Mathematics · Year 9 · Financial Mathematics and Proportion · Term 4

Percentages and Fractions Review

Students will review converting between percentages, fractions, and decimals, and calculating percentages of amounts.

• ACARA Content Descriptions
• AC9M9N04

About This Topic

Percentage change and profit are essential financial literacy skills in the Year 9 curriculum. Students learn to calculate markups, discounts, and the resulting profit or loss in business transactions. A key challenge at this level is 'reverse percentages', finding the original price after a change has occurred. These skills are vital for making informed consumer decisions and understanding the economic world.

In the Australian context, this unit can be linked to GST (Goods and Services Tax) and local business scenarios. ACARA emphasises the application of these concepts to real-life financial situations. This topic comes alive when students can run a 'mock market' or simulate business decisions, where they must calculate margins to ensure their 'business' stays afloat. Students grasp this concept faster through collaborative problem-solving where they have to 'justify' their pricing strategies to their peers.

Key Questions

1. Explain the relationship between percentages, fractions, and decimals.
2. Differentiate between finding a percentage of an amount and finding an amount as a percentage of another.
3. Construct a real-world scenario where converting between these forms is essential.

Learning Objectives

• Convert between percentages, fractions, and decimals with 90% accuracy.
• Calculate the percentage of a given whole number or decimal amount.
• Explain the multiplicative relationship between percentages, fractions, and decimals.
• Identify the percentage that one number represents of another number.
• Construct a word problem requiring conversion between percentages, fractions, and decimals to solve.

Before You Start

Basic Fractions and Decimals

Why: Students need a solid understanding of what fractions and decimals represent before they can convert between them and percentages.

Multiplication and Division of Whole Numbers

Why: Calculating percentages of amounts relies on multiplying the amount by the decimal or fractional equivalent of the percentage.

Key Vocabulary

Percentage	A ratio or fraction out of 100, represented by the symbol '%'. It signifies a part of a whole.
Fraction	A number that represents a part of a whole. It is written as one number over another, separated by a line.
Decimal	A number expressed in the scale of tens. It uses a decimal point to separate whole numbers from fractional parts.
Percentage of an amount	Calculating a specific portion of a total value, expressed as a percentage. For example, finding 25% of $200.
Amount as a percentage	Determining what percentage one value is of another. For example, finding what percentage $50 is of $200.

Watch Out for These Misconceptions

Common Misconception

Thinking that a 10% increase followed by a 10% decrease returns you to the original price.

What to Teach Instead

This is a very common error. Because the second percentage is calculated on a new, larger amount, the decrease is actually larger than the increase. Using a simple $100 example in a peer-discussion task helps students 'see' the missing dollar.

Common Misconception

Confusing 'profit' with 'revenue'.

What to Teach Instead

Students often think the total money taken in is the profit. Using a 'money bucket' analogy in a simulation, where they have to pay back the 'cost of goods' first, helps them understand that profit is only what is left over after all costs are covered.

Active Learning Ideas

Simulation Game: The Classroom Market Stall

Students are given 'wholesale' prices for items and must decide on a percentage markup to cover 'rent' and make a profit. They then have to react to a 'flash sale' (percentage discount) and calculate if they are still making a profit. This makes the maths of business tangible.

60 min·Small Groups

Think-Pair-Share: The 10% Trap

Ask students: 'If a $100 item increases by 10% and then decreases by 10%, is it back to $100?' Pairs calculate the answer and then explain the result to the class. This is a powerful way to surface the misconception that percentage changes are always additive.

20 min·Pairs

Inquiry Circle: The GST Detective

Students are given receipts where the GST has been 'smudged'. They must use their knowledge of reverse percentages (dividing by 1.1) to find the original pre-tax price. They then compare their methods and check each other's work. This applies maths to a standard Australian tax scenario.

35 min·Small Groups

Real-World Connections

• Retailers use percentages daily to calculate discounts on items, such as a '20% off sale' on clothing in a department store. This requires converting the percentage to a decimal to find the sale price.
• Financial advisors use percentages to explain investment returns and fees to clients. For example, explaining that a fund has grown by 8.5% over the year, or that management fees are 1.25% annually.
• When comparing prices at the supermarket, consumers often mentally convert unit prices or discounts to understand the best value. For instance, comparing 'buy one get one half price' offers requires percentage calculations.

Assessment Ideas

Quick Check

Present students with three cards: one with a fraction (e.g., 3/4), one with a decimal (e.g., 0.75), and one with a percentage (e.g., 75%). Ask students to hold up the cards that represent the same value. Follow up by asking them to explain the conversion process for one pair.

Exit Ticket

On a slip of paper, ask students to: 1. Convert 4/5 to a decimal and a percentage. 2. Calculate 15% of $80. 3. Write one sentence explaining why knowing these conversions is useful for shopping.

Discussion Prompt

Pose the following scenario: 'A store is offering a 30% discount on all items. You want to buy a game that originally costs $50. What is the sale price?' Ask students to share their methods for solving this, encouraging them to use different approaches (e.g., calculating the discount amount first, or calculating the remaining percentage directly).

Frequently Asked Questions

What is the difference between a markup and a profit?

A markup is the amount a business adds to the cost price to determine the selling price. Profit is the actual money made after the item is sold and all expenses are paid. While they are related, markup is a pricing strategy, and profit is the final financial result.

How do I calculate a 'reverse percentage' (finding the original price)?

If an item has increased by 10%, the new price is 110% of the original. To find the original, you divide the new price by 1.10. If it decreased by 10%, the new price is 90%, so you divide by 0.90. It's all about finding what 'percentage of the original' you currently have.

Why is it important to learn about percentage profit and loss?

It's a fundamental life skill. Whether you're running a small business, selling something on eBay, or just trying to understand if a 'sale' is actually a good deal, understanding how percentages affect value is key to financial independence.

How can active learning help students understand percentage change and profit?

Active learning, like the 'Classroom Market Stall', forces students to see percentages as dynamic changes rather than static numbers on a page. When they have to adjust prices in real-time to make a profit, they develop a 'feel' for how percentages work. Collaborative investigations into GST also show them that these aren't just school problems, they are the same calculations every Australian adult does every day.