💯Math for Non-Math Majors Unit 13 – Math and...

Key Concepts

• Mathematical concepts provide a foundation for understanding and solving problems in various fields
• Algebra involves using variables, equations, and inequalities to represent and solve problems
  • Variables are symbols (usually letters) that represent unknown quantities
  • Equations are mathematical statements that show two expressions are equal (3x + 5 = 14)
  • Inequalities are mathematical statements that compare two expressions using symbols like $<$, $>$, $\leq$, or $\geq$ (2x - 1 > 7)
• Geometry deals with the properties, measurement, and relationships of points, lines, angles, shapes, and solids
  • Points are exact positions or locations on a plane
  • Lines are straight paths that extend infinitely in both directions
  • Angles are formed when two lines or segments meet at a point (vertex)
• Trigonometry studies the relationships between the sides and angles of triangles
  • Sine, cosine, and tangent are trigonometric functions used to calculate unknown sides or angles in triangles
• Probability is the likelihood of an event occurring, expressed as a number between 0 and 1
  • Events with a probability of 0 are impossible, while events with a probability of 1 are certain
• Statistics involves collecting, analyzing, interpreting, and presenting data
  • Measures of central tendency (mean, median, mode) describe the center or typical value of a dataset
  • Measures of dispersion (range, variance, standard deviation) describe the spread or variability of a dataset

Formulas and Equations

• The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): $a^2 + b^2 = c^2$
• The quadratic formula is used to solve quadratic equations in the form $ax^2 + bx + c = 0$, where a, b, and c are constants and a ≠ 0: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• The distance formula calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a coordinate plane: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• The slope formula determines the steepness of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$
• The compound interest formula calculates the future value (FV) of an investment with an initial principal (P), annual interest rate (r), compounded n times per year, over t years: $FV = P(1 + \frac{r}{n})^{nt}$
• The area of a circle is calculated using the formula $A = \pi r^2$, where r is the radius of the circle
• The volume of a rectangular prism is found by multiplying its length (l), width (w), and height (h): $V = lwh$

Real-World Applications

• Budgeting and financial planning use mathematical concepts to manage income, expenses, savings, and investments
  • Creating a budget involves adding income sources and subtracting expenses to determine net income
  • Calculating compound interest helps in understanding the growth of investments over time
• Construction and engineering rely on geometry and trigonometry to design and build structures
  • The Pythagorean theorem is used to ensure buildings are square and level
  • Trigonometric functions help determine angles and distances in construction projects
• Cooking and baking require an understanding of fractions, ratios, and proportions to follow recipes and scale ingredients
  • Doubling or halving a recipe involves multiplying or dividing ingredient quantities by a factor of 2
• Sports analytics use statistics to evaluate player performance, team strategies, and game outcomes
  • Batting averages in baseball are calculated by dividing the number of hits by the number of at-bats
  • Field goal percentages in basketball are determined by dividing the number of successful shots by the total number of attempts
• Medicine and pharmacology employ mathematical concepts to determine proper dosages and treatment plans
  • Dosage calculations involve using patient weight, drug concentration, and desired dosage to determine the volume of medication to administer
• Polling and surveys use probability and statistics to gather and interpret data on public opinion
  • Random sampling helps ensure that a survey represents the population of interest
  • Margin of error calculations determine the accuracy of survey results

Problem-Solving Strategies

• Read and understand the problem carefully, identifying the given information, the unknown, and the desired outcome
• Break down complex problems into smaller, more manageable steps
  • Solving a multi-step equation may involve isolating variables on one side of the equation and then simplifying
• Look for patterns or similarities to previously solved problems
  • Recognizing that a word problem involves the Pythagorean theorem can help guide the solution process
• Use visualizations, such as diagrams or graphs, to represent the problem and clarify relationships
  • Drawing a sketch of a geometric problem can help identify the necessary measurements and formulas
• Work backwards from the desired outcome to determine the steps needed to reach the solution
  • In a problem involving compound interest, starting with the future value and working backwards can help determine the required initial investment
• Check the reasonableness of the solution by estimating or plugging the answer back into the original problem
• Reflect on the problem-solving process and consider alternative approaches or potential improvements for future problems

