LESSON 1: EXPLORING RELATIONSHIPS

Transcription

1 LESSON 1: EXPLORING RELATIONSHIPS 1. Write what ou alread know about ratios. Share our work with a classmate. Did our classmate understand what ou wrote?. Write what ou know about relationships involving rate and graphing. 3. What does the word proportional mean to ou? Share our thoughts with a classmate. Compare our understandings of the word proportional.. Write a goal stating what ou plan to accomplish in this unit. 5. In the exercises for Lesson 1 of Unit 1: Working With Rational Numbers, ou were asked to respond to this prompt: Based on our previous work in math, write three things that ou will do during this unit to increase our success. For example, consider was ou will participate in classroom discussions, our stud habits, how ou will organize our time, what ou will do when ou have a question, and so on. Think about the three things ou planned to do. If ou can t recall, look back at what ou wrote. a. Did ou do the things ou planned? Wh or wh not? b. Which of the three things most helped ou to be successful in the previous unit? c. Write one thing ou will do differentl or something ou will do more of during this unit to increase our success. Copright 015 Pearson Education, Inc. 5

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3 LESSON : PROPORTIONAL RELATIONSHIPS 1. Luc walks 7 mi in hr. How man miles can she walk in 1 hr? A 5 mi B 3.5 mi C 3 mi D mi. This table represents a proportional relationship. Reccled Glass (lb) Value ($) Express the proportional relationship between the two variables as a ratio. : 3. Which table shows a proportional relationship? A x B x C x D x Copright 015 Pearson Education, Inc. 7

4 LESSON : PROPORTIONAL RELATIONSHIPS. Which table shows a proportional relationship? A x B x C x D x Jack wants to bu strawberries to make smoothies. He plans to use 5 big strawberries to make smoothies. Use this information to complete the table for Jack. Smoothies 1 50 Strawberries This table represents a proportional relationship. Fill in the missing numbers. x Copright 015 Pearson Education, Inc.

5 LESSON : PROPORTIONAL RELATIONSHIPS 7. This dollhouse belongs to Maa s twin sisters. 0 in. in. For the twins birthda, Maa made chairs for the dollhouse. She made the model chairs to match the real chairs the twins have at their stud table. The height of the model chairs is in. If the height of the real chairs is 3 ft, what is the scale Maa used for the model chairs? Show our work. Challenge Problem. A square has a side length of s cm and a perimeter of p cm. a. Make a table showing the perimeter in relationship to the side lengths. b. Is this relationship a proportional relationship? Explain wh or wh not. Copright 015 Pearson Education, Inc. 9

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7 LESSON 3: CONSTANT OF PROPORTIONALITY 1. What is the constant of proportionalit for =.3x? The constant of proportionalit is.. Find the constant of proportionalit for this table. x k = 3. Sophie bought ears of corn for $1.50. She wants to know how much she will have to pa for 3 ears of corn at the same rate. What is the constant of proportionalit? k = $. These are the two formulas for a proportional relationship. = kx x = 1 k Which pair lists the constant of proportionalit for the first formula and the constant of proportionalit for the second formula? A and x B x and C k and 1 k D k and k 5. Use the constant of proportionalit k = 5. to complete this table. x Copright 015 Pearson Education, Inc. 11

8 LESSON 3: CONSTANT OF PROPORTIONALITY. What are the two constants of proportionalit for x = 1 3? Show how ou know. 7. Karen is building a scale model of the Taj Mahal for an integrated art and social studies project. The height of the Taj Mahal is 73 m. The height of the scale model is 0.73 m. a. Karen uses a ratio table to help her convert from the actual measurements of the Taj Mahal to the measurements of the scale model. Complete this ratio table. Actual (m) Scale Model (m) b. What constant of proportionalit describes the relationship? c. Write a formula to represent the proportional relationship between the dimensions of the actual Taj Mahal and those of the scale model. Challenge Problem. These model Spitfire planes were built according to different scales. The scales are (in random order) 1 :, 1 :, 1 : 3, and 1 : 7. a. Which plane is built to each scale? How do ou know? b. Explain how scale is related to the constant of proportionalit. Copright 015 Pearson Education, Inc. 1

9 LESSON : GRAPHS 1. Which graph shows a proportional relationship? A C x x B D x x Copright 015 Pearson Education, Inc. 13

10 LESSON : GRAPHS. Which graph represents this proportional relationship? x A B 0 0 x 0 0 x C D 0 0 x 0 0 x Copright 015 Pearson Education, Inc. 1

11 LESSON : GRAPHS 3. Marcus went on a road trip. Ever 30 min, he checked to see how far he had traveled. This table shows his observations. Time (hr) Distance (mi) Draw a graph of Marcus s road trip data with time (hr) as the independent variable and distance (mi) as the dependent variable.. Consider the table of Marcus s road trip (from the previous exercise), which shows the total distance traveled each 30 min. Recall the graph ou created from the table data. Time (hr) Distance (mi) On Marcus s road trip, distance A is B is not proportional to time. Copright 015 Pearson Education, Inc. 15

12 LESSON : GRAPHS 5. Here is the table of Marcus s road trip, showing the total distance traveled each 30 min. Time (hr) Distance (mi) During which half hour did Marcus travel the fastest?. Compare these two graphs Distance (mi) 0 0 Distance (mi) x x Time (hr) Time (hr) a. Which point(s) do the two graphs have in common? b. Which point(s) are different? Copright 015 Pearson Education, Inc. 1

