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Teacher Guide: Introduction to Percentages

Learn what percentages are and how they relate to fractions and decimals.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Percentages. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define percent as a ratio out of 100
  • Convert between percents, fractions, and decimals
  • Calculate a percent of a given number
  • Apply percentages to real-world contexts like discounts and test scores
Prerequisites
  • • Understanding of fractions and equivalent fractions
  • • Ability to convert between fractions and decimals
  • • Basic multiplication and division skills
Discussion Starters
  • 1. Where have you seen percentages used in your daily life this week?
  • 2. Why do you think stores advertise '50% off' instead of 'half off'?
  • 3. If you scored 90% on a test with 10 questions versus 90% on a test with 100 questions, did you get the same number correct?
  • 4. Can something be more than 100%? When might that happen?
Common Misconceptions

Thinking larger percent always means larger amount

Remediation: Compare: 50% of 20 versus 10% of 200. Calculate both to show that the whole matters.

Confusing the direction of decimal conversion

Remediation: Use memory device: 'Percent to Decimal - Point moves Left' (P to D - Left). Practice with familiar benchmarks like 50% = 0.50.

Differentiation Ideas

For Struggling Students:

  • • Focus on benchmark percents: 10%, 25%, 50%, 75%, 100%
  • • Use visual models like hundred grids
  • • Connect to money: 25% = 25 cents per dollar

For On-Level Students:

  • • Convert between all three forms: fractions, decimals, percents
  • • Solve percent problems involving discounts and tips
  • • Compare percentages in different contexts

For Advanced Students:

  • • Calculate percent increase and decrease
  • • Work with percents greater than 100%
  • • Solve multi-step percent problems (tax + tip)
Standards Alignment
  • 6.RP.A.3c (CCSS.MATH.CONTENT.6.RP.A.3.C)

    Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent

  • 7.RP.A.3 (CCSS.MATH.CONTENT.7.RP.A.3)

    Use proportional relationships to solve multistep ratio and percent problems

Lesson Resources
  • visualInteractive Percent Bar

    Students drag to see percent as parts of 100

  • activityDiscount Calculator Game

    Calculate sale prices in a virtual shopping scenario

  • worksheetPercent Conversions Practice

    Convert between fractions, decimals, and percents

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A percentage (or percent) means "per hundred" or "out of 100." The symbol % represents percent.
50%=50100=0.5050\% = \frac{50}{100} = 0.50
When we say "50%", we mean 50 out of every 100, or half of something.
Key relationships:
  • 1%=1100=0.011\% = \frac{1}{100} = 0.01
  • 25%=25100=14=0.2525\% = \frac{25}{100} = \frac{1}{4} = 0.25
  • 50%=50100=12=0.5050\% = \frac{50}{100} = \frac{1}{2} = 0.50
  • 100%=100100=1100\% = \frac{100}{100} = 1 (the whole thing)

Worked Examples

Write 75%75\% as a fraction in lowest terms.
1

Write as a fraction over 100

75%=7510075\% = \frac{75}{100}
2

Find the GCD of 75 and 100

Factors of 75: 1, 3, 5, 15, 25, 75\nFactors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100\nGCD = 25
3

Divide both by GCD

75÷25100÷25=34\frac{75 \div 25}{100 \div 25} = \frac{3}{4}

Common Mistakes

Confusing 0.50.5 with 5%5\%

Why it's wrong: When converting, students sometimes forget to multiply by 100. 0.5=50%0.5 = 50\%, not 5%5\%.

Correct: To convert decimal to percent, multiply by 100: 0.5×100=50%0.5 \times 100 = 50\%

Forgetting that 100%100\% means the whole thing

Why it's wrong: Students sometimes think percentages can't reach 100 or go beyond.

Correct: 100%=1100\% = 1 (all of it). You can even have more than 100%100\%: 150%150\% means 1.5 times the original.

Moving the decimal the wrong direction

Why it's wrong: Confusion about which way to move when converting between decimals and percents.

Correct: Percent to decimal: divide by 100 (move left 2 places). Decimal to percent: multiply by 100 (move right 2 places).

Why It Matters

Percentages are everywhere in daily life:
  • Shopping: Sales and discounts ("25% off")
  • School: Test scores ("85% correct")
  • Sports: Statistics ("Made 70% of free throws")
  • News: Statistics and polls ("60% of voters agree")
  • Health: Nutrition labels ("15% of daily calcium")
Understanding percentages helps you make better decisions about money, health, and information!

Real World Applications

Shopping Discounts

Stores use percentages to advertise sales and help customers calculate savings.

Example:

A 25% discount on a 40 dollar item saves you 10 dollars.

1Try It Yourself

You're buying a video game that normally costs 80 dollars. The store has a 15% off sale.

How much will you pay after the discount?

Step 1: Write the mathematical expression

First find 15% of 80, then subtract:

Test Scores

Teachers use percentages to show how well you performed on tests and assignments.

Example:

Getting 17 out of 20 questions correct means you scored 1720=85%\frac{17}{20} = 85\%.

2Try It Yourself

You answered 42 questions correctly out of 50 on a math test.

What is your percentage score?

Step 1: Write the mathematical expression

Divide correct by total, then multiply by 100:

Nutrition Labels

Food packages show the percent of daily nutrients in each serving.

Example:

If a snack provides 20% of your daily fiber, eating 5 servings would give you 100% of your daily fiber needs.

3Try It Yourself

A glass of milk provides 30% of your daily calcium. You drink 2 glasses.

What percentage of your daily calcium did you get?

Step 1: Write the mathematical expression

Multiply the percent by the number of servings:

Key Takeaways

  • 1Percent means "per hundred" - the symbol % represents this relationship
  • 2To convert percent to decimal: divide by 100 (25%=0.2525\% = 0.25)
  • 3To convert decimal to percent: multiply by 100 (0.75=75%0.75 = 75\%)
  • 4To convert percent to fraction: write over 100 and simplify (50%=1250\% = \frac{1}{2})
  • 5To find a percent of a number: convert to decimal and multiply (20%20\% of 50=0.20×50=1050 = 0.20 \times 50 = 10)

Frequently Asked Questions

Can a percentage be more than 100%?

Yes! 150%150\% means 1.5 times the original amount. For example, if a price increased by 150%150\%, it's now 2.5 times what it was (original + 1.5 times original).

What's the difference between percent and percentage point?

If an interest rate goes from 5%5\% to 8%8\%, it increased by 3 percentage points. But it increased by 60%60\% (because 35=0.6=60%\frac{3}{5} = 0.6 = 60\%).

Why do we use percentages instead of fractions?

Percentages make comparisons easier. It's simpler to compare 75%75\% vs 80%80\% than to compare 34\frac{3}{4} vs 45\frac{4}{5}. Having a common denominator (100) helps!

Glossary

Percent
A ratio expressed as a fraction of 100, using the symbol %
Percentage
Another word for percent; a proportion expressed out of 100
Percent of a number
A fraction of a quantity found by converting the percent to a decimal and multiplying
Discount
A reduction in price, often expressed as a percentage off the original price

Formula Card

Percent to Decimal

Decimal=Percent100\text{Decimal} = \frac{\text{Percent}}{100}

Divide the percent by 100 to convert it to a decimal

Decimal to Percent

Percent=Decimal×100\text{Percent} = \text{Decimal} \times 100

Multiply the decimal by 100 to convert it to a percent

Percent of a Number

Part=Percent100×Whole\text{Part} = \frac{\text{Percent}}{100} \times \text{Whole}

Convert the percent to a decimal, then multiply by the whole

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