DRAFT
Investigating Student Learning: 6th Grade
Chapter 1: Whole Numbers and Decimals
Standard NS 1.1 (Gr. 5): Manipulate very large (e.g. millions) and very small (e.g.,
thousandths) numbers.
Lesson 1.1: Whole Numbers and Decimals
Concepts:
A numeration system is a plan for naming numbers.
Our numeration system is called the Hindu-Arabic Numeration System.
The Hindu-Arabic Numeration System for whole numbers has the following attributes:
• There are 10 digits (0-9).
• Each group of three digits in a written number is called a period.
• Every period has a ones, tens, and hundreds place. Place value tells the place and period the digit is
in (e.g. In the number 45,238, the place value of the digit 4 is the ten thousands place).
• Commas separate one period from another.
• Position tells the value of a digit or how much the digit represents. (e.g. In the number 45,238, the
value of the digit 4 is forty thousand).
• A place value chart can help tell the value of each digit in a number.
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One Millions
Hundred Thousands
Ten Thousands
One Thousands
Hundreds
Tens
Ones
Tenths
Hundredths
One-thousandths
Ten-thousandths
Hundred-thousandths
One-millionths
4
9
1
6
8
2
7
3
5
4
7
0
8
6
9
The value of 7 is 7 hundred or 700
The value of 7 is seven hundredths or
0.07
The value of 0 is zero thousandths or
0.000
The value of 8 is eight ten-thousandths
or 0.0008
The value of 6 is six hundredthousandths or 0.00006
The value of 9 is nine millionths or
0.000009
Ten Millions
3
The value of 4 is four tenths or 0.4
Hundred Millions
5
1
DECIMALS
The value of 5 is five ones or 5
One Billions
ONES
The value of 3 is three tens or 30
Ten Billions
2
The value of 2 is 200 billion or
200,000,000,000
The value of 5 is 50 billion or
50,000,000,000
The value of 3 is 3 billion or
3,000,000,000
The value of 4 is 400 million or
400,000,000
The value of 9 is 90 million or
90,000,000
The value of 1 is 1 million or
1,000,000
The value of 6 is 600 thousand or
600,000
The value of 8 is 80 thousand or
80,000
The value of 2 is 2 thousand or 2,000
MILLIONS
Hundred Billions
THOUSANDS
BILLIONS
DRAFT
•
•
•
Place value can be used to write numbers in different but equivalent forms (Standard Form,
Expanded Form, Word Form, and Short Word Form)
Zero is a place-holder.
Hindu-Arabic Numeration System is referred to as a Base-Ten System because groups of 10 are
used. Each position to the left is 10 times the one to its right.
The Hindu-Arabic Numeration System for decimal numbers are an extension of our base 10, whole number
place value system:
• To the right of the ones place are the decimal places or parts of a whole.
• The whole and parts are separated by a decimal point.
• All decimals have the suffix ths.
• Decimals are fractions with a special set of denominators (tenths, hundredths, thousands…etc.) and a
special written form.
• A whole number can also be expressed as a decimal.
e.g. 235 can also be written as 235.0 or 235.00, etc.
• As with whole numbers, each place has ten times the value of the place to its right.
• Place values to the right of the decimal point follow the same patterns as place value to the left
(whole numbers) with one major exception—there is not a corresponding decimal place value
for ones.
• The location of a digit in a number determines the value of the digit. (e.g. In the number 3.48, the 3
is in the ones place, and its value is 3. The 4 is in the tenths place, and its value is 0.4 or four
tenths. The 8 is in the hundredths place, and its value is 0.08 or eight hundredths.
• To read a decimal:
o Read the number to the right of the decimal point as you would a whole number.
o Read the place value of the last digit, unless it is a zero. (6.5824 = six and five thousand
eight hundred twenty-four ten-thousandths).
• When reading a number, say “and” only for the decimal point.
• In the decimal number 0.4, the zero is used as a place holder to remind you that the number is less
than 1.
Essential Question(s):
How do you read and write whole numbers to the hundred billions and decimal numbers to the hundredthousandths?
How do you determine the value of a digit in a given number?
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DRAFT
th
ISL Item Bank: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard NS 1.1 (Gr. 5): Manipulate very large (e.g. millions) and very small (e.g.,
thousandths) numbers.
Lesson 1.1: Whole Numbers and Decimals
How do you read and write whole numbers to the hundred billions and decimal numbers to the hundredthousandths?
Use the place value chart to write the number in standard form.
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Ten
Millions
One
Billions
Hundred
Millions
Ten
Billions
Hundred
Billions
forty-eight thousand, two hundred sixty-one
Standard form: __________________________________
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Ten
Millions
One
Billions
Hundred
Millions
Ten
Billions
Hundred
Billions
two hundred seven million, five thousand, one hundred twenty-nine
Standard form: __________________________________
Standard form: __________________________________
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Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Ten
Millions
One
Billions
Hundred
Millions
Ten
Billions
Hundred
Billions
thirty million, five hundred fifty thousand, sixty-eight
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Standard form: __________________________________
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Ten
Millions
One
Billions
Hundred
Millions
Ten
Billions
Hundred
Billions
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Ten
Millions
One
Billions
Hundred
Millions
Ten
Billions
Hundred
Billions
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Ten
Millions
One
Billions
Hundred
Millions
Ten
Billions
Hundred
Billions
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Ten
Millions
One
Billions
Hundred
Millions
Ten
Billions
Hundred
Billions
DRAFT
sixteen billion, twenty-five million, seven hundred fourteen thousand, two hundred twelve
Standard form: __________________________________
eighty billion, five hundred six thousand, twenty-four
Standard form: __________________________________
Thirty billion, seventeen million, four hundred ninety- nine
Standard form: __________________________________
five hundred billion, thirty-four
DRAFT
Find the missing digits.
one hundred thirty-six thousand, two hundred thirty-one = 136,__31
five hundred one thousand, eighty-six = 5__ __, 086
eight hundred thirty thousand, eleven = 8__ __, __ __ __
Six million, five hundred forty-three thousand, two hundred seventy-two = 6, 543, __ __ 2
twenty-nine million, seven hundred forty-three thousand, two hundred fifty = 29,__ __ __, 250
six hundred two million, three hundred sixteen thousand, five hundred sixty-two = __ __ __, 316, __ __2
five billion, six hundred twenty-one million, fifty-six thousand, four hundred nine
= 5, 6 __ __, __ 56, 409
sixty billion, five hundred nine million, twenty-two thousand, five hundred eighty =
__ __,509,__ __ __,580
two hundred ninety-seven billion, six million, seven hundred thousand, one hundred four =
__ __ __, __ __ __,70__,104
four hundred billion, two million, eighteen thousand, one =
4__ __, __ __ __, __ 1 __, __ __ __
nine hundred twelve million, one thousand, seven = __ __ __, __ __ __, __ __ __
one hundred billion, five hundred thirty-four = ______,________,________,_______
three hundred one million, fifty-nine thousand = ______,______,______
eighty-eight billion, five thousand, one hundred four = ______________________________
nine hundred billion, six thousand = _____________________________________
twelve billion, ninety million, seventy-eight = _____________________________________
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DRAFT
Find the missing digits.
four hundred billion, two million, eighteen thousand, one =
4__ __, __ __ __, __ 1 __, __ __ __
nine hundred twelve million, one thousand, seven = __ __ __, __ __ __, __ __ __
one hundred billion, five hundred thirty-four = ______,________,________,_______
three hundred one million, fifty-nine thousand = ______,______,______
eighty-eight billion, five thousand, one hundred four = ______________________________
nine hundred billion, six thousand = _____________________________________
twelve billion, ninety million, seventy-eight = _____________________________________
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DRAFT
Write the number of shaded parts as a decimal in the place value chart.
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The number of shaded parts in
The number of shaded parts in
standard form is ____________
standard form is ____________
The number of shaded parts in
The number of shaded parts in
standard form is ____________
standard form is ____________
The number of shaded parts in
The number of shaded parts in
standard form is ____________
standard form is ____________
The number of shaded parts in
The number of shaded parts in
standard form is ____________
standard form is ____________
7
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Standard form: __________________________________
Hundredthousandths
One-millionths
ONES
Ten-thousandths
Hundredthousandths
One-millionths
Ten-thousandths
ONES
One-thousandths
Hundredths
Tenths
Ones
Hundredthousandths
One-millionths
Ten-thousandths
One-thousandths
Hundredths
Tenths
Ones
Tens
Hundreds
One Thousands
Ten Thousands
ONES
One-thousandths
Hundredths
Tenths
THOUSANDS
Ones
THOUSANDS
Tens
Hundreds
One Thousands
Ten Thousands
Hundred Thousands
One Millions
Ten Millions
THOUSANDS
Tens
Hundreds
One Thousands
MILLIONS
Ten Thousands
MILLIONS
Hundred Thousands
One Millions
Ten Millions
Hundred Millions
One Billions
Ten Billions
Hundred Billions
MILLIONS
Hundred Thousands
One Millions
BILLIONS
Ten Millions
BILLIONS
Hundred Millions
One Billions
Ten Billions
Hundred Billions
BILLIONS
Hundred Millions
One Billions
Ten Billions
Hundred Billions
DRAFT
Use the place value chart to write the number in standard form.
one hundred seven thousand, thirty-four and eight tenths
DECIMALS
Standard form: __________________________________
six hundred fifty million, nine and forty-three hundredths
DECIMALS
Standard form: __________________________________
twenty billion, sixteen thousand, one hundred twenty-two and one hundred seven thousandths
DECIMALS
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Standard form: __________________________________
Hundredthousandths
One-millionths
ONES
Ten-thousandths
Hundredthousandths
One-millionths
Ten-thousandths
ONES
One-thousandths
Hundredths
Tenths
Ones
Hundredthousandths
One-millionths
Ten-thousandths
One-thousandths
Hundredths
Tenths
Ones
Tens
Hundreds
One Thousands
Ten Thousands
ONES
One-thousandths
Hundredths
Tenths
THOUSANDS
Ones
THOUSANDS
Tens
Hundreds
One Thousands
Ten Thousands
Hundred Thousands
One Millions
Ten Millions
THOUSANDS
Tens
Hundreds
One Thousands
MILLIONS
Ten Thousands
MILLIONS
Hundred Thousands
One Millions
Ten Millions
Hundred Millions
One Billions
Ten Billions
Hundred Billions
MILLIONS
Hundred Thousands
One Millions
BILLIONS
Ten Millions
BILLIONS
Hundred Millions
One Billions
Ten Billions
Hundred Billions
BILLIONS
Hundred Millions
One Billions
Ten Billions
Hundred Billions
DRAFT
Use the place value chart to write the number in standard form.
ninety-one and four thousand two hundred eighteen ten-thousandths
DECIMALS
Standard form: __________________________________
four million and five hundredths
DECIMALS
Standard form: __________________________________
Thirty-one billion, sixty and nine thousandths
DECIMALS
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Standard form: __________________________________
Hundredthousandths
One-millionths
ONES
Ten-thousandths
Hundredthousandths
One-millionths
Ten-thousandths
ONES
One-thousandths
Hundredths
Tenths
Ones
Hundredthousandths
One-millionths
Ten-thousandths
One-thousandths
Hundredths
Tenths
Ones
Tens
Hundreds
One Thousands
Ten Thousands
ONES
One-thousandths
Hundredths
Tenths
THOUSANDS
Ones
THOUSANDS
Tens
Hundreds
One Thousands
Ten Thousands
Hundred Thousands
One Millions
Ten Millions
THOUSANDS
Tens
Hundreds
One Thousands
MILLIONS
Ten Thousands
MILLIONS
Hundred Thousands
One Millions
Ten Millions
Hundred Millions
One Billions
Ten Billions
Hundred Billions
MILLIONS
Hundred Thousands
One Millions
BILLIONS
Ten Millions
BILLIONS
Hundred Millions
One Billions
Ten Billions
Hundred Billions
BILLIONS
Hundred Millions
One Billions
Ten Billions
Hundred Billions
DRAFT
Use the place value chart to write the number in standard form.
Ten million, two thousand and fifty-six ten-thousandths
DECIMALS
Standard form: __________________________________
twenty-six billion, one million, nine and two hundred seven hundred-thousandths
DECIMALS
Standard form: __________________________________
two hundred billion, seventy-six thousand, four hundred and five hundred six hundred-thousandths
DECIMALS
DRAFT
Write the number in standard form. Find the missing digits.
four and eighty-five hundredths = 4 . __ __
twenty and five hundredths = 20.__ __
sixty-eight and nine thousandths = 6 __ . 0 __ __
fifty-two and sixty-one ten-thousandths = 52. __ __ __ __
six hundred nineteen and seven ten-thousandths = __ 19. __ __ __ __
thirty-six and fifty-four hundred-thousandths = __ __. __ __ __ __ __
eighty-six thousand two hundred five and five hundred sixty-six hundred-thousandths
= __ __, 205. __ __ __ __ __
six hundred twenty-three thousand and 5 tenths = 6 __3,000. __
fifty-nine million, seventy five and thirty-seven hundredths = 59,000, __ __5.__ __
twenty-seven billion, nine thousand, two hundred ten and fourteen thousandths
= 27,__ __ __, 009, 210. __ __ __
sixty-one million, eight thousand, four and five ten-thousandths = 61, __ __ __, 004 . __ __ __ __
three hundred billion, eight and forty-two hundred-thousandths
= 3__ __, __ __ __, 000, __ __ __ . __ __ __ __ __
nine hundred thousand and three hundred sixty-nine ten-thousandths = __ __ __ , __ __ __ . __ __ __ __
twenty-two million, nineteen and seventy-one hundred-thousandths
= 2__ , __ __ __ , 01 __ . __ __ __ __ __
five hundred billion, one million, nine hundred five thousand, eighty and six hundred forty-one tenthousandths = 500, __ __ __ , 905, __ __ __ . __ __ __ __
ninety-one million, twelve thousand and eleven hundred-thousandths
= 91, __ __ __ , __ __ __ . __ __ __ __ __
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DRAFT
Write the number in standard form. Find the missing digits.
seven hundred thirty-six and twelve ten-thousandths = _______ . ________
two hundred fifty thousand eighteen and thirty-five hundred-thousandths
= ______,________ . ________
sixteen million, four hundred thousand, three hundred forty-one and one hundred seven ten-thousandths
= ______,______,______ . ________
nine billion, fourteen thousand, sixty-nine and three ten-thousandths
= ______,________,________,_______ . __________
ten billion, seven hundred million, five and two hundred eight hundred-thousandths =
______,________,________,_______ . __________
forty-three million seven hundred eleven and twenty-two thousandths
= _________________________________________
six hundred nine billion, forty million, five thousand and two hundred six hundred-thousandths
= _________________________________________
four hundred billion eighty and seventy-seven ten-thousandths
= _________________________________________
three hundred billion, one hundred six million, forty-two thousand, nine hundred five and four thousandths
= _________________________________________
seventy million, eighty and fifteen hundred-thousandths
= _________________________________________
five hundred billion, three and two ten-thousandths
= _________________________________________
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DRAFT
Fill in the blanks to complete the word form for each number.
25,353
twenty five ____________________, three hundred fifty-three
708,215,531
seven hundred eight ____________________, two hundred __________________ thousand,
_____________________ thirty-one
900,301,256,604
___________________ billion, three _______________ one _________________,
two hundred fifty-six ____________________, _______ hundred four
89.023
____________________ and _____________________ thousandths
90,065. 0004
______________ thousand, sixty- __________ and four ______________________
68,038, 009.00078
_________________ ____________________, thirty-_________ thousand nine _________ seventyeight ____________________________
Write the following numbers in word form.
70,462.302
12,067,392.0008
6,093,700.0085
7,000,000,463.00009
20,970,000,012.00415
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DRAFT
Write the number in standard form. Find the missing digits.
236 million, 926 = 236,__ __ __
50 million, 244 = 5__, __ __ __, __ __ __
961 billion, 18 million, 362 = 9__ __, __ __ __, 362
44 million, five and 34 thousandths = 44, __ __ __, __ __ __. __ __ __
167 billion, 80 million, 14 thousand and eighteen ten-thousandths
= 167,__ __ __, 014, __ __ __ . __ __ __ __
506 billion, 2 million and 245 hundred-thousandths
= __ __ __, __ __ __, __ __ __, __ __ __ . __ __ __ __ __
21 billion, 9 thousand, 123 and 24 hundred-thousandths = ______, _______, _______, _______.________
16 million, 10 thousand, 429 and 16 thousandths = _______, _______, _______.________
415 billion, 7 million, eighteen and 56 ten-thousandths = ______, _______, _______, _______.________
8 billion, 924 and 4,126 hundred-thousandths______, _______, _______, _______.________
600 billion, 200 thousand, 300 and 504 ten-thousandths =______, _______, _______, _______.________
21 billion, 3 million, 5 thousand, and 11 hundred-thousandths = _________________________________
867 million, 40 thousand 8 and 1 ten-thousandth = _________________________________________
636 billion, 573 and 219 ten-thousandth = _________________________________________
504 billion, 22 million and 7,226 ten-thousandth = _________________________________________
33 billion, 19 million, 657 thousand 400 and 254 hundred-thousandth = __________________________
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DRAFT
Fill in the blanks to complete the short word form for each number.
7,028.67
_____ thousand, 028 and ____ hundredths
135,632.025
_____ thousand, 632 and 25 _______________
54,013,395.0718
54 __________________, _____ thousand, ______ and 178 _____________________
4,000,862,005.00009
____ billion, _____ million, _____ thousand, 5 and 9 _____________________
687,004,000,502.07,216
687________________, _____ million, 502 _____ _______ hundred-thousandths
901,000,000,534.00244
_____ __________________, _____ and _____ ___________________________
Write the following numbers in short word form.
9,213,861,111.78
42,805,000.256
89,000,705,003.0005
900,000,406.07,982
67,000,000,004.00603
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DRAFT
Fill in the blanks to complete the expanded form for each number.
53,164 = 50,000 + 3,000 + 100 + 60 + _______
57,695,543 = 50,000,000 + 7,000,000 + 600,000 + ____________ + 5,000 + _______+ 40 + 3
93,041,724,367 = __________________ + 3,000,000,000 + 40,000,000 + ________________
+ 700,000 + 20,000 + _______+ 300 + 60 + 7
385.24 = 300 + ________ + _________ + 0.2 + 0.04
67,204.056 = ______________ + _______________ + 200 + 4 + _____________ + 0.006
84,000,046.0102 = 80,000,000 + _______________ + ___________ + __________ +
0.01 + ________
906,000,004,030.01893 = 900,000,000,000 + ________________ + _________________ +
_________ + ________+ 0.008 + ________ + ________
Write the number 45,031.059 in expanded form.
Write the number 100,920,000.4 in expanded form.
Write the number 9,754,001.02 in expanded form.
Write the number 11,070,936.56 in expanded form.
Write the number 403,000,010,042.0009 in expanded form.
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DRAFT
Write the following numbers in standard form.
50,000 + 3,000 + 200 + 6 + 0.8
300,000 + 2,000 + 900 + 5 + 0.9 + 0.004
90,000 + 4,000 + 300 + 60 + 4 + 0.3 + 0.05
60,000,000 + 70,000 + 1,000 + 800 + 9 + 0.06
400,000,000 + 20,000,000 + 70,000 + 5 + 0.2 + 0.007
9,000,000,000 + 60,000,000 + 2,000,000 + 50 + 2 + 0.009 + 0.00007
800,000,000,000 + 3,000,000,000 + 40,000 + 500 + 0.3 + 0.0008
Fill in the blanks to complete the expanded form for each number.
