Mathematics (US Middle School)
Common Core mathematics for US middle school (grades 6-8): ratios and proportions, the number system, expressions and equations, functions, geometry, and statistics and probability.
Ämne: Matematik · Nivå: Högstadium (13–15) · 457 kort
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- A ratio is a comparison of two quantities. The ratio of a to b can be written as a:b, a/b, or 'a to b'.
- A unit rate is a ratio where the second quantity is 1, such as 60 miles per 1 hour (60 mph).
- Percent means 'per hundred'. 25% means 25 per 100, which equals 25/100 or 0.25.
- Equivalent ratios are ratios that represent the same relationship. 2:3 and 4:6 are equivalent because multiplying both terms of 2:3 by 2 gives 4:6.
- To convert a fraction to a percent, multiply by 100. To convert a percent to a decimal, divide by 100 (or move the decimal point two places to the left).
- The greatest common factor (GCF) of two numbers is the largest number that divides both of them evenly. The GCF of 12 and 18 is 6.
- The least common multiple (LCM) of two numbers is the smallest positive number that is a multiple of both. The LCM of 4 and 6 is 12.
- To divide fractions, multiply the first fraction by the reciprocal of the second: a/b ÷ c/d = a/b × d/c.
- Integers are the set of whole numbers and their opposites: ..., −3, −2, −1, 0, 1, 2, 3, ...
- The absolute value of a number is its distance from zero on the number line. |−5| = 5 and |5| = 5.
- On a number line, numbers to the right are greater than numbers to the left. −2 > −5 because −2 is to the right of −5.
- A coordinate plane has two perpendicular number lines: the horizontal x-axis and the vertical y-axis. They intersect at the origin (0, 0).
- The coordinate plane is divided into four quadrants. Quadrant I has both x and y positive; II has x negative, y positive; III has both negative; IV has x positive, y negative.
- An exponent tells how many times to multiply the base by itself. In 3⁴, 3 is the base and 4 is the exponent, so 3⁴ = 3 × 3 × 3 × 3 = 81.
- A variable is a symbol (usually a letter) that represents an unknown or changing quantity. In 3x + 5, x is the variable.
- A coefficient is the number multiplied by a variable. In the expression 7x, the coefficient is 7.
- Like terms have the same variable raised to the same power. 3x and 7x are like terms; 3x and 3x² are not.
- The distributive property states that a(b + c) = ab + ac. For example, 3(x + 4) = 3x + 12.
- The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
- An equation states that two expressions are equal, separated by an equals sign (=). Example: 2x + 3 = 11.
- An inequality uses symbols <, >, ≤, or ≥ to compare quantities. x > 5 means x is greater than 5.
- An open circle on a number line means the value is NOT included (used for < or >). A closed circle means the value IS included (used for ≤ or ≥).
- The area of a rectangle is length × width: A = lw.
- The area of a triangle is one-half base times height: A = ½ bh.
- The area of a parallelogram is base times height: A = bh. The height is the perpendicular distance between the parallel sides.
- The area of a trapezoid is one-half times the sum of the two parallel sides times the height: A = ½(b₁ + b₂)h.
- The volume of a rectangular prism is length × width × height: V = lwh.
- Surface area is the total area of all the faces of a three-dimensional shape, measured in square units.
- A net is a two-dimensional pattern that can be folded to form a three-dimensional shape. Nets are used to find surface area.
- The mean (average) of a data set is found by adding all values and dividing by the number of values.
- The median is the middle value of a data set when values are arranged in order. If there are two middle values, take their mean.
- The mode is the value that appears most often in a data set. A data set can have one mode, multiple modes, or no mode.
- The range is the difference between the greatest and least values in a data set: range = maximum − minimum.
- Mean absolute deviation (MAD) measures how spread out data is from the mean. It is the average of the absolute differences between each value and the mean.
- A dot plot shows each data value as a dot above a number line. It is useful for small data sets and shows frequency at each value.
- A histogram shows the frequency of data grouped into intervals (bins). Unlike a bar graph, the bars touch because the data is continuous.
- A box plot (or box-and-whisker plot) displays the five-number summary: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
- The interquartile range (IQR) is the difference between Q3 and Q1: IQR = Q3 − Q1. It measures the spread of the middle 50% of data.
- A statistical question is one that anticipates variability in the data. 'How tall are sixth graders at this school?' is statistical; 'How tall is my friend?' is not.
- A proportion is an equation stating that two ratios are equal: a/b = c/d. Cross-multiplying gives ad = bc.
- Two quantities are proportional if their ratio is constant. This constant ratio is called the constant of proportionality, often denoted k, so y = kx.
- On a graph, a proportional relationship is a straight line passing through the origin (0, 0). The slope of this line equals the constant of proportionality.
- Percent change is calculated as: (new value − original value) / original value × 100%. A positive result is an increase; negative is a decrease.
- Simple interest is calculated as I = Prt, where P is the principal, r is the annual interest rate (as a decimal), and t is time in years.
- When adding integers with the same sign, add absolute values and keep the sign. −3 + (−5) = −8.
- When adding integers with opposite signs, subtract absolute values and keep the sign of the larger absolute value. −8 + 3 = −5.
- Subtracting an integer is the same as adding its opposite. a − b = a + (−b). So 5 − (−3) = 5 + 3 = 8.
- When multiplying or dividing two integers with the same sign, the result is positive. With opposite signs, the result is negative.
- A rational number is any number that can be written as a fraction a/b where a and b are integers and b ≠ 0. Examples: 3/4, −2, 0.75, 0.333...
- When dividing by a negative number when solving an inequality, flip the inequality sign. −2x < 6 becomes x > −3.