Learning Resources

Fractions

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 08 Pathway: N/A

First Principles

Objective: Understand fractions as equal shares of a unit whole, compare their relative magnitudes using common denominators, and perform combined arithmetic operations accurately in real-life contexts.

(a) Concrete Scenario: Imagine two farmers in Kiambu sharing bags of fertilizer. Farmer A uses \(\frac{3}{4}\) of a bag for maize, while Farmer B uses \(\frac{5}{6}\) of a bag for cabbages. We cannot compare \(3\) and \(5\) directly because each fraction divides the standard bag into pieces of different sizes!

Interactive Fraction Comparison Wall

Select two fractions below to visually compare their sizes on the benchmark unit strip:

Fraction A: 3/4 (0.75)

Fraction B: 5/6 (0.833)

\(\frac{5}{6} > \frac{3}{4}\) because \(\frac{10}{12} > \frac{9}{12}\)

(b) Geometric Insight: A fraction is a partitioned segment of a fixed unit length \(1\). Denominators dictate the granularity of partition (more pieces \(\rightarrow\) smaller unit size). Numerators count the number of chosen units.

(c) Algebraic Rule: To compare, add, or subtract fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), we re-scale both to a shared lowest common denominator \(m = \text{LCM}(b, d)\):

\[\frac{a}{b} = \frac{a \cdot (m/b)}{m}, \quad \frac{c}{d} = \frac{c \cdot (m/d)}{m}\]

Key Formulas

Addition & Subtraction (Common Denominator): \[\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}\] Always express in simplest form by dividing numerator and denominator by their greatest common divisor (GCD).

Multiplication: \[\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\] Direct product of numerators divided by product of denominators.

Division (Multiply by Reciprocal): \[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \quad (c \neq 0)\]

Mixed Numbers to Improper Fractions: \[A\frac{b}{c} = \frac{A \times c + b}{c}\]

Order of Operations (BODMAS / PEMDAS):

Brackets \(\rightarrow\) Orders/Of \(\rightarrow\) Division & Multiplication (left to right) \(\rightarrow\) Addition & Subtraction (left to right).

Worked Examples

Example 1 (Easy): Ordering Simple Fractions

Problem: Arrange \(\frac{2}{3}, \frac{3}{5},\) and \(\frac{7}{10}\) in ascending order (smallest to largest).

Find the \(\text{LCM}\) of denominators \(3, 5, 10\): \(\text{LCM}(3, 5, 10) = 30\).
Convert each fraction to an equivalent fraction with denominator 30: \[\frac{2}{3} = \frac{2 \times 10}{30} = \frac{20}{30}\] \[\frac{3}{5} = \frac{3 \times 6}{30} = \frac{18}{30}\] \[\frac{7}{10} = \frac{7 \times 3}{30} = \frac{21}{30}\]
Compare numerators: \(18 < 20 < 21\).
Final Order: \(\frac{3}{5} < \frac{2}{3} < \frac{7}{10}\).

Example 2 (Medium): Combined Addition and Subtraction

Problem: Evaluate \(2\frac{1}{4} - 1\frac{2}{3} + \frac{5}{6}\).

Convert mixed numbers to improper fractions: \[2\frac{1}{4} = \frac{9}{4}, \quad 1\frac{2}{3} = \frac{5}{3}\]
Find \(\text{LCM}(4, 3, 6) = 12\). Convert each term: \[\frac{9}{4} = \frac{27}{12}, \quad \frac{5}{3} = \frac{20}{12}, \quad \frac{5}{6} = \frac{10}{12}\]
Combine the numerators over 12: \[\frac{27 - 20 + 10}{12} = \frac{7 + 10}{12} = \frac{17}{12} = 1\frac{5}{12}\]

Example 3 (Hard): Multi-Step BODMAS Real Context

Problem: A dairy farmer in Eldoret collects \(60\) litres of milk. She sells \(\frac{2}{5}\) of the milk in the morning and \(\frac{1}{3}\) of the remainder in the evening. How many litres of milk are left?

