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Copyright © Wright Group/McGraw-Hill Teaching Masters and Study Link Masters 1 Teaching Masters and Study Link Masters STUDY LINK117 Volume and Surface Area 341 Name Date Time Copyright © Wright Group/McGraw-Hill 1.Kesia wants to give her best friend a box of chocolates. Figure out the least number of square inches of wrapping paper Kesia needs to wrap the box. (To simplify the problem, assume that she will cover the box completely with no overlaps.) Amount of paper needed: Explain how you found the answer. Sample answer: I found the area of each of the 6 sides and then added them together. 2.Could Kesia use the same amount of wrapping paper to cover a box with a larger volume than the box in Problem 1? Explain. A 4 in. 4 in.  3 1 2in. box has a volume of 56 in 3and a surface area of 88 in 2. Find the volume and the surface area of the two figures in Problems 3 and 4. 3.Volume: 4.Volume: Surface area: Surface area: 216 in 2 351.7 cm 2 216 in 3 502.4 cm 3 Yes 88 in 2 6 in. 2 in. 4 in. 10 cm 8 cm 6 in. cube 197 198 200 201 Area of rectangle: Alº w Volume of rectangular prism: Vlº wº hCircumference of circle: cº d Area of circle: Aº r 2 Volume of cylinder: Vº r 2º h Name Date Time 10 Copyright © Wright Group/McGraw-Hill A magic square is an array of positive whole numbers. The sum of the numbers in each row, column, and diagonal will be the same. 1.Complete this magic square.A heterosquare is like a magic square, except that the sum of the numbers in each row, column, and diagonal are different. A 3-by-3 array for a heterosquare will have an arrangement of the numbers 1–9. 2.Complete this heterosquare, and write the sum for each row, column, and the two diagonals. LESSON12 Magic Square and Heterosquare Arrays A rectangular array is an arrangement of objects in rows and columns. The objects in an array can be numbers or numerical expressions. The Multiplication/Division Facts Table on the inside front cover of your journal is an example of numbers arranged in an array. The objects can also be words or symbols that represent elements of a given situation. For example, a plan for after-school snacks could be arranged in a 1-by-5 array, using Afor apple, Bfor banana, and so on. 2 1010 910 9 12 1434 3434 11 85 31 85 3 3.Create a magic square or heterosquare for your partner to solve. Teaching Masters and Study Link Masters

2 Copyright © Wright Group/McGraw-Hill Name Date Time STUDY LINK 11 Number Poetry Many poems have been written about mathematics. They are poems that share some of the ways that poets think about numbers and patterns. 1. Read the examples below. 2. The ideas in the examples are some of the ideas you have studied in Everyday Mathematics.Subtraction is one of these ideas. Name as many other ideas from the examples as you can on the back of this page. Examples: Arithmetic is where numbers fly like pigeons in and out of your head. Arithmetic tells you how many you lose or win if you know how many you had before you lost or won. from “Arithmetic” by Carl Sandburg A square is neither a line nor circle; it is timeless. Points don’t chase around a square. Firm, steady, it sits there and knows its place. A circle won’t be squared. from “Finding Time” by JoAnne Growney from “Marvelous Math” by Rebecca Kai Dotlich 3. Use a number pattern to make your own poem on the back of this page.Second Poem: “123” . 1 12 123 1-32 1-21 1-10 2 21 21-31 2131 21-31-231 121 1 . from “Asparagus X Plus Y” by Ken Stange How many seconds in an hour? How many in a day? What size are the planets in the sky? How far to the Milky Way? How fast does lightning travel? How slow do feathers fall? How many miles to Istanbul? Mathematicsknows it all!

3 Copyright © Wright Group/McGraw-Hill Name Date Time STUDY LINK 11 Unit 1: Family Letter Introduction to Fifth Grade Everyday Mathematics Welcome to Fifth Grade Everyday Mathematics.This curriculum was developed by the University of Chicago School Mathematics Project to offer students a broad background in mathematics. The features of the program described below are to help familiarize you with the structure and expectations of Everyday Mathematics. A problem-solving approach based on everyday situationsStudents learn basic math skills in a context that is meaningful by making connections between their own knowledge and experience and mathematics concepts. Frequent practice of basic skillsStudents practice basic skills in a variety of engaging ways. In addition to completing daily review exercises covering a variety of topics and working with multiplication and division fact families in different formats, students play games that are specifically designed to develop basic skills. An instructional approach that revisits concepts regularly Lessons are designed to take advantage of previously learned concepts and skills and to build on them throughout the year. A curriculum that explores mathematical content beyond basic arithmeticMathematics standards around the world indicate that basic arithmetic skills are only the beginning of the mathematical knowledge students will need as they develop critical-thinking skills. In addition to basic arithmetic, Everyday Mathematicsdevelops concepts and skills in the following topics—number and numeration; operations and computation; data and chance; geometry; measurement and reference frames; and patterns, functions, and algebra. Everyday Mathematicsprovides you with ample opportunities to monitor your child’s progress and to participate in your child’s mathematical experiences. Throughout the year, you will receive Family Letters to keep you informed of the mathematical contentyour child is studying in each unit. Each letter includes a vocabulary list, suggested Do-Anytime Activities for you and your child, and an answer guide to selected Study Link (homework) activities. Please keep this Family Letter for reference as your child works through Unit 1.

