HESI Mathematics Preparation: Data Interpretation, Geometry, and Conversions

Overview of HESI Mathematics

• Subject Areas:     * Basic Operations     * Fractions, Percentages, and Related Concepts     * Algebra     * Basic Concepts (Tables, Charts, and Graphs)     * Geometry (Triangles, Quadrilaterals, and Circles)     * Conversions (Using Conversion Fractions)

Tables, Charts, and Graphs: Interpreting Relevant Information

• Bivariate Data:     * Definition: Data involving two variables ($x$ and $y$).     * Relation: The variables may or may not be related.     * Examples:         * Height and weight of students.         * Temperature and amount of ice cream sold each day.         * Hours of study and grade on an exam.         * Shoe size and number of siblings.

• Types of Data Presentation:     * List: A straightforward sequence of data values.         * Example: Ryan’s grades on ten quizzes: 93, 81, 95, 72, 87, 85, 98, 88, 93, 79.     * Table: Data organized in rows and columns.         * Example: Population of Largest Countries (Source: worldpopulationreview.com, 2018):             1. China: 1,415,045,928             2. India: 1,354,051,854             3. United States: 326,766,748             4. Indonesia: 266,794,980             5. Brazil: 210,867,954             6. Pakistan: 200,813,818     * Line Graph: Displays two numeric parameters where the independent variable is on the horizontal (x) axis and the dependent variable is on the vertical (y) axis.     * Scatterplot: Displays two parameters to show if data is correlated. It can illustrate relationships but cannot prove cause and effect.

• Critical Elements to Observe in Visual Data:     * Title: Identifies what the data represents.     * Legend (Key): Explains the meaning of shapes, colors, or symbols.     * Axes: Requires checking for the scale, specific labels, and units of measurement.     * Notes or Captions: Provides additional context or source information.

• Statistical Metrics:     * Mean Value: The average of a set of data points.         * Notation: $\bar{x}$ or \textmu.         * Formula: $\bar{x} = \frac{x_1 + x_2 + x_3 + \dots + x_n}{n}$         * Example Case: Find the mean of 1, 3, 4, 4, 6, 8, 9 ($n = 7$).             * $\bar{x} = \frac{1+3+4+4+6+7+8+9}{7}$             * $\bar{x} = \frac{42}{7} = 6$     * Mean Rate of Change: Used for bivariate data to find the slope of the line.         * Formula: $\frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1} = \text{slope of the line}$

• Estimation Techniques:     * Interpolation: Estimating a data point that falls between known data points.         * Example: Estimating plant height on day 4 when measurements exist for days 2 and 5 (result: $20\,mm$).     * Extrapolation: Estimating data points that lie beyond the known data set.         * Example: Estimating plant height on day 10 based on a trend ending at day 7 (result: $50\,mm$).

Variables and Evaluating Information

• Variable Distinction:     * Independent Variable ($x$): The "Cause." This variable does not depend on the other.     * Dependent Variable ($y$): The "Effect." This variable depends on the independent variable.     * Case Study: Ice cream sales versus daily high temperature. The temperature affects sales; therefore, Temperature is independent, and Sales are dependent.

• Measures of Central Tendency:     * Mean: The average calculated by summing all values and dividing by the count ($n$).         * Example: Mean of 3, 4, 11, 15, and 17 is $\frac{50}{5} = 10$.     * Median: The middle value when numbers are placed in ascending order.         * Example: For 3, 8, 5, 2, 6, order them as 2, 3, 5, 6, 8. The median is 5.     * Mode: The data point that occurs most frequently.         * Example: In the set 3, 3, 0, 8, 16, 8, 3, 5, 2, the mode is 3.

• Measures of Dispersion:     * Range: The difference between the largest and smallest values.         * Example: In the set 28, 29, 98, 26, 51, 87:             * Largest = 98, Smallest = 26.             * Range = $98 - 26 = 72$.

• Data Distributions:     * Normal Distribution: A bell-shaped curve where Mean, Median, and Mode are typically centered.     * Uniform Distribution: A constant distribution where data values are spread evenly between points $a$ and $b$.     * Symmetric Distribution: The right and left sides are mirror images of each other.     * Skewness: Data can be Skewed Left, Normal, or Skewed Right.     * Modality:         * Unimodal: A single clear peak (one mode).         * Bimodal: Two clear peaks (two modes).         * Neither: No modes or more than two modes.

• Trends and Outliers:     * Trend Analysis: Observing if data is always increasing/decreasing, changing speed, or staying constant.     * Unexpected Values (Outliers): Data points that lie far from the expected cluster. These may be included or ignored depending on the purpose of the data usage.

Relationship Between Variables

• Correlation (Covariance):     * Correlated: Parameters are related.     * Positive Correlation: Both variables increase or decrease together.     * Negative Correlation: One variable increases as the other decreases.     * No Correlation: No discernible relationship between the two parameters.

• Linear vs. Non-Linear Relationships:     * Linear: The scatter plot follows a relatively straight line.     * Non-Linear: Parameters are related, but the scatter plot does not follow a straight line.

