ACCELERATED MATHEMATICS CHAPTER 7 PART II NON-PROPORTIONAL LINEAR RELATIONSHIPS TOPICS COVERED:

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1 ACCELERATED MATHEMATICS CHAPTER 7 PART II NON-PROPORTIONAL LINEAR RELATIONSHIPS TOPICS COVERED: Representing Linear Non-Proportional Equations Slope & y-intercept Graphing Using Slope & y-intercept Proportional vs. Non-Proportional Systems of Linear Equations Linear Equations from Situations and Graphing Linear Equations from Tables Linear Relationships and Bivariate Data Input/Output and Functions Describing & Comparing Functions Contents modified from my.hrw.com, 8 th grade Edition online practice

2 NONPROPORTIONAL RELATIONSHIPS CHAPTER 7 MODULE 4, 5, 6 (BOOK) Module 4. Relationships can have a constant rate of change but not be proportional. When graphed a linear equation forms a line. Linear equations form y = mx + b. When b the relationship is non-proportional. m = slope These graphs are straight lines that do not go through the origin. Module 4. y-intercept = the y-coordinate of the point where the graph intersects the y-axis. (when x=) y = mx + b is called slope-intercept form. m = slope b = y-intercept Module 4.3/4.4 A line with positive slope will rise from left to right. A line with negative slope will fall from left to right. Linear relationships can be either proportional or non-proportional. Non-linear relationships are all non-proportional. Module 5./5. Word problems, Graphs, Verbal descriptions, Tables to: y = mx + b Linear, proportional: y = mx Linear, non-proportional: y = mx + b Module 5.3 Points on a graph to y = mx + b Bivariate data is a set of data that is made up of two paired variables. Non-linear relationships have a rate of change that varies between pairs of points. Module 6. A function is like a machine. It has exactly one output for each input. The value that is put into the machine is the input. The result is the output. You can determine functions from: Mapping diagrams, Coordinate pairs, Tables, Graphs Module 6. A linear equation is represented by a straight line on a graph. Linear equations are linear functions when their graphs are non-vertical lines. Equations that cannot be written as y = mx + b are not linear equations and therefore not linear functions. Module 6.3 Comparing functions in the form of: Tables, Equations, Graphs, Descriptions

3 Activity 7-: Writing Linear Equations from Situations and Graphs (5.) Writing an Equation in Slope-Intercept Form Greta makes clay mugs and bowls as gifts at the Crafty Studio. She pays a membership fee of $5 a month and an equipment fee of $3. an hour to use the potter s wheel, table, and kiln.. What is the input variable, x, in this situation?. What is the output variable, y, in this situation? 3. Write an equation in the form y = mx + b that Greta can use to calculate her monthly cost. 4. During April Greta does not use the equipment at all. What will her cost be for April? 6. What is the y-intercept, b, in the equation? 7. What is the slope, m, in the equation? 8. A video club charges a one-time membership fee plus a rental fee for each DVD borrowed. Use the graph to write an equation in slope-intercept form. 9. What does the value of the slope represent in this context?. Describe the meaning of the y-intercept. $ Rentals. The cash register subtracts $.5 from a $5 Starbucks gift card for every medium coffee the customer buys. Use the graph to write an equation in slope-intercept form to represent this situation. $ Number of coffees

