Coached Instruction Supplement

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1 Practice Coach PLUS Coached Instruction Supplement Mathematics 8 Practice Coach PLUS, Coached Instruction Supplement, Mathematics, Grade 8 679NASP Triumph Learning Triumph Learning, LLC. All rights reserved. No part of this publication ma be reproduced in whole or in part, stored in a retrieval sstem, or transmitted in an form or b an means, electronic, mechanical, photocoping, recording or otherwise, without written permission from the publisher. v1

2 Table of Contents Domain 1 The Number Sstem Lesson 1 Rational Numbers... 4 Lesson Irrational Numbers... 8 Lesson Compare and Order Rational and Irrational Numbers... 1 Lesson 4 Estimate the Value of Epressions Domain Epressions and Equations Lesson 5 Eponents Lesson 6 Square Roots and Cube Roots... Lesson 7 Scientific Notation... 5 Lesson 8 Solve Problems Using Scientific Notation... 1 Lesson 9 Linear Equations in One Variable... 4 Lesson 10 Use One-Variable Linear Equations to Solve Problems Lesson 11 Slope... 4 Lesson 1 Slopes and -intercepts Lesson 1 Proportional Relationships Lesson 14 Direct Proportions Lesson 15 Pairs of Linear Equations... 6 Lesson 16 Solve Sstems of Equations Graphicall Lesson 17 Solve Sstems of Equations Algebraicall Lesson 18 Use Sstems of Equations to Solve Problems Domain Functions Lesson 19 Introduction to Functions Lesson 0 Work with Linear Functions Lesson 1 Use Functions to Solve Problems Lesson Use Graphs to Describe Relationships Lesson Compare Relationships Represented in Different Was

3 Domain 4 Geometr Lesson 4 Congruence Transformations Lesson 5 Dilations Lesson 6 Similar Triangles Lesson 7 Interior and Eterior Angles of Triangles.. 10 Lesson 8 Parallel Lines and Transversals Lesson 9 The Pthagorean Theorem Lesson 0 Distance... 1 Lesson 1 Appl the Pthagorean Theorem Lesson Volume Domain 5 Statistics and Probabilit Lesson Scatter Plots Lesson 4 Trend Lines Lesson 5 Interpret Linear Models Lesson 6 Patterns in Data Answer Ke

4 Domain 1 Lesson 1 Rational Numbers Getting the Idea Integers include the set of whole numbers (0, 1,,, ) and their opposites (1,,, ). The number line below shows integers from 5 to 5. Positive numbers are located to the right of zero, and negative numbers are located to the left of zero. negative numbers positive numbers A rational number is an real number that can be epressed as the ratio of two integers a b, where b is not equal to zero. Some eamples of rational numbers are shown below. 6, 5, 7, 16%, 4, 0. 7, A rational number can be epanded to form a terminating decimal or a repeating decimal. To convert a fraction to a decimal, ou can divide the numerator b the denominator. Eample 1 Is 5 a rational number? If so, write it as a decimal. 11 Decide if 5 is rational. Then divide to write it as a decimal. 11 Is 5 a rational number? 11 5 and 11 are both integers. 5 So, shows the ratio of two integers. It is rational. 11 Step Divide the numerator, 5, b the denominator, 11. Step Solution Insert zeros after the decimal point in 5 as needed The decimal repeats. Write a bar to show the repeating digits is rational. It can be epressed as the repeating decimal

5 Eample Is a rational number? If so, write it as a decimal. 5 Decide if is rational. Then divide to write it as a decimal. 5 Is a rational number? 5 Convert to an improper fraction. 5 (? 5) Since 1 5 is the ratio of two integers, is rational. 5 Step Divide the numerator, 1, b the denominator, 5. Since the signs are different, the quotient will be negative. For now, drop the negative sign The actual quotient is.6. Solution is rational. It can be epressed as

6 All rational numbers can be represented on a number line. To plot rational numbers on a number line, it is helpful to convert them to the same form. You can convert a percent to a decimal b dividing the percent b 100 and dropping the percent sign. This is the same as moving the decimal point in the percent two places to the left. Some square roots are also rational. An number that has a whole-number square root is a perfect square. If the number under a radical smbol ( ) is a perfect square, its value is an integer and a rational number. For eample, 9 is rational because it is equal to. Eample Plot and label a point for each rational number below on a number line., 1 1, 0.5, 7.5%, 4 Write the numbers in an equivalent form. Step Rewrite each number as a decimal or integer , so is alread in decimal form. 7.5% 5 7.5% 4 100% , because 5 4. Plot and label each number on a number line. Draw a number line from 1 to and divide it into tenths % Solution The number line with the rational numbers labeled is shown in Step. 6

7 Lesson 1: Rational Numbers Coached Eample What decimal is represented b point P on the number line below? 0 1 The number line is divided into siths. Starting at the tick mark for 1, ou count tick marks from 1 to point P. So, point P represents the mied number. Instead of converting the mied number to an improper fraction, just convert the fractional part,, to a decimal. Divide the numerator,, b the denominator,. P 6 5 If the decimal repeats, write it with a bar over the repeating digit: Add 1 to the decimal: The decimal represented b point P is. 7