4 Copyright © Wright Group/McGraw-Hill Fifth Grade Everyday Mathematics emphasizes the following content: Number and Numeration Understand the meanings, uses, and representations of numbers; equivalent names for numbers, and common numerical relations. Operations and Computation Make reasonable estimates and accurate computations; understand the meanings of operations. Data and Chance Select and create appropriate graphical representations of collected or given data; analyze and interpret data; understand and apply basic concepts of probability. Geometry Investigate characteristics and properties of 2- and 3-dimensional shapes; apply transformations and symmetry in geometric situations. Measurement and Reference Frames Understand the systems and processes of measurement; use appropriate techniques, tools, units, and formulas in making measurements; use and understand reference frames. Patterns, Functions, and Algebra Understand patterns and functions; use algebraic notation to represent and analyze situations and structures. Unit 1: Number Theory In Unit 1, students study properties of whole numbers by building on their prior work with multiplication and division of whole numbers. Students will collect examples of arrays to form a class Arrays Museum. To practice using arrays with your child at home, use any small objects, such as beans, macaroni, or pennies. Unit 1: Family Letter cont. STUDY LINK 11 Building Skills through Games In Unit 1, your child will practice operations and computation skills by playing the following games. Detailed instructions for each game are in the Student Reference Book. Factor BingoThis game involves 2 to 4 players and requires a deck of number cards with 4 each of the numbers 2–9, a drawn or folded 5-by-5 grid and 12 pennies or counters for each player. The goal of the game is to practice the skill of recognizing factors. Factor CaptorSeeStudent Reference Book, page 306. This is a game for 2 players. Materials needed include a Factor CaptorGrid, 48 counters the size of a penny, scratch paper, and a calculator. Thegoal of the game is to strengthen the skill of finding the factors of a number. Multiplication Top-ItSeeStudent Reference Book,page 334. This game requires a deck of cards with 4 each of the numbers 1–10 and can be played by 2–4 players. Multiplication Top-Itis used to practice the basic multiplication facts. Name That NumberSeeStudent Reference Book,page 325. This game involves 2 or 3 players and requires a complete deck of number cards.Name That Numberprovides practice with computation and strengthens skills related to number properties.

5 Copyright © Wright Group/McGraw-Hill composite number A counting number greater than 1 that has more than two factors.For example, 4 is a composite number because it has three factors: 1, 2, and 4. divisible by If the larger of two counting numbers can be divided by the smaller with no remainder, then the larger is divisible by the smaller. For example, 28 is divisible by 7 because 28 / 74 with no remainder. exponent The small, raised number in exponential notation that tells how many times the base is used as a factor. Example: 52Òexponent 525º525. 10 3Òexponent 10 310º10º101,000. 2 4Òexponent 242º2º2º216. factor One of two or more numbers that are multiplied to give a product. factor rainbow A way to show factor pairs in a list of all the factors of a number. A factor rainbow can be used to check whether a list of factors is correct. Factor rainbow for 16: number model A number sentence or expression that models a number story or situation. For example, a number model for the array below is 4 º312. prime number A whole number that has exactly two factors: itself and 1. For example, 5 is a prime number because its only factors are 5 and 1. product The result of multiplying two or more numbers, called factors. rectangular array A rectangular arrangement of objects in rows and columns such that each row has the same number of objects and each column has the same number of objects. square number A number that is the product of a counting number multiplied by itself. For example, 25 is a square number, because 25 5º5. 4816 2 1 3 º 5  15 Factors Product 15 º 1  15 Factors Product Vocabulary Important terms in Unit 1: Unit 1: Family Letter cont. STUDY LINK 11