Common Mistakes to Avoid

• Misreading or misinterpreting the problem statement, leading to incorrect assumptions or solving the wrong problem
• Forgetting to include units or using inconsistent units throughout the problem-solving process
  • Mixing inches and centimeters in a measurement problem can result in significant errors
• Incorrectly applying formulas or using the wrong formula for a given problem
  • Using the area formula for a circle when the problem requires the circumference formula
• Making arithmetic errors, such as incorrect calculations or sign errors
  • Forgetting to distribute a negative sign when multiplying or dividing both sides of an equation
• Rounding too early in the problem-solving process, leading to a loss of precision in the final answer
  • Rounding intermediate calculations to two decimal places when the final answer requires four decimal places
• Neglecting to check the reasonableness of the solution or failing to interpret the results in the context of the problem
  • Obtaining a negative value for a measurement that should always be positive (length, area, volume)
• Rushing through the problem without taking the time to fully understand the concepts or verify the solution

Visual Aids and Diagrams

• Number lines can be used to represent integers, fractions, and real numbers, helping to visualize mathematical relationships and perform operations
  • Placing fractions on a number line can help compare their relative sizes and determine which is greater
• Coordinate planes (also called Cartesian planes) are two-dimensional spaces defined by a horizontal x-axis and a vertical y-axis, used to graph points, lines, and curves
  • Plotting points on a coordinate plane can help visualize the slope and y-intercept of a line
• Venn diagrams use overlapping circles to illustrate the relationships between sets, such as union, intersection, and complement
  • Venn diagrams can help solve probability problems involving multiple events
• Flowcharts are diagrams that represent a process or algorithm, using shapes and arrows to show the sequence of steps and decision points
  • Creating a flowchart can help break down a complex problem into a series of logical steps
• Bar graphs display data using rectangular bars of varying heights, allowing for easy comparison of categories or groups
  • Comparing the heights of bars in a graph can help identify trends or differences in data
• Pie charts use a circular graph divided into sectors to show the relative sizes of different categories as parts of a whole
  • Each sector's angle represents the proportion of the category relative to the total

Practice Problems

1. A rectangular garden has a length of 12 feet and a width of 8 feet. What is the area of the garden?
2. Solve for x: $3(2x - 5) = 24$
3. A bag contains 4 red marbles, 6 blue marbles, and 2 green marbles. If a marble is drawn at random, what is the probability of selecting a blue marble?
4. The height of a tree was 20 feet in 2010 and has been growing at a rate of 1.5 feet per year. What will be the height of the tree in 2025?
5. A store sells shirts for $15 each and pants for$25 each. If a customer buys 3 shirts and 2 pants, what is the total cost before tax?
6. Simplify the expression: $\frac{6a^2b^3}{3ab^2}$
7. Convert 3/5 to a decimal.

Additional Resources

• Khan Academy (https://www.khanacademy.org/) offers free online courses, instructional videos, and practice exercises covering a wide range of mathematical topics
• Wolfram MathWorld (https://mathworld.wolfram.com/) is an online resource providing definitions, formulas, and explanations for various mathematical concepts
• Desmos (https://www.desmos.com/) is an online graphing calculator that allows users to plot functions, create tables, and explore mathematical relationships
• Math is Fun (https://www.mathsisfun.com/) provides explanations, examples, and interactive tools for learning mathematical concepts in a simple and engaging way
• The Math Forum (https://mathforum.org/) is an online community where students, teachers, and math enthusiasts can ask questions, share ideas, and find resources
• Brilliant (https://brilliant.org/) offers challenging problem-solving courses and puzzles in mathematics, science, and engineering, designed to develop critical thinking skills
• YouTube channels like 3Blue1Brown (https://www.youtube.com/c/3blue1brown), numberphile (https://www.youtube.com/user/numberphile), and MathBFF (https://www.youtube.com/c/MathBFF) provide engaging and informative videos on various mathematical topics