13 LESSON : GRAPHS 7. Luc is getting read for the science fair. In her project, she is studing mold growth under various conditions. This table shows the data she recorded for one of her test food items. Time (hr) Surface Area Covered in Mold (cm ) a. Draw a graph of Luc s data with time (hr) as the independent variable and surface area covered b mold (cm ) as the dependent variable. b. Is the growth of mold proportional over time? Explain wh or wh not.. Consider this graph x Explain wh the constant of proportionalit for the graph is Copright 015 Pearson Education, Inc. 17

14 LESSON : GRAPHS Challenge Problem 9. Consider this graph, which graphs this equation. = 37.5x x a. Draw the graph again, but with distance (mi) as the independent variable and time (hr) as the dependent variable. b. What is the constant of proportionalit for this new graph? How does it compare to the original graph? Explain. Copright 015 Pearson Education, Inc. 1

15 LESSON 5: A STRAIGHT LINE GRAPH 1. Which graph shows a proportional relationship? A B x x C D x x Copright 015 Pearson Education, Inc. 19

16 LESSON 5: A STRAIGHT LINE GRAPH. Which of these statements is true about proportional relationships? There ma be more than one true statement. A A proportional relationship can be expressed b the formula x = k. B A proportional relationship alwas includes the point (0, 0). C If a graph is a straight line that passes through the origin, then the graph does not represent a proportional relationship. D If the ratios between the varing quantities are not constant, then the relationship between the quantities is proportional. E The graph of a proportional relationship will alwas be a straight line that passes through the origin. 3. Consider this equation. = 1 x a. Draw a graph of the equation. b. Does this graph represent a proportional relationship? Explain wh or wh not.. The table represents a proportional relationship. a. Complete the table. x b. Then draw a graph of the data in the table. 5. This table represents a proportional relationship with a constant of proportionalit of 3. x Explain wh the constant of proportionalit is alwas equal to the value of the dependent variable when the independent variable is equal to 1. Copright 015 Pearson Education, Inc. 0

17 LESSON 5: A STRAIGHT LINE GRAPH. Suppose that a stack of identical coins has a height of 1 in. 1 in. The graph showing the relationship between the number of coins to the height of the stack a straight line that passes through the origin. A is B is not 7. This stack of identical coins has a height of 1 in. Write a formula for the number of coins in terms of the height of the stack. 1 in.. Consider these two stacks of coins. For one tpe of coin, a stack of coins has a height of 1 in. For a different tpe of coin, a stack of coins has a height of 1.5 in. 1 in. 1.5 in. Stack A Stack B How would a formula for Stack B for the number of coins in terms of the height of the stack differ from the formula for Stack A? Explain. Copright 015 Pearson Education, Inc. 1

18 LESSON 5: A STRAIGHT LINE GRAPH 9. Which of these graph(s) represents a proportional relationship? (There ma be more than one graph.) Explain how ou know Graph A Graph B x 1 3 x 1 Graph C 7 Graph D x x Challenge Problem. When one variable in a proportional relationship is equal to zero, the other variable must also equal zero. Explain wh. Copright 015 Pearson Education, Inc.

19 LESSON : RATES AND PROPORTIONS 1. Karen bought 1 can of peaches for $. How man cans of peaches could she bu for $? Karen could bu cans of peaches for $.. Jack read 0 pages of a book in 50 min. How man pages could he read in 75 min at the same rate? Jack could read pages in 75 min. 3. Maa worked for 9 months and earned 1 das of vacation time. How man vacation das will she have earned after working exactl ears? Maa will have earned das of vacation after working exactl ears.. Luc used 3 gal of paint to cover 900 ft. She needs to cover a total of 1,0 ft. How man more gallons of paint does she need? Show our work. 5. Marcus has shirts for ever 5 pairs of jeans in his closet. How man pairs of jeans does he have if he has 0 shirts? Show our work.. The scale on a map is given b 1 in. = 5 mi. What is the distance between two towns that are.5 in. apart on the map? Show our work. Copright 015 Pearson Education, Inc. 3

20 LESSON : RATES AND PROPORTIONS 7. This problem appeared on the unit assessment in Sophie s class. Biologists studing the local nature reserve have recorded that for ever 5 square feet of grassland there are 7 frogs. If the nature reserve has a total of 3,50 square feet of grassland, how man total frogs are there? If Sophie decides to solve the problem b setting up a proportion and solving for the missing value, which of these equations could she use? A 5 x = 7 3, 50 B 7 frogs x = 5 ft 3, 50 ft C number offrogs = D = 7x 3, Suppose there are large art tables for ever 30 students in Ms. Muñoz s after-school art class. a. What is the constant of proportionalit for the relationship of art tables to students? Write our answer as a fraction in simplest form. b. What is the unit rate of students per art table? Express our rate as a decimal rounded to two decimal places. : c. What is the unit rate of art tables per student? Express our rate as a decimal rounded to three decimal places. Challenge Problem : 9. Maa built a robot that can evaluate math equations. Her robot takes.5 min to evaluate 57 equations. How man equations can the robot evaluate in 1 hr? a. Find the number of equations the robot can evaluate in 1 hr using two different methods. Show our work for each method. b. What is similar and what is different between the two methods ou selected? Copright 015 Pearson Education, Inc.

