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 Lesson Plan: Percents and Measurement  

 

Lesson Summary

This lesson introduces sixth-grade students to the concept of percents, emphasizing their relationship to ratios and their applications in real-world scenarios. Students will learn to interpret percents as parts per hundred, convert between percents, fractions, and decimals, and solve practical problems involving percents, such as discounts, interest rates, and data interpretation.

Lesson Objectives

  • Understand percents as ratios
  • Solve problems involving percents
  • Use ratio reasoning to convert measurement units
  • Apply percent and measurement concepts to real-world scenarios in science, art, and business

Common Core Standards

  • 6.RP.A.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
  • 6.RP.A.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Prerequisite Skills

  • Understanding of ratios and unit rates
  • Basic fraction and decimal knowledge

Key Vocabulary

  • Percent: A ratio that represents a number out of 100. For example, 45% means 45 out of 100. The term comes from the Latin "per centum," meaning "by the hundred."
  • Percentage: The result obtained by multiplying a quantity by a percent. It refers to a portion of 100 and is often used to compare ratios or proportions in various contexts. 
  • Unit Conversion: The process of changing a measurement from one unit to another. In the context of percents, this often involves converting between fractions, decimals, and percents to facilitate calculations and comparisons. 
  • Discount: A reduction from the usual cost of something, often expressed as a percent. For example, a 20% discount on a $50 item reduces the price by $10. 
  • Commission: Earnings based on the amount of sales made, typically calculated as a percent of total sales. For instance, a salesperson earning a 5% commission on $1,000 in sales would receive $50. 
  • Interest: The cost of borrowing money or the return on investment, usually expressed as a percent per period of time. Simple interest is calculated using the formula I = PRT, where P is the principal amount, R is the rate, and T is time. 
  • Markup: The amount added to the cost price of goods to cover overhead and profit, often expressed as a percent of the cost price. For example, a store might mark up a product by 25% before selling it. 
  • Percent Increase: The percent by which an original amount is increased. It is calculated by finding the difference between the new and original amounts, dividing by the original amount, and multiplying by 100. 
  • Percent Decrease: The percent by which an original amount is decreased. It is calculated by finding the difference between the original and new amounts, dividing by the original amount, and multiplying by 100. 

Multimedia Resources

 

 

Warm Up Activities

Choose from one or more of these activities.

Activity 1: Brief Review of Equivalent Fractions

Before introducing percents, students will review equivalent fractions, focusing on writing fractions with denominators that are powers of 10. This will help them transition to converting fractions into decimals and percents.

Instructions:

  1. Write the following fractions on the board and ask students to find an equivalent fraction with a denominator of 10, 100, or 1,000:
    • \( \frac{1}{2} \)
    • \( \frac{3}{5} \)
    • \( \frac{7}{20} \)
    • \( \frac{9}{25} \)
  2. Guide students through the process of scaling up the denominator:
    • \( \frac{1}{2} = \frac{50}{100} \)
    • \( \frac{3}{5} = \frac{60}{100} \)
    • \( \frac{7}{20} = \frac{35}{100} \)
    • \( \frac{9}{25} = \frac{36}{100} \)
  3. Discuss why using a denominator of 100 makes it easy to convert fractions to percents.

Activity 2: Brief Review of Benchmark Fractions

Students will review common benchmark fractions that have simple percent equivalents. This will help them estimate and recognize percents more easily.

Instructions:

  1. Write these fractions on the board and ask students to recall their percent equivalents:
    • \( \frac{1}{10} \) → 10%
    • \( \frac{1}{4} \) → 25%
    • \( \frac{1}{2} \) → 50%
    • \( \frac{3}{4} \) → 75%
  2. Have students place these benchmarks on a number line from 0% to 100%.
  3. Ask guiding questions:
    • Which fraction represents the halfway point?
    • Which fraction represents a quarter of a total amount?
    • How would knowing these benchmarks help in estimating percents?
  4. Discuss how these benchmark fractions are commonly used in real-world scenarios, such as sales discounts and test scores.

Activity 3: Percent Visualizations

Here is a slide show that can be used to show percent visualizations:

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

 You can follow-up with this activity:

Display a large 10x10 grid on the board or using a digital projector. Each square represents 1% of the total area. Present the following percentages: 25%, 50%, 75%, 80%, and 120%.

For each percentage:

  1. Ask students to visualize and describe how many squares would be filled in the 10x10 grid.
  2. Fill in the squares on the grid to represent the percentage.
  3. Have students express the percentage as a fraction out of 100.

For 120%, extend the grid by adding two more columns, emphasizing that percentages can exceed 100%.

Discuss how the grid visually represents percentages and reinforces the concept of "per hundred."

