,,,, 2 1 Large Numbers. Name Date Time. trillions billions millions thousands ones 10,000,000,000,000 1,000,000 10,000 1,000,000,000,000

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1 2 1 Large Numbers 100,000,000,000,000,,,, trillions billions millions thousands ones 10,000,000,000,000 1,000,000,000, ,000,000,000 10,000,000,000 1,000,000, ,000,000 10,000,000 1,000, ,000 10,000 1, Write the digit in each place of the number 6,812,507,439. a. millions b. hundred thousands c. ten millions d. billions e. hundred millions f. ten thousands 2. Write each of the following numbers in standard form. a. four hundred thirty thousand b. ninety million, one hundred five thousand c. one hundred seventy million, sixty-five d. nine billion, five hundred million, two hundred forty-three thousand 3. Write each number in expanded form. Example: 235 (2 º 100) (3 º 10) (5 º 1) a. 321,000 b. 7,300,000,000,000 c. 2,510, Use extended facts to complete the following. a. 1 million 1,000 º b. 1 billion 1,000 º c. 1 trillion 1,000 º 41

2 2 1 Large Numbers continued trillion,,,, billion million thousand Because the orbits of the planets are elliptical in shape, the distance between two planets changes over time. The least distances of Mercury, Venus, Saturn, and Neptune from Earth appear in the table at the right. The distances are approximations. Least Distance from Earth Planet Distance (in miles) Mercury 48,000,000 Venus 25,700,000 Saturn 850,000,000 Neptune 2,680,000, Write each planet s least distance from Earth in number-and-word notation. a. Mercury b. Venus c. Saturn d. Neptune 6. Write the following numbers in standard notation. a billion b. 6.5 trillion c. 0.9 million d. 0.7 hundred Practice Round each number to the given place ,254; hundreds ,989; ten thousands 9. 1,857,000; hundred thousands 10. 6,593,278; millions 42

3 LESSON 2 1 Walking Away with a Billion Dollars Suppose you inherit one billion dollars. The bank pays you the entire amount of money in $100 bills. About how much will your payment weigh in tons? Use the information below to solve the problem. You can cover a sheet of paper with about six $100 bills. There are 500 sheets in a ream of paper. There are 10 reams in 1 carton of paper. One ream of paper weighs about 5 pounds. One ton equals 2,000 pounds. Show all your work. Write an explanation that is clear and easy to follow. 43

4 2 2 Writing Decimals 1. Build a numeral. Write: 2. Build a numeral. Write: 9 in the thousandths place, 3 in the tenths place, 4 in the tenths place, 6 in the ten-thousandths place, 8 in the ones place, 4 in the hundredths place, 3 in the tens place, and 0 in the thousandths place, and 6 in the hundredths place. 1 in the ones place Answer: Answer:.. Write the following numbers in words Write a decimal place value in each blank space. 8. Bamboo grows at a rate of about , or four, kilometer per hour. 9. The average speed that a certain brand of catsup pours from the mouth of the bottle is about 0.003, or three, mile per hour. 10. A three-toed sloth moves at a speed of about to 0.098, or sixty-eight to ninety-eight, mile per hour. 44

5 2 2 Writing Decimals continued hundreds tens ones and tenths hundredths thousandths ten-thousandths hundred-thousandths millionths Write each of the following numbers in expanded notation. Example: (2 º 1) (7 º 0.1) (5 º 0.01) (6 º 0.001) Using the digits 0, 3, 6, and 8, write the greatest decimal number possible Using the digits 0, 3, 6, and 8, write the least decimal number possible.. Try This C A D B Name the point on the number line that represents each of the following numbers Refer to the number line above. Round to the nearest hundredth. Practice

6 LESSON 2 2 Modeling and Comparing Decimals One way to compare decimals is to model them with base-10 grids. The flat is the whole, or 1.0. The long is worth 0.1. Another way to compare decimals is to draw pictures. The cube is worth The fractional part of the cube is worth The flat is the whole, or 1.0. The long is worth 0.1. The cube is worth The fractional part of the cube is worth Use decimal models to complete the following º º º Model the decimal numbers in each pair. Draw a picture to record each model. Then compare the decimal numbers using,, or Model and record a decimal number that is between 0.41 and

