Factors, Fractions, and Decimals

Transcription

1 CHAPTER 7 Factors, Fractions, and Decimals connected.mcgraw-hill.com Investigate Animations Vocabulary Math Songs The BIG Idea How do prime and composite numbers, factors, and multiples relate to fractions? Multilingual eglossary 290 Learn Personal Tutor Virtual Manipulatives Audio Foldables Practice Self-Check Practice egames Worksheets Assessment Make this Foldable to help you organize information about factors and multiples. Lesson 1 Prime Factorization and Exponents Review Vocabulary fraction fracción A number that represents part of a whole or part of a set Key Vocabulary English Español greatest common máximo común factor (GCF) divisor (MCD) least common mínimo común multiple múltiplo (mcm) composite number número compuesto prime number número primo exponent exponente 3 4

2 When Will I Use This? Your Turn! You will solve this problem in the chapter. Factors, Fractions, and Decimals 291

3 Are You Ready for the Chapter? You have two options for checking Prerequisite Skills for this chapter. Text Option Take the Quick Check below. Write all of the factors of each number List the first four multiples of each number Find a fraction that is equivalent to each fraction. 13. _ _ _ _ _ _ Online Option Take the Online Readiness Quiz. 292 Factors, Fractions, and Decimals

4 Multi-Part Lesson 1 Prime Factorization and Exponents PART A Main Idea I will explore using models and divisibility rules to identify prime and composite numbers. Vocabulary V prime number composite number Materials color tiles Get ConnectED GLE Write natural numbers (to 50) as a product of prime factors and understand that this is unique (apart from order). B C D E Prime and Composite Numbers Three bass drums are stored on shelves in these two arrangements These rectangular arrangements show that the only factors of 3 are 1 and When a number, like 3, has exactly two factors, the number is prime. 1 You can store 4 drums in any of the three ways shown at the right. What are the factors of 4? When a number, like 4, has more than two factors, the number is composite The numbers 0 and 1 are neither prime nor composite. Use models to determine whether 6 is prime or composite. U You can arrange the 6 color tiles in four different ways. So, 6 is a composite number. Lesson 1A Prime Factorization and Exponents 293

5 Use models to determine whether 5 is prime or composite You can arrange the 5 tiles in only 2 ways: 5 1 and 1 5. So, 5 is a prime number. You can also use divisibility rules to determine if a number is prime or composite. If a number is divisible by another number (besides 1 and itself), then it is composite. Divisibility Rules Even numbers are divisible by 2. A number is divisible by 3 if the sum of the digits is divisible by 3. Numbers divisible by 5 will end in a 0 or a 5. There is no common rule for numbers divisible by 7. Check by dividing. Examples 4, 8, 12, 16, 18, 182 are all divisible by = = is divisible by 3. 10, 15, 85, and 375 are all divisible by 5. 14, 21, 70, 140 are all divisible by 7. Use divisibility rules to determine if 117 is prime or composite. Try Is it a factor? 2 No, 117 is not even. 3 Yes, because = 9 and 9 3 = 3. So, 117 is a composite number because it is divisible by 3. Check = Factors, Fractions, and Decimals

6 Use divisibility rules to determine if 61 is prime or composite. Try Is it a factor? 2 No, 61 is not even. No, because = 7 3 and 7 is not divisible by 3. 5 No, 61 does not end in 0 or 5. 7 No, 61 7 has a remainder. The divisibility rules show that 61 is a prime number. About It 1. Are all even numbers composite? Use a drawing in your explanation. 2. Are all odd numbers prime? Support your explanation with a drawing. and Apply It Use color tiles or divisibility rules to determine whether each number is prime or composite Bruce made 12 dinner rolls. He placed the rolls in 3 rows of 4 on a table. In what other ways could he have arranged the rolls in equal rows? 10. Write a number between 20 and 30. Then use objects or pictures to show whether the number is prime or composite. 11. E WRITE MATH Is there a connection between the number of rectangular arrangements that are possible when modeling a number and the number of factors the number has? Explain your reasoning. Lesson 1A Prime Factorization and Exponents 295

7 Multi-Part Lesson 1 Prime Factorization and Exponents PART Main Idea I will identify prime and composite numbers. Vocabulary even number odd number A B C D E Prime and Composite Numbers In Lesson 1A, you learned that a composite number has more than two factors. So, 12 is a composite number because its factors are 1, 2, 3, 4, 6, and 12. Get ConnectED GLE Write natural numbers (to 50) as a product of prime factors and understand that this is unique (apart from order). The number 3 has only two factors: 1 and 3. So, 3 is a prime number. The numbers 1 and 0 are neither prime nor composite. 1 has only one factor: 1 0 has a never-ending number of factors: 0 1, 0 2,... Tell whether the number 10 is prime or composite. The model shows 2 rows of 5 squares. The squares could also be arranged in 5 rows of 2 squares, 10 rows of 1 square, or 1 row of 10 squares. The number 10 is a composite number because it has more than 2 factors. Use Divisibility Rules Tell whether 91 is prime or composite. Use divisibility rules. Try Is it a factor? 2 No, 91 is not even. No, because = 10 3 and 10 is not divisible by 3. 5 No, 91 does not end in 0 or 5. 7 Yes, 91 7 = 13 So, 91 is a composite number because it has more than two factors. 296 Factors, Fractions, and Decimals

