Fractions and Decimals

Transcription

1 estimate the sum or difference of fractions multiply mentally a fraction by a whole number and vice versa rename fractions and mixed numbers to decimal numbers rename repeating decimals to fractions compare and order fractions, mixed numbers, and decimal numbers use estimation to justify the reasonableness of results when calculating with decimal numbers use mental math when working with decimal numbers solve problems involving whole numbers and/or decimals apply the order of operations Key Words benchmark unit fraction repeating decimal period length of the period terminating decimal

2 Fractions and Decimals Do you like amusement parks? The rides can be terrifying and exciting at the same time. On a hot day, there is nothing more refreshing than splashing around in a water park. In the food court, you can satisfy the raging appetite you have built up while playing and walking around. How are fractions and decimals useful in an amusement park? Chapter Problem In this chapter, you will go to Funderland! Funderland is a fictitious amusement park in Nova Scotia with rides, a water park, and a food court. Along the way, you will encounter and solve a number of problems involving fractions and decimals. Food Court Water Park Wild Ride Sling Shot Mind Warp Cannonball

3 A N N A N N A A Understand Fractions Fractions are a way to compare a part of a whole. This means out of equal parts. You can represent fractions visually as equal partitions of a region, a set, or a line segment. The whole is the black Fraction Factory piece. The whole is the entire circle. The whole is the entire grid. The whole is the entire line segment. A whole can be a unit or a set of things. The whole is the two circles in the box. The whole is the set of eggs.. Identify or estimate the fraction represented by each diagram. Assume that one complete figure represents a whole, or. a) b) c) d). Identify or estimate the fraction represented by each diagram. Assume that one complete set of figures represents a whole, or. a) b) c) d) A D A cents 00 A D A cents 00 A D A cents 00 A D A cents 00 C C C C MHR Chapter

4 . Make two different representations for each fraction one representation in which the unit is a whole, and another in which the unit is a set of things. a) b) c) d) e) f) 0 6 Mixed Numbers and Improper Fractions You can represent a mixed number as an improper fraction and vice versa Write as a mixed number. a) b) c) 8. Write as an improper fraction. a) b) c) Compare Numbers Using a Number Line The number line is a tool you can use to compare and order numbers. The value of a number increases as you move to the right Place the following numbers on a number line. You may need to estimate or measure distances to place them. a) b) c) d) e) 0.7 f) 0. g) 0.9 h).8 i).9 7. Explain your reasons for placing the numbers where you did on the number line in question Place >, <, or between the two numbers to make a true statement. a) b) 8 c) d) 0.7 Get Ready MHR

5 Represent Decimals Decimal numbers are another way to represent parts of a whole. For example, 0. means. 0 You can represent this visually using base-0 materials. Let the large cube represent. 0. can be represented by: 0. can be represented by: 0.8 can be represented by: 9. What decimal is represented by each of these? a) b) c) d) 0. Using base-0 materials, draw a diagram to represent each decimal. Label your diagram with the decimal and write the decimal in words. a) 0.6 b) 0. c) 0.07 d) 0.8 e) 0.00 f) Use base-0 materials to represent each fraction. 7 a) b) c) d) e) f) MHR Chapter

6 Equivalent Fractions Fractions that name the same amount in different ways are called equivalent fractions. For example, In both circles, the same part of the whole circle is shaded, so. 6 In both Fraction Factory models, the same area of the whole Fraction Factory piece is covered, so. 6 The paper is folded into thirds. Two thirds are shaded. The same paper is folded into sixths. Four sixths are shaded. In both paper models, the same area of the whole paper is shaded, so. 6 Base-0 materials can also be used to model fractions Are the following fractions equal? Explain why or why not, using a diagram. a) and b) and 0 c) and d) and Write two equivalent fractions for each. 6 7 a) b) 8 0 c) d) Get Ready MHR 7

7 Focus on estimating the sum or difference of fractions Estimate Sums and Differences of Fractions Gilles is baking a batch of cookies for a school fundraising bake sale. The recipe calls for cups of flour. Gilles finds two bags of flour in the kitchen, each partially full. Will he have enough flour for his cookies? Materials Fraction Factory grid paper ruler coloured pencils Strategies What problem solving strategy are you using to check your estimate? How can I use benchmarks to estimate sums and differences of fractions? Part A: Estimate the Location of a Fraction on a Number Line Work with a partner. Each of you should pick a different coloured pencil.. Draw a number line. Mark the numbers 0 and exactly 8 cm apart.. For each of the following fractions: take turns estimating the position on the number line mark your position with your coloured pencil check using Fraction Factory pieces The person who was closer to the actual position wins a point. Example Locate. Mia s estimate is closer than Chris s. Mia wins a point. 8 cm 0 Estimates Chris Mia 0 Check: 8 MHR Chapter

8 Take turns going first. If your number line starts getting cluttered, draw a new one. If it is a tie, no one gets a point. a) b) c) 8 7 d) e) f) 6. Challenge round! This time, you take turns making up your own fractions. The person who picks the fraction marks their estimate first. Use these guidelines when picking fractions: use proper fractions only pick fractions with denominators of,,, 6, 8, or only The first person to earn 0 points is the winner.. Reflect Describe the strategies you used to estimate where you should place the fractions on the number line. Did your estimating skills improve as you played the game? Explain. Part B: Round Fractions to Benchmark Values To add or subtract fractions, it is useful to round to values that are easy to work with, such as 0,, and. These are called benchmarks.. Copy this number line into your notebook. 8 cm 0. a) Draw seven more number lines of length 8 cm directly under the first one. b) Divide your second number line into thirds and label the fractions and. c) Mark successive number lines with fourths, fifths, sixths, eighths, tenths, and twelfths respectively. (You may find it useful to think of 8 cm as 80 mm when you place the markings.) The Unit Fractions. a) Draw a light curve connecting all of the fractions on your number lines that have a numerator of. These are called the unit fractions. Each of these fractions is one piece of a whole. Which is closest to 0? Explain how you know. b) Look at the denominators of these fractions. What happens to the position on the number line as you increase the denominator? benchmarks numbers that are easy to work with to be used when ordering and comparing numbers Strategies Use a diagram. unit fraction a fraction with a numerator of. Estimate Sums and Differences of Fractions MHR 9

9 c) Predict which of these fractions would lie closest to 0 on the number line.,,, 99 8 The Nearly Ones. a) Draw a light curve connecting all of the fractions on your number lines that are closest to. You will notice that each is one space less than a whole. Which is closest to? Explain how you know. b) Look at the numerators and denominators of these fractions. What pattern do you see? What happens to the position on the number line as you increase both the numerator and the denominator by? c) How are they related to the unit fractions? d) Predict which of these lies closest to. Justify your answer. 8 99,,, 9 00 The Close-to-Halves. a) Use your ruler to draw a vertical line segment through the one-half point on all of your number lines. Write the fractions that the line segment passes through. What pattern do you notice? b) Which of these fractions are greater than? How do you know? 7,,, 0 8 c) What fraction would you write at the midpoint of these number lines? sixteenths, twentieths, hundredths, twenty-fourths d) Predict which of these fractions are less than. Justify your answer. 7 0,,,, e) If you were to write a name for the halfway point on the thirds and fifths number line, what would it be? (Hint: Use a decimal in the numerator.) What fraction would you write at the midpoint of these number lines? sevenths, elevenths, fifteenths, twenty-fifths f) Which of these fractions are greater than? 7 8,,,, Rewrite each as a mixed number. Arrange the numbers from least to greatest. 9, and,,,,. 60 MHR Chapter