4 Activity 7-3: Writing Linear Equations from Situations and Graphs (5.) Writing an Equation from a Description The rent charged for space in an office building is a linear relationship related to the size of the space rented. Write an equation in slope-intercept form for the rent at West Main Street Office Rentals. West Main Street Office Rentals Monthly Rates 6 square feet for $75 9 square feet for $5. What are the input (x) and output (y) variables?. Write the information given in the problem as ordered pairs. 3. What is the slope? 4. What is the y-intercept? Use the slope and one of the ordered pairs to determine the intercept. 5. Substitute the slope and y-intercept into the final equation. 6. Hari s weekly allowance varies depending on the number of chores he does. He received $6 in allowance the week he did chores and $4 in allowance the week he did 8 chores. Write an equation for his allowance in slope-intercept form. 7. Li is making beaded necklaces. For each necklace she uses 7 spacers plus 5 beads per inch of necklace length. Write an equation in slope-intercept form to find how many beads Li needs for each necklace. 8. Kate is planning a trip to the beach. She estimates her average speed to graph her expected progress on the trip. Write an equation in slope-intercept form that represents the situation. 9. At 59 degrees Fahrenheit crickets chirp at a rate of 76 times per minute and at 65 degrees Fahrenheit they chirp times per minute. Write an equation in slope-intercept form that represents the situation.. A dragonfly can beat its wings 3 times per second. Write an equation in slope-intercept form that shows the relationship between flying time in seconds and the number of times the dragonfly beats it wings.. A balloon is released from the top of a platform that is 5 meters tall. The balloon rises at the rate of 4 meters per second. Write an equation in slope-intercept form that tells the height of the balloon above the ground after a given number of seconds. Distance to beach (mi) Driving time (hr)

5 Activity 7-4: Writing Linear Equations from Situations and Graphs (5.) The graph shows a scuba diver s ascent over time.. What is the slope of the line?. What does the slope mean in this problem? 3. What is the y-intercept of the line? 4. What does the y-intercept mean in this problem? 5. Write an equation in slope-intercept form that represents the diver s depth over time. Depth (m) Time (sec) 6. The formula for converting Celsius to Fahrenheit temperatures in a linear equation. Water freezes at C, or 3 F, and it boils at C, or F. Write an equation in slope-intercept form that converts degrees Celsius into degrees Fahrenheit. 7. The cost of renting a sailboat at a lake is $ per hour plus $ for lifejackets. Write an equation in slope-intercept form that can be used to calculate that total amount you would pay for using a sailboat. The graph shows that activity in a savings account. 8. What was the amount of the initial deposit that started this savings account? 9. Find the slope and y-intercept of the graphed line.. Write an equation in slope-intercept form for the activity in this savings account.. Explain the meaning of the slope in this graph. $ saved Months in plan. Explain how you decide which part of any problem will be represented by the variable x and which part will be represented by the variable y in a graph of a situation.

6 Activity 7-5: Writing Linear Equations from Words and Graphs (5.) For each situation described, first write an equation in the form y = mx + b. Then solve each problem.. A sales associate is given a $5 hiring bonus with a new job. She earns an average commission of $5 per week for the next weeks. How much does she earn?. A farm has a reserve of 5 pounds of wheat in his silo. The farmer can harvest the wheat at a rate of pounds of wheat per day. In how many days will the farmer have a total of 3,5 pounds of wheat? 3. Mr. Mangham is going on a long run with 75 ml of Gatorade. Each hour he drinks ml of Gatorade. How long will the Gatorade last? 4. A water tank is currently holding 8, gallons. The water will be used at a rate of 45 gallons per day. How long will it take for the water level to reach 6, gallons? 5. Elsa collects snowmen. At the end of 4 months she had 7 snowmen and at the end of 7 months she had 6 snowmen. Find out how many snowmen she had at month by calculating b and then determine how many months it will take her to have 89 snowmen. 6. (4,75) (9,4) Use these two points to write an equation in the form of y = mx + b. 7. (3,4) (7,3) Use these two points to write an equation in the form of y = mx + b. Write an equation in the form y = mx + b for each situation Height (m) y Time (s) x

7 Activity 7-6: Writing Linear Equations from a Table (5.) The table shows the temperature of a fish tank during an experiment. Time (hr) Temperature (F) Graph the ordered pairs.. Draw a line through the points. 3. Calculate the slope. 4. Find the y-intercept. 5. Write the equation. Temperature Tank Temperature Time (hr) 6. The table shows the volume of water released by Hoover Dam over a certain period of time. Graph the data and then write the equation for the graph in slope-intercept form. Time (sec) 5 5 Volume of water (m 3 ) Water Released 3 5 Volume Time (sec)

