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Buckle Down North Carolina EOG 5 Mathematics Number and Operations Lesson 1: Rational Numbers Lesson 2: Converting and Comparing Rational Numbers Lesson 3: Computing with Rational Numbers Lesson 4: Estimation and Problem Solving Unit 2 Patterns, Relationships, and Algebra Lesson 5: Patterns and Relationships Lesson 6: Algebra Unit 3 Geometry and Spatial Sense Lesson 7: Geometric Figures Lesson 8: Geometric Concepts Unit 4 Measurement Lesson 9: Units of Measurement Lesson 10: Geometric Measurement Unit 5 Data Analysis and Probability Lesson 11: Data Analysis Lesson 12: Probability P.O. Box 2180 Iowa City, Iowa 52244-2180 PHONE: 800-776-3454 FAX: 877-365-0111 www.BuckleDown.com EMAIL: [email protected] Catalog # 4BDNC05MM01 4TH EDITION 5 MATHEMATICS Check out our complete line of EOG/Comprehensive materials for Grades 3–8 and 10 READING • WRITING • MATHEMATICS North Carolina North Carolina EOG The cover image combines geometry and art to illustrate the properties of polygons, such as the sum of the measures of interior angles and the parallelism or perpendicularity of sides. Unit 1 5 Mathematics EOG 4BDNC05MM01 FM 7/8/05 2:01 PM Page iii TABLE OF CONTENTS Introduction ..................................................................................... 1 Testwise StrategiesTM ........................................................ 2 Unit 1 – Number and Operations ............................................... 3 Lesson 1: Rational Numbers.............................................. 4 EOG Standards: 1.01a, 1.01b Lesson 2: Converting and Comparing Rational Numbers ........................................................... 21 EOG Standard: 1.01c Lesson 3: Computing with Rational Numbers................ 29 EOG Standard: 1.02a Skills to Maintain: whole number computation Lesson 4: Estimation and Problem Solving .................... 43 EOG Standards: 1.01d, 1.02b, 1.02c, 1.03 Unit 2 – Patterns, Relationships, and Algebra....................... 55 Lesson 5: Patterns and Relationships............................. 56 EOG Standards: 5.01, 5.03 Lesson 6: Algebra.............................................................. 68 EOG Standard: 5.02 Unit 3 – Geometry and Spatial Sense....................................... 77 Lesson 7: Geometric Figures............................................ 78 EOG Standards: 2.02, 3.01, 3.02a–c, 3.04a–c © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Lesson 8: Geometric Concepts ......................................... 99 EOG Standard: 3.03 Skills to Maintain: transformations, coordinate grid Unit 4 – Measurement ................................................................ 109 Lesson 9: Units of Measurement ................................... 110 EOG Standard: 2.01 Lesson 10: Geometric Measurement ............................. 133 Skills to Maintain: perimeter and area iii 7/8/05 2:01 PM Page iv Table of Contents Unit 5 – Data Analysis and Probability ................................. 139 Lesson 11: Data Analysis ............................................... 140 EOG Standards: 4.01, 4.02, 4.03 Skills to maintain: line graphs, median, mode, and range Lesson 12: Probability .................................................... 164 iv © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 4BDNC05MM01 FM 4BDNC05MM01 L1 7/6/05 3:35 PM Page 4 4 Unit 1 – Number and Operations Lesson 1 Rational Numbers Rational numbers include the whole numbers, some decimal numbers, and all the fractions and mixed numbers. In this lesson, you will read, write, estimate, order, and compare rational numbers. Place Value A number such as 820,305 might look hard to work with, but it’s easy if you understand place value. The following table helps you see the values of the digits so you can write the number in standard form. Hundred Thousands Ten Thousands 8 2 Thousands Hundreds 0 3 Tens