Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematics of radio engineering wikipedia , lookup
Large numbers wikipedia , lookup
Law of large numbers wikipedia , lookup
History of trigonometry wikipedia , lookup
History of logarithms wikipedia , lookup
Elementary arithmetic wikipedia , lookup
Location arithmetic wikipedia , lookup
Approximations of π wikipedia , lookup
Review of 7th Grade Math Concepts Fractions Addition with fractions and mixed numbers Find a common denominator and then rewrite the fractions using this denominator. Add the numerators and keep the same denominator. Simplify if necessary Subtraction with fractions and mixed numbers Find a common denominator and then rewrite the fractions using this denominator. When borrowing, the denominator tells how many you borrowed and the numerator tells how many you already have. So, add them to find the new amount. Subtract the numerators and keep the same denominator. Simplify if necessary. Change a mixed number to an improper fraction Multiply the denominator and whole number. Add the numerator. Write over the denominator. Change an improper fraction to a mixed number Divide the numerator by the denominator. The quotient is the whole number part. The remainder is the numerator and you keep the same denominator. Multiplication with fractions and mixed numbers Rewrite any mixed numbers as improper fractions. Cross cancel if possible Multiply the numerators and the denominators. Simplify if necessary. Reciprocal Definition: Two numbers whose product is 1. To find the reciprocal of a number, exchange the numerator and the denominator. If it is a mixed number first write it as an improper fraction. 5 5 * 2 10 12 12 * 2 24 3 3*3 9 8 8 * 3 24 19 24 + 5 - 4 1 6 1 7 4 4 6 6 6 5 5 6 6 2 1 4 4 6 3 2 4 * 3 2 14 3 3 3 23 3 5 4 4 6 2 6 * 2 12 4 * 7 3 7 * 3 21 7 Number 2 5 Reciprocal 5 2 3 1 = 13 4 4 4 13 Division with fractions and mixed numbers 2 1 8 16 2 3 Rewrite any mixed numbers as improper fractions. 3 5 3 5 Keep the first fraction, change the division sign to a multiplication sign, and 8 5 8 * 5 40 5 * use the reciprocal of the next fraction. 3 16 3 *16 48 6 Follow multiplication rules for fractions. Change a fraction to a decimal Divide the numerator by the denominator. Keep dividing until the decimal portion repeats or terminates. Change a fraction to a percent Make the denominator be 100 or change the fraction to a decimal and then change it to a percent. 7 .875 8 terminates 5 .45 11 repeats 4 4 * 20 80 * 80% 5 5 * 20 100 7 .875 87.5% 8 1 Decimals Addition with decimals Line up the decimal points. Add zeros to make sure both numbers have the same amount of decimal places. Add and then move down the decimal point. Subtraction with decimals Line up the decimals points. Add zeros to make sure both numbers have the same amount of decimal places. Subtract and then move down the decimal point. Multiplication with decimals Multiply as if they were whole numbers. Place the decimal point in the product so that it has the same number of decimals places as the sum of the decimals places in the factors. 4.56 + 3.9 = 12 – 3.456 = 12.000 - 3.456 8.544 6.26 * 4.3 = 6.26 4.3 1878 25040 26.918 * Division with decimals Move the decimal point the same number of places to the right in both the divisor and dividend until the divisor is a whole number. Divide as you divide with a whole number. Move up the decimal point. Change a decimal to a fraction Read the decimal and write it as a fraction. Simplify if needed. Change a decimal to a percent Move the decimal point two places to the right. 4.56 + 3.90 8.46 11.124 ÷ 2.7 2.7 11.124 4.12 27 111.24 4.7 is four and seven tenths 7 4 10 2.35 = 235% .24 = 24% .9 = 90% .007 = .7% Percents Change a percent to a decimal Move the decimal point two places to the left. Change a percent to a fraction Write the percent with a denominator of 100. Finding the percent of a number mentally Find 1% by moving the decimal point 2 places to the left. Find 10% by moving the decimal point 1 place to the left. Find 50% by dividing the number by 2. Use those percents to find other percents. Steps to use when solving a proportion Set the cross products equal to each other. Multiply Undo multiplication by using division Solving percent problems using equations Remember: IS means equal to OF means multiply Change a percent to a decimal before using it in the equation. 