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

To add or subtract numbers in scientific notation, first adjust one or both numbers so they have the same exponent. Then, add or subtract their coefficients (mantissas) and keep the common exponent.

• Exponents must be the same before adding/subtracting
• Adjust mantissa if exponent is changed (e.g., 3.5 x 10^-5 = 0.35 x 10^-4)
• Add/subtract mantissas, keep common exponent

Memory trick: Scientific: Shift the dot, match the power, calculate.

When a total quantity is divided into parts according to a ratio, sum the ratio parts to find the total parts. Divide the total quantity by the total parts to find the value of one part, then multiply by each ratio component.

• Ratios express relative proportions.
• Summing ratio components gives the total 'parts'.
• Each part's quantity is found by multiplying its ratio component by the 'value per part'.

Memory trick: Divide total by parts, then multiply for each share.

To divide a decimal by a fraction, convert the decimal to a fraction or convert the fraction to a decimal. Then perform the division (or multiply by the reciprocal if using fractions).

• Convert decimal to fraction: 3.5 = 7/2.
• Convert fraction to decimal: 1/4 = 0.25.
• Division by fraction is equivalent to multiplication by its reciprocal.

Memory trick: Decimal to fraction, then flip and multiply; or fraction to decimal, then simply divide!

Tolerance defines an acceptable deviation from an ideal value. A quantity is within tolerance if its absolute difference from the ideal value is less than or equal to the specified tolerance. Outside tolerance means rejection.

• Tolerance creates an upper and lower bound.
• Upper Bound = Ideal + Tolerance.
• Lower Bound = Ideal - Tolerance.
• Values outside this range are rejected.

Memory trick: Ideal plus/minus tolerance, defines the acceptance fence!

To find individual quantities when given a ratio and a total amount, sum the ratio parts, divide the total by this sum to find the value of one 'part', then multiply by each ratio number.

• The sum of ratio parts represents the total 'units' of the mixture.
• Divide the total quantity by the sum of parts to find the value of one part.
• Multiply the value of one part by each ratio number to find individual quantities.

Memory trick: Parts add up, total divides, then multiply for real-world sizes!

To convert a small decimal (less than 1) to scientific notation, move the decimal point to the right until there is exactly one non-zero digit to its left. The number of places moved is the negative exponent of 10.

• Move decimal right for negative exponent
• Count places moved to determine exponent value
• Mantissa (coefficient) must be between 1 and 10 (inclusive of 1)

Memory trick: Scientific: Single digit, then power of ten, for big and small.

Comparing the efficiency or speed of different entities by converting their performance to a consistent unit rate (e.g., units per hour, miles per gallon).

• Ensure all quantities are in comparable units.
• Calculate the 'per unit' value for each item.
• Directly compare the calculated unit rates.

Memory trick: Standardize the time, then rate the climb!

To convert a decimal to a fraction, write the decimal over a power of 10 and simplify. To express with a specific denominator, find an equivalent fraction.

• Terminating decimals can always be written as fractions.
• The denominator is a power of 10 corresponding to the decimal places.
• Equivalent fractions maintain the same value by multiplying numerator and denominator by the same number.

Memory trick: Decimal's journey to a specific fraction-land.

To convert a length from meters to centimeters, multiply the value in meters by 100, as there are 100 centimeters in 1 meter.

• 1 meter (m) = 100 centimeters (cm).
• Conversion factor is 100.
• Used for expressing lengths in smaller units.

Memory trick: Meter to Centimeter, multiply by a hundred, make it grander!

The distance between two numbers on a number line, regardless of their sign, is found by taking the absolute value of their difference, or by adding their absolute values if they are on opposite sides of zero.

• Distance is always a non-negative value.
• Can be calculated using absolute values.
• For points on opposite sides of zero, add their absolute values.

Memory trick: From here to there, ignore the sign, just count the space.

To multiply fractions, multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator. Simplify the resulting fraction if possible.

• Numerator * Numerator, Denominator * Denominator.
• Simplify before or after multiplication for easier calculation.
• Often used to find 'a fraction of a fraction'.

Memory trick: Top times top, bottom times bottom, then simplify, and the answer's got 'em!

Compound interest is interest calculated on the initial principal and also on the accumulated interest from previous periods. Annually compounded means interest is calculated once per year.

• A = P(1 + r)^t is the formula.
• Interest is added to the principal before the next period's calculation.
• Leads to exponential growth over time.

