Lesson Plan: Solving Percent Problems and Using Proportional Reasoning

Lesson Summary

In this lesson, students will learn how to solve a variety of percent problems using proportional reasoning, equations, and real-world applications. This lesson builds on their understanding of ratios and proportional relationships by applying these concepts to percent increase, percent decrease, and multi-step percent problems. Students will develop problem-solving strategies to calculate discounts, tax, interest, and commission, reinforcing the practical use of percents in everyday life.

Students will practice:

• Using proportions to solve percent problems, including finding the part, whole, or percent.
• Solving percent increase and decrease problems in real-world contexts.
• Applying the percent equation: \( \text{Part} = \text{Percent} \times \text{Whole} \).
• Working with multi-step percent problems involving discounts, tax, and markup.
• Interpreting and solving problems related to simple interest.

Through hands-on exercises and word problems, students will gain confidence in applying percents in various financial and mathematical contexts.

Lesson Objectives

• Use proportional relationships to solve multi-step percent problems
• Apply percent calculations to real-world situations
• Approximate irrational numbers using rational numbers (ratios)
• Use square root and cube root symbols to represent solutions to equations
• Apply proportional relationships to scale models

Common Core Standards

CCSS.MATH.CONTENT.7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

CCSS.MATH.CONTENT.8.NS.A.2
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π^2).

Prerequisite Skills

• Understanding of percents
• Basic algebra skills
• Knowledge of square roots and cube roots
• Understanding of ratios and proportions

Key Vocabulary

• Percent: A ratio that compares a number to 100, often represented using the symbol "%."
  • Multimedia Resource: Definition
• Percent Proportion: A proportion that represents percent problems in the form \( \frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100} \).
• Percent Equation: A formula used to solve percent problems: \( \text{Part} = \text{Percent} \times \text{Whole} \).
• Percent Increase: The amount a value has increased, expressed as a percentage of the original value.
  • Multimedia Resource: Definition
• Percent Decrease: The amount a value has decreased, expressed as a percentage of the original value.
  • Multimedia Resource: Definition
• Discount: A reduction in price, usually given as a percent off the original price.
• Markup: The amount added to the cost of an item to determine its selling price.
• Sales Tax: A percentage of the cost of an item added to the final purchase price.
  • Multimedia Resource: Definition
• Commission: A percentage of a sale earned by a salesperson.
  • Multimedia Resource: Definition
• Simple Interest: Interest calculated using the formula \( I = P \times r \times t \), where \( P \) is the principal, \( r \) is the interest rate, and \( t \) is the time in years.

 Multimedia Resources

• A collection of definitions on the topic of ratios, proportions, and percents: https://www.media4math.com/Definitions--RatiosProportionsPercents
• A student tutorial slide show on definitions on the topic of ratios, proportions, and percents: https://www.media4math.com/library/slideshow/student-tutorial-ratios-proportions-and-percents-definitions

Warm Up Activities

Choose from one or more activities.

Activity 1: Review of Percents

Review strategies for the following percent calculations:

• Percent of a number: https://www.media4math.com/library/64365/asset-preview 
• Find the whole given the percent: https://www.media4math.com/library/64368/asset-preview
• The percent one number is of another: https://www.media4math.com/library/75772/asset-preview

Activity 2: Review of Percent Concepts

Begin by reviewing the meaning of percent and how to convert between fractions, decimals, and percents.

Example: Convert 0.45 and \( \frac{3}{5} \) to a percent.

Solution:

• 0.45 as a percent: \( 0.45 \times 100 = 45\% \).
• \( \frac{3}{5} \) as a percent: \( \frac{3}{5} \times 100 = 60\% \).

Activity 3: Review of Percent Increase

Introduce percent increase as a method of finding how much a value has grown relative to its original amount.

Formula:

\[ \text{Percent Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \]

Example: The price of a jacket was \$40 but increased to \$50. What is the percent increase?

Solution:

1. Find the difference: \[ 50 - 40 = 10 \]
2. Divide by the original value: \[ \frac{10}{40} = 0.25 \]
3. Convert to a percent: \[ 0.25 \times 100 = 25\% \]

Thus, the price increased by 25%.

Activity 4: Review of Percent Decrease

Introduce percent decrease as a way to measure how much a value has been reduced relative to its original amount.

Formula:

\[ \text{Percent Decrease} = \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100 \]

Example: A phone originally cost \$600 but is now on sale for \$450. What is the percent decrease?

Solution:

1. Find the difference: \[ 600 - 450 = 150 \]
2. Divide by the original value: \[ \frac{150}{600} = 0.25 \]
3. Convert to a percent: \[ 0.25 \times 100 = 25\% \]

Thus, the phone price decreased by 25%.

