Place Value Basics: Decimals for Dixon Elementary School’s 5th-Grade Math Classes

What Is a Decimal? Decimals are numbers LESS THAN one whole. They are simply FRACTIONS of one whole. Decimals are places that appear to the RIGHT of the ONES PLACE. A DECIMAL POINT (.) separates whole numbers from decimals.

What Is a Decimal? Let’s look at the number: 123.456 The digits 123 are in the ONES PERIOD. The digits 456 are in DECIMAL PLACES.

What Is a Decimal? Let’s run over to CoolMath.com Click here!

What Is a Decimal? Just like with whole numbers, decimals are expressed in STANDARD, WORD, and EXPANDED forms (and notations).

Standard Form In STANDARD form, decimals are expressed with numbers: 0.1 0.12 0.123 0.1234

Word Form In WORD form, decimals are expressed with the same terms we use in FRACTIONS: 0.1 = one-tenth = 1/10 0.01 = one hundredth = 1/100 0.001 = one thousandth = 1/1000

Word Form To properly put decimals in word form, we first need to understand how to READ and SAY them. Click here for CoolMath! Click here for a Math Coach video!

Standard & Word Forms So, we need to be able to go between expressing decimals in STANDARD and WORD forms, but we also want to express them as FRACTIONS: 0.6 = six-tenths = 6/10 0.35 = thirty-five hundredths = 35/100 2.572 = two AND five hundred seventy-two thousandths = 2-572/1000

Standard & Word Forms The HYPHEN rules are tricky. Use a hyphen when expressing all TENTHS (four-tenths, eight-tenths, etc.) Only use hyphens for two-digit numbers between 21-99 with hundredths or thousandths (thirty-four hundredths, sixty-five thousandths, eight hundredths).

Standard & Word Forms: Your Turn Write each in 3 forms: standard, word, and fractional forms. Example: 2.5 = two and five-tenths = 2-5/10 0.3 1.43 sixteen hundredths five and eighty-one thousandths 2/100 16/1000

Standard & Word Forms: Your Turn ANSWERS: 0.3 = three-tenths = 3/10 1.43 = one and forty-three hundredths = 1-43/100 sixteen hundredths = 0.16 = 16/100 five and eighty-one thousandths = 5.081 = 5-81/1000 2/100 = two hundredths = 0.02 16/1000 = sixteen thousandths = 0.016

Tomorrow… …we EXPAND!

(EXPANDED NOTATION with EXPONENTS). Expanding 3 Ways We will learn to expand decimals in the same 3 ways we expanded whole numbers: EXPANDED FORM, EXPANDED NOTATION, and SCIENTIFIC NOTATION (EXPANDED NOTATION with EXPONENTS).

Expanded Form When we EXPANDED whole numbers, we simply found the SUM (+) of the values of a number’s digits: 567 = 500 + 60 + 7

Expanded Form We do the same when we EXPAND decimals. We simply found the SUM (+) of the values of a number’s digits: 5.67 = 5 + 0.6 + 0.07 -OR- 5 + 6/10 + 7/100

Expanded Form Let’s look at another one: 324.167 324.167 = 300 + 20 + 4 + 0.1 + 0.06 + 0.07 -OR- 300 + 20 + 4 + 1/10 + 6/100 + 7/1000

Expanded Form Here’s one more: 609.208 609.208 = 600 + 9 + 0.2 + 0.08 600 + 9 + 2/10 + 8/1000

Expanded Form: Your Turn Let’s practice. Write the expanded form of each using DECIMALS and FRACTIONS: Example: 9.35 = 9 + 0.3 + 0.05 and 9 + 3/10 + 5/100 4.6 32.89 467.391 6.407 18.003

Expanded Form: Your Turn ANSWERS: 1) 4.6 = 4 + 0.6 4 + 6/10 2) 32.89 = 30 + 2 + 0.8 + 0.09 30 + 2 + 8/10 + 9/100 3) 467.391 = 400 + 60 + 7 + 0.3 + 0.09 + 0.001 400 + 60 + 7 + 3/10 + 9/100 + 1/1000 4) 6.407 = 6 + 0.4 + 0.007 6 + 4/10 + 7/1000 5) 18.003 = 10 + 8 + 0.003 10 + 8 + 3/1000

Expanded Notation In EXPANDED NOTATION, we found the SUM (+) of each digit multiplied by its place: 684 = (6 x 100) + (8 x 10) + (4 x 1)