8 You can use models to identify 24 as prime or composite. Twenty-four counters can be arranged in equal rows in more than two ways. So, 24 is composite. DINING A banquet hall has 24 square tables that are to be placed together to form a rectangle. Is 24 prime or composite? What does this mean in the problem? What would happen if the banquet hall had only 23 tables? factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Since 24 has more than two factors, it is a composite number. This means that there are more than two ways to arrange the 24 tables. Some of the ways are listed below. 1 row of 24 tables 2 rows of 12 tables 3 rows of 8 tables 4 rows of 6 tables If the banquet hall had only 23 tables, there could be only two possible arrangements, since 23 has only two factors. This is because 23 is a prime number. 1 row of 23 tables 23 rows of 1 table each Tell whether the number represented by each model is prime or composite. See Example Tell whether each number is prime or composite. Use objects or models to justify your answer. See Examples 1 and Tell whether each number is prime or composite. Use divisibility rules. See Example Is there more than one way for Mark to display 21 model cars if each row has the same number of cars? Explain. E 12. TALK MATH Is 33 prime or composite? Explain how you know. Lesson 1B Prime Factorization and Exponents 297

9 Tell whether the number represented by each model is prime or composite. See Example PRACTICE EXTRA Begins on page EP Tell whether each number is prime or composite. Use objects or models to justify your answer. See Examples 1 and Tell whether each number is prime or composite. Use divisibility rules. See Example , , A mountain range has 90 mountains that are one mile or more in height. Is 90 a prime or composite number? 34. Brian s birthday is February 29. Is 29 a prime or composite number? 35. FIND THE ERROR Rico is determining whether 119 is prime or composite using the divisibility rules. Help find and correct his mistake. Since 119 is not divisible by 2, 3, or 5, it must be prime. 36. CHALLENGE Two prime numbers that have a difference of 2 are called twin primes. For example, 5 and 7 are twin primes. Find all pairs of twin primes less than E WRITE MATH Explain how you can use objects or models to tell if a number is prime or composite. 298 Factors, Fractions, and Decimals

10 Test Practice 38. The table shows how many Calories you can burn in 10 minutes for certain activities. Activity Number of Calories Basketball 64 Dancing 35 Hiking 47 Roller skating 57 For which activity is the number of Calories a prime number? A. basketball B. dancing C. hiking D. roller skating 39. Rachel s birthday is a date in November that is a prime number. Which of the following could be her birthday? F. November 21 H. November 15 G. November 19 I. November Which group of numbers below are all composite? A. 6, 15, 28, 100 B. 3, 14, 37, 115 C. 13, 40, 52, 63 D. 13, 18, 77, The table shows the number of students participating in each activity. Activity Number of Students Baseball 24 Science Club 19 Student Council 17 Ski Club 23 For which activity is the number of students a composite number? F. baseball G. science club H. student council I. ski club Odd and Even Numbers An even number is a whole number that is divisible by 2. An odd number is not divisible by 2. An even number of objects can be divided into two equal sets; an odd number of objects cannot. 10 is an even number. Classify each number as even or odd , MAKE A CONJECTURE Is the sum of two odd numbers always, sometimes, or never an odd number? Use a model to explain your reasoning. 11 is an odd number. Lesson 1B Prime Factorization and Exponents 299

11 Multi-Part Lesson PART 1 Prime Factorization and Exponents C A B D E Main Idea I will find the prime factorization of numbers. Vocabulary prime factorization square numbers square root Prime Factorization You can write every composite number as a product of prime factors. This is called the prime factorization of a number. A factor tree is a diagram that shows the prime factorization of a composite number. Get ConnectED GLE Write natural numbers (to 50) as a product of prime factors and understand that this is unique (apart from order). SPI Write the prime factorization of numbers through 50 using both exponential and standard notation. Also addresses SPI AGE Mr. Dempsey tells his class that he is 36 years old. Find the prime factorization of 36. One Way Another Way 36 Write the number to be factored at the top Choose any pair of whole number factors of Continue to factor any number that is not prime Except for the order, the prime factors are the same The prime factorization of 36 is Check Check your answer by working backward. Multiply all the prime factors. Then compare your product with the composite number. 2 2 = 4, 4 3 = 12, 12 3 = Factors, Fractions, and Decimals

12 Prime Factorization Find the prime factorization of You can choose any pair of whole number factors, such as 6 4 or Except for the order, the prime factors of the number are the same. Choose any pair of whole number factors of 24. Continue to factor any number that is not prime The prime factorization of 24 is Prime Factorization Find the prime factorization of 37. Use divisibility rules. Neither 2, 3, 5, or 7 are factors of 37. So, 37 is a prime number. The prime factorization is 37. Find the prime factorization of each number. See Examples The state of Pennsylvania has 67 counties. Write the prime factorization of There are 45 students in the gymnasium. Find the prime factorization of 45. Pennsylvania 11. E TALK MATH What are the first ten prime numbers? Lesson 1C Prime Factorization and Exponents 301

13 Find the prime factorization ti of each number. See Examples PRACTICE EXTRA Begins on page EP Use the table that shows the average weights of popular dog breeds. 24. Which weight(s) have a prime factorization of exactly three factors? 25. Which weight(s) have a prime factorization with factors that are all the same number? 26. Which dog breeds have weights that are prime numbers? Breed Weight (lb) Cocker Spaniel 20 German Shepherd 81 Labrador Retriever 67 Beagle 25 Golden Retriever 70 Siberian Husky 50 Boxer 60 Rottweiler 112 Dalmatian 55 Poodle Of the Beagle, Golden Retriever, Siberian Husky, Rottweiler, and Dalmatian breeds, which have weights that are composite numbers? 28. CHALLENGE Find the prime factorization of 2, WHICH ONE DOESN T BELONG? Which of the numbers below is not a prime factor of 70? REASONING Explain why the prime factorization is for the same number as the prime factorization E WRITE MATH Explain how tree diagrams help you find the prime factorization of a number. 302 Factors, Fractions, and Decimals

