Decimals

Vocabulary To Know Place Value Tenths Period Hundredths Powers of Ten Thousandths Decimal Ten Thousandths Decimal Point Millionths Fraction Ten Thousandths Percent

Representing Decimals Standard Form Written Form Expanded Form Pictures Fractions Percent

Decimals In our place value system, the value of a digit depends on its place, or position, in the number. Each place has a value of 10 times the place to its right. A number in standard form is separated into groups of three digits using spaces (USA uses commas). Each of these groups is called a period. Not every number is a whole number. Our decimal system lets us write numbers of all types and sizes, using a symbol called the decimal point. Money,

Place Value Chart Numbers Get Numbers Get Bigger Smaller Millions Thousands Units/Ones ● Billions Hundred Millions Ten Millions Hundred Thousands Ten Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths Millionths Numbers Get Numbers Get Bigger Smaller

Decimal Place Value • Ones Decimal • Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths Millionths 1 • 0.1 0.01 0.001 0.0001 0.00001 0.000001 10 100 1 000 10 000 100 000 1 000 000 102 103 104 105 106

125 . 578 As you move right from the decimal point, each place value is divided by 10. 125 . 578 hundreds tens ones tenths hundredths thousandths

Reading Decimal Numbers Example: Read 38.7425 Solution: Step 1: Values to the left of the decimal point are greater than one. 38 means 3 tens and 8 ones. Step 2: The word name of the decimal is determined by the place value of the digit in the last place on the right. The last digit (5) is in the ten-thousandth place. 38.7425 is read as thirty-eight and seven thousand four hundred twenty-five ten thousandths Reading Decimal Numbers

decimal number as above How to Read and Write Decimal Numbers Millions Thousands Hundreds • Hundred Million Ten Million Hundred Thousand Ten Thousand Thousand Ten One Tenths Hundredths Thousandths 4 5 1 2 Read the entire number first, then add the last right digit’s place value to the end E.g “four hundred fifty one” thousandths 2. When there is a whole number too, read the whole number first then add “AND” the decimal number as above e.g. “ one hundred fifty AND twelve hundredths *** Notice Decimal Number Names end in “–ths” ***

The value of a digit is determined by its place value. When the decimal point of a number is not shown (for example, in whole numbers), then it is assumed to be at the end of the number on the right hand side Example : 321 = 321. 4 = 4. Number Place Value (of underlined digit) Value of the digit (as a decimal) Value of the digit (as a fraction) 3 .145 Ones 3 3. 145 Tenths 0.1 1 10 3.1 45 Hundredths 0.04 4 100 3.14 5 Thousandths 0.005 5 1000

Reading Place Values of Decimals A decimal number is a number that has digits before and after a decimal point. The decimal point is placed after the ones digit. Example : 3.145 Each digit in a decimal number has a place value depending on its position. Tens Ones Decimal point Tenths Hundredths Thousandths 3 . 1 4 5

Written Form of Decimals Read the whole number before the decimal without the use of “and.” Write that number down. At the decimal point use the word “and” Read the entire number after the decimal as if it were a whole number. Add the name of the place value of the last digit on the right hand side (after the decimal of course)

Written Form of Decimals Example: Write the following decimal in words 8 243.67 Eight thousand, two hundred forty-three AND sixty-seven hundredths

Standard Form of Decimals Example: Write the following decimal number in standard form: two hundred six and fifty-four ten-thousandths . 206 __ __ __ __ 0 0 5 4 The word “ten-thousandths” indicates that we need four decimal places. When we clean it up, the answer is 206.0054

A Few Examples of Reading & Writing Decimal Numbers Fifty eight hundredths 0.58 0.854 Eight hundred fifty four thousandths 12.5 Twelve and five tenths 1.777 One and seven hundred seventy seven thousandths 0.0005 Five ten thousandths One hundred and ten hundredths seven thousand 100.10 0.351 Three hundred fifty one thousandths Twenty three and six tenths 23.6

A Few Examples of Reading & Writing Decimal Numbers Five hundredths 0.05 Twenty three thousandths 0.023 Thirty nine and six tenths 39.6 30.0001 Thirty and one ten thousandths Seven tenths 0.7 nine thousand and one hundredth 9 000.01 80.541 Eighty and five hundred four one thousandths Twenty one ten thousandths 0.0021

Expanded Form of Decimals Similar to writing whole numbers in expanded form. Write the number that appears before the decimal point in expanded form. For decimals, place a zero in the ones place. Also, substitute zeroes for all spaces after the decimal point that come before the digit that you are working with.