432% = 4.32 65% = .65 60% = .6 .9% = .009 35 7 7 35% 7% 100 20 100 Number: 56 1% = .56 10% = 5.6 50% = 28 15% = 10% + 5% = 5.6 + 2.8 = 8.4 30% = 3 * 10% = 3 * 5.6 = 16.8 4 10 9 n 4 * n = 10 * 9 4n = 90 ÷ 4 = ÷4 n = 22.5 What ____ is ____% of _____? What number is 18% of 117? n = .18 * 117 n = 21.06 2 Solving percent problems using proportions Use the proportion at the right. To decide what goes where in the proportion remember the following: The % number has a percent sign with it. The OF number is ALWAYS after the word OF The is number can be before the word IS or after the word IS Follow the steps to solve a proportion. is % of 100 36 is 30% of what number? 36 30 n 100 36 * 100 = 30n 3600 = 30n ÷ 30 ÷ 30 120 = n Integers Absolute Value The absolute value of a number is its distance from zero on a number line. Adding Integers with the same sign 1. Find the absolute value of each integer 2. Add the absolute values. 3. Keep the sign of the integers for your answer Adding Integers with different signs 1. Find the absolute value of each integer 2. Subtract the smaller absolute value from the larger absolute value. 3. Use the sign of the integer with the greater absolute value for you answer. (BIG GUY RULES) Subtracting Integers 1. Keep the first integer. 2. Change the subtraction sign to an addition sign. 3. Change the second integer to its opposite. 4. Add using the addition rules for integers with the same sign or different signs. Multiplying and Dividing Integers 1. Multiply or divide using the absolute value of each integer (or ignoring the signs) 2. Count the number of negative signs in the original problem. If there are: - an odd number of negative signs the answer is negative - an even number of negative signs the answer is positive |5| = 5 |-5| = 5 - 5 + -7 = -12 -20 + -11 = -31 -5 + 6 = 1 -5 + 2 = -3 6 + -10 = -4 -3 + 11 = 8 -5 - -6 becomes -5 + 6 5 – 10 becomes 5 + -10 -5 – 2 becomes -5 + -2 8 – (-8) becomes 8 + 8 -8 – (-8) becomes -8 + 8 which equals 1 which equals -5 which equals -7 which equals 16 which equals 0 3 * -8 * -2 = 48 5 * -6 = -30 -36 ÷ -4 = 9 -50 ÷ 10 = -5 Number Sense and Operations Expression A mathematical phrase made up variables and/or numbers and operations. A dentist earns twice as much as she did last year. If her salary last year was p, write an expression for her current salary. 3x -11 ½x y+8 2p 3 Equation A mathematical statement that two expressions are equal. Two-step equations Whatever you do to one side of the equation you MUST do to the other side. Undo any addition or subtraction first Undo any multiplication or division Unit Rates To find a unit rate make the second quantity be one. Ratios Use a ratio and a proportion to solve a problem. Ratio: - the relationship between two quantities; expressed as a fraction, or quotient of one number divided by the other. The ratio of 3 to 2 can be written 3 / 2 or 3 : 2. Standard Form to Scientific Notation To write a number in scientific notation, count how many places you must move the decimal point to get a number greater than or equal to 1 and less than 10. The number of decimal places you move the decimal point is the power of 10. Scientific Notation to Standard Form To write a number in standard form move the decimal point to the right the number of places indicated by the power of 10. Order of Operations The rules for evaluating expressions using order of operations: P – Parentheses – simplify inside the parentheses E – Exponents M D – multiplication and division from left to right A S – addition and subtraction from left to right Multipyling by ½ Remember that multiplying by ½ is the same as multiplying by 0.5 or dividing by 2. Division Symbols All of these symbols mean to divide. 