Ones 0 5 When you write large numbers in standard form, put a comma to the left of every third digit, starting at the ones place and moving to the left. 8 2 0,3 0 5 2 1 This number can also be written in expanded form and word form. Expanded Form: 800,000 20,000 300 5 Word Form: eight hundred twenty thousand, three hundred five Notice that the place values with zero as a placeholder are not written out in the expanded form of the number. They are also not written out in the word form of the number. TIP: Sometimes it is hard to remember where to put commas when you are writing in word form. Look at the commas in standard form. Keep them in the same place when you write the words out. © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 3 start here 4BDNC05MM01 L1 7/6/05 3:35 PM Page 5 5 Lesson 1: Rational Numbers Practice Directions: Write the following numbers in the table. Then write the numbers in expanded and word form in Numbers 1 and 2. 710,051 164,980 Hundred Ten Thousands Thousands Thousands Hundreds Tens Ones 1. 710,051 Expanded Form: _______________________________________________________ Word Form: ___________________________________________________________ 2. 164,980 Expanded Form: _______________________________________________________ Word Form: ___________________________________________________________ 3. Which digit is in the ten thousands place in the number 271,839? _______ © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 4. Which digit is in the hundreds place in the number 510,436? A 0 B 4 C 5 D 6 5. Which of the following is the standard form of forty-two thousand, one hundred ten? A 4,211 B 42,110 C 420,110 D 421,010 4BDNC05MM01 L1 7/6/05 3:35 PM Page 6 6 Unit 1 – Number and Operations Comparing and ordering whole numbers Understanding place value helps you compare and order whole numbers. Start at the far left place value of the numbers, adding zeros as placeholders if necessary, and compare the digits of each place value from left to right. Stop at the place value where the digits are different. The way those digits compare is the way the whole numbers compare. Use (greater than), (less than), or (equal to) when comparing numbers. Example Compare 23,948 and 23,961. Both numbers’ first digits are in the ten-thousands place. Compare the digits of each place value from left to right using a place-value table until you notice a different digit. Both numbers’ digits are the same until the tens place. Ten Thousands Thousands Hundreds Tens Ones 2 3 9 4 8 2 3 9 6 1 Compare 4 and 6: 4 6. Another way to compare and order whole numbers is to use a number line. A number to the right of another number on a number line is greater in value. A number to the left of another number is lesser (or smaller) in value. Example Order 938, 916, 960, 899, and 927 from least to greatest. Write the numbers in their correct places on a number line. The order of the numbers from least to greatest will be from left to right on the number line. 899 890 900 916 910 920 927 930 938 940 960 950 960 The order from least to greatest is 899, 916, 927, 938, and 960. © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Therefore, 23,948 23,961. 4BDNC05MM01 L1 7/6/05 3:35 PM Page 7 7 Lesson 1: Rational Numbers Practice Directions: For Numbers 1 through 4, write the correct symbol (, , or ) to compare the numbers. 1. 121,150 __________ 121,105 3. 30,544 __________ 300,455 2. 61,219 __________ 61,219 4. 801,760 __________ 810,760 5. Write the following numbers in their correct places on the number line. Then write the numbers in order from greatest to least. 86,573 83,000 87,635 84,000 87,356 85,000 86,735 86,000 87,000 83,765 88,000 6. Write the following numbers in order from least to greatest. © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 210,798 220,376 208,139 201,934 7. Write the following numbers in order from greatest to least. 