Memory trick: Principal plus rate, power of time, makes your money climb!

Converting volume units requires cubing the linear conversion factor. For example, if 1 linear unit A = x linear units B, then 1 cubic unit A = x^3 cubic units B.

• Volume = Length * Width * Height (or Area * Height).
• 1 foot = 12 inches, so 1 cubic foot = 1728 cubic inches.
• 1 yard = 3 feet, so 1 cubic yard = 27 cubic feet.
• Ensure all dimensions are in the same linear unit before calculating volume.

Memory trick: Same unit first, then cube the conversion, for volume's true version!

A set of rules that dictate the sequence in which mathematical operations should be performed to ensure a consistent result.

• Parentheses/Brackets first.
• Exponents/Orders next.
• Multiplication and Division (from left to right).
• Addition and Subtraction (from left to right).

Memory trick: Power up, then dig deep for the root!

To find a fraction of a whole number, multiply the fraction by the whole number. To find the remainder, subtract this result from the whole.

• Represents a part of a total.
• Numerator is the part, denominator is the whole.
• Can be used to calculate specific quantities within a larger group.

Memory trick: Slice the whole, then count what's left.

Compound growth occurs when a quantity increases by a certain percentage over a period, and that percentage is then applied to the new, larger quantity in the next period. It is calculated with the formula P(1+r)^n.

• Growth is applied to the accumulated total, not just the initial amount.
• Formula: Future Value = Present Value * (1 + rate)^time.
• Used for population growth, compound interest, inflation.

Memory trick: Growth that keeps growing on top of itself, year after year.

A ratio compares two quantities, while a proportion states that two ratios are equal. Ratios can be used to divide a total quantity into parts.

• Ratios can be written as a:b, a/b, or 'a to b'.
• The sum of the parts in a ratio represents the total number of 'shares' or 'units'.
• To find the value of one part, divide the total quantity by the sum of the ratio parts.

Memory trick: Parts add up, then multiply for the final cup!

To add or subtract numbers in scientific notation, their exponents must be the same. Adjust one number's coefficient and exponent if necessary, then add/subtract the coefficients.

• Exponents must match before addition/subtraction.
• Adjusting the exponent requires moving the decimal point in the coefficient.
• The final answer should be in standard scientific notation (coefficient between 1 and 10).

Memory trick: Align the power, then add the number!

When a quantity undergoes multiple percentage changes, each subsequent change is applied to the *new* value, not the original value.

• Percentage increases are calculated as (1 + percentage) * current value.
• Percentage decreases are calculated as (1 - percentage) * current value.
• The order of operations matters for sequential changes.

Memory trick: Each step's new sum, for the next change to come!

To add mixed numbers, you can either add the whole number parts and fractional parts separately (finding a common denominator for the fractions), or convert all mixed numbers to improper fractions, add them, and then convert back to a mixed number.

• Option 1: Add whole parts, add fractional parts (with common denominator), combine.
• Option 2: Convert to improper fractions, add, convert back.
• Always simplify fractional part at the end.

Memory trick: Mixed: Whole and part, combine with common ground.

The effective annual rate (EAR) is the actual annual rate of return earned on an investment, considering the effect of compounding over a year. It allows for comparison of investments with different compounding frequencies.

• EAR = (1 + Nominal Rate / Compounding Periods)^(Compounding Periods) - 1
• Used to compare investments with different compounding frequencies
• Higher EAR means better return

Memory trick: Compound: Interest on interest, money grows like a snowball.

Fractions are used to represent parts of a whole in various real-world scenarios, such as recipes, measurements, or proportions. Calculations involve multiplying quantities by their fractional parts.

• Fractional parts are calculated by multiplication.
• Results often need to be summed for a total.
• Contextual understanding is key to setting up the problem correctly.

Memory trick: Each slice of pie is a fraction, add them up for the whole meal.

Compound interest is interest calculated on the initial principal and also on the accumulated interest from previous periods. The more frequently interest is compounded, the faster the investment grows.

• Formula: A = P(1 + r/n)^(nt)
• P = principal, r = annual interest rate, n = compounding periods per year, t = time in years.
• Used for investments, loans, and financial growth.

Memory trick: Principal times (one plus rate over periods) to the power of (periods times years), that's the wealth it clears!

A percentage increase is calculated by finding a percentage of an original value and then adding that amount to the original value. It shows a growth in quantity.

