Comprehensive Study Guide for Mathematics: Algebra, Data, and Geometry

PURE MATHEMATICS AND ALGEBRAIC CONCEPTS

Powers of Two and Exponential Form

• Definition of Exponential and Expanded Form: Numbers can be expressed as powers based on repeated multiplication. Table 1 provides the foundation for powers of 2:   - Exponential Form: The representation of a number using a base and an exponent (e.g., $2^{2}$).   - Expanded Form: The full multiplication process (e.g., $2 \times 2$).   - Power of 2: The resultant value (e.g., $4$).

• Table 1 Data Values:   - $2^{2} = 2 \times 2 = 4$   - $2^{3} = 2 \times 2 \times 2 = 8$   - $2^{4} = 2 \times 2 \times 2 \times 2 = 16$   - $2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32$

• Case Study: The Number 1024:   - To show 1024 is a power of 2, continue multiplying 2 by itself: $32 \times 2 = 64$, $64 \times 2 = 128$, $128 \times 2 = 256$, $256 \times 2 = 512$, $512 \times 2 = 1024$.   - Exponential Form of 1024: The number 1024 is expressed as $2^{10}$.

• Selection Criteria for Specific Powers of 2:   - Problem: Find a power of 2 that is a multiple of 16, greater than 50, and less than 200.   - The powers of 2 in this range are: $2^{6} = 64$ and $2^{7} = 128$.   - Check for multiples of 16: $64 = 16 \times 4$ and $128 = 16 \times 8$. Therefore, both 64 and 128 meet the conditions.

Real Number Intervals and Decimals

• Intervals and the Density of Numbers: There are infinitely many numbers between any two real numbers.   - Between 0.998 and 0.999: Yes, numbers exist, such as 0.9985 or 0.9981.

• Subtraction of Decimals:   - Process for $0.999 - 0.998$:     - Align the decimal points.     - Subtract from right to left: $9-8=1$, $9-9=0$, $9-9=0$, leading to $0.001$.

• Rational Number Intervals: Fractions also exist between any two values.   - Between $\frac{3}{4}$ and 1: Yes, a fraction like $\frac{7}{8}$ exists. This can be conceptualized by converting to decimals ($0.75$ and $1.0$) or finding a common denominator (e.g., $\frac{6}{8}$ and $\frac{8}{8}$).

Algebraic Equations and Transformations

• Box 2 Analysis: The equation is provided as $17 + \_\_ = \_\_ + 3$.   - Variable definitions: Let $a$ be the first blank and $b$ be the second blank.   - Equation form: $17 + a = b + 3$.   - Re-arranging for Relationship: $a + 14 = b$ or $b - a = 14$.   - General Rule: The difference between $b$ and $a$ ($b - a$) is always 14.

• Solving Linear Equations - Detailed Step-by-Step:   - Equation 1: $5y - 8 = 14 - 3y$   - Step 1 (Transformation to Equation 2): Add $3y$ to both sides. $5y + 3y - 8 = 14$.   - Result (Equation 2): $8y - 8 = 14$.   - Step 2: Add 8 to both sides. $8y = 14 + 8$, which simplifies to $8y = 22$.   - Final Solution: $y = \frac{22}{8} = \frac{11}{4} = 2.75$.

• Integer Representation: The form $2r - 1$:   - In this expression, where $r$ is an integer, $2r - 1$ represents odd numbers.   - Potential values for $2r - 1$ from the list (-5, -27, -82, 99, 46, 122): -5, -27, and 99 are all possible because they are odd integers.

DATA INTERPRETATION AND STATISTICS

Relationship between Absences and Academic Performance

• Figure 1 (Masipag Section): A scatter plot representing student data for two academic quarters.

• Correlation Analysis:   - Observation: Students with fewer absences consistently have higher overall academic grades.   - Trend: As the number of absences increases, the overall academic grade decreases (Negative Correlation).

• Data Analysis Example:   - Finding students with grades below 84: Count the points vertically below the 84 threshold on the Y-axis. The position of each point corresponds to an individual student.

Socio-Economic Data Analysis in Barangay San Mateo

• Figure 2 Distribution (Purok 1 and Purok 2):   - Diversity: Refers to the spread or variability in income. If the data points in one Purok (e.g., Purok 2) are spread across a wider range (from P8 to P40) compared to the other, that Purok shows more economic diversity.   - Financial Aid Policy: Even if the average (mean) monthly income is equal between two groups, financial aid decisions might be based on distribution. If one Purok has more families below a certain poverty threshold, it may require more assistance regardless of the mean.

Participation Tables and Probabilities

• Table 2: Grade 7 Music and Sports Activities:   - Total students in Grade 7: 110.   - Total participating in sports: 60.   - Total participating in music: 49.   - Students participating in both: 18.   - Students participating in neither: 19 (calculated from the intersection of "did not participate in sports" and "did not participate in music").

