saying?
- Can you write an equation or expression that represents the relationship you are describing?
- Did you use the context of the problem (that is, what it was about—money, attendance, rainfall, or snowfall) to help you check that your answer made sense? If not, how do you think the context of a problem might help you determine whether your answer makes sense?

# Percents Greater Than 100%

# A Possible Summary

A percent greater than 100% represents a comparison between one value that is greater than another value, with the second value being the base, represented by 100%. In situations like rainfall and revenue, these kinds of comparisons are common.

# Additional Discussion Points

Additional things you might want to discuss are:

- What a percent greater than 100% means
- In what types of situations percents greater than 100% can be used

## Formative Assessment

# Summary of the Math: Percents Greater Than 100%

Write a summary of what you learned about percents greater than 100%.

Check your summary.

- Do you explain what it means for a percent to be greater than 100%?
- Do you give examples of the types of situations in which percents greater than 100% make sense?

# Using Tools to Solve Ratio Problems

# Lesson Guide

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery problems and which students would benefit from review before the assessment. Have students work on the Self Check individually.

# Assessment

Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

SWD: Provide clear feedback to students as they attempt to solve problems or articulate concepts. This type of feedback guides students explicitly as they develop their thinking about mathematics.

# Interventions

**Student creates an incomplete, inaccurate, or incorrectly labeled diagram, resulting in an incorrect solution.**

- Identify the quantities in the situation, both known and unknown, and then create a new diagram to represent how they are related to one another.
- Consider a similar problem with simpler values, such as a ratio of 2:1 with the waiter keeping $10. Now apply your reasoning to the given problem.
- Make a conjecture about the amount the waiter gave the busboy. Will it be less than or greater than $45?
- Create a tape diagram showing a ratio of 5:2 to represent the quantities in this situation, and label each quantity and each unit of your diagram carefully.

**Student gets a correct answer, but does not create an explicit model or explain the solution.**

- How can you show that your answer is correct? Create a diagram that illustrates the relationship between quantities.
- Create a tape diagram, a double number line, a table, or a graph to model the situation, and label your answer.

**Student creates an accurate and complete model, with the correct answer.**

- Use another tool to create a different kind of model to represent the relationship between quantities (e.g., a tape diagram, a double number line, a table, or a graph).
- Explain how the parts of your first model relate to the parts of your second model.

# Possible Answers

- The waiter gave the busboy $18.
- The tape diagram shows a ratio of 5:2, with the 5 units in the upper tape representing the waiter’s $45 and the 2 units in the lower tape representing the busboy’s unknown amount. If the upper tape represents $45, then each unit represents $9, and the busboy’s amount is 2 ⋅ $9 = $18.

## Formative Assessment

# Using Tools to Solve Ratio Problems

Complete this Self Check by yourself. Use a tool such as the Ratio Table, Double Number Line, or a graph to solve the problem.

A waiter at a restaurant shares his tips with the busboy in the ratio 5:2 (5 parts for himself, 2 for the busboy).

- If the waiter had $45 after sharing his tips, how much did he give the busboy?
- Explain how you got your answer.

INTERACTIVE: Double Number Line Tool

INTERACTIVE: Ratio Table Tool

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to find out why students think percents greater than 100% are useful.

## 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.

**Percents greater than 100% are useful because …**