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Growing and Shrinking Quantities

Percent change describes how a quantity increases or decreases compared to its original amount. A percent increase means the new amount is larger than the original, while a percent decrease means the new amount is smaller. In both cases, the original amount represents 100%, and the change is measured in relation to that starting value.

In this topic, we use bar models to make percent change visible. For a percent increase, the bar begins with 100% and then grows by the additional percent. For example, a 20% increase means the new amount is 120% of the original. For a percent decrease, the bar begins with 100% and then shrinks by the percent removed. For example, a 30% decrease means the new amount is 70% of the original.

We will also compare multiple discounts. Two discounts applied one after the other are not always the same as adding the percentages together. For example, taking 20% off and then 10% off is not the same as taking 30% off, because the second discount is applied to the already-reduced price. As future teachers, it is important to help students understand percent change as a relationship between the original amount, the change, and the new amount.

 

Student Learning Goals

By the end of this topic, students should be able to:

  • Define percent change.

  • Identify the original amount, amount of change, and new amount.

  • Distinguish between percent increase and percent decrease.

  • Use bar models to represent percent change problems.

  • Solve percent increase and percent decrease problems.

  • Explain why the original amount represents 100%.

  • Compare multiple discounts and explain why order and base amount matter.

  • Interpret percent change in real-world contexts.

Key Vocabulary

  • Percent Change - The percent increase or decrease from an original amount.

  • Original Amount - The starting amount, usually represented by 100%.

  • New Amount - The amount after the increase or decrease.

  • Percent Increase - A change that makes the original amount larger.

  • Percent Decrease - A change that makes the original amount smaller.

  • Discount - A percent decrease from the original price.

  • Markup - A percent increase from the original price.

  • Bar Model - A visual model used to show the original amount, change, and new amount.

Teacher Connection

Percent change is a powerful topic because it connects math to shopping, taxes, tips, population growth, grades, salaries, science, and data interpretation. Students often want to apply quick rules, but percent change depends heavily on the original amount.

Bar models help students see the relationship clearly. They show that an increase builds on 100%, while a decrease removes part of 100%. They also help students understand why repeated percent changes are based on changing amounts, not one fixed number.

Helpful teacher questions include:

  • What is the original amount?

  • What represents 100%?

  • Is the quantity increasing or decreasing?

  • What percent of the original remains?

  • What is the new percent total?

  • Is this second discount based on the original price or the reduced price?

  • Does the answer make sense compared to the starting amount?

Quick Reflection Question

Why is a 20% discount followed by a 10% discount not the same as a single 30% discount? How could a bar model help students see the difference?