6 Copyright © Wright Group/McGraw-Hill Unit 1: Family Letter cont. STUDY LINK 11 1.11; 2.18; 3.24; 4.28; 5.36; 6.49; 7.50; 8.70; 9. 100; Study Link 1 2 1. 2.1 º 14 14; 14 º114 2 º 7  14; 7 º 2 14 3.1 º 18 18; 18 º 1 18; 2 º 9 18; 9 º 2 18; 3 º 6 18; 6 º 3 18 4.7955.2716.987.9848.5 Study Link 1 3 1.24; 24 3.24; 3, 8; 24 6.1 º5 = 5; 1, 5 7.48.3,9199.2,76310.159 Study Link 1 4 1.The next number to try is 5, but 5 is already listed as a factor. Also, any factor greater than 5 would already be named because it would be paired with a factor less than 5. 2.1, 5, 253.1, 2, 4, 7, 14, 28 4.1, 2, 3, 6, 7, 14, 21, 42 5.1, 2, 4, 5, 10, 20, 25, 50, 100 6.9,5517.488.41,5449.44110.7 Study Link 1 5 1.Divisible by 2: 998,876; 5,890; 36,540; 1,098 Divisible by 3: 36,540; 33,015; 1,098 Divisible by 9: 36,540; 1,098 Divisible by 5: 5,890; 36,540; 33,015 2.Divisible by 4: 998,876; 36,540 3.1,7504.8,7535.2506.13 Study Link 1 6 10.9,82211.23412.21,44813.9 R3 Study Link 1 7 1.162.493.64.645.25 6.817.4 º9 368.5 º5 25 9. a.5 º5 25 b.5 º5 25 shows a square number because there are the same number of rows and columns. A square can be drawn around this array. Study Link 1 8 1.36: 1, 2, 3, 4, 6, 9, 12, 18, 36; 6 236 The square root of 36 is 6. 3.11 2= 121; the square root of 121 is 11. 5.6,2196.3,0608.8 R29.42 Study Link 1 9 1. b.7 27 º7 49 c.20 320 º20 º20 8,000 2. a.11 2 b.9 3 c.50 4 3. a.2 º3 3º5 22 º3 º3 º3 º5 º5 1,350 b.2 4º4 22 º2 º2 º2 º4 º4 256 4. a.40 2 º2 º2 º5 2 3º5 b.90 2 º3 º3 º5 2 º3 2º5 5.5,0416.7207.50 R48.99,140 9.1210.47,668 5 º 1  5 1 º 5  5 As You Help Your Child with Homework As your child brings assignments home, you might want to go over the instructions together, clarifying them as necessary. The answers listed below will guide you through this unit’s Study Links. 6 9 12 18 36 4 3 2 1 p c c c c c c c c 1, 11 ; 1, 2 , 3 , 6, 9, 18; 1, 2, 3, 4, 6, 8, 12, 24; 1, 2, 4, 7, 14, 28; 1, 2, 3, 4, 6, 9, 12, 18, 36; 1, 7, 49; 1, 2, 5, 10, 25, 50; 1, 2, 5, 7, 10, 14, 35, 70; 1, 2, 4, 5, 10, 20, 25, 50, 100;

7 Copyright © Wright Group/McGraw-Hill Copyright © Wright Group/McGraw-Hill LESSON 11 Name Date Time Following Written Directions Read the directions carefully.Do notdo anything until you have read all ten instructions. 1. Draw a square inside of a rectangle on this page. 2. Find the sum of the student fingers and toes in your class. 3. Stand up. Cover your eyes with your hands, and turn 90 degrees to the right. 4. Pat the top of your head with your right hand and, at the same time, rub your stomach in a clockwise direction with your left hand. Sit down. 5. As loudly as you can, count backwards from 10. 6. Find the sum of the digits for today’s date. 7. Estimate how many miles you walked in the last 2 months. 8. Try to touch the tip of your nose with your tongue. 9. If you reach into a bag where there is a $1 bill, a $5 bill, and a $10 bill, what is the chance that, without looking, you will pull a $10 bill? Whisper your answer to a neighbor. 10. Do not do any of the first 9 activities. Instead, turn over your paper and wait for your teacher’s instructions. Read the directions carefully.Do notdo anything until you have read all ten instructions. 1. Draw a square inside of a rectangle on this page. 2. Find the sum of the student fingers and toes in your class. 3. Stand up. Cover your eyes with your hands, and turn 90 degrees to the right. 4. Pat the top of your head with your right hand and, at the same time, rub your stomach in a clockwise direction with your left hand. Sit down. 5. As loudly as you can, count backwards from 10. 6. Find the sum of the digits for today’s date. 7. Estimate how many miles you walked in the last 2 months. 8. Try to touch the tip of your nose with your tongue. 9. If you reach into a bag where there is a $1 bill, a $5 bill, and a $10 bill, what is the chance that, without looking, you will pull a $10 bill? Whisper your answer to a neighbor. 10. Do not do any of the first 9 activities. Instead, turn over your paper and wait for your teacher’s instructions. LESSON 11 Name Date Time Following Written Directions