14 Factors Numbers that have two identical factors are called square numbers. For example, 9 is a square number. 3 3 = 9 A square root of a number is one of two identical factors of a number. The square root of 9 is 3. The table shows other examples of square numbers and square roots. Model Multiplication Fact Square Number Square Root 2 2 = = Name the square number and square root shown in each model Use the multiplication fact 7 7 = 49 to name a square number and its square root. 35. What is the largest square number less than 200? 36. MAKE A CONJECTURE The prime factorization of 100 is Explain how to find a square root of 100 using the prime factorization. Lesson 1C Prime Factorization and Exponents 303

15 Multi-Part Lesson PART 1 Prime Factorization and Exponents A B C D E Exponents Main Idea I will explore using exponents. Materials hole punch construction paper Get ConnectED GLE Use mathematical language, symbols, and definitions while developing mathematical reasoning. SPI Write the prime factorization of numbers through 50 using both exponential and standard notation. Also addresses GLE Any number can be written as a product of prime factors. Step 1 Fold a piece of paper in half and make one hole punch. Open the paper and count the number of holes. Copy the table below and record the results. Number of Folds 1 5 Number of Holes Prime Factorization Step 2 Find the prime factorization of the number of holes and record the results in the table. Step 3 Fold another piece of paper in half twice. Then make one hole punch. Complete the table for two folds. Step 4 Complete the table for three, four, and five folds. About It 1. What prime factors did you record? 2. How does the number of folds relate to the number of factors in the prime factorization of the number of holes? 3. Write the prime factorization of the number of holes made if you folded it eight times. 304 Factors, Fractions, and Decimals

16 Multi-Part Lesson PART 1 Prime Factorization and Exponents A B C D E Main Idea I will use powers and exponents in expressions. Vocabulary exponent base power squared cubed Get ConnectED GLE Use mathematical language, symbols, and definitions while developing mathematical reasoning. SPI Recognize equivalent representations for the same number. Also addresses SPI Powers and Exponents A product of identical factors can be written using an exponent and a base. The base is the number used as a factor. The exponent indicates how many times the base is used as a factor = factors base exponent Numbers expressed using exponents are called powers. Numbers raised to the second or third power have special names. Powers Words to the fifth power to the second power or 3 squared to the third power or 10 cubed Use Exponents Write using an exponent. The base is 3. Since 3 is used as a factor four times, the exponent is = 3 4 Write as a power. FOOD The number of Calories in two pancakes can be written as 7 3. Write 7 3 as a product of the same factor. Then find the value. Write 7 3 as = 343 Two pancakes have 343 Calories. Lesson 1E Prime Factorization and Exponents 305

17 You can calculate expressions with a base of 10 mentally = 10,000 4 zeros ENVIRONMENT In a recent year, about 10 4 youth across the United States participated in activities and events to care for Earth s environment. What is this number? 10 4 = Write 10 4 as a product. = 10,000 Multiply. About 10,000 youth participated in these events. Prime Factorization Using Exponents Write the prime factorization of 72 using exponents So, 72 = Order factors from least to greatest. Write products of identical factors using exponents. Write each product using an exponent. See Examples Write each power as a product of the same factor. Then find the value Write the prime factorization of each number using exponents There are nearly 3 5 species of monkeys on Earth. What is the value of 3 5? E 9. TALK MATH Explain how a factor tree helps you to write the prime factorization of a number using exponents. 306 Factors, Fractions, and Decimals

18 Write each product using an exponent. See Examples PRACTICE EXTRA Begins on page EP Write each power as a product of the same factor. Then find the value A single tusk that weighed just over 2 8 pounds from an African elephant is the largest tooth ever recorded from any modern animal. About how many pounds did the tusk weigh? Write the prime factorization of each number using exponents , To find the amount of space a cube-shaped bird cage occupies, find the cube of the measure of one edge of the bird cage. Express the amount of space occupied by the bird cage shown as a power. Then find the amount in cubic units. 18 units 18 units 18 units 33. OPEN ENDED Write a power whose value is greater than NUMBER SENSE Which is greater: 3 5 or 5 3? Explain your reasoning. 35. E WRITE MATH Explain how to find 10 6 mentally. Lesson 1E Prime Factorization and Exponents 307

19 Multi-Part Lesson PART 2 Main Idea I will find common factors using Venn diagrams. Fractions in Simplest Form A B C Common Factors A Venn diagram uses circles to display elements of different sets. Overlapping circles show common elements. Get ConnectED SUMMER CAMP The Venn diagram shows which activities each camper participated in on Monday. Who participated in both swimming and crafts? GLE Write natural numbers (to 50) as a product of prime factors and understand that this is unique (apart from order). SPI Write the prime factorization of numbers through 50 using both exponential and standard notation. This circle represents swimming. Swimming Crafts This circle represents crafts. This part represents both swimming and crafts. Owen and Isabel are in both circles. So, they participated in both swimming and crafts. Factors that are shared by two or more numbers are called common factors. The common factors of 12 and 20 are: 1, 2, and 4. About It 1. Use a Venn diagram to find the common factors of 30 and E TALK MATH Explain what it means if a factor is in both circles of a Venn diagram. 308 Factors, Fractions, and Decimals

