Place Value: Expanding Whole Numbers 3 Ways For Dixon Elementary School’s 5th-Grade Math Classes

Expanded Forms When expanding numbers, there are 3 different methods we can use: expanding by place value, expanding by base-10 products, and expanding by base-10 exponents. You might see 1, 2, or all 3 methods on standardized tests.

Expanded Form by Place Value Method #1 of expanding shows numbers expanded into the sum (+) of each digit’s place value. This method just involves decomposing numbers. The number 652 would expand like this: 600 + 50 + 2 The value of the 6 = 600, the value of the 5 = 50, and the value of the 2 = 2.

Expanded Form by Place Value Here’s another example: 4,365 There is a 4 in the one thousands place, so the value of the 4 = 4,000 (4 x 1,000). There is a 3 in the hundreds place, so the value of the 3 = 300 (3 x 100). There is a 6 in the tens place, so the value of the 6 = 60 (6 x 10). There is a 5 in the ones place, so the value of the 5 = 5 (5 x 1). 4,000 + 300 + 60 + 5

Expanded Form by Place Value When expanding numbers with 0s in a place, we can skip that place, and move on to the next: 60,503 = 60,000 + 500 + 3

Expanded Form by Place Value: Your Turn! Expand each number by its place value: 79 893 1,265 20,482 304,007 ANSWERS: 70 + 9 800 + 90 + 3 1,000 + 200 + 60 + 5 20,000 + 400 + 80 + 2 300,000 + 4,000 + 7

Expanded Notation with Base-10 Products Method #2 of expanding involves using base-10 products with expanded notation. Base-10 is the number system we use in math. The system uses 10 numerical symbols to form all numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, & 9 When expanding with base-10 products, we call it expanded notation.

Expanded Notation with Base-10 Products In expanded notation, we write the number as the sum (+) of the products of each digit multiplied by the value of its place. The value of the ONES place = 1. The value of the TENS place = 10. The value of the HUNDREDS place = 100. What is the value of the THOUSANDS place?

Expanded Notation with Base-10 Products Let’s look at the number, 359. The 3 is in the hundreds place, so we multiply the 3 by 100 (3 x 100). The 5 is in the tens place, so we multiply the 5 by 10 (5 x 10). The 9 is in the ones place, so we multiply the 9 by 1 (9 x 1). So the expanded notation is: (3 x 100) + (5 x 10) + (9 x 1)

Expanded Notation with Base-10 Products Here are some more examples: 6,219 = (6 x 1,000) + (2 x 100) + (1 x 10) + (9 x 1) 54,678 = (5 x 10,000) + (4 x 1,000) + (6 x 100) + (7 x 10) + (8 x 1)

Expanded Notation with Base-10 Products Let’s do this one together: 3,462 = The 3 is in the place, so we multiply the 3 by . The 4 is in the place, so we multiply the 4 by . The 6 is in the place, so we multiply the 6 by . The 2 is in the place, so we multiply the 2 by . thousands ? ? 1,000 hundreds ? ? 100 ? tens 10 ? ? ones 1 ?

Expanded Notation with Base-10 Products So… 3,462 = (3 x ) + (4 x ) + (6 x ) + (2 x ) ? 1,000 100 ? 10 ? 1 ?

Expanded Notation with Base-10 Products: Your Turn! Use expanded notation to expand each number. 1) 48 2) 387 3) 6,912 4) 52,376

Expanded Notation with Base-10 Products: Your Turn! ANSWERS: 1) 48 = (4 x 10) + (8 x 1) 2) 387 = (3 x 100) + (8 x 10) + (7 x 1) 3) 6,912 = (6 x 1,000) + (9 x 100) + (1 x 10) + (2 x 1) 4) 52,376 = (5 x 10,000) + (2 x 1,000) + (3 x 100) + (7 x 10) + (6 x 1)

Expanded Notation with Base-10 Exponents Method #3 of expanding is expanded notation with exponents. An exponent shows the number of times a number is multiplied by itself. An exponent is attached to its base number like this: 103 exponent base