STUDY LINK 12 More Array Play Copyright © Wright Group/McGraw-Hill 8 Name Date Time 10 A rectangular arrayis an arrangement of objects in rows and columns. Each row has the same number of objects, and each column has the same number of objects. We can write a multiplication number model to describe a rectangular array. For each number below, use pennies or counters to make as many different arrays as possible. Draw each array on the grid with dots. Write the number model next to each array. 1. 5 2. 14 3. 18 4 º 3  12 4. 487 308  5. 679 408  6. 14 º 7  7. 164 º 6  8. 45 9  Practice

LESSON 12 Name Date Time Rows and Columns 9 Copyright © Wright Group/McGraw-Hill A rectangular array is an arrangement of objects in rows and columns. Each row has the same number of objects, and each column has the same number of objects. Work with a partner to build arrays. For each array, take turns rolling dice. The first die is the number of rows. Write this number in the table under Rows. The second die is the number of cubes in each row. Write this number under Columns. Then use centimeter cubes to build the array on the dot grid. How many cubes are in the array? Write this number under Array Total on the dot grid table. Rows Columns Array Total Rows Columns Array Total

Name Date Time 10 Copyright © Wright Group/McGraw-Hill A magic square is an array of positive whole numbers. The sum of the numbers in each row, column, and diagonal will be the same. 1. Complete this magic square.A heterosquare is like a magic square, except that the sum of the numbers in each row, column, and diagonal are different. A 3-by-3 array for a heterosquare will have an arrangement of the numbers 1–9. 2. Complete this heterosquare, and write the sum for each row, column, and the two diagonals. LESSON 12 Magic Square and Heterosquare Arrays A rectangular array is an arrangement of objects in rows and columns. The objects in an array can be numbers or numerical expressions. The Multiplication/Division Facts Table on the inside front cover of your journal is an example of numbers arranged in an array. The objects can also be words or symbols that represent elements of a given situation. For example, a plan for after-school snacks could be arranged in a 1-by-5 array, using A for apple, B for banana, and so on. 2 1010 910 9 12 1434 3434 11 85 31 85 3 3. Create a magic square or heterosquare for your partner to solve.

LESSON 13 Name Date Time Multiplication Facts 11 Copyright © Wright Group/McGraw-Hill B List 6 º 3 18 7 º 3 21 8 º 3 24 9 º 3 27 6 º 4 24 7 º 4 28 8 º 4 32 9 º 4 36 Bonus Problems 11 º 11 121 11 º 12 132 5º1260 12 º 672 7º1284 12 º 896 9º12108 10 º 12 120 5º1365 15 º 7 105 12 º 12 144 6º1484 A List 6 º 7 42 6 º 8 48 6 º 9 54 8 º 7 56 8 º 9 72 7 º 6 42 8 º 6 48 9 º 6 54 7 º 8 56 9 º 8 72 6 º 6 36 7 º 7 49 8 º 8 64 9 º 9 81

STUDY LINK 13 Number Models for Arrays Copyright © Wright Group/McGraw-Hill 12 10 Name Date Time Array Number Model Factors Product 1 6 º 4 6, 4 22, 12 3 3 º 8  41, 15 5 6 Reminder:Look for examples of arrays and bring them to school. Complete the chart. You will need to find each missing part and write it in the correct space. 7. 12 / 3  8. 1,288 2,631  9. 307 º 9  10. 306 147  Practice5 15

LESSON 13 Name Date Time Factoring Numbers with Cube Arrays 13 Copyright © Wright Group/McGraw-Hill Use centimeter cubes to build arrays for the following numbers. With each array write the factor pair.Remember that the number of rows in the array is one factorand that the number of columns in the array is the other factor. Continue to build every possible array until you have all of the factors for the number. 1. 14 2. 8 Factors: Factors: 3. 10 4. 20 Factors: Factors: 5. 33 Factors: 6. Can you tell when you have all of the factors for a number before you have built every possible array? Explain. 7. Write three true statements about factors. Try This 10