7 2 3 Sports Records Solve. 1. The fastest winning time for the New York Marathon (Tesfay Jifar of Ethiopia, 2001) is 2 hours, 7.72 minutes. The second fastest time is 2 hours, minutes (Juma Ikangaa of Tanzania, 1989) How much faster was Jifar s time than Ikangaa s? 2. In the 1908 Olympic Games, Erik Lemming of Sweden won the javelin throw with a distance of meters. He won again in 1912 with a distance of meters. How much longer was his 1912 throw than his 1908 throw? 3. Driver Buddy Baker (Oldsmobile, 1980) holds the record for the fastest winning speed in the Daytona 500 at miles per hour. Bill Elliott (Ford, 1987) has the second fastest speed at miles per hour. How much faster is Baker s speed than Elliott s? 4. The highest scoring World Cup Soccer Final was in Teams played 26 games and scored 140 goals for an average of 5.38 goals per game. In 1950, teams played 22 games and scored 88 goals for an average of 4 goals per game. What is the difference between the 1954 and the 1950 average goals per game? Estimate Estimate Practice Solve mentally. 7. $0.36 $0.29 $0.64 $ $3.69 $8.31 $6.25 $25 47

8 LESSON 2 3 Modeling Subtraction of Decimals You can model subtraction of decimals using base-10 grids or pictures. For example, to solve , first represent 1.237, adjust by trading, and then subtract. trade trade Subtract left Use base-10 grids or pictures to find each difference. Show your work s 0.1s 0.01s 0.001s s 0.1s 0.01s 0.001s s 0.1s 0.01s 0.001s 48

9 2 4 Multiplying by Powers of 10 Some Powers of º 10 º 10 º º 10 º º º º º º 1 10 º 1 10 º ,000 1, Multiply º º º º º º º º Mathematician Edward Kasner asked his 9-year-old nephew to invent a name for the number represented by The boy named it a googol. Later, an even larger number was named a googolplex. This number is represented by 10 googol, or a. How many zeros are in the standard form of a googol, or ? b. One googolplex is 1 followed by how many zeros? 10. The speed of computer memory and logic chips is measured in nanoseconds. A nanosecond is one-billionth of a second, or 10 9 second. Write this number in standard form. 11. Light travels about 1 mile in seconds. If a spacecraft could travel at this speed, it would travel almost 10 6 miles in 5 seconds. About how far would this spacecraft travel in 50 seconds? miles Practice Mentally calculate your change from $ Cost: $4.75; Change: 13. Cost: $3.98; Change: 14. Cost: $0.89; Change: 15. Cost: $8.46; Change: 49

10 LESSON 2 4 What s My Rule? For each problem, complete the table and find the rule. Use Problem 4 to write your own What s My Rule? problem. 1. Rule: 2. Rule: in out in out $10 $100 $25 $1,450 $7,985 $2,300,000 $0.10 $3.00 $30.00 $ $88.50 $ Rule: 4. Rule: in out in out $0.90 $0.09 $5.00 $0.50 $2.00 $760 $1,000 50

11 2 5 Multiplying Decimals: Part 1 Multiply º87 º 23 º Use your answer for Problem 1 5. Use your answer for Problem 3 to place the decimal point in to place the decimal point in each product. each product. a. 2.3 º 8.7 a. 124 º 9.6 b. 23 º 0.87 b º 9.6 c. 2.3 º 87 c º 0.96 Two new U.S. nickels were issued in A likeness of Thomas Jefferson remained on the front of the nickels. The reverse side featured images commemorating either the Louisiana Purchase or the Lewis and Clark expedition. 6. A U.S. nickel is 1.95 mm thick. a. Estimate the height of a stack of 25 nickels. Estimate mm b. Calculate the actual height of the stack in mm. mm c. How much is a stack of 25 nickels worth? Practice Multiply by 0.10 to find 10% of each number % of $ % of $ % of % of