10 7. Reflect a) Describe the problem that arises when you try to round to one of the benchmarks. b) Identify another fraction that has this problem. Explain the problem. 8. John draws a small downward arrow ( ) beside the fraction whenever he rounds down and a small upward arrow ( ) when he rounds up. In the following number statement, how would John likely round? Why? Part C: Use Benchmarks to Estimate Sums and Differences Gilles finds two bags of flour, already opened. He estimates that he has 7 of a cup in one bag and of a cup in the other. The recipe calls for 8 cups of flour.. a) Use benchmarks to estimate how much flour Gilles has altogether: round each fraction to the nearest benchmark add the rounded values b) Do you think Gilles has enough flour? Explain. Strategies What problem solving strategies did you use in questions,, and?. Use Fraction Factory pieces to check your estimate. Explain how you can do this.. Gilles mother finds him working in the kitchen and suggests that the flour is old and should be thrown out. She buys him a new package of flour to use instead. a) The new package contains cups of flour. How much flour will remain after Gilles bakes his cookies? Explain how you found your answer. b) Draw a diagram to illustrate the difference you calculated. Use Fraction Factory pieces or another model.. Reflect a) How do you think you can use benchmarks to estimate a sum of two fractions? Provide an example. b) How do you think you can use benchmarks to estimate a difference of two fractions? Provide an example.. Estimate Sums and Differences of Fractions MHR 6

11 Example : Estimate Sums of Fractions Gilles also needs some milk for his recipe. He finds two containers already opened. Estimate the total volume of milk. Solution Method : Add the wholes and fractions separately. 8 8 is the same as +. L Strategies Use the frontend strategy for addition. L 8 Method : Round mixed numbers, then add. is almost. 8 is the same as When I add all of these numbers together, I can change the order of the addends = I am applying the associative property of addition. I can use and as benchmarks. 0 Add the whole numbers: Add the fraction parts using benchmarks. 8 is almost. 8 is almost. So plus almost is almost. So almost almost. Gilles has almost L of milk. Add the fractional parts: is almost. 8 Add the sum of the whole parts and the fractional parts. almost almost Gilles has just under L of milk. 6 MHR Chapter

12 Example : Estimate Differences of Fractions Sheila and her brother share a 0 cc off-road bike. They agree to keep enough fuel in the bike to keep from using the unit of reserve fuel. The fuel tank holds units of gasoline including the reserve. Sheila uses 6 units 0 of gas to fill the fuel tank before she goes riding. Had her brother Jack brought the bike home using the reserve fuel? Solution Estimate the fuel level in the tank before Sheila filled it with gas. This is the difference between the full capacity and the amount she needed to add: 6. 0 Method : Use benchmarks. 6 is just over. 0 Method : Use a physical model. Let the black Fraction Factory piece represent unit of gas. The capacity of Sheila s tank is units: If I subtract, I get unit. = So, if I subtract just over, Sheila had to add 6 0 units. I get just under unit. 6 is just under. 0 Compare the two quantities. difference The difference appears to be about unit. Use a black Fraction Factory piece to check. difference is almost 6 just under. 0 The fuel tank contained just under unit of gas. Jack was using the reserve fuel.. Estimate Sums and Differences of Fractions MHR 6

13 . a) What are fraction benchmarks? b) Why are benchmarks useful?. Explain how you can use benchmarks to estimate each sum. Then find the estimated sum. a) b) c) Explain how you can use benchmarks to estimate each difference. Then find the estimated difference. a) b) c) Write two mixed numbers whose sum is close to but not exactly equal to.. Write two mixed numbers whose difference is close to but not exactly equal to. 6. How can you add? Are benchmarks useful in this case? If so, explain how you can use them. If not, explain a different method that works. 7. John had to estimate the answer to 6. He realized that 8 is halfway between 0 and and he reasoned that he could round up or down, depending on how he had rounded the other numbers in the question. How should he round in this case and why? 6 MHR Chapter

14 . Which benchmark a0,, or b is each fraction closest to? Use a diagram to explain. a) b) c) d) 6 8. Which benchmark a0,, or b is each fraction closest to? Use a diagram to explain. a) b) c) 9 d) e) f) 7. a) Is greater than or less than? 7 Use estimation to explain. b) Is greater than or less than? Use estimation to explain.. Is this sum greater than or less than? Explain how you know. 7. Pick one of the expressions from question 6 and explain how you estimated the sum. 8. Estimate each difference. 9 8 a) b) c) d) 8 e) f) Pick one of the expressions from question 8 and explain how you estimated the difference. 0. One day, Franz and Leah went travelling in their spaceship. They travelled 8 light-years to Hector s planet, and then light-years to Benny s planet. Approximately how far did they travel, in total? 6 8. Is this difference greater than or less than? Explain how you know Estimate each sum. 7 a) b) c) d) 6 e) f) Did You Know? A light-year is the distance that light can travel in one year. It takes about 8 min for the light of our sun, which is about km away, to reach Earth! How many times this distance can light travel in one year? Does this raise any questions about the possibility of Franz and Leah s journey?. Estimate Sums and Differences of Fractions MHR 6

15 . During one week, Roberto spent h practising the guitar, h playing hockey, and 8 h on the telephone. Estimate how much time he spent, in total, on all of these activities.. Danielle usually reads 6 pages of her current novel during breakfast. Today she woke up late and was able to read only pages while she ate. She read pages on the school bus. She read before school started, and reached 6 pages in total. Estimate how many pages she read before the start of class.. Write a story problem for this expression:. Estimate the answer to your 8 7 story problem. 6. One day, Hillary went to buy her cats favourite brand of cat food, Lipsmackers. Unfortunately the store was all out of Lipsmackers, and Hillary had to buy Hairball Deluxe instead. That evening, Hillary noticed that: Flipflop ate about of a bowl. Underfoot ate about of a bowl. 8 Skidmark ate about of a bowl. 6 Normally Hillary s cats all finish their supper. Their bowls are all the same size.. On their journey through space, Franz and Leah suddenly encountered an asteroid field. Franz blasted about of the 6 asteroids, and Leah blasted about of the asteroids. a) Who blasted more asteroids? How do you know? b) About what fraction of the asteroid field remains? Explain how you know.. a) Make up a problem involving a sum or difference of fractions in which the answer is close to but not equal to. b) Explain how you know the answer is close to. a) Which cat seemed most unhappy with the change in menu? Explain how you know. b) Which cat seemed to be the least bothered by the change? Explain how you know. c) About how many bowls of cat food did the cats eat in total? d) How much less is this total than their normal total dinner consumption? 66 MHR Chapter

16 Use this information to answer questions 7 and 8. The following partial park layout of Funderland shows four rides, a water park, and a food court. Estimated travel times by walking are shown. The average waiting time for each ride is about h. Each ride takes about h. h Food Court 8 h Water Park Wild Ride 6 h 8 h h h h Sling Shot h Mind Warp h Cannonball h Marie and Chantal are just getting into line at Slingshot. They want to ride Slingshot, and then go on two more rides before hitting the water park. They plan to splash around the water park for about h and then break for lunch at the food court. Chapter Problem 7. a) About how much time will they spend waiting in line and riding the two closest rides? b) About how much time will they spend walking from now until lunchtime? c) It is 9:00 A.M. Estimate when lunchtime will begin for Marie and Chantal. Extend 8. Refer to question 7. a) Describe the shortest possible path for Marie and Chantal to visit all four rides, starting at Slingshot, go to the water park, and then go to the food court. b) Estimate the total time to travel this path, assuming that they ride all the rides once. c) Explain how you found your estimate.. Estimate Sums and Differences of Fractions MHR 67

17 Focus on mentally multiplying a fraction by a whole number and vice versa comparing and ordering proper and improper fractions, mixed numbers, and decimal numbers Multiply a Whole Number by a Fraction and Vice Versa Did You Know? Biology is the study of living things, such as plants and animals. A biologist can specialize in many different areas, including plants, animals, the environment, health, and medicine. Some biology students go on to become doctors or research scientists. Materials coloured counters pattern blocks Strategies Can you suggest another way to model the situation other than using concrete materials? Nancy loves working with animals. She wants to become a biologist and specialize in animal behaviour. She is currently conducting an experiment with her hamster. Each trial takes between 0 minutes and half an hour. How can Nancy use her understanding of fraction operations to plan her work time wisely? To learn more about a career in animal behaviour studies, go to and follow the links. How can I multiply a fraction by a whole number, and vice versa? Part A: Multiply a Whole Number by a Fraction Nancy conducts a survey of her class and finds that picked science as their favourite subject. There are students in Nancy s class.. Use coloured counters to find the number of students who love science. a) Count out counters of the same colour. b) Divide the counters into three equal groups. c) One-third of the class picked science as their favourite subject. Change one of the groups to another colour to show this. d) How many students picked science as their favourite subject? Copy and complete this equation. 68 MHR Chapter

