Square Numbers Exponentials

Transcription

1 Student Page Domain: Expressions and Equations Focus: Square Numbers and Roots Lesson: #1 Standard: 8.EE.: Use square root and cube root symbols to represent solutions to equations of the form x = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational. Example: Tables of exponentials and square numbers. Exponentials Square Numbers Exponentials Square Numbers Square numbers end in 0, 1, 4, 6, 9, or 5. Evaluate the following. If the number is irrational, simplify and keep answer in exact form Explain how you simplified 8. Directions: Evaluate the following. If the number is irrational, simplify and keep answer in exact form Explain how you simplified the irrational number. 18

2 =88.3x=.x=40=0.83.x=0 Domain: Expressions and Equations Focus: Using Square Roots Lesson: # Standard: 8.EE.: Use square root and cube root symbols to represent solutions to equations of the form x = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational. Example: Find the following xStudent Page Directions: Solve the following equations x 6.x5.x=4=57.x47Directions: Solve the following equations. Show each step. Simplify irrational numbers if possible and leave answer in exact form. x1.= x1x 19

3 0.0Student Page Domain: Expressions and Equations Focus: Evaluate Cube Roots Lesson: #3 Standard: 8.EE.: Use square root and cube root symbols to represent solutions to equations of the form x = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational. Example: Table of exponentials and cube numbers. Exponentials Cube Numbers n Cube root of n The number inside the check mark of the radical sign is called the index. The index for a cube root is 3. Evaluate the following. If the number is irrational, simplify and keep answer in exact form Explain how you simplified Directions: Evaluate the following. If the number is irrational, simplify and keep answer in exact form Explain how you simplified problem 4. 0

4 303=51Student Page Domain: Expressions and Equations Focus: Using Cube Roots Lesson: #4 Standard: 8.EE.: Use square root and cube root symbols to represent solutions to equations of the form x = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational. Example: Find the following x Directions: Solve the following equations. x x 16 Directions: Solve the following equations. Show each step. Simplify irrational numbers if possible and leave answer in exact form x 8 4. x x 5. x x 3,000 1

5 .7833.x==18=1=6.x0.01Common Core Standards Plus is not licensed for duplication. Copying is illegal. Student Page Domain: Expressions and Equations Evaluation: #1 Focus: Square roots and cube roots Directions: Complete the following problems independently. You may use your tables of perfect squares and cubes. Simplify. Keep answers in exact form Solve for vertically. x. Show all work Line up equal signs. 04.5x=035.x= x88.x

6 Domain: Expressions and Equations Focus: Properties of Exponents Lesson: #5 Standard: 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. Definition and Properties of Exponents Definition of Exponents: ; 3 is called the base and 4 is called the exponent. First Power Property: Any base raised to an exponent of one is equal to itself. 1 Rule: a a Example: 18 1 = 18 Zero Power Property: Any non-zero base raised to a zero exponent equals 1. 0 Rule: a 1 Examples: 57 0 =1, Product Property: When multiplying two or more powers with the same base, add the exponents. m n m n Rule: a a a Example: Proof: Quotient Property: When dividing two powers with the same base, subtract the exponents. m a mn Rule: a n a Example: Proof: Negative Exponent Property: Any non-zero base raised to a negative exponent is equal to the reciprocal of the base raised to the positive exponent. m m 1 1 a m Rules: a and a m m a a Examples: 3 and Example: Simplify the following expressions. Write the property of integer exponents used at each step. Student Page Directions: Simplify the following expressions. Write the property of integer exponents used at each step

7 Student Page Domain: Expressions and Equations Focus: Properties of Exponents Lesson: #6 Standard: 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. Example: Simplify the following expressions x x x Directions: Simplify the following expressions. Show your work x x 4

8 Domain: Expressions and Equations Focus: Properties of Exponents Lesson: #7 Standard: 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. 0 Example: 0 is undefined. Power to a Power Property: When raising any power to an exponent, multiply the exponents. Rule: ( a m ) n a mn Example: ( ) Proof: ( ) ( ) ( ) ( ) Product to a Power Property: When the product of a base is powered by the same exponent, then both the factors are powered by the same exponent. Rule: ( ab) n a n b n Example: ( 3) Proof: ( 3) ( 3) ( 3) Quotient to a Power Property: When the quotient of a base is powered by the same exponent, then both the numerator and denominator are powered by the same exponent. n n Rule: a a n b b Example: Proof: Directions: Simplify the expressions. Show two different approaches. Student Page 3 1. (7 ) Directions: Simplify the expressions. Show two different approaches (10 ) 3. ( 4) Using the Properties of Exponents, write 4 unique expressions that simplify to

9 Student Page Domain: Expressions and Equations Focus: Properties of Exponents Lesson: #8 Standard: 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, = 3 3 = 1/3 3 = 1/7. Example: Simplify the expressions ( 4) (3 ) 4. ( xy ) 6 Directions: Simplify the following expressions. Show your work ( ) a 5 1 b 6

10 Student Page Domain: Expressions and Equations Focus: Properties of Exponents Evaluation: # Directions: Complete the following problems independently. Simplify the expressions using the Properties of Exponents. You may reference the properties from previous lessons. Be sure to show your work (3) (3) ( x y) 7

11 Common Core Standards Plus Mathematics Grade 8 Performance Lesson # Domain: Expressions and Equations Student Page 1 of 3 Square root: A divisor of a quantity that when squared gives the quantity. Perfect square number: The product of an integer multiplied by itself. Principal root: The positive square root of a number. Radicand: The number under the radical sign. Cube root: A divisor of a quantity that when cubed gives the quantity. Perfect cube number: The product of an integer multiplied by itself three times. Index: The small number we insert in the check mark of the radical sign to indicate the root we will take (e.g., in a cube root, the index is 3.). Properties of exponents: The rules we follow when performing operations with exponents. Exponential form: A number written with an exponent (e.g., ). The large number (4) is the base. The small number ( 5 ) is the exponent. The exponent indicates the number of times a base multiplies itself (4 5 = ). First Power Property: Any base raised to an exponent (power) of one is equal to itself. Zero Power Property: Any non- zero base raised to a zero exponent (power) equals 1 (e.g., 33 0 = 1). Product Property: When multiplying two or more powers with the same base, add the exponents (e.g., = = 7 9 ). Quotient Property: When dividing two powers with the same base, subtract the exponents (e.g., = 4 7- = 4 5 ). Negative Exponent Property: Any non- zero base raised to a negative exponent (power) is equal to the reciprocal of the base raised to the positive exponent (e.g., 7-3 = ). Power to a Power Property: When raising any power to an exponent, multiply the exponents (e.g., (5 ) 4 = 5 4 = 5 8 ). Product to a Power Property: When the product of a base is powered by the same exponent, both the factors are powered by the same exponent (e.g., (5 3) 4 = ). Quotient to a Power Property: When the quotient of a base is powered by the same exponent, both 4 4 the numerator and denominator are powered by the same exponent (e.g., ( 3 ) = 3 ) How are exponents and square roots related? 9