Expanded Notation In EXPANDED NOTATION with decimals, we do the same: 6.84 = (6 x 1) + (8 x 0.1) + (4 x 0.01) -OR- (6 x 1) + (8 x 1/10) + (4 x 1/100)

(2 x 10) + (8 x 1) + (7 x 1/10) + (6 x 1/100) + (3 x 1/1000) Expanded Notation Here’s another example: 28.763 28.763 = (2 x 10) + (8 x 1) + (7 x 0.1) + (6 x 0.01) + (3 x 0.001) -OR- (2 x 10) + (8 x 1) + (7 x 1/10) + (6 x 1/100) + (3 x 1/1000)

Expanded Notation Let’s look at one more: 502.096 502.096 = -OR- (5 x 100) + (2 x 1) + (9 x 0.01) + (6 x 0.001) -OR- (5 x 100) + (2 x 1) + (9 x 1/100) + (6 x 1/1000)

Expanded Notation: Your Turn Write each number in EXPANDED NOTATION using DECIMALS and FRACTIONS. Example: 4.23 = (4 x 1) + (2 x 0.1) + (3 x 0.01) (4 x 1) + (2 x 1/10) + (3 x 1/100) 9.2 17.45 8.947 603.020 29.307

Expanded Notation: Your Turn ANSWERS: 1) 9.2 = (9 x 1) + (2 x 0.2) (9 x 1) + (2 x 1/10) 2) 17.45 = (1 x 10) + (7 x 1) + (4 x 0.1) + (5 x 0.01) (1 x 10) + (7 x 1) + (4 x 1/10) + (5 x 1/00) 3) 8.947 = (8 x 1) + (9 x 0.1) + (4 x 0.01) + (7 x 0.001) (8 x 1) + (9 x 1/10) + (4 x 1/100) + (7 x 1/1000) 4) 603.020 = (6 x 100) + (3 x 1) + (2 x 0.01) (6 x 100) + (3 x 1) + (2 x 1/100) 5) 29.307 = (2 x 10) + (9 x 1) + (3 x 0.1) + (7 x 0.001) (2 x 10) + (9 x 1) + (3 x 1/10) + (7 x 1/1000)

Scientific Notation SCIENTIFIC NOTATION expands numbers just like expanded notation, but we use base-10 exponents to identify each digit’s place: 268 = (2 x 102) + (6 x 101) + (8 x 100)

Scientific Notation In SCIENTIFIC NOTATION with decimals, we will learn two new concepts: 1) NEGATIVE exponents and 2) FRACTIONAL exponents: 2.68 = (2 x 100) + (6 x 10-1) + (8 x 10-2) -OR- (2 x 100) + (6 x 1/101) + (8 x 1/1002)

Scientific Notation Holy smokes! Let’s look at another: 31.57 = (3 x 101) + (1 x 100) + (5 x 10-1) + (7 x 10-2) -OR- (3 x 101) + (1 x 100) + (5 x 1/101) + (7 x 1/102)

Scientific Notation Here’s one more: 9.547 = (9 x 100) + (5 x 10-1) + (4 x 10-2) + (7 x 10-3) -OR- (9 x 100) + (5 x 1/101) + (4 x 1/102) + (7 x 1/103)

Scientific Notation: Your Turn Let’s practice. Use SCIENTIFIC NOTATION to expand each number using base-10 exponents and fractional exponents. EXAMPLE: 7.58 = (7 x 100) + (5 x 10-1) + (8 x 10-2) (7 x 100) + (5 x 1/101) + (8 x 1/102) 4.23 9.675 32.04 2.605

Scientific Notation: Your Turn ANSWERS: 1) 4.23 = (4 x 100) + (2 x 10-1) + (3 x 10-2) (4 x 100) + (2 x 1/101) + (3 x 102) 2) 9.675 = (9 x 100) + (6 x 10-1) + (7 x 10-2) + (5 x 10-3) (9 x 100) + (6 x 1/101) + (7 x 1/102) + (5 x 1/103) 3) 32.04 = (3 x 101) + (2 x 100) + (4 x 10-2) (3 x 101) + (2 x 100) + (4 x 1/102) 4) 2.605 = (2 x 100) + (6 x 10-1) + (5 x 10-3) (2 x 100) + (6 x 1/101) + (5 x 1/103)

Whew! You did it! COMPARING decimals! Coming up next… COMPARING decimals!

Comparing Decimals Like we did with whole numbers, we express inequalities (numbers that are NOT equal) with these symbols: < (less than) > (greater than) ≠ (not equal to) But when we move into decimals and fractions, this symbol is important, too: = (equal to).