8 Domain 1 Lesson Irrational Numbers Getting the Idea The set of real numbers includes both rational numbers and irrational numbers. An real number that cannot be written as the ratio of two integers a, where b is not equal to zero, b is irrational. An irrational number can be epressed as a non-terminating, non-repeating decimal. Eample 1 The value of the number p (pi) is shown below. p Eplain wh p is an irrational number. Step Use the definition of an irrational number. Recall the characteristics of irrational numbers. The value of an irrational number is a non-terminating, non-repeating decimal. Describe the decimal value of pi. The ellipsis ( ) shows that the digits continue on forever and do not repeat. So, p is a non-terminating non-repeating decimal. Solution p (pi) is an irrational number because it is a non-terminating, nonrepeating decimal. p is not the onl irrational number. The square roots of positive numbers that are not perfect squares are also irrational. You can estimate the value of a square root b determining which two perfect squares it lies between and then using guess and check to approimate its value more precisel. The smbol means approimatel. 8

9 Eample Approimate the value of. Use the definition of an irrational number. Then use guess and check to approimate the value of. Find which perfect squares the number lies between. You can find perfect squares b squaring consecutive whole numbers ? ? 5 4 is between 1 and. Step Determine if the value is closer to 1 or. is closer to 1 than to 4. So, is closer to 1 than to. Step Use guess and check to estimate to the nearest tenth. Tr ? close, but slightl less than. Solution Tr ? close, but not as close as Since is closer to 1.4 than to 1.5, < 1.4. The number has a value close to

10 Eample Graph the approimate location of 4 on a number line. Step Step Step 4 Use guess and check to approimate the value of 4 to the nearest tenth. Then graph the decimal on a number line. Find which perfect squares 4 lies between ? ? is between the perfect squares 5 and 6, so 4 is between 5 and 6. Find which whole number 4 is closer to. 4 is closer to 6 than to 5, so 4 is closer to 6 than to 5. Use guess and check to estimate 4 to the nearest tenth. Tr ? close, but more than 4. Tr ? close, and less than 4. Since 4 is closer to 5.8 than to 5.9, 4 < 5.8. Graph 4 on a number line divided into tenths. Plot 4 between 5.8 and 5.9, but closer to Solution is graphed on the number line in Step 4. 10

11 Lesson : Irrational Numbers Coached Eample The area of a square is 67 square meters. Find the eact length, in meters, of one side of the square. Then graph that approimate value on a number line. The area, A, of a square is found using the formula A 5 s, where s shows the length of one side. So, the length of one side, s, can be found b taking the square root of. The eact length of each side of the square is meters. To graph that number on a number line, first approimate its value as a decimal. 67 lies between the perfect squares 64 and. 64 5, and the square root of the other perfect square is. So, 67 lies between the whole numbers and, but is closer to. Use guess and check to approimate its value to the nearest tenth. Tr 8.1: ? close, but than 67. Tr 8.: ? 8. 5 close, and than 67. Which is closer to 67: 8.1 or 8.? So, 67 is between 8.1 and 8., but is closer to. Graph 67 on the number line below The eact length of one side of the square is meters. The number line above shows the approimate decimal value. 11

12 Domain 1 Lesson Compare and Order Rational and Irrational Numbers Getting the Idea To compare and order numbers, use the following smbols:. (is greater than), (is less than) 5 (is equal to) To compare an irrational number to another number, approimate its decimal value. Convert the other number to a decimal also. Then compare the digits to determine which decimal is greater. Eample 1 Which smbol makes this sentence true? Use.,,, or Estimate the values of the irrational numbers. Step Step Solution Round to the nearest hundredth Approimate 59 to the nearest whole number. 49, 59, 64, so: 7, 59, is closer to 64 than to 49, so 59 is closer to 8. Continue estimating and compare ? close, but more than 59 Since 59 is less than 7.7, 59 is less than , so The smbol. makes the sentence true

13 When ordering a set of numbers with different signs, know that a positive number is alwas greater than a negative number. Eample Order the numbers below from least to greatest. 8%, 1, 7, 9 Separate the negative numbers from the positive numbers. Then convert each group of numbers to the same form. Step Step Solution Write the negative numbers as decimals and compare , 1 Write the positive numbers as decimals and compare. 8% , 0.9, so 8%, 7. Order all four numbers. 9, 1, 8%, 7 From least to greatest, the numbers are 9, 1 7., 8%, Eample Order these numbers from greatest to least. p, 1, 14 Approimate the value of each number. Approimate the value of p. p <.14 Step Write the value of

14 Step Step 4 Solution Estimate 14 to the nearest tenth. 9, 14, 16, so:, 14, is closer to 16 than to 9, so 14 is closer to ? close.8 5.8? not as close as.7 14 <.7 Order the decimals and then the numbers p From greatest to least, the order of the numbers is: 14, 1, p. Coached Eample Order the following numbers from least to greatest. 1, 5, 0.8,.5 9 Separate the negative numbers from the positive numbers. The negative numbers are: 1 and. 9 Write 1 9 as a decimal: The other negative number is a decimal. Compare the decimals. On a number line, is farther to the left than. So,,. The positive numbers are: 5 and. Approimate 5 to the nearest tenth. 4, 5, 9, so:, 5,. Since the value of 5 is less than,.5 must be than 5. From least to greatest, the order is:,,,. 14

15 Domain 1 Lesson 4 Estimate the Value of Epressions Getting the Idea Sometimes, ou ma need to estimate the value of an epression that includes an irrational number. To do that, estimate the value of the irrational number. Then use that estimate to find the value of the entire epression. When estimating the value of an epression that includes p, it is helpful to remember that p can be approimated as.14 or 7. Eample 1 Estimate the value of p. Substitute.14 for p. Step Write an epression that could be used. p 5 p? p <.14?.14 Multipl. Since each factor has decimal places, the product will have 1, or 4, decimal places..14? < 9.86 Solution The value of p can be estimated as When a number appears to the left of a radical, it means to multipl the number outside the radical b the square root. For eample 5 means to multipl 5. 15