36 thousand, 456 = 30,000 + ___________+ 400 + ________ + 6
207 million, 8 thousand, 13 and 5 tenths =
________________ + 7,000,000 + ________________ + ________ + 3 + 0.5
52 million, 209 thousand, 7 and 9 hundredths =
50,000,000 + ________________ + 200,000 + ___________ + 7 + ______
61 billion, 15 million, 600 and 32 thousandths =
60,000,000,000 + _____________________ + ________________ +5,000,000 + 600 + ______ + 0.002
301 billion, 45 thousand, 8 and 67 ten-thousandths =
________________ + 1,000,000,000 + ________________ + 5,000 + ________ + 0.006 + _________
15 million, 90 thousand and 679 hundred-thousandths =
__________________ + 5,000,000 + _______________ + ___________ + __________ + 0.00009
200 billion, 66 and 2,478 hundred-thousandths =
__________________ + 60 + ________ + ________ + 0.004 + ________ + _________
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DRAFT
Write the expanded form for the number 63 thousand, 182 and 6 hundredths.
Write the expanded form for the number 502 million, 8 thousand and 56 thousandths.
Write the expanded form for the number 9 billion, 60 thousand, 21 and 405 ten-thousandths.
Write the expanded form for the number 18 billion, 490 million, 5 thousand and 14 hundredthousandths.
Write the expanded form for the number 508 billion, 6 and 7,001 hundred-thousandths.
Fill in the blanks to complete the short word form for each expanded number.
80,000 + 7,000 + 500 + 60 + 2 + 0.6 = 87 thousand, ________ and 6 tenths
7,000,000 + 80,000 + 4,000 + 20 + 8 + 0.6 + 0.005 =
____ million, 84 _____________, ______ and 605 ________________
400,000,000,000 + 20,000,000,000 + 60,000,000 + 8,000,000 + 700 + 4 + 0.09 + 0.0003 =
420 _____________, ______ million, ______ and _______ ten-thousandths
6,000,000,000 + 700,000 + 3,000 + 0.002 + 0.0008 =
_____ billion, ______ ________________ and ______ ten-thousandths
70,000,000,000 + 800,000 + 10 + 6 + 0.07 + 0.001 + 0.00003 =
70 _____________, ______ thousand ______ and __________ _____________-thousandths
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DRAFT
Fill in the blanks to complete the expanded form using powers of ten for each number.
53,764 = (5 x 10,000) + (3x 1,000) + (__ x _____) + (6 x 10) + (4 x 1)
57,695,843 =
(5 x 10,000,000) + (__ x ____________) + (6 x 100,000) + (___ x _________) + (5 x 1,000) +
(__ x _____)+ (4 x 10) + (3 x 1)
93,041,724,367 =
(___ x _______________) + (3 x 1,000,000,000) + (4 x 10,000,000) + (___ x _____________)
+ (7 x 100,000) + (___ x __________) + (4 x 1,000) + (__ x _____) + (6 x 10) + (7 x 1)
385.24 =
(3 x 100) + (___ x ______) + (__ x __) + (2 x 0.1) + (4 x ____)
67,204.056 =
(___ x ___________) + (___ x ___________) + (2 x 100) + (4 x 1) + (5 x _____) + (__ x 0.001)
84,000,046.0102 =
(8 x 10,000,000) + (___ x ____________) + (___ x 10) + (__ x ___) + (1 x 0.01) + (___ x _______)
906,000,004,030.02893 =
(9 x 100,000,000,000) + (___ x _______________) + (___ x ______________) +
(___ x ____) + (2 x ______)+ (__ x 0.001) + (9 x 0.0001) + (___ x _______)
Write the number 4,539.05 in expanded form using powers of ten.
Write the number 7,920,000.42 in expanded form using powers of ten.
Write the number 92,704,001.027 in expanded form using powers of ten.
Write the number 50,000,936.006 in expanded form using powers of ten.
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DRAFT
What number is equal to fifty-eight million, three thousand twelve and 62 thousandths?
A. 58,003,012.62
B. 58,300,012.062
C. 5,803,012.0062
D. 58,003,012.062
What number is equal to four hundred two billion, fifty-three thousand, fifteen and six ten-thousandths?
A. 402,000,53,015.0006 B. 402,053,015.0006 C. 402,000,053,015.0006 D. 400,002, 053,015.006
What number is equal to twelve billion, six hundred one million and twenty-four hundred-thousandths?
A. 12,601,000.00024
B. 12,601,000,000.00024
C. 12,601,000,000.0024 D. 12,600,000,001.00024
What number is equal to forty-six million, eight hundred thousand, two hundred and fifty-nine tenthousandths?
A. 46,800,200.0059
B. 46,000,800,200.0059
C. 46,800,200.059
D. 46,802,000.0059
What number is equal to 304 billion, 72 million, 6 thousand 401 and 16 thousandths
A. 304,072,006,401.16
B. 304,072,006,401.016
C. 304,72,6,401.16
D. 304,072,006,401.0016
What number is equal to 94 million, 613 thousand 13 and 702 ten-thousandths
A. 94,000,613,013.0702
B. 94,613,013.702
C. 94,613,013.0702
D. 94,613,013.00702
What number is equal to 700 billion, 812 million, 29 thousand and 2,654 hundred-thousandths
A. 700,812,029,000.2654
B. 700,812,029,000.02654
C. 700,812,029.02654
D. 700,812,29,000.02654
Which number in standard form is equal to
500,000,000 + 60,000,000 + 100,000 + 5,000 + 30 + 5 + 0.03?
A. 56,105,035.03
B. 560,105,035.03
C. 560,150,035.03
D. 560,105,35.03
Which number in standard form is equal to
40,000,000,000 + 900,000,000 + 50,000,000 + 4,000 + 700 + 6 + 0.002 + 0.0007?
A. 40,950,004,706.0027 B. 40,954,000,706.0027
C. 40,950,004,700.6027 D. 40,900,054,706.0027
What is 9,000,000,000 + 3,000,000 + 500,000 + 400 + 20 + 8 + 0.3 + 0.008 in standard form?
A. 9,003,500,428.38
B. 9,003,500,428.308
C. 93,500,428.308
D. 9,003,500,400,028.308
What is 500,000,000 + 20,000,000 + 5,000,000 + 10 + 0.07 + 0.00006 in standard form?
A. 500,025,000,010.07006
7/7/2008
6th 1-1
B. 525,000,010.70006
C. 525,010.07006
20
D. 525,000,010.07006
DRAFT
Which number is the expanded form for 62,560,090,501.023?
A.
B.
C.
D.
60,000,000,000 + 2,000,000 + 500,000,000 + 60,000,000 + 90,000 + 500 + 1 + 0.02 + 0.003
60,000,000,000 + 2,000,000,000 + 500,000,000 + 6,000,000 + 90,000 + 500 + 1 + 0.02 + 0.003
60,000,000,000 + 2,000,000,000 + 500,000,000 + 60,000,000 + 90,000 + 500 + 1 + 0.02 + 0.003
60,000,000,000 + 2,000,000,000 + 500,000,000 + 60,000,000 + 90,000 + 500 + 1 + 0.2 + 0.003
How many periods are in 4,784,309?
How many periods are in 713,607,000,304?
What digits are in the millions period in the number 285,001,391?
What digits are in the billions period in the number 787,095,745,824?
In the number 956,783,254,109 which three digits are in the one’s period?
How many digits do you need to write a million?
How many digits do you need to write a billion?
How are the one’s period and the million’s period alike and different in the number 867,425,867?
What does the comma after the 6 represent in the number 96,370,978,112.35?
What does the decimal represent in the number 798,345,902.056?
How many commas do you use to write one million?
How many commas do you use to write one billion?
What is a place value chart? How does it help you read numbers?
Write an eight digit number with a 4 in the ten millions place and a 8 in the hundreds place.
Write a nine digit number with a 7 in the one millions place and a 3 in the hundred thousands place.
Write a ten digit number with 2 in the one billions place and a 1 in the ten millions place.
7/7/2008
6th 1-1
21
DRAFT
Fill in the blanks.
Standard Form
Example:
528,637,212.038
946,032,549.0007
Word Form
Expanded Form
five hundred twenty-eight million, six
hundred thirty-seven thousand, two
hundred twelve and thirty-eight
thousandths
500,000,000 + 20,000,000 + 8,000,000
+ 600,000 + 30,000 + 7,000 + 200 + 10
+ 2 + 0.03 + 0.008
nine hundred forty-six million,
thirty-two thousand,
five hundred forty-nine and seven
ten-thousandths
three hundred sixty-one million,
nine hundred seventy-five and forty-six
ten-thousandths
809,000,496.0203
800,000,000 + 9,000,000 + 400 + 90 +
6 + 0.02 + 0.0003
six hundred million, seven hundred
thousand, ninety-four and 19 hundredthousandths
40,000,100,002.37
600,000,000 + 700,000 90 + 4 +
0.0001 + 0.00009
forty billion, one hundred thousand, two
and thirty seven hundredths
18, 005,000,213.0012
7/7/2008
6th 1-1
300,000,000 + 60,000,000 + 1,000,000
+ 900 + 70 + 5 + 0.004 + 0.0006
10,000,000,000 + 8,000,000,000 +
5,000,000 + 200 + 10 + 3 + 0.001 +
0.0002
22
DRAFT
How do you determine the value of a digit in a given number?
Use the place value chart to write the value of the digits. Fill in the blanks.
Ones
8
5
1
7
The value of 3 is 3 million or 3,000,000
The value of 8 is ___________________
The value of 2 is 200 thousand or 200,000
The value of 5 is 5 hundred or 500
The value of 6 is 60 thousand or 60,000
The value of 1 is ___________________
Hundredthousandths
One-millionths
Tens
6
Ten-thousandths
Hundreds
2
Hundredths
One Thousands
3
Tenths
Ten Thousands
DECIMALS
Hundred Thousands
ONES
One-thousandths
THOUSANDS
One Millions
Ten Millions
MILLIONS
Hundred Millions
One Billions
Ten Billions
Hundred Billions
BILLIONS
The value of 7 is 7 ones or 7
Ten Thousands
One Thousands
Hundreds
Tens
Ones
Tenths
0
5
2
7
9
4
1
3
4
7
Hundredthousandths
One-millionths
Hundred Thousands
6
Ten-thousandths
One Millions
8
Hundredths
Ten Millions
DECIMALS
Hundred Millions
ONES
One-thousandths
THOUSANDS
One Billions
MILLIONS
Ten Billions
Hundred Billions
BILLIONS
The value of 8 is 80 billion or 80,000,000,000 The value of 9 is _____________________
The value of 6 is 6 billion or 6,000,000,000
The value of 4 is 4 thousand or 4,000
The value of 5 is 50 million or 50,000,000
The value of 1 is ______________________
The value of 2 is ____________________
The value of 3 is 3 tens or 30
The value of 7 is 700 thousand or 700,000
The value of 4 is ______________________
The value of 7 is 7 tenths or 0.7
7/7/2008
6th 1-1
23
DRAFT
Use the place value chart to write the value of the digits. Fill in the blanks.
Ten Millions
One Millions
Hundred Thousands
Ten Thousands
One Thousands
Hundreds
Tens
Ones
Tenths
Hundredths
One-thousandths
5
0
3
7
9
1
8
2
4
0
0
5
0
6
7
Hundredthousandths
One-millionths
Hundred Millions
DECIMALS
One Billions
ONES
Ten-thousandths
THOUSANDS
Ten Billions
MILLIONS
Hundred Billions
BILLIONS
The value of 5 is 500 billion or 500,000,000,000 The value of 8 is ___________________
The value of 3 is _________________________ The value of 2 is 20 thousand or 20,000
The value of 7 is 700 million or 700,000,000
The value of 4 is ___________________
The value of 9 is ______________________
The value of 5 is ___________________
The value of 1 is ______________________
The value of 6 is 6 hundredths or 0.06
The value of 7 is ___________________
Hundred Millions
Ten Millions
One Millions
Hundred Thousands
Ten Thousands
One Thousands
Hundreds
Tens
Ones
Tenths
Hundredths
One-thousandths
Ten-thousandths
DECIMALS
One Billions
ONES
7
0
0
1
3
4
5
7
0
9
8
6
0
2
0
8
Hundredthousandths
One-millionths
THOUSANDS
Ten Billions
MILLIONS
Hundred Billions
BILLIONS
5
The value of 7 is _________________________ The value of 9 is ______________________
The value of 1 is 100 million or 100,000,000
The value of 8 is 8 tens or 80
The value of 3 is _________________________ The value of 6 is ______________________
The value of 4 is ______________________
The value of 2 is ______________________
The value of 5 is ______________________
The value of 8 is 8 ten-thousandths or 0.0008
The value of 7 is 70 thousand or 70,000
The value of 5 is _______________________
7/7/2008
6th 1-1
24
2
7/7/2008
6th 1-1
0
0
,
7
8
Tens
Ones
9, 6
5
8
8
,
4
0
3
8
1
,
3
5
0
7
1
,
25
.
.
.
6
2
4
7
4
7
3
6
2
7
0
6
Hundredthousandths
Tenthousandths
Hundredths
Hundredthousandths
Tenthousandths
Onethousandths
Tenths
8
Hundredthousandths
0
3
Hundredthousandths
0
.
Tenthousandths
1
Onethousandths
0
Onethousandths
4
,
Hundredths
9
Tenths
Ones
8
Hundredths
Ones
Tens
Tens
Ones
Hundredthousandths
Tenthousandths
Onethousandths
Hundredths
Tenths
Hundreds
3
Tenths
Tens
Hundreds
One
Thousands
0
Tenthousandths
Ones
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
2
,
Onethousandths
Tens
One
Thousands
Ten
Thousands
Hundred
Thousands
4
Hundredths
Hundreds
Ten
Thousands
Hundred
Thousands
Ten
Millions
One
Millions
Hundred
Millions
One Billions
Ten Billions
Hundred
Billions
6
Tenths
One
Thousands
4, 4
Ten
Thousands
7, 0
Hundreds
1
8, 7
One
Thousands
0
3, 2
Ten
Thousands
0
Hundred
Thousands
4
One
Millions
Ten
Millions
2
One
Millions
Ten
Millions
Hundred
Millions
One Billions
1
One
Millions
Hundred
Millions
One Billions
Ten Billions
Hundred
Billions
4
Hundred
Thousands
,
7
One
Millions
6
,
Ten
Millions
8
,
Ten
Millions
0
0
,
Hundred
Millions
6
One Billions
2
8
Hundred
Millions
9
5
One Billions
3
Ten Billions
Hundred
Billions
4
Ten Billions
Hundred
Billions
2
Ten Billions
Hundred
Billions
DRAFT
Use the place value chart to write the value of the underlined digit in standard form.
.
value of the underlined digit in standard form = _________________________________
value of the underlined digit in standard form = _________________________________
4
value of the underlined digit in standard form = _________________________________
2
value of the underlined digit in standard form = _________________________________
9
value of the underlined digit in standard form = _________________________________
2
7/7/2008
6th 1-1
0
0
,
7
8
Tens
Ones
9, 6
5
8
8
,
4
0
3
8
1
,
3
5
0
7
1
,
26
.
.
.
6
2
4
7
4
7
3
6
2
9
7
0
6
Hundredthousandths
9
Tenthousandths
Hundredths
Hundredthousandths
Tenthousandths
Onethousandths
Tenths
8
Hundredthousandths
0
3
Tens
Ones
Hundredths
Hundredthousandths
Tenthousandths
Onethousandths
Tenths
7
Hundredthousandths
0
.
2
Tenthousandths
1
9
Onethousandths
0
8
Onethousandths
4
,
Hundredths
9
Tenths
Ones
8
.
Hundredths
Ones
Tens
Hundreds
3
Tenths
Tens
Hundreds
One
Thousands
0
Tenthousandths
Ones
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
2
,
Onethousandths
Tens
One
Thousands
Ten
Thousands
Hundred
Thousands
4
Hundredths
Hundreds
Ten
Thousands
Hundred
Thousands
Ten
Millions
One
Millions
Hundred
Millions
One Billions
Ten Billions
Hundred
Billions
6
Tenths
One
Thousands
4, 4
Ten
Thousands
7, 0
Hundreds
1
8, 7
One
Thousands
0
3, 2
Ten
Thousands
0
Hundred
Thousands
4
One
Millions
Ten
Millions
2
One
Millions
Ten
Millions
Hundred
Millions
One Billions
1
One
Millions
Hundred
Millions
One Billions
Ten Billions
Hundred
Billions
4
Hundred
Thousands
,
7
One
Millions
6
,
Ten
Millions
8
,
Ten
Millions
0
0
,
Hundred
Millions
6
One Billions
2
8
Hundred
Millions
9
5
One Billions
3
Ten Billions
Hundred
Billions
4
Ten Billions
Hundred
Billions
2
Ten Billions
Hundred
Billions
DRAFT
Use the place value chart to write the value of the underlined digit in short word form.
1
value of the underlined digit in short word form = ______________________________
9
value of the underlined digit in short word form = ______________________________
4
value of the underlined digit in short word form = ______________________________
2
value of the underlined digit in short word form = ______________________________
9
value of the underlined digit in short word form = ______________________________
DRAFT
Write the value of the underlined digit in standard form.
824,731.98
296,426.037
4,345,790.005
72,035,668.203
859,140,932.78
536,891,343.00007
14,001,674,397.6804
235,859,400,416.07561
541,943,672,232.00506
506,256,341.20136
400,000,563,251.0098
378,026,138,446.36402
786,830,212,435.36598
238,463,156,999.23746
654,658,963,001.35946
Write the value of the underlined digit in short word form.
824,731.98
296,426.037
4,345,790.005
72,035,668.203
859,140,932.78
536,891,343.00007
14,001,674,397.6804
235,859,400,416.07561
541,943,672,232.00506
506,256,341.20136
400,000,563,251.0098
378,026,138,446.36402
786,830,212,435.36598
238,463,156,999.23746
654,658,963,001.35946
7/7/2008
6th 1-1
27
2
7/7/2008
6th 1-1
0
0
,
7
8
Tens
Ones
9, 6
5
8
8
,
4
0
3
8
1
,
3
5
0
7
1
,
28
.
.
.
6
2
4
7
4
7
3
6
2
0
7
0
6
The place value of the underlined digit = _________________________________
Hundredthousandths
1
Tenthousandths
Hundredths
Hundredthousandths
Tenthousandths
Onethousandths
Tenths
8
Hundredthousandths
0
3
Tens
Ones
Hundredths
Hundredthousandths
Tenthousandths
Onethousandths
Tenths
9
Hundredthousandths
0
.
8
Tenthousandths
1
7
Onethousandths
0
2
Onethousandths
4
,
Hundredths
9
Tenths
Ones
8
.
Hundredths
Ones
Tens
Hundreds
3
Tenths
Tens
Hundreds
One
Thousands
0
Tenthousandths
Ones
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
2
,
Onethousandths
Tens
One
Thousands
Ten
Thousands
Hundred
Thousands
4
Hundredths
Hundreds
Ten
Thousands
Hundred
Thousands
Ten
Millions
One
Millions
Hundred
Millions
One Billions
Ten Billions
Hundred
Billions
6
Tenths
One
Thousands
4, 4
Ten
Thousands
7, 0
Hundreds
1
8, 7
One
Thousands
0
3, 2
Ten
Thousands
0
Hundred
Thousands
4
One
Millions
Ten
Millions
2
One
Millions
Ten
Millions
Hundred
Millions
One Billions
1
One
Millions
Hundred
Millions
One Billions
Ten Billions
Hundred
Billions
4
Hundred
Thousands
,
7
One
Millions
6
,
Ten
Millions
8
,
Ten
Millions
0
0
,
Hundred
Millions
6
One Billions
2
8
Hundred
Millions
9
5
One Billions
3
Ten Billions
Hundred
Billions
4
Ten Billions
Hundred
Billions
2
Ten Billions
Hundred
Billions
DRAFT
Use the place value chart to write the place value of the underlined digit.
