Whole Numbers and Place Value

How Many Briskets?

Franklin Barbecue in Austin, Texas serves about 1,200 pounds of brisket every day. They are open 6 days a week, 52 weeks a year. How many pounds of brisket do they serve in a year?

To answer that, you need to multiply 1,200 by 6 by 52. That is a multi-digit multiplication problem, and getting it right depends on understanding place value -- the idea that a digit's position determines what it is worth.

This unit sharpens the arithmetic you already know and introduces the order of operations, which is the grammar of math: the rules that tell you which calculations to do first.

Place Value: Position Is Everything

In our number system, every digit has a value determined by its position. The number 5 means five. But move it one place to the left and it means fifty. One more place: five hundred.

Consider the number 3,472,905,168.

That number is read: "three billion, four hundred seventy-two million, nine hundred five thousand, one hundred sixty-eight."

Each place is ten times the value of the place to its right. This is why we call it a base-ten system.

Rounding and Estimation

Not every problem needs an exact answer. When you are estimating the cost of groceries, checking if you have enough money for gas, or guessing how many people attended a football game, rounding gets you close enough.

How to Round

1. Find the digit in the place you are rounding to.
2. Look at the digit to its right.
3. If that digit is 5 or greater, round up. If it is less than 5, round down.

Example: Round 47,832 to the nearest thousand.

The thousands digit is 7. The digit to its right is 8. Since 8 is 5 or greater, round up: 48,000.

Example: Round 3,241,506 to the nearest hundred thousand.

The hundred thousands digit is 2. The digit to its right is 4. Since 4 is less than 5, round down: 3,200,000.

Estimation in Action

Back to Franklin Barbecue: 1,200 pounds per day, 6 days per week, 52 weeks per year.

Estimate: 1,200 x 6 is about 1,200 x 6 = 7,200 per week. And 7,200 x 50 (rounding 52 down) = 360,000.

The actual answer: 1,200 x 6 x 52 = 374,400 pounds per year. Our estimate of 360,000 was within 4% -- close enough to plan a supply chain.

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Concept Check

The AT&T Stadium in Arlington, Texas holds 80,000 fans. If 73,428 fans attended last Sunday's game, how would you round that number to the nearest ten thousand? To the nearest thousand?

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To the nearest ten thousand: the ten thousands digit is 7, and the next digit is 3, which is less than 5. Round down to 70,000.
To the nearest thousand: the thousands digit is 3, and the next digit is 4, which is less than 5. Round down to 73,000.

Multi-Digit Multiplication

You have been multiplying since third grade. In sixth grade, the numbers get bigger and the problems get more complex. The key is organizing your work so you do not lose track of place values.

The Standard Algorithm

Multiply 348 x 26.

Step 1: Multiply 348 by 6 (the ones digit of 26).

348 x 6 = 2,088

Step 2: Multiply 348 by 20 (the tens digit of 26).

348 x 20 = 6,960

Step 3: Add the partial products.

2,088 + 6,960 = 9,048

Checking with Estimation

Before you accept your answer, estimate: 348 is about 350, and 26 is about 25. 350 x 25 = 8,750. Our answer of 9,048 is in the right neighborhood.

Multi-Digit Division

Division is the inverse of multiplication. When you divide 9,048 by 26, you are asking: "How many groups of 26 fit into 9,048?"

The Long Division Algorithm

Divide 7,854 by 18.

Step 1: 18 does not go into 7 (too small). Try 78: 18 x 4 = 72. Write 4 above the 8. Subtract: 78 - 72 = 6.

Step 2: Bring down the 5 to get 65. 18 x 3 = 54. Write 3. Subtract: 65 - 54 = 11.

Step 3: Bring down the 4 to get 114. 18 x 6 = 108. Write 6. Subtract: 114 - 108 = 6.

Answer: 436 remainder 6, or 436 R6.

Check: 436 x 18 = 7,848. Add the remainder: 7,848 + 6 = 7,854. Correct.

Order of Operations

What does 3 + 4 x 2 equal? If you add first, you get 14. If you multiply first, you get 11. Only one answer is correct: 11.

Mathematics needs a universal agreement about which operations to do first. That agreement is the order of operations:

1. Parentheses (do what is inside first)
2. Exponents (powers)
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

The memory device is PEMDAS: "Please Excuse My Dear Aunt Sally."

Worked Examples

Evaluate: 5 + 3 x (8 - 2)

Step 1: Parentheses: 8 - 2 = 6

Step 2: Multiplication: 3 x 6 = 18

Step 3: Addition: 5 + 18 = 23

Evaluate: 48 / 6 + 2 x 5 - 1

Step 1: Division: 48 / 6 = 8

Step 2: Multiplication: 2 x 5 = 10

Step 3: Addition and subtraction (left to right): 8 + 10 - 1 = 17

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Concept Check

Evaluate the expression: 2 x (15 - 3) + 36 / 6

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Parentheses first: 15 - 3 = 12.
Then multiplication: 2 x 12 = 24.
Then division: 36 / 6 = 6.
Then addition: 24 + 6 = 30.

Looking Ahead

Whole numbers and place value are the bedrock of everything you will study this year. In Unit 2, you will extend these same ideas to fractions and decimals -- numbers that live between the whole numbers. The multiplication and division algorithms you practiced here will adapt to work with decimal points, and the order of operations will appear in every unit from here to the end of the course.