Fractions, Decimals and Ratio
Overview
This unit develops students' understanding of fractions, decimals, percentages, and ratios as interconnected ways of representing quantities. Students will learn to convert between these forms, perform operations, and apply them to real-world situations. Through practical problem-solving, students will appreciate how different representations are useful for different purposes.
Key Concept
Relationships — Fractions, decimals, percentages, and ratios are different ways of expressing the same underlying mathematical relationships.
Related Concepts
- Equivalence — Different representations can express the same value or relationship.
- Quantity — The amount or number that these different forms represent.
- Representation — The form in which a mathematical relationship is expressed.
Global Context
Identities and Relationships — How do we use fractions, decimals, and ratios to understand and communicate about our world?
Statement of Inquiry
Numbers can be represented in multiple equivalent forms, each useful for different purposes.
Inquiry Questions
Factual Questions
- How do you convert between fractions, decimals, and percentages?
- What is a ratio and how is it written?
- How do you perform operations with fractions?
Conceptual Questions
- What does it mean for two fractions to be equivalent?
- Why might one representation be more useful than another in a given situation?
- How is ratio different from fraction, and how are they related?
Debatable Questions
- Is it better to think in fractions or decimals?
- Should all calculations be done in exact form rather than rounded?
- Is mathematics discovered in the world or invented to describe it?
ATL Skills
Thinking Skills
- Recognise equivalence between different number representations.
- Apply proportional reasoning to solve problems.
- Analyse which representation is most appropriate for a given situation.
Communication Skills
- Explain reasoning clearly when solving problems.
- Use correct mathematical notation for fractions, decimals, and ratios.
- Present solutions with clear steps and justifications.
Research Skills
- Investigate real-world uses of fractions, decimals, and ratios.
- Gather and analyse data involving proportions.
- Explore historical development of number systems.
Self-Management Skills
- Set goals for mastering conversion skills.
- Organise practice materials for skill development.
- Reflect on learning progress and areas for improvement.
Content
Week 1: Understanding Fractions
- What is a fraction? Numerator and denominator.
- Equivalent fractions: simplifying and expanding.
- Comparing and ordering fractions.
- Fractions on a number line.
Week 2: Operations with Fractions
- Adding and subtracting fractions with like and unlike denominators.
- Multiplying fractions.
- Dividing fractions.
- Mixed numbers and improper fractions.
Week 3: Decimals and Their Place Value
- Decimal place value: tenths, hundredths, thousandths.
- Reading, writing, and comparing decimals.
- Operations with decimals.
- Rounding decimals.
Week 4: Percentages
- Understanding percentage as "out of 100."
- Converting between fractions, decimals, and percentages.
- Finding percentages of quantities.
- Real-world applications: discounts, interest, statistics.
Week 5: Ratio and Proportion
- Understanding ratio as a comparison.
- Writing ratios in simplest form.
- Sharing quantities in a given ratio.
- Direct proportion and the unitary method.
Week 6: Real-World Applications
- Fractions and decimals in measurement and money.
- Percentages in business, media, and daily life.
- Ratios in recipes, maps, and scale drawings.
- Problem-solving with mixed representations.
Summative Assessment
Real-World Mathematics Project: Students will investigate a real-world scenario (such as planning a budget, analysing survey data, or scaling a recipe) that requires the use of fractions, decimals, percentages, and ratios. The project must include calculations in multiple forms, justification of representation choices, and a written report explaining the mathematics.
Comprehensive Problem-Solving Test: Students will complete a test demonstrating their ability to convert between forms, perform operations, and solve word problems involving fractions, decimals, percentages, and ratios.
Formative Assessment
- Fraction equivalence and simplification drills.
- Decimal place value and operation practice.
- Percentage calculation exercises.
- Ratio and proportion problem-solving.
- Conversion practice between all forms.
- Quizzes on key concepts and procedures.
Interdisciplinary Connections
- Science: Using fractions and decimals in measurements and data analysis.
- Design: Using ratios in scale drawings and model making.
- Economics: Understanding percentages in finance and business.
- Cooking and Nutrition: Scaling recipes using fractions and ratios.
Service as Action
- Create a mathematics help resource for younger students on fractions.
- Organise a school fundraising event requiring budget calculations.
- Develop a community information sheet explaining percentages in everyday contexts.
- Tutor peers struggling with fraction and decimal concepts.
IB Learner Profile
- Thinkers: Students apply logical reasoning to solve proportional problems.
- Communicators: Students express mathematical relationships clearly using appropriate notation.
- Inquirers: Students explore how mathematics is used in real-world contexts.
- Knowledgeable: Students understand the connections between different number representations.
- Reflective: Students reflect on their problem-solving strategies and mathematical growth.
Self-Test
- Simplify the fraction 18/24 to its lowest terms.
- Convert 3/5 to a decimal and a percentage.
- Add: 2/3 + 1/4 = ?
- What is 25% of 200?
- Write the ratio 12:16 in its simplest form.
- Convert 0.375 to a fraction in simplest form.
- Share $60 in the ratio 2:3.
- If 5 apples cost $3, how much do 8 apples cost?
- Order from smallest to largest: 0.6, 3/5, 65%, 0.55.
- A shirt costs $40 and is on sale for 20% off. What is the sale price?