4
The place value of the underlined digit = _________________________________
3
The place value of the underlined digit = _________________________________
4
The place value of the underlined digit = _________________________________
2
The place value of the underlined digit = _________________________________
9
DRAFT
What is the place value of the underlined digit?
323,751.23
467,382.011
695,373.984
569,304.6301
390,210.98754
218,700.3214
2,387,904.214
4,876,264.03267
9,641,855.0008
4,037,789.39
5,209,371.025
2,340,857.10128
589,329,621,369
95,278,453.21
32,238,782.88
3,684,438,767.1147
56,215,196,001.23698
345,452,000,447.0326
231,345,765,857.36952
589,125,023,556.02145
458,122,024,115.00217
What is the value of the 3 in 43,856,710.24?
A. 3,000
B. 30,000
C. 300,000
D. 3,000,000
What is the value of the 2 in 527,600,148.789?
A. 200,000,000
B. 20,000,000 C. 2,000,000 D. 200,000
What is the value of the 8 in 575,301,244.0381?
A. 0.008
B. 0.08
C. thousandths D. thousands
What is the value of 4 in 8,402,679,677.2589?
A. Four hundred thousand
B. Four hundred billion
7/7/2008
6th 1-1
C. Four hundred million
D. Four hundred two million
29
DRAFT
What is the value of the 6 in 592,875,301,244.0346?
A. 6 thousandths
B. 6 ten-thousandths
C. thousandths
D. ten-thousandths
What is the value of 5 in 605,208,402,679?
A. 5 hundred thousand B. 5 million C. 5 Billion
D. 50 Billion
What is the value of the 0 in 7,620,345.298?
What is the place of the 9 in 296,455,230,007.3589?
What is the value of the 7 in 589,665,457,235.20046?
What is the place of the 3 in 456,863,114.2036?
What is the value of the 6 in 560,123,242.3584?
What is the place of the 7 in 634,123,622,569.0307?
What is the value of the 5 in 45,263.309?
What is the place of the 8 in 993,295,164,753.32618?
7/7/2008
6th 1-1
30
DRAFT
Mathematical Reasoning:
What is the largest number you can write using 9 digits?
What is the smallest number you can write using all 9 digits?
What is the largest number you can write using all the digits that are a multiple of 3?
What is the smallest number you can write using all the digits that are multiples of 2 and all the digits
that are multiples of 3?
What is the largest number you can write using only the odd digits?
Using only the even digits between the numbers 0 and 10, what is the smallest number you can write?
If the following pattern continues, what are the next two numbers?
700,000 7,000,000
70,000,000 ________
_______
Why does the digit 5 have a value of 50,000 in 351,204 and 500,000 in the number 531,204?
7/7/2008
6th 1-1
31
DRAFT
th
Investigating Student Learning: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard NS 1.1: Compare and order…decimals…
Lesson 1.2: Comparing and Ordering Decimals
Concepts:
Place value is used to help us compare and order numbers.
You compare digits to order numbers beginning on the left because those digits have the greatest place
value.
A number line can also be used to compare and order numbers.
On a number line, numbers to the right are greater than numbers to the left of it.
All numbers can be represented by a point of the number line.
The symbols > and < from the number line, can be used to compare numbers.
The symbol > is read as greater than.
The symbol < is read as less than.
Comparing decimals is similar to comparing whole numbers.
The symbols > and < from the number line, can be used to compare decimal numbers.
In our numeration (decimal) system, each place has ten-times the value of the place to its right. So when
comparing 2 numbers, the number with its front digit in the greatest place or place furthest to the left,
is the greatest number.
To compare decimal numbers, the digits with the greatest place value are compared first. If the greatest
place value is not the same in both numbers, such as 23.4 and 3.4, the number with a digit in the
greatest place value is the greater number.
e.g. Compare 23.4 and 3.4
23.4 is the greater number because it has 2 tens and 3.4 has 0 tens.
A systematic way to compare decimals is to
1) line up the decimal point first (in essence, each place is “lined up.”)
2.86
2.34
Line up the decimal points
2) Start with the greatest place value at the left and compare the digits.
2.86
2.34
The ones digits are the same
3) Continue comparing the next digits to the right until the digits are different. Then compare
the values of the different digits.
2.86
2.34
0.8 or 8 tenths is greater than 0.3 or 3 tenths. So, 2.86 is a greater number than 2.34.
7/7/2008
6th 1-2
32
DRAFT
Writing zeros to the right-hand end of a mixed decimal will not change the value of a number.
Hundred Thousandths
Ten Thousandths
One Thousandths
Hundredths
Tenths
Ones
Tens
0.3 = 0.30 = 0.300 = 0.3000 = 0.30000
3
3
0
3
0
0
3
0
0
0
3
0
0
0
0
Equivalent decimals may need to be written so the two decimal numbers can be compared more easily.
e.g. Compare
42.3 and 42.28
At first glance it appears 42.28 is larger than 42.3 because 28 > 3.
However, 42.3 = 42.30. Now that both numbers have the same number of places, compare:
42.3 and 42.28
42.30 > 42.28.
Essential Question(s):
How do you compare decimals numbers?
How do you put decimal numbers in order from greatest to least or least to greatest?
7/7/2008
6th 1-2
33
DRAFT
th
ISL Item Bank: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard NS 1.1: Compare and order…decimals…
Lesson 1.2: Comparing and Ordering Decimals
How do you compare decimals numbers?
Use the drawings to help you compare the decimal numbers.
Fill in the blanks. Write >, <, or = for each.
7/7/2008
6th 1-2
.
0.1
0.2
0.36
0.34
.
1.5
1.3
1.76
.
.
2.8
.
.
.
34
.
.
1.86
2.9
.
DRAFT
Use the drawings to help you compare the decimal numbers.
Fill in the blanks. Write >, <, or = for each.
0.25
.
0.2
.
.
.
.
.
Use the number line to help you compare the decimal numbers.
Fill in the blanks. Write >, <, or = for each.
3.0
3.6
3.4
3.4
7.0
7.5
.
.
8.0
7.8
9.0
.
10.0
.
.
.
26.0
26.05
7/7/2008
6th 1-2
4.0
3.6
26.03
26.05
26.03
35
26.1
DRAFT
Use the number line to help you compare the decimal numbers.
Fill in the blanks. Write >, <, or = for each.
8.4
8.47
8.47
8.5
8.48
52.3
52.33
.
52.4
.
532
14.8
.
7.1
.
.
7.2
.
.
8.43
8.434
8.432
8.44
8.434
8.432
37.6
37.652
37.635
37.7
37.652
37.635
3.82
3.828
14.9
.
.
.
7/7/2008
6th 1-2
8.48
.
3.826
36
.
3.83
DRAFT
Use the number line to help you compare the decimal numbers.
Write >, <, or = for each.
346.3
346.338
346.4
.
.
346.352
35.67
.
.
35.68
.
.
27.1
.
27.2
.
.
.
Use the number line to help you compare the decimal numbers.
Write >, <, or = for each.
18.3
18.358
18.35
694.68
694.6
702.1
702.135
702.16
702.2
91.062
91.1
.
.
91
7/7/2008
6th 1-2
694.7
.
.
.
18.4
18.35 18.358
91.04
.
37
DRAFT
Use the drawings to help you compare the decimal numbers. Then look at the decimal
numbers written lined up by place value. Write >, <, or = for each.
1.5
Look:
1.8
1.5
1.8
1.37
Look:
5 < 8 so 1.5 < 1.8
same
Look:
2.3
2.2
3 > 2 so 2.3
2.2
3.54
3.56
4
6 so 3.54
2.23
3.9
Look:
3.8
3.9
8
9 so 3.8
3.9
3.56
2.17
2.23
2.17
2
1 so 2.23
same
same
.
1.37
1.3
.
.
Look:
7
0 so 1.37
1.3
same
7/7/2008
6th 1-2
3.56
same
3.8
Look:
7 > 4 so 1.37 > 1.34
3.54
2.2
same
Look:
1.37
1.34
same
2.3
Look:
1.34
1.5
1.55
same
38
.
0
5 so 1.5
1.55
2.17
DRAFT
Use the numberlines to help you compare the decimal numbers. Then look at the
decimal numbers written lined up by place value. Write >, <, or = for each.
35.8
35.85
35.9
35.85 35.86
35.86
Look:
35.85
35.86
5 < 6 so 35.85 < 35.86
same
57.3
57.36
57.32
57.4
57.36
57.32
Look:
57.36
57.32
6 > 2 so 57.36
57.32
same
124.567
124.567
124.564
124.56
124.564
Look:
124.567
124.564
7
4 so 124.567
124.57
124.564
same
61.4
61.425
61.467
61.425
61.467
Look:
61.425
61.467
5
7 so 61.425
61.5
61.467
same
.
983.2
983.248
983.233
Look:
.
983.3
983.248
983.233
same
7/7/2008
6th 1-2
39
8
3 so 983.248
983.233
DRAFT
Line up the decimal numbers by place value. Then compare the digits by writing >, <, or = for each.
Example:
Line up
32.56
32.59
<
32.56
32.59
6<9
Example:
Line up
same
237.86
237.84
6
4
82.5
Line up
365.3
Line up
2
3
Line up
8,024.6741
Line up
58,013.20198
58,013.20189 ___ ___
58,013.20198
4,356.3
4,356.298
4,356.3
___ ___
7/7/2008
6th 1-2
__________
__________
8
24,9967.7843
24,9967.7863
24,9967.7843
93,175.5436
Line up
93,175.5436
93,711.5437
426.927
Line up
same???
Line up
6
___ ___
93,711.5437
___ ___
same???
4,356.298
5,669.0187
1,882.564
1,882.586
same
same???
Line up
4
1,882.586
24,9967.7863
same
Line up
5
same
8,024.6301 ___ ___
8,024.6741
58,013.20189
82.5
82.4
82.4
1,882.564
same
8,024.6301
2>1
same
365.2
365.2
365.3
156.429
156.417
237.84
same
Line up
> 156.417
same
237.86
Line up
156.429
426.927
426.9269
426.9269
___ ___
same???
5,669.01
___ ___
678.2574
Line up
40
__________
__________
658.26
___ ___
DRAFT
Underline the first place where the digits in the two numbers are different.
72.5 and 72.4
365.23 and 365.41
945.679 and 945.692
5,646.890 and 5,647.892
235.846 and 235.845
6,549.0729 and 65,491.742
41,296.8247 and 41,276.3597
975,907.283 and 97,590.283
476,332.0634 and 476,332.07
9932,486.19423 and 9932,486.18421
Which place would you use to compare 6,758.4 and 6,759.3?
Which place would you use to compare 438.12 and 428.41?
Which place would you use to compare 2,204.850 and 2,204.803?
Which place would you use to compare 13,998.036 and 13,098.782?
Which place would you use to compare 61,982.3756 and 61,982.385?
Which place would you use to compare 122, 589.261 and 12,258.922
Which place would you use to compare 429,635.5 and 429,635.54?
Which place would you use to compare 21,856,023.887 and 21,856,023.9
Compare. Write >, <, or = for each
6.5
6.3
.
27.5
28.3
38.04
41.05
32.78
27.49
27.486
39.523
39.518
32.84
876.54
876.53
763.142
763.139
338.753
338.699
4,256.128
4,256.206
9,103.429
9,103.419
7,116.455
2,654.621
2,654.593
53,896.746
53,896.751
72,681 .148
72,681.143
5,761.21
5,761.156
17,936.1
48,003.388
48,030.40
237,866.7
7/7/2008
6th 1-2
237,866.625
17,936.100
111,756.8
111,756.845
41
2,963.400
7,016.445
2,963.4
DRAFT
Which number sentence is correct?
A) 0.5 > 0.8
B) 0.3 > 0.2
C) 0.7 < 0.5
D) 0.7 = 0.6
C) 72.5 < 72.42
D) 80.1 = 80.10
C) 19.4 > 19.456
D) 21.01 > 21.100
C) 546.094 < 546.9
D) 8.78 > 9.000
A) 7.3 = 7.300
B) 53.7 > 53.5
Which of the following is not true?
C) 8.7 < 8.60
D) 2.200 = 2.2
A) 45.004 < 45.4
B) 923.348 < 923.0
Which of the following is not true?
C) 1,001.4 < 1,001.426 D) 124.006 > 124.0
Which number sentence is correct?
A) 61.8 = 61.08
B) 93.4 > 93.8
Which number sentence is correct?
A) 24.876< 24.92
B) 47.47 = 46.404
Which number sentence is correct?
A) 893.300 < 893.3
B) 67.2 > 67.261
Which of the following is not true?
A) 8,345.05 > 8,345.0
7/7/2008
6th 1-2
B) 318.6 = 318.600
C) 4,673.600 > 4,673.6
42
D) 907.3 > 907.28
DRAFT
How do you put decimal numbers in order from greatest to least or least to greatest?
Which number is greater than 87.692
A. 87.599
B. 87.692
C. 8.999
D. 87.695
Which number is greater than 964.023
A. 98.967
B. 1,002.1
C. 964.018
D. 959.633
Which number is greater than 6,278.641
A. 6,278.640 B. 6,278.563 C. 60,278.641
D. 999.999
Which number is greater than 87,328.1964
A. 87,328.2
B. 9,678.639
C. 87,327.875
D. 9,000.685
Which number is greater than 707,235,148.2319
A. 707,299,865.2319
B. 88,654,256.2319 C. 707,235.0999
D. 706,253,100.2319
Which number is greater than 900,000,000.000
A.899,999,9999.999 B.100,000,000.000
C.100,000,000.001 D. 9,000,000,000.00
Which number is less than 78.456
A. 78.459
B. 80.000
C. 78.546
D. 78.399
Which number is less than 6,650.127
A. 6,297.003
B. 7,235.014
C. 6,659.901
D. 6,900
Which number is less than 625,014.895
A. 626,010.321
B. 625,238.754
C. 7,968.475
D. 670,258.694
Which number is less than 824,683,201
A. 824,694.001
B. 1,824,693.002
C. 819,876.081
D. 824,700.
Which number is less than 398,214,785.7543
A. 99,247,014.3213
B. 398,877,584.2473 C. 398,216.2133 D. 400,000,000.
Which number is less than 100,956,248.78913
A. 100,956,209.78921 B. 999,876,984.778569 C. 100,978,369.356214 D. 200,548,247.24558
7/7/2008
6th 1-2
43
DRAFT
Write a decimal number greater than 0.03 with 0 as the whole number.
Write a decimal number greater than 5.08 with 5 as the whole number.
Write a decimal number greater than 96.45 with 96 as the whole number.
Write a decimal number less than 317.003 with 317 as the whole number.
Write a decimal number less than 4,261.8 with 4,261 as the whole number.
Write a decimal number less than 23,698.56 with 23,698 as the whole number.
Write a decimal number larger than 0.6 with 0 as the whole number.
Write a decimal number larger than 9.007 with 9 as the whole number.
Write a decimal number larger than 78.061 with 78 as the whole number.
Write a decimal number smaller than 0.454 with 0 as the whole number.
Write a decimal number smaller than 4,006.770.with 4,006 as the whole number.
Write a decimal number smaller than 91,826.205 with 91,826 as the whole number.
Write a number greater than 0.05 but less than 0.179.
Write a number greater than 267.47 but less than 267.508.
Write a number greater than 10,335.45 but less than 10,335.453.
Write a number greater than 498,263.42 but less than 498,263.432.
Line up the decimal numbers by place value. Then order the decimals from least to greatest.
Example:
32.56
32.59
Step 1: Compare all 3 numbers
32.54
Step 2:
for greatest or least.
Line up
32.56
32.59
32.54
Example:
Compare other 2 numbers
for greater or less.
32.56
32.59
less
47.28
Step 1: Compare all 3 numbers
Line up
least
47.28
47.26
47.3
47.3
Compare other 2 numbers
for greater or less.
47.28
47.26
greatest
same
same
same
32.54 < 32.56 < 32.59
7/7/2008
6th 1-2
Step 2:
for greatest or least.
same
(least)
47.26
47.26 < 47.28 < 47.3
(less)
(less)
44
(greatest)
less
DRAFT
Line up the decimal numbers by place value. Then order the decimals from least to greatest.
467.019
Step 1:
Line up
467.02
467.023
Step 2:
467.019
467.02
467.023
3.452
467.02 = 467.020
_______
467.020
467.023
Step 1:
Line up
_______
same
Step 1:
Line up
_____
Step 1:
Line up
______
same
_________ < __________ < ___________
(less)
772.246
772.251
772.197
Step 2:
772.246
772.251
772.197
772.246
772.251
501.861
______
501.816
______
(less)
48.801
Step 1:
Line up
same
48.8
48.811
Step 2:
48.801
48.8
48.811
48.801
48.8
same
same
________ < _________ < _________
643.537
745.862
________ < _________ < _________
643.59
Step 2:
________ < _________ < _________ < _________
7/7/2008
6th 1-2
Step 2:
same
(least)
same
Step 1:
Line up
501.8 = 501.800
501.861
501.800
501.816
501.816
________ < _________ < _________
(greatest)
643.6
501.8
same
same
Step 1:
Line up
(greatest)
501.861
79.376
79.281
79.209
less
same
(less)
Step 2:
79.281
79.209
79.376
3.433
3.431
_________ < __________ < ___________
(less)
79.209
________
same
________ < 467.02 < _________
79.281
3.431
Step 2:
3.452
3.433
3.431
same
(least)
3.433
32.824
Step 1:
Line up
3.2925
32.9
32.821
Step 2:
________ < _________ < _________ < _________
45
DRAFT
Line up the decimal numbers by place value. Then order the decimals from greatest to least.
12.932
Step 1:
Line up
12.867
8.2976
12.941
Step 1:
Line up
Step 2:
12.932
12.867
12.941
12.932
______
12.941
greater
Step 1:
Line up
44.69
Step 2:
44.694
44.690
Step 1
Line up
______
0.69 = 0.690
_________ > __________ > ___________
Step 1:
Line up
554.3724
Step 2:
Step 1:
Line up
554.3604
544.3619
same
221.48
221.42
220
Step 2:
_______
same
(least)
184.7
184.768
Step 2:
184.763
184.70
184.768
184.763
0.7 = 0.70
184.768
same
________ > _________ > _________
635.864
Step 1:
Line up
635.9
635.872
635.856
Step 2:
________ >_________ >_________ >_________
________ >_________ >_________ >_________
7/7/2008
6th 1-2
604.108
604.106
same
________ > _________ > _________
Step 1:
Line up
______
184.763
same
221.43
Step 2:
604.039
604.108
604.106
(greater)
554.3619
554.3604
554.3619
554.3724
604.106
________ > _________ > _________
(greater)
554.3604
604.108
same
same
(greatest)
(greatest)
604.039
______
same
_______
______
(least)
(least)
44.698
44.694
44.690
8.2976
8.2981
_________ > __________ > ___________
12.941 > ___________ > _________
44.694
Step 2:
8.2976
8.2981
8.2499
same
same
44.698
8.2499
same
same
(greater)
8.2981
46
DRAFT
Order the numbers from least to greatest.
517.23
517.2
517.3
73.284
73.296
73.3
73.29
34.8063
34.9
34.0866
35
8,756.325
8,76.235
8,756.410
4,203.16
4,203.106
4,203.016
3,874.302
3,874.099
3,873.3
26,597.3614
26,597.37
26,596
26,597.3611
547.2389
547.2379
549
547.32
662,624.378
662,701.372
662,624.102
869,211.523
869,211.6
869,212.009
3,968,460.458
3,981,385.614
3,981,703.630
639,323.311
693,323.753
639,323.301
639,323.31
171.679
171.675
171.6
172.01
7/7/2008
6th 1-2
47
34.812
547.4
869,210
170.99
DRAFT
Order the numbers from greatest to least.