Morning sales: \(\frac{2}{5} \times 60 = 24\text{ litres}\).
Remainder after morning: \(60 - 24 = 36\text{ litres}\).
Evening sales: \(\frac{1}{3} \times 36 = 12\text{ litres}\).
Milk remaining: \(36 - 12 = 24\text{ litres}\).
Alternative fraction approach: Fraction left after morning \(= 1 - \frac{2}{5} = \frac{3}{5}\). Fraction left after evening \(= \frac{3}{5} \times \left(1 - \frac{1}{3}\right) = \frac{3}{5} \times \frac{2}{3} = \frac{2}{5}\). Remaining milk \(= \frac{2}{5} \times 60 = 24\text{ litres}\).

Common Mistakes

Mistake 1: Adding denominators directly

Error: \(\frac{1}{2} + \frac{1}{3} = \frac{1+1}{2+3} = \frac{2}{5}\).

Why it happens: Intuition tells students to apply whole-number addition to both top and bottom independently.

Correction: Denominators represent partition sizes, not countable items. You must convert to equal sized pieces first: \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).

Mistake 2: "Bigger denominator means bigger fraction"

Error: Thinking \(\frac{1}{8} > \frac{1}{4}\) because \(8 > 4\).

Why it happens: Applying natural number magnitude rules directly to fractions.

Correction: The denominator divides the whole into parts. More parts mean each piece is smaller: sharing an ugali among 8 people gives smaller pieces than sharing among 4 people!

Mistake 3: Confusing fraction of a remainder with fraction of total

Error: In word problems, taking "\(\frac{1}{4}\) of the rest" as \(\frac{1}{4}\) of the total initial quantity.

Correction: Calculate the remaining quantity first before multiplying by the subsequent fraction.

Real World

Coffee & Tea Cooperative Grading: Farmers' cooperative societies in Nyeri grade coffee yields. If \(\frac{1}{2}\) of a harvest is Grade AA, \(\frac{3}{10}\) is Grade AB, and the rest is Grade C, finding the exact proportion requires common denominator reduction: \(1 - \left(\frac{5}{10} + \frac{3}{10}\right) = \frac{2}{10} = \frac{1}{5}\).

Posho Mill Rationing: Balancing animal feed formulation requires mixing \(2\frac{1}{2}\) bags of maize germ, \(\frac{3}{4}\) bag of cotton seed cake, and \(\frac{1}{4}\) bag of mineral salt to achieve balanced protein ratios.

Land Partitioning (Inheritance & Title Sub-division): A 12-hectare family parcel subdivided such that each of 3 siblings receives \(\frac{1}{4}\) and the community school receives the remaining fraction \(\frac{1}{4}\) (3 hectares).

Practice

A fruit vendor in Marikiti market has 120 bananas on display. A customer buys \(\frac{3}{5}\) of all the bananas. How many bananas does the customer take? (Type only the number, e.g., 42)

Review the concepts above.

Bosibori, a tailor in Nairobi, has a piece of fabric 5 metres long. She uses \(\frac{3}{8}\) of the fabric to make a dress. What fraction of the original fabric is left? (Type only the fraction, e.g., 5/8)

Review the concepts above.

A maize farmer had a full bag of seeds. He first planted \(\frac{5}{8}\) of the bag on Monday and later used another \(\frac{1}{4}\) of the bag on Tuesday. What fraction of the bag of seeds is left? (Type only the fraction in simplest form, e.g., 1/8)

Review the concepts above.

A tailor buys decorative lace measuring \(\frac{2}{3}\) metre per piece and uses \(\frac{3}{4}\) of a piece on a dress. How many metres of lace were used? (Type the answer as a decimal number, e.g., 0.5)

Review the concepts above.

A container initially holds 2 litres of juice. A student drinks \(\frac{5}{6}\) litre in the morning and \(\frac{1}{3}\) litre in the afternoon. What fraction of a litre of juice remains in the container? (Type only the fraction in simplest form, e.g., 5/6)

Review the concepts above.

A SACCO member had 1,000 shillings in an emergency wallet. She deposited \(\frac{5}{6}\) of a thousand shillings and later withdrew \(\frac{1}{3}\) of a thousand shillings from the deposit. If she only had the net deposit remaining, how many shillings are represented by \(\left(\frac{5}{6} - \frac{1}{3}\right)\) of 1,000 shillings? (Type only the number, e.g., 500)

Review the concepts above.