Geometry: Perimeter, Circumference, and Area

• Perimeter ($P$): The distance around the edge of a polygon.     * Square: $P = 4s$ (where $s$ is side length).     * Rectangle: $P = 2l + 2w$ (where $l$ is length and $w$ is width).     * Case Study (Garden Fencing): For a garden with segments of $7\,ft$, $7\,ft$, $14\,ft$, $4\,ft$, $7\,ft$, and $3\,ft$, the perimeter is $P = 42\,ft$.

• Circumference ($C$): The distance around the edge of a circle.     * Formula: $C = 2\pi r$ (where $r$ is radius).     * Diameter ($d$): $d = 2r$.     * Example (Circular Park): If radius $r = 500\,ft$:         * Path across center (Diameter) = $1000\,ft$.         * Path around edge (Circumference) = $2 \times 3.14 \times 500 \approx 3140\,ft$.

• Area ($A$): The amount of space a shape covers.     * Triangle: $A = \frac{1}{2}bh$         * Example: A triangle with base and height of $20\,ft$ has an area of $\frac{1}{2}(20\,ft)(20\,ft) = 200\,ft^2$.     * Square: $A = s^2$     * Rectangle: $A = l \times w$         * Example: A garden with area $35\,ft^2$ and length $7\,ft$ must have a width of $\frac{35}{7} = 5\,ft$.     * Circle: $A = \pi r^2$         * Example: A $12\text{"}$ pizza has a diameter of $12\text{"}$, so $r = 6\text{"}$.         * $A \approx (3.14)(6\,in)^2 = 113\,in^2$.

• Irregular Shapes:     * Additive Method: Break the shape into multiple rectangles and add their areas together (e.g., $A = 6 \times 1 + 5 \times 2 + 6 \times 3 = 34$).     * Subtractive Method: Find the area of a large shape (like a square) and subtract the area of a missing piece (like a circle).

Geometry: Volume, Surface Area, and Arcs

• Volume ($V$): The space inside a 3D solid.     * Cylinder Formula: $V = \pi r^2 h$     * Example: A cylinder with height $3\,in$ and diameter $4\,in$ ($r = 2\,in$).         * $V \approx 3.14 \times (2\,in)^2 \times (3\,in) = 37.68\,in^3$.

• Surface Area ($SA$): The total area of the exterior surface.     * Sphere Formula: $SA = 4\pi r^2$     * Example: A sphere with radius $5\,cm$.         * $SA \approx 4 \times 3.14 \times (5\,cm)^2 = 314\,cm^2$.

• Units of Measurement:     * Length (1D): $ft, in, cm, m$     * Area (2D): $ft^2, in^2, cm^2, m^2$     * Volume (3D): $ft^3, in^3, cm^3, m^3$ (Note: $cm^3 = cc$).

• Arcs and Central Angles:     * Arc: A portion of the circumference.     * Central Angle: An angle with its vertex at the center of the circle.     * Proportion Formula: $\frac{\text{Angle Measure}}{360^\circ} = \frac{\text{Arclength}}{\text{Circumference}}$     * Problem 1: Circle with circumference $25\,cm$ and angle $36^\circ$.         * $\frac{36^\circ}{360^\circ} = \frac{\text{Arclength}}{25\,cm}$         * $\text{Arclength} = 2.5\,cm$     * Problem 2: Circle with circumference $9\,in$ and arclength $3\,in$.         * $\frac{\text{Angle}}{360^\circ} = \frac{3\,in}{9\,in} = \frac{1}{3}$         * $\text{Angle} = 120^\circ$

Measurement and Data: Conversions

• Common Units:     * Length: Metric (Meters); US (Inches, feet, yards, miles).     * Mass/Weight: Metric (Grams); US (Ounces, pounds, tons).     * Volume: Metric (Liters); US (Teaspoons, tablespoons, fluid ounces, cups, pints, quarts, gallons).

• Conversion Fractions Method:     * Start with a conversion fact (e.g., $1\text{ gallon} = 4\text{ quarts}$).     * Create a fraction where the numerator equals the denominator (equals 1).     * Set up the fraction so that the original units are canceled out.     * Example: Convert $32\text{ quarts}$ to gallons.         * $\frac{32\,quarts}{1} \times \frac{1\,gallon}{4\,quarts} = \frac{32}{4}\,gallons = 8\,gallons$.

• Complex multi-step conversion:     * Problem: Convert $100\,cc$ to ounces.     * Facts:         * $1\,cc = 1\,ml$         * $1\,liter \approx 1.06\,quarts$         * $1\,quart = 4\,cups$         * $1\,cup = 8\,ounces$     * Calculation:         * $100\,cc \times \frac{1\,ml}{1\,cc} \times \frac{1\,L}{1000\,ml} \times \frac{1.06\,qt}{1\,L} \times \frac{4\,c}{1\,qt} \times \frac{8\,oz}{1\,c}$         * $\frac{100 \times 1 \times 1 \times 1.06 \times 4 \times 8}{1000} \approx 3.392\,oz$