For Example: Write the following decimal 13.361 in expanded form. Expanded Form of Decimals For Example: Write the following decimal 13.361 in expanded form. There are two ways to write 13.361 in expanded form. 1. Use the decimals 13.361 = 10 + 3 + 0.3 + 0.06 + 0.001 2. Use the fractions 13.361 = 10 + 3 + 3 + 6 + 1 10 100 1000

Decimals In Pictures Decimal numbers can also be represented by pictures using the base 10 blocks Unit = 1 How many small boxes make up the whole grid? (100) Read more on TeacherVision: http://www.teachervision.fen.com/decimals/activity/3153.html#ixzz23SEVYwiL Rod = 10 Cube = 1000 Flat = 100

Decimals In Pictures There are 100 units in this flat 50 are grey Using the following pictures write a fraction and decimal for the grey area. There are 100 units in this flat 50 are grey So the fraction is 50 100 and the decimal is 0.5

Decimals In Pictures There are 100 units in this flat 50 are coloured Using the following pictures write a fraction and decimal for the coloured area. There are 100 units in this flat 50 are coloured So the fraction is 75 100 and the decimal is 0.75

Decimals In Pictures There are 100 units in this flat 6 are coloured Using the following pictures write a fraction and decimal for the coloured area. There are 100 units in this flat 6 are coloured So the fraction is 6 100 and the decimal is 0.06

Equivalent Decimals Basic Rules: Add as many zeros LEFT of digits that are BEFORE a decimal. Add as many zeros as you want RIGHT of the digits AFTER the decimal. Equivalent Decimals Examples: 0.5 = 0.50 = 0.500 = 0.500000 2.4 = 02.40 = 2.400 = 0002.4000. 034 is the same as 34. The number still has no hundredths, 3 tens and 4 ones. 1.5 is the same as 1.50. The number still has 1 one, 5 tenths, and no hundredths.    000032.456000 = 032.456 = 32.456 = 32.4560000 These all have 3 tens,, 2 ones, 4 tenths, 5 hundredths and 6 thousandths.

Comparing Numbers < (less than) > (greater than) = (equals) ≤ (less than or equal to) ≥ (greater than or equal to) Basic Steps Compare the whole number parts first. Start and move LEFT TO RIGHT Compare the next most significant digit of each number in the same place. If they are equal, move onto the next place to the right. ** Think of it as PAC –MAN, his mouth is ALWAYS open towards the LARGER number. Examples: a). Compare 1123 and 1126 1123 < 1126 b). Compare 567 and 497 567 > 497 c). Compare 1 and 0.002 1 > 0.002 d). Compare 0.402 and 0.412, 0.402 < 0.412 e). Compare 120.65 and 34.999 120.65 > 34.999 f). Compare 12.345 and 12.097. 12.345 > 12.09.

