Overview
This lecture covers the fundamentals of fractions, including definitions, representations, ways to compare fractions, their relationship to percentages, and upcoming assignments.
Definition and Basics of Fractions
- A fraction represents a part of a whole divided into equal parts.
- The numerator (top number) indicates how many parts are taken.
- The denominator (bottom number) shows into how many equal parts the whole is divided.
- Example: 3/4 means three out of four equal parts of a whole.
Key Components and Units
- "A over B" means A parts, each of size 1/B of the unit.
- The definition of the "whole" is essential when interpreting fractions.
- Comparing fractions depends on the size of the whole; fractions of different wholes can’t be directly compared.
Ways to Model Fractions
- Area model: Shading regions (e.g., pizza slices, pattern blocks).
- Length model: Using rulers, folded paper, or Cuisenaire rods to show partitioning.
- Set model: Representing fractions as subsets within a group of objects.
Improper Fractions and Multiple Representations
- Proper fractions: numerator < denominator; improper: numerator ≥ denominator.
- Some visual models can represent more than one fraction based on how the whole is defined.
Equivalent Fractions
- Fractions are equivalent if they represent the same value (e.g., 1/2 = 2/4 = 3/6).
- Multiplying numerator and denominator by the same number gives an equivalent fraction.
Comparing Fractions
- Decimals: Convert fractions to decimals to compare (caution: not always suitable for all students).
- Common denominators: Make denominators the same, then compare numerators.
- Cross multiplication: Use only when comparing fractions with an equal sign between them.
- Common numerators: If numerators are the same, the fraction with a smaller denominator is larger.
- Benchmarks: Compare to familiar fractions like 1/2 or 3/4.
Fractions and Percentages
- Percent means "per one hundred"; 35% = 35/100.
- Common conversions: 25% = 1/4, 50% = 1/2, 75% = 3/4, 10% = 1/10.
- Percent problems can be written as p/100 = portion/whole.
Key Terms & Definitions
- Fraction — A part of a whole expressed as a/b.
- Numerator — The number of parts taken (top number).
- Denominator — The number of equal parts the whole is divided into (bottom number).
- Equivalent fractions — Fractions that represent the same value.
- Improper fraction — Numerator is greater than or equal to the denominator.
- Proper fraction — Numerator is less than the denominator.
- Percent — A ratio out of 100.
Action Items / Next Steps
- Complete "What is a Fraction?" activity (separate file; uses Cuisenaire rods or pattern blocks, digital or physical).
- Watch all linked section videos (2A, 2F, 2H, 2J).
- Reflect on activity 2E.
- Finish Problem Set 2 and "Chapter 2: What Were They Thinking?"
- Review due dates; ensure chapter 2 work is submitted by the deadline.