12 LESSON 2 5 Estimating and Calculating Cost Suppose you have $25.00 to spend on snacks for your basketball team. You need to purchase 25 pieces of fruit and 25 beverages. The table below shows the food items available and the cost of each item. Fruit Cost Beverages Cost Banana $0.42 Fruit punch $0.65 Apple $0.28 Orange juice $0.50 Orange $0.41 Bottled water $ Make a table of the items you will buy, how many of each item, and the cost. Remember that you can spend up to $25.00 but not more than $ Your table might have four columns with these headings: Food Item, Number of Items, Cost per Item, and Subtotals. 2. Explain how you decided which items to buy and how many of each item. 52

13 LESSON 2 5 Whole Number Multiplication Use your favorite multiplication algorithm to find the following products. Show your work in the computation grid below or on a separate sheet of paper º º º º º º º 5, º

14 2 6 Multiplying Decimals: Part 2 Place a decimal point in each problem º º º º 5.1 3, Multiply. Show your work on a separate sheet of paper or on the back of this page º º º º 53.9 Solve each problem. Then write a number model. (Hint: Change fractions to decimals.) 9. Janine rides her bike at an average speed of 11.8 miles per hour. At that speed, about how many miles can she ride in hours? Number Model 10. Kate types at an average rate of 1.25 pages per quarter hour. If she types for hours, about how many pages can she type? Number Model 11. Find the area in square meters of a rectangle with length 1.4 m and width 2.9 m. Number Model Practice Multiply mentally by 0.10 to find 10%. Then mentally calculate the percent that has been assigned to each number % of $ % of $ % of % of 80 = 54

15 LESSON 2 6 A Mental Multiplication Strategy The same strategy was used to solve both example problems below. This strategy can also be used to multiply numbers mentally. Example 1: Example 2: 16 º º / º 2 72 / º 2 8 º 5 36 º º º / º 2 18 º º º Explain the strategy. 2. Use the strategy to solve the problems below. Show your work. a. 16 º 1.5 b. 18 º 3.5 c. 20 º 0.75 d º Solve these problems mentally. a. 8 º 7.5 b. 24 º Make up two problems that can be solved using a mental multiplication strategy. a. b. 55

16 LESSON 2 6 Modeling Decimal Multiplication You can use an area model to find a product. Example: 0.3 º 0.5 Shade 0.3 of Next, shade 0.5 of The product is the area that is the grid this way: the grid this way: double-shaded this way: Since 0.15 of the grid is double-shaded, º Shade each factor. Then find the product º º º º º Write your own problem º 0.5 º

17 2 7 Dividing Numbers 3 Ways to Write a Division Problem R R6 246 / R Ways to Express a Remainder R , 2 or When estimating quotients, use close numbers that are easy to divide. Example: 346 / 12 Estimate 35 How I estimated: 350 / 10 = / 6 Estimate How I estimated: / 12 Estimate How I estimated: / 9 Estimate How I estimated: 4. 1,270 / 7 Estimate How I estimated: / 34 Estimate How I estimated: Solve using a division algorithm. Show your work on a separate sheet of paper or a computation grid / ,024 / , Practice Multiply mentally notebooks at $1.99 each pens at $2.96 each books at $24.98 each gifts at $99.99 each 57

18 2 8 Dividing Decimals For each problem, follow the steps below. Show your work on a separate sheet of paper or a computation grid Estimate the quotient. Use numbers that are close to the numbers given and that are easy to divide. Write your estimate. Then write a number sentence to show how you estimated. Ignore any decimal points. Divide as if the numbers were whole numbers. Use your estimate to insert a decimal point in the final answer Estimate / 4 Estimate How I estimated Answer How I estimated Answer / 13 Estimate / 7 Estimate How I estimated Answer How I estimated Answer 5. Find Give the answer as a decimal with 2 digits after the decimal point. Practice Divide mentally to find the price for 1 pound (lb). 6. Five people sent a $36 arrangement of flowers to a friend. Divide $36 into 5 equal shares. How much is 1 share, in dollars and cents? 7. $3.98 for 2 lb $ per 1 lb 8. $16.88 for 4 lb $ per 1 lb 9. $45.80 for 5 lb $ per 1 lb 10. $ for 10 lb $ per 1 lb 58