12 Student Page of 3 Common Core Standards Plus Mathematics Grade 8 Performance Lesson # Domain: Expressions and Equations. How are square roots and cube roots related? 3. Write seven squared, the square root of eight, and the cube root of 1 in the area below. Label the following: base, exponent, radicand, index. 4. Evaluate this expression and name the property used: 1 (7 3 ) 5. Evaluate this expression and name the property used: Evaluate this expression and name the property used: Evaluate this expression and name the property used:

13 Common Core Standards Plus Mathematics Grade 8 Performance Lesson # Domain: Expressions and Equations Student Page 3 of 3 8. Evaluate this expression and name the property used: Explain the negative exponent property using sentences and a numerical representation. 31

14 Student Page Domain: Expressions and Equations Focus: Scientific Notation Lesson: #9 Standard: 8.EE.3: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. Example: Scientific Notation Meaning ,000 Example ,000 1,000 10, (not usually used) (not usually used) (not usually used) , , ,000 5,00 Translate into standard notation. Translate into scientific notation Directions: Complete the following problems. 1. Explain why we use scientific notation. 5. Explain the meaning of scientific notation using the number Explain why is not correctly written in scientific notation. Write it correctly in scientific notation. Directions: Translate to either standard or scientific notation. 4. 6,31,000, ,

15 Domain: Expressions and Equations Focus: Scientific Notation Lesson: #10 Standard: 8.EE.3: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. Example: Table of Large Numbers Number in Words Number in Scientific Notation One Thousand One Million One Billion One Trillion One Quadrillion One Quintillion One Sextillion One Septillion One Octillion One Nonillion The corresponding numbers less than one are written with negative exponents. Write the following numbers in scientific notation: 1. The mass of the lightest atom is 1.67 octillionths of a kilogram.. There are 400 billion stars in the Milky Way. 3. A dollar bill is 43 ten thousands of an inch thick. Directions: Complete the following problems. 1. According to the website The United States Outstanding Public Debt in June 01 was about 15.8 trillion dollars. Write this number in standard and scientific notation.. The distance from Earth to the Moon is 400 million meters. Write this number in scientific notation. 3. The estimated population of the United States on 4 Jun 01 was 313,001,39. Round this number to 3 significant figures. Write the number in words. Write the number in scientific notation. 4. The Great Lakes and their connecting channels contain 6 quadrillion gallons. Write this number in standard and scientific notation. Student Page 5. The mass of an electron is nonillionths of a kilogram. Write this number in scientific notation. 33

16 Student Page Domain: Expressions and Equations Focus: Scientific Notation Lesson: #11 Standard: 8.EE.3: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. 8.EE.4: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Example: Complete the following problem. Show your work. The Great Lakes and their connecting channels hold 6 quadrillion gallons of water. Seneca Lake, the largest of the Finger Lakes in New York holds 4. trillion gallons of water. How many times more gallons of water do the Great Lakes hold than Seneca Lake? Directions: Complete the following problems. Show your work. 1. Scientists believe that almost all of the stars in the Universe are collected together in galaxies. Small dwarf galaxies hold about 10 million stars. Large elliptical galaxies hold up to 10 trillion stars. How many times more stars are in the large galaxies compared to the small galaxies? Write your answer in scientific notation and in words.. The population of the planet is about 7.05 billion people. The population of the United States is about 313 million people. How many times more people make up the population of the planet than of the United States? 3. Scientists believe that there are 1.5 million ants on the planet for every human being on the planet. If the population of the planet is 7.05 billion people, how many ants are on the planet? Write your answer in scientific notation and in a number/word combination. 4. An ant weighs 3 thousandth of a gram. An ant can lift 50 times its body weight over its head. How many ants does it take to lift an elephant that weighs 6,500 kilograms if all the ants work together as a team? Write your answer in scientific notation, standard notation, and in a number/word combination. 5. Modern humans have been on Earth for 00 thousand years. Earth is 4.6 billion years old. How many times older is Earth than the existence of modern humans? Write your answer is scientific notation, standard notation, and in a number/word combination. 34

17 Student Page Student Page Domain: Expressions and Equations Focus: Scientific Notation Lesson: #1 Standard: 8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. 8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Example: Dimensions of a dollar bill: Length = 156 mm; Width = 66 mm; Thickness = mm If 1 trillion one-dollar bills are placed end-to-end, as shown below, how many meters long will the string of dollar bills reach? Directions: Complete the following problems. Round to 3 significant figures. Show all work. 1. The Moon is 380 million meters from Earth on average. How many times can the string of 1 trillion one-dollar bills go back and forth between Earth and the Moon?. The circumference of Earth is 40,075 km. How many times can the string of 1 trillion one-dollar bills placed end-to-end go around Earth? 3. If 1 trillion one-dollar bills were placed flat in a rectangle with no gaps and no overlap, how many square meters would the dollar bills cover? 4. The area of Washington D.C. is approximately 177 million square meters. How many Washington D.C.s could be covered with 1 trillion one-dollar bills placed flat with no gaps and no overlap? 5. If you could perfectly stack 1 trillion one-dollar bills, what volume, in cubic meters, would the dollar bills fill? 6. If a classroom is 9 m 9 m 3 m, how many classrooms could be filled by 1 trillion one-dollar bills perfectly stacked? 35 35

18 Student Page Domain: Expressions and Equations Focus: Scientific Notation Evaluation: #3 Directions: Complete the following problems independently. Translate into scientific notation: 1. 3,190, million Translate into standard notation: millionths The National Debt in June 01 was about 15.8 trillion dollars. The estimated population of the United States at the same time was about 313 million. What was the debt for each citizen if the debt is divided equally? 8. In 011, the United States Post Office processed billion pieces of mail. If the post office processes mail 6 days a week, on average, how many pieces of mail were processed each day in 011? 36

19 Domain: Expressions and Equations Focus: Operations Using Scientific Notation Lesson: #13 Standard: 8.EE.4: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Example: 1. Find the difference. Write the answer in scientific notation given problem converted to the same power of 10 3 ( ) 10 grouped the digit terms together by factoring subtracted the digit terms rewrote into scientific notation Find the sum. Write the answer in scientific notation Student Page Directions: Find the sum or difference. Write answers in scientific notation ,510,

20 Student Page Domain: Expressions and Equations Focus: Operations Using Scientific Notation Lesson: #14 Standard: 8.EE.4: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology). Example: The Great Lakes and the connecting channels contain 6 quadrillion gallons of water. The number of gallons in each of the Finger Lakes in New York is listed in the table below. The lakes are listed in the order they lie from West to East geographically. Finger Lakes Name Gallons of Water Conesus Hemlock Canadice Honeoye Canandaigua Keuka Seneca Cayuga Owasco Skaneateles Otisco Onondaga Oneida Use the information in the Finger Lakes table above to answer the question. How many more gallons of water are in the Great Lakes than in Lake Owasco? Directions: Use the information in the Finger Lakes table above to answer the questions. Write answers in scientific notation as appropriate. 1. Write out in a word/number combination the number of gallons in Hemlock.. What is the difference of the number of gallons in Canandaigua and Keuka? 3. Which lake is the smallest (fewest number of gallons)? Explain how you know. 4. What is the total number of gallons in the 4 largest lakes? 5. What is the difference in the number of gallons of water between the sum of Honeoye and Oneida and the gallons in Cayuga? 6. How many times greater is the number of gallons in the largest lake compared to the smallest lake? 7. Explain the process you would use to determine if the total gallons of water in all the Finger Lakes is less than or more than the Great Lakes. 38

