Chapter Numerical Expressions and Factors Information Frame 9. Sample answers are given.. Ke Words: the sum of, the total of Real-Life Application : On a video game, Jacob got 68 points and earned two bonuses worth 9 and 70 points. What is his total score? Answer: 8 points Adding whole numbers Estimate and Check: 68 + 9 + 70 700 + 00 + 00 00 Because 8 00, the answer is reasonable. Numbers: 68 9 + 70 8. Ke Words: the difference of, how man more, how man less, the change in Real-Life Application : On the wa to school, the temperature was 8 F. On the wa home, it was 7 F. How much did the temperature increase? Answer: F Subtracting whole numbers Estimate and Check: 7 8 70 60 0 Because 0, the answer is reasonable. Numbers: 6 7 8. Real-Life Application : How man cartons of milk should the cafeteria manager at Riverdale Middle School order so that all of the 6 students can get one carton per da in Februar (8 das)? Answer: 808 cartons Ke Words: the product of, total of equal-size groups Multipling whole numbers Estimate and Check: 6 8 0 0 00 Because 808 00, the answer is reasonable. Numbers: 6 8 088 + 7 808 Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced
Chapter (continued). Ke Words: Real-Life Application : Sukon has a package of 87 stickers and wants to give each of his classmates the same number of stickers. What is the greatest number of stickers that he can give to each classmate? How man will be left over? Numbers: the quotient of, divide into equal-size groups, dividend quotient divisor Answer: stickers each with left over Dividing whole numbers Estimate and Check: R 87 6 7 The quotient is. 87 600 0 0 Because 0, the answer is reasonable.. Procedure: : 0 + 0 6 + + 8. Perform operations in Parentheses.. Evaluate numbers with Exponents.. Multipl or Divide from left to right.. Add or Subtract from left to right. : Order of operations ( )³ 6 + ³ 6 + 8 6 + 6 + + : 6 ² + 7 6 + 7 + 9 6. : 0 0 6 Words: Write a composite number as a product of its prime factors. Use a factor tree to find a prime factorization. Prime factorization : Find the greatest perfect square that is a factor of. The prime factorization is 7. 9 ( ) ( ) 6 6 6 So, the greatest perfect square factor of is 6. : 99 9 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter (continued) 7. : Find the GCF of 6 and. 6:,,, 8, 6 :,,,, 6, 8,, The GCF is 8. Words: Factors that are shared b two or more numbers are called common factors. The greatest of the common factors is called the greatest common factor (GCF). Greatest Common Factor (GCF) : 6 Find the GCF of 7 and 90. 7 7 90 90 8 9 9 0 The GCF is 8. : Find the GCF of and 0. 0 6 The GCF is 8. : Find the LCM of 6 and 8. Words: Multiples that are shared b two or more numbers are called common multiples. The least of the common multiples is called the least common multiple (LCM). : Find the LCM of 8 and. 6 8 6 8 Least Common 0 6 8 : 8, 6,,, 0, 8,... 8 Multiple (LCM) :,, 6, 8, 60,... 6 8 0 60 7 6 : 66 6 The LCM is. Find the LCM of 9,, and. The LCM 9 is. The LCM is 80. 9. : + 8 8 The LCD is 9 0 + + 8 9 Words: The least common denominator (LCD) of two or more fractions is the least common multiple (LCM) of the denominators. Least Common Denominator (LCD) : 9 : 9 0 9 6 9 6 The LCD is 7 and 6 So, >. 6 Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced
Chapter Fractions and Decimals Notetaking Organizer 8. Sample answers are given.. a c a c b d b d (b 0 and d 0) Multipling fractions Multipl the numerators and multipl the denominators. The denominators cannot be 0. : 8 8 How do ou multipl two mixed numbers?. Multipling mixed numbers. Write each mixed number as an improper fraction. a b c d a c b d (b 0 and d 0). Multipl as ou would with fractions. : 7, or Wh can t b and d equal 0? How do ou divide mixed numbers? Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter (continued). a c a d b d b c (b, c, and d 0) Dividing mixed numbers. Write each mixed number as an improper fraction.. Divide as ou would with fractions. : 9 9 9 9, or 9 0 0. Be sure to line up the decimal points so that ou add or subtract onl the digits that have the same place value. You ma have to insert zeros. addend + addend sum Adding and subtracting decimals. Line up the decimal points.. Bring down the decimal point.. Add or subtract as ou would with whole numbers. : 7.6 + 08. 7.6 + 08.00.9 Insert zeros. How do ou add or subtract a decimal and a fraction? How do ou divide decimals?. Remember to check that our product has the same number of decimal places as the decimal factor. You ma have to insert zeros in our product. You do not have to line up the decimal points when multipling. factor factor product Multipling decimals b whole numbers. Multipl as ou would with whole numbers.. Count the number of decimal places in the decimal factor.. The prduct has the same number of decimal places. : 6.6 6.6 9.9 two decimal places 6. Remember to check that our product has the same number of decimal places as the sum of the decimal places in the factors. You ma have to insert zeros in our product. You do not have to line up the decimal points when multipling. factor factor product Multipling decimals b decimals. Multipl as ou would with whole numbers.. Add the number of decimal places in the factors.. The sum is the number of decimal places in the product. :.6.07.07.6.6 decimal places decimal places decimal places How do ou multipl two decimals? Wh don t ou line up the decimal points when multipling? Wh don t ou line up the decimal points when multipling? How do ou multipl a decimal b a fracton? Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced
Chapter (continued) 7. dividend quotient divisor quotient divisor dividend Remember to place the decimal point in the quotient first. Dividing decimals b whole numbers. Place the decimal point in the quotient above the decimal point in the dividend.. Divide as ou would with whole numbers.. If needed, insert a zero in the dividend, and continue to divide. :.7 6.9 6.70 6 7 0 0 0 Wh can ou insert a zero after a decimal? How do ou divide a decimal b another decimal? 8. dividend quotient divisor quotient divisor dividend Multipling the divisor and dividend b a power of 0 does not change the quotient. Dividing decimals b decimals. Multipl the divisor and the dividend b a power of 0 to make the divisor a whole number.. Place the decimal point in the quotient.. Divide as ou would with whole numbers.. If needed, insert a zero in the dividend and continue to divide. : 0.9 0. 0. 0.9.8.90 9 70 70 0 Wh can ou insert a zero after a decimal? How do ou divide a fraction b a decimal? 6 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter Algebraic Expressions and Properties Information Wheel 8. Sample answers are given.. Substitute a number for each variable. Evaluate a + b when a 7 and b. a + b (7) + () + 9 Evaluating algebraic expressions Use order of operations to find the value of the expression. You get 6n dollars for helping with n chores. How much do ou earn for chores? 6n 6() $0. Addition: added to, plus, sum of, more than, increased b, total of, and Multiplication: multiplied b, times, product of, twice, of : The sum of and twice a number r. Answer: + r Writing algebraic expressions Subtraction: subtracted from, minus, difference of, less than, decreased b, fewer than, take awa Division: divided b, quotient of : 6 less than the quotient of a number and 0. Answer: 6 0. Numbers: + + Algebra: a + b b + a a b b a Commutative Properties of Addition and Multiplication Changing the order of the addends does not change the sum. Changing the order of the factors does not change the product. s: 8 + + 8 + + + x x x Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced 7
