Praxis Core Academic Skills for Educators: Mathematics (5733) flashcards

Modeling exponential growth when the growth period is different from the unit of time used for the independent variable.

• The exponent must represent the number of growth periods that have occurred.
• If the growth period is 'k' units and the time variable is 't' in those units, the exponent is t/k.
• Ensure consistency in time units (e.g., if growth is every 30 min, and 't' is in hours, convert 30 min to 0.5 hours, so the exponent is t/0.5 or 2t).

Memory trick: Match the units! Doubling period must fit the time variable!

For a quadratic function f(x) = ax^2 + bx + c with a > 0, the minimum value occurs at the vertex, whose x-coordinate is -b/(2a).

• Parabola opens upwards (a > 0).
• Vertex coordinates: (-b/2a, f(-b/2a)).
• The y-coordinate of the vertex is the minimum value.

Memory trick: Positive 'A' means a Valley, Vertex is the Lowest Point.

When a quantity grows by a constant factor over a fixed period (e.g., doubles every X hours), the exponent in the exponential model is the total time divided by that period.

• Formula: N = N_0 * (factor)^(t / period)
• N_0 is the initial amount.
• Factor is the multiplier (e.g., 2 for doubling, 3 for tripling).
• Period is the time it takes for one growth cycle.

Memory trick: Time Over Period is the Exponent Key.

Mixture problems involve combining two or more quantities with different concentrations to achieve a desired final concentration; solved using a system of linear equations.

• One equation for total quantity.
• Another equation for total amount of specific component (e.g., acid, solute).
• Solve using substitution or elimination.

Memory trick: Volume Plus Concentration Equals Solution.

To find the difference between function values at two points, evaluate the function at each point separately and then subtract the results.

• Substitute the first input value into the function.
• Substitute the second input value into the function.
• Subtract the two resulting output values.

Memory trick: Calculate Each, Then Subtract.

For a quadratic function in factored form f(x) = a(x - r1)(x - r2), the x-coordinate of the vertex is the average of the roots (r1 + r2)/2.

• Roots (x-intercepts) are r1 and r2.
• Axis of symmetry is x = (r1 + r2)/2.
• Substitute the x-coordinate of the vertex into the function to find the maximum/minimum y-value.

Memory trick: Roots Average to Vertex X, then Plug for Y.

A linear trend describes a pattern in data where the relationship between two variables can be approximated by a straight line, showing a consistent rate of change.

• Represents a constant rate of increase or decrease.
• Can be used for prediction within the observed range or slightly beyond.
• Often visualized with a scatter plot and a line of best fit.

Memory trick: Line up your points, then project your future score!

For a linear function, the average rate of change between any two points is constant and equal to the slope of the line.

• Formula: (y2 - y1) / (x2 - x1)
• For y = mx + b, rate of change is 'm'.
• Represents how much the dependent variable changes per unit change in the independent variable.

Memory trick: Linear Means Slope is Always the Rate.

Evaluating a polynomial involves substituting a specific numerical value for the variable and then performing the indicated arithmetic operations to find the polynomial's value.

• Substitute the value for every instance of the variable.
• Follow the order of operations (PEMDAS/BODMAS).
• The result is the value of the polynomial at that specific point.

Memory trick: Test Each Option, Plug and Play.

When a polynomial function is evaluated at x=0, the result is always the constant term of the polynomial.

• All terms with 'x' become zero.
• The constant term is the y-intercept.
• It represents the initial value or starting point when x signifies time or distance.

Memory trick: Plug in value, then calculate sums.

In an exponential growth model P(t) = P₀(1+r)^t, the growth factor (1+r) indicates the multiplicative increase per time period.

• It's the base of the exponent.
• If > 1, it represents growth; if < 1, it represents decay.
• It is 1 + the growth rate (as a decimal).

Memory trick: Initial Ants Grow by Rate Times Time.

For a quadratic function f(x) = ax^2 + bx + c with a < 0, the maximum value occurs at the vertex.