18 . Use this method to find the number of students who picked other subjects as their favourite. Write an equation for each. a) of the class picked physical education as their favourite subject. b) 6 of the class picked math as their favourite subject. Communicating Mathematically of means to multiply.. Look at your answers. Explain how you can use division to find the answers mentally.. Calculate mentally. a) b) 0 c) 8 6. Levi knows that 0. Explain how he could use this fact to help him mentally calculate Reflect Create an example and use it to explain how you can multiply a whole number by a fraction: a) using counters b) using mental calculation Part B: Multiply a Fraction by a Whole Number Suppose the hexagon block represents one unit: Hexagon Trapezoid Rhombus Triangle unit. Copy each shape into your notebook and write the fraction that each block represents.. Use pattern blocks to evaluate each multiplication. Write an improper fraction to show the number of parts you have brought together. Then write your answer as a whole number. a) b) 9 c) d) e) 6 6. Multiply a Whole Number by a Fraction and Vice Versa MHR 69

19 . Look at your answers to question. Explain how you can use improper fractions to find the answers mentally. Strategies Look for a pattern.. Calculate mentally. a) 6 b) c) 8 0. Reflect Create an example and use it to explain how you can multiply a fraction by a whole number: a) using pattern blocks b) using mental calculation 6. Reflect Compare the questions and answers in question above with those in question of Discover, part A, page 69. What is the difference in meaning between 6 and 6? What do you notice about the product in each case? Example : Multiply a Whole Number by a Fraction and Vice Versa Multiply. a) 0 b) 6 c) d) Solution a) 0 Method : Use a model. Suppose you have $0. How can you divide this amount evenly among four friends? How much money will each friend receive? Method : Use a fraction strip. You can show by shading in one-quarter of a rectangle: = Each friend will receive $. Therefore 0. To find what of 0 represents, divide the rectangle into 0 equal parts. How many parts are covered by the part? 70 MHR Chapter

20 b) To multiply a fraction by a whole number, you can think of repeatedly adding the fraction. 6 Strategies Use a concrete model or picture. Let the hexagon represent whole, so the trapezoid represents. 6 You can think of 6 as 6 halves or wholes. 6 0 When 6 trapezoids are placed over the hexagons, three complete hexagons are filled. c) You can model the product using Fraction Factory pieces. d) 9 I know that of is = or one set of stars. So, must be sets of stars or 9 stars. This is the same as saying: of is, so of must be three times as much. = 9. Multiply a Whole Number by a Fraction and Vice Versa MHR 7

21 Example : Estimate the Time Needed to Conduct an Experiment Nancy is conducting a science experiment involving her pet hamster. To get more accurate results, she decides to repeat the experiment 8 times. Each trial takes between 0 minutes and half an hour. Approximately how many hours will Nancy need to spend conducting her experiment? Solution Nancy needs to conduct 8 trials. Each trial takes between 0 minutes and half an hour. To find the length of time needed, multiply the minimum and maximum times by 8. Minimum time: 0 min is the same as h. 0 The minimum time needed for Nancy s trials is: = 8 is 8 thirds or 6 wholes. The minimum time needed for Nancy to complete her trials is 6 h. Maximum time: The maximum time for each trial is half an hour. To find the maximum time required for the entire experiment, multiply: is 8 halves or 9 wholes. The maximum time needed for Nancy to complete her trials is 9 h. It will take between 6 h and 9 h for Nancy to complete her experiment. 7 MHR Chapter

22 . Use a diagram to show each multiplication statement. a) b) c) 8 d) e) 6 f) 6. Explain how you can find each product in question mentally.. Use coloured counters to show that 8 7. Draw a diagram to illustrate.. The diagram shows multiplication of a fraction by a whole number. One hexagon represents. Write the multiplication equation that the diagram represents.. Multiplication of a fraction by a whole number is the same as repeated addition. Create your own example to show this. Write an equation. Use a diagram to support your explanation. 6. Explain how you can calculate the product mentally. 8. Write a multiplication sentence to describe each diagram. Let one hexagon represent. a) b) 0 8 c) d). Multiply a Whole Number by a Fraction and Vice Versa MHR 7

23 . Multiply. Include a diagram to support your answer. a) 8 b) c) d) 8 6. Pick one expression from question. Explain how you can solve it mentally.. Multiply. Include a diagram to support your answer. a) b) 6. Pick one expression from question. Explain how you can solve it mentally. 6. Use Fraction Factory pieces to multiply. Show your work. a) 8 b) 8 7. Use Fraction Factory pieces to multiply. Record your work in your 8 notebook. 8. Two products are given. Use Fraction Factory pieces to show which is greater. a) or 6 b) or 9. Vera s lemonade recipe calls for cup of sugar per batch. How many cups of sugar will Vera need in order to prepare eight batches of lemonade? Write a number sentence to show your thinking. 0. George takes about h to eat each meal. Assuming that George eats breakfast, lunch, and dinner every day, how many hours does he spend eating: a) per day? b) per week?. a) Write a story problem that can be solved by multiplying. 8 b) Write a story problem that can be solved by multiplying 8. c) Explain why the products and 8 8 are the same.. Cynthia lives of a block from school. How many blocks does she walk to school and back in a typical week? Explain your reasoning.. Andrew has started a new summer job mowing lawns. He gets paid $0 for every day he works. a) How much does Andrew earn if he works only of a day? b) If Andrew worked an entire week in which he worked only of each day, what would his pay be at the end of the week? c) Explain how you solved part b). 7 MHR Chapter

24 Chapter Problem. The Funderland Funhouse has a chicken door through which terrified people can escape before reaching the really scary part.. Nell spends between min and h per night on math homework, from Monday to Thursday. a) Approximately how many hours per week does Nell spend on math homework? Provide a minimum and maximum amount. b) At the end of the year, Nell increases her math homework time from days to 6 days a week. By how much does her math homework time increase? Provide a minimum and maximum amount. Extend 6. Can you draw a diagram to illustrate? What about your diagrams.? Show Suppose the students at your school go to Funderland, and everyone agrees to enter the Funderland Funhouse. How many students do you think will chicken out: a) in your class? b) in your entire school? c) Describe any assumptions you made. 7. Write a problem that requires the multiplication of a fraction by a fraction. Solve the problem, and draw a diagram to support your solution. 8. A B C D E F G H I J K L M N O P Q R 0 a) If the fraction represented by point A and the number were multiplied, what point on the line would best represent the product? b) If the fraction represented by point C and the number were multiplied, what point on the line would best represent the product? c) If the fraction represented by point F and the number were multiplied, what point on the line would best represent the product? d) If the improper fraction represented by point I and the number were multiplied, what point on the line would best represent the product?. Multiply a Whole Number by a Fraction and Vice Versa MHR 7

25 Focus on renaming numbers from fractions and mixed numbers to decimal numbers by applying patterns renaming decimals to fractions by applying patterns, and using these patterns to make predictions Changing Form: Fractions and Decimals Suppose you made a square pan of brownies to share with some friends. Is it possible to share them equally? Does it matter how many friends there are? Materials centimetre grid paper (optional) terminating decimal a decimal number whose digits end Examples:.6,.0 Part A: How can I write a fraction as a terminating decimal? Suppose you share a pan of brownies equally among four friends plus yourself.. a) How many people are sharing the brownies? b) Draw a diagram to show how you would divide the pan of brownies. c) What fraction of the pan of brownies does each person receive? d) Draw another diagram to show what part of the pan each person would get if it were divided into 0 equal parts. e) Write the fraction as an equivalent fraction with a denominator of 0. f) Compare your diagrams for parts b) and d). What do you notice? g) Write a decimal to represent each person s share of the pan of brownies. 76 MHR Chapter