16 Eample Estimate the value of 7. Step Step Solution Estimate the value of 7 to the nearest tenth. Approimate 7 to the nearest whole number. 4, 7, 9, so, 7,. 7 is slightl closer to 9 than to 4, so 7 is slightl closer to. Use guess and check to approimate 7 to the nearest tenth ? close.7 5.7? not as close as.6 7 <.6 Use that decimal approimation to estimate the value of the epression. 7 5? 7 <?.6 < 5. A good estimate of the value of 7 is 5.. Eample Estimate the value of this epression Step Approimate the value of each number. Then subtract. Approimate 4 to the nearest whole number. 6, 4, 49, so 6, 4, 7. 4 is slightl closer to 49 than to 6, so 4 is slightl closer to 7. Use guess and check to approimate 4 to the nearest tenth ? close ? closer than < 6.6 Step Approimate 7 to the nearest tenth < Step 4 Subtract the estimated values of the numbers. 4 7 < < 6 11 Solution The value of 4 7 is approimatel

17 Lesson 4: Estimate the Value of Epressions Coached Eample Approimate the value of p. Plot a point to represent that value on a number line. p < 0 1 Use the space below to divide that approimate value for p b. Now, plot and label a point representing the value of p on the number line above. The value of p is approimatel and is represented on the number line above. 17

18 Domain Lesson 5 Eponents Getting the Idea A number in eponential form has a base and an eponent. The eponent indicates how man times the base is used as a factor. In a s, the base is a and the eponent is s. In 5 4, 5 is used as a factor 4 times: The epression 5 4 can also be called a power of 5 and is read as five to the fourth power. The value of a nonzero epression in which the eponent is 0 is 1, so Eample 1 What is 4 written in standard form? Multipl the base b itself the number of times shown b the eponent Solution in standard form. A base raised to a negative eponent is equal to the reciprocal of the epression with a positive eponent. Look at the eamples below. 5 5 ( 1 5 ) ( 6 1 ) 5 6 To change the sign of an eponent, move the epression to the denominator of a fraction. a n 5 1, if a 0 a n To change the sign of an eponent in a denominator, move the epression to the numerator. 1 a n 5 a n 1 5 a n, if a 0 18

19 Eample What is 8 written in standard form? Write the reciprocal of the eponential epression with a positive eponent, then simplif. Step Write the reciprocal of the eponential epression to eliminate the negative eponent. 8 5 ( 1 Simplif. 1 8 ) Solution in standard form. 51 Numbers in eponential form are sometimes called powers. There are some properties ou can appl to simplif powers. In the table below, a and b are real numbers, and m and n are integers. Properties of Powers Product of Powers To multipl two numbers with the same base, add the eponents. Quotient of Powers To divide two numbers with the same base, subtract the eponents. Power of a Power To raise a power to a power, multipl the eponents. Power of Zero An nonzero number raised to the power of zero is 1. Power of a Product To find a power of a product, find the power of each factor and multipl. Power of a Quotient To raise a quotient to a power, raise both the numerator and denominator to that power. Eamples a m a n 5 a m 1 n a m 4 a n 5 a m n ( a m ) n 5 a m n ( 6 4 ) a 0 5 1, if a (ab) m 5 a m b m ( ) 5 ( a b ) m 5 a m, if b 0 b m 19

20 Eample What is 5 0 written in standard form? Use the power of zero propert. 5 0, so Solution in standard form. Eample 4 What is 4? Write the product in standard form. Step Use the product of powers propert. The eponential terms have the same base,. Add the eponents Evaluate. Write the number in standard form Solution You could have solved the problem b evaluating each eponent and then multipling. Since 5 9 and , Eample 5 What is ? Write the quotient in standard form. Use the quotient of powers propert. The eponential terms have the same base, 5. Subtract the eponents Step Evaluate. Write the number in standard form Solution

21 Lesson 5: Eponents Eample 6 What is 7 7 5? Write the product in standard form. Use the product of powers propert. Step Step The eponential terms have the same base. Add the eponents (5) 5 7 Write the eponential epression without the negative eponent Evaluate. Write the number in standard form Solution Coached Eample What is ( 10 ) in standard form? To raise a power to a power, ou must the eponents. ( 10 ) Use 10 as a factor times. Multipl to find the product in standard form. ( 10 ) 5 1

22 Domain Lesson 6 Square Roots and Cube Roots Getting the Idea Squaring a number means raising it to the power of. For eample, 7 is equivalent to 7 7, or 49. So, 49 is a perfect square. The opposite, or inverse, of squaring a number is taking its square root. The radical smbol ( 0 ) is used to represent square roots. To find the square root of a perfect square, think about what number, when multiplied b itself, will result in that perfect square. Eample 1 Solve for Determine what number, multiplied b itself, results in 196. Take the square root of both sides of the equation Step Tr squaring numbers until ou find one that results in Too low Too low Step Solve for , so Solution 5 14.

