Mathematics 7

Equations

Vocabulary

The following terms are important to understand when working with algebraic equations.

Term	Definition
Mathematical Term	A mathematical expression that does not contain +, - or equality symbols.
Coefficient	The number being multiplied to a variable in a mathematical term.
Constant Term	A mathematical term with no variable or exponents.
Like Terms	Two or more mathematical terms having the same variable and the same exponent.
Equation	Two mathematical expressions that have been set equal to each other, such as $2 + 5 = 7$.
Equivalent Equations	Two equations that look different, but are mathematically identical.
Inequality	An equation in which the two sides are not equal.
Variable	A symbol (or placeholder) for a value we don’t yet know.
Exponent	The number of times the base of a power should be multiplied.
Literal Equation	Equations made up of alphabet variables, such as $E = MC^2$.
Inverse Operation	An operation that reverses another operation, such as addition and subtraction.
Ratio	TWo values that have been set in relation to each other.
Proportion	Two ratios that are set in relation to each other.
Rate	A proportion in which one value has been set to 1.
Cross Multiplication (or Cross Product)	Multiplying the numerators of fraction with each other’s denominators.
General Mathematical Principle	A core principle of mathematics.
Addition Property of Equality	If we add or subtract the same number to both sides of an equation, the sides remain equal.
Multiplication Property of Equality	If we multiply or divide the same number to both sides of an equation, the sides remain equal.
Commutative Property of Addition	The order in which numbers are added does not change the sum. (This property does not apply to subtraction.)
Commutative Property of Multiplication	The order in which numbers are multiplied does not change the sum. (This property does not apply to division.)
Associative Property of Addition	The order in which numbers are grouped when adding does not change the sum. (This property does not apply to subtraction.)
Associative Property of Multiplication	The order in which numbers are grouped when multiplying does not change the product. (This property does not apply to division.)
Distributive Property
Identity Property of Addition	When a number is added to zero, the result is the original number.
Identity Property of Multiplication	When a number is multiplied by 1, the result is the original number.
Justification	The mathematical principle used to transform an equation into another form.
Solution	The answer to a mathematical problem.
Solution Set	The set of all possible solutions to a mathematical problem.

About

Logical reasoning is central to all mathematics, and is at the heart of algebraic thinking. A proper justification for each step in the solution of an equation is not simply a description of what was done (“I added 4 to both sides.”), but rather a general principle, such as The Addition Property of Equality.

Ron

Decimal System

The Decimal Number System, also called Hindu-Arabic number system, is a positional numeral system that uses 10 as the base and requires 10 different symbols which are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It also uses a dot (the decimal point) to represent fractions. In this system, the numerals used to show a value take different place values depending upon position. In a base-10 system the number 543.21 represents the sum:

Ron

Squares and Square Roots

A headlong plunge into the mighty story of perfect squares and their roots.

Ron

Area in the Coordinate Plane

Finding the area of polygons defined by linear equations.

Ron

Metric Measurements

Metric Linear

\begin{align} 10,mm &= 1,cm \\ 100,cm &= 1,m \\ 1000,m &= 1,km \\ \end{align}

Metric Area

\begin{align} 100 mm^2 &= 1 cm^2) \\ 10000 cm^2 &= 1 m^2) \\ &= 1000000 m^2 \\ \end{align}

Volume

10 milliliters (ml) = 1 centiliter (cl) 10 centiliters = 1 deciliter (dl) = 100 milliliters 10 deciliters = 1 liter (l) = 1,000 milliliters 10 liters = 1 dekaliter (dal) 10 dekaliters = 1 hectoliter (hl) = 100 liters 10 hectoliters = 1 kiloliter (kl) = 1,000 liters

Ron

Metric to Us Measurement Conversions

	US	Metric
Length
	1 inch (in)	2.54 centimeters (cm)
	1 foot (ft)	30.48 centimeters
	1 yard (yd)	0.914 meters (m)
	1 mile (mi)	1.609 kilometers (km)
Weight
	1 ounce	28.3 grams
	1 pound	0.45 kg
Area
	1 in²	6.45 cm²
	1.196 yd²	1 m²
	2.47 acres	1 ha
Volume
	1 in³	16.39 mL
	1.06 qt	1 liter
	1 gallon	3.79 liters
	1 m³	35.31 ft³
	1 quart	0.95 liters (L)
Temperature	$ C^\circ \text{ to } F^\circ$	$ F^\circ \text{ to } C^\circ$
	$ F^\circ = \frac{9}{5}C + 32$	$C^\circ = \frac{5}{9}(F - 32)$

Ron

Multiplying and Dividing Monomials

Vocabulary

Power: A base raised to an exponent, such as $2^8$.
Base: The number that is being raised. In $2^8$, 2 is the base.
Exponent: The number of times to multiply the base to itself. $2^8 = (2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2) = 256$.

Tips

When multiplying powers, add the exponents.
When dividing powers, subtract the exponents.
When converting a decimal into scientific notation, move the decimal point to the right of the first non-zero digit, and count the number of digits moved. Each digit is a multiple of 10.
Positive exponents are a form of multiplication: $ 3200 = 3.2 \times 10^3$
Negative exponents are a form of division: $ 0.00058 = 5.8 \times 10^{-4}$

Ron

Prime Numbers

What is the Prime Number?

Prime numbers are all integers greater than 1 that have only two factors, the number itself and 1. For example, the only factors of 3 are 1 and 3, so 3 is prime.

The smallest prime number is 2 because it is only divisible by itself and 1. The number 2 also happens to be the only even prime number.

A number that has more than two factors is called a composite number. Such numbers can be made by combining more than two factors. For example, $2 \times 2 \times 3 = 12$, therefore 12 iS a composite number.