STUDY LINK 14 Factors Copyright © Wright Group/McGraw-Hill 14 10 Name Date Time To find the factors of a number, ask yourself: Is 1 a factor of the number? Is 2 a factor? Is 3 a factor?Continue with larger numbers. For example, to find all the factors of 15, ask yourself these questions. 1. You don’t need to go any further. Can you tell why? So the factors of 15 are 1, 3, 5, and 15. List as many factors as you can for each of the numbers below. 2. 25 3. 28 4. 42 5. 100 6. 8,417 1,134  7. 73 25  8. 6,924 º 6  9. 634 193  10. 56 / 8  Yes/No Number Sentence Factor Pair Is 1 a factor of 15? Yes 1 º 15 15 1, 15 Is 2 a factor of 15? No Is 3 a factor of 15? Yes 3 º 5 15 3, 5 Is 4 a factor of 15? No Practice

STUDY LINK 15 Divisibility Rules 15 11 Name Date Time Copyright © Wright Group/McGraw-Hill All even numbers are divisible by 2. A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 6 if it is divisible by both 2 and 3. A number is divisible by 9 if the sum of its digits is divisible by 9. A number is divisible by 5 if it ends in 0 or 5. A number is divisible by 10 if it ends in 0. 1. Use divisibility rules to test whether each number is divisible by 2, 3, 5, 6, 9, or 10. A number is divisible by 4 if the tens and ones digits form a number that is divisible by 4. Example:47,836is divisible by 4 because 36 is divisible by 4. It isn’t always easy to tell whether the last two digits form a number that is divisible by 4. A quick way to check is to divide the number by 2 and then divide the result by 2. It’s the same as dividing by 4, but is easier to do mentally. Example:5,384is divisible by 4 because 84 / 2 42 and 42 / 2 21. 2. Place a star next to any number in the table that is divisible by 4. 3. 250 º 7  4. 1,931 4,763 2,059  5. (20 30) º 5  6. 78 6  NumberDivisible… by 2? by 3? by 6? by 9? by 5? by 10? 998,876 5,890 36,540 33,015 1,098 Practice

LESSON 15 Name Date Time Divisibility by 4 16 Copyright © Wright Group/McGraw-Hill 1. What number is shown by the base-10 blocks? 2. Which of the base-10 blocks could be divided evenly into 4 groups of cubes? 3. Is the number shown by the base-10 blocks divisible by 4? 4. Circle the numbers that you think are divisible by 4. 324 5,821 7,430 35,782,916 Use a calculator to check your answers. 5. Use what you know about base-10 blocks to explain why you only need to look at the last two digits of a number to decide whether it is divisible by 4. 1,000 cubes 100 cubes 10 cubes 1 cube

STUDY LINK 16 Prime and Composite Numbers 17 12 Name Date Time Copyright © Wright Group/McGraw-Hill A prime numberis a whole number that has exactly two factors—1 and the number itself. A composite numberis a whole number that has more than two factors. For each number: List all of its factors. Write whether the number is prime or composite. Circle all of the factors that are prime numbers. Number Factors Prime or Composite? 111 218 324 428 536 649 750 870 9100 10. 4,065 2,803 2,954  11. 392 158  12. 1,532 º 14  13. 39 / 4 ∑ 14. 48 º 15  Practice

LESSON 16 Name Date Time Goldbach’s Conjecture 18 Copyright © Wright Group/McGraw-Hill 1. Write each of the following numbers as the sum of two prime numbers. Examples:56 26  a. 6  b. 12  c. 18  d. 22  e. 24  f. 34  The answers to these problems are examples of Goldbach’s Conjecture.A conjectureis something you believe is true even though you can’t be certain that it is true. Goldbach’s Conjecture might be true, but no one has ever proven it. Anyone who can either prove or disprove Goldbach’s Conjecture will become famous. 2. Work with a partner. Find and write as many of the addition expressions as you can for the numbers in the grid on page 19. 3. Can any of the numbers in the grid be written as the sum of two prime numbers in more than one way? If so, give an example. Show all possible ways. 13+13 43+13 Try This 4. Write 70 as the sum of two primes in as many ways as you can.

LESSON 16 Name Date Time Goldbach’s Conjecture continued 19 Copyright © Wright Group/McGraw-Hill Write each number below as the sum of two prime numbers. 4681012 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 102 2 + 2

STUDY LINK 17 Exploring Square Numbers Copyright © Wright Group/McGraw-Hill 20 6 Name Date Time A square numberis a number that can be written as the product of a number multiplied by itself. For example, the square number 9 can be written as 3 º 3. Fill in the missing numbers. 1. 4 º 4  2. 7 º 7 3. º 6 36 4. 82 5. 52 6. 9 2 Write a number model to describe each array. 7. Number model: 8. Number model: 9. a. Which of the arrays above shows a square number? b. Explain your answer. 9  3 º 3  3 2 10. 97 º 43  11. 4,006 2,675  12. 1,416 8,348  13. 725 414  Practice