21 Domain: Expressions and Equations Focus: Operations Using Scientific Notation Lesson: #15 Standard: 8.EE.4: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Example: Below is a table of the area, in square miles of nine selected states and the District of Columbia. Area of Selected States and the District of Columbia Name Square Miles Alaska Arizona California Connecticut District of Columbia Florida Hawaii Illinois Rhode Island Texas What is the total area of the District of Columbia and the smallest state? Write your answer in both scientific and standard notation. Student Page Directions: Use the data in the table of the Area of Selected States and the District of Columbia to answer the following questions. Show your work. 1. What is the difference in the area of Rhode Island and the District of Columbia? Write your answer in standard notation.. How much larger, in square miles, is California than Arizona? Write your answer in standard notation. 3. What is the total area of Florida and Illinois? Write your answer in scientific notation. 4. How many times larger is Alaska than Texas? 5. Is the area equal to the size of two Connecticut s greater or lesser than the area of one Hawaii? What is the difference? Write your answer in standard notation. 39

22 Student Page Domain: Expressions and Equations Focus: Using Technology w/ Scientific Notation Lesson: #16 Standard: 8.EE.4: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Example: How to use your scientific calculator to work with numbers in scientific notation: 1. Punch the number (the digit number) into your calculator.. Push the EE or EXP button. Do NOT use the x (times) button!! 3. Enter the exponent number. Use the +/- button to change its sign. 4. Treat this number normally in all subsequent calculations. 5 3 To check yourself, multiply 8 10 times 4 10 on your calculator. Your answer should 9 be If the result is a very large or very small number, your calculator may represent the number 9 as 3.E9 which means Directions: Complete the following problems using the exponent button on your calculator The state of California has an area of m. The United States has an 1 area of m. What percent of area, to the nearest tenth, is California of the United States?. The United States national debt in June 01 was about 15.8 trillion dollars. The population of the United States at the same time was estimated at 313 million. What was each citizen s share of the debt? 3. How many more gallons of water does Lake Skaneateles contain than Lake Otisco? Finger Lake Name Number of Gallons Skaneateles Otisco Light travels 3 10 m/s. If you travel at 30 m/s (about 67 miles per hour), how long will it take you, in months, to cover the distance light travels in one second? 5. If people blink on average once every 3 seconds while awake, how many times has an 85 year old blinked in her lifetime? Assume she sleeps for 8 hours a day. Write your answer in scientific notation to 1significant figure and write your answer in number/word form. 40

23 Student Page Domain: Expressions and Equations Focus: Radicals, Integer Exponents, Scientific Notation Evaluation: #4 Directions: Complete the following problems independently. Simplify. Keep answers in exact form Solve for x. Show all work vertically. Line up equal signs x 3 Simplify the following expressions using the Properties of Exponents Ants are 130 million years old. Humans are 00 thousand years old. How many times older are ants than humans? Convert the numbers to scientific notation, show work, and write answer in standard form. 10. Which state from the table below has more water surface area and by how many square meters? Write answer in scientific notation. Water Area in State Square Meters 9 Indiana Ohio

24 Common Core Standards Plus Mathematics Grade 8 Performance Lesson #3 Domain: Expressions and Equations Student Page 1 of 3 Vocabulary: Properties of exponents: The rules we follow when performing operations with exponents. Exponential form: A number written with an exponent (e.g., ). The large number (4) is the base. The small number ( 5 ) is the exponent. The exponent indicates the number of times a base multiplies itself 4 5 = Scientific notation: Powers of ten are used to show very large or very small numbers. Scientific notation is the product of two numbers: the digit term and the exponential term equals 7,500,000,000,000 because we moved the decimal point to the right 1 times equals because we moved the decimal point to the left 1 times. 1. When do we use scientific notation? Why not just use standard notation?. Explain the difference between these two numbers: and Why is it helpful to use a calculator when computing with numbers in scientific notation? 43

25 Student Page of 3 Common Core Standards Plus Mathematics Grade 8 Performance Lesson #3 Domain: Expressions and Equations 4. Complete the table using numbers in scientific notation. Number in Words Number in Scientific Notation One Thousand One Million One Billion One Trillion One Quadrillion One Quintillion One Sextillion One Septillion One Octillion One Nonillion Directions: Find the sum or difference. Write the answers in scientific notation. 5. 8,900, ,000, ,000, Directions: Write the following distances using scientific notation. 9. The Earth is 38,900 miles from the moon. 10. The Earth is 9,960,000 miles from the sun. 11. The Earth is 34,000,000 miles from Mars. 1. The Earth is 483,000,000 miles from Jupiter. 44

26 Common Core Standards Plus Mathematics Grade 8 Performance Lesson #3 Domain: Expressions and Equations Student Page 3 of 3 Directions: Use the information in problems 9-1 to write three word problems. Trade your problems with a partner and solve each other s problems. Then check each other s answers

27 Student Page 1 of Domain: Expressions and Equations Focus: Graph Proportional Relationships & Determine Unit Rate Lesson: #17 Standard: 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: A helium balloon starts from the ground and ascends at a constant rate of 100 feet every 40 seconds. 1. Graph this relationship on the grid below. Title your graph. Determine the dependent and independent variables. Draw and label the axes and define the scale. Place at least 3 points on your graph that represent the relationship. Connect your points with a line.. Does your line contain the point (0, 0)? 3. What is the unit rate of the helium balloon s ascent? Where can you find this information on the graph? 4. What is the slope of your line? How is the slope related to the unit rate? A second helium balloon ascends at a constant rate of 300 feet every 80 seconds. Is the second balloon ascending at a slower or faster rate than the first balloon? Explain how you know. 46

28 Domain: Expressions and Equations Focus: Graph Proportional Relationships & Determine Unit Rate Lesson: #17 Standard: 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Directions: Complete the following problems. There are 56 cups in 3.5 gallons of liquid. 1. Graph this relationship on the grid below. Title your graph. Determine the dependent and independent variables. Draw and label the axes and define the scale. Place at least 3 points on your graph that represent the relationship. Connect your points with a line. Student Page of. Does your line contain the origin? 3. How many cups are in one gallon? Where can you find this information on your graph? 4. What is the slope of your line? How is the slope of the line related to the number of cups in one gallon? 5. Explain 3 methods to determine the number of gallons in one cup. 47

29 Student Page 1 of Domain: Expressions and Equations Focus: Graph Proportional Relationships & Determine Unit Rate Lesson: #18 Standard: 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: Two runners each set their speed on a treadmill in miles per hour and the distance they want to run in miles. The time it took each runner, rounded to the nearest half minute, for each of their predetermined distance is given in the table below. Runner Distance (miles) Time (minutes) Graph both relationships on the same grid below. Place at least points on your graph that represent each relationship. Connect your points with a line. Label each line as Runner 1 or Runner.. Does each line contain the origin? 3. What does the slope of the line mean in this scenario? What is another expression for slope? 4. Write an equation to represent the relationship between the time and distance for each runner. Runner 1: Runner : 5. Which runner ran at a faster pace? Justify your answer. 48

