Conceptual Understanding and Early Applications

Theory: A fraction represents a part of a whole. It is written in the form \(\frac{a}{b}\), where:
a is the numerator (number of parts taken),
b is the denominator (total number of equal parts the whole is divided into).
Each part must be equal in size for it to be a valid fraction.

Example:
Question: A pizza is divided into 4 equal parts. What fraction is one slice?
Solution: Each part is \(\frac{1}{4}\)​. So, one slice = \(\frac{1}{4}\)​ of the whole pizza.

Theory: In a fraction \(\frac{a}{b}\)​, the numerator (a) shows how many parts are taken. The denominator (b) shows how many equal parts the whole is divided.

Example:
Question: What do the numerator and denominator represent in \(\frac{3}{5}\)​?
Solution: Numerator = 3 (parts taken), Denominator = 5 (equal parts in the whole).

Theory: Fractions are used in everyday scenarios like measuring time, sharing food, and dividing money. They show how much of a whole has been used or left.

Example:
Question: If you drink half of a glass of milk, what fraction remains?
Solution: If \(\frac{1}{2}\)​ is drunk, then remaining = 1−12=121 – \(\frac{1}{2}\) = \(\frac{1}{2}\)​.

Theory: To represent a fraction visually, divide a shape (like rectangle or circle) into equal parts, and shade the number of parts shown by the numerator.

Example:
Question: Shade \(\frac{3}{4}\)​ of a rectangle divided into 4 equal parts.
Solution: Shade 3 out of 4 parts. Each part is equal, so the shaded area represents \(\frac{3}{4}\)​.

Theory: Looking at shaded or colored parts of shapes helps us name the fraction as \(\frac{\text{shaded parts}}{\text{total parts}}\).

Example:
Question: A circle has 8 equal parts, with 5 shaded. What is the fraction?
Solution: Shaded parts = 5, total parts = 8 ⇒ Fraction = \(\frac{5}{8}\)​.

Theory: Fractions appear when sharing cakes, chocolate bars, or plotting positions on a clock, indicating portions of a whole object.

Example:
Question: A chocolate bar has 6 pieces. You eat 2. What fraction did you eat?
Solution: Fraction eaten = \(\frac{2}{6}\) = \(\frac{1}{3}\)​ after simplification.

Theory: When two fractions have the same denominator, the one with the greater numerator is the larger fraction.

Example:
Question: Which is greater: \(\frac{3}{8}\) or \(\frac{5}{8}\)​?
Solution: Same denominator (8), compare numerators: 5 > 3 ⇒ \(\frac{5}{8}\) > \(\frac{3}{8}\)​

Theory: Two fractions are equal if they represent the same area or quantity, even if their numerators and denominators differ.

Example:
Question: Show that \(\frac{1}{2}\) = \(\frac{2}{4}\)​ using shapes.
Solution: Shade half of a rectangle (2 out of 4 equal parts) and also one-half. Both show the same area ⇒ Equal fractions.

Theory: By folding paper into equal parts, and then again into more parts, we can observe how fractions remain equivalent even when subdivided.

Example:
Question: Fold a strip into 2 parts, shade 1 part. Then fold again into 4 parts. How many parts are shaded now?
Solution: 1 out of 2 parts = 2 out of 4 parts ⇒ \(\frac{1}{2}\) = \(\frac{2}{4}\)​

Theory: These special fractions represent common divisions:
Half = \(\frac{1}{2}\)
One-fourth = \(\frac{1}{4}\)
Three-fourth = \(\frac{3}{4}\)

Example:
Question: A ribbon is cut into 4 equal parts. You use 3. What fraction did you use?
Solution:
Fraction used = \(\frac{3}{4}\)

Theory: Comparing shaded regions with equal-sized holes can help determine which fraction is larger.

Example:
Question: In two same-sized pizzas, one has \(\frac{2}{3}\)​ eaten, the other \(\frac{3}{4}\)​. Which is more?
Solution: Convert to decimal: \(\frac{2}{3}\)​≈0.67,\(\frac{3}{4}\)
= 0.75 ⇒ \(\frac{3}{4}\) > \(\frac{2}{3}\)

Theory: Fractions can be plotted on a number line by dividing the space between 0 and 1 into equal parts (as per the denominator).

Example:
Question: Mark \(\frac{3}{4}\) on a number line from 0 to 1.
Solution: Divide segment between 0 and 1 into 4 parts. Count 3 parts from 0 ⇒ Point shows \(\frac{3}{4}\)

Theory: Common errors include uneven part divisions or incorrect numerator/denominator placement.

Example:
Question: A circle is divided into 3 unequal parts, and 1 is shaded. Is it \(\frac{1}{3}\)​?
Solution: No. For a fraction to be \(\frac{1}{3}\)​, all parts must be equal. This is not a valid representation.

Theory: Every fraction ab\(\frac{a}{b}\) means “a divided by b”. It connects fractions with the operation of division.

Example:
Question: Express the fraction \(\frac{3}{5}\)as a division.
Solution: \(\frac{3}{5} = 3 \div 5\) = 0.6

Theory: Fractions also apply to groups of items. If a portion of a group is selected, that part is a fraction of the total group.

Example:
Question: Out of 12 students, 4 are girls. What fraction are girls?
Solution: Girls = 4 out of 12 ⇒ 412=13\(\frac{4}{12}\) = \(\frac{1}{3}\)