Final Study Guide

Student Learning Outcomes

1. Real Numbers and Algebraic Properties

• Identify and use algebraic terminology

  • Understanding basic terms such as variables, coefficients, constants, expressions, equations, and terms.

• Apply properties of the Real Number System

• Recognize properties such as closure, commutative, associative, distributive, identity, and inverse properties.

• Perform binary operations on the set of real numbers

  • Operations include addition, subtraction, multiplication, and division among real numbers.

• Perform the standard order of operations

  • Use the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to evaluate expressions accurately.

• Classify real numbers

  • Distinguish between:

  • Natural numbers (e.g., 1, 2, 3, …)

  • Whole numbers (e.g., 0, 1, 2, 3, …)

  • Integers (e.g., -2, -1, 0, 1, 2)

  • Rational numbers (e.g., 1/2, 3.5)

  • Irrational numbers (e.g., $rac{ ext{Pi}}{4}$, $ext{e}$)

• Order the set of real numbers

  • Utilize symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

• Evaluate and simplify algebraic expressions

  • Solve expressions by substituting values and performing operations.

• Apply the properties of exponents

  • Know the laws of exponents for simplifying expressions, such as:

  • $a^m imes a^n = a^{m+n}$

  • $rac{a^m}{a^n} = a^{m-n}$

  • $(a^m)^n = a^{mn}$

  • $a^0 = 1$ if $a <br>eq 0$.

2. Solve and Apply Equations and Inequalities

• Solve linear equations with rational coefficients

  • Techniques involving isolating the variable to find its value.

• Solve linear inequalities

  • Similar processes to solve equations but require reversing the inequality symbol when multiplying or dividing by a negative number.

• Express solutions to linear inequalities

  • Use interval notation (e.g., $(- ext{inf}, 2]$) and graph solutions on number lines.

• Solve applications involving ratios, rates, and percent

  • Applications that may require setting up proportion equations or percentage calculations.

• Translate English sentences into algebraic expressions/equations

  • Convert word problems into mathematical form.

• Solve compound inequalities

  • Handle inequalities that involve two or more conditions simultaneously,

  • For example: x > 2 AND x < 5.

3. Graphing and the Cartesian Coordinate System

• Use the terminology and notation of the Cartesian coordinate system

  • Understanding coordinates (x, y), quadrants, and axes.

• Plot and interpret graphs without graphing technology

  • Learn how to manually graph equations and interpret their results.

• Determine and interpret the slope of a line

  • Calculate slope using the formula: $m = rac{y_2 - y_1}{x_2 - x_1}$.

  • Understand the implications of positive, negative, zero, and undefined slopes, including concepts of parallel (same slope) and perpendicular lines (negative reciprocal slopes).

• Graph and write linear equations using slope-intercept and point-slope forms

  • Slope-intercept form: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

  • Point-slope form: $y - y_1 = m(x - x_1)$.

• Determine and recognize x- and y-intercepts

  • Calculate intercepts by setting x = 0 for y-intercept and y = 0 for x-intercept.

• Determine whether the graph of an equation is a function

  • Use the vertical line test: if a vertical line intersects the graph at more than one point, the graph is not a function.

• Identify domain and range

  • Domain: all possible x-values (inputs) of a function.

  • Range: all possible y-values (outputs) of a function.

• Create and interpret scatter plots and regression lines

  • Develop skills to represent data visually and analyze relationships.

  • Identify trends through regression analysis to predict outcomes.

4. Numerical Reasoning

• Read, interpret, and make decisions based upon data from line graphs, bar graphs, and charts

  • Analyzing visual data representations for insights and decision-making.

• Solve applications involving area, perimeter, volume, and surface area

  • Utilize formulas appropriately, such as:

  • Area of rectangle: $A = l imes w$

  • Perimeter of rectangle: $P = 2(l + w)$

  • Volume of rectangular prism: $V = l imes w imes h$

  • Surface area of rectangular prism: $SA = 2lw + 2lh + 2wh$.

• Find mean, median, and mode of a data set

  • Mean: average value, calculated as $rac{ ext{sum of values}}{ ext{number of values}}$.

  • Median: middle value when data set is ordered.

  • Mode: value that appears most frequently in the data set.

• Convert within and between English and metric units

  • Familiarity with conversions (e.g., inches to centimeters, pounds to kilograms).

• Compute experimental and theoretical probabilities

  • Experimental probability: ratio of successful outcomes to the total number of trials.

  • Theoretical probability: ratio of favorable outcomes to the total number of possible outcomes.

• Translate between standard and scientific notation

  • Understanding how to express large or small numbers concisely: e.g., $3000 = 3 imes 10^3$.

• Perform operations with numbers in scientific notation

  • Rules include adding/subtracting coefficients while keeping the exponent, or multiplying coefficients and adding exponents when multiplied.

5. Modeling

• Evaluate functions

  • Assess functions through various means including substitution and graphical analysis.

• Determine and interpret the rate of change

  • Rate of change measures how a quantity changes concerning another; in linear functions, it corresponds to the slope.

• Model with linear and exponential equations

  • Develop equations that represent real-world situations mathematically.

• Solve applications using linear and exponential models

  • Apply learned models to practical problems to derive solutions, including business, science, and engineering contexts.

Student Learning Outcomes

1. Real Numbers and Algebraic Properties

• Identify and use algebraic terminology

  • Example: Variables (e.g., x, y), coefficients (e.g., in 3x, 3 is the coefficient), constants (e.g., 5 in x + 5), expressions (e.g., 2x + 3), equations (e.g., 2x = 6), and terms (e.g., 3x and 4 in 3x + 4).

• Apply properties of the Real Number System

  • Example: Closure: 2 + 3 = 5 (closed under addition), 4 * 5 = 20 (closed under multiplication).