• New Value = Original Value + (Percentage Increase * Original Value)
• Can also be calculated as Original Value * (1 + Percentage Increase as Decimal)
• Essential for financial growth, population changes, etc.

Memory trick: To grow by percent, multiply and then add, or use the 1-plus-decimal method, it's not bad!

To add or subtract fractions, they must have a common denominator. Once common denominators are established, add or subtract the numerators and keep the denominator the same.

• Find the Least Common Multiple (LCM) for the denominators.
• Convert each fraction to an equivalent fraction with the common denominator.
• Perform the addition or subtraction on the numerators.
• Simplify the resulting fraction if possible.

Memory trick: Common ground for the bottom, then add/subtract the top!

Converting a mixed number (whole number and a fraction) into its decimal equivalent involves converting the fraction part to a decimal and adding it to the whole number.

• Fractions can be converted to decimals by dividing the numerator by the denominator.
• Mixed numbers combine a whole number and a proper fraction.
• Decimal form is often easier for multiplication and division.

Memory trick: Mix and match, then multiply the batch!

Exponential doubling occurs when a quantity repeatedly multiplies by 2 over fixed intervals. The formula is Initial Quantity * 2^(number of doubling periods).

• The quantity increases very rapidly.
• Number of doubling periods must be calculated first.
• Applies to populations, compound interest, and certain decay processes.

Memory trick: Double, double, toil and trouble, population grows at a bubble.

Multi-step unit conversion involves converting a quantity from one unit to another by passing through intermediate units using a series of conversion factors.

• Use conversion factors as fractions (e.g., 100 cm / 1 m)
• Arrange factors so unwanted units cancel out
• Perform multiplications and divisions sequentially

Memory trick: Units: Link them, cancel them, convert them.

To add mixed numbers and fractions, convert all to improper fractions with a common denominator. Add the numerators, keep the common denominator, then convert the result back to a mixed number.

• Find the least common denominator (LCD) for all fractions.
• Convert mixed numbers to improper fractions.
• Rewrite all fractions with the LCD.
• Add the numerators, keeping the common denominator.

Memory trick: Common ground for all, then add 'em all!

The process of converting a measurement from one unit of length to another, often using conversion factors.

• 1 foot = 12 inches.
• Consistency in units is crucial for accurate calculations.
• When dividing to find 'how many', only whole numbers (full items) are considered.

Memory trick: Feet to inches, multiply by twelve; inches back to feet, divide yourself!

To find the overall effect of multiple sequential percentage changes, multiply the original value by (1 + or - percentage change) for each step. The final value can then be compared to the original.

• Each percentage change is applied to the *new* value, not the original.
• Increase: multiply by (1 + decimal_percent).
• Decrease: multiply by (1 - decimal_percent).
• The order of application matters if the changes are not expressed as multiplicative factors.

Memory trick: Growth and shrink, step by step, not all at once.

When subtracting numbers in scientific notation that share the same exponent, simply subtract their coefficients and keep the common exponent. Ensure the result is in standard scientific notation.

• Exponents must be identical for direct subtraction.
• Subtract the coefficients (the numerical parts).
• The exponent of 10 remains unchanged.
• Resulting coefficient should be between 1 and 10 (normalize if needed).

Memory trick: Exponents match, then subtract the batch!

When multiple percentage discounts are applied sequentially, each subsequent discount is calculated on the *already discounted* price, not the original price.

• A discount of x% means multiplying by (1 - x/100).
• The order of discounts does not change the final price, but it's crucial to apply them one after another.
• Simply adding percentages will overstate the total discount.

Memory trick: Discount, then discount again, on the new price, my friend!

Real numbers that cannot be expressed as a simple fraction (a/b) where 'a' and 'b' are integers and 'b' is not zero. Their decimal representations are non-terminating and non-repeating.

• Examples include pi (π) and square roots of non-perfect squares (e.g., √2, √5).
• Rational numbers include integers, fractions, and terminating or repeating decimals.
• Irrational numbers are a subset of real numbers.

Memory trick: Rational is neat and complete, irrational never repeats!

When an ideal value has a tolerance, the acceptable range is found by subtracting the tolerance from the ideal value for the lower bound and adding the tolerance to the ideal value for the upper bound.

• Lower Bound = Ideal Value - Tolerance
• Upper Bound = Ideal Value + Tolerance
• The range is expressed as Lower Bound ≤ Variable ≤ Upper Bound

Memory trick: Tolerance: Give or take, a little wiggle room.