26 . Suppose one of your friends does not like brownies. Repeat question but divide the pan of brownies into 00 equal parts.. a) What is as a decimal? b) What is as a decimal?. a) Compare your answers to question to the decimals you calculated in questions and. b) Suggest a method for using a calculator to find the decimal equivalent of a fraction. c) Try your method using two fractions of your choice. Find the decimal equivalent by paper-and-pencil first, and then by using a calculator. d) Compare your results with those of some of your classmates. Does your method seem to work? Explain.. Reflect Describe how you can express a fraction as a decimal: mentally using a hundred grid using paper-and-pencil using a calculator Create an example to illustrate your explanations. Example : Convert a Mixed Number to a Decimal Each pan of brownies requires cups of sugar. Write as a decimal. Solution The recipe requires full cup plus part of a cup. The part is. Write as an equivalent fraction with a denominator of 0, 00, or 000. I could also use a calculator. Divide 00 the numerator by the denominator. 0. = 0. Strategies Use procedures you learned previously. Combine the whole cup and the part to express the complete number in decimal form: 0.. Each pan of brownies requires. cups of sugar.. Changing Form: Fractions and Decimals MHR 77

27 Example : Convert a Terminating Decimal to a Fraction Lynnette and her sister together ate 0.6 of a pan of brownies. What fraction of the brownies did they eat? Solution means 6 tenths or. 0 You can express this fraction in a simpler form by dividing the numerator and the denominator by the same number Together the two girls ate of a pan of brownies. Part B: How can I express a fraction as a repeating decimal? One of your friends had a big lunch, and does not want any brownies. That leaves just three of you. How can you write each person s share as a decimal?. Write the fraction of the brownies that each of you can have.. a) Cut the thousandths grid paper into ten equal strips. Divide the strips evenly among the members of your group. Leave any remaining strips separate. Glue your strips into your notebook. b) Cut each remaining strip into ten equal strips. Divide the strips evenly among the members of your group. Leave any remaining strips separate. Glue your strips into your notebook. c) Repeat part b) two more times.. a) Write the decimal number that you have glued in your notebook. b) Consider what will happen if you continue to make smaller strips and share them. Will there always be a strip left over? Explain how you can show more decimal places. Materials 0 cm 0 cm thousandths grid scissors glue stick coloured pencils repeating decimal a decimal number that has one or more digits that repeat without end Examples:.6,.0, 0. Strategies Use a model. Act the situation out. 78 MHR Chapter

28 . Reflect a) Compare the two numbers from this activity: the fraction of the whole tray of brownies that each friend receives the decimal that each friend receives b) How does the decimal number differ from the decimals in part A? c) Convert to a decimal number by dividing the numerator by the denominator. How does the decimal number differ from the decimals in part A? d) Predict the decimal equivalent of, then calculate. What do you notice about the last digit of your decimal? Explain why it is so.. Your younger sibling wants to know why you sometimes use fractions and sometimes use decimals. Explain when it is better to use a fraction and when it is better to use a decimal number. Some fractions, when expressed as a decimal number, appear as repeating decimals. A repeating decimal occurs when one or more digits in the decimal repeat forever. The sequence of repeating digits is called the period. The length of the period is the number of digits in the repeating sequence. Example : Repeating Decimals Use a calculator to convert each fraction to a decimal. Identify the period and the length of the period, and write the decimal using repeating notation a) b) c) d) 9 0 Solution To write a repeating decimal, put a bar over the period. Fraction Decimal (calculator output) Period Length of Period Decimal (repeating notation) When you write repeating decimals using repeating notation, write the period once only. Place the bar over the repeating digits only. Communicating Mathematically Three dots at the end of a decimal mean that the sequence repeats forever. You can add the dots after the pattern has been clearly established. For example, 0. can be written as can be written as Technology Tip Some calculators round repeating decimals. For this reason the last digit in the display may appear to break the repeating pattern. For example, If your calculator window could show more digits, the pattern would continue.. Changing Form: Fractions and Decimals MHR 79

29 Part C: What patterns can I recognize in repeating decimals?. a) Use a calculator to express each fraction as a repeating decimal b) Predict the decimal form of. Use a calculator to check 9 your prediction. c) Use the pattern in parts a) and b) to write a fraction for each repeating decimal d) Write two equivalent fractions for 0.6. e) Write 0.9 as a fraction. What is this fraction equal to? Materials calculator Strategies Look for patterns.. a) Convert to a decimal and explain whether you think this 7 is a repeating decimal or not, and why. b) Convert each fraction to a decimal. Use a calculator c) Describe any patterns that you see in the decimals. d) Predict the decimal equivalent for each fraction e) Check your predictions with a calculator. f) Do the results of this activity suggest that produces 7 a repeating decimal or not? Explain.. Reflect Use your calculator to explore the repeating pattern in the decimal equivalents of and. Write a brief report of your findings.. Reflect Use your calculator to search for and explore another pattern of repeating decimals of your choice. Write a brief report of your findings. Technology Tip You can display a decimal in fraction form using your graphing calculator by pressing k, then selecting : Frac. 80 MHR Chapter

30 . Explain two methods for expressing a fraction as a decimal. Use examples to show how each method works.. Explain how you can convert a terminating decimal to a fraction. Use an example to support your explanation.. a) Describe what is meant by a repeating decimal. Give an example to support your explanation. b) Describe what is meant by a terminating decimal. Give an example to support your explanation.. Use the repeating decimal to explain what is meant by the period and the length of the period of a repeating decimal.. These repeating decimals have been written incorrectly. Identify the error and write the decimal correctly using repeating notation. a) b) c) d) Who do you think is right? Explain why. Roberto, I think that 6 produces a repeating decimal. I don t think so, Denzel. Look carefully at the last digit of my calculator display. 0.. Convert each fraction to its decimal equivalent. Is the decimal terminating or repeating? 7 a) b) c) d) 0 8. Write each number as a decimal, without using a calculator. 9 9 a) b) c) d) e) f) g) h) 0 0. Check your answers to question using a calculator.. Changing Form: Fractions and Decimals MHR 8

31 . Write each decimal as a fraction. a) 0.7 b) 0. c) d) 0.0 e).9 f).07. Write each decimal as a fraction in simplest form. a) 0. b) Identify the period and the length of the period of each repeating decimal. a) 0... b) c) d) Write each decimal in question 6 using repeating notation. 8. Classify each decimal as either a terminating or a repeating decimal, and explain your choice. a) 0... b) 0. c) 0. d) e) f) Convert each fraction to a decimal, and write it using repeating notation. a) b) c) d) 90 e) f) g) h) 0 0. Refer to your answer to question 9, part f). How can you obtain more evidence that this decimal repeats? Check some other fractions to verify that this fraction repeats. Record the fractions you checked and their decimal equivalents to support your explanation.. a) Convert each fraction to a decimal b) Predict the decimal equivalent of each fraction c) Predict the fraction equivalent of each decimal d) Check your predictions to parts b) and c) with a calculator.. a) Refer to question. Predict the decimal 8 equivalent of. 90 b) Check your prediction with a calculator. Does the result fit the pattern? Explain why or why not. c) Predict the decimal equivalent of each fraction d) Check your predictions using a calculator. Do the results fit the pattern? Explain.. Refer to question. Use your pattern to predict the fraction equivalent of each decimal. a) b) 0.8 c) 0. d) Verify your predictions using a calculator.. Who is right? Give an example to help explain your answer. Aceena, every fraction produces a repeating decimal. That s not true, Danielle. 8 MHR Chapter

32 . Who is right, Lei Mei or Denzel? Conduct an investigation and write a brief report of your findings. Include examples. I don t get it. Can you give me some examples? Well, think about or. divides evenly into 0, and divides evenly into 00, so I know both of these will produce terminating decimals. On the other hand, will produce a repeating 7 decimal because 7 does not divide evenly into a power of 0. Pretty cool, huh? That s a pretty good method, Lei Mei, but it doesn t always work. Communicating Mathematically I can tell if a fraction will produce a repeating decimal or not by looking at its denominator. If it divides evenly into a power of 0 (for example, 0, 00, 000), then the decimal will terminate. If it does not, then the decimal will repeat. You studied divisibility rules in Chapter Number Sense. How can these strategies be useful in your investigation? 6. a) Explore the pattern of decimals produced by these fractions. b) Predict the decimal equivalent of. c) Check your prediction using a calculator. Does this seem to fit the pattern? Explain. d) Explore the following decimal equivalents. 6 e) Which of these fractions do you expect to give: terminating decimals? repeating decimals? f) Predict the decimal equivalent of each fraction above, without calculating it. Hint: look at the difference between pairs of consecutive decimals. g) Use a calculator to check your predictions in parts e) and f). 7. Find two equivalent fractions for each fraction. Use a diagram of a pan of brownies to help you determine the equivalent fractions and to show your reasoning. 6 Extend 6 8. Use long division to verify whether is a 7 repeating decimal or not. Use mathematical reasoning to justify your conclusion. Hint: What are the possible remainders when you divide by 7?. Changing Form: Fractions and Decimals MHR 8