• The x-coordinate of the vertex is given by -b/(2a).
• Substitute this x-value into the function to find the maximum y-value.
• If a > 0, the vertex represents a minimum value instead.

Memory trick: Vertex is the peak or pit, -b/2a is the right fit.

For a quadratic function in vertex form, f(x) = a(x - h)^2 + k, the vertex is at the point (h, k).

• The 'h' value represents the x-coordinate of the vertex.
• The 'k' value represents the y-coordinate of the vertex, which is the maximum or minimum value of the function.
• If 'a' is negative, 'k' is the maximum; if 'a' is positive, 'k' is the minimum.

Memory trick: Vertex is home, (h, k) is the known.

To find the initial value (N_0 or P_0) in an exponential growth model (N(t) = N_0 * b^t), given a future value N(t) at a specific time t, rearrange the formula to N_0 = N(t) / b^t.

• The initial value is the amount at time t=0.
• The base 'b' is the growth factor.
• Requires knowing a value at a time t > 0.

Memory trick: Go BACK in Time, DIVIDE by the growth!

This concept involves calculating a total change over time by repeatedly subtracting a fixed amount for each discrete time interval.

• Determine the total elapsed time.
• Identify the duration of each interval.
• Calculate the number of intervals that occur within the total time.
• Multiply the change per interval by the number of intervals.

Memory trick: Track the time, then decrease, to see the final temperature's ease.

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10. To convert to standard notation, move the decimal point according to the exponent of 10.

• Positive exponent means move decimal to the right (larger number).
• Negative exponent means move decimal to the left (smaller number).
• The number of places moved equals the absolute value of the exponent.

Memory trick: Exponent positive, decimal right. Exponent negative, decimal left.

The distance formula calculates the length of a line segment between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane.

• Derived from the Pythagorean theorem.
• Formula: d = √((x₂ - x₁)² + (y₂ - y₁)²).
• Always yields a non-negative value.

Memory trick: Coordinates are far when you square the differences, add, then root.

A proportion is an equation stating that two ratios are equal. When one or more terms in a proportion are fractions, cross-multiplication can be used to solve for an unknown value.

• A ratio compares two quantities.
• In a proportion a/b = c/d, the cross products are equal (ad = bc).
• Fractions can be terms in a ratio or the ratio itself.

Memory trick: Ratios equal, cross-multiply and solve.

Evaluating a polynomial function means substituting a specific numerical value for the variable and then performing the indicated arithmetic operations (following order of operations) to find the function's output.

• Substitute the value for every instance of the variable.
• Follow PEMDAS/BODMAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
• Careful calculation is crucial, especially with powers and negative signs.

Memory trick: Plug It In, Power Up, Multiply, Then Add/Subtract!

To calculate a fraction of a whole number, multiply the fraction by the whole number. This operation determines what portion of the total quantity the fraction represents.

• The word 'of' in math often indicates multiplication.
• Multiply the numerator by the whole number, then divide by the denominator.
• Can be used to find parts of quantities, percentages, or ratios.

Memory trick: Fraction Times Whole, Find the Part!

A unit rate describes how many units of the first quantity correspond to one unit of the second quantity. It is often expressed as 'quantity per one unit of another quantity'.

• Calculated by dividing the first quantity by the second quantity.
• Useful for comparing different rates or values.
• Examples: miles per hour, cost per item, units per hour.

Memory trick: Divide to Find the One!

To divide mixed numbers, first convert them into improper fractions. Then, multiply the first fraction by the reciprocal of the second fraction. Finally, simplify the result if necessary.

• Convert mixed numbers to improper fractions: a b/c = (ac + b)/c.
• Dividing by a fraction is equivalent to multiplying by its reciprocal.
• The reciprocal of a/b is b/a.

Memory trick: Improper Flip Multiply!

For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex represents the maximum or minimum value of the function. If a > 0, the parabola opens upwards and the vertex is a minimum. If a < 0, the parabola opens downwards and the vertex is a maximum.