8 Activity 7-7: Writing Linear Equations from a Table (5.). Elizabeth s cell phone plan lets her choose how many minutes are included each month. The table shows the plan s monthly cost y for a given number of included minutes x. Write an equation in slopeintercept form to represent the situation. Minutes included Cost of the plan ($) What is the base price for the cell phone plan, regardless of how many minutes are included? What is the cost per minute? 3. Elizabeth s cell phone company changes the cost of her plan as shown below. Write an equation in slope-intercept form to represent the situation. How did the plan change? Minutes included Cost of the plan ($) A salesperson received a weekly salary plus a commission for each computer sold. Write an equation in slope-intercept form to represent the situation. Number of computers sold Total pay ($) To rent a van a moving company charges a daily fee plus a fee per mile. Write an equation in slopeintercept form to represent the situation. Number of miles driven Total cost ($) Jaime purchased a $ bus pass. Each time he rides the bus a certain amount is deducted from the pass. Write an equation in slope-intercept form to represent the situation. Number of rides $ left The table below shows the temperature at different altitudes. Write an equation in slope-intercept form to represent the situation. Altitude (ft) Temperature (F)

9 Activity 7-8: Writing Linear Equations from a Table (5.). The table shows how much an air-conditioning repair company charges for different numbers of hours of work. Graph the data and then write the equation for the graph in slope-intercept form. Number of hours Amount charged Amount charged ($) Hours. The tables below show linear relationships between x and y. Write an equation in slope-intercept form for each relationship. x x 4 6 y 3 y Desiree starts a savings account with $5. Every month she deposits $53.5. Complete the table to model the situation. Month, x Amount in savings, y 4. Write an equation in slope-intercept form for the question above. 5. Use the equation to find how much money Desiree will have in savings after months.

10 Activity 7-9: Writing Linear Equations from a Table (5.) Graph the data and find the slope and y-intercept from the graph. Then write an equation for the graph in slope-intercept form... Weight (oz), x 4 8 Cost ($), y Time (min), x Elevation (ft), y slope: y-intercept: equation: slope: y-intercept: equation: Write an equation in slope-intercept form that represents the data. 3. Sales Per Day, x 3 Daily Pay ($), y 5 5 equation: 4. Time Since Turning Oven Off (min), x 5 5 Temperature of Oven ( F), y equation: The table shows the linear relationship of the height y (in inches) of a tomato plant x weeks after it was planted. 5. Write an equation that shows the height of the tomato plant. 6. Use the equation to find the height of the tomato plant 6 weeks after it was planted. Weeks After Planting, x Height (in.), y 4 3 7

11 Activity 7-3: Linear Relationships and Bivariate Data (5.3) You can use an equation of a linear relationship to predict a value.. The graph shows the cost for taxi rides. Predict the cost of a taxi ride that covers a distance of.5 miles.. Suppose a regulation changes the cost of the taxi ride to $.8 per mile, plus a fee of $4.3. How does the price of a.5 mile ride compare to the original price? Cost ($) Distance (mi) Paulina s income from a job that pays her a fixed amount per hour is shown in the graph. Use the graph to find the predicted value. 3. Income earned for working hours. 4. Income earned for working 3.5 hours. 5. Total income earned for working for five 8-hour days all at the standard rate. Income ($) Time (hr) 6. The graph shows the relationship between the number of hours a kayak is rented and the total cost of the rental. Write an equation of the relationship and then predict the cost of a rental that lasts 5.5 hours. Cost ($) Time (hr) 7. Louis says that if the differences between the values of x are constant between all the points on a graph, then the relationship is linear. Do you agree? Explain.

