Ready to Go On? Skills Intervention Integers in Real-World Situations

Transcription

1 11-1 Ready to Go On? Skills Intervention Integers in Real-World Situations Positive numbers are greater than 0. They can be shown with or without the positive sign. For example: 25 or 25 Negative numbers are less than 0. They are always written with a negative sign. For example: 25 Identifying Positive and Negative Numbers in the Real World Name a positive or negative number to represent each situation. A. earning $65 numbers are used to represent gains or increases. 65 What sign do you use to represent earnings? B. a decrease of 46 points numbers are used to represent losses or decreases. 46 What sign do you use to represent a decrease? On a number line, opposites are the same distance from 0 but on different sides of 0. Integers are the set of all whole numbers and their opposites. Graphing Integers Graph 2 and its opposite on a number line. Plot 2 on the number line. What other number is the same distance from 0 as 2? Plot the opposite of 2 on the number line. Vocabulary positive number negative number opposite integer absolute value The absolute value of an integer is its distance from 0 on a number line. The symbol for absolute value is. Writing Integer Expressions to Represent Situations At a football game, the Tigers gain 7 yards on the first play of the drive. On the next play, they lose 13 yards. On the next play, they gain 15 yards. Write an expression to represent this situation. The Tigers gained and yards. Gains are. The Tigers lost yards. Losses are. Write the expression with the integers in the same order as the events that occurred. 216 Holt Mathematics

2 11-1 Ready to Go On? Problem Solving Intervention Integers in Real-World Situations You can use integers to represent changes in conditions in a real world situation. This may help you solve problems. A college library has 4 floors of books above ground and 5 floors of archives below ground. The librarian goes up to the third floor to get some history books and takes them down to the history archives on the fourth underground level to have the books stored. How many floors does she have to go to get back to the main lobby on the first floor? Understand the Problem 1. What are you asked to find? Make a Plan 2. How might a number line help you solve this problem? 3. What integer can you use to represent to the 3 rd floor? The history archives? 4. Should you find the difference between these integers? Explain. 5. Should 0 be on your number line? Explain. Solve 6. Sketch a number line. Graph each integer for the 3 rd floor, history archives, and main lobby and label the points. 7. Count the units between the history archives and the main lobby. Answer the question. Check History Archives 8. Make sure you have correctly graphed the points. Main Lobby 3 rd Floor 217 Holt Mathematics

3 11-2 Ready to Go On? Skills Intervention Comparing and Ordering Integers The numbers on a number line increase in value moving from left to right. Comparing Integers Use the number line to compare each pair of integers. Write or A Is 3 to the left or right of 1 on the number line? B Is 2 to the left or right of 4 on the number line? Ordering Integers Order the integers in each set from least to greatest. A. 3, 1, 0 Graph the integers on the same number line Read the numbers from left to right. You read:,,. B. 8, 5, 1, 8, 7 Graph the integers on the same number line Read the numbers from left to right. You read:,,,,. Weather Application On which day was the temperature the highest? Graph the temperatures on a number line. Which temperature is farthest to the right? Day Temperature Tuesday 2 F Thursday 0 F Saturday 3 F The temperature was the highest on. 218 Holt Mathematics

4 11-2 Ready to Go On? Problem Solving Intervention Comparing and Ordering Integers When you compare and order integers, pay close attention to negative signs and absolute value signs. List these three integers in order from greatest to least Understand the Problem 1. What does the absolute value sign mean? 2. What is 8 equal to? 3. Besides the numeral 6, what two types of symbols are in 6? Make a Plan 4. Why might a number line help in solving this problem? Solve 5. Draw a number line from 10 to 10. Then draw and label a dot at 8 and write what 8 is equal to. 6. What is the absolute value of 4? What is that with a negative sign in front of it? Show 4 on the number line. 7. What is the absolute value of 6? What is that with a negative sign in front of it? Show 6 on the number line. 8. What is the order of the three integers from greatest to least? Check 9. Does the problem ask for greatest to least order or least to greatest order? 219 Holt Mathematics