• Perform binary operations on the set of real numbers

  • Example: Addition: 3 + 4 = 7, Subtraction: 10 - 7 = 3, Multiplication: 6 * 2 = 12, Division: 8 / 2 = 4.

• Perform the standard order of operations

  • Example: Evaluate 3 + 5 * 2. Step 1: Multiply first, 5 * 2 = 10. Step 2: Add, 3 + 10 = 13. So, 3 + 5 * 2 = 13.

• Classify real numbers

  • Example: Natural numbers: 1, 2, 3, …; Whole numbers: 0, 1, 2, 3, …; Integers: -2, -1, 0, 1, 2; Rational numbers: 1/2, 3.5; Irrational numbers: $\frac{ ext{Pi}}{4}$, $ext{e}$.

• Order the set of real numbers

  • Example: For 3 and 5, since 3 < 5, we write 3 < 5. For -1 and 2, -1 < 2.

• Evaluate and simplify algebraic expressions

  • Example: If x = 2, evaluate 3x + 4. Step 1: Substitute, 3(2) + 4. Step 2: Multiply, 6 + 4 = 10.

• Apply the properties of exponents

  • Example: Simplifying $a^2 imes a^3$: Use the property $a^m imes a^n = a^{m+n}$, so it becomes $a^{2+3} = a^5$.

2. Solve and Apply Equations and Inequalities

• Solve linear equations with rational coefficients

  • Example: Solve 2x + 3 = 11. Step 1: Subtract 3 from both sides: 2x = 8. Step 2: Divide by 2: x = 4.

• Solve linear inequalities

  • Example: Solve 3x - 5 < 1. Step 1: Add 5: 3x < 6. Step 2: Divide by 3: x < 2.

• Express solutions to linear inequalities

  • Example: For x < 2, it can be written in interval notation as (-∞, 2) and graphed on the number line.

• Solve applications involving ratios, rates, and percent

  • Example: If 20% of a number is 60, find the number. Set up the equation: 0.2x = 60. Step 1: Divide by 0.2: x = 300.

• Translate English sentences into algebraic expressions/equations

  • Example: “Three times a number increased by 2” translates to 3x + 2.

• Solve compound inequalities

  • Example: Solve 2 < x < 5. This means x is greater than 2 and less than 5. The solution is (2, 5).

3. Graphing and the Cartesian Coordinate System

• Use the terminology and notation of the Cartesian coordinate system

  • Example: The point (2, 3) means move 2 units right along the x-axis and 3 units up along the y-axis.

• Plot and interpret graphs without graphing technology

  • Example: To plot y = 2x, if x = 0, then y = 0; if x = 1, then y = 2; plot points (0,0) and (1,2) and connect them.

• Determine and interpret the slope of a line

  • Example: For points (1, 2) and (3, 4), slope m = (y_2 - y_1) / (x_2 - x_1) = (4-2)/(3-1) = 2/2 = 1.

• Graph and write linear equations using slope-intercept and point-slope forms

  • Example: For slope-intercept form y = 2x + 3 (slope 2, y-intercept 3), start at (0, 3) and go up 2 for every 1 unit right.

• Determine and recognize x- and y-intercepts

  • Example: For y = 2x - 6, find x-intercept by setting y = 0: 0 = 2x - 6 gives x = 3. The x-intercept is (3, 0).

• Determine whether the graph of an equation is a function

  • Example: The equation y^2 = x is not a function since it fails the vertical line test (a vertical line can intersect it at two points).

• Identify domain and range

  • Example: For the function f(x) = x^2, the domain is all real numbers, and the range is y >= 0 (only non-negative outputs).

• Create and interpret scatter plots and regression lines

  • Example: Plot data points for hours studied vs. exam scores, draw a line that best fits through the points to see if there's a trend.

4. Numerical Reasoning

• Read, interpret, and make decisions based upon data from line graphs, bar graphs, and charts

  • Example: In a bar graph showing sales over months, see which month had the highest sales and make decisions accordingly.

• Solve applications involving area, perimeter, volume, and surface area

  • Example: For a rectangle with length 5 and width 2, Area = l * w = 5 * 2 = 10. Perimeter = 2(l + w) = 2(5 + 2) = 14.

• Find mean, median, and mode of a data set

  • Example: For data 3, 5, 3, 8, Mean = (3 + 5 + 3 + 8) / 4 = 4.75, Median (when ordered 3, 3, 5, 8) = 3.5, Mode = 3.

• Convert within and between English and metric units

  • Example: To convert 10 inches to centimeters: 10 in * 2.54 cm/in = 25.4 cm.

• Compute experimental and theoretical probabilities

  • Example: Experimental probability of rolling a 3 on a die after 50 rolls with 7 threes is 7/50. Theoretical probability is 1/6 since there is one 3 in six total outcomes.

• Translate between standard and scientific notation

  • Example: The number 5000 in scientific notation is $5 imes 10^3$.

• Perform operations with numbers in scientific notation

  • Example: $(2 imes 10^3) + (3 imes 10^3) = 5 imes 10^3$.

5. Modeling

• Evaluate functions

  • Example: For the function f(x) = x^2, evaluate f(3) = 3^2 = 9.

• Determine and interpret the rate of change

  • Example: In a linear function where output increases from 2 to 5 when input increases from 1 to 4, Rate of Change = (5-2) / (4-1) = 3/3 = 1.

• Model with linear and exponential equations

  • Example: Linear: y = 2x + 1, Exponential: y = 3(2^x); use them to model growth in populations or financial risks.

• Solve applications using linear and exponential models

  • Example: If a car depreciates 20% each year, use an exponential model to find its value after n years: V = V_0 * (0.8)^n where V_0 is the initial value.