Contents. Introduction Place Value up to Billions... 6 Powers and Exponents... 9 Place Value... 12

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1 Contents Introduction... 4 Place Value up to Billions... 6 Powers and Exponents... 9 Place Value Counting and Adding Large Numbers Rounding Rounding Scientific Notation Calculator Multiples, Estimation, and the Calculator Review Answers More from Math Mammoth

2 Introduction Math Mammoth Place Value 5 is a short worktext that covers value up to the trillions that is, numbers up to 15 digits rounding them, and using a calculator. It is suitable for 5th and 6th grades. The lessons are taken from the complete curriculum I have written for the 5th and 6th grades (Math Mammoth Light Blue Series). The first lesson covers place value up to billions. Then, we study exponents, and right after that, place value up to trillions, writing the numbers in the expanded form using exponents. After working with addition a little, there are two lessons about calculator usage. I have received numerous comments on the harm that indiscriminate calculator usage can cause. In a nutshell, if children are allowed to use calculators freely, their minds get lazy, and they will start relying on calculators even for simple things. It s just human nature! As a result, students enter college without even knowing their multiplication tables by heart. Then they have tremendous trouble if they are required to use mental math to solve simple problems. So we educators need to limit calculator usage until the students are older. Children can not decide this for themselves, and definitely not in fifth grade. However, I realize that the calculator is extremely useful, and students do need to learn to use it. In this curriculum, I strive to show the students not only how to use a calculator, but also when to use it and when not to use it. This book includes problems where calculator usage is appropriate. We also practice estimating the result before calculating it with a calculator. In the last lesson, students need to choose whether mental math or a calculator is the best tool for the calculation. I wish you success in your math teaching! Maria Miller, the author 4

3 Helpful Resources on the Internet Use these free online resources to supplement the bookwork as you see fit. You can access an up-to-date online version of this list at Naming Numbers These pages teach number naming skills covered in K8 math courses. Each page has an explanation, interactive practice and challenge games about naming numbers. Megapenny project Visualizes big numbers with pictures of pennies. Powers of ten Illustrates the dramatic changes of scale when zooming in or out is by powers of ten. Cookie dough Practices naming big numbers. Keep My place Fill in the big numbers to this cross-number puzzle. Estimation Exercises about rounding whole numbers and decimals, front-end estimation, estimating sums and differences. Estimation at AAA Math Exercises about rounding whole numbers and decimals, front-end estimation, estimating sums and differences. Each page has an explanation, interactive practice, and games. Place Value Game Create the largest possible number from the digits the computer gives you. Unfortunately, the computer will give you each digit one at a time and you won't know what the next number will be. 5

4 Place Value up to Billions This number is read: four hundred [and] nine billion, three hundred [and] eighty-two million, forty-three thousand, five hundred and fifty-nine We separate the digits in large numbers in groups of three. These groupings are called periods. Learn their names from the chart. The letters h t o over the top of the digits stand for hundreds, tens, ones. Once you remember the names for these groupings, it is very easy to read large numbers. Simply read each three digits as if it were a number by itself, and when you come to the comma, say the word billion, million, or thousand. Look at the billions period. The digit 2 is in the hundred billions place. The digit 7 is in the ten billions place. The digit 3 is in the billions place. Look at the millions period. The digit 5 is in the hundred millions place. The digit 1 is in the ten millions place. The digit 3 is in the millions place. Look at the thousands period. The digit 4 is in the hundred thousands place. The digit 0 is in the ten thousands place. The digit 0 is in the thousands place. Lastly look at the ones period. The digit 0 is in the hundreds place. The digit 2 is in the tens place. The digit 1 is in the ones place. 1. Study the number 85,359,204,031. Read it aloud. a. Write the digit in the hundred thousands place. b. Write the digit in the ten billions place. c. Write the digit in the millions place. d. Write the digit in the billions place. 6 Math Mammoth Place Value 5 (Blue Series)

5 2. Arrange the digits into groups of three by writing the commas where they belong. Practice reading them until you can do so fluently. a c b Follow the example to fill in the missing numbers. a. 308,067,008,307 = 308 billion, 67 million, 8 thousand, 307 b. 45,038,300,820 = billion, million, thousand, c. 915,008,360,000 = billion, million, thousand, d. 9,000,004,000 = billion, million, thousand, 4. Write the numbers. You will need to use some zeros! a. 159 billion, 372 million, 932 thousand, 2 b. 7 billion, 372 million, 40 thousand, 20 c. 372 million, 150 d. 607 billion, 43 thousand, 17 e. 390 billion, 430 thousand, 89 f. 50 billion, 302 million d Write these numbers. Now the periods are mixed up! a thousand + 32 million = b. 500 billion thousand = c. 4 billion + 9 million thousand = d million billion + 2 thousand = e. 200 billion + 53 thousand million = f billion + 3 thousand = 7 Math Mammoth Place Value 5 (Blue Series)