30 Student Page of Domain: Expressions and Equations Focus: Graph Proportional Relationships & Determine Unit Rate Lesson: #18 Standard: 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Directions: Complete the following problems using the same information from the table in the example. 1. The axes labels are switched on the grid below. Re-plot the points from the table. Connect your points with a line. Label each line as Runner 1 or Runner.. How can you use the graph to determine the rate of each runner in miles per hour? Write the rate of each runner in miles per hour. Runner 1: Runner : 3. Which runner ran at a faster pace? Justify your answer. Does your answer confirm your answer in problem 5 of the example? Modify your answer if needed. 4. How does switching the meaning of the axes change how you interpret the relative rates (comparing rates of two or more proportional relationships on the same graph)? Does a steeper line always represent a faster runner? Explain. 5. How does defining the independent and dependent variables affect how you interpret relative rates? 49

31 Student Page 1 of Domain: Expressions and Equations Focus: Comparing Proportional Relationships Lesson: #19 Standard: 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: Use the chart below to complete the problems. Produce Prices Produce Type Price Unit Price Bananas 5 lbs for $.40 Peaches $1.48/lb Strawberries lbs for $3.56 Yellow Squash $0.98/lb Zucchini 3 lbs for $ Each line on the graph below represents one of the produce types listed in the chart. Label each line. Also label the axes and title the graph. You do not need to add a scale.. Avocados cost more than peaches but less than strawberries. Draw in a line on the graph that could represent the price of x pounds of avocados. Label the line. 3. By looking at the graph, how can you determine which produce type is the least expensive? 4. What is the unit rate of the least expensive produce type? 5. Write an equation for bananas that represents the relationship between the price, y, and the weight, x. 50

32 Student Page of Domain: Expressions and Equations Focus: Comparing Proportional Relationships Lesson: #19 Standard: 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Directions: The graph below shows the distance three planes have traveled over a period of time. Plane B travels at 45 meters per second (m/s). 1. Label the axes and title the graph. You do not need to include a scale.. Does Plane C travel slower or faster than Plane B? Explain. 3. Write an equation for Plane B that represents the relationship between the distance traveled, y, and the time, x. 4. Write an equation that could represent Plane A. 5. Plane D s data is shown in the table below. Time (seconds) Plane D Distance (meters) 5 1, , ,435 If you graph Plane D s data on the same graph, where would you place the line relative to plane B? Explain. 51

33 Student Page 1 of Domain: Expressions and Equations Focus: Comparing Proportional Relationships Lesson: #0 Standard: 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Example: The coverage area for different brands of paint are shown in the displays below. Brand A Paint Brand B Paint Number of Gallons Coverage Area (square feet) Brand C Paint a = 75g where, a is the coverage area in square feet g is the number of gallons of paint 1. Which brand covers more with 10 gallons?. What is the unit rate for each brand of paint? Brand A: Brand B: Brand C: 3. If you graph the lines and number of gallons for Brands A and C paints on the same graph as Brand B, how can you determine which brand has the greatest coverage? 4. Order the paint brands from greatest coverage to least coverage. Justify your answer. 5

34 Domain: Expressions and Equations Focus: Comparing Proportional Relationships Lesson: #0 Standard: 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Directions: Four friends each have summer jobs that pay an hourly rate. Use the information below to complete the problems. Melissa s Summer Pay Student Page of Tanya s Pay Stub Information Hours worked this pay period = 36 Earnings = $3.0 Steve s Summer Pay p=8.85h where, p is Steve s pay h is the number of hours worked David s Summer Pay Day Time Worked Money Earned (hours) (dollars) Who makes more money for working for 10 hours? Justify your answer.. What is the unit rate for each friend? 3. Write an equation for each friend that represents the relationship between money earned, p, and the time worked, h. Melissa: Tanya: Steve: p=8.85h David: 4. If you graph Tanya s, Steve s, and David s data on the same graph with Melissa s, how can you determine who makes more per hour? 5. Order the friends from least earned per hour to greatest earned per hour. 53

35 Student Page 1 of Domain: Expressions and Equations Focus: Graphing and Comparing Proportional Relationships Evaluation: #5 Directions: Complete the following problems independently. The prices of two types of juices in the same size bottles are shown in the table below. Juice Type A B Price $5.96 for bottles $13.40 for 5 bottles 1. Graph both relationships on the same grid below. Determine the dependent and independent variables. Draw and label the axes and define the scale. Title your graph. Place at least points on your graph that represent each relationship. Connect your points with a line. Label each line as Juice A or Juice B.. Which juice is more expensive? How do you know by looking at the graph? 3. What does the slope represent in this scenario? 4. Write an equation to represent the proportional relationship between the price and the number of bottles for each juice. Juice A: Juice B: 5. What is the unit rate? Juice A: Juice B: 54

36 Student Page of Domain: Expressions and Equations Focus: Graphing and Comparing Proportional Relationships Evaluation: #5 Krista is driving one car and Jorge is driving another car. They are traveling at a constant rate of speed. The data for each person are shown below. Krista s Trip Jorge s Trip Time (minutes) Distance (miles) Who travels further in hours? Justify your answer. 7. What is the average speed of Krista s and Jorge s car in miles per hour? Show your work. Krista: Jorge: 8. If you graphed Krista s data on the same graph with the line representing the relationship between the distance traveled and time for Jorge s trip, how can you determine which person travels at a greater speed? 55

37 Student Page Domain: Expressions and Equations Focus: Similar Triangles and Slope Lesson: #1 Standard: 8.EE.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Example: Therefore, 13 is similar to 456 and corresponding sides are proportional. slope vertical change 3 4 horiztonal change 6 8. In the coordinate plane below, ABC is similar to ADE. What is the value of x? Directions: In the coordinate plane below, y? Show your work. ABC is similar to ADE. What is the value of 57

38 Student Page Domain: Expressions and Equations Focus: Similar Triangles and Slope Lesson: # Standard: 8.EE.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Example: On the coordinate plane below, ABC and ARS are similar. 1. What is the coordinate (x, y) that defines point C?. What is the slope of AC? 3. How can you use the slope to determine the coordinate of another point on the line? Directions: On the coordinate plane below, ALM, ABC, and ADE are similar. Show your work. 1. What is the coordinate (x, y) that defines point M?. What is the slope of AE? 3. What is the coordinate (x 1,y 1 ) that defines point E? 58

39 Domain: Expressions and Equations Focus: Derive the Equation y = mx Lesson: #3 Standard: 8.EE.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Example: Derive the equation y = mx for a line through the origin using similar triangles. For a line through the origin, the right triangle whose hypotenuse is the line segment from (0, 0) to a point (x,y) on the line is similar to the right triangle from (0,0) to the point (1,m) on the line. m = slope or unit rate m is the y-coordinate when x = 1 on a line through the origin Student Page Set up a proportion to describe this situation and solve for y. Directions: Write an equation in the form y = mx to describe the following lines