• The x-coordinate of the vertex is given by the formula x = -b/(2a).
• The y-coordinate of the vertex is found by substituting the x-coordinate back into the function: y = f(-b/(2a)).
• This concept is crucial for optimization problems in various fields like business, engineering, and physics.

Memory trick: Vertex Vibe: 'a' decides the curve, -b/2a finds the peak!

To find the percent of a number, convert the percentage to a decimal by dividing by 100, then multiply that decimal by the given number.

• Percent means 'per hundred' or 'out of 100'.
• To convert a percent to a decimal, divide by 100 (move decimal two places left).
• The word 'of' in math problems often indicates multiplication.

Memory trick: Percent to decimal, then multiply the whole.

The Zero Product Property states that if the product of two or more factors is zero, then at least one of the individual factors must be zero. It is commonly used to solve quadratic equations in factored form.

• If A * B = 0, then A = 0 or B = 0 (or both).
• This property is fundamental for solving equations by factoring.
• It helps find the x-intercepts (roots/zeros) of polynomial functions.

Memory trick: Factors to zero, means each part's a hero!

The area of a rectangle is the measure of the two-dimensional space it occupies, calculated by multiplying its length and width.

• Units for area are always square units (e.g., square feet, square meters).
• Area is used to calculate surface coverage.
• Formula: Area = Length × Width.

Memory trick: Rectangles love 'L' and 'W' for their space.

Evaluating a linear function means substituting a given input value into the function's equation to find the corresponding output value.

• Linear functions are of the form f(x) = mx + b.
• The input is typically represented by 'x' (or 'd' in this case).
• The output is represented by 'f(x)' (or 'H(d)').

Memory trick: Plug in the day, get the height display!

Exponential functions (with base > 1) exhibit growth that increases over time, while linear functions exhibit constant growth. Exponential growth ultimately surpasses linear growth.

• Linear functions change by a constant amount per unit interval.
• Exponential functions change by a constant ratio (or percentage) per unit interval.
• For positive growth, exponential functions always eventually exceed linear functions.

Memory trick: Exponential ALWAYS Wins the Long Race!

Function evaluation is the process of determining the value of a function for a given input value. It involves substituting the input into the function's expression.

• Replace the variable with the given input value.
• Follow the order of operations (PEMDAS/BODMAS) to simplify.
• The result is the output value of the function for that input.

Memory trick: Input goes IN, Output comes OUT!

Division with decimals involves dividing a number by another number that contains a decimal point. This often requires moving the decimal point in both the divisor and dividend to make the divisor a whole number.

• Can be conceptualized as repeatedly subtracting the divisor from the dividend.
• Moving decimal points in both numbers by the same amount does not change the quotient.
• Results in a quotient and potentially a remainder or a repeating decimal.

Memory trick: Total length, divided by piece, gives how many pieces.

The volume of a cube is the amount of three-dimensional space it occupies, calculated by raising its side length to the power of three.

• A cube has equal length, width, and height.
• Units for volume are always cubic units (e.g., cubic inches, cubic meters).
• Formula: Volume = side³.

Memory trick: Cubes hold volume by multiplying their side, side, side.

Proportional reasoning involves understanding how quantities relate to each other in a consistent ratio. It is used to solve problems where one quantity changes in direct relation to another, often by setting up and solving proportions.

• A proportion is an equality between two ratios.
• Cross-multiplication is a common method to solve for an unknown in a proportion.
• Unit rates are a specific form of proportional reasoning.

Memory trick: Ratio up, cross-multiply down, solve around.

To multiply mixed numbers, first convert each mixed number into an improper fraction. Then, multiply the numerators together and multiply the denominators together. Finally, simplify the resulting improper fraction back into a mixed number or simplest form.

• A mixed number is a whole number and a fraction.
• An improper fraction has a numerator greater than or equal to its denominator.
• Multiplying fractions involves multiplying numerators and denominators straight across.

Memory trick: Improper fractions first, then multiply straight across, finally simplify.

To convert a number from scientific notation (a × 10ⁿ) to standard notation, move the decimal point in the coefficient 'a' by 'n' places. Move right for a positive 'n' and left for a negative 'n'.