21 LESSON 7: UNIT RATES 1. Sophie can swim laps in 5 min. What is the unit rate of laps per minute? laps per minute. Karen noticed while pumping gas that it took 1.5 min to pump 3. gal of gas. What is the unit rate in gallons per minute? 3. Jack spends $1.31 on 1 lb of carrot raisin salad. What is the price per pound of 3 carrot raisin salad? The price of carrot raisin salad is $ per pound.. Marcus drank 1 gal of juice in 1 min. Suppose he drank at a constant rate. How much juice did he drink in 1 min? 5. Luc and Jack participated in a fundraising walk-a-thon held on the track at school. Each lap of the track is 1 of a mile. During the event, the raised $0 for each lap the completed. Which of these formulas could be used to represent the relationship between mone raised and laps completed? There are two correct formulas. A = 0x B 0 = 5x C x = 1 0 D = 1 x E = 1 00 x. It takes Maa 0 min to walk to school. Her school is 3 What is Maa s walking speed in miles per hour? mi from her house. 7. If it takes Maa 0 min to walk to school, which is 3 could Maa walk in 1 hr? mi from her house, how far Copright 015 Pearson Education, Inc. 5

22 LESSON 7: UNIT RATES. Maa s school is 3 mi from her house. It takes her 0 min to walk to school. Write a formula to describe the distance,, Maa can walk in an amount of time, x. 9. Maa s school is 3 mi from her house. It takes her 0 min to walk to school. Sketch a graph of the relationship.. Maa s school is 3 mi from her house. It takes her 0 min to walk to school. What is the constant of proportionalit of this relationship? 11. The owners of Nift Mart advertised that the would not change their unit price of gas for one week. Karen wanted to make sure that the kept their promise, so each da she recorded one gas purchase. Da Volume of Gas (gal) Price ($) Monda Tuesda Wednesda 11.0 Thursda 1.0 Frida Unfortunatel, Karen lost some of her data. Fill in the missing values in the table, assuming the price of gas is constant. 1. The owners of Nift Mart advertised that the would not change their unit price of gas for one week. Karen wanted to make sure that the kept their promise, so each da she recorded one gas purchase. Da Volume of Gas (gal) Price ($) Monda Tuesda Wednesda 11.0 Thursda 1.0 Frida Write a formula describing the price,, in terms of the number of gallons, x. Copright 015 Pearson Education, Inc.

23 LESSON 7: UNIT RATES Challenge Problem 13. At the farmer s market, the price for tomatoes is $1.50 for lb. a. Write a formula that shows the proportional relationship between cost and weight for tomatoes at the farmer s market. b. Use the formula to determine the cost of lb of tomatoes. c. Set up an equation as a proportion using lb of tomatoes to determine the cost. d. Which method do ou prefer? Explain. Copright 015 Pearson Education, Inc. 7

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25 LESSON : NON-PROPORTIONAL RELATIONSHIPS 1. Which of these relationships shows a proportional relationship? There ma be more than one proportional relationship. A A wind turbine functioning at a consistent rate will generate 500 kilowatts of power each hour. B x C Time Earnings $1.75 $.5 $.50 $11.00 $ D = x E x. A muffin recipe calls for 1 c of mashed bananas per 1 3 c of oats. In this recipe, the amount of bananas used proportional to the amount of oats. A is B is not Copright 015 Pearson Education, Inc. 9

26 LESSON : NON-PROPORTIONAL RELATIONSHIPS 3. Consider this equation. =. x 3 The equation describes a A proportional B non-proportional relationship.. Decide whether this scenario describes a proportional or non-proportional relationship. Explain wh. A 1 in. candle burns at a constant rate of 3 in. per hour. Is the height of the candle proportional to the number of hours the candle burns? 5. Decide whether this graph represents a proportional or non-proportional relationship. Explain wh. 5 x. An amusement park offers a $30 summer pass that allows customers to enter the park as man times as the like between the months of June and August for just $ per visit. Is the amount a customer spends on admission fees proportional to the number of times he or she visits the park? 7. Consider the relationships shown in each equation. Determine if each equation is showing a proportional or non-proportional relationship. = 1 x 1 p = l C = πr p 3 = x = x 1 Copright 015 Pearson Education, Inc. 30

27 LESSON : NON-PROPORTIONAL RELATIONSHIPS. Consider this non-proportional scenario. Jack went to the local arcade. He paid the student admission fee of $5. Then he plaed 5 arcade games, paing $0.50 per game. He spent a total of $ Explain how the scenario could be changed so that it represents a proportional relationship between the cost and the number of games plaed. Challenge Problem 9. Maa and her mom went to visit her grandmother, who lives in another part of the state. To get there, Maa s mom drove her van. Compare these two graphs, which both represent information about the trip Maa and her mom took. Gas (gal) Graph A x Distance (mi) Gas (gal) Graph B x Distance (mi) a. Graph A shows the relationship between the volume of gas remaining in the tank of the van and the number of miles Maa and her mom traveled. Describe the relationship that Graph B shows. b. Do these graphs show proportional or non-proportional relationships? c. Wh do these graphs look so different? Copright 015 Pearson Education, Inc. 31

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29 LESSON 9: USING PROPORTIONS 1. This graph represents the relationship between the volume of sand in a clindrical container and the height of the sand. 0 Volume (cm 3 ) 15 5 v = h V = h h Height (cm) 5 x Which of these statements correctl describes whether the points (, 7) and (, 1) lie on the graph? A Both the point (, 7) and the point (, 1) lie on the graph. B Neither the point (, 7) nor the point (, 1) lies on the graph. C The point (, 7) does lie on the graph. The point (, 1) does not lie on the graph. D The point (, 7) does not lie on the graph. The point (, 1) does lie on the graph.. This graph represents the relationship between the volume of sand in a clindrical container and the height of the sand. 0 Volume (cm 3 ) 15 5 v = h V = h h Height (cm) 5 x Explain what the plotted points (.5, ) and (3, 1) on the graph represent. Copright 015 Pearson Education, Inc. 33