 

 

Teach

Definitions

  • Percent: A ratio that compares a number to 100
  • Percentage: The conversion of fractions and decimals to percents.
  • Unit conversion: The process of changing a measurement from one unit to another

You can also review these definitions and others with this slide show:

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

Instruction

Percents. Use this slideshow to provide an overview of percents:

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

Demonstrate how to solve percent problems:

  1. Finding a percent of a number. Use this video to develop this concept. This video includes three detailed examples:
    https://www.media4math.com/library/1807/asset-preview
  2. Finding what percent one number is of another. Use this video, which also includes three detailed examples:
    https://www.media4math.com/library/1810/asset-preview
  3. Finding the whole when given a part and percent. Use this video:
    https://www.media4math.com/library/1808/asset-preview

Real-world applications in science. Here are real-world applications of percent problems.

  • Concentration in solutions: If a saline solution is 0.9% salt, how many grams of salt are in 500 mL of solution?
  • Genetic inheritance: If a trait has a 75% chance of being passed on, what's the probability of it appearing in 200 offspring?
  • Color mixing: If an artist needs to lighten a color by 20%, how much white paint should be added to 100 mL of the original color?
  • Scaling artwork: If a 24-inch painting needs to be reduced to 75% of its original size, what will the new dimensions be?

If time allows, introduce these applications:

  1. Discount Calculation:

    A store is offering a 25% discount on a $80 jacket. Calculate the discount amount and the final price of the jacket.

    Solution:

    • Discount amount: 25% of $80 = 0.25 × $80 = $20
    • Final price: $80 - $20 = $60

  2. Simple Interest:

    If you invest $1000 at a 6% annual interest rate, how much interest will you earn after 2 years?

    Solution:

    • Interest formula: I = P × r × t (where I = interest, P = principal, r = rate, t = time in years)
    • I = $1000 × 0.06 × 2 = $120

Discuss how percentages are crucial in business and finance for calculating discounts, interest rates, profit margins, and tax rates.

Measurement. Show how to use ratio reasoning for measurement unit conversion. This slide show provides a dozen examples of converting measurements using rates:

https://www.media4math.com/library/slideshow/math-examples-measurement-conversion

Example 1: Finding a Percent of a Number

Problem: A store is offering a 25% discount on a jacket that costs $80. How much will the customer save?

Solution:

  1. Convert 25% to a decimal: \[ 25\% = \frac{25}{100} = 0.25 \]
  2. Multiply by the original price: \[ 0.25 \times 80 = 20 \]
  3. Final Answer: The customer will save $20.

Example 2: Finding What Percent One Number is of Another

Problem: A student answered 18 out of 24 questions correctly on a test. What percent did they get correct?

Solution:

  1. Set up the ratio: \[ \frac{18}{24} \]
  2. Convert to a fraction with a denominator of 100 by setting up a proportion: \[ \frac{18}{24} = \frac{x}{100} \]
  3. Solve for \( x \) by cross-multiplying: \[ 18 \times 100 = 24x \] \[ 1800 = 24x \] \[ x = \frac{1800}{24} = 75 \]
  4. Final Answer: The student got a 75% on the test.

Example 3: Finding the Whole When Given a Part and a Percent

Problem: A restaurant bill includes a tip of $12, which is 20% of the total bill. What was the total bill before the tip?

Solution:

  1. Set up the equation: \[ 0.20 \times \text{Total Bill} = 12 \]
  2. Divide both sides by 0.20: \[ \text{Total Bill} = \frac{12}{0.20} = 60 \]
  3. Final Answer: The total bill before the tip was $60.

Example 4: Percent Increase

Problem: A laptop originally costs $800. Due to high demand, the price increases by 15%. What is the new price?

Solution:

  1. Calculate the increase: \[ 0.15 \times 800 = 120 \]
  2. Add the increase to the original price: \[ 800 + 120 = 920 \]
  3. Final Answer: The new price of the laptop is $920.

Example 5: Percent Decrease

Problem: A bicycle was originally priced at $500. It is now on sale for 20% off. What is the sale price?

Solution:

  1. Calculate the discount: \[ 0.20 \times 500 = 100 \]
  2. Subtract the discount from the original price: \[ 500 - 100 = 400 \]
  3. Final Answer: The sale price of the bicycle is $400.

Example 6: Simple Interest

Problem: A person deposits $1,500 in a savings account that earns 4% simple interest per year. How much interest will be earned after 3 years?

Solution:

  1. Use the simple interest formula: \[ I = P \times r \times t \] where:
    • \( I \) = Interest earned
    • \( P \) = Principal (initial deposit)
    • \( r \) = Interest rate (as a decimal)
    • \( t \) = Time in years
  2. Substitute the values: \[ I = 1500 \times 0.04 \times 3 \]
  3. Calculate: \[ I = 1500 \times 0.12 = 180 \]
  4. Final Answer: The total interest earned after 3 years is $180.

 

 

Review

Lesson Summary

In this lesson, students explored the concept of percents and their relationship to fractions and decimals. They learned how to convert between these representations and apply percents to real-world situations. Key takeaways include:

  • Understanding Percents: Percents are ratios that compare a number to 100.
  • Solving Percent Problems: There are three main types of percent problems:
    • Finding a percent of a number.
    • Finding what percent one number is of another.
    • Finding the whole when given a part and a percent.
  • Real-World Applications: Percent concepts are commonly used in calculating discounts, tax, interest, percent increase/decrease, and other financial contexts.