• Positive exponent (n > 0) means a large number; move decimal right.
• Negative exponent (n < 0) means a small number; move decimal left.
• The number of places to move the decimal is equal to the absolute value of the exponent.

Memory trick: Exponent's Sign Shows Decimal's Direction!

A compound inequality combines two or more inequalities. To solve it, apply inverse operations to all parts (left, middle, right) simultaneously to isolate the variable in the middle.

• Operations must be applied to all three parts of the inequality.
• Multiplying or dividing by a negative number reverses the inequality signs.
• The goal is to isolate the variable in the middle.

Memory trick: Balance the middle, move to all sides, then divide!

A unit rate expresses how much of one quantity there is per one unit of another quantity. It is typically calculated by dividing the total amount of the first quantity by the total amount of the second quantity.

• Unit rates simplify comparisons between different situations.
• Common unit rates include miles per hour, cost per item, and residents per square mile.
• The word 'per' often indicates division.

Memory trick: Divide total quantity by total units to get 'per one'.

The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) are the coordinates of the vertex. If a > 0, the parabola opens upwards and (h, k) is the minimum point. If a < 0, it opens downwards and (h, k) is the maximum point.

• Immediately reveals the vertex (h, k).
• h is the x-coordinate of the vertex (axis of symmetry).
• k is the y-coordinate of the vertex (minimum or maximum value).

Memory trick: Vertex is (H, K), just like a King's Home!

For a quadratic function in standard form f(x) = ax^2 + bx + c, the vertex is the point (h, k) where h = -b/(2a) and k = f(h). This vertex represents the maximum point if a < 0 or the minimum point if a > 0.

• The x-coordinate of the vertex is -b/(2a).
• The y-coordinate of the vertex is f(-b/(2a)).
• If 'a' is negative, the parabola opens downward, and the vertex is a maximum.
• If 'a' is positive, the parabola opens upward, and the vertex is a minimum.

Memory trick: Vertex formula: find the peak, then the height you seek!

A recursive sequence defines each term based on one or more preceding terms. To evaluate a specific term, you must calculate all preceding terms starting from the initial term(s).

• Requires an initial term (or terms) to start the sequence.
• Each term is generated by applying a rule to the previous term(s).
• Calculation must be done step-by-step, in order.

Memory trick: Previous term is your NEXT step!

Exponential growth occurs when a quantity increases at a rate proportional to its current value, often characterized by a constant doubling or multiplying period.

• Involves repeated multiplication by a constant factor.
• Can be modeled by N = N₀ * (factor)^(t/T), where N₀ is initial amount, factor is growth factor, t is total time, and T is growth period.
• Leads to very rapid increases over time.

Memory trick: Grow More Powerfully Every Time!

To divide a mixed number by a fraction, first convert the mixed number to an improper fraction. Then, multiply the improper fraction by the reciprocal of the divisor fraction.

• A mixed number combines a whole number and a fraction.
• An improper fraction has a numerator greater than or equal to its denominator.
• The reciprocal of a fraction a/b is b/a.

Memory trick: Keep, Change, Flip (KCF): Keep the first, change to multiply, flip the second.

To compare two different rates, calculate the unit rate for each scenario (amount per one unit of time, quantity, etc.). Then, directly compare these unit rates to determine which is higher or lower.

• A unit rate expresses a quantity per single unit of another quantity.
• Comparison requires both rates to be in the same 'per unit' format.
• The difference between unit rates indicates the magnitude of the disparity.

Memory trick: Each rate, then compare. Divide, then subtract.

Exponential growth occurs when a quantity increases by a constant factor over equal time intervals. The formula is P(t) = P₀ * (1 + r)ᵗ or P(t) = P₀ * (factor)ᵗ, where 't' is the number of growth intervals.

• P₀ is the initial amount.
• 'factor' is the multiplier for each interval (e.g., 2 for doubling, 3 for tripling).
• 't' must represent the number of growth periods, not necessarily raw time units.

Memory trick: Initial times factor to the power of intervals.