• Probability Calculation:   - Probability of selecting a student participating in both music and sports: $P(\text{Both}) = \text{Joint Frequency} / \text{Total} = \frac{18}{110}$.

COORDINATE GEOMETRY AND SPATIAL ANALYSIS

Number Line Positioning

• Figure 3 Analysis:   - The number line shows increments of 100.   - Point F is located at $-300$.   - Point G is located at $200$.

Cartesian Plane and Triangle Area

• Coordinate System: Positioned as $(x, y)$.

• Point Coordinates (Figure 4):   - D is located at $(0, 2)$.   - C has specific coordinates determined by its location relative to the origin.

• Linear Line Analysis:   - A line passing through Points B and C can be evaluated for other points (e.g., $(3, 2)$, $(5, 6)$).   - A line through Points A and B can be represented by equations like $y = -x$ or $y = -x + 2$.

• Area of Triangle ABC:   - Calculated using the formula: $Area = \frac{1}{2} \times \text{base} \times \text{height}$.   - The coordinates of the vertices are used to find segment lengths.

• Distance Analysis:   - Point A (House), Point B (School), Point C (Barangay Hall).   - The shorter walk is determined by calculating the distance between coordinates using the distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.

Real-World Linear Modeling

• Fruit Stand Pricing: 3 apples for Php100.   - Cost expression for $n$ apples: $\text{cost} = \frac{100n}{3}$.   - This can also be stated as the ratio $3 : 100 = n : \text{cost}$.

• Tricycle Rental (Figure 5):   - Formula: $c = 250 + 200d$   - The value 250: Represents the fixed cost (y-intercept), which is the base fee regardless of the number of days.   - The value 200: Represents the daily rate (slope), describing the change in cost per day ($\Delta c / \Delta d$).   - Calculation for 5 days: $c = 250 + 200(5) = 250 + 1000 = 1250$.

GEOMETRY AND MEASUREMENT

Triangle Properties and Angular Geometry

• Triangle Interior Sum: The sum of interior angles $p + q + r = 180^{\circ}$.

• Exterior Angle Theorem:   - The exterior angle (e.g., at Point Q = $130^{\circ}$) is equal to the sum of the two remote interior angles ($p + r$).   - Interior and adjacent exterior angles form a linear pair, summing to $180^{\circ}$. Therefore, the interior angle at Q is $180 - 130 = 50^{\circ}$.

• Specific Scenario Analysis:   - If $r = 60^{\circ}$ and exterior at Q is $130^{\circ}$, then $p + r = 130$, implying $p = 70^{\circ}$.   - Because $p + q + r = 180$, then $70 + q + 60 = 180$, meaning $q = 50^{\circ}$.

Similarity and Ratios (Dog House Example)

• Ratio of Sides: The triangular dog house has sides in a ratio of $3 : 3 : 2$.

• Scaling Calculations:   - The shortest side (base) is $1\,m$.   - Using the ratio $x : x : 1 = 3 : 3 : 2$, we find the other sides by solving $\frac{2}{1} = \frac{3}{x}$, resulting in $x = 1.5\,m$.

• Similarity and Proportionality:   - The toy storage triangle is proportional to the main dog house triangle.   - If the toy storage base is $25\,cm$ and the scale ratio is $3 : 3 : 2$, the other two sides are computed: $\text{side} = \frac{3 \times 25}{2} = 37.5\,cm$.

Circular Geometry and Volume

• Fundamental Formulas:   - Circumference ($C$): $C = 2\pi r$   - Area ($A$): $A = \pi r^2$

• Pool and Sidewalk (Figure 8):   - Pool diameter = $10\,m$ (radius $r_1 = 5\,m$).   - Sidewalk width = $1\,m$ (outer radius $r_2 = 6\,m$).   - Area of Sidewalk: $A_{\text{outer}} - A_{\text{inner}} = \pi(6)^2 - \pi(5)^2 = 36\pi - 25\pi = 11\pi \, m^{2}$.   - Applying $\pi = 3.14$: $11 \times 3.14 = 34.54\,m^{2}$.

• Volume of Water (Figure 9):   - Volume ($V$) for a cylinder: $V = A \times h$.   - Pool area $A = 25\pi\,m^{2}$.   - Pool is split into two halves ($\frac{25\pi}{2}$ each).   - Adult part depth = $1.5\,m$; Children part depth = $0.6\,m$.   - Total Volume = $(\frac{25\pi}{2} \times 1.5) + (\frac{25\pi}{2} \times 0.6) = \frac{25\pi(2.1)}{2}$.

Rotational Mechanics

• Wheel Displacement (Figure 10):   - Diameter = $60\,cm$. Circumference = $60\pi \, cm$.   - After 5 rolls, distance traveled = $5 \times 60\pi = 300\pi \, cm$.   - Applying $\pi \approx 3.1416$ results in roughly $942.48\,cm$.

• Angular Rotation:   - Each full roll equals $360^{\circ}$.   - After 5 rolls, the total rotation of the pin is $5 \times 360 = 1800^{\circ}$.