5 11-3 Ready to Go On? Skills Intervention The Coordinate Plane A coordinate plane is formed by two number lines in a plane that intersect at right angles. The two number lines are called the axes and they divide the coordinate plane into four quadrants. The point where the axes intersect is called the origin. Identifying Quadrants Name the quadrant where each point is located. A. A Place your pencil on Point A. In which quadrant is it located? Quadrant B. B Place your pencil on Point B. In which quadrant is it located? Quadrant 4 Vocabulary coordinate plane axes quadrants origin A 4 2 y 2 O B 4 x Locating Points on the Coordinate Plane Give the coordinates of point L. From the origin, how many units to the right is point L? From the origin, how many units up is point L? 4 2 y 4 2 L O 2 4 x Point L is located at (, ). 2 Graphing Points on the Coordinate Plane Graph each point on the coordinate plane. 4 A. P( 1, 0) Start at the origin. How many units to the left or right 4 y will you move? How many units up or down will you move? Plot and label point P. B. J(0, 4) Start at the origin O x How many units to the left or right will you move? How many units up or down will you move? Plot and label point J. 220 Holt Mathematics

6 SECTION 11A Ready to Go On? Quiz 11-1 Integers in Real-World Situations Name a positive or negative number to represent each situation. 1. a gain of 12 yards feet below sea level degrees above zero 4. 3 degrees below zero 5. an elevator travels down 8 floors 6. a toy rocket shoots 16 feet up Write the opposite of each integer The average depth of the Pacific Ocean is 4,270 meters. Write the depth in meters of the Pacific Ocean as an integer. 12. The mean temperature of the water in Lake Tahoe is 40.3 degrees. Write the temperature as an integer Comparing and Ordering Integers Compare. Write or Order the integers in each set from least to greatest , 4, , 3, 21, , 1, 23, The high school football team gained 16 yards. On the next play, they lost 9 yards. Write integers for each of these events and compare them. 221 Holt Mathematics

7 SECTION 11A Ready to Go On? Quiz continued 11-2 Comparing and Ordering Integers (continued) 23. The scores of the players on the high school golf team in a single game were 6, 5, 7, 2, 1, and 10. Put these scores in order from least to greatest. 24. During an archaelogical dig, the closer to the surface of the earth an object is found, the newer it is. If pieces of pottery are found at 8 ft, 22 ft, 13 ft and 11 ft, which piece is the newest? 11-3 The Coordinate Plane Use the coordinate plane for problems Name the quadrant where each point is located. 25. R 26. H 27. S 28. B Give the coordinates of each point. 29. Y 30. X 31. A 32. W y S 4 W B A 2 Y X 2 R 4 H x Graph each point on a coordinate plane. 33. Q (0, 4) 34. R (2, 5) 4 y R 35. S ( 2, 2) 36. T (0.5, 3.5) 2 x S 2 4 T Q 222 Holt Mathematics

8 SECTION 11A Ready to Go On? Enrichment National Arbor Day Students at different schools decided to plant trees for National Arbor Day. Write an equation for the function for each situation and solve each problem. 1. Students at Southeastern Middle School decided to plant Norfolk pines. If 30 6 th graders planted 3 trees each and 20 7 th graders planted 4 trees each, how many trees would they plant in all? 2. Students at Southwestern Middle School decided to plant palm trees. If 25 6 th graders planted 4 palm trees each and 20 7 th graders planted 3 trees each, how many would they plant in all? 3. Students at Northeastern Middle School decided to plant maple trees. If the 6 th and 7 th graders planted 50 maple saplings in all, and the 7 th graders planted 22, how many did the 6 th graders plant? 4. Students at Northwestern Middle School decided to plant cedar saplings. If 20 6 th graders planted 2 trees each, 30 7 th graders each planted 1 tree, and 23 8 th graders planted 3 trees each, how many trees were planted in all? 5. The schools in Exercises 1, 2, and 4 are in a national contest to see which school plants the most trees. If each 6 th grader at each school plants one more tree, which school will plant the most trees? 6. Midwestern Middle School decides to plant oak saplings to enter the contest. If 20 6 th graders plant 5 trees each, 15 7 th graders plant 3 trees each, and 20 8 th graders plant 4 trees each, can Midwestern Middle School beat the winner from Exercise 5? By how many trees will the winner beat the loser? 223 Holt Mathematics