40 Student Page Domain: Expressions and Equations Focus: Derive the Equation y = mx + b Lesson: #4 Standard: 8.EE.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Example: The right triangle whose hypotenuse is the line segment from (0,b) to (x,y) is shown below. Derive the equation y = mx + b for a line intercepting the vertical axis at b. Write the equation to each line below in the form y = mx + b. You may use the points to help you determine the slope. Directions: Write the equation to each line below in the form y = mx + b. Use the points to help you determine the slope

41 Student Page 1 of Domain: Expressions and Equations Focus: Proportional Relationships, Lines, and Linear Equations Evaluation: #6 Directions: Complete the following problems independently. 1. In the coordinate plane below, right ABC and right ADE are similar. What is the coordinate (x,y) that defines vertex E? Show your work. Write the equation to the lines graphed below

42 Student Page of Domain: Expressions and Equations Focus: Proportional Relationships, Lines, and Linear Equations Evaluation: #6 4. Derive the equation y = mx for a line through the origin. Use the points (0,0), (1,m), and (x,y) as points that lie on the line. June and Ricardo each ride bicycles. The relationship between their distance and time is shown in the displays below. Each person travels at a constant rate. June s Bike Ride Time (hours) Distance (miles) Ricardo s Bike Ride 5. Who would cover more distance in.5 hours? 6. Who has a faster average speed? 7. Write an equation in the form y = mx that represents the distance, y, each covers after x hours. June: Ricardo: 8. If you graph June s data on the same graph as Ricardo s, how can you determine who has a faster average speed? 63

43 Common Core Standards Plus Mathematics Grade 8 Performance Lesson #4 Domain: Expressions and Equations Student Page 1 of Vocabulary: Proportional Relationship: A relationship in which the ratio of y/x is constant. Equivalent: Having the same value; the same size. Constant of Proportionality: The mathematical term for the unit rate. Origin: The point at which the x- and y- axes intersect (0, 0). Slope: The measure of how steep a line is, or vertical change divided by horizontal change; slope may be positive, moving up from left to right; negative, moving downward from left to right; or zero, moving neither up nor down from left to right. 1. Use the grid below to draw the x- and y- axes, label the tic lines, and graph and label a line with a positive slope and a line with a negative slope: 65

44 Student Page of Common Core Standards Plus Mathematics Grade 8 Performance Lesson #4 Domain: Expressions and Equations. Analyze the graph below. Consider the labels and scale. Write a description of each of the triangles. Then analyze how the triangles are related. What is the slope of each? Are they similar? Explain your thinking. Show the proportional relationships between triangles A and B, triangles B and C, and triangles A and C

45 Domain: Expressions and Equations Focus: Types of Solutions to a Linear Equation Lesson: #5 Standard: 8.EE.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Example: Without solving the following equations, determine if the solution is positive, negative, zero, no solution, or infinite solutions. Justify your answer in words. 1. 6x Student Page. x x x 14 7( x ) Directions: Without solving the following equations, determine if the solution is positive, negative, zero, no solution, or infinite solutions. Justify your answer in words. 1. 3x 3 3x x x x x 6x 5. 5x 1 Write an equation of your own with infinite solutions. 67

46 Student Page Domain: Expressions and Equations Focus: Linear Equations Lesson: #6 Standard: 8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Example: 1. Without solving the following equation, determine if the solution is positive, negative, zero, no solution, or infinite solutions. Justify your answer in words. x 3. Write an equation that has no solution. How can you determine if the equation has no solution? Directions: Without solving the equations in questions 1-5, determine if the solution is positive, negative, zero, no solution, or infinite solutions. Justify your answer in words. 1. x 4 x. 1 3x 3 3. x ( x ) 3(x 4) 5. 4x 3x 6. Write an equation that has a negative solution. 68

47 Domain: Expressions and Equations Focus: Solving 1-step and -step Equations Lesson: #7 Standards: 8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Example: Before solving, determine if the solution is positive, negative or zero. Then solve each equation using the properties of operations and the properties of equality. Show each step. Be sure to line up your equal signs vertically. Student Page A z B. y Directions: Before solving, determine if the solution is positive, negative or zero. Then solve each equation using the properties of operations and the properties of equality. Show each step. Be sure to line up your equal signs vertically. 1. x x 0. y x z x x 8. 1 x

48 Student Page Domain: Expressions and Equations Focus: Solving 1-step and -step Equations Lesson: #8 Standards: 8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Example: Before solving, determine if the solution is positive, negative or zero. Then solve each equation using the properties of operations and the properties of equality. Show each step. Be sure to line up your equal signs vertically. A. y 5 3 B. 3x Directions: Before solving, determine if the solution is positive, negative or zero. Then solve each equation using the properties of operations and the properties of equality. Show each step. Be sure to line up your equal signs vertically x 5. 5x 1. 1 x x 3. x x z x 4 70

49 Student Page Domain: Expressions and Equations Focus: Finding Solutions to 1- and -step Linear Equations Evaluation: #7 Directions: Complete the following problems independently. Without solving the following equations, determine if the solution is positive, negative, zero, no solution, or infinite solutions. Justify your answer in words x x 3. 5 x x.5 7x x 0x Directions: Before solving, determine if the solution is positive, negative or zero. Then solve each equation using the properties of operations and the properties of equality. Show each step. Be sure to line up your equal signs vertically. z x x z 71

50 Student Page Domain: Expressions and Equations Focus: Distributive Property Lesson: #9 Standard: 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Distributive Property: a( b c) ab ac Things to Remember when Applying the Distributive Property: 1. Distribute to all terms inside the parentheses, not just the first term.. The sign in front of the factor being distributed goes with the number when distributing. 3. If only a negative sign is in front of the parentheses, it is understood to represent a negative one. 4. Do not distribute the factor to any number outside the parentheses. 5. Always follow the order of operations. Example: Apply the Distributive Property to the following expressions. 1 A. (10 x 1) B ( 5 6 x) C. (3x ) 5 5 Directions: Apply the Distributive Property to the following expressions (6 10 x ) 5. 1 ( x 8) 1 8. (8x 3) 6. 4(3 x 11) x x ( x 9 x) 73

51 Student Page Domain: Expressions and Equations Focus: Simplifying Expressions Lesson: #30 Standard: 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Like terms Terms that contain the same variable raised to the same power. Only the coefficients of like terms may be different. For example, x and 3 x are like terms. 3xy and 7xy are like terms. All constants are like terms. Distributive Property a( b c) ab ac We will write the terms of an expression in decreasing order. For example, we write 5x + 8 not 8 + 5x. Example: Simplify the expressions by combining like terms. A. 8 x x 8x B. 5 3( x 4) Directions: Simplify the expressions by combining like terms z ( z 8) 4. 4y 6 6 y ( y 1). 10 6( x ) 5. 4( z 1) 4 z 3. 1 (15 x 18) 1 6. (5 11 x) 6 ( x 1) 3 74

52 Student Page Domain: Expressions and Equations Focus: Multi-step Linear Equations Lesson: #31 Standard: 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Example: Solve the following equations. Show your work. Be sure to work vertically and line up equal signs. A. ( x 15) 1 0 B. 6 0.(5 4 ) x Directions: Solve the following equations. Show your work. Be sure to work vertically and line up equal signs ( 9) 10 6 x. (5 4 x ) 3 x ( x 1) (3x 1.) 75

