PRAXIS Math

Numbers & Operations

tests your knowledge of the foundations of the base-10 number system; skills include knowing basic terminology associated with the number system, knowing how to represent and move in the place value system, how to use the place value system to compose & decompose numbers, how to represent numbers in different forms (including fractions, decimals, percents, & scientific notation), & how to solve problems using all four operations.

Cardinal Number

a number that says how many of something there are; used specifically for counting (ex: 1, 2, 3…).

Ordinal Number

number that tells the position of something in a list; used specifically when referring to the order of an object (ex: 1st, 2nd, 3rd…).

Base-10 Number System

our number system; based on the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (ex: snap cubes).

Place Value

the value of a certain digit is determined by the spot where it resides within a number.

Example of Place Values

millions, hundred thousands, ten thousands, thousands, hundreds, tens, ones, (“.” read as “AND”), tenths, hundredths, thousandths.

To Round…

look at the number to the right of the place value to which you are rounding; if between 0-4, the number doesn’t change; if between 5-9, round up by 1.

Expanded Form

a number written so that the place value of each number is represented as part of a sum (ex: 872 → 800 + 70 + 2).

Addend

any number that is added to another in an addition equation to form a sum.

Sum

the result of adding two or more numbers together.

Difference

the result of subtracting one number from another.

Product

the result of multiplying two or more numbers.

Quotient

the result of dividing one number by another.

Expanded Notation

where each digit is written as a product of its place value (ex: 872 → (8 x 100) + (7 x 10) + (2 x 1)); extension of expanded form.

Scientific Notation

a number greater than or equal to 1 and less than 10 multiplied by a power of 10; the number multiplied by the power of ten must have only 1 number in front of the decimal point.

Add To Problem

where a number is given, and more is being added to the number to find a sum; result, change, & start unknown

Take From Problem

where a number is given, and some is being taken from this number to find a difference; result, change, & start unknown.

Put Together/Take Apart Problem

often referred to as part, part whole; where part of the number is given, then another part is given to make a total amount; total, addend, & both addends unknown.

Compare Problem

where two values are given for a total, with the size of one being compared to the size of the other; difference, bigger, & smaller unknown; unknown, group size, & number of groups unknown.

Equal Groups Problem

can be used with multiplication and division with arrays and area; product, group size, & number of groups unknown.

Manipulatives (Physical Models)

attribute blocks, base-10 block, bar diagrams, counters, geoboards, fraction strips, snap cubes, & tiles.

Attribute Blocks

come in five different geometric shapes and with different colors; used for sorting, patterns, & teaching characteristics of geometric figures.

Base-10 Block

visual models in powers of 10 that represent ones, tens, hundreds, and thousands; used to teach place value, regrouping with addition/subtraction, fractions, decimals, percents, & area/volume.

Bar Diagrams

used to represent parts and whole; often used with finding a missing value in a number sequence (ex: 5 + ? = 12).

Counters

come in different shapes and colors (like bears, bugs, chips); used for sorting and counting.

Geoboards

pegboard grids on which students stretch rubber bands to make geometric shapes; used to teach basic shapes, symmetry, congruency, perimeter, & area.

Fraction Strips

help to show the relationship between the numerator and denominator of a fraction & how parts relate to a whole.

Numerator

represents part of a whole; top number of a fraction.

Denominator

represents the whole; bottom number of a fraction.

Snap Cubes

pieces that come in various colors that can be snapped together from any face; used to teach number sense, basic operations, counting, patterns, & place value.

Tiles

1-inch squares that come in different colors; used to teach counting, estimating, place value, multiplication, fractions, & probability.

Order of Operations

Please Excuse My Dear Aunt Sally (PEMDAS); Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Commutative Property of Addition

a + b = b + a; changing the order of two numbers being added does not change their sum.

Commutative Property of Multiplication

a x b = b x a; changing the order of two numbers being multiplied does not change their product.

Associative Property of Addition

(a + b) + c = a + (b + c); changing the grouping of the addends does not change their sum.

Associative Property of Multiplication

a x (b x c) = (a x b) x c; changing the grouping of the factors does not change their product.

Additive Identity property of 0

a + 0 = 0 + a = a; adding 0 to a number does not change the value of that number.

Multiplicative Identity Property of 1

a x 1 = 1 x a = a; multiplying a number by 1 does not change the value of that number.

Inverse Property of Addition

for every a, there exists a number -a, such that a + (-a) = (-a) + a = 0; adding a number and its opposite results in a sum equal to 0.

Inverse Property of Multiplication

for every a, there exists a number 1/a, such that a x 1/a = 1/a x a = a/a = 1; multiplying a number and its multiplicative inverse results in a product equal to 1.

Distributive Property of Multiplication over Addition

a x (b + c) = a x b + a x c; multiplying the sum is the same as multiplying each addend by that number, then adding their products.

Distributive Property of Multiplication over Subtraction

a x (b - c) = a x b - a x c; multiplying the difference is the same as multiplying the minuend and subtrahend by that number, then subtracting their products.

Opposite of a Number

is the same number with a different sign (ex: for 5, it’s -5); the sum of a number and its ___ equals 0; 0 does not have one of these.

Reciprocal of a Number

is what the number is multiplied by to get 1 (ex: for 3, it’s ⅓); the product of a number and its ___ is 1; 0 does not have one of these.

Decompose a Fraction

break the fraction into parts.

To Find the Percent of a Number

change the percent to a decimal and multiply the decimal by the number (ex: 12% of 40 is 0.12 x 40 = 4.8).

