# Lesson Plan: Solving Percent Problems and Using Proportional Reasoning

This lesson plan is designed for two 50-minute class periods.

## Lesson Objectives

- Solve various types of percent problems
- Use proportional relationships in percent calculations
- Apply percent concepts to financial situations
- Compare simple and compound interest

## TEKS Standards

- 8.4D: Use proportional relationships to solve multi-step ratio and percent problems.
- 8.5A: Represent linear proportional situations with tables, graphs, and equations.
- 8.5E: Solve problems using direct variation.
- 8.12D: Calculate and compare simple interest and compound interest earnings.

## Prerequisite Skills

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

## Key Vocabulary

- Percent
- Markup
- Markdown
- Commission
- Interest
- Irrational number
- Square root
- Cube root
- Rational approximation
- Scale model

## Warm-up Activity (10 minutes)

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__

## Teach (70 minutes)

### 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:

1. Set up the proportion: | 45/100 = x/80 |

2. Cross multiply: | 45 * 80 = 100x |

3. Solve for x: | x = (45 * 80) / 100 = 36 |

4. The markup amount is | \$36 |

5. Add the markup to the original price: | \$80 + \$36 = \$116 |

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:

1. Set up the proportion: | 6/100 = x/250,000 |

2. Cross multiply: | 6 * 250,000 = 100x |

3. Solve for x: | x = (6 * 250,000) / 100 = 15,000 |

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:

1. Find the perfect squares on either side of 20: | 16 (4^{2}) and 25 (5^{2}) |

2. √20 is between | 4 and 5 |

3. Use a calculator to find | √20 ≈ 4.472135... |

4. Round to the nearest tenth: | √20 ≈ 4.5 |

5. Express as a ratio: | 45:10 or 9:2 |

### 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:

1. Set up the proportion: | 15 cm / 45 m = 2 cm / x m |

2. Convert 45 m to cm: | 45 m = 4500 cm |

3. Rewrite the proportion: | 15 / 4500 = 2 / x |

4. Cross multiply: | 15x = 2 * 4500 |

5. Solve for x: | x = (2 * 4500) / 15 = 600 |

6. Convert 600 cm to meters: | 600 cm = 6 m |

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:

C-14 | C-12 | Age |

1 | 1.00 • 10^{12} | -- |

1 | 5.00 • 10^{11} | 5730 |

1 | 2.50 • 10^{11} | 11,460 |

1 | 1.25 • 10^{11} | 17,190 |

1 | 6.25 • 10^{10} | 22,920 |

1 | 3.13 • 10^{10} | 28,650 |

1 | 1.56 • 10^{10} | 34,380 |

1 | 7.81 • 10^{9} | 40,110 |

1 | 3.91 • 10^{9} | 51,570 |

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 (30 minutes)

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__

## Assess (10 minutes)

Administer this 12-question quiz.

## Quiz Questions

- A store buys a television for $400 and marks it up by 35%. What is the selling price?

- 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?

- Approximate √18 to the nearest tenth and express your answer as a ratio.

- 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?

- A shirt originally priced at $50 is on sale for 20% off. What is the sale price?

- If the population of a city increased from 50,000 to 57,500, what was the percent increase?

- A car's value depreciates from $25,000 to $21,250 after one year. What is the percent decrease?

- Estimate √8 by finding two perfect squares it falls between, then narrow it down to a range of tenths.

- A salesperson earns an 8% commission. How much will they earn on a $1500 sale?

- In a scale model, 2 cm represents 5 m. How many centimeters would represent 12.5 m?

- Approximate π to two decimal places and express your answer as a ratio.

- Which is greater: √13 or 3.7? Justify your answer using rational approximations.

## Answer Key

- $540
- $15,400
- 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.
- 4.5 m or 450 cm
- $40
- 15%
- -15%
- 2.8 < √8 < 2.9 (between 2^2=4 and 3^2=9)
- $120
- 5 cm
- 314:100 or 157:50
- √13 is greater. 3.6^2 = 12.96, 3.7^2 = 13.69, so 3.6 < √13 < 3.7

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