In $\frac{2}{3}$, which number is the numerator and what does it mean?

The numerator is 2. It means you have 2 of the 3 equal parts that make up the whole.

What does the denominator tell you in a fraction?

The denominator tells you how many equal parts the whole is divided into. For $\frac{3}{5}$, the whole is split into 5 parts.

A fraction is part of a whole, written as one number over another (numerator over denominator), representing a portion of a whole quantity.

How do you shade $\frac{3}{5}$ of a grid with 25 squares?

Divide 25 by 5 to get 5, then multiply by 3 to get 15. Shade 15 squares.

Why must the parts of a whole be equal when defining a fraction?

If parts are unequal, the fraction would not have a consistent meaning. Equal parts ensure that each part represents the same proportion of the whole.

Which is larger: $\frac{1}{3}$ or $\frac{1}{5}$?

$\frac{1}{3}$ is larger. When the whole is split into fewer parts (3), each part is bigger than when split into more parts (5).

Compare $\frac{2}{3}$ and $\frac{3}{4}$: which represents more of the whole?

$\frac{3}{4}$ represents more. As equivalent fractions, $\frac{2}{3} = \frac{8}{12}$ and $\frac{3}{4} = \frac{9}{12}$, so $\frac{3}{4}$ is greater.

What is the difference between the numerator and denominator?

The numerator (top) counts how many parts you have. The denominator (bottom) counts how many equal parts the whole is divided into.

How does $\frac{1}{2}$ of a circle differ from $\frac{1}{4}$ of a circle?

$\frac{1}{2}$ is one half (2 equal parts, 1 shaded). $\frac{1}{4}$ is one quarter (4 equal parts, 1 shaded). A half is twice the size of a quarter.

What error occurs when you shade 5 squares for $\frac{3}{5}$ of 20?

Incorrect count. $\frac{3}{5}$ of 20 = $20 \div 5 \times 3 = 12$ squares. Shading only 5 would be $\frac{5}{20} = \frac{1}{4}$.

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Introduction to Fractions

Summary

A fraction represents part of a whole. The numerator (top) indicates how many parts you have; the denominator (bottom) indicates how many equal parts the whole is divided into. Understanding this structure enables shading fractions on grids and circles, and forms the foundation for all fraction operations.

1. Definition & Core Concepts

A fraction is part of a whole, written as $\frac{a}{b}$ where $a$ (numerator) and $b$ (denominator) are integers.

The denominator tells you how many equal parts the whole is split into; the numerator tells you how many of those parts you have.

For example, $\frac{2}{3}$ means the whole is divided into 3 parts and you have 2 of them.

Fractions can represent quantities less than 1 (proper fractions), equal to 1, or greater than 1 (improper fractions).

Visual representation of ¼ of a circle and ⅗ of a bar model, showing numerator and denominator.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls

7. Connections

Introduction to Fractions

Summary

1. Definition & Core Concepts

A fraction is part of a whole, written as $\frac{a}{b}$ where $a$ (numerator) and $b$ (denominator) are integers.

The denominator tells you how many equal parts the whole is split into; the numerator tells you how many of those parts you have.

For example, $\frac{2}{3}$ means the whole is divided into 3 parts and you have 2 of them.

Fractions can represent quantities less than 1 (proper fractions), equal to 1, or greater than 1 (improper fractions).

Visual representation of ¼ of a circle and ⅗ of a bar model, showing numerator and denominator.

2. Underlying Principles

The whole must be divided into equal parts for a fraction to be meaningful.

The size of each part depends on the denominator: more parts means smaller pieces.

The numerator counts how many of those equal parts are being considered.

3. Methods & Techniques

Shading a fraction on a grid: Divide total squares by the denominator, then multiply by the numerator. For $\frac{3}{5}$ of 20 squares: $20 \div 5 = 4$ , then $4 \times 3 = 12$ squares to shade.

Shading a fraction of a circle: Divide the circle into the number of equal sectors given by the denominator (e.g. 4 quarters with perpendicular lines through the centre), then shade the number of sectors given by the numerator.

Group shaded squares neatly when possible for clarity.

4. Key Distinctions

Term	Position	Meaning
Numerator	Top	Parts you have
Denominator	Bottom	Total equal parts

A larger denominator does not mean a larger fraction: $\frac{1}{2}$ is greater than $\frac{1}{4}$ because you have more of the whole when it is split into fewer parts.

5. Exam Strategy & Tips

When shading grids, count total squares first, then apply divide-by-denominator, multiply-by-numerator.

For circles, use a protractor if needed to divide into equal sectors, or use symmetry (e.g. perpendicular diameters for quarters).

Check that your shaded count matches the fraction: e.g. 12 out of 20 is $\frac{12}{20} = \frac{3}{5}$ .

6. Common Pitfalls

Confusing numerator and denominator: The denominator is always the total number of parts; the numerator is the count you have.

Unequal parts: When drawing, ensure sectors or segments are equal; otherwise the fraction is incorrect.

Wrong count: Double-check your multiplication: for $\frac{5}{7}$ of 21, $21 \div 7 = 3$ , $3 \times 5 = 15$ .