The Foundation of Sharing: Fractions, Equivalence, and Operations | Core Mathematics SHS 1 SEM 1 WEEK 4 (WASSCE & NaCCA Aligned)

Understanding the Whole and the Part

A fraction is fundamentally a numerical representation of a part compared to a whole. Think about sharing a single, large ball of **Kenkey** among four friends in Accra. The whole is the one ball, and the part each friend receives is one out of the four equal divisions. This is why the denominator (the whole, 4) is crucial—it defines the standard size of the piece. The numerator (the part, 1) tells us how many of those pieces we are considering.

• **Proper Fractions:** The numerator is smaller than the denominator (e.g., 3/5). The value is strictly less than one whole.
• **Improper Fractions:** The numerator is greater than or equal to the denominator (e.g., 7/4). The value is one whole or more.
• **Mixed Numbers:** A combination of a whole number and a proper fraction (e.g., 1 ¾). This form offers a more intuitive understanding of quantities greater than one.

Conversion between improper fractions and mixed numbers is essential for clarity and operational ease. To move from the improper form (like 7/4) to the mixed form (1 ¾), you divide the numerator by the denominator, finding how many full wholes (the quotient, 1) and how many remaining parts (the remainder, 3) are left over the original denominator (4). This skill is particularly useful when presenting a final answer in a real-life context.

Equivalence: Representing the Same Quantity Differently

The idea of equivalent fractions is vital because it reveals that a quantity can have multiple numerical expressions. If you fold a paper strip into two and shade one side (1/2), and then fold it again into four equal sections, you now have two shaded sections out of four (2/4). The physical size of the shaded area has not changed; only the description has. Thus, $1/2 = 2/4 = 4/8$.

Finding equivalent fractions relies on multiplying or dividing both the numerator and the denominator by the exact same non-zero number. This operation is effectively multiplying the fraction by a form of 1 (like 2/2 or 4/4), which preserves the intrinsic value of the quantity but changes the unit size (the denominator).

Addition and Subtraction: The Necessity of Common Ground (LCM)

You cannot add or subtract parts unless they reference the same sized whole. Imagine trying to add an amount measured in Ghana Cedis to an amount measured in Nigerian Naira without a conversion rate—the figures are incomparable. When adding $1/3$ and $2/5$, we cannot simply add the denominators (3 and 5) because the pieces are of unequal sizes.

We must find the **Lowest Common Multiple (LCM)** of the denominators (3 and 5), which is 15. The LCM acts as our standard unit of comparison, creating new, equal-sized pieces (fifteenths) out of the original thirds and fifths.

• **Step 1: Find the LCM.** For 3 and 5, the LCM is 15.
• **Step 2: Convert to Equivalent Fractions.** Convert $1/3$ to fifteenths ($5/15$) and $2/5$ to fifteenths ($6/15$).
• **Step 3: Perform the Operation.** Now that the pieces are the same size (‘like’ fractions), you simply add or subtract the numerators: $5/15 + 6/15 = 11/15$.

Subtraction follows this identical conversion principle. The denominator must remain the same once the fractions are ‘like’ fractions, as we are only counting the total number of common units.

Multiplication and Division: The Algorithm and the Rationale

Unlike addition, multiplication and division of fractions do not require finding the LCM because the operation inherently changes the size of the whole.

Multiplication

When we multiply fractions, we are essentially finding ‘a fraction *of* a fraction’ (e.g., $1/2 \times 1/2$ means half of a half). This naturally yields a smaller result. The rule is simply:
$$\frac{\text{Numerator} \times \text{Numerator}}{\text{Denominator} \times \text{Denominator}}$$
If you multiply $3/4$ by $9/10$, the result is $27/40$. Emphasize the efficiency of simplifying (cross-cancellation) before multiplying to manage large numbers.

Division (Keep, Change, Flip)

Division involves determining how many times one fraction fits into another. To simplify this, we use the **’Keep, Change, Flip’ (KCF)** mnemonic, which relies on the mathematical principle of multiplying by the reciprocal.

• **Keep:** Keep the dividend (the first fraction).
• **Change:** Change the division sign ($\div$) to multiplication ($\times$).
• **Flip:** Use the reciprocal of the divisor (invert the second fraction).

For example, $2/3 \div 4/5$ is equivalent to $2/3 \times 5/4$, transforming the division into a standard multiplication problem ($10/12$, simplified to $5/6$).

Operating on Mixed Numbers in Context

When tackling complex problems, such as calculating the total cement used by a mason ($2/3 + 3/4 + 1/2$), the crucial first step is to convert all mixed numbers to improper fractions before applying the operation rules. This ensures algorithmic consistency and prevents errors common in handling mixed number arithmetic directly. The LCM of 3, 4, and 2 is 12, leading to the sum $8/12 + 9/12 + 6/12 = 23/12$, which is then converted back to the contextual answer of $1 \ 11/12$ bags.

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