MATHEMATICS EDUCATION (ME) YEAR TWO (2) 2025 ASSIGNMENT ONE (1) 1. Find a fraction between each pair of given fractions below through two methods. [24] a. 1 and \( 1 \frac{1}{8} \) b. \( \quad \frac{1}{2} \) and 1 c. \( \frac{2}{7} \) and \( \frac{3}{7} \) d. \( \frac{1}{4} \) and \( \frac{1}{3} \) 2. Evaluate the following through diagrams [12] a) \( \frac{2}{5} \times \frac{3}{4} \) b) \( \frac{3}{4} \div \frac{3}{10} \) 3. a) Give three facts about direct proportion [4] b) come up with a limited table and a well labelled graph of two quantiles on direct proportion. [10] 4. a) Write down three facts facts about inverse proportion. [6] b) come up with a limited table and a well labelled graph on inverse proportion. [10]

Let's break down the assignment step by step. ### 1. Finding a Fraction Between Given Fractions For each pair of fractions, we will find a fraction using two methods: the average method and the mediant method. #### a. Between 1 and \( 1 \frac{1}{8} \) **Method 1: Average Method** 1. Convert \( 1 \frac{1}{8} \) to an improper fraction: \[ 1 \frac{1}{8} = \frac{8 + 1}{8} = \frac{9}{8} \] 2. Find the average: \[ \text{Average} = \frac{1 + \frac{9}{8}}{2} = \frac{\frac{8}{8} + \frac{9}{8}}{2} = \frac{\frac{17}{8}}{2} = \frac{17}{16} \] **Method 2: Mediant Method** 1. Find the mediant: \[ \text{Mediant} = \frac{1 + \frac{9}{8}}{2} = \frac{1 + \frac{9}{8}}{2} = \frac{17}{16} \] **Fraction Found:** \( \frac{17}{16} \) #### b. Between \( \frac{1}{2} \) and 1 **Method 1: Average Method** 1. Find the average: \[ \text{Average} = \frac{\frac{1}{2} + 1}{2} = \frac{\frac{1}{2} + \frac{2}{2}}{2} = \frac{\frac{3}{2}}{2} = \frac{3}{4} \] **Method 2: Mediant Method** 1. Find the mediant: \[ \text{Mediant} = \frac{1 + 2}{2} = \frac{3}{4} \] **Fraction Found:** \( \frac{3}{4} \) #### c. Between \( \frac{2}{7} \) and \( \frac{3}{7} \) **Method 1: Average Method** 1. Find the average: \[ \text{Average} = \frac{\frac{2}{7} + \frac{3}{7}}{2} = \frac{\frac{5}{7}}{2} = \frac{5}{14} \] **Method 2: Mediant Method** 1. Find the mediant: \[ \text{Mediant} = \frac{2 + 3}{7 + 7} = \frac{5}{14} \] **Fraction Found:** \( \frac{5}{14} \) #### d. Between \( \frac{1}{4} \) and \( \frac{1}{3} \) **Method 1: Average Method** 1. Find the average: \[ \text{Average} = \frac{\frac{1}{4} + \frac{1}{3}}{2} = \frac{\frac{3}{12} + \frac{4}{12}}{2} = \frac{\frac{7}{12}}{2} = \frac{7}{24} \] **Method 2: Mediant Method** 1. Find the mediant: \[ \text{Mediant} = \frac{1 + 1}{4 + 3} = \frac{2}{7} \] **Fraction Found:** \( \frac{7}{24} \) ### 2. Evaluate the Following Through Diagrams #### a) \( \frac{2}{5} \times \frac{3}{4} \) To visualize this, we can represent \( \frac{2}{5} \) as a rectangle divided into 5 equal parts, shading 2 parts. Then, we take that shaded area and divide it into 4 equal parts, shading 3 of those parts. The area of the shaded region will be: \[ \frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10} \] #### b) \( \frac{3}{4} \div \frac{3}{10} \) To visualize this, we can represent \( \frac{3}{4} \) as a rectangle divided into 4 equal parts, shading 3 parts. Then, we want to see how many \( \frac{3}{10} \) fit into that shaded area. The calculation will be: \[ \frac{3}{4} \div \frac{3}{10} = \frac{3}{4} \times \frac{10}{3} = \frac{10}{4} = \frac{5}{2} \] ### 3. Direct Proportion #### a) Three Facts About Direct Proportion 1. In a direct proportion, as one quantity increases, the other quantity also increases at a constant rate. 2. The relationship can be expressed as \( y = kx \), where \( k \) is a constant. 3. The graph of a direct proportion is a straight line that passes through the origin. #### b) Table and Graph of Direct Proportion | \( x \) | \( y \) | |---------|---------| | 1 | 2 | | 2 | 4 | | 3 | 6 | | 4 | 8 | The graph will be a straight line through the origin with points plotted at (1,2), (2,4), (3,6), and (4,8). ### 4. Inverse Proportion #### a) Three Facts About Inverse Proportion 1. In an inverse proportion, as one quantity increases, the other quantity decreases. 2. The relationship can be expressed as \( y = \frac{k}{x} \), where \( k \) is a constant. 3. The graph of an inverse proportion is a hyperbola. #### b) Table and Graph of Inverse Proportion | \( x \) | \( y \) | |---------|---------| | 1 | 10 | | 2 | 5 | | 5 | 2 | | 10 | 1 | The graph will show a hyperbolic curve approaching the axes but never touching them. This completes the assignment. If you need further assistance or clarification on any part, feel free to ask!