30 LESSON 9: USING PROPORTIONS 3. This graph represents the relationship between the volume of sand in a clindrical container and the height of the sand. 0 Volume (cm 3 ) 15 5 v = h V = h h Height (cm) 5 x Explain wh the volume of sand is proportional to the height of the sand.. This graph represents the relationship between the volume of sand in a clindrical container and the height of the sand. 0 Volume (cm 3 ) 15 5 v = h V = h h Height (cm) 5 x What is the constant of proportionalit, also known as the unit rate? The constant of proportionalit, or unit rate, is. Copright 015 Pearson Education, Inc. 3

31 LESSON 9: USING PROPORTIONS 5. This graph represents the relationship between the volume of sand in a clindrical container and the height of the sand. 0 Volume (cm 3 ) 15 5 v = h V = h h Height (cm) 5 x Suppose the total volume of the container is 0 cm 3. What is the value of h? h = cm. This graph represents the relationship between the volume of sand in a clindrical container and the height of the sand. 0 Volume (cm 3 ) 15 5 v = h V = h h Height (cm) 5 x Can ou think of a different container in which the volume of the sand is not proportional to the height of the sand? Explain. Copright 015 Pearson Education, Inc. 35

32 LESSON 9: USING PROPORTIONS 7. A team of graduate students collected data to show the relationship between the number of hours a piece of wood is submerged (completel covered) in a certain chemical and the portion of the wood that is dissolved b the chemical. The results are shown in this graph. Portion of Wood Dissolved (%) x Time Wood Is Submerged (hr) Maa claims that this graph represents a proportional relationship between how long the wood is submerged and the percent of the wood dissolved. Is Maa correct?. This graph represents the relationship between the number of hours a piece of wood is submerged (completel covered) in a certain chemical and the percentage of the wood that is dissolved b the chemical. Portion of Wood Dissolved (%) x Time Wood Is Submerged (hr) What is the constant of proportionalit expressed as percent per hour? k = Copright 015 Pearson Education, Inc. 3

33 LESSON 9: USING PROPORTIONS 9. This graph represents the relationship between the number of hours a piece of wood is submerged (completel covered) in a certain chemical and the percentage of the wood that is dissolved b the chemical. Portion of Wood Dissolved (%) x Time Wood Is Submerged (hr) Write an equation to express the relationship. Use p for portion of wood dissolved in percent and t for time in hours. Challenge Problem. a. A relationship can have two variables and not be proportional. Explain wh. b. Write a definition of a proportional relationship in our own words. Copright 015 Pearson Education, Inc. 37

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35 LESSON : ANALYZING GRAPHS 1. This coordinate grid shows the graphs of five proportional relationships. 1 1 a b c d e 1 1 x Which of these statements is true? There ma be more than one true statement. A The constant of proportionalit for line c is less than the constant of proportionalit for line d. B The constant of proportionalit for line d is greater than the constant of proportionalit for line c. C The constant of proportionalit for line b is greater than the constant of proportionalit for line c. D The constant of proportionalit for line e is less than the constant of proportionalit for line d. E The constant of proportionalit for line a is less than the constant of proportionalit for line b. Copright 015 Pearson Education, Inc. 39

36 LESSON : ANALYZING GRAPHS. Compare these graphs, both of which represent proportional relationships Graph B Graph A x 5 15 x Which graph shows a relationship with a greater constant of proportionalit? 3. Consider these graphs of proportional relationships Graph B Graph A x 5 15 x Approximate, as closel as possible, the constant of proportionalit for each graph. a. Graph A: k = b. Graph B: k = Copright 015 Pearson Education, Inc. 0

37 LESSON : ANALYZING GRAPHS. Consider this graph x Make a table that lists at least three pairs of values that define the points on the graph. 5. Consider this graph x Write a formula for the line shown in the graph. Copright 015 Pearson Education, Inc. 1

38 LESSON : ANALYZING GRAPHS. Mr. Valdez bought lumber to remodel his restaurant. He made a stack of 5 pieces of plwood. Sophie measured the height of the stack and found that it was in. Mr. Valdez also made a stack of 5 two-b-fours. Sophie measured the height of this stack as well and found that it was in. In both cases, there is a proportional relationship between the height of the stack and the amount of lumber. Here is a graph of these two proportional relationships. 1 1 Height (in.) A B 5 15 x Number of Pieces Which line represents the plwood? represents the plwood. 7. Sophie measured the height of the stacks of plwood and two-b-fours Mr. Valdez bought to remodel his restaurant. Here are her results. Plwood: 5 pieces are in. high Two-b-fours: 5 pieces are in. high Find the constant of proportionalit for the plwood. Write an appropriate formula for the height, h, in terms of the number of pieces, n. Copright 015 Pearson Education, Inc.

39 LESSON : ANALYZING GRAPHS. Sophie measured the height of the stacks of plwood and two-b-fours Mr. Valdez bought to remodel his restaurant. Here are her results. Plwood: 5 pieces are in. high Two-b-fours: 5 pieces are in. high Find the constant of proportionalit for the two-b-fours. Write an appropriate formula for the height, h, in terms of the number of pieces, n. 9. Here is a graph of the measurements Sophie took of the materials Mr. Valdez bought to remodel his restaurant. Sophie found the height of a stack of 5 sheets of plwood to be in. and a stack of 5 two-b-fours to be in. 1 Height (in.) 1 A B 5 15 Number of Pieces x Mr. Valdez also bought 5 sheets of drwall. Sophie measured the height of this stack and found that it was.5 in. Which of these statements describes where a graph of the proportional relationship of the drwall would be located in the coordinate grid? A The drwall graph would be between line A and line B. B The drwall graph would be steeper than line A; it would be between line A and the -axis. C The drwall graph would not be as steep as line B; it would be between line B and the x-axis. D The drwall graph would have the same steepness as line B; it would be in exactl the same location as line B. Copright 015 Pearson Education, Inc. 3

40 LESSON : ANALYZING GRAPHS Challenge Problem. Both of these graphs represent the relationship between gallons of gas and miles traveled. Distance (mi) d g Volume of Gas (gal) Volume of Gas (gal) g d Distance (mi) a. Do these graphs represent the same data? Explain how ou know. b. Compute the constant of proportionalit in each case. What do ou notice? Copright 015 Pearson Education, Inc.