20 Multi-Part Lesson PART 2 Fractions in Simplest Form A B C Main Idea Find the greatest common factor of two or more numbers. Vocabulary common factor greatest common factor (GCF) Greatest Common Factor Factors shared by two or more numbers are called common factors. The greatest of the common factors of two or more numbers is the greatest common factor (GCF) of the numbers. Get ConnectED GLE Write natural numbers (to 50) as a product of prime factors and understand that this is unique (apart from order). SPI Write the prime factorization of numbers through 50 using both exponential and standard notation. Also addresses GLE Identify Common Factors Identify the common factors of 16 and 24. First, list the factors by pairs Factors of 16 for each number. Then, 1 16 circle the common factors. The common factors are 1, 2, 4, and Factors of Find the GCF by Listing Factors Find the GCF of 60 and 54. Make an organized list of the factors for each number. factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 factors of 54: 1, 2, 3, 6, 9, 18, 27, 54 The common factors are 1, 2, 3, and 6. So, the greatest common factor, or GCF, of 60 and 54 is 6. Check Use a Venn diagram to show the factors. The common factors are 1, 2, 3, and 6. Lesson 2B Fractions in Simplest Form 309

21 You can use prime factorization to determine the greatest common factor. Divisibility tests are a good way to find factors. 18 and 30 are even numbers. They are both divisible by 2. Find the GCF by Using Prime Factorization Find the GCF of 18 and 30. Write the prime factorization and 3 are common factors. The common prime factors are 2 and 3. So, the GCF of 18 and 30 is 2 3 or 6. FOOD A bakery arranges Muffins three different types of Type Number muffins in a display case. blueberry 40 There should be an equal cinnamon raisin 24 number of muffins in chocolate chip 32 each row in the case. What is the greatest possible number of muffins in each row? prime factorization of 40: prime factorization of 24: prime factorization of 32: The common prime factors are 2, 2, and 2. The GCF of 40, 24, and 32 is or 8. So, the greatest number of muffins that could be placed in each row is 8. How many rows of muffins are there if there are 8 in each row? There are a total of , or 96 muffins. So, the number of rows of muffins is 96 8, or Factors, Fractions, and Decimals

22 Identify the common factors of each set of numbers. See Example , , 21, 30 Find the GCF of each set of numbers. See Examples , , , 12, , 10, 14 Solve. See Example 5 7. Oliver has 14 chocolate cookies and 21 iced cookies. Oliver gives each of his friends an equal number of each type of cookie. What is the greatest number of friends with whom he can share his cookies? E 8. TALK MATH Refer to Exercise 7. Explain how you could find how many cookies each friend would receive. Then solve. Identify the common factors of each set of numbers. See Example , , 90 PRACTICE EXTRA Begins on page EP , 21, , 24, 40 Find the GCF of each set of numbers. See Examples , , , , , 35, , 18, , 52, , 30, Annika is placing photos in a scrapbook. Each page will have only one size of photo. She also wants to place the same amount of photos on each page. What is the greatest number of photos that could be on each page? Justify your response. Scrapbooking PHOTO SIZE Large Medium Small AMOUNT Lesson 2B Fractions in Simplest Form 311

23 22. A grocery store sells boxes of juice in equal-size packs. Carlos bought 18 boxes, Rico bought 36 boxes, and Winston bought 45 boxes. What is the greatest number of boxes in each pack? How many packs did each person buy? 23. The table shows the number of each type of toy in a store. The toys will be placed on shelves so that each shelf has the same number of each type of toy. How many shelves are needed for each type of toy so that it has the greatest number of toys? Toy Number dolls 45 footballs 105 small cars The table shows the amount of money Ms. Ayala made over three days selling 4-by-6-inch prints at an arts festival. Each print costs the same amount. What is the most each print could have cost? Ms. Ayala s Artwork Day Amount ($) Friday 60 Saturday 144 Sunday What is the GCF of all the numbers in the pattern 9, 18, 27, 36,...? Explain your reasoning. Use the information to solve the problem. 26. What is the length of the longest piece of sandwich that can be cut so that all 16 guests get the same-sized sandwich? Explain. 312 Factors, Fractions, and Decimals

24 27. CHALLENGE Determine whether each statement is true or false. If true, explain why. If false, give a reason. a. The GCF of any two even numbers is always even. b. The GCF of any two odd numbers is always odd. c. The GCF of an odd number and an even number is always even. 28. WHICH ONE DOESN T BELONG? Which number can you take away so that 8 will be the GCF? E WRITE MATH Which method would you prefer to use to find the GCF of 48, 64, and 144? Explain your reasoning. Test Practice 30. SHORT RESPONSE Find the greatest common factor of the numbers below. 28, 42, Which number is NOT a common factor of 24 and 36? A. 2 B. 6 C. 12 D Jeremiah will share his collection with his brother so that they each have the same number of each type of card. What is the greatest number of baseball cards they will each have? Sports Cards Type Number baseball 32 football 48 F. 4 cards H. 12 cards G. 8 cards I. 16 cards 33. California has 5 2 area codes. What is the value of 5 2? (Lesson 1E) Find the prime factorization of each number. (Lesson 1C) Lesson 2B Fractions in Simplest Form 313

25 Multi-Part Lesson PART 2 Fractions in Simplest Form A B C Main Idea I will use the GCF to write a fraction in simplest form. Vocabulary simplest form Simplest Form A fraction is written in simplest form when the GCF of the numerator and the denominator is 1. The simplest form of a fraction is one of its many equivalent fractions. Get ConnectED GLE Use mathematical language, symbols, and definitions while developing mathematical reasoning. SPI Recognize equivalent representations for the same number. MEASUREMENT A praying mantis is 12 centimeters long, and a walking stick is 22 centimeters long. So, a praying mantis is 12_ 22 of the length of a walking stick. Write the fraction in simplest form. Step 1 Find the GCF of the numerator and the denominator. factors of 12: 1, 2, 3, 4, 6, 12 factors of 22: 1, 2, 11, 22 The GCF of 12 and 22 is 2. Step 2 Divide both the numerator and the denominator by the GCF. Dividing both the numerator and the denominator by the same number is equivalent to dividing by one. _ = _ = _ 6 11 The GCF of 6 and 11 is 1. So, a praying mantis s length is _ 6 of the length of a 11 walking stick. Check Use models. So, _ = _ Factors, Fractions, and Decimals