19 2 9 Using Scientific Notation Write each number in standard notation º º º º 10 8 Change the numbers given in standard notation to scientific notation. Change the numbers given in scientific notation to standard notation. 5. Light travels about 11,802,000,000, or, inches per second. 6. A bacterium can travel across a table at a speed of 1.6 º 10 4, or, km per hour. 7. One dollar bill has a thickness of , or, inches. 8. The mass of 1 million pennies is approximately º 10 6, or, grams. Use,, or to compare each pair of numbers º , º º º º Explain how you can tell whether a number written in scientific notation is less than 1. Practice Solve mentally ,625 3, º ,000 /

20 LESSON 2 9 Ground Areas of Buildings The approximate ground areas of some famous buildings are given below in scientific notation. To the left of the photograph of each building is its ground plan. Convert the scientific notation to standard notation. 1. Great Pyramid of Giza (Egypt; c B.C.) º 5.7 º 10 5, or, ft 2 2. Roman Colosseum (Rome, Italy; ) º 2.5 º 10 5, or, ft 2 3. St. Peter s Basilica (Vatican City; ) º 3.9 º 10 5, or, ft 2 4. Taj Mahal (Agra, India; ) º 9.8 º 10 4, or, ft 2 º Location and date(s) of construction 60

21 LESSON 2 9 Ground Areas of Buildings continued 5. Pentagon (Arlington, Virginia, United States; ) º 1.3 º 10 6, or, ft 2 º Location and dates of construction Source: Comparisons 6. Use the information in Problems 1 5 to write two comparisons. a. Ratio comparison (The area of one building is x times larger [or smaller] than the area of another building.) b. Difference comparison (The area of one building is x square feet more [or less] than the area of another building.) 7. Try to find out the ground area of a large building, such as your school, a shopping mall, an historic landmark, a sports arena, or a factory. How does that building s ground area compare to the ground area of each building pictured in Problems 1 5? 61

22 LESSON 2 9 Patterns and Powers of 10 Use any patterns you notice to fill in the blanks , , What do you notice about the number of digits after the decimal point and the negative powers of 10? LESSON 2 9 Patterns and Powers of 10 Use any patterns you notice to fill in the blanks , , What do you notice about the number of digits after the decimal point and the negative powers of 10? 62

23 2 10 Exponential Notation Use your calculator to write each number in standard notation (0.25) (0.41) (2.5) 3 Use digits to write each number in exponential notation. 7. three to the ninth power 8. eight to the seventh power 9. eleven to the negative third power 10. five-tenths to the negative sixth power Write each number as a product of repeated factors. Example: º 5 º ( 1 2 ) You can find the total number of different 4-digit numbers that can be made using the digits 1 through 9 by raising the number of choices for each digit (9) to the number of digits (4), or 9 4. Based on this pattern, how many different 5-digit numbers could you make from the digits 1 through 8? Practice Solve mentally ,200 / º525 63

24 LESSON 2 10 Binary Numbers The table below shows how to write whole numbers 1 through 10 as binary numbers. A binary number is written with a subscripted two to distinguish it from a base-ten number. Powers of Base-Ten Number Binary Number two 1 º two 1 º º two 1 º º two 1 º º º two 1 º º º two 1 º º º two 1 º º º two 1 º º º º two 1 º º º º two 1 º º º º 2 0 To write a binary number as a base-ten number, first write the binary number in expanded notation. Then convert to standard form. Example: two (1 º 2 4 ) (1 º 2 3 ) (1 º 2 2 ) (1 º 2 1 ) (1 º 2 0 ) (1 º 16) (1 º 8) (1 º 4) (1 º 2) (1 º 1) Use the table and example above to write each binary number as a base-ten number two two two two two two Try This Use patterns in the table to write the binary number for each number