LESSON 17 Name Date Time Completing Patterns 21 Copyright © Wright Group/McGraw-Hill Build these patterns with counters. Draw the dot pattern that comes next and record the number of dots in the pattern. Example: 1. 2. 3. 4. Write a description of the pattern in Problem 3. 1 13 6 14 9 142835 7

STUDY LINK 18 Copyright © Wright Group/McGraw-Hill 22 Name Date Time Factor Rainbows, Squares, and Square Roots 1. List all the factors of each square number. Make a factor rainbowto check your work. Then fill in the missing numbers. Reminder:In a factor rainbow, the product of each connected factor pair should be equal to the number itself. For example, the factor rainbow for 16 looks like this: Example: 4: 2  4 The square root of 4 is . 25: 2 25 The square root of 25 is .9: 2 9 The square root of 9 is . 36: 2 36 The square root of 36 is . 2. Do all square numbers have an odd number of factors? Unsquare each number. The result is its square root. Do not use the square root key on your calculator. 3. 2 121 4. 2 2,500 The square root of 121 is .The square root of 2,500 is . 5. 4,318 6. 36 7. 2,852  1,901  85  5 8. 50 6 ∑ 9. 333 291  1, 2, 4 22 4816 2 1 24 1 271 Practice 1 º 16 16 2 º 8 16 4 º 4 16

LESSON 18 Name Date Time Comparing Numbers with Their Squares 23 Copyright © Wright Group/McGraw-Hill 1. a. Unsquare the number 1. 2 1 b. Unsquare the number 0. 2 0 2. a. Is 5 greater than or less than 1? b. 52 c. Is 5 2greater than or less than 5? 3. a. Is 0.50 greater than or less than 1? b. Use your calculator. 0.50 2 c. Is 0.50 2greater than or less than 0.50? 4. a. When you square a number, is the result always greater than the number you started with? b. Can it be less? c. Can it be the same? 5. Write 3 true statements about squaring and unsquaring numbers.

STUDY LINK 19 Exponents Copyright © Wright Group/McGraw-Hill 24 16 Name Date Time An exponentis a raised number that shows how many times the number to its left is used as a factor. Examples:5 2 Ò exponent 52 means 5 º 5, which is 25. 10 3 Ò exponent 10 3 means 10 º 10 º 10, which is 1,000. 2 4 Ò exponent 24 means 2 º 2 º 2 º 2, which is 16. 1. Write each of the following as a factor string. Then find the product. Example:2 3 a. 10 4 b. 72 c. 20 3 2. Write each factor string using an exponent. Example:6 º 6 º 6 º 6  a. 11 º 11  b. 9 º 9 º 9  c. 50 º 50 º 50 º 50  3. Write each of the following as a factor string that does nothave any exponents. Then use your calculator to find the product. Example:2 3º 3  a. 2 º 3 3º 5 2 b. 24º 4 2 4. Write the prime factorization of each number. Then write it using exponents. Example:18  a. 40   b. 90  2º 3 2 2º 3 º 3 24 2º 2º 2º 3 6 4 8 2º 2º 2 5. 6,383 1,342  6. 48 15  7. 7354 ∑ 8. 50,314 48,826  9. 84 7  10. 701 68  Practice

LESSON 19 Name Date Time Using Factor Trees 25 Copyright © Wright Group/McGraw-Hill Factor Trees One way to find all the prime factors of a number is to make a factor tree.First write the number. Then, underneath, write any two factors whose product is that number. Then write factors of each of these factors. Continue until all the factors are prime numbers. Below are three factor trees for 36. It does not matter which two factors you begin with. You always end with the same prime factors — for 36, they are 2, 2, 3, and 3. The prime factorizationof 36 is 2 º 2 º 3 º 3. Make a factor tree for each number. Then write the prime factorization for each number. 24 50  48 100  6 3 2 º º º 6 36 3 2 º 3 º 12 36 3 3 4 º º 3 º 3 º 2 º 2 9 3 3 º º º 4 36 2 2 º 24 50 48 100

LESSON 19 Name Date Time The Sieve of Eratosthenes 26 Copyright © Wright Group/McGraw-Hill The mathematician Eratosthenes, born in 276 B.C., developed this method for finding prime numbers. Follow the directions below for Math Masters,page 27. When you have finished, you will have crossed out every number from 1 to 30 in the grid that is not a prime number. 1. Since 1 is not a prime number, cross it out. 2. Circle 2 with a colored marker or crayon. Then count by 2, crossing out all multiples of 2—that is, 4, 6, 8, 10, and so on. 3. Circle 3 with a color different from Step 2. Cross out every third number after 3 (6, 9, 12, and so on). If a number is already crossed out, make a mark in a corner of the box. The numbers you have crossed out or marked are multiples of 3. 4. Skip 4 on the grid because it is already crossed out, and go on to 5. Use a new color to circle 5 and cross out the multiples of 5. 5. Continue. Start each time by circling the next number that is not crossed out. Cross out all multiples of that number. If a number is already crossed out, make a mark in a corner of the box. If there are no multiples for a number, start again. Use a different color for each new set of multiples. 6. Stop when there are no more numbers to be circled or crossed out. The circled numbers are the prime numbers from 1 to 30. 7. List the prime numbers from 1 to 30.