53 Student Page Domain: Expressions and Equations Focus: Multi-step Linear Equations Lesson: #3 Standard: 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Example: Solve the following equations. Show your work. Be sure to work vertically and line up equal signs. A. 4 x B. ( 9 15 x ) x 9 3 Directions: Solve the following equations. Show your work. Be sure to work vertically and line up equal signs ( 8) 4 10 x x 3. 6 ( x 1) (6 x 1) 18x x x 76

54 Student Page Domain: Expressions and Equations Directions: Complete the following problems independently. Define the following words: 1. variable Focus: Solving Multi-step Linear Equations Evaluation: #8. constant 3. coefficient 4. Give an example of like terms. Directions: Solve the following equations. Show your work. Be sure to work vertically and line up equal signs x x x

55 Student Page Domain: Expressions and Equations Focus: Multi-step Linear Equations Lesson: #33 Standard: 8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Example: Solve the following equations. Show your work vertically and line up equal signs. 1 3 A. 5x 4 4x 8 B. (6 5 x ) 3 x C x x Directions: Solve the following equations. Show your work vertically and line up equal signs. 1. 8x 6x 4. 3x 1 7x (x 4) 7 x 4. (7x 3) 4 x 8 79

56 Student Page Domain: Expressions and Equations Focus: Multi-step Linear Equations Lesson: #34 Standards: 8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Example: Solve the following equations. Show your work vertically and line up equal signs. A. 7( x ) 7x 14 B. x 4 9(x 3) 15 Directions: Solve the following equations. Show your work vertically and line up equal signs x 3x x 1 3(1 x) 4. 9 (8x 9) 11x 4. 5(x 7) x 80

57 Domain: Expressions and Equations Focus: Multi-step Linear Equations Lesson: #35 Standard: 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Example: Solve the following equation. Show your work vertically and line up equal signs. 4 (6 x 15) ( x 4) 3 Student Page Directions: Solve the following equations. Show your work vertically and line up equal signs (8x 10) 3 x (x 5) 4 x (3x ) (15x ) 5 81

58 -3()()x-(+1)--(-510x-(x-)()Student Page Domain: Expressions and Equations Focus: Multi-step Linear Equations Lesson: #36 Standard: 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Example: Solve the following equation. Show your work vertically and line up equal signs. 4( x ) 1 3(x 5) 8 Directions: Solve the following equations. Show your work vertically and line up equal signs x+5=-5x-1x3. 38x3=5xx+6x). 1+x37=54x+93Common Core Standards Plus is not licensed for duplication. Copying is illegal. 8

59 Student Page Domain: Expressions and Equations Focus: Solve Multi-step Linear Equations Evaluation: #9 Directions: Complete the following problems independently. Solve the following equations. Show your work vertically and line up equal signs. Keep answers in fraction form. 1. 4x 1x (4x 10) ( x 7) x (3x 4) 83

60 Domain: Expressions and Equations Focus: Systems of Equations Lesson: #37 Standard: 8.EE.8a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Example: Four classmates have a tricycle race. Use the data in the chart to identify which graph represents each cyclist. Fill in the empty cells of the chart. Then answer the questions. Race Details Cyclist 1 Cyclist Cyclist 3 Cyclist 4 Speed meters in 3 1 meter in 3 meters in 3 5 meters in 3 seconds seconds seconds seconds Start Delay (in seconds) Line Tricycle Race Student Page 1. How did you determined which line represents which cyclist?. Line A intersects the other three lines. What is the meaning of these points of intersection in terms of the race? 3. If the race is 4 meters long, who wins? Who comes in last? Explain your reasoning. Directions: Use the Tricycle Race graph above to answer the following questions. 1. How does the rate affect the graph?. How does the delay start affect the graph? 3. If the race is 8 meters long, who wins? Explain your reasoning. 4. When will the cyclist represented by line C catch up with the cyclist represented by line B? Explain. 85

61 Student Page Domain: Expressions and Equations Focus: Systems of Equations Lesson: #38 Standard: 8.EE.8a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Example: Sparkle Wash and Car Shine both offer an annual car wash membership. With this membership, customers pay a one-time fee that entitles them to a reduced price on car washes for one year with no limit to the number of car washes. Let x represent the number of car washes in a year and y represent the total charged in dollars. Sparkle Wash: $0 annual membership Car Shine: $60 annual membership $10 per wash $5 per wash Sparkle Wash Equation: Car Shine Equation: Number of Car Washes, x Sparkle Wash Total Charged, y, (in dollars) Number of Car Washes, x Car Shine Total Charged, y, (in dollars) By looking at the tables, what is the point of intersection of the graphs of the equations? Explain. Directions: Complete the following problems. 1. Using the tables or the equations, graph the lines (with dashes) that represent the relationship between the total charged and the number of car washes for Sparkle Wash and Car Shine on the same coordinate plane below. Define axes and scale. Label each line.. What is the solution to the system of equations? Circle the solution on your graph. 3. What is the meaning of the solution to the system of equations in this scenario? 4. Which company is a better choice for someone who washes their car once a month? Explain. 86

62 Student Page Domain: Expressions and Equations Focus: System of Equations Lesson: #39 Standard: 8.EE.8a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Example: Burt and Jillian both have motorcycles. Burt s motorcycle has a gas tank that holds 5 gallons of gasoline and averages 35 miles per gallon. Let x represent the number of miles driven and y represent the gallons of gas left in the tank. 1 Burt s equation: y 5 x 35 1 Why is the coefficient of x and not 35? 35 Complete Burt s column on the table below, and graph Burt s data on the coordinate plane provided to show the amount of gasoline in the motorcycle tank, after filling up and then taking a road trip. Motorcycle Trip Distance driven (in miles) Burt s amount of gas left in tank (in gallons) Motorcycle Trip Jillian s amount of gas left in tank (in gallons) Student Page Directions: Complete the following problems. Jillian s motorcycle has a gas tank that holds 3 gallons of gas and averages 70 miles per gallon. She also fills her take before taking a road trip. 1. What is the equation that represents the amount of gallons left in Jillian s tank after x miles of the road trip?. Complete Jillian s column in the table above. 3. Graph the line that represents this relationship on the same graph as Burt s line. 4. What is the solution to the system of equation? Circle the solution on your graph. 5. What is the meaning of the point of intersection in the context of this scenario? 6. Which motorcycle would you choose to use for a 00 mile road trip? Why? 87 87

63 Student Page 1 of 3 Domain: Expressions and Equations Focus: Systems of Equations Lesson: #40 Standard: 8.EE.8a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Example: Two walkers walk up or down a staircase. They may start at different locations and have different rates. Use the given scenario to complete the table, write the system of equations, graph the system of equations, and answer the questions. Walker A begins on step 10 and walks down the stairs at a rate of 1 step per seconds. Walker B begins on step 0 and walks up the stairs at a rate of steps per second. Let x represent the number of seconds and y the step number. Location of Staircase Walkers Seconds Step # of Walker A Step # of Walker B System of Equations Walker A: Walker B: Graph the system of equations below. Label the lines. Staircase Walkers 1. What is the solution to the system of equations? Circle the solution on your graph.. What is the meaning of the point of intersection of the lines? 3. What is the description of the lines? Circle one. Intersecting lines Coinciding lines Parallel lines 88