25 2 11 Scientific Notation Write the following numbers in scientific notation , thousand Write the following numbers in standard notation º º º º 10 2 Write the next two numbers in each pattern º 10 1 ; 0.1; 1 º 10 2 ; 0.01; ; , 0.002, ,, Solve the following problems. Write each answer in scientific notation. 11. (4 º 10 3 ) (2 º 10 1 ) 13. (5 º 10 1 ) (7 º 10 4 ) Use a calculator to complete the table. Problem Calculator Display Scientific Notation Standard Notation 5,000, º / 5 16 Practice Find the missing digits to complete each number sentence. 16., 63 3,9 9 2, , 4 4,8 6 6,270 65

26 LESSON 2 11 Practicing Calculator Skills Use your calculator to complete the table. Problem Scientific Notation Standard Notation 100, º ,000,000,000,000,000 20, º LESSON 2 11 Practicing Calculator Skills Use your calculator to complete the table. Problem Scientific Notation Standard Notation 100, º ,000,000,000,000,000 20, º

27 2 12 Unit 3: Family Letter Variables, Formulas, and Graphs In Unit 3, students will be introduced to variables symbols such as x, y, and m that stand for a specific number or any number in a range of values. The authors of Everyday Mathematics believe that work with variables is too important to be delayed until highschool algebra courses. The problem Solve 3x might be difficult for some high-school students because they see it as merely symbol manipulation. Problems such as these are posed to Everyday Mathematics students as puzzles that can be unraveled by asking, What number makes the equation true? I need to add 12 to 40 to get 52. Three times what number yields 12? The answer is x 4. In addition to being used in algebraic equations, variables are also used to describe general patterns, to form expressions that show relationships, and to write rules and formulas. Unit 3 will focus on these three uses of variables. In this unit, your child will work with What s My Rule? tables like the one below (introduced in early grades of Everyday Mathematics). He or she will learn to complete such tables following rules described in words or by algebraic expressions. Your child will also determine rules or formulas from information given in tables and graphs. Rule: y (4 º x) 3 x y ln addition, your child will learn how to name cells in a spreadsheet and write formulas to express the relationships among spreadsheet cells. If you use computer spreadsheets at work or at home, you may want to share your experiences with your child. The class will play Spreadsheet Scramble, in which students practice computation and mental addition of positive and negative numbers. Encourage your child to play a game at home. See the Practice through Games section of this letter for some suggestions A B C D E F Total 4 5 Total Please keep this Family Letter for reference as your child works through Unit 3. 67

28 2 12 Unit 3: Family Letter cont. Math Tools Your child will be using spreadsheets, a common mathematics tool for the computer. The spreadsheet, similar to the one shown here, gets its name from a ledger sheet for financial records. Such sheets were often large pages, folded or taped, that were spread out for examination. Vocabulary Important terms in Unit 3: algebraic expression An expression that contains a variable. For example, if Maria is 2 inches taller than Joe, and if the variable m represents Maria s height, then the algebraic expression m 2 represents Joe s height. cell In a spreadsheet, a box formed where a column and a row intersect. A column is a section of cells lined up vertically. A row is a section of cells lined up horizontally. column A B C D A Class picnic ($$) B budget for class picnic quantity food items unit price cost 6 packages of hamburgers packages of hamburger buns bags of potato chips quarts of macaroni salad bottles of soft drinks subtotal % tax 4.23 total general pattern In Everyday Mathematics, a number model for a pattern or rule. special case In Everyday Mathematics, a specific example of a general pattern. For example, is a special case of y y 2y and 9 = 4.5 º 2 is a special case of A = l º w. Same as instance of a pattern. time graph A graph representing a story that takes place over time. For example, the time graph below shows the trip Mr. Olds took to drive his son to school. The line shows the increases, decreases, and constant rates of speed that Mr. Olds experienced during the 13-minute trip. C D row cell formula A general rule for finding the value of something. A formula is often written using letters, called variables, that stand for the quantities involved. For example, the formula for the area of a rectangle may be written as A b º h, where A represents the area of the rectangle, b represents its base, and h represents its height. Speed (mph) Time (min) variable A letter or symbol that represents a number. A variable can represent one specific number, or it can stand for many different numbers. 68