Teach

Introduction

Understanding percent problems is essential for solving real-world financial and mathematical problems, such as calculating discounts, taxes, interest, and commission. In this lesson, students will explore different methods for solving percent problems using proportions and equations. They will also learn how to apply percent increase and decrease formulas to analyze changes in value over time.

Key Concepts Covered

• Using proportions to solve percent problems.
• Applying the percent equation: \( \text{Part} = \text{Percent} \times \text{Whole} \).
• Determining percent increase and decrease in real-world contexts.
• Solving multi-step percent problems, including discounts, tax, and markup.
• Understanding simple interest and how it applies to savings and loans.

Using Proportions to Solve Percent Problems

One way to solve percent problems is by setting up a proportion using the percent proportion formula:

\[ \frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100} \]

Example: What is 30% of 250?

Solution:

1. Set up the proportion: \[ \frac{x}{250} = \frac{30}{100} \]
2. Use cross-multiplication: \[ x \times 100 = 250 \times 30 \]
3. Solve for \( x \): \[ x = \frac{250 \times 30}{100} = \frac{7500}{100} = 75 \]

Thus, 30% of 250 is 75.

Using the Percent Equation

Another method to solve percent problems is using the equation:

\[ \text{Part} = \text{Percent} \times \text{Whole} \]

Example: 18 is what percent of 60?

Solution:

1. Set up the equation: \[ 18 = p \times 60 \]
2. Divide both sides by 60 to solve for \( p \): \[ p = \frac{18}{60} = 0.3 \]
3. Convert to a percent: \[ 0.3 \times 100 = 30\% \]

Thus, 18 is 30% of 60.

Percent Increase and Decrease

Students should also understand how to calculate percent increase and decrease using the following formulas:

• Percent Increase: \[ \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \]
• Percent Decrease: \[ \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100 \]

Example: The price of a phone increased from \$500 to \$650. What is the percent increase?

Solution:

1. Find the difference: \[ 650 - 500 = 150 \]
2. Divide by the original price: \[ \frac{150}{500} = 0.3 \]
3. Convert to a percent: \[ 0.3 \times 100 = 30\% \]

Thus, the phone price increased by 30%.

Example: A TV originally cost \$1,200 but is now on sale for \$900. What is the percent decrease?

Solution:

1. Find the difference: \[ 1200 - 900 = 300 \]
2. Divide by the original price: \[ \frac{300}{1200} = 0.25 \]
3. Convert to a percent: \[ 0.25 \times 100 = 25\% \]

Thus, the TV price decreased by 25%.

Multi-Step Percent Problems

Many percent problems involve multiple steps, such as calculating a discount followed by tax.

Example: A jacket originally costs \$80 and is on sale for 25% off. After the discount, a 7% sales tax is applied. What is the final price?

Solution:

1. Find the discount amount: \[ 80 \times 0.25 = 20 \]
2. Subtract the discount from the original price: \[ 80 - 20 = 60 \]
3. Calculate the sales tax: \[ 60 \times 0.07 = 4.20 \]
4. Add the tax to the discounted price: \[ 60 + 4.20 = 64.20 \]

Thus, the final price of the jacket is \$64.20.

Simple Interest

Students will also learn how to calculate simple interest using the formula:

\[ I = P \times r \times t \]

where:

• \( I \) = Interest earned
• \( P \) = Principal (initial amount)
• \( r \) = Interest rate (as a decimal)
• \( t \) = Time in years

Example: If you deposit \$1,500 in a savings account that earns 5% interest per year, how much interest will you earn after 3 years?

Solution:

1. Convert the interest rate to a decimal: \[ 5\% = 0.05 \]
2. Use the formula: \[ I = 1500 \times 0.05 \times 3 \]
3. Calculate: \[ I = 225 \]

Thus, the total interest earned after 3 years is \$225.

Definitions 

• Percent: A ratio that compares a number to 100
• Markup: An increase in the price of a product
• Markdown: A decrease in the price of a product
• Commission: A fee paid to an agent or employee for conducting a transaction
• Interest: Money paid regularly at a particular rate for the use of borrowed money
• Irrational number: A number that cannot be expressed as a simple fraction
• Square root: A value that, when multiplied by itself, gives the number
• Cube root: A value that, when multiplied by itself twice, gives the number
• Rational approximation: An estimate of an irrational number using a ratio of integers
• Scale model: A representation of an object that is larger or smaller than the actual size

You can also use this slide show of definitions, which include examples of the relevant term:

https://www.media4math.com/library/slideshow/definitions-solving-percent-problems

Instruction 

Demonstrate how to set up proportions to solve problems. Use this slide show to review examples of solving different proportions:

https://www.media4math.com/library/slideshow/math-examples-solving-proportions-algebraically

Use this slide show to give an overview of percents:

https://www.media4math.com/library/slideshow/overview-percents

 Next, review these examples.