The vertex of a parabola, which is the graph of a quadratic function, represents the maximum or minimum point of the function.

• For f(x) = ax^2 + bx + c, x-coordinate of vertex is -b/(2a).
• If a > 0, the parabola opens upward, and the vertex is a minimum.
• If a < 0, the parabola opens downward, and the vertex is a maximum.

Memory trick: Vertex formula finds the peak or valley of the curve.

Finding the range(s) of x-values for which a quadratic expression is greater than or less than zero.

• First, find the roots (x-intercepts) of the quadratic equation by setting it to zero.
• These roots divide the number line into intervals.
• Test a value from each interval in the original inequality.
• Alternatively, consider the parabola's shape (opens up/down) to determine where it's above/below the x-axis.

Memory trick: Find the roots, check the parabola's smile/frown!

The constant term 'b' in a linear equation y = mx + b, representing the value of the dependent variable when the independent variable is zero.

• Often interpreted as the initial amount, starting value, or fixed cost.
• Graphically, it's the point where the line crosses the y-axis.
• It's the value of y when x = 0.

Memory trick: B is for Beginning, where the line crosses Y!

The highest or lowest point on the graph of a quadratic function, representing the maximum or minimum value of the function.

• For f(x) = ax^2 + bx + c, the x-coordinate of the vertex is x = -b / (2a).
• If a > 0, the parabola opens upward, and the vertex is a minimum.
• If a < 0, the parabola opens downward, and the vertex is a maximum.
• The y-coordinate of the vertex is found by substituting the x-coordinate back into the function.

Memory trick: Vertex formula: x equals negative b over two a, finds the peak!

The process of converting a real-world scenario described in words into a mathematical equation or set of equations.

• Assign variables to unknown quantities.
• Identify keywords that indicate mathematical operations (e.g., 'is' for equals, 'more than' for addition).
• Break down complex sentences into smaller, manageable parts.
• Formulate equations that accurately represent the given relationships.

Memory trick: Read, define, translate, equation: the four steps to problem creation.

The formula a_n = a_1 + (n-1)d allows you to calculate any term (a_n) in an arithmetic sequence given the first term (a_1), the term number (n), and the common difference (d).

• a_1 is the first term.
• d is the common difference (each term is d more/less than the previous).
• n is the position of the term you want to find.
• The (n-1) factor accounts for the number of 'steps' from the first term.

Memory trick: Starting point plus steps times difference, that's the nth term's essence.

The starting amount or quantity of a substance or value at the beginning of a process, represented by the coefficient in an exponential model.

• Found when the exponent (time) is zero.
• In P(t) = P_0 * b^t, P_0 is the initial value.
• Represents the y-intercept on a graph.

Memory trick: Initial is 'I' for 'Isolated coefficient' – the number standing alone, multiplying the power.

For an absolute value function of the form f(x) = -a|x - h| + k (where a > 0), the maximum value is 'k', occurring at the vertex (h, k).

• Graph is a 'V' shape opening downwards.
• Maximum value is the y-coordinate of the vertex.
• Vertex is (h, k) for f(x) = a|x - h| + k.

Memory trick: Absolute value functions point up or down like a V. The 'k' is the peak or lowest point!

The highest or lowest value that a quadratic function attains, occurring at the vertex of its parabolic graph.

• If the leading coefficient (a) is negative, the parabola opens downward, and the vertex is a maximum.
• If the leading coefficient (a) is positive, the parabola opens upward, and the vertex is a minimum.
• The y-coordinate of the vertex is the maximum or minimum value.

Memory trick: Find x with -b/2a, then plug it in to get the height!

A function in which the variable appears in the exponent, representing quantities that grow or decay at a constant percentage rate.

• General form: f(x) = a * b^x, where a is initial value, b is growth/decay factor.
• Used to model population growth, compound interest, radioactive decay.
• Graphs show rapid increase or decrease.

Memory trick: Linear, quadratic, exponential: each tells a different story of change.

An algebraic equation in which each term has an exponent of 1, resulting in a straight line when graphed.

• Can be written as Ax + By = C or y = mx + b.
• Used to model situations with a constant rate of change.
• Solving involves isolating the variable.

Memory trick: Isolate the variable, balance the equation, find the value.

The process of converting a real-world scenario described in words into a mathematical linear equation or system of equations to solve for unknown quantities.