Expanded Notation with Base-10 Exponents We say 103 like this: 10 to the power of 3 OR 10 to the 3rd power. What 103 means is: 1 2 3 10 x 10 x 10 (10 multiplied by itself 3 times)

Expanded Notation with Base-10 Exponents Here are some more examples: 1) 102: Say, “10 to the power of 2” or “10 to the 2nd power.” It means, 10 x 10 2) 104: Say, “10 to the power of 4” or “10 to the 4th power.” It means, 10 x 10 x 10 x 10

Expanded Notation with Base-10 Exponents: Your Turn! How do we SAY these bases & exponents in 2 different ways? 1) 106 2) 101 3) 104 4) 105 5) 107 10 to the power of 6 & 10 to the 6th power 10 to the power of 1 & 10 to the 1st power 10 to the power of 4 & 10 to the 4th power Complete this practice orally. 10 to the power of 5 & 10 to the 5th power 10 to the power of 7 & 10 to the 7th power

Expanded Notation with Base-10 Exponents: Your Turn! What does each MEAN? 1) 106 2) 101 3) 104 4) 105 5) 107

Expanded Notation with Base-10 Exponents: Your Turn! ANSWERS: 106 = 10 x 10 x 10 x 10 x 10 x 10 2) 101 = 10 3) 104 = 10 x 10 x 10 x 10 4) 105 = 10 x 10 x 10 x 10 x 10 5) 107 = 10 x 10 x 10 x 10 x 10 x 10 x 10

Expanded Notation with Base-10 Exponents To calculate the value of a base and its exponent, just complete the multiplication: 102 = 10 x 10, and 10 x 10 = 100, so 102 = 100; 103 = 10 x 10 x 10, and 10 x 10 x 10 = 1,000, so 103 = 1,000.

Expanded Notation with Base-10 Exponents Here are some more examples: 1) 104 = 10 x 10 x 10 x 10 = 10,000 2) 105 = 10 x 10 x 10 x 10 x 10 = 100,000 Do you notice a relationship between the exponent and its number’s value? 104 = 10,000 105 = 100,000

Expanded Notation with Base-10 Exponents The relationship between the exponent and its number’s value makes finding its value easy! 100 = 1 101 = 10 102 = 100 103 = 1,000 104 = 10,000 105 = 100,000

Expanded Notation with Base-10 Exponents When we expand using base-10 exponents, we just write the number as the sum (+) of each place’s base-10 exponent products: 64 = (6 x 101) + ( 4 x 100 ) 643 = (6 x 102) + ( 4 x 101 ) + (3 x 100)

Expanded Notation with Base-10 Exponents: Your Turn! Expand each number by its base-10 exponents. 72 193 5,408 62,937

Expanded Notation with Base-10 Exponents: Your Turn! ANSWERS: 1) 72 = (7 x 101) + (2 x 100) 2) 193 = (1 x 102) + (9 x 101) + (3 x 100) 3) 5,408 = (5 x 103) + (4 x 102) + (8 x 100) 4) 62,937 = (6 x 104) + (2 x 103) + (9 x 102) + (3 x 101) + (7 x 100)

POP QUIZ! Expand each number 3 ways: 461 5,387 90,256 a) by its place value: 50 + 4; b) by its expanded notation with base-10 products: (5 x 10) + (4 x 1); and c) by its expanded notation with base-10 exponent products: (5 x 101) + (4 x 100). 461 5,387 90,256

POP QUIZ! ANSWERS: 1) 461 = a) 400 + 60 + 1 b) (4 x 100) + (6 x 10) + (1 x 1) c) (4 x 102) + (6 x 101) + (1 x 100) 2) 5,387 = a) 5,000 + 300 + 80 + 7 b) (5 x 1,000) + (3 x 100) + (8 x 10) + (7 x 1) c) (5 x 103) + (3 x 102) + (8 x 101) + (7 x 100) 3) 90,256 = a) 90,000 + 200 + 50 + 6 b) (9 x 10,000) + (2 x 100) + (5 x 10) + (6 x 1) c) (9 x 104) + (2 x 102) + (5 x 101) + (6 x 100)