Chapter (continued). Numbers: ( + ) + + ( + ) Algebra: (a + b) + c a + (b + c) (a b) c a (b c) Associative Properties of Addition and Multiplication Changing the grouping of the addends does not change the sum. Changing the grouping of the factors does not change the product. s: 9 + (7 + m) (9 + 7) + m 6 + m x x x. Numbers: + 0 : 7 + 0 + k 7 + k Algebra: a + 0 a Addition Propert of Zero The sum of an number and 0 is that number. 6. Numbers: 0 0 Algebra: a 0 0 a a Multiplication Properties of Zero and One The product of an number and 0 is 0. is that number. s: 0 k 0 k 0 m m m 8 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter (continued) 7. To multipl a sum or difference b a number, multipl each number in the sum or difference b the number outside the parentheses. Then evaluate. : 8 6 8(60 + ) 8 60 + 8 80 + 6 96 Numbers: ( + ) + ( ) Algebra: a(b + c) a b + a c a(b c) a b a c Distributive Propert : 7(x ) 7(x) 7() x : ( + ) ( ) + ( ) + 8. Use the Distributive Propert to write an expression as a product of factors. Algebra: ab ac a b c ab ac a b c Factoring expressions : Factor x using the GCF. Find the GCF of x : x x Solution: x x x Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced 9
Chapter Areas of Polgons Four Square. Sample answers are given.. Words The area A of a triangle is one-half the product of its base b and its height h. Algebra A bh. Words The area A of a trapezoid is one-half the product of its height h and the sum of its bases b and b. Algebra A h(b + b ) A mm bh ()(6) The area of the triangle is square millimeters. Area of a triangle 6 mm A bh ()() ft ft The area of the triangle is square feet. ft ft 7 ft A h(b + b ) ()( + 7) The area of the trapezoid is square feet. Area of a trapezoid 9 m A h(b + b ) (6)(7 + 9) 8 6 m The area of the trapezoid is 8 square meters. 7 m. Definition A composite figure is made up of triangles, squares, rectangles, and other two-dimensional figures. 6 cm rectangle cm 8 cm Words Area of a composite figure trapezoid cm To find the area of a composite figure, separate it into figures with areas ou know how to find. Then find the sum of the areas of those figures. Solution Area of trapezoid A h(b + b ) ()(8 + ) 8 Area of rectangle A lw (8) 96 So the area of the composite figure is 8 + 96 square centimeters. 0 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter (continued).. Words You can use ordered pairs to represent vertices of polgons. To draw a polgon in a coordinate plane, plot and connect the ordered pairs. Quadrilateral D(, ), E(, 9), F(8, 7), G(7, ) 8 6 D E 0 0 6 8 x Words Drawing a polgon in a coordinate plane G You can find the length of a horizontal or vertical line segment in a coordinate plane using the coordinates of the endpoints. F Triangle A(, ), B(, 8), C(7, ) 8 6 B A C 0 0 6 8 x Pentagon P(, ), Q(, 6), R(,9), S(8, 6 ), T(7, ) 8 6 Q R S P T 0 0 6 8 x Vertical Lines When the x-coordinates are the same, find the difference of the -coordinates. (a, b) b c (a, c) Finding distances x in the first quadrant Horizontal Lines Find the perimeter. width When the -coordinates are the same, find the difference B(, ) C(6, ) of the x-coordinates. (d, e) (f, e) f d x A(, ) D(6, ) x length 6 The perimeter of the rectangle is () + () units. Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced
Chapter Ratios and Rates Definition and Chart 0. Sample answers are given.. Equivalent ratios: two ratios that describe the same relationship. : and 9 : 0 to 0 and to : and to 60. Ratio table: a table used to find and organize equivalent ratios. Girls Bos 0 6 0 Equivalent ratios: :, 0 : 6, 0 : Flour (cups) Water (cups) 6 9 Equivalent ratios: :, 6 :, 9 :, : Red Yellow 7 8 Equivalent ratios: : 7, 8 :. Rate: a ratio of two quantities using different units.. Unit rate: a rate that compares a quantit to one unit of another quantit. $ hours 60 miles hour 6 feet seconds. feet second 8 hits at bats 7 words minute Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter (continued). Equivalent rates: rates that have the same unit rate. Points 0 6 8 0 6 6. Percent: a part-to-whole ratio where the whole is 00. So, ou can write a percent as a fraction with a denominator of 00. 0 0% 0 out of 00 00 0 Touchdowns 6 % 00 00 Distance (feet) Time (seconds) 6 96 8 6 ft ft 96 ft Equivalent rates: sec sec 8 sec 9 9 % 0 0 00 Cost (dollars) Apples (pounds) 7.. 