41 LESSON 11: PUTTING IT TOGETHER 1 1. Look back at the notes ou took during this unit, and think about the strategies ou have developed for working on problems involving proportional and non-proportional relationships. Write three things ou have learned about proportional relationships.. Consider the differences between proportional and non-proportional relationships. Create a graphic organizer to show these differences and explain how to identif whether a relationship is proportional or not. Organize our visual in a wa that will allow ou to use it as a reference throughout the rest of the school ear. Use a chart similar to the one shown. Start b thinking of two real-world situations, one that represents a proportional relationship and one that represents a nonproportional relationship. In the Word Problem row, describe the situations. Then complete the rest of our graphic organizer based on the two situations. Word Problem Table Graph Equation Proportional Relationship Non-Proportional Relationship 3. Read our work on the Self Check. What would ou do differentl if ou were starting the Self Check task now?. During this unit so far, ou have examined different methods for solving problems involving proportional relationships. For example, ou have solved proportional relationship problems b: Setting up a proportion and solving for the missing value Finding the unit rate and multipling Writing and solving a formula using the constant of proportionalit Has the method ou now prefer to use changed from the method ou preferred earlier in the unit? If so, wh has our preferred method changed? If not, wh do ou still prefer our initial method? 5. Complete an exercises from earlier lessons that ou have not finished. Copright 015 Pearson Education, Inc. 5

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43 LESSON 15: CONNECTION TO PERCENT 1. Maa is doing some back-to-school shopping. She purchased a backpack for $35. The sales tax is 7%. What is the total amount Maa paid for the backpack, including the sales tax? $. Another item Maa purchased while back-to-school shopping is a pair of shoes. The price of the shoes was $9. The sales tax is 7%. What is the total amount Maa paid for the pair of shoes, including the sales tax? $ 3. Jack is also doing some back-to-school shopping. He purchased a box of pencils. The price was $3 and the sales tax is 7%. Write and evaluate an equation to find the total amount Jack paid for the pencils.. While doing some back-to-school shopping, Jack wanted to figure out whether he could afford a calculator. The price was $15. The sales tax is 7%. Write and evaluate an equation to find the total amount Jack would pa for the calculator. 5. What is 0% of 50? A B 15 C 30 D 0. Write a formula to determine 0% of an amount. 7. Consider the formula to determine 0% of an amount, x. Make a graph that shows this relationship for amounts from Copright 015 Pearson Education, Inc. 7

44 LESSON 15: CONNECTION TO PERCENT. What is 0% of $1.09 to the nearest cent? A $0.31 B $3.1 C $3. D $ Consider a situation in which sales tax is 0%. Write a formula to determine the total cost of an amount with 0% sales tax.. Consider the formula to determine the total cost of an price, x, and a constant sales tax of 0%. Make a graph that shows this relationship for dollar amounts from $ to $0. Challenge Problem 11. You wrote and evaluated equations to determine the total amount Jack paid for some back-to-school items. Box of pencils: price is $3; sales tax is 7% Calculator: price is $15; sales tax is 7% Explain how ou could use the distributive propert to help simplif the total cost formulas ou wrote about this situation. Copright 015 Pearson Education, Inc.

45 LESSON 1: MORE ABOUT SALES TAX 1. Sophie bought a present for her friend s birthda. The price of the present was $15. The sales tax in her cit is.75%. Which of these formulas could be used to determine the total cost, t, of the present? There ma be more than one correct formula. A t B t C 75. t = D t E t. Jack is figuring out how much it would cost him to purchase a new skateboard. The price of the skateboard is $ The sales tax is 7%. Which table correctl represents this situation? A Starting Amount Percent Change Final Amount $ % increase $1.33 B Starting Amount Percent Change Final Amount $ % decrease $17.7 C Starting Amount Percent Change Final Amount $ % increase $13.30 D Starting Amount Percent Change Final Amount $ % increase $0.33 Copright 015 Pearson Education, Inc. 9

46 LESSON 1: MORE ABOUT SALES TAX 3. Maa would like to purchase some pens. Each pen costs $0.75. The sales tax is %. Which graph correctl represents the proportional relationship in this situation? A Cost of Pens 1 B Cost of Pens 1 Total Cost ($) 1 Total Cost ($) Number of Pens x 5 15 Number of Pens x C Cost of Pens D Cost of Pens Total Cost ($) Total Cost ($) 5 15 Number of Pens x 5 15 Number of Pens x. Marcus bought a new smart phone for $5.0. The price of the smart phone before sales tax was $0. Complete the table to show the percent change in terms of an increase or decrease. Starting Amount Percent Change Final Amount $0 $ What formula would ou use to find the sales tax if ou know the prices of an item before and after sales tax? Explain. Copright 015 Pearson Education, Inc. 50