29 2 12 Unit 3: Family Letter cont. Do-Anytime Activities Try these ideas to help your child with the concepts taught in this unit. 1. If you are planning to paint or carpet a room, consider having your child measure and calculate the area using the area formula for rectangular surfaces: Area base º height. If the room is irregular in shape, divide it into rectangular regions, find the area of each region, and add all the areas to find the total area. If a room has a cathedral ceiling, imagine a line across the top of the wall to form a triangle. Your child can use the area formula for triangles: Area 1 2 º (base º height), to calculate the area of the triangle. height base 2. If you use a spreadsheet program on a home computer, help your child learn how to use it. You might help your child set up a spreadsheet to keep track of his or her math scores and to figure out the mean. 3. Practice renaming fractions, which is a prerequisite skill for Unit 4. Examples: Rename as Fractions Rename as Mixed or Whole Numbers Building Skills through Games The concepts learned in Unit 3 will be reinforced through several math games included in this unit that are fun to play in class and at home. Detailed game instructions for all sixth-grade games are available in the games section of the Student Reference Book. Here is a list and a brief description of some of the games in this unit: Getting to One See Student Reference Book, page 321 Two players can play this game using a calculator. The object of the game is to divide a number by a mystery number and to find the mystery number in as few tries as possible. Players apply place-value concepts of decimal numbers to determine which numbers to play. Division Top-It (Advanced Version) See Student Reference Book, page 336 Two to four people can play this game using number cards 1 through 9. Players apply place-value concepts, division facts, and estimation strategies to generate whole-number division problems that will yield the largest quotient. 69

30 2 12 Unit 3: Family Letter cont. As You Help Your Child with Homework As your child brings assignments home, you may want to go over the instructions together, clarifying them as necessary. Some of the answers listed below will guide you through the unit s Study Links. Study Link 3 1 Sample answers (1 7): 1. a. The sum of any number and 0 is equal to the original number. b ; (2 º 24) 24 3 º 24; (2 º 10) 10 3 º ; x 2 º x 3 x 5 5. s º 0.1 s m , , , , 0.70 Study Link 3 2 Sample answers (1 7): 1. (6 º 2) º 3 6 º (2 º 3); (6 º 1) º 5 6 º (1 º 5) ( 6 2 ) (2 º 12) 6; 10 ( 4 ) (2 º 10) º ; º a b a ( b) 5. m n m n 6. s t s 2 c 1 7. º t 2 d 2 c º 1 d º Study Link º 3 º 3 1. x 7 2. d c 12, c 12, or 1 12 c 4. 2 º h, or 2h; r 8, or (3 º r) 8; Study Link a. Subtract 0.22 from m. b. n m a. Multiply r by 1 or divide r by 2. 2 b. r º 0.5 t 3. q (2 º p) , ,040 Study Link in: ; out: , 9 1 2, 3, in: 6, 1 ; out: 48, 1.2, 2 3. in: 7, 0; out: 0, Divide the in number by 3; d b 3 5. Answers vary Study Link Perimeter (in.): 4, 8, 12, 16, 20; Area (in. 2 ): 1, 4, 9, 16, in in in in Study Link January 2. $ A5 4. C3 5. Column E: $118.75; $152.95; $2, E3 B3 C3 D3 7. E5 B5 C5 D5 8. $ Study Link , 2, 9 a. Sample answer: Add 6 to x. b. x ( 6) y 12. a. 25 b. 32 c. 50 d a b. 1 c. 2 d. 9 2 Study Link Sample answer: People are getting on the Ferris wheel sec sec 4. 4 times sec Study Link Jenna s Profit: $3, $6, $9, $12, $15; Thomas s Profit: $6, $8, $10, $12, $14 2. $18, $16 3. Jenna 4. Jenna s 5. $3, $2 6. (4,12) 7. a. 81 b. 8,000 c. 76 d. 875 e. 3