12 Activity 7-3: Linear Relationships and Bivariate Data (5.3) Andrew has two options in which to invest $. Option A earns simple interest of 5% and Option B earns 5% compounded interest. Option A Option B Year, x Total ($) Total ($) Graph the data from the table.. Find the rate of change for Option A. What type of relationship is this? Aount of Investment Year 3. Find the rate of change for Option B between the following points. What type of relationship is this? Option B From -5 years Rate of Change From 5- years From -5 years 4. Why are the graphs drawn as lines or curves and not as discrete points? 5. Explain whether or not you think each relationship is linear. a) the cost of equal-priced DVDs and the number purchased b) the height of a person and the person s age c) the area of a square quilt and its side length d) the number of miles to the next service station and the number of kilometers Does each of the tables represent a linear relationship? Explain why or why not Months Account balance ($) 4 6 Time (sec) 3 4 Distance (ft) 8 5

13 Activity 7-3: Linear Relationships/Bivariate Data (5.3) Write an equation for each linear relationship... Weight (lb), x Total cost ($), y The graph shows the relationship between the number of rows in a friendship bracelet and the time it takes Mia to make the bracelet, including the time it takes to prepare the threads. 3. Determine whether the relationship is linear. If so, write an equation for the relationship. 4. How long will it take for Mia to complete 4 rows? 5. A weightlifter is adding plates of equal weight to a bar. The table shows the total weight, including the bar, that he will lift depending on the total number of plates on the bar. Based on this information what is the weight of the bar without any plates? Number of plates Total Weight (lbs) A 7-inch candle burns at a rate of inches per hour. Which equation represents the relationship between y, the height of the candle in inches, and x, the number of hours the candle burns? y = x + 7 y = 7 x y = 7 x y = 7x + 7. An architect is designing an office building with n floors that will have an FM radio antenna 5.85 m tall on its roof. Each floor of the building will be 3.9 m high. Which equation can be used to find the total height of the building in meters, including the FM antenna? y = 5.85x y = 3.9x y = 3.9x 5.85 y = 9.75x

14 Activity 7-33: Identifying and Representing Functions (6.) Carlos needs to buy some pencils and he asks his classmates how much they spent on pencils at the school supply store. Angela says she bought pencils for $.5. Paige bought 3 pencils for $.75 and Spencer bought 4 pencils for $.. Carlos thinks about the rule for the price of a pencil as a machine. When he puts the number of pencils he wants to buy into the machine, the machine applies a rule and tells him the total cost of that number of pencils.. Use the price in the problems to fill in total cost in the rows of the table. Number of pencils Rule Total Cost 3 4 x. Describe any pattern you see. Use the pattern to determine the cost of pencil. 3. Use the pattern you identified to write the rule applied by the machine. Write the rule as an algebraic expression and fill in the Rule column row of the table. 4. Carlos wants to buy pencils. Use the rule to show how much Carlos will pay for pencils. A function assigns exactly one output to each input. The value that is put into a function in the input. The result is the output. A mapping diagram can be used to represent a relationship between input vales and output values. A mapping diagram represents a function if each input value is paired with only one output value. 5. Determine whether each relationship is a function

15 Activity 7-34: Identifying and Representing Functions (6.). Is it possible for a function to have more than one input value but only one output value? Provide an illustration of your answer.. Determine whether each relationship is a function Identifying Functions from Tables Relationships between input and output values can also be represented using tables. The values in the first column are the input values. The values in the second column are the output values. The relationship represents a function if each input value is paired with only one output value. 3. Determine whether each relationship is a function. Input Output Input Output What is always true about the numbers in the first column of a table that represents a function? Why must this be true?