9 11-4 Ready to Go On? Skills Intervention Adding Integers Writing Integer Addition Write the addition modeled on the number line The addition modeled is. Adding Integers Find the sum. 5 ( 3) Start at 0 and move 5 units right to represent. Then move 3 units left to represent. Where do you stop? ( 3) Evaluating Integer Expressions Evaluate x 2 for each value of x. A. x 3 B. x 4 x Write the expression. What will you substitute for x? Add. x Write the expression. What will you substitute for x? Add. Earth Science Application A tree s roots reach 7 ft below the ground. The trees highest branch is 15 ft above the root. How much of the tree is above ground? How far below the ground is the root? How far above the root is the branch? Add. of the tree is above the ground. 224 Holt Mathematics

10 11-4 Ready to Go On? Problem Solving Intervention Adding Integers When you add integers, you can tell a lot about the sum without calculating it exactly. List these four sums in order from greatest to least. x is an integer x ( x) Understand the Problem 1. Are you asked to find the exact sums? What are you asked? Make a Plan 2. Why might estimation and number sense be useful for solving this problem? Solve 3. Think about x ( x). What is the sum of any integer and its opposite? 4. Which of the other sums is greater than 0? Explain. 5. Now compare the other two sums. Which is greater, or ? Explain. 6. What is the order of the four sums from greatest to least? Check 7. Make rough estimates by sketching the sums on a number line. 225 Holt Mathematics

11 11-5 Ready to Go On? Skills Intervention Subtracting Integers Writing Integer Subtraction Write the subtraction modeled on the number line The subtraction modeled is. Subtracting Integers Find each difference. A. 8 3 Start at 0 and move 8 units right to represent. Then move 3 units left to represent. Where do you stop? 8 3 B. 4 ( 1) Start at 0 and move 4 units left to represent. Then move 1 unit right to represent. Where do you stop? 4 ( 1) Evaluating Integer Expressions Evaluate x ( 3) for each value of x. A. x 3 B. x 2 x Write the expression. What will you substitute for x? Subtract. x Write the expression. What will you substitute for x? Subtract Holt Mathematics

12 11-5 Ready to Go On? Problem Solving Intervention Subtracting Integers You can subtract to find a change. Use the information in the table to find the change in temperature from 6 P.M. to 10 P.M. Understand the Problem 1. What was the temperature at 6 P.M.? At 10 P.M.? 2. What are you supposed to find? Time Temperature C 6 P.M. 2 8 P.M P.M midnight 14 Make a Plan 3. Which subtraction expression is correct for the change in temperature from 6 P.M. to 10 P.M.? Explain. 9 C 2 C 2 C 9 C Solve 4. Label the marks on the number line. Then draw arrows to subtract 9 C 2 C. Mark and label the result What was the change in temperature from 6 P.M. to 10 P.M.? Check 6. Why does it make sense that the answer is negative? Solve 7. Compare the change from 6 P.M. to 10 P.M. with the change from 8 P.M. to midnight. 227 Holt Mathematics

13 11-6 Ready to Go On? Skills Intervention Multiplying Integers Rules for Multiplying Integers If the signs are the same, the product is positive. If the signs are different, the product is negative. The product of any number and 0 is 0. Multiplying Integers Find each product. A. 3 2 How many groups of 2 are there? 3 2 Is the product positive or negative? B. 4 ( 3) How many groups of 3 are there? 4 ( 3) The signs are different so the product is. C is the opposite of how many groups of 2? 6 2 The signs are different so the product is. D. 2 ( 5) 2 ( 5) is the opposite of how many groups of 5? 2 ( 5) The signs are the same so the product is. Evaluating Integer Expressions Evaluate 4x for each value of x. A. x 7 x Write the expression. What will you substitute for x? Are the signs different or the same? Multiply. The product is. B. x 0 x Write the expression. What will you substitute for x? Multiply. 228 Holt Mathematics