538,312 539,082 8. Which of these numbers is greater than 83,621 but less than 83,802? 541,293 538,311 9. Which list of numbers is in order from least to greatest? A 74,429 75,013 74,216 73,912 A 83,548 B 75,013 74,429 73,912 74,216 B 83,912 C 74,216 73,912 75,013 74,429 C 83,795 D 73,912 74,216 74,429 75,013 D 83,617 4BDNC05MM01 L1 7/6/05 3:35 PM Page 8 8 Unit 1 – Number and Operations Decimals A decimal expresses a whole as divided into ten equal parts (tenths), into one hundred equal parts (hundredths), and so on. A decimal point separates the whole-number part of the decimal from the part that is less than 1. Read the number from left to right. The decimal point is read and. When you get to the decimal side, read the number as a whole number, followed by the last digit’s place value. Ones Decimal Point Tenths Hundredths Thousandths 5 • 7 3 4 5.734 is read . . . “five and seven hundred thirty-four thousandths.” The last digit is 4, which is in the thousandths place. 0.07 is read . . . “seven hundredths.” 1.5 is read . . . “one and five tenths.” 2.654 is read . . . “two and six hundred fifty-four thousandths.” Practice Directions: Write the decimal for Numbers 1 through 5. 1. thirty-two hundredths: _______________________________ 2. six tenths: _______________________________ 3. one hundred fifty-four thousandths: _______________________________ 4. two tenths: _______________________________ 5. ninety-nine thousandths: _______________________________ © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Examples 4BDNC05MM01 L1 7/6/05 3:35 PM Page 9 9 Lesson 1: Rational Numbers Comparing and ordering decimals Just like whole numbers, decimals can be compared and ordered by looking at their place values. Example Which of the following decimals is the greatest? 3.458 3.526 3.5 3.59 The comparison is easier if you write the numbers in a place-value table. Put zeros in as placeholders where necessary, and compare as you would whole numbers, looking at each value from left to right. Ones Decimal Point Tenths Hundredths Thousandths 3 • 4 5 8 3 • 5 2 6 3 • 5 0 0 3 • 5 9 0 3.59 is the greatest. Practice © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Directions: For Numbers 1 through 4, compare the decimals. (Use , , or .) 1. 0.76 ______________ 0.763 3. 8.2 ______________ 8.12 2. 5.99 ______________ 5.9 4. 43.38 ______________ 42.96 Directions: For Numbers 5 through 7, order the decimals from greatest to least. 5. 3.08, 3.9, 3.75, 3.801 ___________________________________ 6. $38.50, $38.06, $39.99, $37.75 ___________________________________ 7. 19.88, 18.98, 19.8, 19.08 ___________________________________ 4BDNC05MM01 L1 7/6/05 3:35 PM Page 10 10 Unit 1 – Number and Operations Fractions A fraction expresses a whole divided into a number of equal parts. The numerator, the top number, tells you how many equal parts of the whole you have. The denominator, the bottom number, tells you how many equal parts the whole is divided into. numerator ’ 1 2 { denominator The fraction above tells you that a whole is divided into 2 parts and that you have 1 of those parts. Example Don’s family has 7 pets. They have 3 dogs, 2 cats, and 2 birds. What fraction of the family’s pets are dogs? In this problem, the whole is the total number of pets the family has. Therefore, the denominator of the fraction is 7. The numerator of the fraction is the number of dogs the family has, which is 3. 3 Three out of the family’s 7 pets are dogs. The fraction form is . 