K-12 Math (US): 6-8 Number & Operations K-12 Math Percents Percent increasedecrease

Grade 6–8 Math Study Guide: Percent Increase & Decrease

1. The Driving Question

"If a $50 video game goes on sale for 20% off, why isn’t the new price just $30—and how do stores actually calculate discounts, markups, or even the ‘percent change’ in your test scores? When numbers go up or down, how do you describe that change as a single percentage, and why does it matter more than just saying ‘it got cheaper by $10’?"

2. The Core Idea — Built, Not Listed

Imagine you’re running a lemonade stand. On Monday, you sell 40 cups. On Tuesday, you sell 50 cups. You know sales went up by 10 cups—but that doesn’t tell you how big the jump was compared to Monday. Was 10 cups a huge deal or just a small bump? Percent increase answers that by asking: "What fraction of the original amount did the change represent?"

Here’s how it works: 1. Find the change: 50 cups (new) – 40 cups (original) = 10 cups.
2. Compare the change to the original: 10 ÷ 40 = 0.25.
3. Convert to a percent: 0.25 × 100 = 25% increase.

The same logic works for decreases. If you sold 30 cups on Wednesday, the change is 30 – 40 = –10 cups. –10 ÷ 40 = –0.25, or a 25% decrease. The negative sign tells you the number went down, but the percent is always written as positive (just say "decrease").

Key Vocabulary:
- Percent change: The amount of increase or decrease expressed as a percent of the original value.
Example: If your phone battery drops from 80% to 60%, the percent change is a 25% decrease (not 20%!).
Note (Grades 9–12): In calculus, percent change becomes "relative change," and the original value can be the starting or ending amount, depending on context (e.g., economics vs. physics).

• Markup: The percent increase a store adds to the cost of an item to set the selling price.
  Example: A store buys a hoodie for $20 and marks it up 50%. The selling price is $30 (not $25!).
  Note: In business, "markup" is based on cost, while "margin" is based on selling price—confusing but important later.

• Discount: The percent decrease applied to the original price of an item.
  Example: A $120 jacket is 30% off. The discount is $36, so the sale price is $84 (not $90!).
  Note: Some discounts are "stacked" (e.g., 20% off, then 10% off the reduced price), which doesn’t equal 30% off the original.

• Original value: The starting amount before any increase or decrease.
  Example: If a population grows from 200 to 250, the original value is 200, not 250.
  Note: In statistics, the "original value" can be ambiguous (e.g., pre-test vs. post-test scores), so always clarify.

3. Assessment Translation

How This Appears on State Tests (Grades 6–8):
- Multiple Choice: Questions often ask for the percent change or the new amount after a percent increase/decrease. Distractors typically: - Use the wrong original value (e.g., calculating 20% off $50 as 20% of $40).
- Forget to add/subtract the change (e.g., saying 20% off $50 is $10, not $40).
- Mix up increase and decrease (e.g., a 15% increase becomes a 15% decrease).
- Short Answer/Grid-In: You might see: - "A shirt originally costs $24. It’s on sale for 25% off. What is the sale price?" (Answer: $18) - "A town’s population grew from 8,000 to 8,400. What was the percent increase?" (Answer: 5%) - Evidence-Based Writing (Less Common): "Explain why a 50% increase followed by a 50% decrease doesn’t return you to the original amount. Use an example."

Proficient vs. Developing Responses:
| Proficient | Developing | |----------------|----------------| | Shows all steps: finds change, divides by original, converts to percent. | Skips steps or uses the wrong original value. | | Labels answers (e.g., "$40" or "10% increase"). | Forgets units or labels. | | Explains reasoning in words (e.g., "The change was 20, and 20 ÷ 100 = 0.2, so 20%"). | Just writes the answer without context. |

Model Proficient Response:
Prompt: A bike costs $150. It goes on sale for 30% off. What is the sale price? Response: 1. Find 30% of $150: 0.30 × 150 = $45.
2. Subtract the discount: $150 – $45 = $105.
OR 1. If it’s 30% off, you pay 70% of the price: 0.70 × 150 = $105.
Why this works: Both methods show understanding of the relationship between the percent and the original price.

4. Mistake Taxonomy

Mistake 1: Using the New Value as the Original
Prompt: A video game’s price dropped from $60 to $45. What was the percent decrease? Common Wrong Answer: 25% decrease.
Why It Loses Credit: The student calculated 15 ÷ 60 = 0.25 (correct) but then used the new price ($45) as the original value: 15 ÷ 45 = 0.33. The question asks for the change relative to the original price.
Correct Approach: 1. Change = $60 – $45 = $15.
2. Percent decrease = ($15 ÷ $60) × 100 = 25%.

Mistake 2: Forgetting to Add/Subtract the Change
Prompt: A store marks up a $25 item by 40%. What is the new price? Common Wrong Answer: $10.
Why It Loses Credit: The student found 40% of $25 ($10) but forgot to add it to the original price. The question asks for the new price, not just the markup.
Correct Approach: 1. Markup = 0.40 × $25 = $10.
2. New price = $25 + $10 = $35.

Mistake 3: Misreading "Percent Of" vs. "Percent Change"
Prompt: A test score increased from 70 to 84. What was the percent increase? Common Wrong Answer: 14%.
Why It Loses Credit: The student calculated 84 – 70 = 14 and wrote "14%," confusing the amount of change with the percent of change. The question asks for the percent relative to the original score.
Correct Approach: 1. Change = 84 – 70 = 14.
2. Percent increase = (14 ÷ 70) × 100 = 20%.

5. Connection Layer

1. Within Math: Percent increase/decrease → Exponential growth/decay
   Why it matters: Percent change is the foundation for understanding how quantities grow or shrink repeatedly over time (e.g., compound interest, population growth). A 5% annual increase isn’t just adding 5% of the original amount each year—it’s 5% of the new amount, leading to faster growth.

2. Across Subjects: Percent change → Science (error analysis, concentration changes)
   Why it matters: In chemistry, if a solution’s concentration changes from 10% to 15%, the percent increase is 50% (not 5%!). This helps students interpret lab results, like how much a measurement deviated from the expected value.

3. Outside School: Percent change → Sports statistics (player performance, team records)
   Why it matters: When a basketball player’s free-throw percentage improves from 70% to 77%, that’s a 10% increase (not 7%!). Sports analysts use percent change to compare players across eras or adjust for rule changes—now you’ll notice it in every post-game highlight.

6. The Stretch Question

"If a store raises the price of a $100 jacket by 20% and then later discounts it by 20%, is the final price $100? If not, why—and what percent change would actually return it to $100?"

Pointer Toward the Answer: - The first change: $100 + (20% of $100) = $120.
- The second change: $120 – (20% of $120) = $96.
- The final price is not $100 because the 20% discount is applied to a larger amount ($120) than the 20% markup ($100). To return to $100, you’d need a 16.67% discount (because $20 ÷ $120 ≈ 0.1667). This shows why "undoing" a percent change isn’t as simple as reversing the percent!