45.3
45.306
45
74.318
74.3
74.321
560.02
560.1
560.07
372.148
372.5
372.514
806.234
806.324
806.243
806.4
14.936
14.9
14.939
14.891
14.94
5,986
5,986.1001
5,992
5,986.118
5,986.21
2,172.500
2,172.501
2,171.499
6,283.65
6,283.7
6.283.651
54,613.327
54.631.723
54,613.732
28,412.897
28,400.998
28,412.9
189,563.721
189,396.986
18,989.719
302,271,847.200
304,271,748.200
30,271,825.200
312,234,114.863
40,368,114.642
314,002,681.162
451.67
451.69
451.674
7/7/2008
6th 1-2
48
6,283.655
28,420.19
451.7
452
DRAFT
How do you put decimal numbers in order from greatest to least or least to greatest?
Which number is greater than 3.05?
A) 3.057
B) 3.009
Which number is greater than 59.067?
C) 3.049
D) 3.03
A) 59.062
B) 59.06
Which number is greater than 312.863?
C) 59.007
D) 59.07
A) 312.86
B) 312.87
Which number is greater than 29.5641?
C) 312.799
D) 312.861
A) 29.5639
B) 29.5099
Which number is greater than 19.7984?
C) 29.5
D) 29.6
C) 19.7981
D) 19.799
A) 478.006
B) 478.086
Which number is less than 2,618.053?
C) 477.04
D) 478.008
A) 2,618.054
B) 2,618.06
Which number is less than 566.108?
C) 2,618.072
D) 2,618.040
A) 566.1
B) 566.12
Which number is less than 3,405.259?
C) 566.109
D) 567.001
A) 3,405.261
B) 3,405.3
Which number is less than 674.338?
C) 3,405.187
D) 3,405.26
C) 674.4
D) 674.3357
A) 19.79
B) 19.7
Which number is less than 478.007?
A) 674.3381
7/7/2008
6th 1-2
B) 674.34
49
DRAFT
Mathematical Reasoning:
Using the following information, order the towns from smallest land area to greatest land area:
Aston
Land Area: 405,560.758
Bay City
Land Area: 450,620.426
Connley
Land Area: 405,560.369
Dormer
Land Area: 452,060.014
A. Aston, Bay City, Connley, Dormer
C. Connley, Aston, Dormer, Bay City
B. Connley, Aston, Bay City, Dormer
D. Dormer. Bay City, Aston, Connley
Use the digits 1,2,6,7,5,4, and 3 to write the greatest possible 7-digit decimal number to the
thousandths. Each digit must be used exactly once.
Use the digits 1,2,6,7,5,4, and 3 to write the least possible 7-digit decimal number to the
thousandths. Each digit must be used exactly once.
Use the digits 0,1,2,3,4,5,6,7,8, and 9 to write the greatest 10-digit decimal number to the hundredthousandths. Each digit must be used exactly once.
Use the digits 0,1,2,3,4,5,6,7,8, and 9 to write the least 10-digit decimal number to the hundredthousandths. Each digit must be used exactly once.
Write the above number in short word form.
Which is greater-- a 7 or 8 digit whole number?
Which number would be farther to the left on a number line, 138,232,000.58 or 128,300,000.63?
Write two numbers greater than 310,000.000 but less than 320,000,000.
Write two numbers greater than 0.365124 but less than 0.365218
Is 1,695,142.8 miles less than 1,686,639.5 miles?
Tad and Myra were playing a video game. Tad scored 907,238.692,0486 points and Myra scored
907,238.7 points. Who won?
What digit do you change to make the numbers equal?
9,865,143.583 and 9,865,143.58
7/7/2008
6th 1-2
50
DRAFT
th
Investigating Student Learning: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard AF 1.3: Apply…the commutative, associative properties to evaluate
expressions….
Lesson 1.4: Addition Properties
Concepts:
Mental math is a technique used to find exact solutions to problems without paper and pencil or a
calculator.
Mental math is an important part of computation and is a prerequisite for estimation.
Compatible numbers are numbers that are easy to compute mentally.
Compatible numbers often are ones that result in multiples of ten or one hundred when the operations are
completed.
e.g. 5 + 26 + 15
5 and 15 are compatible numbers because their sum is 20.
Addition and subtraction are inverse operations.
Compensation is a mental math technique used to “create” compatible numbers.
When using compensation, a small value is added to or subtracted from one of the numbers to make that
number easier to compute with. After computing, the sum or difference is adjusted by that same small
value.
Proficient mental math calculations involves thinking about numbers more flexibly:
• “see” pairs of numbers that are compatible or easy to compute to make sums of 10 or 100
e.g. 18 + 24 +12
See that 8 and 2 make 10. So first add 18 + 12 =30.
Then 30 + 24 = 54
•
break apart numbers to make adding easier
e.g. 136 + 52
50 + 2 Break apart
= (136 + 50) + 2
=
186 + 2
=
188
•
adjust numbers by compensating
e.g. 246 + 18
+4
–4
250 + 14
= 264
The Addition Properties show why this works:
246 + 18 = 246 + 18 + 0
(Identity Property)
= 246 + 4 + 18 – 4
(Commutative Property)
= (246 + 4) + (18 – 4) (Associative Property)
=
250 + 14
=
264
7/7/2008
6th 1-4
51
DRAFT
e.g.
e.g. 46 – 38
+2
+ 2 Add 2 to 38 to get the compatible number 40.
48 – 40
=8
Then compensate by adding 2 to 46 to get 48. (You “took away” 2 more than the actual
minuend so you have to add those 2 back to the subtrahend so the difference is constant.) Now
it is easy to subtract 48 – 40 without having to regroup.
A number line shows why this works:
38
46
The difference is 8.
40
48
The difference is 8.
Essential Question(s):
How can compatible numbers be used to compute mentally?
How can the “break apart” strategy be used to compute mentally?
How can the compensation strategy be used to compute mentally?
7/7/2008
6th 1-4
52
DRAFT
th
ISL Item Bank: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard AF 1.3: Apply…the commutative, associative properties to evaluate
expressions….
Lesson 1.4: Addition Properties
How can compatible numbers be used to compute mentally?
Why are 2 and 8 compatible numbers?
Why are 13 and 7 compatible numbers?
Why are 15 and 5 compatible numbers?
Why are 75 and 25 compatible numbers?
Why are 70 and 30 compatible numbers?
Why are 94 and 6 compatible numbers?
Find a single digit compatible number by filling in the blank.
4 + ____
2 + ____
____ + 6
____ + 9
7 + ___
5 + ____
3 + ____
____ + 1
____ + 4
8 + ___
Find a two-digit compatible number by filling in the ones digit.
7/7/2008
6th 1-4
6 + 4__
2__ + 3
5 + 8__
4__ + 1
7__ + 2
4 + 7__
6__ + 7
9 + 2__
9__ + 8
5__ + 3
53
DRAFT
Circle the two numbers that are compatible.
6+2+8
6+4+3
8+9+1
13 + 4 + 7
6 + 2 + 14
5 + 28 + 2
5 + 13 + 45
46 + 7 + 33
68 + 27 + 2
75 + 33 + 25
30 + 70 + 21
8 + 91 + 9
25 + 37 + 15
46 + 83 + 24
39 + 22 + 58
11 + 13 + 29
26 + 83 + 17
36 + 18 + 14
Find a single digit compatible number by filling in the blank.
8 + ____ = 10
____ + 6 = 10
3 + ___ = 10
4 + ____ = 10
____ + 2 = 10
7 + ___ = 10
Find a two-digit compatible number by filling in the blank.
7/7/2008
6th 1-4
40 + ____ = 100
____ + 70 = 100
80 + ___ = 100
____ + 64 = 90
32 + ____ = 50
47 + ___ = 80
____ + 37 = 60
19 + ____ = 30
____ + 55 = 80
83 + ___ = 100
24 + ____ = 100
48 + ___ = 100
39 + ____ = 100
42 + ___ = 100
56 + ___ = 100
54
DRAFT
Add mentally.
4+6
7+3
8+2
9+1
5+5
40 + 40
30 + 60
20 + 30
70 + 10
50 + 30
40 + 60
90 + 10
80 + 20
70 + 30
50 + 50
21 + 9
36 + 4
8 + 52
37 + 3
44 + 6
32 + 18
11 + 49
17 + 23
54 + 16
25 + 15
27 + 33
46 + 24
25 + 15
39 + 21
32 + 48
Circle the two numbers that are compatible. Then find the sum x.
3+4+6=x
9+8+1=x
2+6+8=x
x = _____
x = _____
x = _____
8+9+2=x
7+6+3=x
5+5+4=x
x = _____
x = _____
x = _____
2 + 58 + 5 = x
33 + 7 + 4 = x
2 + 6 + 14 = x
x = _____
x = _____
x = _____
49 + 7 + 1 = x
7 + 5 + 65 = x
8 + 76 + 4 = x
x = _____
x = _____
x = _____
What Addition Properties help you order the addends so that the compatible numbers are added first in the
problem below?
36 + 8 + 4
= 36 + 4 + 8
__________________ (property)
= (36 + 4) + 8
___________________ (property)
=
40 + 8
=
48
7/7/2008
6th 1-4
55
DRAFT
Circle the two numbers that are compatible. Then find the sum x.
23 + 7 + 45 = x
68 + 29 + 2 = x
45 + 12 + 5 = x
x = _____
x = _____
x = _____
47 + 29 + 3 = x
44 + 42 + 6 = x
37 + 9 + 21 = x
x = _____
x = _____
x = _____
9 + 3 + 91 = x
25 + 46 + 75 = x
30 + 70 + 74 = x
x = _____
x = _____
x = _____
60 + 97 + 40 = x
10 + 35 + 90 = x
20 + 80 + 78 = x
x = _____
x = _____
x = _____
28 + 22 + 39 = x
15 + 27 + 35 = x
14 + 43 + 26 = x
x = _____
x = _____
x = _____
25 + 33 + 25 = x
26 + 34 + 17 = x
18 + 27 + 53 = x
x = _____
x = _____
x = _____
What Addition Properties help you order the addends so that the compatible numbers are added first
in the problem below?
43 + 18 + 37
= 43 + 37 + 18
__________________ (property)
= (43 + 37) + 18
___________________ (property)
=
80 + 18
=
98
7/7/2008
6th 1-4
56
DRAFT
Use compatible numbers to add mentally.
4+7+6
A) 11
B) 13
C) 17
D) 16
C) 52
D) 35
38 + 5 + 2
A) 45
B) 43
6 + 45 + 14
A) 55
B) 65
C) 61 D) 59
25 + 37 + 23
A) 60
B) 48
C) 82
D) 85
C) 60
D) 96
38 + 34 + 22
A) 94
B) 93
21 + 46 + 39
A) 67
B) 60
C) 106
D) 105
Use compatible numbers to add mentally.
7/7/2008
6th 1-4
6+4+7
8+3+2
5+4+5
2 + 58 + 9
8 + 1 + 39
7 + 57 + 3
8 + 55 + 5
44 + 3 + 6
6 + 8 + 84
57
DRAFT
Use compatible numbers to add mentally.
7/7/2008
6th 1-4
34 + 48 + 2
34 + 6 + 27
7 + 53 + 24
5 + 26 + 25
34 + 51 + 9
22 + 43 + 7
24 + 57 + 13
22 + 38 + 19
13 + 27 + 14
18 + 11 + 49
21 + 37 + 39
22 + 19 + 48
60 + 27 + 40
80 + 20 + 48
82 + 30 + 70
25 + 36 + 64
63 + 48 + 52
97 + 26 + 74
81 + 34 + 19
58 + 23 + 77
62 + 96 + 38
35 + 82 + 65
86 + 43 + 14
72 + 28 + 59
58
DRAFT
How can the “break apart” strategy be used to compute mentally?
Break apart each number so that addition is simpler.
Example:
43
29
37
62
58
56
49
40 + 3
72
94
85
Fill in the blanks. Break apart one addend and then add to find the sum.
Example:
62 + 35
56 + 33
= 62 + (30 + ___)
= 56 + (30 + 3)
= (62 + 30) + 5
= (56 + 30) + 3
=
92 + __
=
86 + 3
=
97
=
89
44 + 24
= 44 + (20 + __)
= (44 + __) + 4
=
64 + __
=
___
86 + 33
= 86 + (__ + __)
= (86 + __) + 3
= 116 + __
=
___
94 + 52
= 94 + (__ + __)
= (94 + __) + __
= ___ + __
=
___
77 + 51
= 77 + (__ + __)
= (__ + __) + __
= ___ + __
=
___
84 + 37
= 84 + (__ + __)
= (__ + __) + __
= 114 + __
=
___
95 + 48
= __ + (__ + __)
= (__ + __) + __
= ___ + __
=
___
67 + 75
= __ + (__ + __)
= (__ + __) + __
= ___ + __
=
___
What Addition Properties help you group the addends so the numbers are easier to add in the
problem below?
73 + 26
= 73 + (20 + 6)
= (73 + 20) + 6
___________________ (property)
=
93 + 6
=
99
7/7/2008
6th 1-4
59
DRAFT
Fill in the blanks. Break apart one addend and then add to find the sum.
Example:
135 + 23
134 + 43
=
135 + (20 + ___)
= 134 + (40 + 3)
=
(135
+ __) + 3
= (134 + 40) + 3
=
155 + __
=
174 + 3
=
___
=
177
256 + 42
= 256 + (40 + ___)
= (___ + __) + __
=
___ + __
=
___
337 + 54
= ___ + (__ + __)
= (___ + __) + __
=
___ + __
=
___
424 + 68
= ___ + (__ + __)
= (___ + __) + __
=
___ + __
=
___
739 + 53
= ___ + (__ + __)
= (___ + __) + __
=
___ + __
=
___
343 + 77
= ___ + (__ + __)
= (___ + __) + __
=
___ + __
=
___
847 + 66
= ___ + (__ + __)
= (___ + __) + __
=
___ + __
=
___
286 + 44
= ___ + (__ + __)
= (___ + __) + __
=
___ + __
=
___
What Addition Properties help you group the addends so the numbers are easier to add in the
problem below?
164 + 33
= 164 + (30 + 3)
= (164 + 30) + 3
___________________ (property)
=
194 + 3
=
197
Break apart one addend and then add to find the sum.
7/7/2008
6th 1-4
62 + 34
24 + 53
82 + 17
27 + 64
57 + 36
73 + 24
126 + 44
337 + 42
435 + 36
60
DRAFT
Choose the best strategy that helps you compute mentally. Then explain why you chose this strategy.
46 + 14
(circle one)
compatible
numbers
36 + 42
break
apart
Explanation:
(circle one)
compatible
numbers
Explanation:
74 + 23
(circle one)
compatible
numbers
56 + 23
break
apart
Explanation:
(circle one)
compatible
numbers
compatible
numbers
62 + 18
break
apart
Explanation:
(circle one)
compatible
numbers
compatible
numbers
19 + 31
break
apart
Explanation:
(circle one)
compatible
numbers
compatible
numbers
Explanation:
7/7/2008
6th 1-4
break
apart
Explanation:
45 + 35
(circle one)
break
apart
Explanation:
38 + 45
(circle one)
break
apart
Explanation:
47 + 33
(circle one)
break
apart
46 + 27
break
apart
(circle one)
compatible
numbers
Explanation:
61
break
apart
DRAFT
How can compensation be used to compute mentally?
What small value can be added to make the number easier to use? Fill in the blank.
9+
= 10
19 +
= 20
48 +
= 50
57 +
= 60
87 +
= 90
57 +
= 60
79 +
= 80
28 +
= 30
What small value can be added to make the number easier to use? Fill in the blanks.
69 +
=
18 +
=
99 +
=
27 +
=
46 +
=
58 +
=
76 +
=
37 +
=
What small value can be added to make the number easier to use AND THEN adjusted by subtracting to get
the original value? Fill in the blanks.
18 +
7/7/2008
6th 1-4
= 20
78 +
= 80
49 +
= 50
20 -
80 -
50 62
= _____
= _____
= _____
DRAFT
What small value can be added to make the number easier to use AND THEN adjusted by subtracting to get
the original value? Fill in the blanks.
98
+
=
100
100
adjust
compensate
26
+
=
30
30
-
+
=
_____
=
_____
adjust
compensate
78
_____
=
=
80
80
adjust
compensate
Compensating in addition involves adding a small amount to one addend to make an
easier number to add. Then the same small amount is subtracted so the amount
actually added remains unchanged. Fill in the blanks.
Example:
74 + 27
-
3
+ 3
71
+ 30
=
101
=
101
Compensating in addition involves adding a small amount to one addend to make an
easier number to add. Then the same small amount is subtracted so the amount
actually added remains unchanged. Fill in the blanks.
17
+ 8
-
+
+
7/7/2008
6th 1-4
=
=
63
DRAFT
Compensating in addition involves adding a small amount to one addend to make an
easier number to add. Then the same small amount is subtracted so the amount
actually added remains unchanged. Fill in the blanks.
37
+
-
19
=
+
+
=
Compensating in addition involves adding a small amount to one addend to make an
easier number to add. Then the same small amount is subtracted so the amount
actually added remains unchanged. Fill in the blanks.
66
+
+
32
=
-
+
=
Compensating in addition involves adding a small amount to one addend to make an
easier number to add. Then the same small amount is subtracted so the amount
actually added remains unchanged. Fill in the blanks.
59
+
+
24
=
-
+
=
Compensating in addition involves adding a small amount to one addend to make an
easier number to add. Then the same small amount is subtracted so the amount
actually added remains unchanged. Fill in the blanks.
23
-
+
47
+
+
7/7/2008
6th 1-4
=
=
64
DRAFT
Use compensation to find 55 + 79.
Example:
55
55
+
1
1
+ 80
+ 79
=
compensate
135
- 1 =
134
adjust
So 55 + 79 =
Use compensation to find 39 + 23.
39
+ 23
+
=
+
compensate
23
-
=
adjust
So 39 + 23 =
Use compensation to find 84 + 58.
84
+ 58
84
+
=
+
compensate
-
=
adjust
So 84 + 58 =
Use compensation to find 69 + 73.
69
+ 73
+
compensate
=
+
73
-
=
adjust
So 69 + 73 =
7/7/2008
6th 1-4
65
134
DRAFT
Use compensation to find 57 + 28.
57
+ 28
57
+
=
+
compensate
-
=
adjust
So 57 + 28 =
Use compensation to find 28 + 44.
28
+ 44
+
=
+
compensate
44
-
=
adjust
So 28 + 44 =
Use compensation to find 42 + 56.
42
+ 56
42
+
compensate
=
+
-
=
adjust
So 42 + 56 =
Use compensation to find 97 + 29.
97
97
+
+ 29
+
=
compensate
-
=
adjust
So 97 + 29 =
7/7/2008
6th 1-4
66
DRAFT
Use compensation to add mentally.
19 + 34
59 + 23
38 + 43
63 + 28
56 + 29
77 + 26
99 + 53
69 + 34
34 + 58
47 + 39
62 + 19
48 + 43
67 + 24
16 + 28
37 + 45
Use compensation to add mentally.
39 + 8
A) 48
B) 47
C) 317
D) 57
C) 96
D) 94
88 + 6
A) 82
B) 104
37 + 52
A) 99
B) 89
C) 95 D) 85
B) 96
C) 85
36 + 59
A) 95
D) 86
35 + 77
A) 118
B) 115
C) 102
D) 112
B) 118
C) 112
D) 102
67 + 45
A) 115
7/7/2008
6th 1-4
67
DRAFT
Choose the best strategy that helps you compute mentally. Then explain why you chose this strategy.