7 1. Mrs. O’Leary bought 12 boxes of cereal at the grocery store, 5 of which are Choco-bran Monster Pops. What fraction of the cereal boxes Mrs. O’Leary bought are Choco-bran Monster Pops? __________ 2. Fred threw a total of 9 pitches, and 6 of them were strikes. What fraction of the pitches were not strikes? __________ © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Practice 4BDNC05MM01 L1 7/6/05 3:35 PM Page 11 11 Lesson 1: Rational Numbers Comparing and ordering fractions with the same denominator Fractions with the same denominator can be compared and ordered by looking at their numerators. The fraction with the lesser numerator will have the lesser value. The fraction with the greater numerator will have the greater value. Example Greg and Heather ordered two 10-inch pizzas. They cut each pizza into 6 equal slices. Greg has 2 slices remaining from his pizza Heather has 3 slices remaining from her pizza . Who has 3 6 . 2 6 the smaller amount of pizza remaining? Compare the numerators: 2 3. 2 6 3 6 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. Greg has the smaller amount remaining from his pizza: 2 6 3 . 6 Practice Directions: For Numbers 1 through 4, compare the fractions. (Use , , or .) 1. 1 8 2. 7 10 ______________ 2 8 ______________ 6 10 3. 2 3 4. 5 12 ______________ 1 3 ______________ 5 12 Directions: For Numbers 5 and 6, order the fractions from least to greatest. 5. 6 4 7 2 , , , 8 8 8 8 ________________________________ 6. 21 20 23 19 , , , 100 100 100 100 ________________________________ 4BDNC05MM01 L1 7/6/05 3:35 PM Page 12 12 Unit 1 – Number and Operations Comparing and ordering unit fractions You can compare and order fractions with the same numerator by looking at their denominators. The fraction with the lesser denominator will have the greater value. The fraction with the greater denominator will have the lesser value. Example Arthur and Agatha ordered two 10-inch pizzas. They cut 1 pizza into 2 equal slices and the other pizza into 6 equal slices. Which is larger, a slice from the pizza cut into 2 slices into 6 slices ? 1 6 or a slice from the pizza cut 1 2 Compare the denominators: 2 6. 1 6 A slice from the pizza cut into 2 slices is larger: 1 2 1 . 6 Practice Directions: For Numbers 1 through 4, compare the fractions. (Use , , or .) 1. 1 6 2. 1 10 ______________ 1 5 ______________ 1 12 3. 4. 1 4 ______________ 1 100 1 4 ______________ 1 1,000 Directions: For Numbers 5 and 6, order the fractions from greatest to least. 5. 1 1 1 1 , , , 10 6 3 12 ________________________________ 6. 1 1 1 1 , , , 100 10 1,000 5 ________________________________ © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 1 2 4BDNC05MM01 L1 7/6/05 3:35 PM Page 13 13 Lesson 1: Rational Numbers Improper Fractions An improper fraction has a numerator that is equal to or greater than its denominator. You can write an improper fraction as a whole number or a mixed number by dividing its numerator by its denominator. When the numerator can be divided evenly by the denominator, the quotient is a whole number. When the numerator cannot be divided evenly by the denominator, the improper fraction is written as a mixed number. The quotient becomes the whole number part of the mixed number. The remainder becomes the numerator of the fraction part. The denominator of the fraction part is the same as the denominator of the original improper fraction. Make sure the fraction part is in lowest terms. Example How is 28 8 written as a mixed number? Divide 28 by 8. 3 8 82 24 4 The quotient is 3. This is the whole number part of the mixed number. The remainder is 4 and the denominator of the improper fraction is 8. 4 Therefore, the fraction part of the mixed number is . This can be 8 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. simplified to 1 . 2 The mixed number for 28 8 1 is 3 . 2 Practice Directions: For Numbers 1 through 3, write the improper fraction as a whole number or a mixed number. 1. 13 6 __________ 2. 40 8 __________ 3. 17 5 __________ 4BDNC05MM01 L1 7/6/05 3:35 PM Page 14 14 Unit 1 – Number and Operations Directions: For Numbers 4 and 5, make a model to represent each improper fraction. 4. 8 5 5. 