26 18_ Simplest Form Write in simplest form. 30 Equivalent fractions are fractions that have the same value. 18_ 30 = 3_ 5 These fractions are equivalent. One Way: Divide by Common Factors 18_ 30 = 18 2 _ 30 2 = 9_ 15 9_ 15 = 9 3 _ 15 3 = 3_ 5 Divide 18 and 30 by the common factor 2. Divide 9 and 15 by the common factor 3. Since 3 and 5 have no common factors other than 1, stop dividing. Another Way: Divide by the GCF factors of 18: 1, 2, 3, 6, 9, 18 factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 The GCF of 18 and 30 is 6. 18_ 30 = _ = 3_ 5 Divide by the GCF 6. Using either method, _ written in simplest form is _ 3 5. Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. See Examples 1 and 2 1. _ 4 2. _ 2 3. _ _ _ _ _ _ Kara buys 24 bagels. Ten are whole wheat. What fraction of the bagels are whole wheat, in simplest form? E 10. TALK MATH Use at least two sentences to explain how to find the simplest form of any fraction. Lesson 2C Fractions in Simplest Form 315

27 Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. See Examples 1 and _ _ _ _ 5 PRACTICE EXTRA Begins on page EP _ _ _ _ _ _ _ _ _ _ _ A basket of fruit has 10 oranges, 12 apples, and 18 peaches. Express in simplest form the fraction of fruit that are oranges. 28. Measurement Andeana is 4 feet tall. Her brother Berto is 38 inches tall. What fractional part of Andeana s height is Berto s height? 29. In a typical symphony orchestra, 16 out of every 100 musicians are first and second violin players. Express the fraction of the orchestra that are violinists in simplest form. 30. The table shows the results of a survey about favorite movie theater snacks. Write a fraction in simplest form that compares the number of people who chose popcorn to the total number of people surveyed. Favorite Movie Snack Snack Frequency popcorn 24 hot dog 12 nachos 11 chocolate 8 licorice OPEN ENDED Write a real-world problem that uses _ 14 in the 18 problem. Write the fraction in simplest form. 32. WHICH ONE DOESN T BELONG? Identify the fraction that does not belong with the other three. Explain your reasoning. 3_ 12 4_ 16 5_ 25 6_ E WRITE MATH Explain how you would write _ 24 in simplest form Factors, Fractions, and Decimals

28 Test Practice 34. Gil s aunt cut his birthday cake into 32 equal pieces, as shown below. Eighteen pieces were eaten at his birthday party. What fraction of the cake was left? 36. The fractions _ 2 8, _ 3 12, _ 4 16, and _ 5 20 can all be simplified to _ 1. What is the 4 relationship between the numerator and denominator in each fraction? F. The numerator is 4 times the denominator. A. _ 7 16 B. _ 9 16 C. _ 7 12 D. _ 9 14 G. The denominator is 4 times the numerator. H. The numerator is 4 more than the denominator. I. The denominator is 4 more than the numerator. 35. GRIDDED RESPONSE Amelia rode _ 12 mile on the bike trail. 20 What is the greatest common factor of 12 and 20? 37. SHORT RESPONSE Joshua answered 95 out of 100 test questions correctly. Express the fraction of correct answers in simplest form. 38. Thirty-six fourth graders, 48 fifth graders, and 24 sixth graders will attend a play. An equal number of students must sit in each row, and only students from the same grade can sit in a row. What is the greatest number of fifth graders that can sit in each row? (Lesson 2B) Write each product using an exponent. (Lesson 1E) Grant has $225 in his savings account. Write the prime factorization of 225. (Lesson 1C) 43. A tangerine has about 37 Calories. Is 37 prime or composite? (Lesson 1B) To assess mastery of SPI , see your Tennessee Assessment Book. 317

29 Mid-Chapter Check Tell whether each number is prime or composite. (Lesson 1B) MULTIPLE CHOICE Which model does NOT represent a composite number? (Lesson 1B) A. Find the GCF of each set of numbers. (Lesson 2B) 15. 9, , , 30, , 24, MULTIPLE CHOICE Devin recorded the shirt color of the 30 students who rode his bus on Monday. The results are shown below. B. C. D. Find the prime factorization of each number. (Lesson 1C) Identify the common factors of each set of numbers. (Lesson 2B) 10. 5, , , 32, , 22, MULTIPLE CHOICE Which group shows all the numbers that are common factors of 24 and 40? (Lesson 2B) F. 1, 2, 4 H. 1, 2, 4, 6 G. 1, 2, 4, 8 I. 1, 2, 4, 6, 8, 12 Which fraction of shirts were red? (Lesson 2C) A. _ 1 3 B. _ 1 5 C. 1_ 4 D. _ 1 6 Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. (Lesson 2C) _ 24 9_ _ _ E WRITE MATH Explain how to write 4 3 as a product of its factors. Then find its value. (Lesson 1E) 318 Mid-Chapter Check