47 LESSON 1: MORE ABOUT SALES TAX. Mrs. Luna is a driving instructor. On average, 75% of her students pass their driving test on the first tr. How man students should pass on the first tr if Mrs. Luna has 3 students? Complete the table. Starting Amount Percent Change Final Amount 7. On average, 75% of the students in Mrs. Luna s driving instruction course pass their driving test on the first tr. What formula would ou use to find the number of students in a particular class who pass the driving test on their first tr? Be sure to define each variable in our formula.. In Mrs. Luna s driving instruction course, an average of 75% of her students pass their driving test on the first tr. Make a graph that shows the formula for class sizes of 0 0 students. Click in the coordinate grid to add a point. 9. Luc s step-brother owns a coffee shop. He estimates that 5% of his customers order regular coffee and the rest order decaffeinated coffee. What formula could Luc use to determine the number of customers who will order decaffeinated coffee on an given morning? Explain.. An estimated 5% of the customers at Luc s step-brother s coffee shop order regular coffee. The rest of the customers order decaffeinated coffee. Make a graph that shows the formula for the relationship of the number of customers ordering decaffeinated coffee to the total number of customers. Show the formula for 0 to 00 total customers. Click in the coordinate grid to add a point. Challenge Problem 11. Consider Luc s step-brother s coffee shop. An estimated 5% of the customers order regular coffee and the rest order decaffeinated coffee. a. Explain how ou would use a formula to determine the total number of customers Luc s step-brother has if he sells 5 cups of decaffeinated coffee. b. Describe how ou would use a graph to determine the total number of customers Luc s step-brother has if he sells 5 cups of decaffeinated coffee. Copright 015 Pearson Education, Inc. 51

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49 LESSON 17: PERCENT INCREASE 1. Find 55 increased b 30%.. A number increased b 1% is. What is the original number? The original number is. 3. The average length of a business phone call changed from 30 sec to 5 sec. What percent increase is this? Write our response as a complete sentence.. The price of a pair of socks changed from $7.50 to $.0. What percent increase is this? Write our response as a complete sentence. 5. Mrs. Abir s school raised $,50 during last ear s annual fundraising campaign. This ear s target amount is 15% higher than the amount raised last ear. What is this ear s target amount? Write our response as a complete sentence and show our work.. Karen ordered lemon chicken from her favorite Chinese restaurant. She paid $9.00 for a bill of $7.77. What percent tip did Karen pa? Round the percent to the nearest tenth. Karen paid about a % tip. 7. After lunch at her favorite Chinese restaurant, Karen wants to tip her server 0% for his service. What should Karen multipl her bill b to determine the tip she wants to leave? A 0.0 B 1.0 C 1.0 D 0.0 Copright 015 Pearson Education, Inc. 53

50 LESSON 17: PERCENT INCREASE. After lunch at her favorite Chinese restaurant, Karen wants to tip her server 0% for his service. What should Karen multipl her bill b to determine the total amount, including the tip? A 0.0 B 1.0 C 1.0 D The minimum wage in San Francisco was increased to $.55 per hour in 013. This amount is 31.75% greater than the minimum wage in the state of California. What is the minimum wage in California? Write our response as a complete sentence. Challenge Problem. This is a check for a meal shared b ou and three other people. You owe 1.5 (British pounds). Tip: Total: Split Check: Per Contributor: 15% How much of this amount pas for the tip? Explain. Copright 015 Pearson Education, Inc. 5

51 LESSON 1: PERCENT DECREASE 1. Sophie joined a running group to prepare for a 5 mi race. After two months, her time had dropped from 0 min to 3 min. What is the percent decrease? A The decrease is about 9.5%. B The decrease is about 0.05%. C The decrease is about 0.95%. D The decrease is about 5%.. Find 75 decreased b 5%. 3. A number decreased b 1% is 3. What is the original number? The original number is.. Maa bought two boxes of cereal, each marked 0% off. Therefore, she saved % altogether. A % B 0% C 0% 5. A fitness club manager reduces the prices b 50% during a spring sale. The manager raises the prices b 0% after the sale. Therefore, after the sale the prices are the prices before the sale. A the same as B higher than C lower than. Karen answered 0% of the questions on her first quiz correctl and 30% of the questions on her second quiz correctl. Both quizzes had the same number of questions, so for both quizzes she answered of all the quiz questions correctl. A less than 50% B 50% C more than 50% 7. Jack starts a new job making sandwiches at a baker. Soon the amount of time it takes for him to make a sandwich changes from 50 sec to 15 sec. What is the percent decrease? Write our response as a complete sentence.. Maa fills a bucket with 51 kg of sand. The weight of the sand changes to 37.5 kg after she uses some to build a sand castle. What percent decrease is this? Write our response as a complete sentence. Copright 015 Pearson Education, Inc. 55

52 LESSON 1: PERCENT DECREASE 9. A reccling truck makes two separate runs on its dail collection route. It collects 1 tons of recclable materials on its first run, which is 15% more than the amount it collects on the second run. How much reccling does the truck collect on its second run? Write our answer as a decimal to one place. The truck collects tons of recclable materials on its second run.. A dress that normall costs $.00 is on sale for 5% off the original price. What is the sale price? Write our response as a complete sentence. 11. The length of a min video is decreased b 30% when it is edited. What is the length of the edited video? Express our answer in minutes and seconds. The edited version is min and sec. 1. Sophie bought a pair of skates for $03 during a summer sale. What is the original price of the skates if the were on sale for 35% off? Write our response as a complete sentence. 13. Marcus earned 59 out of 0 possible points on his biolog exam. His teacher tells him that his score is % below the class average. What is the class average? Challenge Problem 1. Each spring a clothing store has a sale: 0% off all winter items. An winter items that remain after one month are marked an additional 0% off the first sale price. a. A coat originall sold for $00. What is the price of the coat if it remains after one month? b. Write an expression to find the price of an item after both discounts. c. For an item after both discounts have been applied, what is the percent off the original price? Copright 015 Pearson Education, Inc. 5