7 $7.0 $.0 $7.00 Equivalent rates: lb lb lb 7. U.S. customar sstem: a sstem of measurement that contains units for length, capacit, and weight. 8. Metric sstem: a decimal sstem of measurement, based on powers of 0, that contains units for length, capacit, and mass. Unit of length: foot Unit of length: meter Unit of capacit: quart Unit of capacit: liter Unit of weight: pound Unit of mass: gram 9. Conversion factor: a rate that equals. Length: mi 80 ft m.8 ft cm 0 mm and, and, and 80 ft mi.8 ft m 0 mm cm Capacit: gal qt qt 0.9 L L 000 ml and, and, and qt gal 0.9 L qt 000 ml L Mass: lb 6 oz lb 0. kg kg 000 g and, and, and 6 oz lb 0. kg lb 000 g kg 0. Unit analsis: a process ou can use to decide which conversion factor will produce the appropriate units. lb oz 6 oz 9 oz lb ml L L 0.0 L 000 ml mph ft/sec mi 80 ft h 8,800 ft ( )( ). ft/sec h mi 600 sec 600 sec Time: h 60 min da h h 600 sec and, and, and 60 min h h da 600 sec h Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced
Chapter 6 Integers and the Coordinate Plane Summar Triangle 7. Sample answers are given... Positive integers Definition: The whole numbers that are greater than 0; The can be written with or without a positive sign (+). Ke words: gain, up, earn, climb, rise, ascend, receive, deposit, above zero, above sea level s: +; 9; +6; 00,000 Negative integers Definition: The opposites of the positive integers; The are written with a negative sign ( ). Ke words: lose, down, fall, debt, withdraw, below zero, below sea level, descend s: ; 9; 6; 00,000.. Opposites Definition: Two numbers that are the same distance from 0 on a number line, but on opposite sides of 0 The opposite of 0 is 0. Absolute value Definition: The distance between a number and 0 on a number line The absolute value of a number a is written as lal. : opposites s: l l ll units units 0 0. Coordinate plane Formed b the intersection of a horizontal number line and a vertical number line x-axis: the horizontal number line -axis: the vertical number line Ordered pairs (x, ) are plotted in a coordinate plane. : (, ) O x (, ) (0, ) Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter 6 (continued) 6. Origin Definition: The point, represented b the ordered pair (0, 0), where the axes intersect in a coordinate plane The origin is at (0, 0). O x 7. Quadrants Definition: The four regions of a coordinate plane that are separated b the axes Quadrant II O Quadrant IlI Quadrant I x Quadrant IV Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced
Chapter 7 Equations and Inequalities and Non- Chart 8. Sample answers are given.. Inverse Operations s addition and subtraction multiplication and division Non-s addition and multiplication addition and division subtraction and multiplication subtraction and division. Equations Solved Using Addition or Subtraction s + 9 a 0 x 9 r + Non-s a 6 0.b 6 0 6. Equations Solved Using Multiplication or Division s a 6 6 0.b 6 7 x Non-s x + 9 a 0 x 9. Equations in Two Variables s x 7 x C 0 + w d t x Non-s x 6 x a + 0 0.b. Inequalities s Non-s 6. Graphs of Inequalities s Non-s x 8 + < n + r > 0 k 6 x + 9 7 a 0 c n < x C 00 x 0 0 00 0 0 00 O (, ) x 7. Inequalities Solved Using Addition or Subtraction s Non-s 8. Inequalities Solved Using Multiplication or Division s Non-s x + < 9 8 n a > m + x + 7 8x > 6 0.b n > a < 6 0.b > 6 x 7 a x 7 n 7 > 0 + 9 6 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter 8 Surface Area and Volume Process Diagram. Sample answers are given... Drawing a pramid Hexagonal pramid Finding the surface area of a prism Rectangular prism 7 cm Draw a base and a point. Draw a net. side 7 cm top back bottom 7 cm side front Connect the vertices of the base to the point. Find the areas of the faces. Change an hidden lines to dashed lines. Find the sum of the areas of the faces... Finding the surface area of a pramid Triangular pramid Finding the volume of a rectangular prism 7 in. in. 8 in. Draw a net. Write the formula. V wh Find the areas of the faces. Substitute values. V (7 )()( ) 8 Find the sum of the areas of the faces. Multipl. Be sure to include units. 7 V 66 in. 8 Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced 7