64 Domain: Expressions and Equations Focus: Systems of Equations Lesson: #40 Standard: 8.EE.8a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Directions: Two walkers walk up or down a staircase. They may start at different locations and have different rates. Use the given scenarios to complete the table, write the system of equations, graph the system of equations, and answer the questions. 1. Walker A begins on step 4 and walks up the stairs at a rate of 1 step per second. Walker B also begins on step 4 and walks up the stairs at a rate of steps per seconds. Let x represent the number of seconds and y the step number. Location of Staircase Walkers Seconds Step # of Walker A Step # of Walker B System of Equations Walker A: Walker B: Graph the system of equations below. Label the lines. Staircase Walkers Student Page of 3 a. When will the walkers be on the same step? b. What is the solution to the system of equations? c. What is the description of the lines? Circle one. Intersecting lines Coinciding lines Parallel lines 89

65 Student Page 3 of 3 Domain: Expressions and Equations Focus: Systems of Equations Lesson: #40 Standard: 8.EE.8a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.. Walker A begins on step 0 and walks up the stairs at a rate of 3 steps per second. Walker B begins on step 3 and walks up the stairs at a rate of 3 steps per second. Location of Staircase Walkers Seconds Step # of Walker A Step # of Walker B System of Equations Walker A: Walker B: Graph the system of equations below. Label the lines. Staircase Walkers a. When will the walkers be on the same step? b. What is the solution to the system of equations? c. What is the description of the lines? Circle one. Intersecting lines Coinciding lines Parallel lines 90

66 Student Page 1 of Domain: Expressions and Equations Evaluation: #10 Directions: Complete the following problems independently. Focus: System of Equations Ricky and Matt are brothers. Ricky is 10 years old and Matt is 5 years old. Matt challenged Ricky to several races. Race #1 Ricky moved at a rate of one meter per second and Matt moved at a rate of 0.5 meter per second. Because Matt is so little, Ricky gave him a head start of 1.5 meters. 1. Complete the table. Seconds Distance Traveled by Ricky (in meters) Distance Traveled by Matt (in meters). Write the system of equations. Ricky: Matt: 3. Graph the equations. Label the lines. Race #1 4. What is the solution to the system of equations? Circle the solution on the graph. 5. What is the meaning of the point of intersection in the context of this scenario? 6. Who wins the race if the race is 5 meters long? 91

67 Student Page of Domain: Expressions and Equations Focus: Systems of Equations Evaluation: #10 Race # Ricky moved at a rate of 1.5 meters per second and Matt moved at a rate of 1.5 meters per second. This time Ricky gave Matt a head start of.5 meters. 1. Complete the table. Seconds Distance Traveled by Ricky (in meters) Distance Traveled by Matt (in meters). Write the system of equations. Ricky: Matt: 3. Graph the system of equations. Label the lines. Race # 4. What is the solution to this system of equations? 5. What type of lines make up this system of equations? 93

68 Domain: Expressions and Equations Focus: Systems of Equations Lesson: #41 Standard: 8.EE.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: The Substitution Method Step 1: Identify and define the variables in the scenario, if given a real-world context. Step : Write the system of linear equations. Step 3: Write an equation by substituting the value for the chosen variable in one of the equations into the other equation. Step 4: Solve for the variable. Step 5: Substitute the value of one variable in either equation and find the value of the other variable. Step 6: State the solution to the system of linear equations as an ordered pair. Step 7: If the system of linear equations represents a real-world scenario, explain the meaning of the solution. System of Equations: Sparkle Wash: y 10x 0 Car Shine: y 5x 60 Where y represents the total amount charged for x car washes. Solve the system of equations. Show your work. Student Page Directions: Solve the system of equations by using the substitution method. Explain the meaning of the solution. Show your work. 1. From the motorcycle trip scenario where y represents the gallons of gas left in the tank after traveling x miles: Burt s motorcycle: y x Jillian s motorcycle: 1 y x From the staircase walkers scenario where y represent the step number after x seconds: 1 Walker A: y x 10 Walker B: y x 95

69 Student Page Domain: Expressions and Equations Focus: Systems of Equations Lesson: #4 Standard: 8.EE.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. 8.EE.8c: Solve real-world and mathematical problems leading to two linear equations in two variables. Example: Solve the following systems of equations using the substitution method. Follow the applicable steps above. Show your work. A. x y 10 3x y 18 B. A candy store manager is making sour candy mix by combining sour cherry worms, which cost $.50 per pound, and sour lime rings, which cost $3.50 per pound. How much of each candy should she include if she wants 0 pounds of a mix that costs a total of $65? Directions: Solve the following systems of equations using the substitution method. Follow the applicable steps above. Show your work. 1. 3x y x y 10. x y 6 x y 4 3. A theater sells adult tickets and children s tickets. Adult tickets are $8 each and child tickets are $4 each. If the theater sold 00 tickets for a movie and collected $1304, how many of the tickets sold were adult tickets and how many were child tickets? 96

70 Domain: Expressions and Equations Focus: Systems of Equations Lesson: #43 Standard: 8.EE.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Linear Combination Method Step 1: Identify and define the variables if given a real-world problem. Step : Write the system of equations in standard form (ax + by =c). Step 3: You must eliminate a variable. Decide which variable to eliminate. Multiply one or both equations by a constant so that when the system of equations is added together, you have an equation with one variable (since the other will be eliminated). Step 4: Add the new system of equations together to obtain an equation in one variable. Step 5: Solve for the variable. Step 6: Substitute the number for the variable into either of the original equations and solve for the other variable. Step 7: Write the answer as a coordinate pair. Step 8: Explain the meaning of the solution if given a real-world scenario. Example: A. Solve the system of equations using the linear combination method. x 3y 0 x y 7 Student Page 1 of B. Write the slope-intercept form for the given system of equations. Graph the system on the coordinate plane below to find the solution. C. Does your graphical solution confirm your algebraic solution? 97

71 Student Page of Domain: Expressions and Equations Focus: Systems of Equations Lesson: #43 Standard: 8.EE.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Directions: Complete the following problems. 1. Solve the system of equations using the linear combination method. Show your work. x 4y 8 5x y 4. Write the slope-intercept form for the given system of equations. Graph the system on the coordinate plane below to find the solution. Use the intersection feature on a graphing calculator to check your answer. 3. Does your graphical solution confirm your algebraic solution? 98

72 Student Page 1 of Domain: Expressions and Equations Focus: Systems of Equations Lesson: #44 Standard: 8.EE.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: A. Solve the system of equations using the linear combination method. Show your work. x 3y 9 5x 6y 9 B. Write the slope-intercept form for the given system of equations. Graph the system on the coordinate plane below to find the solution. Slope-intercept form: x 3 y 9 5x 6y 9 C. Does your graphical solution confirm your algebraic solution? Directions: Complete the following problems. 1. Solve the system of equations using the linear combination method. Show your work. x 3y 9 4x y 10 99

73 Student Page of Domain: Expressions and Equations Focus: Systems of Equations Lesson: #44 Standard: 8.EE.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.. Write the slope-intercept form for the given system of equations. Graph the system on the coordinate plane below to find the solution. Use the intersection feature on a graphing calculator to check your answer. Slope-intercept form: x 3y 9 4x y Does your graphical solution confirm your algebraic solution? 100