53 LESSON 19: MISTAKES WITH PERCENTS 1. The students in Ms. Siboani s class are studing nutrition labels on food. The are discussing the calories and calories from fat in 1 serving of a snack the all like. The have highlighted this information in pink on the label. Nutritional Facts Serving Size 1 cup (g) Serving per Container Amount per Serving Calories 0 Calories from Fat % Dail Value* Total Fat 13 g 0% Saturated Fat 5g 5% Trans Fat g Cholesterol 30mg % Sodium 0mg % Total Carbohdrate 31g % Dietar Fiber 0g 0% Sugars 5g Protein 5g Vitamin A % Vitamin C % Calcium 15% Iron % *Percent Dail Values and based ona,000 calorie diet. Your Dail Values ma be higher or lower depending on our calorie needs. Which of these students have reasoned correctl? There ma be more than one student who is correct. A Luc sas, If ou eat 1 serving, ou are eating 0 calories. Of those calories, calories are from fat. So, the percent of calories from fat is 0, or about %. B Marcus sas, No matter how man servings ou have, the ratio of calories from fat to total calories stas the same: : 0 or : 13. C Karen sas, In 1 serving, calories from fat () is approximatel % of the calories (0). So, if ou eat 1 of a serving, the percentage of calories from fat would be 50% of %, which is 3%. D Sophie sas, If ou reduce the serving size b 50%, ou reduce the calories from fat b 50%. E Jack sas, If ou eat 1 serving, ou are eating 0 calories. The percent of fat that ou eat is, + or 31.5%. 0 Copright 015 Pearson Education, Inc. 57

54 LESSON 19: MISTAKES WITH PERCENTS. The students in Ms. Siboani s class are studing nutrition labels on food. The are discussing the total fat in 1 serving of a snack the all like. The have highlighted this information in pink on the label. Nutritional Facts Serving Size 1 cup (g) Serving per Container Amount per Serving Calories 0 Calories from Fat % Dail Value* Total Fat 13 g 0% Saturated Fat 5g 5% Trans Fat g Cholesterol 30mg % Sodium 0mg % Total Carbohdrate 31g % Dietar Fiber 0g 0% Sugars 5g Protein 5g Vitamin A % Vitamin C % Calcium 15% Iron % *Percent Dail Values and based ona,000 calorie diet. Your Dail Values ma be higher or lower depending on our calorie needs. Which of these students has reasoned correctl? A Sophie sas, If ou eat 1 serving, ou are getting 0% of our dail value of total B Jack sas, If ou eat 3 of a serving, instead of getting 0% of our dail value of total fat ou would get 15% of our dail value, because 15% is 75% of 0%. C Luc sas, If ou eat 1 1 value of total fat. servings of the snack, ou are getting 0% of our dail D Maa sas, If ou reduce the serving size b %, the formula for the percent dail value of total fat would be Copright 015 Pearson Education, Inc. 5

55 LESSON 19: MISTAKES WITH PERCENTS 3. The students in Ms. Siboani s class are studing nutrition labels on food. The are discussing the sodium in 1 serving of a snack the all like. The have highlighted this information in pink on the label. Nutritional Facts Serving Size 1 cup (g) Serving per Container Amount per Serving Calories 0 Calories from Fat % Dail Value* Total Fat 13 g 0% Saturated Fat 5g 5% Trans Fat g Cholesterol 30mg % Sodium 0mg % Total Carbohdrate 31g % Dietar Fiber 0g 0% Sugars 5g Protein 5g Vitamin A % Vitamin C % Calcium 15% Iron % *Percent Dail Values and based ona,000 calorie diet. Your Dail Values ma be higher or lower depending on our calorie needs. Which of these students has reasoned correctl? A Maa sas, If ou eat 1 serving, ou are getting % of our dail value of sodium. If ou eat 1 1 servings, ou are still getting % of our dail value of sodium. B Karen sas, If ou eat 3 of a serving, instead of getting % of our dail value of sodium ou would get 75% of %. C Marcus sas, If ou eat servings of the snack, ou would get less than 0% of our dail value of sodium. D Jack sas, If ou eat servings, the formula for the percent dail value of sodium would be. An athletic apparel store is going out of business and needs to sell its entire remaining inventor. Ever item in the store is 75% off the usual price. Sophie wants to bu a new pair of running shoes that usuall cost $0. What is the sale price of the shoes? The sale price of the shoes is $. Copright 015 Pearson Education, Inc. 59

56 LESSON 19: MISTAKES WITH PERCENTS 5. Sophie bought a new pair of running shoes that usuall cost $0. She paid 75% off the usual price. Jack sas, Wow, Sophie, that s a great deal. You onl have to pa three-quarters of the original price! Is Jack correct or incorrect? Explain.. Sophie is talking with Jack about the new pair of running shoes she purchased at 75% off the usual price of $0. Sophie sas, The ratio of the sale price to the original price is 1 :. Is Sophie correct or incorrect? Explain. 7. Yesterda, % of the students watched hocke and 0% watched basketball. A student reporting the school news said that, therefore, 70% of the students did not watch either sport. What mistake in reasoning about percents did the student reporter make?. Karen answered 0% of the questions on her first math test correctl and 0% of the questions on her second math test correctl. Therefore, Karen sas, she answered 0% of the questions correctl. What mistake in reasoning about percents did Karen make? 9. On Blowout Sale Frida, the price of ever item in a store was reduced b 70%. The next da, prices were increased b 50%. Maa sas that the prices are now 0% lower than the were before Blowout Sale Frida. What mistake in reasoning about percents did Maa make? Copright 015 Pearson Education, Inc. 0