LESSON 19 Name Date Time The Sieve of Eratosthenes continued 27 Copyright © Wright Group/McGraw-Hill 12345 678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

LESSON 19 Name Date Time Palindromic Squares 28 Copyright © Wright Group/McGraw-Hill Palindrome numbers are numbers that read the same forward or backward. A single-digit number is also a palindrome. The two-digit palindrome numbers are 11, 22, 33, 44, 55, 66, 77, 88, and 99. The table below lists samples of 3-digit and 4-digit palindromes. 1. Find 3-digit and 4-digit numbers to add to the table. Palindrome Numbers 3-digit 4-digit 101, 111 1,001; 1,111 202, 222 2,002; 2,222 303, 333 3,003; 3,333 Sometimes finding the square of a palindrome number results in a square number that is also a palindrome number?a palindromic square. For example, 111 212,321. 2. Which 3 single-digit numbers have palindromic squares? 3. Which 2-digit numbers have palindromic squares? 4. Find the numbers from the table that have a palindromic square and write the number model. Example:101 210,201

29 Copyright © Wright Group/McGraw-Hill STUDY LINK 110 Unit 2: Family Letter Name Date Time Estimation and Calculation Computation is an important part of problem solving. Many of us were taught that there is just one way to do each kind of computation. For example, we may have learned to subtract by borrowing, without realizing that there are many other methods of subtracting numbers. In Unit 2, students will investigate several methods for adding, subtracting, and multiplying whole numbers and decimals. Students will also take on an Estimation Challenge in Unit 2. For this extended problem, they will measure classmates’ strides, and find a median length for all of them. Then they will use the median length to estimate how far it would take to walk to various destinations. Throughout the year, students will practice using estimation, calculators, as well as mental and paper-and-pencil methods of computation. Students will identify which method is most appropriate for solving a particular problem. From these exposures to a variety of methods, they will learn that there are often several ways to accomplish the same task and achieve the same result. Students are encouraged to solve problems by whatever method they find most comfortable. Computation is usually not the first step in the problem-solving process. One must first decide what numerical data are needed to solve the problem and which operations need to be performed. In this unit, your child will continue to develop his or her problem-solving skills with a special focus on writing and solving equations for problems. Please keep this Family Letter for reference as your child works through Unit 2.

100s 2 3 / 1 110s 14 4 /5/ 6 81s 12 2 / 4 8 Estimation Challenge A problem for which it is difficult, or even impossible, to find an exact answer. Your child will make his or her best estimate and then defend it. magnitude estimate A rough estimate. A magnitude estimate tells whether an answer should be in the tens, hundreds, thousands, and so on. Example: Give a magnitude estimate for 56 º32 Step 1: Round 56 to 60. Step 2: Round 32 to 30. 60º301,800, so a magnitude estimate for 56 º32 is in the thousands. maximum The largest amount; the greatest number in a set of data. mean The sum of a set of numbers divided by the number of numbers in the set. The mean is often referred to simply as the average. median The middle value in a set of data when the data are listed in order from smallest to largest or vice versa. If there is an even number of data points, the median is the meanof the two middle values. minimum The smallest amount; the smallest number in a set of data. partial-sums addition A method, or algorithm, for adding in which sumsare computed for each place (ones, tens, hundreds, and so on) separately and are then added to get a final answer. place value A number system that values a digit according to its position in a number. In our number system, each place has a value ten times that of the place to its right and one-tenth the value of the place to its left. For example, in the number 456, the 4 is in the hundreds place and has a value of 400. range The difference between the maximumand minimumin a set of data. reaction time The amount of time it takes a person to react to something. trade-first subtraction A method, or algorithm, for subtracting in which all trades are done before any subtractions are carried out. Example: 352164 10s 100s 1,000s 10,000s Vocabulary Important terms in Unit 2: 30 268 483 600 140 11 751 1. Add 100s 2. Add 10s 3. Add 1s 4. Add partial sums. Partial-sums algorithm Trade 1 hundred for 10 tens and subtract in each column. 100s 3 110s 4 5 / 61s 12 2 / 4 Trade 1 ten for 10 ones. Copyright © Wright Group/McGraw-Hill Unit 2: Family Letter cont. STUDY LINK 110