14 11-7 Ready to Go On? Skills Intervention Dividing Integers Rules for Dividing Integers If the signs are the same, the quotient is positive. If the signs are different, the quotient is negative. Dividing Integers Find each quotient. A. 18 ( 6) Think: What number times 6 equals 18? ( 6), so 18 ( 6). B. 21 ( 3) Think: What number times 3 equals 21? ( 3), so 21 ( 3). Evaluating Integer Expressions c Evaluate 2 for each value of c. A. c 10 c Write the expression. 2 What will you substitute for c? Divide. Are the signs different or the same? So, the answer is. B. c 16 c Write the expression. What will you substitute for c? Divide. Are the signs different or the same? So, the answer is. 229 Holt Mathematics

15 SECTION 11B Ready to Go On? Quiz 11-4 Adding Integers Write the addition modeled on each number line Find each sum ( 5) ( 2) 6. 6 ( 17) Evaluate 3 x for each value of x. 7. x 4 8. x 3 9. x x x x A treasure hunter digs 5 feet below the surface on the first day. On the second day he digs an additional 2 feet. What depth has he reached at the end of the second day? 11-5 Subtracting Integers Write the subtraction modeled on each number line Holt Mathematics

16 SECTION 11B Ready to Go On? Quiz continued 11-5 Subtracting Integers (continued) Evaluate x ( 3) for x. 16. x x x A diver goes 3 feet below the water on her first day out. On the second day, she goes 10 feet below. How much further does she dive on the second day than on the first? 11-6 Multiplying Integers Evaluate 5x for each value of x. 20. x x x 8 Find each product ( 9) ( 6) 11-7 Dividing Integers Evaluate x 2 for each value of x. 26. x x x 122 Find the quotient Evaluate x 6 for each value of x. 32. x x x Holt Mathematics

17 SECTION 11B Ready to Go On? Enrichment Going Crabbing There are many different shapes, sizes and types of salt-water crabs. Most states with an ocean border have laws that regulate the crabbing season and the number and size of crabs that can be caught by recreational crabbers. Crabs that fall outside of the regulations must be thrown back. 1. If the law states that the length of each crab that is kept must be at least 6 inches and one crabber catches 5 crabs each of whose length is one of: 4 inches, 5 inches, 6 inches, 7 inches and 8 inches long, how many crabs can the crabber keep? 2. If the law limits the crabbing season to the months March through December, during how many months can recreational crabbers put out their pots? 3. Male and female crabs have different markings on their abdomens. In Judi s state, crabbers may keep only the male crabs they catch. If 4 of the 12 crabs that Judi catches are female, how many crabs can she keep? 4. Jon, Marcus, and Todd caught 4 crabs each in April and 3 crabs each in May. How many crabs did they catch in all? 5. If a state issues a crabbing license that allows the crabber to catch 3 crabs a day, how many crabs would a crabber be allowed to keep if he went crabbing every day during the month of December? 6. One week Sergei caught 12 crabs and put back 9, caught 8 crabs and put back 5, caught 3 crabs and put back 1, and caught 7 crabs and put back 4. How many crabs did he end up with? 7. In Natalie s state, crabbers may only keep 2 female crabs per day and only male crabs longer than 5 inches. She made a table of all the crabs she caught in one day. How many crabs will she be allowed to keep? M or F M M F F M F F M M M Length (in.) Holt Mathematics