23 Cubing a number means raising it to the power of. For eample, is equivalent to, or 8. So, 8 is a perfect cube. The opposite, or inverse, of cubing a number is taking its cube root. The smbol is used to represent cube roots. To find the cube root of a perfect cube, think about what number, when multiplied b itself twice, will result in that perfect cube. Eample Solve for r. r 5 15 Determine what number, when cubed, results in 15. Take the cube root of both sides of the equation. r 5 15 r 5 15 Step Tr cubing numbers until ou find one that gives a result of Step Solve for r , so r 5 15 Solution r 5 5. r 5 5 The number under a radical sign is called the radicand. If ou do not have a calculator hand, ou ma need to estimate the value of a square root or a cube root. To estimate a square root, find the two perfect squares between which the radicand lies. Take the square root of each to find the range of our estimate. To estimate a cube root, find the two perfect cubes between which the radicand lies. Then take the cube root of each to find the range of our estimate.

24 Eample Between which two consecutive integers is 500? Solution Find the two perfect cubes between which 500 lies. Then take the cube root of each to make our estimate. Tr cubing consecutive positive integers The radicand, 500, is between the perfect cubes 4 and 51. 7, 500, has a value between 7 and 8. Coached Eample The area of the square garden on the right is 11 square ards. What is the length, s, of each side of the garden? The formula for finding the area, A, of a square is A 5 s, where s is the length of a side. The area of the garden above is square ards. To find the length of one side, take the root of that area. On the lines below, tr squaring numbers until ou find one that results in. That is the value of s. The length of each side, s, of the garden is ards. Garden s s 4

25 Domain Lesson 7 Scientific Notation Getting the Idea Scientific notation is a wa to abbreviate ver large or ver small numbers using powers of 10. A number written in scientific notation consists of two factors. The first factor is a number greater than or equal to 1, but less than 10. The second factor is a power of 10. Here are some guidelines and eamples of numbers written in scientific notation. Standard Form Scientific Notation Numbers $ 10 8,000, Numbers $ 1 and, Numbers. 0 and, A number raised to the power of 0 is equal to 1, so multipling b 10 0 is the same as multipling b 1. Eample 1 Jupiter s minimum distance from the Sun is about 460,100,000 miles. What is that number written in scientific notation? Use the definition of scientific notation to find the two factors. Write the first factor, which must be greater than or equal to 1 and less than 10. Put the decimal point after the first nonzero digit, starting at the left. Drop all zeros after the last nonzero digit The first factor is Step Find the eponent for the power of 10. Count the number of places that the decimal point was moved The decimal point was moved 8 places to the left. Since the original number is greater than 10, the eponent will be positive. 8 is the eponent for the power of 10. 5

26 Step Step 4 Write the second factor. The eponent is positive 8. The second factor is Write the number in scientific notation Solution Jupiter s minimum distance from the Sun is about miles. Eample Mr. Kendall measured a specimen that was millimeter long. What is the specimen s length, in millimeters, written in scientific notation? Use the definition of scientific notation to find the two factors. Write the first factor, which must be greater than or equal to 1 and less than 10. Put the decimal point after the first nonzero digit, starting from the left. Drop the zeros that precede that digit The first factor is 4.5. Step Find the eponent for the power of 10. Count the number of places that the decimal point was moved The decimal point was moved 7 places to the right. Since the original number is less than 1, the eponent will be negative. 7 is the eponent for the power of 10. Step Write the second factor. The eponent is 7. The second factor is Step 4 Write the number in scientific notation Solution The specimen was millimeter long. 6

27 Lesson 7: Scientific Notation When converting from scientific notation to standard form, move the decimal point to the right for a positive power of 10 and to the left for a negative power of 10. Eample What is written in standard form? Step Look at the eponent of the second factor to move the decimal point. Look at the eponent of the second factor. The eponent is negative, so the decimal point will move to the left. The eponent is 6, so move the decimal point 6 places to the left. Move the decimal point in.5 si places to the left. Add zeros as needed Step Use a scientific calculator to check our solution. To find : Tpe.5. Press the multiplication sign ke. Tpe 10. Press the eponent ke. Tpe ^ 6 Press the positive/negative ke to change the sign on the 6. Press the equal sign ke. The screen should show The solution is correct. Solution To multipl numbers in scientific notation, first multipl the decimal factors and then multipl the power-of-10 factors. Use the properties of powers when ou multipl the power-of-10 factors. In the eample below, a and b are the decimal factors. (a 10 m )(b 10 n 1 ) 5 ab 10 m n 7

28 To divide numbers in scientific notation, first divide the decimal factors. Then divide the power-of-10 factors, using the properties of powers. In the eample below, a and b are the decimal factors and b 0. (a 10 m ) (b 10 n ) 5 a b 10 m n When ou multipl or divide numbers in scientific notation, our product or quotient ma not be in scientific notation because the decimal factor is not greater than or equal to 1 and less than 10. To fi this, write the decimal factor in scientific notation and use the properties of powers to simplif the epression. Eample 4 Find the product in scientific notation. ( )( ) Multipl the decimal-number factors. Then multipl the power-of-10 factors. Step Step Use the commutative and associative properties to regroup the factors. ( )( ) 5 ( )( ) Multipl the decimal factors Multipl the power-of-10 factors (7) Step 4 Write the product using the products from Steps and. ( )( ) Step 5 Write in scientific notation. Move the decimal point in 11.7 one place to the left. Since ou moved the decimal point one place to the left, the eponent increases b Solution ( )( )