Example 1: Markup Problem

A store buys a jacket for \$80 and wants to mark it up by 45%. What should the selling price be?

Solution:

The selling price should be \$116.

Example 2: Commission Problem

A real estate agent earns a 6% commission on home sales. If they sell a house for \$250,000, how much commission will they earn?

Solution:

The agent will earn \$15,000 in commission.

Example 3: Approximating Irrational Numbers

Approximate √20 to the nearest tenth and express it as a ratio.

Solution:

Example 4: Scale Model Problem

An architect is creating a scale model of a building. The actual building is 45 meters tall, and in the model, it is 15 centimeters tall. If a window on the model is 2 centimeters tall, how tall is the actual window?

Solution:

The actual window is 6 meters tall.

Example 5: Carbon Dating

Use this slide show to introduce an application of proportional reasoning in the context of carbon dating:

https://www.media4math.com/library/slideshow/applications-proportional-reasoning-carbon-dating

This table summarizes the data:

Make a note of the changing ratios. With each subsequent ratio, the number in scientific notation is reduced by 50% and the age of the artefact is an additional 5730 years old.

Review

Lesson Summary

In this lesson, students explored various strategies for solving percent problems, including the use of proportions, equations, and multi-step calculations. They learned how to determine percent increase and decrease, apply percent-based calculations to real-world scenarios, and solve financial problems such as discounts, tax, commission, and simple interest. By mastering these skills, students developed a deeper understanding of proportional reasoning and its applications in everyday life.

Key takeaways from this lesson include:

• Using Proportions: The percent proportion formula \( \frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100} \) helps solve problems involving missing values.
• Using the Percent Equation: The formula \( \text{Part} = \text{Percent} \times \text{Whole} \) provides an alternative approach to solving percent problems.
• Percent Increase and Decrease: The formulas \[ \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \] and \[ \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100 \] help determine how values change over time.
• Multi-Step Percent Problems: Problems involving multiple percent calculations, such as discounts followed by tax, require breaking down the problem into sequential steps.
• Simple Interest: The formula \( I = P \times r \times t \) allows students to calculate interest earned or owed over time.

By reviewing these concepts, students are now prepared to apply percent-based problem-solving strategies in real-world financial and mathematical contexts.

Refer to the following videos to review key concepts:

• Calculating Tips and Commissions: https://www.media4math.com/library/1819/asset-preview
• Calculating Tax: https://www.media4math.com/library/1818/asset-preview
• Percent Increase: https://www.media4math.com/library/1815/asset-preview
• Percent Decrease: https://www.media4math.com/library/1816/asset-preview

Quiz 

Answer the following questions.

1. A store buys a television for \$400 and marks it up by 35%. What is the selling price?
   
    
2. A real estate agent earns a 5.5% commission on a house sale. How much will they earn if the house sells for \$280,000?
   
    
3. Approximate √18 to the nearest tenth and express your answer as a ratio.
   
    
4. An architect's scale model has a 1:150 ratio. If a door in the model is 3 cm tall, how tall is the actual door?
   
    
5. A shirt originally priced at \$50 is on sale for 20% off. What is the sale price?
   
    
6. If the population of a city increased from 50,000 to 57,500, what was the percent increase?
   
    
7. A car's value depreciates from \$25,000 to \$21,250 after one year. What is the percent decrease?
   
    
8. Estimate √8 by finding two perfect squares it falls between, then narrow it down to a range of tenths.
   
    
9. A salesperson earns an 8% commission. How much will they earn on a \$1500 sale?
   
    
10. In a scale model, 2 cm represents 5 m. How many centimeters would represent 12.5 m?
    
     
11. Approximate π to two decimal places and express your answer as a ratio.
    
     
12. Which is greater: √13 or 3.7? Justify your answer using rational approximations.
    
     

Answer Key

1. \$540
2. \$15,400
3. The square root is between 4 and 4.3. A reasonable estimate is 4.2. As a ratio, it's expressed as 42:10 or 21:5.
4. 4.5 m or 450 cm
5. \$40
6. 15%
7. -15%
8. 2.8 < √8 < 2.9 (between 2^2=4 and 3^2=9)
9. \$120
10. 5 cm
11. 314:100 or 157:50
12. √13 is greater. 3.6^2 = 12.96, 3.7^2 = 13.69, so 3.6 < √13 < 3.7