Lesson Plan: Solving Percent Problems and Using Proportional Reasoning

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

Lesson Objectives

  • Convert between fractions, decimals, and percents
  • Solve problems involving percents, including percent increase and decrease
  • Apply percent calculations to real-world situations, such as discounts and interest

Florida BEST Standards

  • MA.8.NSO.1.1: Extend understanding of rational numbers to define irrational numbers within the real number system.
  • MA.8.AR.2.1: Solve multi-step linear equations in one variable.
  • MA.8.AR.3.3: Given a real-world situation, determine if the relationship between two quantities is linear. Write an equation in slope-intercept form.

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:

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 (42) and 25 (52)
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-14C-12Age
11.00 • 1012--
15.00 • 10115730
12.50 • 101111,460
11.25 • 101117,190
16.25 • 101022,920
13.13 • 101028,650
11.56 • 101034,380
17.81 • 10940,110
13.91 • 10951,570

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:

Assess (10 minutes)

Administer this 12-question quiz.

Quiz 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

 

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