Rounding Numbers BASIC RULES If the number to the RIGHT of the Rounding Number is 5 or more (5, 6, 7 8 or 9) then the LEFT (Rounding Number) number goes UP by ONE place. 2. If the number to the RIGHT of the Rounding Number is 4 or less (4, 3, 2, 1 or 0) then the LEFT (Rounding Number) number remains the SAME. 3. Every number AFTER the Rounding Number becomes 0. 4. Everything BEFORE the Rounding Number remains the same. 5. If the Rounding Number is a 9 and it goes up by one, then a 0 is placed in the Rounding Number Place and 1 is moved to the next place to the LEFT. (same as carrying in adding)   Examples Because ... 3.1416 rounded to hundredths is 3.14 ... the next digit (1) is less than 5 1.2635 rounded to tenths is 1.3 ... the next digit (6) is 5 or more 1.2635 rounded to 3 decimal places is 1.264 ... the next digit (5) is 5 or more 134.9 rounded to tens is 130 ... the next digit (4) is less than 5 12,690 rounded to thousands is 13,000 1.239 rounded to units is 1 ... the next digit (2) is less than 5

Adding Decimals Basic Steps: First, line up the decimal points When one number has more decimal places than another, use 0's as place holders to give them the same number of decimal places. Add. Example #1: 76.69 + 51.37 1) Line up the decimal points: 76.69 +51.37 2). Then add. 76.69 +51.37 128.06 Example #2 : 12.924 + 3.6 Line up the decimal points: 12.924 +  3.600 2) Then add. 12.924 +  3.600 16.524

Subtracting Decimals Basic Steps: 1). Line up the decimal points on all the numbers 2). When one number has more decimal places than another, use 0's as place holders to give them the same number of decimal places 3). Subtract. Example: 18.2 - 6.008 1) Line up the decimal points. 18.2    -  6.008 2) Add extra 0's, using the fact that 18.2 = 18.200 18.200 -  6.008 3) Subtract. 18.200 - 6.008 12.192

Multiplying Decimal Numbers Basic Steps. Line up the NUMBERS, not the decimals. Multiply the same way as you would with whole numbers. After multiplying, add the numbers of digits to the RIGHT of the decimal point in both factors. This is how many places the decimal will move to the LEFT of the last right digit. Example 1: Example 2: 1. Multiply 4.032 × 4 1. Multiply 6.74 × 9.063 2. Line up Numbers 4.032 2. Line up Numbers 6.74 x 4 x 9.063 3. Multiply 4.032 3. Multiply 6.74 x 4 x 9.063 16128 2022 4044 4. Count the Number of Decimal Places 0000 in both Numbers. +6066 . The decimal moves 3 digits from the right: 6108462 4.032 x 4 4. Count the Number of Decimal Places 16.128 in both Numbers. The decimal moves 5 digits from the right: 6.74 x 9.063 61.88462

Dividing A Decimal by A Whole Number Basic Steps. When dividing the decimal goes STRAIGHT UP from where it is in the dividend. Divide the same way as whole numbers. Example 1: Example 2: Divide 15.567 ÷ 3 1. Divide 241.8 ÷ 22 ______ 2. ______ 3 | 15.567 22 | 241.8 3. Place the decimal straight up 3. Place the decimal straight up from the dividend from the dividend 4. Solve 4. Solve 5.189 10.99 3 | 15.567 22 | 241.800 -15 -22 05 21 - 3 - 0 26 218 -24 -198 27 200 -27 -198 0R 2R

Dividing A Decimal by A Decimal Basic Steps. First, it is easier to make the Divisor a whole number. Move the decimal in the Divisor as many places to the RIGHT as to create a whole Number. Move the decimal in the Dividend the same number of places to the RIGHT. Next, the decimal goes STRAIGHT UP from where it is in the dividend. Divide the same way as whole numbers. Example 1: Example 2: Divide 24.808 ÷ .3 1. Divide 250.85 ÷ 0.25 ______ 2. ______ 0.3 | 24.808 0.25 | 250.85 Move the Decimal in the Divisor 3. Move the Decimal in the Divisor and the Dividend 1 places to and the Dividend 2 places to the RIGHT to make the Divisor the RIGHT to make the Divisor a Whole Number a Whole Number 4. The decimal goes straight up now 4. The decimal goes straight up now 5. Solve 5. Solve 82.69 1003.4 3 | 248.08 25 | 25085.0 -24 -25 08 00 - 6 - 0 20 08 -18 - 0 28 85 -27 -75 1R 100 -100 0R