74 Student Page 1 of Domain: Expressions and Equations Focus: Solving Systems of Equations Algebraically Evaluation: #11 Directions: Complete the following problems independently. 1. Solve the system of equations using substitution. Show your work. y y x x 4. Prove your solution to problem 1 by graphing the original equations. Label each line. Circle and label the solution. 101

75 Student Page of Domain: Expressions and Equations Focus: Solving Systems of Equations Algebraically Evaluation: #11 3. The perimeter of a rectangle is 8 yards. The width is 15 yards less than the length. What are the dimensions of the rectangle? a. Identify and define your variables. b. Write a system of equations. c. Solve the system of equations using substitution. 4. Solve the system of equations using linear combination. Show your work. 4x 5y 8 x 3y Prove your solution to #4 by substituting your solution into each original equation. 103

76 Domain: Expressions and Equations Focus: Systems of Equations Lesson: #45 Standard: 8.EE.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Example: Systems can be solved using a variety of methods. We can use tables, graphs, substitution, and linear combination. Indicate which method you would use to solve the system below and why. A. System Method Rationale Student Page 1 of y y x 1 3x 4 Standard Form ax + by = c Slope-Intercept Form y = mx + b No Solution ax + by are the same, different c value y = mx + ; same m value: same slope. Infinitely Many Solutions ax + by = c; n(ax + by = c) Where the second equation is a multiple of the first. y = mx + b; the equations are the same. Type of Lines Parallel Same line; coinciding lines B. Without solving, describe the solution to the following system. Explain. 3x 5y 10 3x 5y 14 C. Create a system of linear equations that has infinitely many solutions. The lines should both have negative slopes. 105

77 Student Page of Domain: Expressions and Equations Focus: Systems of Equations Lesson: #45 Standard: 8.EE.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Directions: Complete the following problems. y 1. Systems can be solved using a variety of methods. We can use tables, graphs, substitution, and linear combination. Indicate which method you would use to solve the system below and why. System Method Rationale 3 x 5x 3y 1 x 3y 4 3x 5y 10. Without solving, describe the solution to the following system. Explain. y x 1 y x 6 3. Create a system of linear equations that has no solution. The lines should both have zero slopes. 106

78 Student Page Domain: Expressions and Equations Focus: Systems of Equations Lesson: #46 Standard: 8.EE.8c: Solve real-world and mathematical problems leading to two linear equations in two variables. Example: Solve the word problem by writing and solving a system of equations. Show your work. Substitute your solutions into both original equations to check your answers. Megan has a total of 4 nickels and dimes. The total value of the coins is $.15. How many of each coin does Megan have? Directions: Complete the following problems choosing any method to solve. Show your work. 1. A website allows users to download individual songs or an entire album. All individual songs cost the same to download and all albums cost the same to download. Robert pays $14.94 to download 5 individual songs and an album. Ramona pays $.95 to download 3 individual songs and albums. a. Identify and define your variables. Write a system of equations and solve. Prove your answers. b. Would it cost more to download 8 individual songs or an entire album?. A team mom ordered 100 pizzas for a total of $155. Cheese pizzas cost $11.50 each and pepperoni pizzas cost $13.00 each. a. Identify and define your variables. Write a system of equations. b. How many of each pizza did the team mom order for the team party? Prove your answer. 107

79 Student Page 1 of Domain: Expressions and Equations Focus: Solving systems of equations Lesson: #47 Standard: 8.EE.8c: Solve real-world and mathematical problems leading to two linear equations in two variables. Example: The price of a sweater is $5 less than twice the price of a jacket. If 4 sweaters and 3 jackets cost $00, what is the price of one jacket? Define variables: System of equations: Work space to solve system: Check solution: Directions: Complete the following problems. 1. Samantha paid $17.75 for 3 hamburgers and sodas. Roger paid $3.75 for 5 hamburgers and 1 soda. What is the cost of 1 hamburger and 1 soda if bought separately? Define variables: System of equations: Work space to solve system: Check solution: Answer: 108

80 Domain: Expressions and Equations Focus: Solving systems of equations Lesson: #47 Standard: 8.EE.8c: Solve real-world and mathematical problems leading to two linear equations in two variables.. Find the solution of the equations whose lines, L and L, go through the following pairs 1 of points. Use the coordinate plane below. Circle and label the solution as a coordinate pair. L( 10, 6)( 5, 5) and L ( 8, 7)( 4, 5) 1 Student Page of 109

81 Student Page 1 of Domain: Expressions and Equations Focus: Systems of Equations Lesson: #48 Standard: 8.EE.8c: Solve real-world and mathematical problems leading to two linear equations in two variables. Example: Below is the solution of the equations whose lines, L 1 and L, go through the following pairs of points. You found the solutions by creating this graph yesterday. L( 10, 6)( 5, 5) and L ( 8, 7)( 4, 5) 1 Write the equations of the lines, L and L, by looking at the graph. 1 Write the equation of the lines algebraically. Use the slope formula and the point-slope formula. y y 1 Slope formula: m Point-slope formula: y y mx ( x) 1 1 x x 1 Solve the system of equations algebraically. Does your solution match the graphical solution above? In general, why would we use the algebraic method over the graphical method? 110

82 Domain: Expressions and Equations Focus: Systems of Equations Lesson: #48 Standard: 8.EE.8c: Solve real-world and mathematical problems leading to two linear equations in two variables. Directions: Find the solution of the equations whose lines, L 1 and L, go through the following pairs of points. L(0, 3)(, 5) and L ( 3, 3)(6, 0) 1 Student Page of 1. Use the coordinate plane below. Circle and label the solution as a coordinate pair.. Write the equation of the lines algebraically. Use the slope formula and the point-slope formula. y y 1 Slope formula: m Point-slope formula: y y mx ( x) 1 1 x x 1 3. Solve the system of equations algebraically. Show your work. 4. Which method was most effective in finding the solution of the equations L 1 and L? Explain your answer. 111

83 4(3x-8)-10=(3x+5)-Student Page 1 of Domain: Expressions and Equations Focus: Systems of Equations Evaluation: #1 Directions: Complete the following problems independently. 1. The distance from Earth to the edge of the observable universe is meters. Earth s diameter through the equator is 1,756 kilometers. How many Earth s diameters would reach from Earth to the edge of the observable universe? Write answer in scientific notation to 4 significant figures. Apply the properties of exponents to simplify each expression What value of x makes the equation below true? Common Core Standards Plus is not licensed for duplication. Copying is illegal. 11

84 Student Page of Domain: Expressions and Equations Focus: Systems of Equations Evaluation: #1 5. An ice cream shop sells shakes for $3.5 and double scoop cones for $4.85 each. On one summer day, the shop sold 10 fewer shakes than 3 times the number of double scoop cones and made $ How many shakes and double scoop cones did the shop sell? Define variables: System of equations: Work space to solve system: Check solution: Answer: 6. Two cars are moving at different average speeds. The data for each car is displayed below. Car 1 Car Time (hours) 1 Distance (miles) a. Which car traveled farther after 1 hours? Explain. b. What is the average speed for each car? 113