39 + 14
14 + 26
A) compatible numbers
B) break apart
C) compensation
Explanation:
A) compatible numbers
B) break apart
C) compensation
Explanation:
53 + 16
31 + 19
A) compatible numbers
B) break apart
C) compensation
Explanation:
A) compatible numbers
B) break apart
C) compensation
Explanation:
37 + 13
36 + 58
A) compatible numbers
B) break apart
C) compensation
Explanation:
A) compatible numbers
B) break apart
C) compensation
Explanation:
24 + 69
54 + 17
A) compatible numbers
B) break apart
C) compensation
Explanation:
A) compatible numbers
B) break apart
C) compensation
Explanation:
38 + 52
47 + 18
A) compatible numbers
B) break apart
C) compensation
Explanation:
7/7/2008
6th 1-4
A) compatible numbers
B) break apart
C) compensation
Explanation:
68
DRAFT
th
Investigating Student Learning: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard NS 1.1 (Gr. 5): Estimate, round,…very large (e.g. millions) and very small
(e.g. thousandths) numbers.
Lesson 1.5: Rounding Whole Numbers and Decimals
Concepts:
Rounding a number is replacing one number with another number that tells about how many or how much.
Rounding is a process for finding the multiple of 10, 100 etc. closest to a given number.
In some situations it is practical to give an approximate number rather than and exact amount.
A number can be rounded to any place value.
When you round a number to the nearest ten or tens place, you decide whether the number is closer to the
ten before (round down) or the ten after the number (round up).
A number line can help you decide whether or not to round up or down.
On a number line, a number is rounded to the closest multiple of thousand, hundred, or ten depending on the
place to which the number is rounded. A number halfway between two multiples is rounded up to the
next multiple.
Rounding decimals is similar to rounding whole numbers.
Like whole numbers, a decimal number can be rounded to any place value.
Rounding decimals is a process for finding the tenths, hundredths, thousandths, etc. closest to a given
number.
Essential Question(s):
How do you use a number line to round whole numbers?
How do you use a number line to round decimal numbers?
How do you round numbers to a given place value?
7/7/2008
6th 1-5
69
DRAFT
th
ISL Item Bank: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard NS 1.1 (Gr. 5): Estimate, round,…very large (e.g. millions) and very small
(e.g. thousandths) numbers.
Lesson 1.5: Rounding Whole Numbers and Decimals
How do you use a number line to round whole numbers?
Round each number to the nearest ten by first determining the closest ten before and after each number.
What is the closest ten before and after 62?
•
•
62
•
62 rounded to the nearest ten is ______.
What is the closest ten before and after 87?
•
87
•
•
87 rounded to the nearest ten is ______.
What is the closest ten before and after 174?
•
•
•
174
174 rounded to the nearest ten is ______.
What is the closest ten before and after 635?
•
•
635
•
635 rounded to the nearest ten is ______.
What is the closest ten before and after 786?
•
•
•
786
786 rounded to the nearest ten is ______.
7/7/2008
6th 1-5
70
DRAFT
Round each number to the nearest hundred by first determining the closest hundred before and after each
number.
What is the closest hundred before and after 187?
•
150
•
•
187
•
187 rounded to the nearest hundred is ______.
What is the closest hundred before and after 323?
•
•
323
•
350
•
323 rounded to the nearest hundred is ______.
What is the closest hundred before and after 505?
505
••
•
550
•
505 rounded to the nearest hundred is ______.
What is the closest hundred before and after 8,462?
•
•
8,450 8,462
•
•
8,462 rounded to the nearest hundred is ________.
What is the closest hundred before and after 3,678?
•
•
3,650
•
3,678
•
3,678 rounded to the nearest hundred is ________.
What is the closest hundred before and after 56,131?
•
•
56,131
•
•
56,150
56,131 rounded to the nearest hundred is _________.
7/7/2008
6th 1-5
71
DRAFT
Round each number to the nearest thousand by first determining the closest thousand before and after each
number.
What is the closest thousand before and after 6,648?
•
6,500
•
6,648
•
•
6,648 rounded to the nearest thousand is ______.
What is the closest thousand before and after 2,284?
•
•
2,284
•
2,500
•
2,284 rounded to the nearest thousand is ______.
What is the closest thousand before and after 76,382?
•
76,382
•
•
76,500
•
76,382 rounded to the nearest thousand is ______.
What is the closest thousand before and after 35,461?
•
35,461
••
35,500
•
35,461 rounded to the nearest thousand is ________.
What is the closest thousand before and after 246,778?
•
•
246,500
•
246,778
•
246,778 rounded to the nearest thousand is ________.
What is the closest thousand before and after 582,871?
•
582,871
•
•
•
582,500
582,871 rounded to the nearest thousand is _________.
7/7/2008
6th 1-5
72
DRAFT
Round each number.
What is the closest ten thousand before and after 86,255?
•
•
85,000
86,255
•
•
86,255 rounded to the nearest ten thousand is _________.
What is the closest ten thousand before and after 2,612,197?
•
2,612,197
•
•
•
2,615,000
2,6162,197 rounded to the nearest ten thousand is _________.
What is the closest hundred thousand before and after 8,691,241?
•
8,691,241
•
•
•
8,650,000
8,691,241 rounded to the nearest hundred thousand is __________.
What is the closest million before and after 7,476,427?
•
7,476,427
••
7,500,000
•
7,476,427 rounded to the nearest million is __________.
What is the closest million before and after 56,873,620?
•
•
56,500,000
37,873,620
•
•
56,873,620 rounded to the nearest million is ___________.
What is the closest ten million before and after 836,502,546?
•
836,502,546
•
•
835,000,000
•
836,502,546 rounded to the nearest ten million is ___________.
7/7/2008
6th 1-5
73
DRAFT
Place 63 on the number line.
•
•
60
65
•
70
63 rounded to the nearest ten is _____.
Find the “halfway” number and then place 567 on the number line.
•
•
•
560
570
567 rounded to the nearest ten is _____.
Place 52,396 on the number line.
•
52,390
•
•
52,395
52,400
52,396 rounded to the nearest ten is _____.
Find the “halfway” number and then place 837,153 on the number line.
•
•
•
837,150
837,160
837,153 rounded to the nearest ten is _____.
Place 684 on the number line.
•
•
•
600
650
700
684 rounded to the nearest hundred is _____.
Find the “halfway” number and then place 745 on the number line.
•
•
•
700
800
745 rounded to the nearest hundred is _____.
7/7/2008
6th 1-5
74
DRAFT
Find the “halfway” number and then place 5,493 on the number line.
•
•
•
5,400
5,500
5,493 rounded to the nearest hundred is _____.
Place 37,736 on the number line.
•
•
•
37,700
37,750
37,800
37,736 rounded to the nearest hundred is _____.
Find the “halfway” number and then place 4,711 on the number line.
•
•
•
4,000
5,000
4,711 rounded to the nearest thousand is _____.
Place 8,378 on the number line.
•
•
•
8,000
8,500
9,000
8,378 rounded to the nearest thousand is _____.
Place 14,674 on the number line.
•
•
•
14,000
14,500
15,000
14,674 rounded to the nearest thousand is _____.
Find the “halfway” number and then place 731,605 on the number line.
•
•
•
731,000
732,000
731,736 rounded to the nearest thousand is _____.
7/7/2008
6th 1-5
75
DRAFT
Place 57,821 on the number line.
•
•
50,000
55,000
•
60,000
57,821 rounded to the nearest ten thousand is _____.
Find the “halfway” number and then place 723,289 on the number line.
•
•
•
700,000
800,000
723,289 rounded to the nearest hundred thousand is _____.
Place 7,889,614 on the number line.
•
•
•
7,000,000
7,500,000
8,000,000
4,889,614 rounded to the nearest million is _____.
Find the “halfway” number and then place 960,534,000 on the number line.
•
•
•
960,000,000
970,000,000
960,534,000 rounded to the nearest ten million is _____.
Place 357,821,776 on the number line.
•
•
300,000,000
350,000000
•
400,000,000
357,821,776 rounded to the nearest hundred million is _____.
Find the “halfway” number and then place 8,365,211,036 on the number line.
•
•
•
8,00,000,000
400,000,000
8,365,211,036 rounded to the nearest billion is _____.
7/7/2008
6th 1-5
76
DRAFT
How do you round numbers to a given place value?
Underline the digit in the tens place. Then fill in the blank.
62
62 is between 60 and _____.
Underline the digit in the tens place. Then fill in the blank.
743
743 is between 740 and_____ .
Underline the digit in the hundreds place. Then fill in the blank.
367
367 is between 300 and ________.
Underline the digit in the hundreds place. Then fill in the blank.
8,803
8,803 is between 8,800 and _____.
Underline the digit in the thousands place. Then fill in the blank.
23,758
23,758 is between 23,000 and ______________.
Underline the digit in the thousands place. Then fill in the blank.
949,425
949,625 is between ____________ and ______________.
Underline the digit in the ten thousands place. Then fill in the blanks.
426,513
426,513 is between 420,000 and __________________.
Underline the digit in the ten thousands place. Then fill in the blank.
8,644,232
8,644,232 is between _______________ and 8,650,000.
Underline the digit in the hundred thousands place. Then fill in the blank.
1,456,558
1,456,558 is between 1,400,000 and _______________________.
Underline the digit in the hundred thousands place. Then fill in the blank.
9,721,645
9,721,645 is between ______________________ and 9,800,000.
Underline the digit in the millions place. Then fill in the blank.
6,256,558
6,256,558 is between 6,000,000 and _______________________.
Underline the digit in the millions place. Then fill in the blank.
65,741,645
7/7/2008
6th 1-5
65,741,645 is between ________________________ and 66,000,000.
77
DRAFT
Underline the digit in the ten millions place. Then fill in the blanks.
489,657,308
489,657,308 is between 480,000,000 and_____________________.
Underline the digit in the ten millions place. Then fill in the blanks.
863,245,286
863,245,286 is between _____________________ and 870,000,000.
Underline the digit in the hundred millions place. Then fill in the blanks.
551,669,487
551,669,487 is between 500,000,000 and_____________________.
Underline the digit in the hundred millions place. Then fill in the blanks.
828,112,036
828,112,036 is between ________________ and_______________.
Underline the digit in the billions place. Then fill in the blanks.
2,399,658,045 2,399,658,045 is between 2,000,000,000 and__________________.
Underline the digit in the ten billions place. Then fill in the blanks.
721,698,005,400
721,698,005,400 is between _____________________
and_______________________.
What number is halfway between 10 and 20?
What number is halfway between 50 and 60?
What number is halfway between 300 and 400?
What number is halfway between 700 and 800?
What number is halfway between 820 and 830?
What number is halfway between 7,000 and 8,000?
What number is halfway between 8,500 and 8,600?
What number is halfway between 6,430 and 6,440?
7/7/2008
6th 1-5
78
DRAFT
What number is halfway between 11,250 and 11,260?
What number is halfway between 8,600 and 8,700?
What number is halfway between 30,000 and 40,000?
What number is halfway between 80,300 and 90,400?
What number is halfway between 47,220 and 48,230?
What number is halfway between 35,800,000 and 35,900,000?
What number is halfway between 62,000,000 and 63,000,000?
What number is halfway between 300,000,000 and 400,000,000?
What number is halfway between 538,600,000 and 538,700,000?
Round each number to the nearest place.
Underline the digit in the tens place. Then fill in the blanks.
26
26 is between 20 and _____.
The “halfway” number is ___________.
26 is closer to ________ which means
26 rounded to the nearest ten is ____________.
Underline the digit in the tens place. Then fill in the blank.
52
52 is between 50 and _____.
The “halfway” number is ___________.
52 is closer to ________ which means
52 rounded to the nearest ten is ____________.
7/7/2008
6th 1-5
79
DRAFT
Round each number to the nearest place.
Underline the digit in the tens place. Then fill in the blank.
833
833 is between _____ and 840.
The “halfway” number is ___________.
833 is closer to ________ which means
833 rounded to the nearest ten is ____________.
Underline the digit in the hundreds place. Then fill in the blank.
345
345 is between _______ and _______.
The “halfway” number is ___________.
345 is closer to ________ which means
345 rounded to the nearest hundred is ____________.
Underline the digit in the hundreds place. Then fill in the blank.
6,892
6,892 is between 6,800 and ___________.
The “halfway” number is ___________.
6,892 is closer to __________ which means
6,892 rounded to the nearest hundred is ______________.
Underline the digit in the thousands place. Then fill in the blank.
41,463
41,463 is between _____________ and _____________
The “halfway” number is _____________.
41,463 is closer to _____________ which means
41,463 rounded to the nearest thousand is ___________________.
7/7/2008
6th 1-5
80
DRAFT
Round each number to the nearest place.
Underline the digit in the thousands place. Then fill in the blanks.
74,788
74,788 is between ________________ and 75,000.
The “halfway” number is ___________________.
74,788 is closer to __________________ which means
74,788 rounded to the nearest thousand is _____________________.
Underline the digit in the ten thousands place. Then fill in the blanks.
63,311
63,311 is between 60,000 and __________________.
The “halfway” number is __________________.
63,311 is closer to __________________ which means
63,311 rounded to the nearest ten thousand is ___________________.
Underline the digit in the ten thousand place. Then fill in the blank.
247,503
247,503 is between _________________ and 250,000.
The “halfway” number is _________________.
247,503 is closer to _________________ which means
247,503 rounded to the nearest ten thousand is ___________________.
Underline the digit in the hundred thousand place. Then fill in the blanks.
467,543
467,543 is between 400,000 and __________________.
The “halfway” number is __________________.
467,543 is closer to ____________________ which means
467,543 rounded to the nearest hundred thousand is _____________________.
7/7/2008
6th 1-5
81
DRAFT
Round each number to the nearest place.
Underline the digit in the hundred thousand place. Then fill in the blank.
4,654,968
4,654,968 is between ____________________ and 4,700,000.
The “halfway” number is ___________________.
4,654,968 is closer to ____________________ which means
4,654,968 rounded to the nearest hundred thousand is ______________________.
Underline the digit in the millions place. Then fill in the blanks.
5,412,001
5,412,001 is between 5,000,000 and ____________________.
The “halfway” number is ___________________.
5,412,001 is closer to ____________________ which means
5,412,001 rounded to the nearest million is ______________________.
Underline the digit in the ten millions place. Then fill in the blanks.
236,330,866
236,330866 is between 230,000,000 and _______________________.
The “halfway” number is ___________________.
236,330,866 is closer to ____________________ which means
236,330,866 rounded to the nearest ten million is ______________________.
Underline the digit in the hundred millions place. Then fill in the blanks.
472,476,002
472,476,002 is between ____________________ and 500,000,000.
The “halfway” number is ___________________.
472,476,002 is closer to ____________________ which means
472,476,002 rounded to the nearest hundred million is ______________________.
Underline the digit in the billions place. Then fill in the blanks.
5,736,330,214
5,736,330,214 is between __________________ and ___________________.
The “halfway” number is _______________________.
5,736,330,214 is closer to _______________________ which means
5,736,330,214 rounded to the nearest billions is _________________________.
7/7/2008
6th 1-5
82
DRAFT
Round to the nearest ten.
54
28
275
6,398
25,432
96,217
356,004
788,274
726
363
5,785
76,938
44,147
32,424
358,465
2,406,197
7,826
4,263
89,785
21,938
65,542
213,424
4,086,465
44,369,830
Round to the nearest hundred.
Round to the nearest thousand.
Round to the nearest ten thousand.
45,833
36,263
149,785
871,938
2,545,547
9,823,424
65,386,465
814,169,897
Round to the nearest hundred thousand.
374,147
422,424
8,178,465
4,756,197
63,041,258
32,184,433
896,775,179
4,637,306,797
5,694,147
6,432,424
32,278,465
19,856,197
74,314,585
35,557,669
203,803,193
899,733,054
Round to the nearest million.
7/7/2008
6th 1-5
83
DRAFT
Round to the nearest ten million.
72,694,147
18,732,424
86,278,465
40,856,197
344,314,585
265,557,669
7,123,803,193
9,359,733,054
Round to the nearest hundred million.
652,541,258
148,364,433
721,175,179
960,306,797
4,958,525,832
6,206,196,517
32,895,376,004
75,418,268,174
8,603,248,695
2,708,693,421
65,214,369,569
49,741,456,357
85,123,369,987
31,658,951,234
852,456,301,147
996,079,360,581
Round to the nearest billion.
Round to the underlined place.
475,294
1,305
2,569,145
5,469,013
21,968
657,359
624,208,574
8,269,428
56,957
45,690,361
658,241
60,238,473
958,525,832
206,196,517
895,376,004
418,268,174
2,584,456,456
8,428,869,983
4,121,003,219
7,669,308,357
Write three numbers that round to 460 when rounded to the nearest ten.
Write three numbers that round to 700 when rounded to the nearest ten.
7/7/2008
6th 1-5
84
DRAFT
Write three numbers that round to 4,000 when rounded to the nearest ten.
Write three numbers that round to 8,300 when rounded to the nearest hundred.
Write three numbers that round to 2,000 when rounded to the nearest hundred.
Write three numbers that round to 50,000 when rounded to the nearest hundred.
Write three numbers that round to 7,000,000 when rounded to the nearest thousand.
Write three numbers that round to 64,000 when rounded to the nearest thousand.
Write three numbers that round to 30,000 when rounded to the nearest thousand.
Write three numbers that round to 700,000 when rounded to the nearest thousand.
To what place value is 5,370,000 rounded?
To what place value is 26,700,000 rounded?
To what place value is 14,350 rounded?
To what place value is 356,400 rounded?
To what place value is 79,727,000 rounded?
To what place value is 59,000,000 rounded?
To what place value is 800,400,000 rounded?
To what place value is 3,000,000 rounded?
7/7/2008
6th 1-5
85
DRAFT
How do you use a number line to round decimal numbers?
=
0.3
=
0.30
Explain why these two decimal numbers are equal in value.
0.6
0.60
Explain why these two decimal numbers are equal in value.
0.9
0.90
Explain why these two decimal numbers are equal in value.
0.4
0.40
Explain why these two decimal numbers are equal in value.
7/7/2008
6th 1-5
86
DRAFT
What number is equal to 8.7?
A) 8.07
B) 8.70
C) 0.807
D) 80.7
What number is equal to 92.6?
A) 92.06
B) 926.0
C) 92.60
D) 920.60
C) 30.062
D) 300.62
C) 75.004
D) 75..040
What number is equal to 30.62?
A) 30.620
B) 3.062
What number is equal to 75.04?
A) 750.04
B) 75.40
What number is equal to 963.5?
A) 9,630.50
B) 963.05
C) 9,630.50
D) 963.500
What number is equal to 280.58?
A) 28.058
B) 280.058
C) 280.580
Write a number that is equal in value to 7.4 using 3 digits.
Write a number that is equal in value to 0.8 using 3 digits.
Write a number that is equal in value to 52.4 using 4 digits.
Write a number that is equal in value to 6.9 using 3 digits.
Write a number that is equal in value to 3.1 using 3 digits.
Write a number that is equal in value to 3.1 using 4 digits.
Write a number that is equal in value to 3.1 using 5 digits.
7/7/2008
6th 1-5
87
D) 25.0580
DRAFT
Round each number to the nearest whole number or one by first determining the closest whole number
before and after each number.
What are the closest whole numbers before and after 6.2?
•
•
6.2
•
6.2 rounded to the nearest whole number is ______.
What are the closest whole numbers before and after 6.7?
•
6.7
•
•
6.7 rounded to the nearest whole number is ______.
What are the closest whole numbers before and after 6.45?
•
6.45
•
•
6.45 rounded to the nearest whole number is ______.
What are the closest whole numbers before and after 874.291?
•
874.291
•
•
874.291 rounded to the nearest whole number is ______.
Round each number to the nearest tenth by first determining the closest tenth before and after each number.
What are the closest tenths before and after 3.68?
•
•
3.68
•
3.68 rounded to the nearest tenth is ______.
What are the closest tenths before and after 78.27?
•
•
•
78.27
78.27 rounded to the nearest tenth is ______.
7/7/2008
6th 1-5
88
DRAFT
Round each number to the nearest tenth by first determining the closest tenth before and after each number.
What are the closest tenths before and after 3.672?