33 Focus on comparing and ordering proper and improper fractions, mixed numbers, and decimal numbers Compare and Order Fractions and Decimals How can I compare and order fractions and decimals? Materials one or more of the following: Fraction Factory linking cubes coloured tiles/counters pattern blocks Cuisenaire rods grid paper calculator The table shows the number of free-throw baskets scored and the number of free-throw attempts for four basketball players. Free-throws are awarded when the other team commits a foul. Player Number of Free-Throw Baskets Made Number of Free-Throw Attempts (shots) Dunbar 8 Jones 6 Singh 7 Matsu 6 0. a) Who scored the most baskets? b) Does this mean this player is the best shooter? Why or why not?. a) Who scored the fewest baskets? b) Does this mean this player is the worst shooter? Why or why not?. Use diagrams or concrete materials to represent the number of free throws made as a fraction of the number of free throws attempted. Make sure you use the same representation for whole each time.. Rank the free-throw shooters starting with the best. Explain your reasoning. State any assumptions that you must make. 8 MHR Chapter

34 . Illustrate your ranking using another method. 6. Reflect a) Explain how you can use fractions or decimals to make sense of sports data. b) Look up another sports statistic or invent one and use it to create an example to support your explanation. c) How do you compare numbers in fraction form? in decimal form? Which do you prefer? Example : Compare and Order Decimals In baseball, a player s batting average is found by dividing the number of hits by the number of times at bat. It is expressed as a decimal, usually rounded to three decimal places. Compare and arrange the following players batting averages from least to greatest. Sanchez 0.0 Lebeau 0.88 O Connor 9 hits out of 00 times at bat Williams hits out of 0 times at bat Solution Convert O Connor s and Williams data to batting averages, expressed to three decimal places. O Connor s batting average: Hits 9 Times at bat = Williams batting average: Hits Times at bat To decimal places. Strategies Use previously learned procedures to change fractions to decimals.. Compare and Order Fractions and Decimals MHR 8

35 Each player s average can be modelled using base-0 materials. Let the largest cube represent. Base-0 Materials Player Batting Average Tenths Hundredths Thousandths Sanchez 0.0 Lebeau 0.88 O Connor 0.90 Sanchez s model has flats, the rest have two, so Sanchez s model represents the greatest batting average. Now compare the models for the remaining players for the hundredths place. Lebeau s model has 8 rods. The other two models have 9 rods. Lebeau has the least batting average. Now compare O Connor s and Williams models for the thousandths place. Williams 0.9 Williams model has one cube. O Connor s model has no cubes. Williams batting average is greater than O Connor s. Lebeau O Connor Williams Sanchez 0.88 < 0.90 < 0.9 < 0.0 You can also compare the digits of the batting averages. Compare the tenths digits: Compare the hundredths digits of the Sanchez 0.0 remaining players: Lebeau 0.88 Lebeau 0.88 O Connor 0.90 O Connor 0.90 Williams 0.9 Williams 0.9 Sanchez s batting average is the greatest. Place this number at the end of the list. Lebeau s batting average is the least. Place this number at the beginning of the list. Compare the thousandths digits for O Connor and Williams: O Connor 0.90 Williams 0.9 Williams average is greater than O Connor s. Place this number before 0.0. Arrange the batting averages for all four players. Lebeau O Connor Williams Sanchez 0.88 < 0.90 < 0.9 < MHR Chapter

36 Example : Comparing Fractions Arrange these fractions from least to greatest using benchmarks. 9 8 a),,,,, b),,,, c),, d),,,, Solution a) These are unit fractions and each is one space away from 0 on a number line. Recall that greater denominators require more divisions on the number line so the divisions must become smaller. Therefore, if the numerators are equal, the greater the denominator, the lesser the fraction. 0,,,,, 8 6 b) These are Nearly Ones. Each fraction is one division unit away from on a number line, so since the greater denominators require smaller divisions on the number line, would be the greatest. It is only 8 9 away from, while is away from.,,,, c) These are Close-to-Halves. The numerator of each is about half of its denominator. Think about the fraction that is equivalent to for each different denominator. There is one part of a whole amount in each of these fractions. As the denominator increases, the size of the part decreases. The midpoint of the fifths number line is., so must be greater than. The midpoint of the fifteenths number line is 7., so 7 must be less than. 7,, d) This set contains a mix of fractions. and are both unit fractions, 8 0 so they are close to 0. must be the lesser one, since its denominator 0 0 is the greater. Put them first. is the only Nearly One in the set, so it should be last. The two fractions remaining are Close-to-Halves. The. midpoint of the sevenths number line is, so is less than. The 7 7. midpoint of the ninths number line is, so is greater than. 9 9 The order is: 0,,,, Compare and Order Fractions and Decimals MHR 87

37 Example : Compare and Order Fractions Order the fractions in each set from least to greatest. a),,, b),,, c),,, Strategies Use a previously learned procedure. Solution a) Rewrite each fraction with a denominator of To compare fractions that have the same denominator, compare the numerators. As the numerator increases, so does the number of parts of the same whole. All of the fractions now have the same denominator. < < < b) Rewrite each fraction with a numerator of All of the fractions now have the same numerator. < < < To compare fractions that have the same numerator, compare the denominators. c) It is not easy to express all fractions with either the same numerator or the same denominator. Convert these fractions to decimals by dividing the numerator by the denominator = 0. Strategies When converting to decimals: use mental math when the denominator divides evenly into a power of 0, use a calculator when it does not. 88 MHR Chapter

38 Compare the tenths digits is the greatest. Compare the hundredths digits to order the remaining numbers Add zero placeholders if necessary < 0. < 0. < 0. 7 Now order the fractions from least to greatest: < < < 0 0 Example : Compare and Order Fractions and Decimals Who ate the most pizza? Who ate the least? All pizzas and slices are the same size. I was starving. I ate of a whole pizza! I hadn t eaten all day. I can t believe I ate slices out of. Well, I ve beaten you both. I ate 0. of a pizza! Solution Method : Convert to decimals. Cherise 0. Azar 0.6 Lei Mei 0. You can arrange the pizza-eaters by comparing the tenths digits. Cherise Lei Mei Azar 0. < 0. < 0.6 Azar ate the most pizza. Cherise ate the least. Method : Convert to fractions with a common denominator. Lei Mei s decimal value is easily recognized as a familiar fraction: 0. Both Cherise s and Lei Mei s fractions can be expressed with a denominator of. Cherise Azar You can arrange the pizza-eaters by comparing the numerators. Cherise Lei Mei Azar < < Lei Mei. Compare and Order Fractions and Decimals MHR 89

39 . a) Explain how you can compare and order fractions that have: the same denominator different numerators and denominators b) Create an example to support your explanation.. a) Explain how you can compare and order these decimals b) Order the decimals from least to greatest.. a) Explain how you can compare and order fractions and decimals. b) Create an example to support your explanation.. Order the fractions in each set from least to greatest. Explain your reasoning. a) 6 b) c) d) e) 6. Order the decimals in each set from least to greatest. Explain your reasoning. a) b) Order the numbers in each set from least to greatest. Explain your reasoning. a) b) MHR Chapter a) Order the numbers from least to greatest. Explain your reasoning b) Place these numbers on a number line.. Increase the numerator of the fraction by. Record your result. Increase the denominator of by. Record your result. Compare each fraction with. Which fraction is the greatest? Which fraction is the least? Use these Cuisenaire rods to answer questions 6 8. cm cm cm cm cm 6 cm 7 cm 8 cm 9 cm 0 cm 6. a) Let the brown rod be one whole. What is the value of a purple rod? Express this as a fraction and a decimal. b) Let the purple rod be one whole. What is the value of the brown rod? Express this as a fraction and a decimal.