30 Multi-Part Lesson PART 3 Write Multiples and Compare Fractions A B C D Least Common Multiple Main Idea I will explore finding the least common multiple of two numbers. Materials color tiles Step 1 Draw a number line from 0 to Step 2 Find the product of 2 and each of the numbers 1, 2, 3, 4, 5, 6, and 7. Place a red tile above each of the products on the number line Get ConnectED GLE Write natural numbers (to 50) as a product of prime factors and understand that this is unique (apart from order). Step 3 Find the product of 3 and each of the numbers 1, 2, 3, 4, and 5. Place a blue tile above each of the products on the same number line About It 1. Which numbers have both a red and a blue tile? 2. What is the least number that has a red and blue tile? and Apply It Use a number line and color tiles to find the least number that is a product of each of the numbers. 3. 2, , , 6 6. E WRITE MATH Explain how you can use color tiles to find the common products of 2, 4, and 5. Lesson 3A Write Multiples and Compare Fractions 319

31 Multi-Part Lesson PART 3 Write Multiples and Compare Fractions A B C D E F G Main Idea Find the least common multiple of two or more numbers. Vocabulary multiple common multiples least common multiple (LCM) Get ConnectED GLE Write natural numbers (to 50) as a product of prime factors and understand that this is unique (apart from order). Least Common Multiple A multiple of a number is the product of the number and any other whole number (0, 1, 2, 3, 4,... ). Multiples that are shared by two or more numbers are common multiples. Identify Common Multiples Identify the first three common multiples of 4 and 8. First, list the nonzero multiples of each number. multiples of 4: 4, 8, 12, 16, 20, 24, , 2 4, 3 4, multiples of 8: 8, 16, 24, 32, 40, 48, , 2 8, 3 8, The first three common multiples of 4 and 8 are 8, 16, and 24. The least common multiple (LCM) is the least multiple, other than 0, common to sets of multiples. FOOD Ben s Burgers gives away a free order of fries every 2 days, a free milkshake every 3 days, and a free hamburger every 4 days. If they gave away all three items today, in how many days will they give away all three items again? Find the LCM of 2, 3 and 4. Fries Shake multiples of 2: 2, 4, 6, 8, 10, , 2 2, 2 3, 2 4, multiples of 3: 3, 6, 9, 12, 15, , 3 2, 3 3, 3 4, multiples of 4: 4, 8, 12, 16, , 4 2, 4 3, 4 4, Notice that 12 is the least common multiple of 2, 3, and 4. So, Ben s Burgers will give away all three items again in 12 days. Draw a number line to check. H H M H M H M F F M F F F M F F F M Day Factors, Fractions, and Decimals

32 Find the LCM You can use a factor tree to find the prime factorization. Find the LCM of 15 and 40. Step 1 Write the prime factorization of each number Step 2 Identify all common prime factors. 15 = = is a common prime factor. Step 3 Find the product of the prime factors using each common prime factor only once and any remaining factors. The LCM is or 120. Identify the first three common multiples of each set of numbers. See Examples , , , 4, , 6, 12 Find the LCM of each set of numbers. 5. 6, , , 7, , 7, 9 9. Juan gets an allergy shot every 3 weeks. Percy gets an allergy shot every 5 weeks. If Juan and Percy meet while getting an allergy shot, how many weeks will it be before they see each other again? 10. E TALK MATH Could the LCM of two numbers be one of the numbers? Explain. Support your answer with an example. Lesson 3B Write Multiples and Compare Fractions 321

33 PRACTICE EXTRA Begins on page EP2. Identify the first three common multiples l of each set of numbers. See Examples , , , , , 8, , 9, 18 Find the LCM of each set of numbers , , , , , 25, , 12, A full moon occurs about every 30 days. If the last full moon occurred on a Friday, how many days will pass before a full moon occurs again on a Friday? 24. The cycles for two different events are shown in the table. Each of these events happened in the year What is the next year in which both will happen? Event Cycle (yr) Summer Olympics 4 United States Census FIND THE ERROR Maria is finding the LCM of 6 and 8. Help find and correct her mistake. 6 = = The LCM of 6 and 8 is CHALLENGE Is the statement below always, sometimes, or never true? Give at least two examples to support your reasoning. The LCM of two numbers is the product of the two numbers. 27. E WRITE MATH Write a real-world problem in which it would be helpful to find the least common multiple. 322 Factors, Fractions, and Decimals

34 Test Practice 28. Micah is buying items for a birthday party. If he wants to have the same amount of each item, what is the least number of packages of cups he needs to buy? Party Supplies Item Number in Each Package cups 6 plates 8 A. 2 packages B. 3 packages C. 4 packages D. 5 packages 29. What is the least common multiple of 5, 9, and 15? F. 3 H. 45 G. 29 I Look at the patterns in each sequence below. Each sequence is an example of which kind of numbers? 3, 6, 12, 24, 48 5, 10, 20, 40, 80 8, 16, 32, 64, 128 A. even numbers C. multiples B. odd numbers D. prime numbers 31. A container of bagels has 10 plain, 5 blueberry, 6 poppy seed, and 3 mixed grain bagels. What fraction of bagels are poppy seed? Write in simplest form. (Lesson 2C) Find the GCF of each set of numbers. (Lesson 2B) 32. 9, , , , , , 28, and 42 Write each product using an exponent. (Lesson 1E) Denzel has a collection of 149 magnets. Is 149 a prime or composite number? (Lesson 1B) Lesson 3B Write Multiples and Compare Fractions 323