• Identify unknown quantities and assign variables.
• Look for keywords indicating operations (sum, product, difference, etc.).
• Formulate equations based on the relationships described.

Memory trick: Read, Define, Translate, Check – like turning a story into math code!

A method to find the roots (solutions) of a quadratic equation by expressing the quadratic as a product of two linear factors and then setting each factor to zero.

• Equation must be in standard form: ax^2 + bx + c = 0.
• Find two numbers that multiply to 'ac' and add to 'b'.
• Set each factor to zero and solve for x.

Memory trick: Factor Fun: Find two numbers that multiply to 'ac' and add to 'b'!

Translating a real-world scenario with a constant rate of change and an initial value into a linear equation (y = mx + b).

• The 'rate' or 'per unit' value typically represents the slope (m).
• The 'initial amount', 'starting value', or 'fixed cost' typically represents the y-intercept (b).
• The variable (x or t) represents the independent quantity (e.g., hours, units).

Memory trick: Rate is the slope, start is the intercept!

A proportional relationship exists when two quantities vary in such a way that one is a constant multiple of the other, meaning their ratio is constant.

• Can be expressed as a/b = c/d.
• Involves direct variation.
• Graphs as a straight line through the origin.

Memory trick: Ratio balance, cross-multiply for an unknown prize.

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!

The numerical difference between the output values of a function when evaluated at two different input values.

• Evaluate the function for each given input separately.
• Subtract the smaller result from the larger result (or as specified).
• Ensure correct order of operations for each evaluation.

Memory trick: Think 'two separate calculations, then one subtraction' – like finding the change between two points on a graph.

The process of finding the value of an unknown variable that satisfies a linear equation by isolating the variable using inverse operations.

• Goal: Isolate the variable.
• Use inverse operations (add/subtract, multiply/divide).
• Maintain balance: do the same operation to both sides.

Memory trick: Balance the scales! Whatever you do to one side, do to the other.

A set of two or more linear equations involving the same variables, whose solutions are points that satisfy all equations simultaneously.

• Can be solved by substitution, elimination, or graphing.
• A unique solution represents the intersection point of lines.
• No solution (parallel lines) or infinite solutions (same line) are possible.

Memory trick: Two lines meet, one solution sweet; substitution or elimination, choose your feat.

The process of substituting a specific input value into a function's expression to determine its corresponding output value.

• Replace every instance of the independent variable with the given input value.
• Follow the order of operations (PEMDAS/BODMAS) to simplify the expression.
• The result is the function's output for that specific input.

Memory trick: Plug it in, work it out!

Using the vertex of a quadratic function to find the maximum or minimum value in real-world scenarios, such as profit maximization or cost minimization.

• If 'a' (leading coefficient) > 0, vertex is a minimum.
• If 'a' < 0, vertex is a maximum.
• x-coordinate of vertex: -b/(2a).
• y-coordinate of vertex: substitute x-value into function.

Memory trick: Downward parabola, profit's peak; upward, cost's lowest streak.

For a quadratic function f(x) = ax^2 + bx + c, the x-coordinate of the vertex, which gives the time or input value at which the maximum or minimum occurs, is found using the formula x = -b / (2a).

• Vertex is the turning point of a parabola.
• Formula: x = -b / (2a).
• If a < 0, vertex is a maximum; if a > 0, vertex is a minimum.

Memory trick: Remember 'negative B over 2A' is the secret key to the peak (or valley) time!

The process of finding the value(s) of the variable(s) that make a linear equation true.

• Involves isolating the variable using inverse operations.
• Operations must be applied equally to both sides of the equation.
• Common steps include combining like terms, distributing, adding/subtracting, and multiplying/dividing.

Memory trick: Translate to math, isolate the unknown, check your work!

A sequence of numbers such that the difference between consecutive terms is constant.

• Common difference (d) is constant.
• Formula: a_n = a_1 + (n-1)d
• Used to find any term in the sequence.

Memory trick: Think of an arithmetic sequence like steps on a ladder, each one a consistent height above the last.

For a quadratic function f(x) = ax^2 + bx + c, the minimum or maximum value is the y-coordinate of the vertex, found by calculating f(-b/(2a)). It's a minimum if a>0 and a maximum if a<0.

• Vertex formula: x = -b / (2a).
• Substitute x-coordinate into f(x) to find y-coordinate.
• Positive 'a' means minimum, negative 'a' means maximum.

Memory trick: Find the X for the peak/valley, then plug it back in for the Y-value height!