40 7. Let one black rod plus one yellow rod equal one whole. Identify the rod that represents: a) b) c) 8. Let one black rod plus one yellow rod equal one whole. Identify the rods that represent: a) b) c) d) 9. Refer to question 8. a) Which fraction is the greatest? b) Order the fractions from least to greatest. c) Explain the pattern. 9 d) Which is greater, or? Explain A hockey goalie s goals against average is the number of goals allowed per game. Data for four goalies is given. Goalie Goals Allowed Number of Games Lachance 60 0 Brown 7 0 Lundergard 0 Stiles Select the two goalies with the best goals against average for the all-star team. Explain your choices.. In a pie-eating contest, Evan ate pies, Dora ate pies, and Gianetta ate 0 pies. a) Who ate the most pies? b) Order the contestants from first place to third place.. Some of these gymnastics scores out of 0 were given as mixed numbers and some as decimals , 6, 6, 6, 6., a) Order the scores from greatest to least. 7 6 b) What strategy did you use to solve this question? What other strategies might you use?. A school spirit award goes to the class with the best attendance at a dance. Teacher Number of Students Number at Dance Mr. Sommers 0 Mrs. Cheng Ms. Morales 6 9 Mr. Ford 8 0 a) Without calculating, predict which class should win. Explain your thinking. b) Which class has won the school spirit award? Justify your decision.. Four friends are in a band. Their average amount of time spent practising is shown. Band Member Average Daily Practice Time Spike Buzz Axe Crawler a) Arrange the bandmates in order of practice time, expressed in fractions of an hour, hours in decimal form, and minutes (Hint: h = 60 min). b) Find the difference between the greatest and least practice times. Express this answer in three different ways. Extend an hour and a half. hour 7 minutes an hour and 0 minutes. Compare and order the following. Explain your reasoning. You may need to do a little research and apply some estimation. the fraction of your life you spend sleeping the decimal equivalent of the fraction of a human body that is water the fraction of Canada s population that live in the Maritime provinces the fraction of students in your school who are in Grade 7. Compare and Order Fractions and Decimals MHR 9

41 Operating With Decimals Focus on using estimation and mental math when calculating with decimal numbers applying order of operations Communicating Mathematically Usually taxes and tips are not included in restaurant prices. Do you know how to calculate these amounts? You will learn more about taxes and tips in Chapter. How much should each friend withdraw from the bank machine to pay for her share of the dinner? Do the girls need to calculate the exact amount or will an estimate be good enough? How can you apply estimation and calculation skills when operating with decimals? Heather s Bill All prices include taxes and tips. Side Caesar Salad $.99 Classic Personal Pizza.99 Juice 0.9 Shannon s Bill All prices include taxes and tips. Side Caesar Salad $.99 Chicken Wrap 6.99 Juice 0.9 Use the information given above.. Assuming the bank machine only allows withdrawals of amounts that are a multiple of $0, what minimum amount of money should each girl withdraw in order to pay for her meal? Explain your reasoning. 9 MHR Chapter

42 . Find the total for each girl s meal.. How much is the total for both bills?. Reflect Describe the strategies you used to: a) estimate the amount each girl needed to withdraw from the bank machine b) calculate the total amounts When working with decimals, sometimes it is necessary to find an exact amount and sometimes an estimate is sufficient. You can apply a variety of strategies to estimate or calculate an amount. Whether you need to find an exact amount or just an estimate often depends on the situation. Example : Estimate the Cost of Cell Phone Service Julia is allowed to get a cell phone, but only if she agrees to pay for the service. Julia asks her friend Sarah how much her monthly bills are. a) Estimate Sarah s cell phone expenses for September to December. b) Julia thinks she can afford to pay up to $00 per year for a cell phone. Assuming her phone habits are similar to her friend s, can she afford to get a cell phone? September $9. October $.76 November $.0 December $8.7 Solution a) Each bill is close to $0. Multiply this amount by to estimate Sarah s cell phone expenses for this time period. $0 $60 Sarah spent about $60 on her cell phone from September to December. b) If Julia has phone habits similar to Sarah s, her monthly bills will be about $0. Multiply this by to find the yearly expense. $0 (0 $0) ( $0) $00 $80 $80 This is less than Julia s budget of $00, so she can probably afford a cell phone. To learn more about the history of the cell phone, go to and follow the links. Strategies This method is called clustering, because all the numbers cluster, or are close to, the same value. You can use multiplication instead of adding a long string of numbers. Did You Know? On April, 97, the first public telephone call on a cellular phone was made. The call was made by Martin Cooper, then manager of Motorola, to rival AT&T.. Operating With Decimals MHR 9

43 Example : Add a Grocery Bill WHEAT BREAD WHEAT WHEAT BREAD WHEAT BREAD WHEAT WHEAT BREAD $.09/dozen $.9/loaf $.99/L $7.7 a) How much money should you withdraw from the bank machine in order to buy the items shown? b) Find the total for this grocery bill. Solution a) To decide how much money to withdraw from the bank machine, an estimate is good enough. Method : Round costs, then add. Method : Use front-end rounding. Estimate the grocery bill by rounding. Estimate dollars and cents separately, then add. Use a table to organize your work. Eggs $ = $ Bread $.0 = $ Milk $ = $ Roast $8 = $8 TOTAL = $9 Item Dollar Portion $ $ $9 Cents Portion Cents Portion in Estimated Dollars Eggs $ = $ 0.09 = 0.8 $0 Bread $ = $ 0.9 =.8 $ Milk $ = $ 0.99 =.98 $ Roast $7 = $7 0.7 = 0.7 $ TOTAL $ $ The total of the grocery bill is about $9. Bank machines usually issue funds in multiples of $0 or $0. Withdraw $0 to pay for this grocery bill. b) To find the total of the grocery bill, calculate the exact amount. Eggs $.09 = $.8 Bread $.9 = $.8 Milk $.99 = $.98 Roast $7.7 = $7.7 TOTAL = $9.09 I can calculate this in one step: , using a scientific calculator. *.09 + *.9 + * = The total of the grocery bill is $ MHR Chapter

44 Example : Calculate the Cost of Hockey Equipment Angus is the coach of a little league hockey team. He bought the following items for his team. Find the cost of each purchase. a) 0 hockey pucks, each costing $.6 b) 0 jerseys, each costing $8.09 c) 6 practice pylons, each costing $.0 Solution a) It is easier to find the cost of 00 pucks than 0 pucks. You can double the number of pucks if you halve the price and you will still get the total cost of the hockey pucks. 0 $.6 00 $. $ It costs $ for 0 hockey pucks. Check: 0 *.6 = 00 *. = To find half of $.6, work by parts: I can divide the dollars and cents by and add the results. $ = $ $0.6 = $0. $.6 = $. I am applying the distributive property of division. b) It is easier to find the cost of 0 jerseys than 0 jerseys. This time, halve the number of jerseys and double the price. 0 $ $6.8 $6.80 The 0 jerseys cost $6.80. To double $8.09, I can break it up into parts, as before: $8 = $6 $0.09 = $0.8 $8.09 = $6.8 Sometimes you can apply the halve/double strategy to get close to a special number. c) This time, you can double the price, and halve the number of practice pylons. 6 $.0 To multiply 8 $0.0, I can multiply 8 by the dollars and cents 8 $0.0 separately and add (8 $0) (8 $0.0) the results. $80 $.60 $8.60 It will cost $8.60 for the practice pylons. Strategies You can apply the halve/double strategy when multiplying two numbers together: by doubling one factor and halving the other, you get the same result as if you were to multiply the two factors. To use this strategy, one of the factors must be an even number. The halve/double strategy can be applied when you are multiplying two numbers together. When dividing two numbers, you can apply a variation of this strategy.. Operating With Decimals MHR 9