29 Lesson 7: Scientific Notation Eample 5 What is 4. ( ) written in standard form? Step Step Step 4 Use the associative propert to regroup the factors. Then write the product in standard form. Use the associative propert to regroup the factors. 4. ( ) 5 (4..5) 10 6 Multipl the decimal factors Rewrite the epression using the result from Step and the power-of-10 factor. (4..5) Write the product in standard form. Look at the power-of-10 factor. The negative eponent means ou move the decimal point to the left. So, 6 means ou move the decimal point in 10.5 si places to the left Solution 4. ( ) Eample 6 Find the quotient in scientific notation Divide the decimal-number factors and divide the power-of-10 factors. Rewrite the epression Step Divide the decimal-number factors Step Divide the power-of-10 factors Step 4 Write the result using the quotients from Steps and Solution

30 Coached Eample In 01, the Hartsfield-Jackson Atlanta International Airport ranked as the world s busiest airport. In that ear, approimatel passengers passed through this airport. What is that number written in standard form? Since the eponent is positive, this is a number greater than. The eponent of the second factor is. The eponent tells ou to move the decimal point in 9.4 places to the. The number in standard form is. About passengers passed through the Hartsfield-Jackson Atlanta International Airport in 01. 0

31 Domain Lesson 8 Solve Problems Using Scientific Notation Getting the Idea Sometimes, ou ma need to multipl or divide numbers written in scientific notation in order to solve real-world problems. Eample 1 A rectangular section of wilderness will be set aside as a new wildlife refuge. Its dimensions are meters b meters. Find the area of the land in square meters. Then convert the area into square kilometers using the conversion below. 1 square kilometer ( km ) square meters ( m ) Which unit is a better choice for measuring the area of the wildlife refuge, and wh? Multipl the dimensions. Convert the area into square kilometers. Compare the two units. Multipl the dimensions to find the area, in square meters A. Step Step Step 4 A 5 ( )( ) Multipl the first factors and then multipl the power-of-10 factors So, A 5 ( )( ) Rewrite the number in scientific notation Convert the area into square kilometers. To convert a smaller unit (square meters) to a larger unit (square kilometers), divide: A (in km ) Divide the first factors and then divide the power-of-10 factors So, A (in km )

32 Step 5 Which is the better unit to use? square meters 5 0,000,000,000 m 10 4 square kilometers 5 0,000 km 0,000 is a more reasonable number to work with in standard form. Also, square kilometers are larger units than square meters. Since the area is large, it is better to use the larger unit. Solution The area of the refuge is square meters or 10 4 square kilometers. Square kilometers is a better unit to use because the area is large. Sometimes, ou ma use technolog, such as a calculator, to generate a number. If the result is a number that is ver large or ver small, man calculators will automaticall give the number in scientific notation. Eample One cubic millimeter of Ms. Murph s blood contains about 5,000,000 red blood cells. There are about 4,900,000 cubic millimeters of blood in her entire bod. Use a calculator to determine approimatel how man red blood cells Ms. Murph has in total. Interpret the number our calculator gives as the final answer. Use a calculator to determine the answer. Interpret the result. Step Step How can ou find the total number of red blood cells? Multipl the number of red blood cells in one cubic millimeter of blood (5,000,000) b the total number of cubic millimeters of blood in the bod (4,900,000). Use a calculator to determine the answer. Tpe Tpe Press Press Interpret the answer shown on the calculator displa. The screen shows this: Solution Ms. Murph has a total of about red blood cells in her bod.

33 Lesson 8: Solve Problems Using Scientific Notation Eample California, the most populous state, has approimatel people living in it. The population of the entire United States is approimatel 10 8 people. About how man times greater is the population of the United States than the population of California? Step Solution Decide if ou should multipl or divide. Then solve the problem. Decide on which operation to use. To find how man times greater, divide 10 8 b Divide the first factors and then divide the power-of-10 factors So, The population of the United States is 7.5 times the population of California. Coached Eample A computer was used to draw a rectangle with an area of square meter. Would it be better to measure the area in square meters or square millimeters? Use the conversion below to help determine our answer. 1 square meter ( m ) square millimeters ( mm ) Rewrite in scientific notation The decimal point was moved places to the right. The original number is less than, so the eponent will be negative Multipl to convert that number of square meters to square millimeters: (7 10 )( ) Multipl the first factors: Multipl the power-of-10 factors: The area is square millimeters. It is better to measure the area in square because it is better to measure a small area using a unit.

34 Domain Lesson 9 Linear Equations in One Variable Getting the Idea An equation is a mathematical sentence that uses an equal (5) sign to show that two quantities are equal in value. A variable is a smbol or letter that is used to represent one or more numbers. A constant is a value that does not change. A linear equation has one or more variables raised to the first power. You can use inverse operations and the properties of equalit to solve a linear equation that has one variable. Addition Propert of Equalit If ou add the same number to both sides of an equation, the equation continues to be true. If a 5 c, then a 1 b 5 c 1 b. Subtraction Propert of Equalit If ou subtract the same number from both sides of an equation, the equation continues to be true. If a 5 c, then a b 5 c b. Properties of Equalit Multiplication Propert of Equalit If ou multipl both sides of an equation b the same number, the equation continues to be true. If a 5 c, then ab 5 cb. Division Propert of Equalit If ou divide both sides of an equation b the same nonzero number, the equation continues to be true. If a 5 c and b 0, then a b 5 c b. Whatever ou do to one side of the equation, ou must also do to the other side. That wa, ou can isolate the variable while still keeping the equation true. Eample 1 Find the value of in this equation Use inverse operations to isolate the variable. Remove the constant. Subtract 9 from both sides

