top of page
Addition
Addition: The process of determining the sum of two or more numbers.
Addend: A number that contributes towards the sum.
Sum: The result of adding addends.
Addition equations should never be completed horizontally, since this disregards place values. Instead, addition equations should be changed to a top-to-bottom format.
To test if the sum is correct, check if the inverse operation (subtraction) is also true.
Application is no longer available.
When adding multidigit numbers, align the numbers using their decimal points. After being aligned, any place value without a digit can be filled in with a zero.
Addition begins with the rightmost digits of the addends and moves leftwards.
Carrying: If the sum of two digits is greater than nine, then carrying is required. The digit in the ones place remains, and the digit in the tens place becomes an additional addend for the place value to the left.
Application is no longer available.
To add or subtract fractions, the fractions must share a common denominator. To find the common denominator, the least common multiple (LCM) of the denominators must be deduced. Then, the fractions are multiplied so each denominator contains the least common multiple.
Multiple: The product of a positive integer of interest multiplied by another positive integer.
Least Common Multiple (LCM): The smallest shared multiple of two or more numbers of interest.
Application is no longer available.
Least Common Denominator (LCD): The least common multiple of the denominators of two or more fractions.
Mixed numerals should be converted into improper fractions before adding or subtracting.
Application is no longer available.
Application is no longer available.
Commutative Law of Addition: When two numbers are added, changing the order of the addends does not affect the sum.
Associative Law of Addition: When three or more numbers are added, changing the order of the addends does not affect the sum.
Subtraction
Subtraction: The process of determining the difference between two or more numbers.
Minuend: The number that is subtracted from.
Subtrahend: The subtracting number.
Difference: The number that results from subtracting one number from another.
Subtraction equations should never be completed horizontally, since this disregards place values. Instead, subtraction equations should be changed to a top-to-bottom format.
To test if the difference is correct, check if the inverse operation (addition) is also true.
Application is no longer available.
When subtracting multidigit numbers, align the numbers using their decimal points. After being aligned, any place value without a digit can be filled in with a zero.
Subtraction begins with the rightmost digits of the minuend and subtrahend and moves leftwards.
Borrowing: A subtraction process that is utilized if the minuend digit of interest is smaller than the subtrahend digit of interest. When borrowing, a neighboring place value to the left is reduced by one, and the minuend digit of interest is increased by ten.
Chain Borrowing: When the place value directly to the left of the digit of interest is occupied by a zero, a distant place value must be borrowed from. In this case, the borrowing process occurs multiple times.
Application is no longer available.
To add or subtract fractions, the fractions must share a common denominator. To find the common denominator, the least common multiple (LCM) of the denominators must be deduced. Then, the fractions are multiplied so each denominator contains the least common multiple.
​Mixed numerals should be converted into improper fractions before adding or subtracting.
Application is no longer available.
Application is no longer available.
Multiplication
Multiplication: The process of determining the product of two or more numbers. Multiplication can be thought of as repeated addition.
Multiplicand: The number that is multiplied.
Multiplier: The number that multiplies the multiplicand.
Product: The number that results when one number is multiplied by another.
To test if the product is correct, check if the inverse operation (division) is also true.
Whenever zero is a multiplicand or multiplier, the product will always be zero
Multiplication can be represented by either the dot symbol or the cross symbol. The dot symbol is preferred since the cross symbol can be confused with the variable "x."
Application is no longer available.
Knowledge of multiplication facts can hasten the speed at which multiplication equations are solved.
Application is no longer available.
Application is no longer available.
Taking advantage of trailing zeros can reduce the multiplication process.
Leading Zeros: Zeros that do not have any nonzero digits ahead of them.
Sandwiched Zeros: Zeros that are between nonzero digits.
Trailing Zeros: Zeros that do not have any nonzero digits behind them.
​Trailing zeros can be ignored during the multiplication process. Once the multiplication process is complete, the number of trailing zeros can then be used to move the decimal to the correct position.
Decimal place trailing zeros can be ignored during the multiplication process; however, decimal place trailing zeros do not contribute to the movement of the decimal when solving for the final product.
Expanding the multiplicand, multiplier, or both can create easier multiplication equations. Then, the products of these equations can be added to determine the product of the original multiplication equation.
.png)
.png)
.png)
​Long multiplication is especially useful for solving multiplication equations containing large numbers or decimal values.
Application is no longer available.
Application is no longer available.
Multiplication equations can be solved by repeatedly adding the multiplicand or multiplier.
.png)
.png)
Application is no longer available.
When multiplying fractions, the numerators and denominators are multiplied directly across.
Mixed numerals should be converted into improper fractions before multiplying.
Application is no longer available.
Commutative Law of Multiplication: When two numbers are multiplied, swapping the multiplicand and multiplier does not affect the product.
Associative Law of Multiplication: When three or more numbers are multiplied, changing the order of the numbers does not affect the product.
Distributive Law of Multiplication: Multiplying a number by a sum or difference is equivalent to multiplying each component separately, then adding or subtracting.