•
•
•
5.672
3.672 rounded to the nearest tenth is ______.
What are the closest tenths before and after 211.937?
•
•
•
211.937
211.937 rounded to the nearest tenth is ______.
Round each number to the nearest hundredth by first determining the closest hundredth before and after
each number.
What are the closest hundredths before and after 0.367?
•
0.365
•
•
0.367
•
0.367 rounded to the nearest hundredth is ______.
What are the closest hundredths before and after 0.923?
•
•
0.925
•
0.923
•
0.923 rounded to the nearest hundred is ______.
What are the closest hundredths before and after 1.604?
•
1.604
•
•
1.604 rounded to the nearest hundredth is ______.
What are the closest hundredths before and after 58.767?
•
•
58.767
•
58.767 rounded to the nearest hundredth is ________.
7/7/2008
6th 1-5
89
DRAFT
Round each number to the nearest thousandths by first determining the closest thousandths before and
after each number.
What are the closest thousandths before and after 0.1827?
•
0.1825
•
0.1820
•
0.1827
•
0.1827 rounded to the nearest thousandths is ______.
What are the closest thousandths before and after 0.4632?
•
•
0.4632
•
0.4635
•
0.4632 rounded to the nearest thousandths is ______.
What are the closest thousandths before and after 2.8704?
•
•
•
2.8704
2.8704 rounded to the nearest thousandths is ______.
What are the closest thousandths before and after 5.2188?
•
•
5.2188
•
5.2188 rounded to the nearest thousandths is ______.
7/7/2008
6th 1-5
90
DRAFT
Place 3.3 on the number line.
3.0
Is 3.3 closer to 3.0 or 4.0?
4.0
3.5
Place 72.3 on the number line.
72.0
Is 72.3 closer to 72.0 or 73.0?
72.5
73.0
6.55
6.6
0.5
1
Place 6.58 on the number line.
6.5
Is 6.58 closer to 6.5 or 6.6?
Place 0.34 on the number line.
0
Is 0.34 closer to 0 or 1?
Place 9.23 on the number line.
9.2
Is 9.23 closer to 9.2 or 9.3?
7/7/2008
6th 1-5
9.3
91
DRAFT
Place 62.32 on the number line.
62.3
62.4
Is 62.32 closer to 62.3 or 62.4?
Place 5.697 on the number line.
5.6
5.7
Is 5.697 closer to 5.6 or 5.7?
Place 7.914 on the number line.
7.91
7.915
7.92
Is 7.914 closer to 7.91 or 7.92?
Place 55.872 on the number line.
55.87
55.88
Is 55.872 closer to 55.87 or 55.88?
Place 9.7863 on the number line.
9.786
9.7865
9.787
Is 9.7863 closer to 9.786 or 9.787?
Place 57.4197 on the number line.
57.419
57.42
Place 57.4197 closer to 57.419 or 57.42?
7/7/2008
6th 1-5
92
DRAFT
How do you round numbers to a given place value?
Underline the digit in the tenths place. Then fill in the blank.
0.42
0.42 is between 0.3 and _____.
Underline the digit in the tenths place. Then fill in the blank.
5.87
5.87 is between 5.8 and _____.
Underline the digit in the tenths place. Then fill in the blank.
6.45
6.45 is between ______ and ________.
Underline the digit in the tenths place. Then fill in the blank.
1.473
1.473 is between __________ and __________ .
Underline the digit in the tenths place. Then fill in the blank.
79.6373
79.6373 is between __________ and __________ .
Underline the digit in the hundredths place. Then fill in the blank.
0.667
0.667 is between 0.66 and ______________.
Underline the digit in the hundredths place. Then fill in the blank.
0.041
0.041 is between ______________ and 0.05.
Underline the digit in the hundredths place. Then fill in the blank.
0.703
0.703 is between ____________ and ______________.
Underline the digit in the hundredths place. Then fill in the blank.
9.495
9.495 is between 9.49 and _______________
Underline the digit in the hundredths place. Then fill in the blanks.
6.3281
6.3281 is between __________________ and __________________.
Underline the digit in the thousandths place. Then fill in the blanks.
0.3657
7/7/2008
6th 1-5
0.3657 is between 0.365 and __________________.
93
DRAFT
Underline the digit in the thousandths place. Then fill in the blanks.
4.2961
4.2961 is between ________________ and 4.297
Underline the digit in the thousandths place. Then fill in the blanks.
69.0356
69.0356 is between __________________ and __________________.
Underline the digit in the thousandths place. Then fill in the blanks.
11.7806
11.7806 is between 11.78 and __________________.
Underline the digit in the thousandths place. Then fill in the blanks.
589.4535
589.4535 is between __________________ and __________________.
What decimal number is halfway between 0.0 and 1.0?
What decimal number is halfway between 0 and 1?
What decimal number is halfway between 15.0 and 16.0?
What decimal number is halfway between 22 and 23?
What decimal number is halfway between 8.00 and 9.00?
What decimal number is halfway between 0.1 and 0.2?
What decimal number is halfway between 3.5 and 3.6?
What decimal number is halfway between 1.4 and 1.5?
What decimal number is halfway between 44.7 and 44.8?
What decimal number is halfway between 247.9 and 247.0?
What decimal number is halfway between 0.27 and 0.28?
What decimal number is halfway between 9.81 and 9.82?
What decimal number is halfway between 45.04 and 45.05?
What decimal number is halfway between 0.564 and 0.565?
What decimal number is halfway between 9.962 and 9.963?
7/7/2008
6th 1-5
94
DRAFT
What are the two closest whole numbers to 1.5?
What are the two closest whole numbers to 64.5?
What are the two closest whole numbers to 247.5?
What are the two closest whole numbers to 7,560.5?
What are the two closest tenths to 18.27?
What are the two closest tenths to 60.63?
What are the two closest tenths to 523.05?
What are the two closest tenths to 7,297.65?
What are the two closest hundredths to 7.435?
What are the two closest hundredths to 37.895?
What are the two closest hundredths to 494.425?
What are the two closest hundredths to 6,110.055?
What are the two closest thousandths to 7.18635?
What are the two closest thousandths to 96.5422?
What are the two closest thousandths to 581.2105?
What are the two closest thousandths to 9,450.3699?
Round each decimal to the nearest tenth.
7.26
0.35
7/7/2008
6th 1-5
6.62
3.10
120.19
21.526
737.864
973.382
1,471.48
8,356.712
4,789.158
62,233.57
95
DRAFT
Round each decimal to the nearest hundredth.
7.269
0.345
13.653
54.104
180.196
214.5257
2,137.863
9,573.3872
8,471.489
3,356.7138
28,789.1543
69,233.571
19.6645
67.2105
Round each decimal to the nearest thousandths.
6.3321
0.8926
753.1364
682.5408
7,136.9762
6,334.0084
7,169.6455
78,213.6215
36,444.0097
78,354.2083
Round 7.38 to the nearest tenth.
A) 7.3
B) 7.39
Round 3.79 to the nearest tenth.
C) 7.4
D) 7
A) 3.7
B) 3.70
Round 59.35 to the nearest tenth.
C) 3.75
D) 3.8
A) 59.3
B) 59.36
Round 8.28 to the nearest tenth.
C) 59.4
D) 59.04
A) 8.0
B) 8.2
Round 869.72 to the nearest tenth.
C) 8.30
D) 8.20
A) 869.7
B) 869
Round 7,367.09 to the nearest tenth.
C) 869.71
D) 869.8
C) 7,367.01
D) 7,368.1
A) 7.10
7/7/2008
6th 1-5
B) 7,367.1
96
DRAFT
Round 3.274 to the nearest hundredth.
A) 3.20
B) 3.3
Round 1.748 to the nearest hundredth.
C) 3.27
D) 3.28
A) 1.75
B) 1.70
Round 21.305 to the nearest hundredth.
C) 1.74
D) 1.8
A) 21.30
B) 21.31
Round 87.287 to the nearest hundredth.
C) 21.36
D) 21.35
A) 87.28
B) 87.29
Round 683.725 to the nearest hundredth.
C) 87.30
D) 87.288
A) 683.73
B) 683.72
Round 5,665.091 to the nearest hundredth.
C) 683.70
D) 683.726
C) 5,665.10
D) 5,665.092
A) 0.923
B) 0.9235
Round 5.3314 to the nearest thousandths.
C) 0.924
D) 0.9237
A) 5.332
B) 5.3310
Round 42.6507 to the nearest thousandths
C) 5.3315
D) 5.3320
A) 42.651
B) 42.650
Round 33.2945 to the nearest thousandths
C) 42.6506
D) 42.649
A) 33.295
B) 33.294
Round 728.0017 to the nearest thousandths
C) 33.2946
D) 33.2940
A) 728 .001
B) 7 2 8.00 15
Round 8,974.3031 to the nearest thousandths
C) 728.016
D) 728.002
C) 8,974.30
D) 8,974.302
A) 5,665
B) 5,665.09
Round 0.9236 to the nearest thousandths.
A) 8,974.3032
B) 8,974.303
When rounded to the nearest tenth, which of these decimals round to 72?
7.9
72.7
77.9
72.2
71.6
24.21
70.2
When rounded to the nearest hundredth, which of these decimals round to 0.56?
0.566
0.561
0.565
0.05
0.519
0.563
0.569
When rounded to the nearest thousandths, which of these decimals round to 9.450?
9.4457
7/7/2008
6th 1-5
9.4491
9.4511
9.5
9.4495
97
9.4498
9.4494
DRAFT
Write three decimal numbers that round to 0.6 when rounded to the nearest tenth.
Write three decimal numbers that round to 4.2 when rounded to the nearest tenth.
Write three decimal numbers that round to 0.34 when rounded to the nearest hundredth.
Write three decimal numbers that round to 4.69 when rounded to the nearest hundredth.
Write three decimal numbers that round to 781.56 when rounded to the nearest hundredth.
Round each decimal number to the underlined place.
2.16
5.8
9.513
24.94
60.28
89.753
21.648
7.274
38.845
97.2916
6215.354
54.5644
729.0231
80.393
746.7766
236.5272
3,896.0304
52,369.365
9,753.7034
To what place value is 76.2 rounded?
To what place value is 958.03 rounded?
To what place value is 300.868 rounded?
To what place value is 5,297.006 rounded?
7/7/2008
6th 1-5
6.33
98
DRAFT
th
Investigating Student Learning: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard NS 2.0: Students calculate and solve problems involving addition….
Lesson 1.6: Adding Whole Numbers and Decimals
Concepts:
Addition is used to join or put together quantities.
Aligning digits by place value allows us to regroup numbers easily.
When adding multi-digit whole numbers, line up the digits starting with the ones column.
When adding multi-digit whole numbers, start at the ones place.
Add one place or column at a time.
Regroup when the sum in a place or column has two digits or more.
Regroup digits from smaller place values to larger place values so there is just one digit in each place.
When adding money values, line up numbers using the decimal point.
When performing column addition, add place by place and regroup if necessary.
Addition is the inverse operation of subtraction.
Estimating the sum before calculating develops number sense.
Check for reasonableness of an answer by estimating the sum.
Check for accuracy of an answer by subtracting one addend from the sum.
Like whole number addition, decimal addition is used to join or put together decimal quantities.
Aligning digits by place value allows us to regroup numbers easily.
When adding decimal numbers, align numbers using the decimal point.
When adding decimals with a different number of decimal places, use equivalent decimals to write place
holders. By filling in zeros to the right of the decimal point, all the decimal numbers will have the
same number of places making it easier to compute.
4.1
4.1 = 4.10 so
4.10
+ 3.37
+ 3.37
7.47
When adding decimal numbers, start with the smallest place value or the place furthest to the right.
Add one place or column at a time.
Regroup when the sum in a place or column has two digits or more.
Regroup digits from smaller place values to larger place values so there is just one digit in each place.
Rewrite the decimal point in the sum directly under the decimal points in the addends.
Check for reasonableness of an answer by estimating the sum.
Essential Question(s):
How do you add multi-digit numbers?
How do you add decimal numbers?
How do you estimate the sum of whole numbers and decimal numbers?
7/7/2008
6th 1-6
99
DRAFT
th
ISL Item Bank: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard NS 2.0: Students calculate and solve problems involving addition….
Lesson 1.6: Adding Whole Numbers and Decimals
How do you add multi-digit numbers?
Rewrite the addition problem in the place-value chart. Then find the sum.
7/7/2008
6th 1-6
100
Ones
Hundreds
Ones
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
Thousands Period
Ones
Tens
One
Thousands
96,387 + 44,697
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
Thousands Period
Ones Period
Tens
52,695 + 26,327
Ten
Thousands
Hundred
Thousands
Thousands Period
Ones
Tens
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
Thousands Period
74,207 + 1,356
Tens
1,224 + 5,643
7/7/2008
6th 1-6
101
Thousands Period
Ones
Ones
Thousands Period
Tens
Hundreds
One
Thousands
Ten
Thousands
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Ten
Millions
Thousands Period
Tens
Hundreds
One
Thousands
Millions Period
Ten
Thousands
Millions Period
Hundred
Thousands
One
Millions
Ten
Millions
Hundred
Millions
One
Billions
Ten
Billions
Hundred
Billions
Millions Period
Hundred
Thousands
One
Millions
Billions Period
Ten
Millions
Billions Period
Hundred
Millions
One
Billions
Ten
Billions
Hundred
Billions
Billions Period
Hundred
Millions
One
Billions
Ten
Billions
Hundred
Billions
DRAFT
45,765 + 6,230,034
Ones Period
7,249,861, 456 + 43,079
Ones Period
20,378,453 + 88,579,937,003
Ones Period
DRAFT
Rewrite the addition problem in the place-value chart. Then find the sum.
Ones
Hundreds
One
Thousands
Ones
Tens
Ones Period
Hundreds
One
Thousands
Hundred
Thousands
Tens
Ones
102
Ten
Thousands
Thousands Period
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
7/7/2008
6th 1-6
Ones Period
7,253 + 24,021 + 186,747
58 + 3,929 + 94,618
Thousands Period
Ten
Thousands
Hundred
Thousands
Thousands Period
Ones
Tens
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
Thousands Period
314 + 72,613 + 2,068
Tens
1,023 + 34 + 601
7/7/2008
6th 1-6
103
Thousands Period
Ones
Ones
Thousands Period
Tens
Hundreds
One
Thousands
Ten
Thousands
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Ten
Millions
Thousands Period
Tens
Hundreds
One
Thousands
Millions Period
Ten
Thousands
Millions Period
Hundred
Thousands
One
Millions
Ten
Millions
Hundred
Millions
One
Billions
Ten
Billions
Hundred
Billions
Millions Period
Hundred
Thousands
One
Millions
Billions Period
Ten
Millions
Billions Period
Hundred
Millions
One
Billions
Ten
Billions
Hundred
Billions
Billions Period
Hundred
Millions
One
Billions
Ten
Billions
Hundred
Billions
DRAFT
23,043 + 272,635 + 321
Ones Period
4,421 + 39,538 + 7,341,032
Ones Period
76,034,217 + 100,465,983 + 9,268,410,485
Ones Period
DRAFT
Use the graph paper to find the sum.
7/7/2008
6th 1-6
3,016 + 62,971
7,338 + 247
836 + 4,255
8,746 + 51,448
83,965 + 4,047
503,649 + 67,283
2,269,740,658 + 35,336,756
4,796,431,590 + 6,034,589,647
104
DRAFT
Use the graph paper to find the sum.
7/7/2008
6th 1-6
5,215 + 713 + 72,061
34 + 87,302 + 2,472
626 + 4,254 + 71,205
3,746 + 351,448 + 37
981 + 238,415 + 4,062
561,398 + 54,896 + 426,301
48,382 + 20,176,523 + 35,042,103
526 + 369,046,928 + 4,523,136,654
105
DRAFT
Find the sum. Then check your answer by subtracting.
Example:
4,678,209 + 56,826
4,678,209
+
56,826
4,735,035
14
6 2 4 10 2 15
–
270,865 + 336,461
26,489 + 364,027
Check:
+
4,735,035
4,678,209
56,826 (Check √)
–
Check:
+
Check:
346,259 + 4,200,367
Check:
+
–
63,287 + 175,733
–
Check:
+
59,367 + 890,733
Check:
+
–
2,368,256 + 659,014
–
8,360,092 + 5,458,611
Check:
Check:
+
+
–
–
23,632,987 + 69,532,145,587
5,668,235,147 + 82,605,479,854
Check:
+
Check:
+
–
7/7/2008
6th 1-6
–
106
DRAFT
Find the sum.
94,263
+ 3,724
52,257 + 3,534
3,162
+ 5,454
82,356 + 4,354
+
2 82,323
3,654
4,365 + 347 +
860,450
60,321
846,098
+ 2,651
7/7/2008
6th 1-6
262,852
+ 13,036
728,633 + 285
42,133
+ 33,584
71,534 + 27,078
447,701
+ 72,178
32,098 + 902,325
+ 1,089
917,701
52,359
+ 3,178
635,425
+ 322,574
853,417 + 5,641
4 85,267
+ 2,470
53,468 + 322,555
4,868,341
+
1,413
724,228 + 3,634
8,365,322
+ 314,294
654,142,466
+
343,232
476,653 + 20,129
4,113,043
+ 2,624,675
6,321,687 + 2,434 312,754 + 5,248,549
3,016,848
242,050
7,524,665
+ 2,362,333
27,956 + 887 +
615,376
212,987 + 850 +
12,386
894,309 + 5,792 +
3,486
9,253,625
154,943
+ 32,476
37,420,420
43,650
+ 2,332,374
+
367,848
23,548
+ 83,052
107
+
18,420,724
6,532,163
DRAFT
How do you estimate the sum of whole numbers and decimal numbers?
Estimate each sum by rounding the addends to the largest place value of the smaller or smallest number.
Example:
Estimate the sum by rounding.
Estimate the sum by rounding.
78,263 + 8,235
78,000 + 8,000 = 86,000
Estimate the sum by rounding.
Round each addend to the
hundreds place because
757 is the smaller addend
and its largest place value is
the hundreds place.
29,681 + 757
+
520,000 + 60,000 =
Estimate the sum by rounding.
+
Round each addend to the
ten thousands place
because both numbers’
largest place value is the ten
thousands place.
46,311 + 87,631
+
23,468 + 5,478
=
Estimate the sum by rounding.
28,655,479 + 8,026,378
=
+
Estimate the sum by rounding.
Estimate the sum by rounding.
4,936 + 703,569
Round each addend to the
Round each addend to the
________________ place.
=
+
Estimate the sum by rounding.
6,521,308 + 66,802
=
Estimate the sum by rounding.
37,652 + 185,364
Round each addend to the
Round each addend to the
________________ place.
+
7/7/2008
6th 1-6
Round each addend to the
one millions place because
8,026,378 is the smaller
addend and its largest place
value is the one millions
place.
=
________________ place.
+
Round each addend to the
hundred thousands place
because 629,854 is the
smaller addend and its
largest place value is the
hundred thousands place.
629,854 + 4,201,365
=
Estimate the sum by rounding.
Round each addend to the
ten thousands place
because 63,214 is the
smaller addend and its
largest place value is the ten
thousands place.
516,263 + 63,214
Round each addend to the
one thousands place because
8,235 is the smaller addend
and its largest place value is
the one thousands place.
________________ place.
=
+
108
=
DRAFT
Estimate each sum by rounding the addends to the largest place value of the smaller or smallest number.
Estimate the sum by rounding.
862 + 72,036
Estimate the sum by rounding.
Round each addend to the
8,035,146 + 67,322,951
________________ place.
________________ place.
+
=
+
Estimate the sum by rounding.
59,357 + 6,450
=
Estimate the sum by rounding.
Round each addend to the
687,231 + 56,231,147
________________ place.
+
Round each addend to the
________________ place.
=
+
Example:
Estimate the sum by rounding.