10 2 10/2 x – 10 = 6 3y = 9 x - 4 = 14 4y + 10 = 74 4 -10 = - 10 +4= +4 4y = 64 x = 18 ÷4 = ÷ 4 4 y = 16 *4 = *4 x = 72 Rate: 504 miles in 12 hrs. Unit Rate: Divide each quantity by 12 to get 42 miles in 1 hr. Ratio of boys to girls is 6:5. If there are 392 students, how many of them are girls? In this sample there are 11 total students, 6 boys and 5 girls. Write a proportion to solve this problem: 5 = n 11 392 5 * 392 = 11n 1960 = 11n ÷ 11 = ÷ 11 178.2 = n There are 178 girls. 245,000 = 2.45 * 105 1,890,000 = 1.89 * 106 45.68 = 4.568 * 101 6,000,000,000 = 6 * 109 6.75 * 104 = 67,500 9 * 106 = 9,000,000 3.4 * 107 = 34,000,000 6 + 5(24 ÷ 3 + 2) – 4 6 + 5(8 + 2) – 4 6 + 5(10) – 4 6 + 50 – 4 56 – 4 52 243 ÷ (-3)2 – 5 + 6 243 ÷ 9 – 5 + 6 27 – 5 + 6 22 + 6 28 36 * ½ = 36 * 0.5 = 36 ÷ 2 18 = 18 = 18 10 = 5 2 10/2 = 5 10 ÷ 2 = 5 10 ÷ 2 4 Data Analysis, Statistics and Probability Venn Diagram A Venn Diagram is used to display data. This diagram shows the following information about 30 students that were surveyed. 3 like basketball only 4 like basketball and soccer 5 like soccer only 3 like basketball and baseball 2 like soccer and baseball 4 like only baseball 7 like all three 2 don't like any of the three It also shows that 18 students like soccer, 16 like baseball, and 17 like basketball Mean, Median, Mode and Range Mean (average) – divide the sum of the values by the number of values. Median – Arrange the values in order from least to greatest. The median is the one in the middle. Mode – The most common data value. If no value occurs more than once there is no mode. If two values occur more than once and equally there are two modes. Range – the difference between the highest and lowest values. Stem-and-Leaf Diagrams Shows how often data values occur. STEM: - the tens digit LEAF: - the ones digit. Example: 43 has a stem of 4 and a leaf of 3 Values: 4, 2, 5, 3, 4, 6, 4 Mean: 28 ÷ 7 = 4 Median: 2, 3, 4, 4, 4, 5, 6 (4 is the middle) Mode: 4 occurs the most often Range: 6 – 2 = 4 Values: 5, 3, 6, 3, 4, 6 Mean: 27 ÷ 6 = 4.5 Median: 3, 3, 4, 5, 6, 6 There are two middle values so add them together and divide by 2. 9 ÷ 2 = 4.5 Mode: 3 and 6 Range: 6 – 3 = 3 Stem 4 5 6 Leaf 23556 012 566 The values represented are 42, 43, 45, 45, 46, 50, 51, 52, 65, 66 and 66. Probability The probability of an event compares the number of ways the event can occur to the number of possible outcomes. It is expressed as a fraction, decimal, percent or ratio: Example: There are 4 white marbles, 3 blue marbles, 6 green marbles and 2 red marbles in a bag. What is the probability you pick a blue marble from the bag? number of ways event can happen = 3 number of possible outcomes 15 Probability (event) = number of ways event can happen number of possible outcomes 5 Tree Diagram A tree diagram is a way to show all the different outcomes for a given situation. Example: How many ways can a burglar describe the length and color of a suspect’s hair? Make a tree diagram to show this: Color Black S T A R T Organized List You can make an organized list to count the number of possible descriptions. Length Short Medium Long Brown Short Medium Long Blonde Short Medium Long So, there are 9 different possible descriptions. Hair Color Hair Length Black Short Black Medium Black Long Brown Short Brown Medium Brown Long Blonde Short Blonde Medium Blonde Long So, there are 9 different possible descriptions. Geometry Congruent Figures If two figures are congruent, then their corresponding parts are congruent. Let's look at the corresponding parts of triangles ABC and DFE. Here, angle A corresponds to angle D, angle B corresponds to angle F, and angle C corresponds to angle E. Side AB corresponds to side DF, side BC corresponds to side FE, and side CA corresponds to side ED. 6 Similar Figures If two objects have the same shape, they are called "similar." When two figures are similar, the ratios of the lengths of their corresponding sides are equal. AB = AC 6 = 10 DE DF 3 5 You can use this information to