11 4 submarine sandwiches © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. cherry pies 4BDNC05MM01 L1 7/6/05 3:35 PM Page 15 15 Lesson 1: Rational Numbers Mixed Numbers A mixed number is the sum of a whole number and a fraction. Both parts are written together. Example What mixed number represents the shaded parts of these figures? How many whole figures are shaded? 3 What fraction of the fourth figure is shaded? 5 6 5 The mixed number 3 represents the shaded parts of the figures. 6 Practice Directions: Use the following information to answer Numbers 1 and 2. © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. The pictures represent how much is left of 3 pies after Cindy’s party. 1. How many whole pies can be made with the slices that are left? __________ 2. Write a mixed number that shows how much pie is left altogether. __________ 4BDNC05MM01 L1 7/6/05 3:35 PM Page 16 16 Unit 1 – Number and Operations Directions: For Numbers 3 and 4, make a model to represent each mixed number. 6 3. 2 10 1 4. 4 5 6. Which mixed number represents the shaded parts of these figures? 1 A 2 4 B 1 2 3 1 C 3 3 D 3 3 4 1 A 3 2 2 B 3 3 1 C 4 3 1 D 4 2 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 5. Which mixed number represents the shaded parts of these figures? 4BDNC05MM01 L1 7/8/05 2:02 PM Page 17 17 Lesson 1: Rational Numbers Equivalent Fractions Equivalent fractions are different fractions that represent the same value. If Tom ate 1 2 of a 10-inch pizza and Bonita ate 3 6 of a 10-inch pizza, who ate more? Neither of them—they ate the same amount. The fractions 1 2 and 3 6 are equivalent. 1 2 3 6 One way to find out whether fractions are equivalent is to draw pictures of them. 1 5 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 1 10 3 5 1 5 1 10 1 10 1 5 1 10 1 10 1 10 6 10 When both the numerator and the denominator are the same number, the fraction is equivalent to 1. 7 7 1 1 1 1 4BDNC05MM01 L1 7/6/05 3:35 PM Page 18 18 Unit 1 – Number and Operations Practice Directions: For Numbers 1 and 2, write equivalent fractions to describe the shaded parts of the figures. 1. 2. © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 4BDNC05MM01 L1 7/6/05 3:35 PM Page 19 19 Lesson 1: EOG Practice EOG Practice 1. What digit is in the hundredthousands place in the following number? 4. What is the value of the 4 in the number 245,991? A 4 B 400 A 2 C 40,000 B 5 D 400,000 254,079 C 7 D 9 2. For track practice, Michael ran 2.32 miles, Jenn ran 2.23, Kayla ran 3.02 miles, and Gabe ran 2.14 miles. Who ran the shortest distance? A Jenn B Gabe C Kayla © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. D Michael 3. Five of the 12 students in Mr. Franklin’s class are boys. What fraction of the students are girls? A 5 17 B 5 12 C 7 12 D 12 17 5. During the summer, Worlds of Fun Amusement Park had a total of two hundred fourteen thousand, fifty-eight visitors. How is this number written in standard form? A 240,580 B 240,058 C 214,580 D 214,058 6. Which list of fractions is in order from least to greatest? A 7 1 8 4 , , , 12 12 12 12 B 1 4 7 8 , , , 12 12 12 12 C 7 4 8 1 , , , 12 12 12 12 D 8 7 4 1 , , , 12 12 12 12 7. How is thirteen-hundredths written as a decimal? A 1,300 B 1.3 C 0.13 D 0.013 4BDNC05MM01 L1 7/6/05 3:35 PM Page 20 20 Unit 1 – Number and Operations 8. Which set of figures shows 7 4 shaded? A 11. Which of the following decimals is the greatest? A 0.63 B 0.52 C 0.60 B C D D 0.59 12. Which of the following is the standard form of one hundred nineteen and twenty-five hundredths? A 119.025 B 119.25 C 190.025 9. What is the expanded form of 300,609? A 300,000 600 9 B 300,000 600 90 C 300,000 6,000 90 D 190.25 13. Which symbol correctly compares the following two fractions? 11 4 D 300,000 6,000 900 _____ 5 4 A A 0.25, 0.025, 0.52, 0.525 B C D B 0.25, 0.52, 0.025, 0.525 C 0.025, 0.525, 0.25, 0.52 D 0.025, 0.25, 0.52, 0.525 14. Which fraction is equivalent to 15 ? 100 A 1 85 B 3 50 C 3 20 D 1 10 © 2006 Buckle Down Publishing. COPYING IS FORBIDDEN BY LAW. 10. Which list shows the decimals in order from least to greatest?