35 Multi-Part Lesson 3 Write Multiples and Compare Fractions PART A B C D Problem-Solving Strategy: Look for a Pattern Main Idea I will solve problems by looking for a pattern. Shawna is saving money to buy an airplane ticket to visit her aunt. Each month she puts money into her savings account. Based on the pattern in the table, determine how much money Shawna will have in July. Month Total in Savings January $35 February $70 March $105 April $140 Understand What facts do you know? We know how much money Shawna has saved for four months. The amount in her account increases according to a pattern. Plan What do you need to find? The amount of money in Shawna s account in July. One way to solve the problem is by looking for a pattern. Then extend the pattern to find the amount of money in her account in July. Solve Use your plan to solve the problem. Jan $35 Feb $70 Mar $105 Apr $140 May June July The amount in Shawna s savings account increases each month by $35. Continue the pattern to find the total in July. Jan $35 Feb $70 Mar $105 Apr $140 May $175 June $210 July $245 In July, Shawna will have $245 in her savings account Check Since July is the 7th month find the first seven multiples of 35. They are 35, 70, 105, 140, 175, 210, and 245. GLE Write natural numbers (to 50) as a product of prime factors and understand that this is unique (apart from order). GLE Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution. Also addresses GLE Factors, Fractions, and Decimals

36 Refer to the problem on the previous page. 1. How much money will Shawna have in her account in August? 2. If the airline ticket costs $315, when can Shawna stop saving? 3. Explain when to use the look for a pattern strategy to solve a problem. 4. Can you always use the look for a pattern strategy when solving a problem? PRACTICE EXTRA Begins on page EP2. Solve. Use the look for a pattern strategy. 5. Draw the next two figures in the pattern. For Exercises 8 10, use the following information. Gavin rode his bike for a longer distance each day while training. Here is his record of the number of miles he rode. Mon Tues Wed Thurs Fri 3.5 mi 4.2 mi 5.0 mi 6.9 mi 6. Stefano is buying a few pencils. The table shows the price of different numbers of pencils. 8. Based on Gavin s pattern, how long did he ride on Thursday? 9. Algebra If the pattern continues, how far will Gavin ride on Saturday? 10. Explain how to find the number of miles Gavin will ride on Sunday, if the pattern continues. What is the relationship between the number of pencils and price? 7. Measurement Cheryl is filling a pool. She measures the depth in feet every 5 minutes. Her measurements are 2.5, 3.6, 4.7, and 5.8. If this pattern continues, how deep will the water be the next time she measures? 11. The Fibonacci sequence is a famous pattern of numbers. The first seven numbers in the Fibonacci sequence are 1, 1, 2, 3, 5, 8, and 13. Find the next three numbers. Explain the pattern. 12. E WRITE MATH Write a real-world problem that uses the look for a pattern strategy. Use the pattern below. 2.45, 2.8, 3.15, 3.5,... Lesson 3C Write Multiples and Compare Fractions 325

37 Multi-Part Lesson PART 3 Write Multiples and Compare Fractions D A B C E Main Idea I will compare fractions using common denominators. Vocabulary least common denominator (LCD) Get ConnectED Compare Fractions If two fractions have the same denominator, you can compare them by comparing the numerators. If the fractions have different denominators, first write equivalent fractions with the least common denominator. The least common denominator (LCD) is the least common multiple of the denominators of the fractions. GLE Use mathematical language, symbols, and definitions while developing mathematical reasoning. SPI Compare whole numbers, decimals, and fractions using the symbols <, >, and =. 3_ Compare 5 and 1_ 2 Compare Fractions using the least common denominator. Step 1 Find the LCM of the denominators. The LCM of 5 and 2 is 10. Step 2 Find equivalent fractions with a denominator of 10. 3_ 5 = _ _ 2 = _ 5 10 THINK 5 2 = 10, 3 2 = 6 THINK 2 5 = 10, 1 5 = 5 Step 3 Compare the numerators. Since 6 > 5, then 6 _ 10 > 5 _ 10. So, 3 _ 5 > 1 _ 2. Check The models show that 3 _ 5 > 1_ You can always multiply the denominators of two fractions to find a common denominator. But, this method does not always give the LCD. 326 Factors, Fractions, and Decimals

38 You can also find a set of equivalent fractions for 7_ 5_ and to find 9 the LCD. 7_ 6 9, 14 18, 21 27, , 10 12, 15 18, 20 24,... 7_ Compare Fractions Using the LCD 5_ Compare and using the least common denominator. 9 6 Step 1 Find the LCM of the denominators. The LCM of 9 and 6 is 18. Step 2 Find equivalent fractions with a denominator of 18. 7_ 9 = _ = THINK 9 2 = 18, 7 2 = 14 THINK 6 3 = 18, 5 3 = 15 Step 3 Compare the numerators. Since 14 < 15, then _ < _ So, _ 7 9 < _ 5 6. SPORTS Trevor made 2 out of 3 field goals and Tyler made 5 out of 6 field goals. Who made a greater fraction of field goals? Step 1 Find the LCM of the denominators. The LCM of 3 and 6 is 6. Step 2 Find equivalent fractions with a denominator of 6. 2_ 3 = _ 4 6 THINK 3 2 = 6, 2 2 = 4 5_ 6 = _ 5 6 THINK 6 1 = 6, 5 1 = 5 Step 3 Compare the numerators. Since 5 > 4, then _ 5 6 > _ 4 6. So, _ 5 6 > _ 2 3. Tyler made a greater fraction of field goals. Check The models show that 5 _ 6 > 2_ Lesson 3D Write Multiples and Compare Fractions 327