Chapter 9 Statistical Measures Concept Circle 0. Sample answers are given.. Mean. Outlier Concept Concept Words The mean of a data set is the sum of the data divided b the number of data values. A mean is a tpe of average. Video game scores: 0,, 9, 0,,, 0, 9 Mean: 9.9 8 Numbers Data:,,,, data values + + + + Mean:. Words When included in a data set, an outlier can affect the mean. A data value that is much greater or much less than the other values Mean with outlier: 6 6.9 9 9 Mean without outlier:. 8 Without the outlier, the mean better represents the data. Number of pets: 0,,,, 0,,,, Outlier:. Measures of Center. Median Vocabular mean median mode Concept A measure that describes the tpical value of a data set Video game scores: 0,, 9, 0,,, 0, Mean: about 9.9 Median:. Mode: 0 Which measure best represents the data? The median best represents the data. The mode is greater than most of the data, and the mean is less than most of the data. Words. Order the data.. For a set with an odd number of values, the median is the middle value.. For a set with an even number of values, the median is the mean of the two middle values. Concept Tells the middle of the data set Video game scores: 0,, 9, 0,,, 0, Order the data:, 9, 0,,,, 0, 0 + Median:. Numbers Data,,,, Median: Data:,,,,, 6 + Median:. Data:,,,, Median: 8 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter 9 (continued). Mode 6. Measures of Variation Concept Concept Words Data can have one mode, more than one mode, or no mode. When all values occur onl once, there is no mode. The value or values that occur most often in a data set Numbers Data:,,,, Mode: Data:,,,, Mode: none Data:,,,,, Modes: and Vocabular range interquartile range (IQR) mean absolute deviation (MAD) A measure that decribes the distribution of a data set Visual Number of Strings on an Instrument Number of strings on an instrument:,,,, 6, 6, 6, Range: 8 IQR:. MAD:. Video game scores: 0,, 9, 0,,, 0, Mode: 0 6 7 8 9 0 All but one of the data values are close together. 7. Range 8. Quartiles Concept Concept Words The range of a data set is the difference between the greatest value and the least value. A simple measure of variation Number of strings on an Instrument:,,,, 6, 6, 6, Range: 8 Numbers Data:,,,, Range: Data:,,,, 99 Range: 99 98 Data:,,, Range: 0 Words. The median (second quartile, Q ) divides the data set into two halves.. The first quartile, Q, is the median of the lower half.. The third quartile, Q, is the median of the upper half. Divide the data into four equal parts Number of strings on an Instrument:,,,, 6, 6, 6, + 6 Median(Q ):. + Q :. 6 + 6 Q : 6 Visual Q 9 Q Q Data:,, 6, 8, 0,,, 6 lower half upper half 9. Interquartile Range (IQR) 0. Mean Absolute Deviation (MAD) Concept Concept Words The interquartile range (IQR) is the difference between the third quartile and the first quartile. A measure of variation that represents the range of the middle half of the data Number of strings on an Instrument:,,,, 6, 6, 6, Q. Q. Q 6 IQR: 6.. Visual Q 9 Q Q Data:,, 6, 8, 0,,, 6 IQR: 8 Process Step : Find the mean of the data. Step : Find the distance between each data value and the mean. Step : Find the sum of the distances in step. A measure of variation that tells the average of how much data values differ from the mean Step : Divide the sum in step b the Numbers total number of data values. + + + + 0 + 0 + 0 + 6 MAD 8 8. The mean absolute deviation is.. Visual Number of strings on an instrument:,,,, 6, 6, 6, Distances from the mean: 0 0 0 6 7 8 9 0 6 8 Mean: 6 8 Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced 9