35 Step Isolate the variable. Multipl both sides b 6. 6? ? Solution The value of is 60. To undo multiplication b a fraction, multipl b the reciprocal of that fraction. Flip the fraction to find its reciprocal. For eample, the reciprocal of 4 5 is 5 4. Eample What is the value of c in this equation? c Use inverse operations to isolate the variable. Step Remove the constant. Add to both sides. 5 c c Isolate the variable. c Give 7 10 and the same denominator. 5 c c Divide both sides b or multipl b the reciprocal,. c? ? 9 1c c Solution The value of c is

36 You ma also need to combine like terms if the same variable is on both sides of the equation. Like terms are terms that contain the same variables raised to the same power. Eample What is the value of z in this equation? 0.8 z z Step Step Combine like terms. Then solve. Combine like terms so there is onl one variable term. Subtract 0.8 z from both sides. 0.8 z z z 0.8 z z 0.8 z z Remove the constant. Subtract 1.5 from both sides z z z Isolate the variable. Divide both sides b z z z Solution The value of z is

37 Lesson 9: Linear Equations in One Variable To solve some equations, ou ma need to use the distributive propert. Distributive Propert of Multiplication over Addition When a factor is multiplied b the sum of two numbers, multipl each of the two numbers b the factor and then add the products. a(b 1 c) 5 ab 1 ac Distributive Propert of Multiplication over Subtraction When a factor is multiplied b the difference of two numbers, multipl each of the two numbers b the factor and then subtract the products. a(b c) 5 ab ac Eample 4 What is the value of in this equation? ( ) Use the distributive propert. Then combine like terms. Appl the distributive propert to evaluate the right side of the equation. Distribute the over ( ) ( ) (? ) (? ) Step Step Combine like terms so there is onl one variable term in the equation. Subtract from both sides Remove the constant. Add 1 to both sides

38 Step 4 Isolate the variable. Divide both sides b Solution The value of is 1 1. In Eamples 1 4, all the equations had one solution. However, linear equations ma also have no solution or infinitel man solutions. One Solution No Solution Infinitel Man Solutions Onl one number, 1, makes the equation below true. Eample: No number makes the equation below true. Eample: never true An number makes the equation below true. Eample: alwas true Eample 5 Does this equation have one solution, no solutions, or infinitel man solutions? 10q (q 1 4) Use the distributive propert. Then combine like terms. Decide how man solutions the equation has. Step Appl the distributive propert to evaluate the right side of the equation. Distribute 5 over (q 1 4). 10q (q 1 4) 10q 15 5 (5? q) 1 (5? 4) 10q q 1 0 Combine like terms. Subtract 10q from both sides. 10q q q 10q q 10q never true 8

39 Lesson 9: Linear Equations in One Variable Step Solution Determine the solution of the equation. Since 15 0, no value of q makes the equation true. The equation has no solutions. The equation has no solutions. Coached Eample Does the equation below have one solution, no solutions, or infinitel man solutions? n (n 1 6) Appl the distributive propert. Distribute 1 over (n 1 6). n (n 1 6) n 1 5 ( 1? n 1 ) 1 ( 1? ) n Subtract from both sides. n 1 5 n 5 Is the equation above alwas true, never true, or sometimes true? The equation is true, so an value of n makes the equation true. Does the equation have one solution, no solutions, or infinitel man solutions? Since an value of n makes the equation true, the equation has solution(s). 9

40 Domain Lesson 10 Use One-Variable Linear Equations to Solve Problems Getting the Idea Sometimes, ou can solve a real-world problem b writing a linear equation to represent it and then solving the equation. Identifing which quantities are equal can help ou as a first step. Writing an epression for each of the equal quantities can also help. Operation Problem Epression addition 4 more than the sum of and 4 4 increased b the total of and 4 4 combined with 4 1 subtraction multiplication division 4 minus fewer than 4 less than 4 subtracted from 4 4 decreased b the difference of from 4 4 times 4 multiplied b the product of 4 and groups of 4 partitioned into 4 equal groups shared b 4 equall

41 Eample 1 Diego has 60 CDs. This is 1 more than Heidi has. How man CDs does Heidi have? Step Solution Write and solve an equation. Write an equation. Let h represent the number of CDs Heidi has h 1 1 Solve the equation h h Subtract 1 from both sides h Heidi has 48 CDs. Eample Simone has si less than of the number of baseball cards that Manuel has. Simone has 14 baseball cards. How man cards does Manuel have? Write and solve an equation. Step Step Solution Identif the quantities that are equal less than the number of baseball cards Manuel has. Write an equation. Let m represent the number of cards Manuel has. m Solve the equation. m m Add 6 to both sides. m m? 5 0 1? 5 0 Multipl both sides b, the 1 reciprocal of. Manuel has 0 baseball cards. 41

42 Eample A tai charges $.50 for each ride plus $1.5 per mile traveled. If the total charge for one ride was $8.75, how man miles were traveled? Step Solution Write and solve an equation. Write an equation. Let m represent the number of miles traveled m Solve the equation m Subtract.5 from each side. 1.5m Divide both sides b 1.5. m 5 5 The tai traveled 5 miles. Coached Eample At a video store, for ever DVD bought at the regular price, a customer can bu a second DVD for half the regular price. Nadia bus two DVDs, each of which regularl costs d dollars, and pas $ in all. What is the regular price of each DVD? Translate the problem into an equation. One DVD at the regular price of d dollars Second DVD at half the regular price of d dollars $ in all Rewrite the equation and solve for d. The regular price of a DVD at the store is $. 4