45 Example : Estimate the Cost of a Ski Trip Five friends rent a ski condo for a week. The total cost is $780. Estimate the cost for each friend. Solution To find the exact cost, you would divide the total cost by the number of friends. $780 $800 $600 0 $60 The cost for each friend is about $60. If I double both the dividend and the divisor, the calculation will be easier and the answer will be the same. This is the same as making an equivalent fraction. I can get a quick estimate by rounding the total cost to Describe a situation in which: a) you would need to find an exact amount involving decimals b) finding an estimate is sufficient. a) Describe a strategy that you can use to estimate the sum. $.9 $. $. $. b) Find the estimated sum. c) Compare your solution with a classmate.. Describe a strategy that you can use to multiply the following mentally. a) $0 b) 00 $0.7. Describe a strategy that you can use to mentally calculate $.0 0. Use the following information to answer questions.. Kendra and her two sisters each get a grilled fish sandwich, a large salad, and a smoothie. a) Estimate the total cost, assuming taxes are included in the price. b) Explain how you found your estimate. c) Show and explain another method to find the estimate.. Zach has $0. If he orders two fish sandwiches, a salad, and a smoothie, will he have enough to pay for it? Do you need an exact answer or will an estimate do? Explain why.. Randall s weekly grocery bills for his family are given, rounded to the nearest dollar: $99 $0 $0 $98 $0 a) Use the clustering strategy to estimate the total for the five weeks shown. b) How can you quickly calculate the total? c) How much is a year s supply of groceries? 96 MHR Chapter

46 . A family of five attends a concert. Each ticket costs $8, including tax. Use the halve/double strategy to calculate the total cost.. Rene sells 00 comic books for $0.7 apiece. Use the halve/double strategy to calculate how much she received. 6. Amanda wants to buy hockey equipment: sticks at $.99 per stick pucks at $. per puck rolls of tape at $.99 per roll a) Estimate the total cost. Explain your method. b) Describe another method for estimating this total. 7. The restaurant bill for a large dinner party is $9.98, including taxes and tips. The 0 members of the party split the bill evenly. a) Show how you can halve both values to estimate each person s share of the bill. b) Find the estimate. c) Calculate each person s share, and verify that this method works. 8. Roberto lives.6 km from school. Estimate how far he walks to school and back a) per week b) per month c) Describe any assumptions you must make. 9. a) Estimate the area of each garden, to the nearest square metre..9 m. m 6. m b) Which garden is larger and by approximately how much, to the nearest square metre? c) Describe the strategies you used..8 m Chapter Problem Use the following information to answer questions 0. Ride tickets at Funderland cost $.0 each. 0. a) Find the cost of each ride. b) How much would it cost, in total, to take each ride once?. You can buy a Value Pack of 0 tickets for $. Is this a good deal? Explain.. a) What is the greatest number of times you could ride Wild Rider on one Value Pack? b) How much will you save if you use a Value Pack instead of individual tickets? Extend Ride Suppose you buy four panels of stickers for $0.99 each. How can you find the total cost, mentally? Consider the following method. panel = cent short of dollar panels = cents short of dollars The total price is $.00 $0.0 = $.96.. Use this method to find the total of each purchase. a) sandwiches costing $.99 each b) pens costing $0.98 each Number of Tickets Cannonball Mind Warp Slingshot Wild Rider 6. Use a variation on the method to calculate each purchase. Describe your method in each case. a) plants costing $.0 each b) 6 books costing $9.98, $9.98, $9.99, $0.00, $0.0, and $0.0. Operating With Decimals MHR 97

47 Focus on using estimation and mental math when calculating with decimal numbers applying order of operations using the most appropriate method for solving problems Solve Problems Involving Decimals Frances received a $00 gift certificate to spend on CDs and DVDs for her birthday. How many CDs and movies can she buy? How can you apply estimation and calculation strategies to plan your spending? Answer the following by estimating only. Do not use a calculator. Assume that taxes are included in the prices.. Suppose Frances spent her entire gift certificate on CDs only. How many CDs could she buy? Explain how you know.. What if, instead, Frances spent her entire gift certificate on DVDs only? How many DVDs could she buy? Explain how you know.. Suppose Frances decides she wants to buy CDs and DVDs. Will her gift certificate cover the entire cost? If yes, about how much will she have left over? If no, about how much additional money will she have to pay? 98 MHR Chapter

48 . What possible combinations of CDs and DVDs could Frances buy, assuming that she does not want to exceed the amount of her gift certificate?. Show how you can mentally calculate the exact cost of: a) CDs b) DVDs 6. Reflect a) Describe the strategies you used to estimate or calculate in this activity. b) Compare your strategies with those used by some of your classmates. Describe any strategies that they used that you did not. The strategies you have learned in this chapter to estimate and calculate amounts involving fractions and decimals can be applied to solve a variety of problems. When solving a problem, first decide if you must find an exact answer or if an estimate is good enough. If you need to find an exact answer, apply a mental estimation strategy to check if your answer is reasonable. 7. Create a list of estimation strategies you have used before. Provide examples of how each strategy can be used. For each strategy, describe the types of numbers that make the strategy the most suitable one to use. Example : Backyard Improvement The McNamaras are replacing the sod and fence in their backyard. Sod costs $.9/m and fencing costs $.0/m. a) Find the cost of replacing the sod. b) Find the cost of replacing the fence, and discuss any assumptions you must make. Solution. m. m a) The length of the yard is. m. The width of the yard is. m. Sod costs $.9/m. I need to find the cost of covering the yard with sod. Determine the area of the yard. Then multiply the area by the price of the sod per square metre. Apply the formula for the area of a rectangle. A l w...08 Strategies The area of the yard is hidden information you need to find. Strategies Use a formula..6 Solve Problems Involving Decimals MHR 99

49 The area of the backyard is.08 m. Check by estimating: The length is a little more than m. The width is a little more than m. = The area is a little more than m, so my answer seems reasonable. m m Strategies What other strategy could you use to estimate the product of. and.? The backyard area is.08 m. Multiply this value by the cost of sod per square metre..08 $.9 $7.6 $7.6 It will cost $7.6 to replace the sod. Check by estimating: The area is about m. The cost is about $/m. Therefore the total cost should be about: $ $8 The answer of $7.6 seems reasonable. I need an exact cost. I will use a calculator to find the required amounts, and round to the nearest cent. Then I ll check the reasonableness of the answer. b) The length of the yard is. m. The width of the yard is. m. Fencing costs $.0/m. I need to find the cost of putting a fence around the yard. Strategies Make an assumption. Find the perimeter of the yard. Multiply this value by the cost of fencing per metre. This is assuming that all four sides of the yard are to be fenced. Apply the formula for the perimeter of a rectangle. A (l w) (..) Remember BEDMAS. Do the operations in brackets 7.6 first and then multiply.. The perimeter of the backyard is. m. Multiply this value by the cost of fencing per metre.. $.0 $90 It will cost $90 to replace the fence. Check by estimating: The perimeter is about m. The cost of fencing per metre is about $ Therefore the total cost should be about $9. The answer of $90 seems reasonable. Strategies Use the distributive property. 00 MHR Chapter

50 Example : Half-Price Sale Suppose that the store in the Discover the Math activity is having a sale. How much would it cost for four DVDs? Solution Method : Find the sale price of one DVD, then multiply. Find the sale price of a DVD by finding half of $6.96. Then multiply by. $6.96 Write as a decimal, $6.96 $.8 $.9 Method : Multiply by half of the number of DVDs. $6.96 I can calculate this mentally: $6.96 DVD costs cents less $.9 than $7, so DVDs will cost 8 cents less than $7 = $. $6.96 = $ $0.08 = $.9 The cost of four DVDs on sale is $.9.. For each situation, decide whether it is important to find an exact amount or if an estimate is sufficient. Explain your reasons. a) You are going to the grocery store to buy some food, and you need to stop at the bank machine to get some money first. b) A cashier is totalling up a grocery bill for a customer. c) You are planning where and when to stop for meals during a long car trip. d) Racing times are measured for each racer in a 00-m dash.. a) Describe a situation of your own in which you would need to estimate an amount involving fractions or decimals. b) Create a problem based on your situation. c) Solve the problem.. a) Describe a situation of your own in which you would need to find an accurate amount involving fractions or decimals. b) Create a problem based on your situation. c) Solve the problem.. Trade the problems you created in questions and with a classmate. Solve the problems and check the solutions.. Compare the methods for solving the problem in Example. Which method did you prefer and why?.6 Solve Problems Involving Decimals MHR 0