Chapter 0 Data Displas Word Magnet. Sample answers are given.. Stem-and-leaf plot Uses the digits of data values to organize a data set The stem is the digit or digits on the left. The leaf is the digit or digits on the right. Tickets sold per da:,,, 9, 7, 8,, 9, 8, 7 Stem 0 Leaf 7 9 8 9 7 8 Ke: 8 8 tickets Shows how data are distributed. Shapes of You can use dot plots and distributions histograms to identif shapes of distributions. Smmetric: Skewed left: 0 Skewed right: The left side is a mirror image of the right side. When a data distribution is smmetric, use the mean to describe the center and use the MAD to describe the variation. 0 When a data distribution is skewed, use the median to describe the center and use the IQR to describe the variation.. Represents a data set along a number line b using the least value, the greatest value, and the quartiles of the data The five numbers that make up the box-and-whisker plot are called the five-number summar of the data set. Shows the variabilit of a data set least Q median(q ) Q greatest Box-and-whisker plot About of the data are in each whisker. About of the data are in the box. A long whisker or box indicates that the data are more spread out. Smmetric: Skewed right: 0 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter 0 (continued). Absolute value Distance between a number and 0 on a number line The absolute value of a number a is written as a. The absolute value of a positive number is positive. The absolute value of a negative number is positive... 6 6 The absolute value of 0 is 0. 0 0 Words Addition: When ou add the same number to each side of an equation, the two sides remain equal. Subtraction: When ou subtract the same number from each side of an equation, the two sides remain equal. Multiplication: When ou multipl each side of an equation b the same nonzero number, the two sides remain equal. Division: When ou divide each side of an equation b the same nonzero number, the two sides remain equal. Properties of Equalit Algebra x 6 + + x x + 6 x x x x x x x 7 Formed b the intersection of a horizontal number line and a vertical number line Coordinate plane Ordered pairs (x, ) are plotted in a coordinate plane. Quadrant II x-axis O Quadrant III Quadrant I -axis x Quadrant IV The origin is at (0, 0). (, ) O (, ) (, ) x Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced
Chapter Integers Idea and s Chart 6. Sample answers are given.. Integers:,,,, 0,,,, 86 0 6 a. Adding integers with the same sign: Add the absolute values of the integers. Then use the common sign. 6 + 7 + ( ) 9 + ( ) 00 b. Adding integers with different signs: Subtract the lesser absolute value from the greater absolute value. Then use the sign of the integer with the greater absolute value. 8 + ( ) 6 8 + 6 97 + 9 78. Additive Inverse Propert: The sum of an integer and its additive inverse, or opposite, is 0. + ( ) 0 00 + 00 0 6 + ( 6) 0. Subtracting integers: To subtract an integer, add its opposite. + ( ) ( ) + ( ) + Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter (continued) a. Multipling integers with the same sign: The product of two integers with the same sign is positive. b. Multipling integers with different signs: The product of two integers with different signs is negative. 6 8 6 ( ) 8 6 ( ) 8 6 8 8( 6) 88 8( 6) 88 6a. Dividing integers with the same sign: The quotient of two integers with the same sign is positive. 00 00 ( ) 98 7 6b. Dividing integers with different signs: The quotient of two integers with different signs is negative. 00 ( ) 00 98 7 Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced
Chapter Rational Numbers Process Diagram. Sample answers are given.. Writing rational numbers as decimals Write the rational number a as a fraction if necessar. b Use long division to find the quotient a b. If the remainder is 0, the rational number is a terminating decimal. If the remainder repeats, the rational number is a repeating decimal. Write as a decimal. 6. 6.0 0 0 0 So,.. Write as a decimal. 0..000 0 0 So, 0... Subtracting rational numbers Add the opposite of the rational number being subtracted. if the addends have the same sign if the addends have different signs Add the absolute values of the rational numbers. Subtract the lesser absolute value from the greater absolute value. Write the sum using the common sign. Write the sum using the sign of the rational number with the greater absolute value. s + 8 8 8 8 s + + 6.8 (.67).8 +.67 6., or 9.0. 