Lesson Plan: Solving Percent Problems and Using Proportional Reasoning

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

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
  • 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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