find the length of a missing side by writing a proportion. Angles formed by intersecting lines Vertical angles – angles on opposite sides of the intersection of two lines. <2 and <3 1 2 Corresponding angles - The angles are in the same 3 4 position from one parallel line to the other parallel line. <1 and <5; <2 and <6; <4 and <8; <3 and <7 5 6 Alternate Interior angles - The angles are on opposite sides of the transversal and on the inside of the lines 7 8 it intersects. <3 and <6; <4 and <5 Alternate Exterior angles - The angles are on opposite sides of the transversal and on the outside of the lines it intersects. <1 and <8; <2 and <7 A nonagon 15-sided figure Sum of the measures of the angles of a polygon To find this use the formula: S = (9 – 2) * 180 S = (15 – 2) * 180 S = (n – 2) * 180 S = 7 * 180 S = 13 * 180 Where n is the number of sides in the polygon. S = 1260 degrees S = 2340 degrees Other Angles Complementary angles – two angles whose sum is 90° Supplementary angles – two angles whose sum is 180° Obtuse angle – an angle greater than 90° and less than 180° Acute angle – an angle less than 90° Straight angle – an angles that measures exactly 180° right 3 down 4 – means to move each vertex of the Translations A translation moves a figure right or left and up or figure 3 places to the right on the x-axis and down 3 on down. The size of the figure does not change. the y-axis. (x, y) (x – 2, y + 3) – means to move each vertex 2 places to the left on the x-axis and up 3 on the y-axis. Example: A triangle whose coordinates are at: A: (4,5); Reflections A reflection flips a figure across a line. Every point B: (2,3) and C: (1, 4) is reflected across on the original figure is the same distance from the line as the matching point on the reflection. 1. the x-axis. What are its new coordinates: When a point is reflected across the x-axis: A: (4,-5) B: (2,-3) C: (1,-4) - its x-coordinate stays the same - its y-coordinate is multiplied by -1 2. the y-axis. What are its new coordinates: When a point is reflected across the y-axis: A: (-4,5) B: (-2,3) C: (-1,4) - its x-coordinate is multiplied by -1 - its y-coordinate stays the same 7 Prisms A polyhedron whose bases are congruent and parallel. It is named by the shape of its bases. hexagonal prism Pyramid A pyramid has a polygon for a base, but comes to a point. It is named by the shape of its base. 3-D Vocabulary face – a flat surface on a solid edge – a segment joining two faces of a polyhedron vertex – a corner where the edges meet vertex face edge Measurement Converting units in the same system of measurement You need to be able to convert between different units by using a conversion factor. Remember in your conversion factor to make sure the unit you want to convert is in the denominator of the conversion factor. See the examples: Convert 15 feet to inches Convert 144 inches to feet 15 ft * 12 inches = 180 inches 144 inches * 1 foot = 12 feet 1 foot 12 inches Converting units in different systems of measurement You will also need to be able to convert between different systems of measurement (i.e. metric to English). One mile is about 1.6 kilometers. Use a conversion factor to find the answer to this type of question. Remember in your conversion factor to make sure the unit you want to convert is in the denominator of the conversion factor. See the examples: How many kilometers are there in 2350 miles? 2350 miles * 1.6 km = 3860 km. 1 mile Scale A scale is used in drawings and the reading of maps. It is written in the following form: actual value scale value Perimeter of a Polygon The perimeter of a shape is the distance around the shape. To find the perimeter add up the length of all the sides. How many miles are there in 3440 km ? 3440 km. * 1 mile = 2150 miles 1.6 km On a map, 1 in = 13 miles. When I measure between two cities it is 2 ¼ inches. What is the actual distance between the cities? 13 * 3.25 = 42.25 miles What is the perimeter of a parallelogram with sides of 5 in, 4in, 8 in. and 4 in.? 5 + 4 + 8 + 4 = 21 inches 8