6 Consider a single digit in a number, such as 6 in the number 3,065,800. The place of that digit and its value are different things. What place is the digit in? Just say the name of the place or the location. The 6 is in the ten thousands place. What is the value of a digit? Imagine setting all the other digits in the number to zero to see the digit s value, or what it is worth, in that number. The value of the digit 6 in our number is 60,000. h t o h t o 1 9 0,2 3 5 The value of the digit 1 is 100,000 The value of the digit 9 is 90,000 The value of the digit 2 is 200 The value of the digit 3 is 30 The value of the digit 5 is 5 If you add all these, you will get the number itself! ht o ht o ht o ht o 4 71,906,238,050 The digit 7 is in the ten billions place. Its value is 70,000,000,000. The digit 6 is in the millions place. Its value is 6,000,000. The digit 2 is in the hundred thousands place. Its value is 200, In what place is the underlined digit? What is its value? a. 293,476,020 b. 3,299,005,392 Place: ten thousands place c. 28,837,402,000 d. 293,476,020 e. 3,299,005,392 f. 28,837,432, Add. Write the numbers with digits. a. 1 billion + 1 million + 40 million billion = b. 50 thousand + 7 million + 7 thousand million + 6 billion = c. 45 million + 8 thousand million + 40 thousand + 35 = 8 Math Mammoth Place Value 5 (Blue Series)

7 Exponents are a kind of shorthand for writing repeated multiplications by the same number. For example, is written is written 5 6. The tiny raised number is called the exponent. It tells us how many times the base number is multiplied by itself. Powers and Exponents The expression 2 5 is read two raised to the fifth power, two to the fifth power, or even just two to the fifth. Similarly, 7 9 is read seven raised to the ninth power, seven to the ninth power, or seven to the ninth. The powers of 6 are simply expressions where 6 is raised to some power: For example, 6 3, 6 4, 6 45, and 6 99 are powers of 6. However, expressions with powers of 2 and 3 are almost always read differently: The expression 11 2 is usually read as eleven squared because it describes the area of a square with sides 11 units long. Similarly, 31 3 is generally read as thirty-one cubed because it gives the volume of a cube with sides 31 units long. 1. Write out these expressions as multiplications, then solve them. a. 3 2 = 3 3 = 9 f b. 1 6 g. 2 3 c. 4 3 h. 8 2 d i. 0 3 e. 5 3 j Rewrite these expressions as multiplication. Then use a calculator to solve them. a. 6 4 c b d Rewrite each expression using an exponent, then solve it. You may use a calculator. a b c. 40 squared d e. nine to the eighth power f. eleven cubed 9 Math Mammoth Place Value 5 (Blue Series)

8 The expression 7 2 is read seven squared because it tells us the area of a square with sides 7 units long. For example, if the sides of a square are 5 cm long, then its area is 5 cm 5 cm = 25 cm 2. Notice that the symbol for square centimeters is cm 2. This means centimeter centimeter. We are, in effect, squaring the measuring unit! In fact, we do the same thing when we use the units square meters and square kilometers. We could also write that expression as (5 cm) 2 or the quantity, five centimeters, squared. This means that both the 5 and the unit cm are squared, which makes 25 cm 2. Without the parenthesis it would be 5 cm 2 and mean five square centimeters, which is something very different. We can do the same thing with the traditional units of inches, feet, and miles. People often write sq. in. for square inches, or sq. ft. for square feet, instead of in 2 and ft 2, but both ways are correct. Similarly, 7 3 is read seven cubed because it gives the volume of a cube with sides 7 units long. For example, if the sides of a cube are 10 cm long, then its volume is (10 cm) 3 = 1,000 cm 3, or one thousand cubic centimeters.. 4. Express the area using exponents and solve. a. A square with a side of 12 kilometers: The area is (12 km) 2 = c. A square with sides each 6 inches long: Its area is b. A square with sides 6 m long: Its area is d. A square with a side with a length of 12 ft: The area is 5. Express the volume using exponents and solve. a. A cube with a side of 2 cm: b. A cube with sides each 10 inches long: The volume is c. A cube with sides 1 ft in length: d. A cube with edges that are all 5 m long: 6. a. The perimeter of a square is 40 cm. What is its area? b. The volume of a cube is 64 cubic inches. How long is its side? c. The area of a square is 121 m 2. What is its perimeter? d. The area of one face of a cube is 64 in 2. What is its volume? 10 Math Mammoth Place Value 5 (Blue Series)