51 . Colin is taking his family out for lunch. He and his wife usually order the Daily Special for $6.99, and each of his three children always gets a Child s Lunch, for $.9. Both prices include tax. Colin s bank machine issues money in multiples of $0. a) How much should Colin withdraw to pay for his family s meal? b) How much change will Colin have left, after paying for lunch?. Brenda takes a taxi to get around town. Home.7 km.80 km Work 0. km. Refer to question. How do your answers change if a) used novels are marked down by half? b) all novels are marked down by half? 6. Raquel has designed a rectangular coffee table.. m 0.8 m a) Estimate the area of the table top, to the nearest square metre. b) Estimate the perimeter of the table top, to the nearest metre. What is the distance for: Bruce s House a) a trip from home to work and back? b) a trip from home to her friend Bruce s and back? c) a trip from home to work, then to Bruce s, and then back home?. Refer to question. Suppose taxi fares are $/km. Brenda works Monday to Friday. On Saturday she visits Bruce and then returns home. Sunday she stays home. Estimate Brenda s weekly taxi expenses. 7. Dennis added his grocery bill using the following strategy. Item Cost Eggs $.0 Milk $.9 Coffee $.90 Ham $. Butter $.7 Mixed vegetables $.8 Strategies How would you add.9 and.8 mentally? Make Make. Sandy goes to her favourite bookstore with $0 to spend. New novels cost $9.9, and used novels cost $.0. All taxes are included in the prices at this bookstore. a) If Sandy buys new novels only, how many can she get? b) If Sandy buys used novels only, how many can she get? c) What other combinations of new and used novels can Sandy purchase? $.0 $. $ $8.00 $7.00 $6.0 $.0 Dennis calls this strategy Make. a) Explain how this strategy works. b) Why is this strategy called Make? 0 MHR Chapter

52 8. Refer to question 7. a) Use the Make strategy to calculate the shopping list total. Item Cost CD $8.9 MP player $.9 Alarm clock $6. DVD $8.90 Headphones $.0 CD cleaner $.0 Chapter Problem. Marc and Fergus are conducting science experiments at Funderland. They want to measure various forces acting on a rider during a ride. To get accurate results, they need to ride each ride twice. Wild Ride 6 h h 8 h Food Court h h 8 h Water Park h b) Explain the steps to your solution. Sling Shot h Mind Warp h Cannonball 9. Refer to question 6. Raquel wants to apply two coats of lacquer to the entire tabletop. One can of lacquer covers. m and costs $.99.. m 0.8 m a) How many tins of lacquer should Raquel buy? Explain why. b) Estimate the cost of the lacquer, to the nearest dollar. c) About what fraction of a tin of lacquer will Raquel have left over, after she finishes the table? Explain your estimate. 0. Deanna s rectangular yard is.8 m wide. The length of fence required to completely surround it is 6 m. How long is the yard?. Choose one of the expressions and write a story that can be solved by evaluating it. (.8.) Provide the solution to the question in your story. h It takes about h to ride each ride once. The average waiting time for each ride is about h. Assume that they are starting at one of the rides. a) What is the shortest amount of time the boys will need to complete their experiment? b) Explain how you found your answer. Extend. Mackenzie is recording her favourite songs on a readable/writable CD that can hold 7 min of music. So far she has songs recorded. The length of each song is given. :, 7:08, :6, :, :9, 6:, 6:, :, :, 6:0, :, 7:9 How much time remains on the CD? Explain how you found your answer..6 Solve Problems Involving Decimals MHR 0

53 . Express each fraction as a decimal. 7 7 a) b) c) d) 0. Write each decimal as a fraction. a) 0. b) 0. c) d) 0.7. Estimate each sum. a) b) c) Create a word problem using the addition statement in part a) or part b).. Estimate each difference. 7 a) b) 8 8 c) Create a word problem using the subtraction statement in part a) or part b).. Multiply. Illustrate your solution with a diagram. a) 6 b) 8 6 c) d) 6 0 e) f) 6. Explain how you can solve the multiplication statements in question mentally Write each decimal using repeating notation. a) 0... b) c) d) Kermit, Hermit, and Zermit all live in 9 King Pond. Hermit is of a giant leap 0 north of Kermit, and of a giant leap 8 south of Zermit. a) Draw a diagram that shows roughly where the three frogs live, and the distances between their lily pads. b) One day, Zermit travelled to Kermit s lily pad. Approximately how far did she travel? c) Whose lily pad is farther from Hermit, and by approximately what distance? 9. Copy and complete the chart. 0. Order the numbers from greatest to least. Show your work Order the fractions from least to greatest. Show your work. 6 Fraction Decimal (calculator output) 6 Terminating or Repeating? 0 6 Period 8 Length of Period 0 Decimal (repeating notation) 0 MHR Chapter

54 . Order the decimals from least to greatest. Show your work Refer to question. Exactly how much money will Oscar have after he pays for his groceries?. The speeds of five yachts were recorded in knots. Order the speeds from slowest to fastest. Show your work Did You Know? 8 0 A knot is a unit of speed used for watercraft. One knot is approximately equal to.8 km/h. 7. Pina needs to run a few errands: pay a bill at the bank drop a book off at the library pick up some things at the store Home 6 h. h Library 7 6 h. Jan s printer has a problem printing out math symbols. Use a calculator to help you fill them in. a) b).. c). a b 0.6 d) Oscar needs to buy some items at the store. The prices are shown. Item Quantity Price per Item Shampoo $.90 Loaf of bread $.9 Steak $8.8 Juice $.9 Bag of potatoes $. Oscar s bank machine issues bills in multiples of $0. Use estimation to determine how much Oscar should withdraw to pay for his groceries. Explain your method. Store h Bank a) Estimate the total time taken. b) She returns home and realizes she left her bank card at the bank. What is the shortest route to and from the bank? 8. Explore the decimal equivalents of proper fractions having a denominator of 8.,,, and so on Identify and describe any patterns that you see. Review MHR 0

55 Selected Response Select the best answer is equal to which fraction? A B C D 6. is equal to which decimal? A 0. B 0.08 C 0. D 0.. The sum is closest to: 8 A B C D. The difference is closest to: 8 A B C D. Which is the correct decimal form of? A 0.0 B 0.0 C D 0. Short Response Provide a complete solution. 6. Draw a diagram to illustrate each multiplication statement. a) 8 b) 8 7. One-quarter of a class of students can ski. a) How many students does this represent? b) How many non-skiers are in the class? Show two methods of finding this answer mentally. 8. a) Write the fraction of the tile pattern that is: red blue green b) Write two equivalent fractions for each fraction from part a). c) Show how you can express these fractions as decimals, without using a calculator. 9. The scores for science projects are shown: If the greatest and least scores are excluded, order the remaining scores from least to greatest. Show your reasoning. 0. Calculate. a) b) a 6b c). 0. a0 b. Copy and complete the table. Fraction 9 0 Decimal Equivalent (calculator output) Terminating or Repeating Period Length of Period Repeating Decimal Equivalent (repeating notation) 06 MHR Chapter

56 . Magic Mountain Water Theme Park near Moncton charges the following: Admission Price Family of $6.6 Person over. m $9. Person under. m $.9 Senior (60+) $.6 Child ( and under) free Francis, her parents, grandparents, and brother want to visit. She, her parents, and her 6-year-old grandparents are over. m tall. Her brother is under. m. a) Which admission option would be the least expensive? State your assumptions. b) Estimate the total cost of admission and explain how you found your estimate. Assume that taxes are included. Extended Response Provide a complete solution.. Sod costs $.9/m.. m.0 m.0 m a) Estimate how much it will cost to replace the sod in the small yard shown. b) Explain how you found your answer.. A hot tub is in the shape of a regular hexagon. A strip of padding goes around the perimeter..8 m.0 m a) Estimate the total length of the padding. b) Describe the strategy you used. Chapter Problem Wrap-Up Funderland is expanding and needs new rides. a) Design a new ride and create a problem based on it. Your problem can have more than one part, and it must require making an estimation and a calculation, each involving fractions or decimals. b) Solve your problem. c) Trade problems with a classmate and solve your classmate s problem. Then check each other s solutions. Practice Test MHR 07