Key Vocabulary

  • Percent: A ratio that compares a number to 100, written with the "%" symbol.
  • Percentage: The actual amount obtained when applying a percent to a quantity.
  • Proportion: An equation that states two ratios are equal, often used in percent problems.
  • Percent Increase: A measure of how much a quantity has grown, calculated as a percentage of the original amount.
  • Percent Decrease: A measure of how much a quantity has been reduced, calculated as a percentage of the original amount.
  • Simple Interest: Interest calculated only on the principal amount using the formula \( I = P \times r \times t \).

Multimedia Resources

Review this video to see visual models of percents:

https://www.media4math.com/library/1811/asset-preview

Review properties of percents using this slide show:

https://www.media4math.com/library/slideshow/fractions-decimals-and-percents

Use this slide show to demonstrate various conversion formulas for converting units:

https://www.media4math.com/library/slideshow/conversion-formulas 

Additional Examples

Example 1: Finding a Percent of a Number

Problem: A laptop is on sale for 30% off its original price of $1,200. How much is the discount?

Solution:

  1. Convert 30% to a decimal: \[ 30\% = \frac{30}{100} = 0.30 \]
  2. Multiply by the original price: \[ 0.30 \times 1200 = 360 \]
  3. Final Answer: The discount is $360.

Example 2: Finding What Percent One Number is of Another

Problem: Out of 250 students in a school, 75 participate in a sports club. What percent of students participate?

Solution:

  1. Set up the proportion: \[ \frac{75}{250} = \frac{x}{100} \]
  2. Cross-multiply: \[ 75 \times 100 = 250x \] \[ 7500 = 250x \]
  3. Divide by 250: \[ x = \frac{7500}{250} = 30 \]
  4. Final Answer: 30% of the students participate in the sports club.

Example 3: Finding the Whole When Given a Part and a Percent

Problem: A worker receives a 20% commission on sales and earned $500 in commission. What was the total sales amount?

Solution:

  1. Set up the equation: \[ 0.20 \times \text{Total Sales} = 500 \]
  2. Divide both sides by 0.20: \[ \text{Total Sales} = \frac{500}{0.20} = 2500 \]
  3. Final Answer: The total sales amount was $2,500.

Example 4: Real-World Application – Tax Calculation

Problem: A customer buys a television for $850. If the sales tax is 7.5%, how much tax does the customer pay?

Solution:

  1. Convert 7.5% to a decimal: \[ 7.5\% = \frac{7.5}{100} = 0.075 \]
  2. Multiply by the purchase price: \[ 0.075 \times 850 = 63.75 \]
  3. Final Answer: The customer pays $63.75 in tax.

Example 5: Real-World Application – Percent Decrease

Problem: A pair of shoes originally cost $90 but is now on sale for 25% off. What is the sale price?

Solution:

  1. Find the discount amount: \[ 0.25 \times 90 = 22.50 \]
  2. Subtract the discount from the original price: \[ 90 - 22.50 = 67.50 \]
  3. Final Answer: The sale price is $67.50.

Example 6: Real-World Application – Simple Interest

Problem: A person deposits $2,000 in a bank account that earns 5% simple interest per year. How much total interest will be earned after 4 years?

Solution:

  1. Use the simple interest formula: \[ I = P \times r \times t \] where:
    • \( I \) = Interest earned
    • \( P \) = Principal (initial deposit)
    • \( r \) = Interest rate (as a decimal)
    • \( t \) = Time in years
  2. Substitute the values: \[ I = 2000 \times 0.05 \times 4 \]
  3. Calculate: \[ I = 2000 \times 0.20 = 400 \]
  4. Final Answer: The total interest earned after 4 years is $400.

 

 

Quiz

Answer the following questions.

 

  1. What is 40% of 80?

     
  2. 24 is what percent of 60?

     
  3. 18 is 75% of what number?

     
  4. Convert 3.5 meters to centimeters.

     
  5. In a 2-liter solution, 5% is alcohol. How many milliliters of alcohol are present?

     
  6. If 30% of a number is 21, what is the number?

     
  7. An artist wants to increase the size of a 15-inch sculpture by 120%. What will be the new height?

     
  8. What percent of 200 is 150?

     
  9. If a gene has a 60% chance of expression, how many individuals in a population of 500 would be expected to show the trait?

     
  10. A paint mixture requires 3 parts red to 5 parts blue. What percentage of the mixture is red?

     
  11. A store is offering a 15% discount on a $120 item. What is the final price after the discount?

     
  12. If you invest $2000 at an annual interest rate of 4.5%, how much interest will you earn after 1 year?

     

Answers:

  1. 32
  2. 40%
  3. 24
  4. 350 cm
  5. 100 mL
  6. 70
  7. 33 inches
  8. 75%
  9. 300 individuals
  10. 37.5%
  11. $102
  12. $90