57 LESSON 19: MISTAKES WITH PERCENTS Challenge Problem. Luc wants to set $300 aside in a savings account. She has two options: set the mone aside at 3% interest for ears, or set the mone aside at % interest for 3 ears. Luc created tables to help her compare the options and determine which option will result in the highest paoff. Then she graphed the table data. Year Starting Amount Plan 1 Percent Change Final Amount Total Gain 1 $ % increase $ $9.00 $ % increase $31.7 $1.7 Year Starting Amount Plan Percent Change Final Amount Total Gain 1 $ % increase $30.00 $.00 $30.00 % increase $31.1 $1.1 3 $31.1 % increase $31.3 $1.3 Comparing Two Different Savings Plans 0 Totlal Gained ($) x Year Luc is confused. She sas, I thought the two plans would give me the same final amount. Since the graph shows these are proportional relationships and since both have a total increase of % from the original amount of $300, I must have done something wrong. Mabe I made an error with m calculations. Explain the mistake(s) in reasoning about percents that Luc made. Copright 015 Pearson Education, Inc. 1

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59 LESSON 0: PUTTING IT TOGETHER PART 1. Which of these decimal multipliers is the same as a % increase? A B. C 1. D 1.0. Which of these fraction multipliers are the same as a % decrease? There ma be more than one equivalent fraction multiplier. A 9 B 1 3 C 3 D E 5 3. Consider the change from the dollar amount of $150 to the dollar amount of $10. $150 $10 Which of these percents, fractions, or decimals indicate the correct increase or decrease between the amounts of mone? There ma be more than one correct value. A % increase B 1. C 0% increase D 1 3 E 5 Copright 015 Pearson Education, Inc. 3

60 LESSON 0: PUTTING IT TOGETHER PART. In a bookstore, 1.5% of all the books sold are hardcover. The bookstore sold books one week. How man of those books were hardcover? books 5. In a bookstore, 1.5% of all the books sold are hardcover. What would ou multipl the amount of hardcover books b to find the total number of books sold? A B C D Maa participated in a km run. She planned to run at a constant speed. She was hoping to run 9.5 min kilometers, which means it takes 9.5 min to run 1 km. Here is the table of her actual run times. Distance (km) Time (min) a. For how man miles did she run at that constant speed? mi b. Write an equation representing the proportional relationship of 9.5 minutes (t) per kilometer (d). 7. Maa participated in a km run. She planned to run at a constant speed. She was hoping to run 9.5 min kilometers, which means it takes 9.5 min to run 1 km. Here is the table of her actual run times. Distance (km) Time (min) The table indicates an increase of time from 9.5 min/km at the beginning of the run to.1 min/km at the end. What is the percent of this increase? Round to a whole percent. % increase Copright 015 Pearson Education, Inc.

61 LESSON 0: PUTTING IT TOGETHER PART. Maa participated in a km run. She planned to run at a constant speed. She was hoping to run 9.5 min kilometers, which means it takes 9.5 min to run 1 km. Here is the table of her actual run times. Distance (km) Time (min) What is the meaning of the point (, 95) in the context of Maa s run? 9. A store manager reduces the price of a tool kit that normall costs $5.00 down to $17.50 during a sale. However, demand for the tool kit increases when the students at the local high school begin a building project, and the manager decides to raise the price back to $5.00. Find the percent of price decrease from $5.00 to $ Write our answer as a whole percent. % decrease. A store manager reduces the price of a tool kit that normall costs $5.00 down to $17.50 during a sale. However, demand for the tool kit increases when the students at the local high school begin a building project, and the manager decides to raise the price back to $5.00. Find the percent of price increase from $17.50 to $5.00. Write our answer as a percent to one decimal place. % increase 11. A store manager reduces the price of a tool kit that normall costs $5.00 down to $17.50 during a sale. However, demand for the tool kit increases when the students at the local high school begin a building project, and the manager decides to raise the price back to $5.00. Wh is the percent increase different from the percent decrease? Explain. Challenge Problem 1. Consider this multiplication equation, where r is the product, n is a number, and p is a percentage. r = p n When is r > n? When is r < n? Explain. Copright 015 Pearson Education, Inc. 5

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63 LESSON 1: PUTTING IT TOGETHER PART 3 1. Read our Self Check and think about our other work about percents in this unit. Write down three things ou have learned. Share our work with a classmate. Does our classmate understand what ou wrote?. In the second part of this unit, ou explored man was that proportional relationships are used in real-world situations such as percent problems. For example, ou looked at how sales tax is calculated from the price of an item. Describe a situation from our everda life in which ou used proportional relationships. (If ou have never before used proportional relationships in our everda life, consider some of the examples from the unit. Imagine a real-world situation in which ou would need to use proportional relationships; describe that imagined situation.)write what ou have learned. 3. In the unit, ou discovered that people often make mistakes with percents. Explain a strateg or method ou like to use to double-check our work and ensure ou have not made a mistake in our reasoning about percents.. Review the notes ou took during the lessons about proportional relationships, percents, and ratios. Add an additional ideas ou have about these topics to our notes. If ou are still confused about something, be sure to research our questions and add more information to our notes. Ask for help from a classmate, review the related lessons, look at the resources in the Concept Corner, talk with our teacher, and so on, to help ou clarif areas of confusion. 5. Complete an exercises from this unit that ou have not finished. Copright 015 Pearson Education, Inc. 7

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