18 11-8 Adding and Subtracting to Solve Equations Solve each equation. A. 5 x 3 5 x 3 What is added to x? What will you subtract from both sides to undo the x addition? B. t 4 2 t 4 2 What is subtracted from t? What will you add to both sides to undo the t subtraction? C. x 4 12 x 4 12 What is subtracted from x? What will you add to both sides to undo the x subtraction? Multiplying and Dividing to Solve Equations Solve each equation. A. 5g 20 5g 20 g Ready to Go On? Skills Intervention Solving Integer Equations What is g multiplied by? What will you divide both sides by to undo the multiplication? f B. 5 6 f 5 What is f divided by? 6 What will you multiply both sides by to undo the f division? C. j 7 8 j 8 What is j divided by? 7 What will you multiply both sides by to undo the j division? 233 Holt Mathematics

19 11-8 Ready to Go On? Problem Solving Intervention Solving Integer Equations You can use equations with integers to model situations and solve problems. The table shows the daily high temperature for 5 days, but the temperature for Thursday is missing. The mean high temperature for the 5 days is 2. What was the high temperature on Thursday? Day of the week Mon Tues Wed Thurs Fri High Temperature ( F) 2 7 9? 10 Understand the Problem 1. How do you find the mean of a group of values? Make a Plan 2. Let x stand for the missing temperature. Suppose you knew S, the sum of the 5 temperatures. Write an equation you could use to find x. 3. You know that the mean high temperature for 5 days is 2. Write an equation you could use to find S, the sum of the 5 temperatures. Solve 4. To find S, the sum of the temperatures, solve the equation you wrote for Exercise Find the missing temperature x by solving the equation you wrote for Exercise 2. Hint: Use the value of S you found in Exercise 4. Check 6. Use the value you found for x and see if the mean is 2. Solve 7. A week earlier, the mean daily high temperature for Monday to Friday was 0 F. The highs for the first four days were 10, 6, 0, and 4. What was the high temperature on Friday? 234 Holt Mathematics

20 11-9 Ready to Go On? Skills Intervention Tables and Functions A function is a rule that relates two quantities so that each input value corresponds exactly to one output value. Writing Equations from Function Tables Write an equation for a function that gives the values in the table. Use the equation to find the value of y for the indicated value of x. Vocabulary function input output x y y is times x. What pattern do you find when you compare x and y? y x Use the pattern to write an equation. y Find the missing value. What will you substitute for x? y Multiply. y Subtract. When x is 12, y is. Use your function rule to find y when x 12. Translating Words into Math Write an equation for the function. Tell what each variable you use represents. Jessica is 5 inches taller than Monica. Monica s height Choose variables to represent each girl s height. Jessica s height 5 Write an equation. Sales Application Brian sells bananas. He charges the same price for each banana. The cost for 2 bananas is $1.50, 4 bananas is $3.00, and 7 bananas is $5.25. Write an equation for the function. Let b be the number of bananas. Let c be the cost. Complete the table. b 2 c $1.50 c is equal to times b. Write an equation: b 235 Holt Mathematics

21 11-9 Ready to Go On? Problem Solving Intervention Tables and Functions You can write an equation to model a situation. You are planning to paint posters for a school election. The posters take you 25 minutes each to paint. It takes you 15 minutes to set up and 20 minutes to clean up no matter how many posters you paint. If you start at 10:30 A.M. and paint 5 posters, at what time will you finish? Understand the Problem 1. What will be the form of your answer? 2. Complete the table. No. of Paint Setup Cleanup Total posters Time Time Time Time 1 2 Make a Plan 3. Let n be the number of posters you paint. Write an expression for the number of minutes it takes just for the painting, without setup or cleanup. 4. Fill in the blanks to write an equation that shows how the total time t depends on n, the number of posters you paint. total time paint time setup time cleanup time t 5. Simplify the equation by combining the setup time and cleanup time. Solve t 6. Use the equation you wrote in Exercise 5 to find the total time it will take for 5 posters. 7. How many minutes will it take in all for 5 posters? What is that in hours and minutes? At what time will you finish? Check 8. Estimate to check your answer. 236 Holt Mathematics