9.0 + (.).6 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter (continued). Multipling rational numbers If the factors have the same sign If the factors have different signs The product is positive. The product is negative. s ( ) ( ) ( ) 9 ( ) 9 6.7. 8 + 7 8.88 The product is 8.88. s 7 ( ) ( ) 7, or.6. The product is.7.. Dividing rational numbers If the dividend and divisor have the same sign If the dividend and divisor have different signs The quotient is positive. The quotient is negative. s. ( 0.7). ( ) 6 ( ) ( ), or s. (0.7). 0 Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced
Chapter Expressions and Equations Four Square 7. Sample answers are given.. Definition An algebraic expression is in simplest form when it has:. no like terms and. no parentheses. Words To write an algebraic expression in simplest form: Step : Rewrite as a sum. Step : Use the Distributive Propert on parentheses, if necessar. Step : Rearrange terms. Step : Combine like terms.. Definition An algebraic expression in which the exponent of the variable is s 7x, x +, 8 x Non-examples: x, x + x, x 7 9 x + 6x x + 8 x x + 6x + ( x ) + 8 + ( x) x + ( x ) + 6x + ( x) + 8 [ + ( )]x + [6 + ( )]x + 8 x + x + 8 Simplest form 9 ( m ) + m 9 + ( )( m + ( )) + m 9 + ( )( m) + ( )( ) + m 9 + ( m) + + m ( m) + m + 9 + ( + )m + (9 + ) m + 0 Adding linear expressions: (7 w) + ( w + ) 7 + ( w) + ( w) + () 7 + ( w) + ( 6w) + ( w) + ( 6w) + 7 + 7w + 9 Linear expression Subtracting linear expressions: ( + 7) ( 8) + 7 ( 8) + 7 + ( ) + 8 +. Words Write the expression as a product of factors. You can use the Distributive Propert. Factor a the GCF. a a 6 a 6(a 6 6(a Factor out of r +. Factor 7 out of p + 8. r r p 7 p 8 7 p + 8 r + r + 7 p (r 7 p. Words Two equations are equivalent equations if the have the same solutions. You can use the Addition, Subtraction, Multiplication, and Division Properties of Equalit to write equivalent equations. s x 7 and x 7 + 7 + 7 d + 7 and d + 7 and c c and Algebra Equivalent equations a b and a + c b + c a b and a c b c a b and a c b c a b a b and, c 0 c c Non-s x + 7 and x + 7 7 + 7 c c and ( ) 7 m + and 7 7 m + x + 7 and x 7 + 6 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter (continued). Words To undo addition, use the Subtraction Propert of Equalit: subtracting the same number from each side of an equation produces an equivalent equation. To undo subtraction, use the Addition Propert of Equalit: adding the same number to each side of an equation produces an equivalent equation. x + + x 6 Algebra Solving equations using addition or subtraction Check x? 6 If a + b c, then a + b b c b. If a b c, then a b + b c + b. x +... x 7.9 Check x +.? 7.9 +. 6. Words Algebra To undo multiplication, use the Division Propert of Equalit: dividing each side of an equation If ab c, b the same number produces ab c then, b 0. an equivalent equation. b b To undo division, use the a Multiplication Propert of If c, b Equalit: multipling each side of a then b c, b 0. an equation b the same number b produces an equivalent equation. Solving equations using multiplication or division Check x x x? ( ) x, or x x ( ) x 6 Check x 6? 7. Words Undo the operations in the reverse order of the order of operations:. Undo addition or subtraction.. Undo multiplication or division. After solving for the variable, check our solution. a 7 7 7 7 a a ( ) ( ) a x + x x x Solving two-step equations Check a 7 7? 7 7 + ( )? 7 7 7? + ( ) 7 7 7 7 7 (x ) x + ( 8) +8 +8 x 0 x Check x +? ( ) +? + Check (x )? ( )? () Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced 7