31 Copyright © Wright Group/McGraw-Hill Unit 2: Family Letter cont. STUDY LINK 110 In Unit 2, your child will practice computation skills by playing these games. Detailed instructions are in theStudent Reference Book. Addition Top-ItSeeStudent Reference Book,page 333. This game for 2 to 4 players requires a calculator and 4 each of the number cards 1–10, and provides practice with place–value concepts and methods of addition. High-Number TossSeeStudent Reference Book, pages 320 and 321. Two players need one six-sided die for this game. High-Number Tosshelps students review reading, writing, and comparing decimals and large numbers. Multiplication Bull’s-EyeSeeStudent Reference Book,page 323. Two players need 4 each of thenumber cards 0–9, a six-sided die, and a calculator to play this game. Multiplication Bull’s Eyeprovides practice in estimating products. Number Top-ItSeeStudent Reference Book,page 326. Two to five players need 4 each of the number cards 0–9 and a Place-Value Mat. Students practice making large numbers. Subtraction Target PracticeSeeStudent Reference Book,page 331. One or more players need 4 each of the number cards 0–9 and a calculator. In this game, students review subtraction with multidigit whole numbers and decimals. Building Skills through Games Do-Anytime Activities To work with your child on the concepts taught in Units 1 and 2, try these activities: 1.When your child adds or subtracts multidigit numbers, talk about the strategy that works best. Try not to impose the strategy that works best for you! Here are some problems to try: 467343_______ ________ 76179 894444_______ 842 59_______ 2.As you encounter numbers while shopping or on license plates, ask your child to read the numbers and identify digits in various places—thousands place, hundreds place, tens place, ones place, tenths place, and hundredths place.

Copyright © Wright Group/McGraw-Hill 32 As You Help Your Child with Homework As your child brings assignments home, you might want to go over the instructions together, clarifying them as necessary. The answers listed below will guide you through this unit’s Study Links. Study Link 2 1 Answers vary for Problems 1-5. 6.7207.90,3618.129.18 Study Link 2 2 Sample answers: 1.571 and 2612.30, 20, and 7 3.19 and 234.533 and 125 5.85.2 and 20.5, or 88.2 and 17.5; Because the sum has a 7 in the tenths place, look for numbers with tenths that add to 7: 85.2 20.5105.7; and 88.217.5105.7. 6.4,5727.4.48.2469.1.918 10.4711.20812.313.8 R2 Study Link 2 3 1.451 and 299 2.100.9 and 75.3 3.Sample answer: 803 and 5,000 4.17 and 155.703 and 1,500 6.25 and 9 7.618.1379.5.8 10.18.8511.612.84,01813.$453.98 14.9815.14 Study Link 2 4 1. a.148 and 127b.Total number of cards c.148127bd.b275 e. 275 baseball cards 2. a.20.00; 3.89; 1.49b.The amount of change c.20.003.891.49c, or 20(3.891.49)c d.c14.62e.$14.62 3. a.0.6; 1.15; 1.35; and 0.925 b. The length of the ribbons c.b0.61.151.350.925 d.b4.025e. 4.025 meters Study Link 2 5 Answers vary for Problems 1–5. 6.5,6227.29,6168.5189.13 Study Link 2 6 1.Unlikely: 30% Very likely: 80% Very unlikely: 15% Likely: 70% Extremely unlikely: 5% 2.30%: Unlikely 5%: Extremely unlikely 99%: Extremely unlikely 20%: Very unlikely 80%: Very likely 35%: Unlikely 65%: Likely 45%: 50-50 chance Study Link 2 7 1.1,000s; 70 º302,100 2.1,000s; 10 º7007,000 3.10,000s; 100 º10010,000 4.10s; 20 º240 5.10s; 3 º412 6.Sample answers: 45 º683,060; 684º53,420; and 864 º54,320 Study Link 2 8 1.152; 100s; 8º20160 2.930; 100s; 150 º6900 3.2,146; 1,000s; 40º602,400 4.21; 10s; 5 º420. 5.26.04; 10s; 9º327 Study Link 2 9 1.6,862; 1,000s2.88.8; 10s3.33.372; 10s 4.100,224; 100,000s5.341.61; 100s 6.9,9897.5 R28.919.$19.00 Study Link 2 10 1.390.7562.3,471.5493.9,340 4.2445.44,6046.19 R2 Unit 2: Family Letter cont. STUDY LINK 110