16 Activity 7-35: Identifying and Representing Functions (6.). Determine whether each relationship is a function. Explain. Input Output Input Output Graphs can be used to display relationships between two sets of numbers. The first or x coordinate is the input values and the second or y coordinate is the output value. The graph represents a function if each input value is paired with only one output value.. The graph shows the relationship between the number of hours students spent studying for an exam and the exam grades. Is the relationship represented by the graph a function? Hours Studied and Exam Grade Exam Grade Hours Studied The points represent the following ordered pairs: (,7), (,7), (,85), (3,75), (5, 8), (6,8), (7,88), (9,9), (9,95), (,97)

17 Activity 7-36: Identifying and Representing Functions (6.). The graph shows the relationship between heights and weights of the members of a basketball team. Is the relationship represented by the graph a function? Weight (lb) Heights and Weights of Team Members Height (in). Complete each table. In the row with x as the input write a rule as an algebraic expression for the output. Input Output Input Output Input Output Tickets Cost ($) Minutes Pages Muffins Cost ($) x x x 3. Determine whether each relationship is a function Input Output

18 Activity 7-37: Identifying and Representing Functions (6.). The graph shows the relationship between weights of 5 packages and the shipping charge for each package. Is the relationship represented by the graph a function? Explain. Weight and Shipping Costs Shipping Cost ($) Weight (lb). What are the four ways of representing functions? How can you tell if a relationship is a function? 3. Determine whether each relationship represented by the ordered pairs is a function. Explain. a. (,), (3,), (5,7), (8,), (9,) b. (,4), (5,), (,8), (6,3), (5,9) 4. Jack receives $.4 per pound for to 99 pounds of aluminum cans he recycles. He receives $.5 per pound if he recycles more than pounds. Is the amount of money Jack receives a function of the weight of cans he recycles? Explain your reasoning. 5. A biologist tracked the growth of a strain of bacteria, as shown in the graph. Explain why the relationship represented by the graph is a function. 6. Suppose there was the same number of bacteria for two consecutive hours. Would the graph still represent a function? Explain. Number of Bacteria Bacteria B Time (hr)

19 Activity 7-38: Identifying and Representing Functions (6.). Give an example of a function in everyday life, and represent it as a graph, a table, and a set of ordered pairs. Describe how you know it is a function x y The graph shows the relationship between weights of six wedges of cheese and the price of each wedge. Cost of Cheese Price ($) Weight (lb) 3. Is the relationship represented by the graph a function? Justify your answer using words such as input and output. 4. Suppose the weights and prices of additional wedges of cheese were plotted on the graph. Is it likely to change your answer to the previous question? Explain your reasoning. 5. A mapping diagram represents a relationship that contains three different input values and four different output values. Is the relationship a function? Explain your reasoning. 6. An onion farmer is hiring workers to help harvest the onions. He knows the number of days it will take to harvest the onions is a function of the number of workers he hires. Explain the use of the word function in this context.

20 Activity 7-39: Identifying and Representing Functions (6.) Tell whether each relationship is a function Input 3 8. Output 4 4 Input Output {(, ), (, 4), (3, 6), (5, 5), (7, 6)}. {(, 8), (, ), (3, 7), (5, 9), (3, 6)} The graph shows the relationship between the hours Rachel studied and the exam grades she earned.. Is the relationship a function? Justify your answer. Use the words input and output in your explanation, and connect them to the context represented by the graph.. Rachel plans to study hours for her next exam. How might plotting her grade on the same graph change your answer to Exercise? Explain your reasoning.

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22 Activity 7-4: Describing Functions (6.). The US Department of Agriculture defines heavy rain as rain that falls at a rate of.5 centimeters per hour. Complete the table below. Time (hr) Total Amount of Rain (cm).5. How much rain falls in 3.5 hours? 3. Plot the ordered pairs from the table on the coordinate plane. 4. Is the total amount of rain that falls a function of the number of hours that rain has been falling? Why or why not? total Amount of Rain (cm) Heavy Rainfall Time (hr) Graphing Linear Functions The relationship in the previous activity could be represented with the equation y =.5x. The graph of the relationship is a line, so the equation is a linear equation. Since there is exactly one value of y for each value of x, the relationship is a function. It is a linear function because its graph is a nonvertical line. The temperature at dawn was 8 F and increased steadily F every hour. Write a slope-intercept equation for this situation and graph the function on the coordinate plane below. Temperature Temperatures Time (hr) 5. Is the relationship between time and temperature proportional or nonproportional? 6. Show that the equation y + 3 = 3(x + ) is linear and that it represents a proportional relationship between x and y. Tell whether each equation can be written in the form y = mx + b. Write yes or no. If yes, write the equation in the form y = mx + b. 7. y = 8 x 8. y = 4 + x 9. y = 3 x