Round each addend to the
=
Estimate the sum by rounding.
6,708 + 2,406 + 1,588
361,429 + 862 + 5,489
Round each addend to the
Round each addend to the
____________________
place.
hundrerds
place.
+
+
=
361,400 + 900 + 5,500 = 367,800
Estimate the sum by rounding.
Estimate the sum by rounding.
22,798 + 389 + 4,185
793,115 + 4,342 + 85,000,674
+
+
Round each addend to the
Round each addend to the
____________________
place.
____________________
place.
=
+
+
=
Estimate the sum by rounding.
Estimate the sum by rounding.
27,556 + 80,411+ 7,661
891 + 564,989 + 2,304
+
7/7/2008
6th 1-6
+
Round each addend to the
Round each addend to the
____________________
place.
____________________
place.
=
+
109
+
=
DRAFT
First estimate the sum, and then find the exact answer. Determine if your computed answer is reasonable.
Find the exact sum.
Compare the computed sum to
Example:
the estimate.
Estimate the sum by rounding.
13,686 + 4,218
Is it a reasonable answer? (circle)
13,686
13,686 + 4,218
+
4,218
Yes
(Explain) 17,904 is
17,904
very close to the estimate 18,000.
14,000 + 4,000 = 18,000
No
(Compute again)
+
Estimate the sum by rounding.
Find the exact sum.
26,769 + 3,962
26,769 + 3,962
+
26,769
3,962
Compare the computed sum to
the estimate.
Is it a reasonable answer? (circle)
Yes
(Explain)
No
(Compute again)
+
Estimate the sum by rounding.
Find the exact sum.
523,716 + 42,803
523,716 + 42,803
Compare the computed sum to
the estimate.
Is it a reasonable answer? (circle)
Yes
(Explain)
No
(Compute again)
+
+
7/7/2008
6th 1-6
110
DRAFT
First estimate the sum, and then find the exact answer. Determine if your computed answer is reasonable.
Find the exact sum.
Compare the computed sum to
the estimate.
Estimate the sum by rounding.
60,752 + 861,459
Is it a reasonable answer? (circle)
60,752 + 861,459
Yes
(Explain)
No
(Compute again)
+
+
Estimate the sum by rounding.
Find the exact sum.
9,246,058 + 54,369
9,246,058 + 54,369
Compare the computed sum to
the estimate.
Is it a reasonable answer? (circle)
Yes
(Explain)
No
(Compute again)
+
+
Find the exact sum.
Estimate the sum by rounding.
6,821,034 + 475,369,564
Compare the computed sum to
the estimate.
Is it a reasonable answer? (circle)
6,821,034 + 475,369,564
Yes
(Explain)
No
(Compute again)
+
+
7/7/2008
6th 1-6
111
7/7/2008
6th 1-6
112
Hundredthousandths
Ten-thousandths
Ones
Period
Hundredthousandths
Ten-thousandths
One-thousandths
Hundredths
Tenths
Ones
Period
One-thousandths
Hundredths
Tenths
755.6234 + 86
Ones
0.5 + 5.69
Ones
Decimals
Tens
Hundreds
Hundredthousandths
Ten-thousandths
Decimals
Tens
Hundreds
Hundredthousandths
Ones
Period
Ten-thousandths
Ones
Period
One-thousandths
Hundredths
Tenths
Ones
Tens
Hundreds
Hundredthousandths
Ten-thousandths
One-thousandths
Hundredths
Tenths
Ones
Tens
Hundreds
Ones
Period
One-thousandths
Hundredths
Tenths
Ones
Tens
Hundreds
DRAFT
How do you add decimal numbers?
Rewrite the subtraction problem in the place-value chart. Line up the decimals. Write zeros as placeholders
if necessary.
Example:
6.453 + 0.2
Decimals
0.2 = 0.200
6 4 5 3
0 2 0 0
1.36 + 49.2587
Decimals
0.69877 + 356.4
Decimals
7/7/2008
6th 1-6
113
Ones
Hundredthousandths
Hundredthousandths
Decimals
Tenthousandths
26.7 + 8.462
Tenthousandths
Decimals
Onethousandths
Hundredthousandths
Tenthousandths
Onethousandths
Hundrredths
Tenths
Ones
Hundredthousandths
Tenthousandths
Onethousandths
Hundrredths
Tenths
Ones
Decimals
Onethousandths
Hundrredths
Tenths
Ones
Hundredthousandths
Tenthousandths
Onethousandths
Hundrredths
0.0248 + 0.7863
Hundredths
Tenths
Thousands
Ones
Tens
Hundreds
One
Thousands
Millions
Ten
Thousands
Hundred
Thousands
One
Millions
Tenths
Ones
DRAFT
Rewrite the addition problem in the place-value chart. Then find the sum.
0.586 + 0.89
0.6 + 0.306
Decimals
0.30068 + 0.9658
Decimals
7/7/2008
6th 1-6
114
Hundredthousandths
Ones
Tenthousandths
Hundredthousandths
Tenthousandths
Ones
Onethousandths
Hundredths
Tenths
Hundredthousandths
Tenthousandths
Onethousandths
Hundredths
Tenths
Ones
Tens
Hundreds
One
Thousands
Ones
Onethousandths
Hundredths
Thousands
Tenths
Thousands
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Thousands
Ones
Tens
Hundreds
Millions
One
Thousands
Millions
Ten
Thousands
Hundred
Thousands
One
Millions
Millions
Ten
Thousands
Hundred
Thousands
One
Millions
DRAFT
Rewrite the addition problem in the place-value chart. Then find the sum.
50.0692 + 0.574
Decimals
6,784.006 + 9.45
Decimals
32.69114 + 364.357
Decimals
7/7/2008
6th 1-6
115
Hundredthousandths
Tenthousandths
Decimals
Hundredthousandths
Tenthousandths
Onethousandths
Hundrredths
Tenths
Ones
Hundredthousandths
Tenthousandths
Onethousandths
Hundrredths
Tenths
Ones
Decimals
Onethousandths
0.93675 + 0.075 + 0.46
Hundredths
Tenths
Ones
Hundredthousandths
Tenthousandths
Onethousandths
Hundredths
Tenths
Ones
DRAFT
Rewrite the addition problem in the place-value chart. Then find the sum.
0.08 + 0.496 + 0.2
0.4398 + 0.6 + 0.703
Decimals
0.2007 + 0.075 + 0.46
Decimals
7/7/2008
6th 1-6
116
Hundredthousandths
Ones
Tenthousandths
Hundredthousandths
Tenthousandths
Ones
Onethousandths
Hundredths
Tenths
Hundredthousandths
Tenthousandths
Onethousandths
Hundredths
Tenths
Ones
Tens
Hundreds
One
Thousands
Ones
Onethousandths
Hundredths
Thousands
Tenths
Thousands
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Thousands
Ones
Tens
Hundreds
Millions
One
Thousands
Millions
Ten
Thousands
Hundred
Thousands
One
Millions
Millions
Ten
Thousands
Hundred
Thousands
One
Millions
DRAFT
Rewrite the addition problem in the place-value chart. Then find the sum.
79.2063 + 856.2 + 0.48
Decimals
2.796 + 842,190 + 57.0079
Decimals
756 + 0.97846 + 43,752.558
Decimals
DRAFT
Use the graph paper to find the sum.
7/7/2008
6th 1-6
25.364 + 0.405
3.5698 + 567
8.6 + 7,951.078
963.485 + 5.34
0.6811 + 4,047.3
569.23781 + 2.445
564.2103 + 27,479.48
0.34567 + 59,256
117
DRAFT
Use the graph paper to find the sum.
7/7/2008
6th 1-6
29.3 + 56.789 + 789.06
75 + 0.4587 + 6.489
0.368 + 8.75 + 299.4
867.036 + 4.008 + 5,331.7
8,236 + 66.398 + 4.1887
32.654 + 94.785 + 7,645
85.369 + 2.0008 + 78.12364
9 + 7.69543 + 26,789
118
DRAFT
Find the sum.
23.89
+ 5.60
798.420
+ 547.436
1.8906
+ 0.5600
956,126.54
+
24.36
+
5,521.780
683.265
758.63 + 46.591
56.79 + 5.96
1,234.6 + 569.5
879.364 + 0.598
1.5587 + 33.5624
26.458 + 5.6
789.4 + 3.479
3,560.76+ 2.8963
824 + 9.655
0.3374 + 89
45.336
879.200
+ 0.585
4,367.48
64.80
+ 997.34
65,234.332
75.300
+ 4,288.017
0.69365
259.00000
+
5.88000
2,377.6 + 5.89 +
733.4
7/7/2008
6th 1-6
826 + 4.31 +
95.282
3.7400
486.2372
+ 81.0050
0.346 + 0.0058 +
667
119
34,000,561.2 +
5.48 + 762.008
500 + 89,000.038
+ 0.9
DRAFT
How do you estimate the sum of whole numbers and decimal numbers?
Estimate each sum by rounding the decimal numbers to the nearest whole number, if any of the addends has
a whole number. If there are no whole numbers in the addends, estimate the decimal numbers by rounding
the addends to the largest place value of the smaller or smallest number.
Example:
Example:
Estimate the sum by rounding.
Estimate the sum by rounding.
Round each addend to the
0.345 + 0.0784
863.281 + 56.42
hundredths place because
Round to the
0.0784 is the smaller
addend and its largest place
value is the hundredths
place.
nearest whole
number
0.35 + 0.08 = 0.43
863 + 56 = 919
Estimate the sum by rounding.
2,459.3 + 7.564
Estimate the sum by rounding.
0.0478 + 0.00296
Round to the
nearest whole
number
0.048 + 0.003 =
2,459 + 8 =
Estimate the sum by rounding.
Estimate the sum by rounding.
Round each addend to the
4,368.4 + 24.061
+
Round each addend to the
thousandths place because
0.0029 is the smaller
addend and its largest place
value is the thousandths
place.
________________ place.
=
+
Estimate the sum by rounding.
________________ place.
=
Estimate the sum by rounding.
Round each addend to the
87,365.4 + 644.398
Round each addend to the
0.03004 + 0.561
Round each addend to the
0.065 + 0.008
________________ place.
+
________________ place.
=
+
Estimate the sum by rounding.
Estimate the sum by rounding.
Round each addend to the
2.64987 + 45.921
=
Round each addend to the
0.493 + 0.0879
________________ place.
+
7/7/2008
6th 1-6
________________ place.
=
+
120
=
DRAFT
Estimate each sum by rounding the decimal numbers to the nearest whole number.
Estimate the sum by rounding.
Estimate the sum by rounding.
42.3 + 3.06 + 541
+
0.51 + 0.023 + 0.8
+
Round each addend to the
Round each addend to the
____________________
place.
____________________
place.
=
+
+
=
Estimate the sum by rounding.
Estimate the sum by rounding.
357.28 + 5,964.3 + 2.586
0.07 + 0.85 + 0.0417
+
+
Round each addend to the
Round each addend to the
____________________
place.
____________________
place.
=
+
+
=
Estimate the sum by rounding.
Estimate the sum by rounding.
3,861.24 + 5.67 + 346.8
0.006 + 0.0589 + 0.00097
Round each addend to the
Round each addend to the
____________________
place.
+
+
____________________
place.
=
+
+
=
Estimate the sum by rounding.
Estimate the sum by rounding.
2.1169 + 58.96 + 500.32
0.74 + 0.009 + 0.03114
Round each addend to the
Round each addend to the
____________________
place.
____________________
place.
+
+
+
+
=
=
Estimate the sum by rounding.
Estimate the sum by rounding.
33,568.6 + 9.465 + 30.254
0.08632 + 0.04 + 0.2978
Round each addend to the
____________________
place.
____________________
place.
+
7/7/2008
6th 1-6
+
Round each addend to the
+
=
121
+
=
DRAFT
Use estimation to determine if the sum is reasonable.
Example:
Is the underlined answer reasonable?
Is the underlined answer reasonable?
36.21 + 7.526 = 43.736
2.39 + 7.1 = 2.462
(estimate)
(estimate)
+
2 + 7 = 9
(circle)
A) reasonable
A) reasonable
B) not reasonable
Is the underlined answer reasonable?
Is the underlined answer reasonable?
5.967 + 8.0126 = 13.9796
9.423 + 71.8 = 10.141
(estimate)
(estimate)
+
=
(circle)
+
=
(circle)
A) reasonable
B) not reasonable
Explain:
A) reasonable
B) not reasonable
Explain:
Is the underlined answer reasonable?
Is the underlined answer reasonable?
0.0258 + 0.366 = 0.3918
0.0478 + 0.0069 = 0.0547
(estimate)
(estimate)
+
=
(circle)
+
=
(circle)
A) reasonable
B) not reasonable
Explain:
A) reasonable
B) not reasonable
Explain:
Is the underlined answer reasonable?
Is the underlined answer reasonable?
0.00729 + 0.117 = 0.00846
0.603 + 0.0584 = 0.6614
(estimate)
(estimate)
+
=
(circle)
A) reasonable
7/7/2008
6th 1-6
B) not reasonable
Explain:
Explain:
Explain:
=
(circle)
+
=
(circle)
B) not reasonable
A) reasonable
Explain:
122
B) not reasonable
DRAFT
First estimate the sum, and then find the exact answer. Determine if your computed answer is reasonable.
Find the exact sum.
Compare the computed sum to
Example:
the estimate.
Estimate the sum by rounding.
23.91 + 5.671
Is it a reasonable answer? (circle)
23.91
23.91 + 5.671
+
5.671
Yes
(Explain)
8.062
24 + 6 = 30
No
(Compute again)
23.91 + 5.671
23.9= 23.910
23.910
5.671
29.581
+
Compare the computed sum to
the estimate.
Find the exact sum.
Estimate the sum by rounding.
366.47 + 5.978
366.47 = 366.470
366.47 + 5.978
+
366.470
5.978
YES!
Is it a reasonable answer? (circle)
Yes
(Explain)
No
(Compute again)
+
Find the exact sum.
Estimate the sum by rounding.
72,563.34 + 809.7
Compare the computed sum to
the estimate.
Is it a reasonable answer? (circle)
72,563.34 + 809.7
Yes
(Explain)
No
(Compute again)
+
+
7/7/2008
6th 1-6
123
Estimate the sum by rounding.
Find the exact sum.
0.589 + 0.04511
0.589 + 0.04511
DRAFT
Compare the computed sum to
the estimate.
Is it a reasonable answer? (circle)
Yes
(Explain)
No
(Compute again)
+
+
Estimate the sum by rounding.
Find the exact sum.
0.03089 + 0.7
0.03089 + 0.7
Compare the computed sum to
the estimate.
Is it a reasonable answer? (circle)
Yes
(Explain)
No
(Compute again)
+
+
Estimate the sum by rounding.
Find the exact sum.
0.00634 + 0.045
0.00634 + 0.045
Compare the computed sum to
the estimate.
Is it a reasonable answer? (circle)
Yes
(Explain)
No
(Compute again)
+
+
7/7/2008
6th 1-6
124
DRAFT
th
Investigating Student Learning: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard NS 2.0: Students calculate and solve problems involving …subtraction…..
Lesson 1.7: Subtracting Whole Numbers and Decimals
Concepts:
Subtraction is used to separate or compare quantities.
Numbers are aligned according to place value so that corresponding place value digits are subtracted.
When subtracting whole numbers, start at the ones place.
Subtract one place or column at a time.
Regroup when trying to subtract a larger digit from a smaller digit.
Regroup from a greater place value to a smaller place value.
When subtracting money values, line up numbers using the decimal.
Subtraction is the inverse operation of addition.
Check for reasonableness of an answer by estimating the difference.
Check for accuracy of an answer by adding the answer (difference) to the number subtracted (subtrahend).
The result should be the beginning number you subtracted from (minuend).
Like whole number subtraction, decimal subtraction is used to separate or compare decimal quantities.
Numbers are aligned according to place value so that corresponding place value digits are subtracted.
When subtracting decimal numbers, align numbers using the decimal point.
When subtracting decimals with a different number of decimal places, use equivalent decimals to write
place holders. By filling in zeros to the right of the decimal point, all the decimal numbers will have
the same number of places making it easier to compute.
4.1
- 3.37
4.1 = 4.10 so
4.10
- 3.37
0.73
Rewrite the decimal point in the difference directly under the decimal points in the minuend and subtrahend.
Estimation can be done by rounding two-place decimals to one decimal or the nearest whole number.
Check for reasonableness of an answer by estimating the difference.
Essential Question(s):
How do you subtract multi-digit numbers?
How do you subtract decimal numbers?
How do you estimate the difference of whole numbers and decimal numbers?
7/7/2008
6th 1-7
125
DRAFT
th
ISL Item Bank: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard NS 2.0: Students calculate and solve problems involving …subtraction…..
Lesson 1.7: Subtracting Whole Numbers and Decimals
How do you subtract multi-digit numbers?
Rewrite the subtraction problem in the place-value chart. Then find the difference.
8,974 – 5,643
64,480 – 1,356
7/7/2008
6th 1-7
Ones
Tens
Hundreds
One
Thousands
42,107 – 17,218
126
Ones
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
Thousands Period
Ones
Tens
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
Thousands Period
Ones Period
Tens
72,395 – 26,327
Ten
Thousands
Ones
Tens
Thousands Period
Hundred
Thousands
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
Thousands Period
DRAFT
Rewrite the subtraction problem in the place-value chart. Then find the difference.
10,294 – 8,396
70,200 – 31,524
7/7/2008
6th 1-7
Ones
Tens
Hundreds
One
Thousands
91,007 – 42,197
127
Ones
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
Thousands Period
Ones
Tens
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
Thousands Period
Ones Period
Tens
83,289 – 75,729
Ten
Thousands
Ones
Tens
Thousands Period
Hundred
Thousands
Ones Period
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
Thousands Period
7/7/2008
6th 1-7
128
Thousands Period
Ones
Ones
Thousands Period
Tens
Hundreds
One
Thousands
Ten
Thousands
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Ten
Millions
Thousands Period
Tens
Hundreds
One
Thousands
Millions Period
Ten
Thousands
Millions Period
Hundred
Thousands
One
Millions
Ten
Millions
Hundred
Millions
One
Billions
Ten
Billions
Hundred
Billions
Millions Period
Hundred
Thousands
One
Millions
Billions Period
Ten
Millions
Billions Period
Hundred
Millions
One
Billions
Ten
Billions
Hundred
Billions
Billions Period
Hundred
Millions
One
Billions
Ten
Billions
Hundred
Billions
DRAFT
Rewrite the subtraction problem in the place-value chart. Then find the difference.
1,645,765 – 230,034
Ones Period
3,249,821,456 – 143,079
Ones Period
214,820,100,453 – 88,579,937,003
Ones Period
DRAFT
Use the graph paper to find the difference.
3,516 – 2,415
7/7/2008
6th 1-7
9,338 – 236
8,836 – 3,247
7,746 – 5,448
93,029 – 4,047
800,649 – 76,283
52,340,675 – 95,231
90,456,353 – 29,369,145
129
DRAFT
Use the graph paper to find the difference.
564,369,429 – 5,874,632
7/7/2008
6th 1-7
702,362,169 – 33,472,825
430,028,065 – 278,964
861,304,215 – 772,405,306
3,269,740,658 – 35,330,456
4,794,436,590 – 936,281,647
6,328,654,953 – 753,219,654
5,408,300,759 – 4,369,211,675
130
DRAFT
Find the difference. Then check your answer by adding.
Example:
752,604 – 42,913
75,239 – 3,017
Check:
Check:
11 15
4 1 5 10
–
752,604
42,913
709,691
–
1 1 1
+
819,247 – 46,128
+
709,691
42,913
752,604 (Check √)
Check:
–
973,064 – 20,287
Check:
–
+
9,267,154 – 538,249
+
Check:
–
6,367,200 – 956,124
Check:
–
+
46,369,456 – 7,335,109
+
Check:
–
33,269,874 – 1,498,928
Check:
–
+
56,002,007 – 9,132,458
+
Check:
–
862,139,080,364 – 7,369,357,045
Check:
–
+
7/7/2008
6th 1-7
+
131
DRAFT
Find the difference.