Equivalent Whole Numbers With whole numbers, there is not any confusion about numbers that are equal: 8 = 8 37 = 37 246 = 246 4,693 = 4,693 20,762 = 20,762

Equivalent Decimals With decimals, it isn’t quite so simple because: .8 = 0.8 3.7 = 3.70 2.46 = 2.4600 4 tenths = 40 hundredths and 4 tenths = 400 thousandths

Equivalent Decimals Just remember: any number of 0s to the right of the last non-0 number in a decimal is equivalent to the same decimal without the ending 0s: 0.72 = 0.720 = 0.7200 = 0.72000 1.6 = 1.60 = 1.600 = 1.6000 84.34 = 84.340 = 84.3400 = 84.34000

Equivalent Decimals To dig a little deeper, let’s watch a TurtleDiary video on equivalent decimals…

Equivalent Decimals Click HERE to begin! Let’s practice a few on IXL. When I call your team number, line up at the SmartBoard in the same order as your seat number (1-4). When it’s your turn, choose the correct answer, click the submit button, and return to your seat. Click HERE to begin!

Comparing Decimals When we compare decimals, we use the same process we used to compare whole numbers: we start with the greatest place value and compare numbers in the same places moving to the right. When comparing decimals, however, it’s easier to do when the numbers we compare have the same number of digits before and after the decimal.

Comparing Decimals For example, if we want to compare12.2 and 1.8536, it will be easier if we stack them, line up the decimals, and even up the places by adding 0s to empty spaces. At first glance, it might seem that 1.8536 is greater because it has more digits, but once we even up the number of digits, we can see more clearly that 12.2 is greater: 12.2000 01.8536

Comparing Decimals Let’s watch a few videos: Video #1 Video #2

Comparing Decimals Click HERE to begin! Let’s practice a few on IXL. When I call your team number, line up at the SmartBoard in the same order as your seat number (1-4). When it’s your turn, choose the correct answer, click the submit button, and return to your seat. Click HERE to begin!

Pop Quiz! Write a decimal equivalent to: 1.2 24.01 3) 965.207 Compare using <, = , or >. 4) 6.57 ___ 6.6 5) 2.034 ___ 2.0034 6) 65.0400 ___ 65.04 7) 821.546 ___ 821.52 8) 0.3000 ___ .3

Pop Quiz ANSWERS! Write a decimal equivalent to: 1.2 = 1.20 or 1.200 or 1.2000… 24.01 = 24.010 or 24.0100… 3) 965.207 = 965.2070 or 965.20700… Compare using <, = , or >. 4) 6.57 < 6.6 5) 2.034 > 2.0034 6) 65.0400 = 65.04 7) 821.516 < 821.52 8) 0.3000 = .3

Great Job! ORDERING and ROUNDING decimals! Coming up next… ORDERING and ROUNDING decimals!

Ordering Decimals When we order decimals, just like with comparing them, it can be easier to do when we line up the decimals and even up the places by adding 0s. But let’s watch Sal on KhanAcademy reason through ordering decimals WITHOUT doing so.

Ordering Decimals Let’s practice. Order these decimals greatest to least: 1.82 1.804 1.8042 1.8 1.825 Which number was greatest? Which number was least? Did you put them in this order? 1.825 1.82 1.8042 1.804 1.8

Rounding Decimals Let’s watch a MathAntics video to review rounding whole numbers and introduce us to rounding decimals. And here’s one of Sal breaking down rounding decimals for us.

Rounding Decimals: Your Turn Try rounding these numbers to the place in parentheses ( ). 1) 34.693 (tenths) Did you get 34.7? 2) 683.873 (hundredths) Did you get 683.87? 3) 5.960 (tenths) Did you get 6? 4) 9.999 (thousandths) Did you get 10?

Pop Quiz Let’s practice comparing, ordering, and rounding! Compare: 9.4 ___ 9.398 68.07 ___ 68.0700 Order greatest to least: 7.5 7.501 7.49 7 7.439 Round to the given place: 4) 85.684 (hundredths) 5) 709.951 (tenths)

Pop Quiz: ANSWERS Let’s practice comparing, ordering, and rounding! Compare: 9.4 > 9.398 68.07 = 68.0700 Order greatest to least: 3) 7.501 7.5 7.439 7.43 7 Round to the given place: 4) 85.684 (hundredths) = 85.68 5) 709.951 (tenths) = 710