39 Compare each pair of fractions using models or the LCD. See Examples _ 5 and _ _ 2 and _ _ 4 and 7 _ _ 3 and 7 _ 10 Algebra Replace each with <, >, or = to make a true statement. See Examples _ 3 5 _ _ 3 7 _ A recipe calls for _ 5 cup of brown sugar 8 and _ 2 cup of flour. Which ingredient is 3 greater? 7. 1 _ 4 1 _ 6 E 8. 2 _ 5 6 _ TALK MATH Explain how the LCM and the LCD are alike. How are they different? Compare each pair of fractions using models or the LCD. See Examples _ 2 3 and _ _ and _ _ 15 6 and _ _ 5 and 5 _ _ 7 8 and _ _ 3 10 and _ 1 12 PRACTICE EXTRA Begins on page EP _ 2 5 and _ _ 6 and 4 _ 9 Algebra Replace each with <, >, or = to make a true statement. See Examples _ 5 3 _ _ 2 6 _ _ 3 4 _ _ _ A trail mix has 0.5 cup of raisins and _ 2 cup of peanuts. Which 3 ingredient is greater? 21. _ 1 5 _ _ 3 8 _ _ 2 6 _ _ _ A survey showed that _ 7 of the class 15 liked soccer and 0.4 liked basketball. Which sport was liked less? 29. The amounts of water four runners drank are shown at the right. Who drank the most? 30. The fifth graders were given sandwiches for lunch during their field trip. Nathan ate _ 5 6 of his sandwich, Leroy ate _ 7 8 of his sandwich, and Sofia ate _ 5 of her 8 sandwich. Who had the least amount of sandwich left to eat? Evita Jack Keisha Sirjo Factors, Fractions, and Decimals

40 31. OPEN ENDED Replace with a number to make _ statement. 24 > _ 1 4 a true 32. NUMBER SENSE Suppose two fractions have the same numerator and different denominators. How can you decide which fraction is greater without finding the LCD? 33. E WRITE MATH Write a real-world problem that can be solved by comparing two fractions with different denominators. Then solve. Support your answer with a model. Test Practice 34. The table shows the cost of renting a bicycle. If the pattern continues, how much will it cost to rent a bicycle for 6 hours? Number of Hours Cost ($) A. $6 C. $36 B. $32 D. $ Eighteen out of 24 of Emil s CDs are country music. Five out of 8 of Imani s CDs are country music. Which is a true statement? F. Both of their CD collections are half country music. G. Both of their CD collections are less than half country music. H. Emil s collection is closer to half country than Imani s collection. I. Imani s collection is closer to half country than Emil s collection. 36. Find the missing number in the pattern 1, 2, 4, 7,, 16,.... (Lesson 3C) Find the first two common multiples of each pair of numbers. (Lesson 3B) 37. 4, , , , The table shows the number of games lost by the girls basketball team in three months. The fraction _ 4 represents the 16 losses. Write this fraction in simplest form. (Lesson 2C) Number Number of Games of Losses 16 4 To assess partial mastery of SPI , see your Tennessee Assessment Book. 329

41 Multi-Part Lesson PART 4 Fractions and Decimals A B C D E Main Idea I will explore using models to write fractions as decimals. Fractions and Decimals You can use models to write fractions in their equivalent decimal form. Get ConnectED GLE Move flexibly between concrete and abstract representations of mathematical ideas in order to solve problems, model mathematical ideas, and communicate solution strategies. Use a model to write 1_ 2 Step 1 Write 1_ 2 1_ 50 _ 2 = as a decimal. Step 2 _ Shade a model of as a fraction with a denominator of 100. Since 2 50 = 100, multiply Since 50 out of the 100 squares are shaded, the model shows 50 hundredths or So, 1 _ 2 = Factors, Fractions, and Decimals About It 1. How would the Activity change if _ 1 was written as a 2 fraction with a denominator of 10? Would the result be the same? Explain. Use a model to write each fraction as a decimal. 2. _ 1 3. _ _ _ 20

42 Multi-Part Lesson 4 Fractions and Decimals PART A C D E B Main Idea I will use equivalent fractions to write fractions as decimals. I will write decimals as fractions. Get ConnectED GLE Use mathematical language, symbols, and definitions while developing mathematical reasoning. SPI Recognize equivalent representations for the same number. Fractions and Decimals Fractions with denominators that are factors of 10, 100, or 1,000 can be written as decimals by writing equivalent fractions. 3_ Write Fractions as Decimals Write as a decimal. 4 Since 4 is a factor of 100, write an equivalent fraction with a denominator of _ 25 4 = _ Since 4 25 = 100, multiply = 0.75 Read 0.75 as seventy-five hundredths. HONEYBEES The average length of a honeybee is 0.8 inch. Write this length as a fraction in simplest form. You can use a place-value chart. The place value of the last decimal is tenths. Ones Tenths Hundredths = _ 8 10 = _ = 4 _ 5 Say eight tenths. Divide the numerator and denominator by the GCF, 2. Simplify. The length of a honeybee is 4 _ 5 inch. Lesson 4B Fractions and Decimals 331

43 Write each fraction as a decimal. See Example 1 1. _ 1 2. _ _ _ 10 Write each decimal as a fraction in simplest form. See Example Yesterday it rained 0.45 inch. Write 0.45 as a fraction in simplest form. E 10. TALK MATH Explain how to write a fraction as a decimal using equivalent fractions. Write Wit each fraction as a decimal. See Example _ _ _ _ _ _ _ _ PRACTICE EXTRA Begins on page EP2. Write each decimal as a fraction in simplest form. See Example At basketball practice, Savannah spent _ 19 of an hour practicing 20 free throws. Write _ 19 as a decimal Paolo made a model of his house that is 0.08 the size of his actual house. What fraction of the actual house length is the model? Write the fraction in simplest form. The smallest known female spider is 0.46 millimeter long. The smallest male spider is 0.37 millimeter long. Write each decimal as a fraction in simplest form Factors, Fractions, and Decimals