23 Activity 7-4: Describing Functions (6.) -. For each function below complete the table and then graph the function. Tell whether the function is linear or nonlinear. y = 5 x y = x Input 3 5 Input Output Output 3. The Fortaleza telescope in Brazil is a radio telescope. Its shape can be approximated with the equation y =.3x. Is the relationship between x and y linear? Is it proportional? Explain. 4. A student claims that the equation y = 7 is not a linear equation because it does not have the form y = mx + b. Do you agree or disagree? Why? Graph each equation. Tell whether the equation is linear or nonlinear. 5. y = 3x 6. Input, x 4 Output, y y = x + Input, x Output, y

24 Activity 7-4: Comparing Functions (6.3) John and Maggie buy MP3 files from different music services. The monthly cost, y dollars, for x songs is linear. The cost of John s service is y =.5x +. The cost of Maggie s service is shown below. Songs, x Cost ($), y Write an equation to represent the monthly cost of Maggie s service.. Which service is cheaper when 3 songs are downloaded? Quentin is choosing between buying books at the bookstore or buying online versions for his tablet. The cost, y dollars, of ordering books online for x books is y = 6.95x +.5. The cost of buying the books at the bookstore is shown in the table. Books, x Cost ($), y Write an equation to represent the total cost of books at the bookstore. 4. Which method of buying books is more expensive if Quentin wants to buy 6 books? The table and graph show how many words Morgan and Brian typed correctly on a typing test. For both students, the relationship between words typed correctly is linear. Time (min) 5. Find Morgan s unit rate. 6. Find Brian s unit rate. Morgan s typing test Words Which student types more correct words per minute? Words Brian's Typing Test Time (min)

25 Activity 7-43: Comparing Functions (6.3) Doctors have two methods of calculating maximum heart rate. With the first method, maximum heart rate, y, in beats per minute is y = x, where x is the person s age. Maximum heart rate with the second method is shown in the table. Age, x Heart rate (bpm), y Which method gives the greater maximum heart rate for a 7-year-old?. Are heart rate and age proportional or nonproportional for each method? The table and graph show the miles driven and gas used for two scooters. Distance (mi) Gallons used Scooter A Which scooter uses fewer gallons of gas when 35 miles are driven? 4. Are gas used and miles proportional or nonproportional for each scooter? Gallons (gal) Scooter B Distance (mi) A cell phone company offers two texting plans to its customers. The monthly cost, y dollars, of one plan is y =.x + 5, where x is the number of texts. The cost of the other plan is shown in the table. Number of texts, x Cost ($), y Which plan is cheaper for under texts? 6. The graph of the first plan does not pass through the origin. What does this indicate? 7. Gym A charges $6 a month plus $5 a visit. The monthly cost at Gym B is represented by y = 5x + 4, where x is the number of visits per month. What conclusion can you draw about the monthly cost of the gyms? 8. The equations of two functions are y = x + 9 and y = 4x + 8. Which function is changing more quickly? Explain.

26 Activity 7-44: Comparing Functions (6.3) Find the slopes of linear functions f and g. Then compare the slopes.. f ( x) = 5x x 3 4 g(x) slope of f slope of g Find the y-intercepts of linear functions f and g. Then compare the two intercepts.. x 3 4 f(x) y-intercept of f: y-intercept of g: Connor and Pilar are in a rock-climbing club. They are climbing down a canyon wall. Connor starts from a cliff that is feet above the canyon floor and climbs down at an average speed of feet per minute. Pilar climbs down the canyon wall as shown in the table. Time (min) 3 Pilar s height (ft) Interpret the rates of change and initial values of the linear functions in terms of the situations that they model. Compare the results and what they mean. Connor Pilar Initial value: Rate of change: Initial value: Rate of change:

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29 Activity 7-45: Eric The Sheep Eric the Sheep is at the end of a line of sheep waiting to be shorn. Being an impatient sheep, Eric sneaks up two places every time the shearer takes a sheep to be shorn. For example, while the first sheep is being shorn, Eric moves ahead so that there are two sheep behind him. If at some point it is possible for Eric to move only one place, he does that instead. You may want to use centimeter cubes, where possible, to act out the line of sheep.. Complete the number part of the attached table for sheep in line to help you answer the questions below.. If there were 6 sheep in front of Eric (7 th in line), Eric was sheep shorn number 3. If there were sheep in front of Eric ( th in line), Eric was sheep shorn number 4. How many sheep get shorn before Eric if he was originally the 5 th sheep waiting to be shorn? 5. Look for pattern between his original position in line and when he gets shorn. Think of our input and output machines. Complete the variable part of the table which has the variables. 6. If Eric was 9 th in line, Eric was sheep shorn number 7. If Eric was 37 th in line, Eric was sheep shorn number 8. If Eric was, st in line, how many sheep were shorn before Eric? 9. If there were 7,695 sheep in front of Eric, Eric was sheep shorn number. If Eric were the 4 th sheep shorn, where did he start in line? (Hint: There is more than one answer.). If Eric were the 5 th sheep shorn, where did he start in line?. If Eric were the 7 th sheep shorn, where did he start in line? 3. If there were 673 sheep shorn before Eric, where did he start in line?

30 Activity 7-46: Eric The Sheep Place In Line Sheep In Front of Eric Sheep Shorn Before Eric Eric was sheep shorn number x ** ** 3 * * * * * * y ** ** a * * * * * * b * Simplify all answers (distribute, combine all like terms). ** These boxes will have rounding as part of the answer.

31 Activity 7-47: Locker Pattern The Emily Hopkins School for the Gifted has, students and, lockers. The lockers are numbered from to,. The students enter the building one at a time. The first student opens all the lockers (multiples of ). The second student begins with the second locker and closes all the lockers with even numbers (multiples of ). The third student changes either opening closed doors or closing open doors all lockers that are multiples of 3. This pattern continues until all students walk past all the lockers. After the last student has gone past all the lockers, which lockers are open? In a question where the numbers are very large, one method to try is Make It Simpler. Try the problem with a smaller number of students and see what you get for the result. If there is some sort of pattern in the answers, you can use this pattern to find all the other answers. For this problem, let s try 5 students with 5 lockers. Using the table keep track of which lockers open and close as the students walk through. From these answers, see if you can determine all the rest of the answers. If not, try 3, 4, or 5. Below in words state how you found your pattern and what the pattern is. Then write ALL the final answers. MAKE SURE YOU SHOW ALL YOUR WORK. S T U D E N T S LOCKERS

32 Activity 7-48: Where did the money go? Three guys in a hotel call for room service and order two large pizzas. The delivery boy brings them up the pizzas with a bill for exactly $3. Each guy gives him a $ bill and he leaves. When he hands the $3 to the cashier, he is told a mistake was made. The bill was only $5, not $3. The cashier gives the delivery boy five $ bills and tells him to take it back to the 3 guys who ordered pizza. On the way back to the room the delivery boy has a thought these guys did not give him a tip. He figures that since there is no way to split $5 evenly three ways anyhow, he will keep $ for himself and give them back $3. He knocks on the door and one fellow answers. He explains about the mix up in the bill and hands the guy three dollars, then departs with his $ tip in his pocket. Remember $3-$5 = $5 Right? $5-$3 = $ Right? So answer this: Each of the three guys originally gave $ each. They each got back $ in change. That means they paid $9 each, which times three is $7. The delivery boy kept $ for a tip. $7 plus $ equals $9. So where in the world is the other dollar???