Chapter Ratios and Proportions Information Wheel 8. Sample answers are given.. A ratio of two quantities with different units : $ hours : 6 feet seconds Rate : 8 hits at bats Unit rate: A rate with a denominator of of a unit rate: 60 miles hour. A rate with a denominator of : 7 words minute :. feet second You can write rates as unit rates. : 00 units 0 units 8 hours hour Unit rate : 60 miles hour You can write unit rates as rates with denominators other than. : 0. mile mile hour hours. An equation stating that two ratios are equivalent : 6 8 You can use a table to write a proportion. : See page 80. Proportion Two quantities that form a proportion are proportional. The cross products of a proportion are equal. : 8 6 To solve a proportion, ou can use mental math. 8 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter (continued). The products a d and b c in the proportion a c b d Numbers: 6 The cross products 8 are 8 and 6. Note that the are both equal to. You can use the Cross Products Propert to solve a proportion. Cross products Cross Products Propert In words: The cross products of a proportion are equal. Cross Products Propert using a c algebra: ad bc in b d, where b 0 and d 0 : x x 6 x. : x The graph has a constant rate of change. + 6 + + + 6 9. : Find the unit rate. Graphing proportional relationships : The graph passes through the origin. The graph is a line that passes through the origin. So, x and are in a proportional relationship. 8 6 0 0 6 8 x 6. Method : Use mental math. Method : Use the Multiplication Propert of Equalit. Method : Use the Cross Products Propert. Solving proportions : 0, so x x 0 : x 0 0 0 x : 0 x 0 x x x 0 x 0 Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced 9
Chapter (continued) 7. Rate of change between an two points on a line vertical change horizontal change Slope A measure of the steepness of a line change in change in x Through (0, 0) and (, ): slope Through (, ) and ( 6, ): slope 6 8 8. Two quantities x and show direct variation when kx, where k is a number and k 0. The graph of kx is a line that passes through the origin. varies directl with x. Direct variation x and are directl proportional. x k x k 0 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.
Chapter Percents Summar Triangle 8. Sample answers are given... Writing a decimal as a percent Multipl b 00, or just move the decimal point two places to the right. Then add a percent smbol. Comparing and ordering fractions, decimals, and percents Write the numbers as all fractions, all decimals, or all percents. Then compare. Use a number line if necessar. n (n 00)% s: Which is greater,.% or? 0 %. So,.% is greater. 0 00 : 0..% Which is greater,.% or 0.0?.% 0.0. So,.% is greater... The percent proportion The percent equation Use a proportion to represent a is p percent of w. Use an equation to represent a is p percent of w. part whole a w p 00 percent percent in fraction or decimal form part of the whole a p w whole : What percent of 0 is? p 0 00 6. p So, 6.% of 0 is. : What number is % of 60? a 00 So, is % of 60. Copright Big Ideas Learning, LLC. All rights reserved. Big Ideas Math Advanced
Chapter (continued). 6. Percent of change The percent that a quantit changes from the original amount percent of change amount of change original amount : Find the percent of change from inches to inches. Because >, the percent of change is a percent of increase. 9 percent of increase 0.6, or 60% Discount A decrease in the original price of an item Discount: a p w Sale price original price discount : An item that originall sells for $ is on sale for 0% off. What is the sale price? a 0. Sale price 0 The sale price is $0. 7. 8. Markup To make a profit, stores charge more than what the pa. The increase from what the store pas to the selling price is called a markup. Markup: a p w Selling price cost to store + markup : The wholesale price of an item is $. The percent of markup is %. What is the selling price? a 0..8 Selling price +.8 6.8 The selling price is $6.80. Simple interest Principal Simple interest Mone paid or earned onl on the principal Annual interest rate (in decimal form) I Prt Time (in ears) : You deposit $000 in an account that earns % simple interest per ear. How much interest will ou earn after ears? I Prt 000(0.0)() $0 Big Ideas Math Advanced Copright Big Ideas Learning, LLC. All rights reserved.