Ratio

a comparison of two numbers using a fraction, a colon, of the word “to”; have the same units.

Rate

when ratios have different units.

Unit Rates

a rate with a denominator of 1; read using the word “per” instead of “to”.

Proportions Given a (Scale)

use a proportion if it contains a ___ (2:15), each of the numbers contains units, there are 3 numbers with units, & the word problem asks for a 4th number with units.

Proportions Given an (Equivalency Statement)

use a proportion if it contains an _____ that represents a rate, each of the numbers contains units, there are 3 numbers with units, and the problem asks for a 4th number with units.

Proportions Given a (Description of Similar Figures)

use a proportion if it contains _____ that can be drawn, each of the numbers contains units, there are 3 numbers with units, and the problem asks for a 4th number with units.

Proportions Given (Similar Figures)

use a proportion if the problem stated the figures are similar, there are 3 numbers with units & the problem asks for a 4th number with units.

To Find Percent of Change

(new number - original number) / original number; negative sign is not included in the answer of a percent decrease

To Find Percent of Total

requires writing numbers as a fraction, converting the numbers to decimals, and then converting the decimal to a percent.

Real Number System

includes counting numbers, whole numbers, integers, & rational numbers.

Counting Numbers

1, 2, 3, 4, 5, 6, …

Whole Numbers

0, 1, 2, 3, 4, 5, 6, …

Integers

…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …

Rational Numbers

any number that can be written as a fraction a/b, where a and b are any integer; include all terminating and repeating decimals (ex: 0.2, 4 ½, 7 ⅓).

Additional Classifications

includes prime numbers, composite numbers, even numbers, & odd numbers.

Prime Number

a positive integer that only has 1 and itself as factors (ex: 2, 3, 13, 29).

Composite Number

a positive integer that has factors other than 1 and itself (ex: 4, 12, 27, 44).

1 is neither ___ nor ___; 2 is the only even ___ number

prime … composite … prime.

Composite Number

a positive integer that has factors other than 1 and itself (ex: 4, 12, 27, 44).

Even Number

a number that is divisible by 2.

Odd Number

a number that is not divisible by 2.

Prime Factorization

refers to finding all the prime numbers, when multiplied together, result in a composite number (ex; using factor tree & list).

Greatest Common Factor (GCF)

the largest number that divides into all numbers in a given set; can only be as large as the smallest number in the set.

Least Common Multiple (LCM)

the smallest multiple that all numbers in a set have in common (for elementary students, found by skip counting).

Estimation

finding a rough calculation or approximation; different strategies are helpful in different situations.

Estimation Strategies

compatible numbers, clustering, & front-end.

Compatible Numbers Estimation

estimating by rounding pairs of numbers to numbers that are easy to add, subtract, multiply, or divide (ex: 31.8 / 5.2 → 30 / 5; est. value is 6).

Clustering Estimation

estimating sums or products when all the numbers are close to a single value (ex: 42 + 38 + 40 + 41 → 40 + 40 + 40 + 40; est. value is 160).

Front-End Estimation

estimating by rounding to the greatest place value or the number in front (ex: 412 + 58 + 1780 → 400 + 60 + 2000; est. value is 2460).

Algebraic Thinking

tests your knowledge of identifying, manipulating, & performing operations on expressions & equations that are found in algebra; includes being able to solve equations & inequalities with one variable using properties of operations, identifying & creating linear equations and functions from a table, & applying properties of arithmetic & geometric sequences to find missing numbers of figures in a pattern.

Algebraic Expression

a mathematical phrase that contains terms that include numbers (constants), variables, or a combination of numbers & variables.

Term (in an Expression)

separated by a + or a - sign (ex: 2a + 7 → 2 ___).

Equation

two algebraic expressions separated by an equal sign; presence of equal sign differentiates ___ from an expression.

Polynomials

are algebraic expressions that do not have a variable in the denominator of a fraction.

Degree of a Polynomial

is the largest sum of exponents in one term (ex: 3a^2b^3 → 1 term, monomial, ___ of 5).

Two Types of Equations

linear & quadratic.

Linear Equation

an equation with all the variables having a power of 1

(ex: y = 2x + 4).

Quadratic Equation

a polynomial equation where the highest exponent on the variable is 2 (ex: (x + 3)(x - 5) = 0 → power of x would be 2).

Independent Variable

represents where numbers are input in order to find the value of the dependent variable; typically represented as x.

Dependent Variable

the output value; typically represented as y; depends on what is substituted in for x.

Addition Words

sum, plus, add, altogether, total, increased by.

Subtraction Words

difference, minus, take away, less, less than, subtracted from, decreased by.

Multiplication Words

product, times, multiply, of, double, twice.

Division Words

quotient, divide, ratio, split into parts.

Equal Words

is, ___, equivalent to.

Grouping Symbols

quantity, 2 operations in a row (ex: times the sum of).

Evaluating an Expression or Equation

means to substitute given values into the variables & simplify the math expression; important to remember to follow the order of operations & be very careful about negative signs.

Equation/Inequality Types

multistep linear equations, multistep linear inequalities, & compound linear inequalities.

Example of Multistep Linear Equations

-2(2x + 3) - 5 = - (5 = x).

Example of Multistep Linear Inequalities

5 - 4(x -2) < 2x - (x + 7).

Example of Compound Linear Inequalities

-3 < 2x - 8 < 10; x + 3 > -4 or

x + 3 ≤ 7.

(To Solve) Multistep Linear Equations…

isolate the variable using inverse operations, remembering what is done to one side has to be done to the other to keep the expressions equal.