43 Domain Lesson 11 Slope Getting the Idea The graph of a linear equation is a straight line. The steepness of the line is called its slope. The slope shows the rate at which two quantities are changing. Specificall, it is the ratio of the vertical change to the horizontal change, or rise run. 1 1 positive slope negative slope slope of 0 undefined slope Look at the lines graphed above. A line that slants up from left to right has a positive slope. For this graph, as the -values increase, the -values also increase. A line that slants down from left to right has a negative slope. For this graph, as the -values increase, the -values decrease. A horizontal line has a slope of 0 because there is no vertical change. A vertical line has an undefined slope, because there is no horizontal change. rise A fraction with a denominator of 0 is undefined, and run would have a denominator of 0. 4

44 Eample 1 What is the slope of this line? Find the vertical change and the horizontal change. Choose two points on the line. (1, 1) and (6, ) Step Step Identif the vertical change and horizontal change. From (1, 1) to (6, ), the line moves 4 units down and 5 units to the right. Write the slope. The line slants down from left to right, so the slope is negative. slope 5 rise run Solution The slope is (1, 1) (6, )

45 Lesson 11: Slope Similar figures have the same shape, but not necessaril the same size. The ratios of the lengths of their corresponding sides are equal. Eample Does the slope of a non-vertical line change depending on which two points ou use to determine it? Use the two similar triangles and the line graphed below to help ou answer the question (8, 11) 6 (4, 5) (, ) For each triangle, write the ratio of the vertical side length to the horizontal side length. Compare those ratios, and compare them to the slope of the line. Step Find the ratio of the vertical and horizontal side lengths for each triangle. vertical side length Smaller triangle: 5 horizontal side length vertical side length Larger triangle: 5 6 horizontal side length 4 5 Compare the ratios. The ratio of the lengths of the vertical and horizontal sides for both triangles is. 45

46 Step Step 4 Solution Compare the ratios to the slope. The slope is the ratio rise run or vertical change. horizontal change rise If ou use the points (, ) and (4, 5), slope 5 run 5. rise If ou use the points (4, 5) and (8, 11), slope 5 run Analze the slope. No matter which two points on the line ou use, the slope is the same. This is because the graph of a line changes at a constant rate. The slope of a non-vertical line stas the same no matter which two points ou use to determine it. Eample demonstrates that the slope of a line represents a constant rate of change. If ou know an two points on a line, ou can determine its slope using the formula below. Slope The slope, m, of a line containing points ( 1, 1 ) and (, ) is: change in m 5 change in Eample What is the slope of a line that passes through (, 6) and (5, 1)? Use the slope formula. Step Identif the points. Let ( 1, 1 ) 5 (, 6). Let (, ) 5 (5, 1). Substitute the numbers into the slope formula. m m () m m Solution The slope is

47 Lesson 11: Slope The slope of a line represents a constant rate of change. For eample, a slope of 50 could represent these rates: $50 hours 50 miles 50 pages gallons minutes Eample 4 A locksmith charges a flat fee for each house call plus an hourl rate, as shown b the graph below. Total Charge (in dollars) House Call (, 70) (1, 50) Number of Hours Find the slope of the graph. What does the slope represent in this problem? Find and interpret the slope of the line. Step Solution Find the slope. Let ( 1, 1 ) 5 (1, 50). Let (, ) 5 (, 70). m Interpret the slope. The slope compares the total charge, in dollars, to the number of hours worked. 0 dollars The slope shows a rate of change of, or $0 per hour. 1 hour So, the slope represents the hourl rate charged b the locksmith. The slope is 0. It shows that the locksmith charges $0 per hour for each job. 47

48 Coached Eample Joanie bought an airplane phone card that charges her a connection fee plus an additional rate for each minute a call lasts. The graph below represents this situation. 4 Airplane Phone Card Costs Total Cost (in dollars) Number of Minutes What is the slope of the graph, and what does it represent? Choose an two points on the graph. Let ( 1, 1 ) 5 (, ). Let (, ) 5 (6, ). m The -ais shows the total cost in. The -ais shows the time in. So, the slope shows a rate of change of dollars to, or per minute. Does the slope represent the connection fee or the rate per minute for the call? The slope is. It shows that Joanie must pa $ per minute for the calls she makes. 48

49 Domain Lesson 1 Slopes and -intercepts Getting the Idea Some linear equations have two variables. For eample, the linear equation includes the variables and. All of the ordered pairs, in the form (, ), that make that equation true are solutions of the equation. The graph of a linear equation is a straight line. The point at which the graph crosses the -ais is called its -intercept. An point (0, b) that is a solution of the equation is the -intercept. Eample 1 Below is a graph of the linear equation Identif its -intercept. Then show that the -intercept is a solution for the equation. Step Solution Identif the coordinates of the -intercept. Then show that those - and -values make the equation true. Identif the -intercept. The graph crosses the -ais at (0, 5). That is the -intercept. Show that (0, 5) is a solution for the equation. Substitute 0 for and 5 for into the equation (0) The -intercept is (0, 5). Those coordinates are a solution for the equation, as shown in Step