Application is no longer available.
Division
Division: The process of determining the quotient of two or more numbers. Division can be thought of as repeated subtraction.
Dividend: The number that is split.
Divisor: The number that splits the dividend.
Quotient: The number that results when one number is divided by another.
To test if the quotient is correct, check if the inverse operation (multiplication) is also true.
Application is no longer available.
​Knowledge of multiplication facts can make division easier.
Application is no longer available.
Application is no longer available.
When the dividend can’t be evenly divided, the quotient will contain a remainder, a fraction, or a decimal.
Remainder: The amount that is left after division.
To check a quotient with a remainder, multiply and then add the remainder.
Application is no longer available.
Instead of a remainder, the quotient may contain a fraction or decimal.
Application is no longer available.
Long division is a process that is especially useful for solving division equations containing large numbers or decimals.
Application is no longer available.
Application is no longer available.
Application is no longer available.
Division may result in a non-terminating decimal or a decimal that reaches a distant place value. Rounding these quotients to three decimal digits gives a reasonable estimate of the actual value.
​To divide fractions, multiply by the reciprocal of the divisor.
Reciprocal: The reciprocal of a fraction can be found by swapping the numerator and denominator. If the reciprocal is multiplied by the original fraction, the product is always one.
To divide mixed numerals, convert them into improper fractions.
Application is no longer available.
Complex Fractions: Fractions in which other fractions occupy the numerator and/or denominator. Complex fractions are solved by multiplying by the divisor's reciprocal.
.png)
Application is no longer available.
When zero is a divisor, it will always result in an undefined or indeterminate quotient. When dividing by zero, the logical result would be a quotient of zero; however, the inverse operation would then result in any desired number. Thus, zero is excluded as a divisor and always results in an undefined or indeterminate quotient.
Exponents
Exponential Notation: A product is represented by a base and an exponent.
Application is no longer available.
Application is no longer available.
Application is no longer available.
Radicals
Radical Notation: A root represented by a radicand under a radical sign.
Root: A number multiplied by itself "x" times, based on the index, that equals the radicand. Roots can either be positive or negative.
Radical Sign (√): Indicates that only the positive root should be solved for.
Application is no longer available.
Application is no longer available.
Square Root: "b" is a square root of "a" if b^2 = a. Many basic square roots can be solved by memorizing multiplication facts.
.png)
Application is no longer available.
Powers of 10
Powers of 10: Any of the integer powers of the number ten.
Multiplying or dividing a number by a power of 10 can be thought of as moving the decimal to the left or right a certain number of times, depending on the number of leading or trailing zeros.
Application is no longer available.
Application is no longer available.
Scientific Notation: A notation that is especially useful for presenting large or small numbers in a reduced format.
Application is no longer available.
Application is no longer available.
Factorization
Factor: A positive integer that divides a larger positive integer evenly, leaving no remainder.
If a number is a factor of a product, it also means that the product is a multiple of the factor.
Factorization: A number expressed as the product of two or more of its factors.
Only positive integers are considered when determining a number's factors.
Being able to determine all the factors of a number is essential. This skill makes multiplying, dividing, and simplifying fractions easier.
.png)
Factors can be determined by either multiplication or division.
The product itself and one will always be factors. One is the multiplicative identity and always equals the number it is multiplied by.
Prime Numbers: Numbers that have only two factors, one and themselves. Two is the only even prime number.
Composite Numbers: Numbers that have more than two factors.
Application is no longer available.
Multiplication Chart: A tool used to determine the factors and multiples of a number.
Application is no longer available.
Application is no longer available.
Divisibility Tests: Tests that circumvent division and allow one to check if an integer is divisible by 2, 3, 5, 6, 9, or 10.
Application is no longer available.
Application is no longer available.
Prime Factorization: The sequential factorization of a composite number and its factors. Prime factorization results in a unique set of prime numbers that, when multiplied, equal the original composite number.
Factor Tree: A branching prime factorization method that involves factoring composite numbers until only prime numbers remain.
Application is no longer available.
Application is no longer available.
​​​Prime factorization can be used as a method to find the least common multiple for a set of numbers.
Application is no longer available.
Application is no longer available.
​​​Prime factorization can be used as a method to simplify fractions.
Application is no longer available.
Application is no longer available.
Order of Operations
Order of Operations: The steps taken to calculate an equation correctly.
Bracketing Method: A way to manage multiple numbers and operations in an equation by using brackets to complete the equation in a step-by-step manner, rather than attempting to solve the equation in a single step.
Application is no longer available.
Application is no longer available.
Rounding
Rounding: Approximating the value of a number.
Application is no longer available.
Rounding numbers in an equation allows for an estimate of the final answer.
Tilde (~): Indicates an approximation of a value.
Application is no longer available.
Inequalities
Inequality: A mathematical sentence using ≤ , ≥ , < , > , or ≠.
Application is no longer available.
Number Patterns
Number Pattern: A sequence of numbers that follow a specific pattern.
Application is no longer available.
Practice Sheets
bottom of page