22 11-10 Ready to Go On? Skills Intervention Graphing Functions When you graph the ordered pairs of some functions, they form a straight line. The equations that express these functions are called linear equations. Vocabulary linear equation Finding Solutions of Equations with Two Variables Use the given x-values to write solutions of the equation as ordered pairs. y 4x for x 1, 1, 2 Complete the function table by using the given values for x to find values for y. Then write the solutions as ordered pairs. x y 4x y (x, y) 1 4( ) 4 ( ) Checking Solutions of Equations with Two Variables Determine whether the ordered pair is a solution to the given equation. ( 1, 2); y 3x 1 y 3x 1 Write the equation. 3( ) 1 Substitute for x and for y. Evaluate both sides of the equation. Is ( 1, 2) a solution of y 3x 1? How do you know? 4( ) ( ) 4( ) ( ) Graphing Linear Functions Graph the function described by the equation above. Graph each ordered pair from the table on the coordinate plane. Draw a line through the points. Is the equation linear? 8 y 8 ( 1, 4) 4 4 O (1, 4) x How do you know? 8 (2, 8) 237 Holt Mathematics

23 SECTION 11C Ready to Go On? Quiz 11-8 Solving Integer Equations Solve each equation. Check your answers x y ( 3) b c k 6. r ( 8) ( 7) s 8. t ( 8) Maria read 3 times as many books in the summer than she read in the spring. If she read 4 books in the spring, how many did she read in the summer? 11-9 Tables and Functions Write an equation for a function that gives the values in each table. Use the equation to find the value of y for each indicated value of x. 10. x y x y x y x y For Exercises 14 19, write an equation for the function. Tell what each variable you use represents. 14. The number of forks is 2 times the number of spoons 15. The time Morgan spends practicing the piano is 5 minutes less than the time he spends practicing soccer 16. The length of a rectangle is 2 times its width 238 Holt Mathematics

24 SECTION 11C Ready to Go On? Quiz continued 11-9 Tables and Functions (continued) 17. A store manager tracked the sale of bicycle helmets. The store charges the same price for each helmet. On Monday, 6 helmets were sold for a total of $150. On Tuesday, 2 helmets were sold for a total of $50. On Wednesday, 5 helmets were sold for a total of $ On Field Day, 100 middle school students received blue ribbons. If 25 6th graders received blue ribbons, and 30 7th graders received blue ribbons, how many 8 th graders received blue ribbons? 19. On Field Day, 25 6 th graders received red ribbons and 35 7 th graders received red ribbons. If the 8 th graders received 2 times more red ribbons than the 6 th graders, how many red ribbons in all were received? Graphing Functions Use the given x-values to write solutions of each equation as ordered pairs. 20. y 3x 4 for x 5, 6, 7, y 9x 6 for x 1, 2, 3, y 8x 2 for x 3, 5, 7, y 3x 4 for x 2, 6, 8, 10 Use the graph of the linear function at the right to find the value of y for each given value of x. 4 y 24. x x x 26. x x Holt Mathematics

25 SECTION 11C Ready to Go On? Enrichment Football Moves The line of scrimmage is an imaginary line stretching the width of the field which no player may cross before the ball is snapped. Each team has its own line of scrimmage, separated by the neutral zone. 1. A player on the offensive team passes the football 25 yards from the line of scrimmage toward the goal post. One of his teammates catches it and runs 5 yards more. How many yards does the offensive team gain? Graph the moves and show the answer. scrimmage yards 2. A player on the Mighty Mountain Football Team has the ball on the line of scrimmage. He throws the ball to one of his teammates, who is 15 yards away, but the football is caught by a member of the opposing team, who races 40 yards down the field in the opposite direction. How far from the line of scrimmage does the ball end up? Graph the moves and show the answer. scrimmage yards 3. In the first play, the Wildcats gain 10 yards. In the next play, they lose 30 yards. How many yards from the original line of scrimmage are they? Graph the moves and show the answer. scrimmage yards 4. In the first 5 minutes, Team A gains 45 yards. Team B gains 30 yards. What is the difference in their yardage gain? 240 Holt Mathematics