50 The equation is written in slope-intercept form. If a linear equation is in slope-intercept form, ou can use the -intercept and the slope to graph it. The slope-intercept form of an equation is: 5 m 1 b, where m represents the slope and b represents the -intercept. Eample Graph the equation Identif the -intercept and slope. Use the slope to find a second point on the line. Step Identif the slope and the -intercept. The equation is in slope-intercept form, 5 m 1 b. m 5, so the slope is. b 5 1, so the -intercept is (0, 1). Use the slope to find a second point. Plot a point at the -intercept, (0, 1). Use the slope to find a second point. rise slope 5 run 5 Start at (0, 1). Since the slope is positive, rise up units and run units to the right. Plot a point at (, ). Step Draw a straight line through the points (0, 1) and (, ). Solution The graph of is shown above (0, 1) (, ) 50

51 Lesson 1: Slopes and -intercepts A linear equation in the form 5 m has b 5 0. This means that its -intercept is at (0, 0), the origin. To graph 5 m 1 b: First, graph 5 m. Shift each point on the graph up or down b units. If b. 0, shift the graph b units up. If b, 0, shift the graph b units down. Eample Graph 5 1. On the same grid, graph 5 1. Compare the graphs. Graph 5 1. Then shift the graph down or up to graph 5 1. Graph 5 1. Start at the -intercept, (0, 0). The slope is negative, so count 1 unit down and units to the right. Plot a point there at (, 1). Draw a line through the points (0, 0) and (, 1). Step Graph 5 1. The slope is the same as for 5 1. Since b, 0, each point on the graph of 5 1 is shifted units down in the graph of 5 1. So, (0, 0) moves units down to (0, ). (, 1) moves units down to (, 4), and so on (0, 0) (, 1)

52 Solution The graphs of 5 1 and 5 1 are shown on the right. Their slopes are the same. Ever point in the graph of 5 1 is shifted three units down from the graph of (0, 0) (, 1) (0, ) (, 4) When ou know the coordinates of one point on a line and its slope, ou can use the point-slope form to write the equation in slope-intercept form. A line that passes through ( 1, 1 ) with a slope m can be written in point-slope form: 1 5 m( 1 ) Eample 4 A line has a slope of 4 and passes through (, 5). Write the equation of the line in slope-intercept form. Solution Use the point-slope form to write the equation. The slope, m, is 4. Let (, ) 5 (, 5) m( 1 ) ( ) Distribute 4 over ( ) Add 5 to both sides A line with a slope of 4 that passes through (, 5) has the equation

53 Lesson 1: Slopes and -intercepts Coached Eample What is the equation of the line graphed below? Find the values of m and b to write an equation in slope-intercept form. The -intercept is the point at which the graph crosses the -ais. The -intercept of this graph is (0, ). So, b 5. Choose two points on the graph to find the slope, m. Use the -intercept, (0, ), and the point (, 0). To move from the -intercept to (, 0), count units up and units to the right. m 5 rise run 5 5 Since the line slants from left to right, the slope is positive. Substitute those values of m and b into 5 m 1 b. 5 The equation of the line is 5. 5

54 Domain Lesson 1 Proportional Relationships Getting the Idea A ratio is a comparison of two numbers. For eample, if there are 1 bos and 1 girls in a class, the ratio of bos to girls is 1 to 1. This can also be written with a colon, 1:1, or as a fraction, 1 1. Since a ratio is a comparison of numbers, the ratio 1 is not an improper 1 fraction and cannot be rewritten as The ratio 1 compares part of a class (the bos) to another part of the class (the girls). 1 You can also use ratios to compare parts to totals. Eample 1 A bouquet contains onl red and white roses. The ratio of red roses to white roses is 1:. What is the ratio of red roses to the total number of roses in the bouquet? Step Solution Use the part-to-part ratio to find the part-to-total ratio. Write the part-to-part ratio as a fraction. red roses white roses 5 1 Use that ratio to write the part-to-total ratio. The total includes all the red roses and all the white roses. red roses 5 1 red roses 1 white roses The ratio of red roses to total roses is 1. A proportion shows that two ratios are equal in value. You can use proportional reasoning to solve for an unknown value in a proportion. Eample The debate team won out of 5 debates it participated in this semester. If the team participated in 0 debates, how man debates did it win? How man did it lose? Set up a proportion and use proportional reasoning. What does the given ratio represent? is the ratio of debates won to total debates. 5 54

55 Step Step Step 4 Step 5 Solution Write a second ratio that includes the same terms. Let represent the number of debates won. There were 0 debates total. debates won 5 total debates 0 Set the ratios equal to each other to form a proportion Use proportional reasoning and think about equivalent fractions to find the value of. The denominators are 5 and 0, and So, multipl the numerator and denominator of the first ratio b is equivalent to 1 5, so the value of is 1. 0 Find the number of debates that the team lost. If the team won 1 out of 0 debates, then the number of debates lost was: The team won 1 debates and lost 8 debates. Another wa to solve for an unknown value in a proportion is to use cross-multiplication. To cross-multipl, multipl the numerator of each ratio b the denominator of the other ratio and set them equal to each other. Then solve for the unknown value. A rate is a ratio that compares quantities that use different units. Use the same strategies to work with rates as ou use with other ratios. Eample It costs $61 for nights at Pavia Pavilion Hotels. At the same rate, how much will it cost to sta for 7 nights? Step Step Set up a proportion and cross-multipl. Write a ratio comparing the cost to the number of nights. cost 5 61 number of nights Write a second ratio that includes the unknown. Let represent the unknown cost. cost 5 number of nights 7 Set the ratios equal to each other to form a proportion