94,785
– 2,724
57,657 – 3,538
8,162
– 5,454
82,362 – 4,354
2 87,323
3,654
–
4,105 – 347
60,320
– 2,651
7/7/2008
6th 1-7
264,857
– 43,036
728,197 – 285
642,585
– 522,574
856,717 – 5,641
47,136
– 33,584
79,534 – 27,078
447,701
– 72,178
32,008 – 902,325
900,701
– 3,178
4 81,567
– 2,470
53,468 – 325,555
–
3,066,848
247,050
27,050 – 9,887
363,008
– 83,052
132
3,861,574
21,433
758,945,466
7343,232
–
–
720,748 – 3,634
476,653 – 20,129
8,365,322
– 314,294
4,817,097
– 2,624,675
6,325,287 – 2,438 4,359,734 – 248,549
7,524,665
– 2,367,383
210,907 – 3,858
9,050,505
– 382,476
–
18,770,724
6,532,963
894,000 – 75,792
–
31,420,420
2,332,374
DRAFT
How do you estimate the difference of whole numbers and decimal numbers?
Estimate each difference by rounding the minuend and subtrahend to the largest place value of the
subtrahend (smaller number).
Example:
Estimate the difference by rounding.
Estimate the difference by rounding.
78,263 – 8,235
78,000 – 8,000 = 70,000
Estimate the difference by rounding.
Round each number to the
hundreds place because the
subtrahend 857 is the
smaller number and its
largest place value is the
hundreds place.
26,681 – 857
–
580,000 – 60,000 =
Estimate the difference by rounding.
–
Round each number to the ten
thousands place because the
subtrahend 87,631 is the
smaller number and its largest
place value is the ten thousands
place.
96,311 – 87,631
–
=
Estimate the difference by rounding.
3,328,655,479 – 2,126,378
=
–
Estimate the difference by rounding.
23,468 – 5,478
Round each number to the
hundred thousands place
because the subtrahend
201,365 is the smaller number
and its largest place value is the
hundred thousands place.
429,854 – 201,365
=
Estimate the difference by rounding.
Round each number to the ten
thousands place because the
subtrahend 63,214 is the
smaller number and its largest
place value is the ten thousands
place.
576,263 – 63,214
Round each number to the one
thousands place because the
subtrahend 8,235 is the smaller
number and its largest place value
is the one thousands place.
=
Estimate the difference by rounding.
841,936 – 703,569
Round each number to the
________________ place.
–
66,521,308 – 66,812
–
=
Estimate the difference by rounding.
533,052 – 185,364
Round each number to the
________________ place.
–
7/7/2008
6th 1-7
Round each number to the
________________ place.
=
Estimate the difference by rounding.
Round each number to the
one millions place because
the subtrahend 2,026,378 is
the smaller number and its
largest place value is the
one millions place.
Round each number to the
________________ place.
=
–
133
=
DRAFT
First estimate the difference, and then find the exact answer. Determine if your computed answer is
reasonable.
Find the exact difference.
Compare the computed
Example:
difference to the estimate.
Estimate the difference by
13,686 – 4,218
rounding.
Is it a reasonable answer? (circle)
13 7 16
13,686
Yes
(Explain) 9,468 is
13,686 – 4,218
–
4,218
close
to
the estimate 10,000.
9,468
No
14,000 – 4,000 = 10,000
(Compute again)
–
Estimate the difference by
rounding.
Find the exact difference.
26,769 – 3,962
26,769
– 3,962
26,769 – 3,962
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
Yes
(Explain)
No
(Compute again)
–
Estimate the difference by
rounding.
Find the exact difference.
523,712 – 42,803
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
523,712 – 42,803
Yes
(Explain)
No
(Compute again)
–
–
7/7/2008
6th 1-7
134
DRAFT
First estimate the difference, and then find the exact answer. Determine if your computed answer is
reasonable.
Find the exact difference.
Compare the computed
Estimate the difference by
difference to the estimate.
618,294 – 56,378
rounding.
Is it a reasonable answer? (circle)
618,294 – 56,378
Yes
(Explain)
–
No
(Compute again)
–
Estimate the difference by
rounding.
Find the exact difference.
4,366,169 – 459,375
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
4,366,169 – 459,375
–
Yes
(Explain)
No
(Compute again)
–
Estimate the difference by
rounding.
Find the exact difference.
72,060,358 – 49,132,499
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
72,060,358 – 49,132,499
Yes
(Explain)
No
(Compute again)
–
–
7/7/2008
6th 1-7
135
Estimate the difference by
rounding.
Find the exact difference.
8,200,356 – 6,345,725
DRAFT
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
8,200,356 – 6,345,725
–
Yes
(Explain)
No
(Compute again)
–
Estimate the difference by
rounding.
Find the exact difference.
546,360,102 – 8,297,614
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
546,360,102 – 8,297,614
Yes
(Explain)
No
(Compute again)
–
–
Estimate the difference by
rounding.
Find the exact difference.
54,020,300 – 6,124,317
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
54,020,300 – 6,124,317
Yes
(Explain)
No
(Compute again)
–
–
7/7/2008
6th 1-7
136
7/7/2008
6th 1-7
137
Hundredthousandths
Ten-thousandths
Ones
Period
Hundredthousandths
Ten-thousandths
One-thousandths
Hundredths
Tenths
Ones
Period
One-thousandths
Hundredths
Tenths
755.6234 – 86
Ones
8.5 – 5.69
Ones
Decimals
Tens
Hundreds
Hundredthousandths
Ten-thousandths
Decimals
Tens
Hundreds
Hundredthousandths
Ones
Period
Ten-thousandths
Ones
Period
One-thousandths
Hundredths
Tenths
Ones
Tens
Hundreds
Hundredthousandths
Ten-thousandths
One-thousandths
Hundredths
Tenths
Ones
Tens
Hundreds
Ones
Period
One-thousandths
Hundredths
Tenths
Ones
Tens
Hundreds
DRAFT
How do you subtract decimal numbers?
Rewrite the subtraction problem in the place-value chart. Line up the decimals. Write zeros as placeholders
if necessary.
Example:
6.453 – 0.2
Decimals
0.2 = 0.200
6 4 5 3
0 2 0 0
21.36 – 9.2587
Decimals
200.69877 – 156.4
Decimals
7/7/2008
6th 1-7
138
Ones
Hundredthousandths
Hundredthousandths
Decimals
Tenthousandths
26.7 – 8.462
Tenthousandths
Decimals
Onethousandths
Hundredthousandths
Tenthousandths
Onethousandths
Hundredths
Tenths
Ones
Hundredthousandths
Tenthousandths
Onethousandths
Hundredths
Tenths
Ones
Decimals
Onethousandths
Hundredths
Tenths
Ones
Hundredthousandths
Tenthousandths
Onethousandths
Hundredths
0.824 – 0.7863
Hundredths
Tenths
Thousands
Ones
Tens
Hundreds
One
Thousands
Millions
Ten
Thousands
Hundred
Thousands
One
Millions
Tenths
Ones
DRAFT
Rewrite the subtraction problem in the place-value chart. Then find the difference.
0.996 – 0.89
0.6 – 0.306
Decimals
0.30068 – 0.2658
Decimals
7/7/2008
6th 1-7
139
Hundredthousandths
Ones
Tenthousandths
Hundredthousandths
Tenthousandths
Ones
Onethousandths
Hundredths
Tenths
Hundredthousandths
Tenthousandths
Onethousandths
Hundredths
Tenths
Ones
Tens
Hundreds
One
Thousands
Ones
Onethousandths
Hundredths
Thousands
Tenths
Thousands
Ones
Tens
Hundreds
One
Thousands
Ten
Thousands
Hundred
Thousands
One
Millions
Thousands
Ones
Tens
Hundreds
Millions
One
Thousands
Millions
Ten
Thousands
Hundred
Thousands
One
Millions
Millions
Ten
Thousands
Hundred
Thousands
One
Millions
DRAFT
Rewrite the addition problem in the place-value chart. Then find the sum.
50.0692 – 0.574
Decimals
6,784.006 – 9.45
Decimals
32.69114 – 24.357
Decimals
DRAFT
Use the graph paper to find the difference.
25.364 – 0.405
7/7/2008
6th 1-7
935 – 567.56
811.6 – 79.078
963.485 – 5.34
0.6811 – 0.00475
569.23781 – 2.445
27,564.2103 – 479.48
34,567 – 0.59256
140
DRAFT
Find the difference..
23.89
– 5.60
798.420
– 547.436
1.8906
– 0.5600
956,126.54
–
24.36
5,521.780
–
683.265
758.63 – 46.591
56.79 – 5.96
1,234.6 – 569.5
879.364 – 0.598
155.87 – 33.5624
26.458 – 5.6
789.4 – 3.479
3,560.76 – 2.8963
824 – 9.655
0.3374 – 0.89
45.336
– 0.585
4,367.48
– 997.34
–
486.2372
81.0050
65,234.332
– 4,288.017
826 – 95.282
0.346 – 0.0058
34,000,561.2 – 5.48
2,377.6 – 5.89
7/7/2008
6th 1-7
141
–
259.00000
5.88003
500 – 0.9
DRAFT
How do you estimate the difference of whole numbers and decimal numbers?
Estimate each difference by rounding the decimal numbers to the nearest whole number, if the subtrahend
has a whole number. If there is no whole number in the subtrahend, estimate the decimal numbers by
rounding the numbers to the largest place value of the subtrahend (smaller number).
Example:
Example:
Estimate the difference by rounding.
Estimate the difference by rounding.
Round each nuber to the
0.345 – 0.0784
863.281 – 56.42
hundredths place because
Round to the
0.0784 is the subtrahend
(smaller number) and its
largest place value is the
hundredths place.
nearest whole
number
0.35 – 0.08 = 0.27
863 – 56 = 807
Estimate the difference by rounding.
2,459.3 – 7.564
Estimate the difference by rounding.
0.0478 – 0.00296
Round to the
nearest whole
number
0.048 – 0.003 =
2,459 – 8 =
Estimate the difference by rounding.
Estimate the difference by rounding.
Round each number to the
4,368.4 – 24.061
–
Round each number to the
thousandths place because
0.0029 is the subtrahend
(smaller number) and its
largest place value is the
thousandths place.
________________ place.
=
–
Estimate the difference by rounding.
________________ place.
=
Estimate the difference by rounding.
Round each number to the
87,365.4 – 644.398
Round each number to the
0.53004 – 0.061
0.00065 – 0.00008
________________ place.
–
________________ place.
=
–
Estimate the difference by rounding.
=
Estimate the difference by rounding.
Round each number to the
92.64987 – 45.921
Round each number to the
0.493 – 0.00879
________________ place.
–
7/7/2008
6th 1-7
Round each number to the
________________ place.
=
–
142
=
DRAFT
Use estimation to determine if the difference is reasonable.
Example:
Is the underlined answer reasonable?
Is the underlined answer reasonable?
36.21 – 7.526 = 4.095
8.39 – 4.1 = 79.8
(estimate)
(estimate)
–
8 – 4 = 5
(circle)
A) reasonable
A) reasonable
B) not reasonable
Explain: 79 ones is not close to 5 ones
Is the underlined answer reasonable?
15.967 – 8.0126 = 7.9544
469.423 – 70.8 = 468.715
(estimate)
(estimate)
–
=
(circle)
–
=
(circle)
A) reasonable
B) not reasonable
Explain:
A) reasonable
B) not reasonable
Explain:
Is the underlined answer reasonable?
Is the underlined answer reasonable?
0.525 – 0.0366 = 0.4884
0.0478 – 0.0069 = 0.0409
(estimate)
(estimate)
–
=
(circle)
–
=
(circle)
A) reasonable
B) not reasonable
Explain:
A) reasonable
B) not reasonable
Explain:
Is the underlined answer reasonable?
Is the underlined answer reasonable?
0.117 – 0.00729 = 0.612
0.603 – 0.0584 = 0.5446
(estimate)
(estimate)
–
=
(circle)
A) reasonable
7/7/2008
6th 1-7
B) not reasonable
Explain:
Is the underlined answer reasonable?
Explain:
=
(circle)
–
=
(circle)
B) not reasonable
A) reasonable
Explain:
143
B) not reasonable
DRAFT
First estimate the difference, and then find the exact answer. Determine if your computed answer is
reasonable.
Find the exact difference.
Compare the computed
Example:
difference to the estimate.
Estimate the difference by
23.91 – 5.671
rounding.
Is it a reasonable answer? (circle)
1 13
23.91
Yes
(Explain)
23.91 – 5.671
–
5.671
4.720
No
24 – 6 = 18
(Compute again)
23.91 – 5.671
–
Find the exact difference.
Estimate the difference by
rounding.
366.47 – 5.978
366.47 – 5.978
366.47 = 366.470
–
366.470
5.978
1 13 8 10 10
23.9 = 23.910
23.910
5.671
18.239
YES!
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
Yes
(Explain)
No
(Compute again)
–
Find the exact difference.
Estimate the difference by
rounding.
72,563.34 – 809.7
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
72,563.34 – 809.7
Yes
(Explain)
No
(Compute again)
–
–
7/7/2008
6th 1-7
144
Estimate the difference by
rounding.
Find the exact difference.
0.589 – 0.04511
DRAFT
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
0.589 – 0.04511
Yes
(Explain)
No
(Compute again)
–
–
Estimate the difference by
rounding.
Find the exact difference.
0.93089 – 0.7
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
0.93089 – 0.7
Yes
(Explain)
No
(Compute again)
–
–
Estimate the difference by
rounding.
Find the exact difference.
0.08634 – 0.045
Compare the computed
difference to the estimate.
Is it a reasonable answer? (circle)
0.08634 – 0.045
Yes
(Explain)
No
(Compute again)
–
–
7/7/2008
6th 1-7
145
Investigating Student Learning: 6th Grade
Chapter 1: Whole Numbers and Decimals
DRAFT
Standard AF 1.4 Solve problems manually by using the correct order of operations
or by using a scientific calculator.
Lesson 1.9: Order of Operations
Concepts:
Mathematicians use the same rules to evaluate expressions to make sure everyone gets the same
answer to a problem. The set of rules are called order of operations.
Order of operations tells the order in which you compute operations in a numerical expression.
Add, subtract, multiply, divide, exponentiation, and grouping are considered mathematical operations.
Parentheses ( ) and brackets [ ] are used to group terms and are sometimes called grouping symbols.
Grouping symbols are evaluated first.
If there is more than one grouping symbol (e.g., parentheses and brackets), you work from the
innermost grouping symbol first, then work on the next grouping symbol.
Powers are evaluated before the four basic operations.
Multiplication and division are evaluated before addition and subtraction.
If an expression has more than one multiplication, more than one division, or a combination of
multiplications and divisions, perform operations in left-to-right order.
Addition and subtraction operations are performed last.
If an expression has more than one addition, more the one subtraction, or a combination of addition
and subtraction, perform operations in left-to-right order.
Essential Questions(s):
How do you use order of operations to evaluate expressions?
7/7/2008
6th 1-9
146
DRAFT
ISL Item Bank: 6th Grade
Chapter 1: Whole Numbers and Decimals
Standard AF 1.4 Solve problems manually by using the correct order of operations
or by using a scientific calculator.
Lesson 1.9: Order of Operations
How do you use order of operations to evaluate expressions?
List the operation(s) to be performed in each of the following numerical expressions. Use
multiplication, addition, subtraction, division, exponentiation, or grouping.
Expression
Operation(s)
Expression
Operation(s)
6x4
Multiplication
62
Exponentiation
12 ÷ 4
(5 + 3)
(25 – 13)
12 • 6
(5)3
–6(10)
48
6
(4 x 3)
74
(17)(2)
(9 • 9)
22 – 4
152
2
3
–5 • 8
−
7/7/2008
6th 1-9
8(–9)
9
13
( 28 ÷ 4 )
6 ÷ 17
( 73 )
17 + 4
147
DRAFT
List the operation(s) to be performed in each of the following numerical expressions.
Expression
7/7/2008
6th 1-9
Operation(s)
Expression
4+x
a2
24y
24
z
15 – c
2(b)
a
12
b–7
5•x
s4
n + (–4)
–5+y
(m + 9)
( x ÷9 )
t 10
(a)3
–3 – x
x+y
x
4
(c ) 3
148
Operation(s)
DRAFT
A common technique for remembering the order of operations is the abbreviation "PEMDAS", which is
turned into the phrase "Please Excuse My Dear Aunt Sally".
It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". This
tells you the order of the operations:
Parentheses outrank exponents, which outrank multiplication and division (but multiplication and
division are at the same rank), and these two outrank addition and subtraction (which are together on
the bottom rank). When you have a bunch of operations of the same rank, you just operate from left to
right. For instance, 15 ÷ 3 × 4 is not 15 ÷ 12, but is rather 5 × 4, because, going from left to right, you
get to the division first.
Fill in the blanks. Use PEMDAS to help you decide what operation to perform first.
Expression
List the operations as they appear left To simplify the expression, what
to right
operation do you perform first?
Addition
3+6x4
Multiplication
Multiplication
5 + (6 – 3)
Add
Parentheses (Grouping)
Subtraction
Exponentiation
7 + 24 – 1
35 – 4 + 2
16 ÷ 4 x 2
(4 – 8) +1
Parentheses (Grouping) - Subtraction
9 x (4 ÷ 2)
(5 – 3 ) + ( 2 x 4)
5 • 7 + 13
7/7/2008
6th 1-9
149
DRAFT
Fill in the blanks. Use PEMDAS to help you decide which operation to perform first.
Expression
List the operations as they appear left To simplify the expression, what
to right
operation do you perform first?
4(7 – 6) + 3
12 • 3 – 6 • 3
8 + 5 • (4 ÷ 2)
6 + 32 – 9 ÷ 3
14 ÷ (5 + 2) x 3
(4 − 2) 3 −5
5 2 −(32 − 1 2 ) + 2
48 + 7 2 ÷49 • 2
100 ÷ 52 x (3 + 2)
3 • 12 – 6 ÷ 2
7/7/2008
6th 1-9
150
DRAFT
Place a 1 above the operation which should be completed first. Then place a 2 above the operation
which should be completed second. Continue placing numbers above the operation signs according to
the Order of Operations.
Example:
7/7/2008
6th 1-9
2 1
3
6 x (4 – 1) ÷ 3
(100 + 50) ÷ 5 – (2 x 9) + 3
64 + 48 ÷ (9 – 15 ) x 3
(9 – 2) x 8 + 20 ÷ 5
22 x (32 ÷ 4) ÷ (17 + 2) – 6
6 x (7 + 32 ÷ 4) – 25
7 + 42 ÷ (3 x 2) – 2
(8 x 8 ÷ 22 – 3 x 4) x 7 + 2
28 – 7 x (4 ÷ 2) + 1
(4 + 28) ÷ (6 x 7) – 5
2 + 14 ÷ (9 – 2) x 3 2
(13 – 5) x (9 ÷ 3) + 41
(38 + 6) ÷ 4 – 3 x 3
5 + (28 + 18 ÷ 6 – 9) x 4
(91 + 63) ÷ 7 2 – 7 x 8
(8 x 3 ÷ 3 – 12 x 5) x 20 + 7
10 x (2 + 23) ÷ (3 + 7) – 4
(15 + 30 ÷ 6) – (1 x 7) + 99
(54 + 12 x 1) ÷ 32 – (2 x 3) + 12
72 – 62 ÷ (18 – 3 x 4) ÷ 8 + 12
151
DRAFT
Simplify each expression by applying the Order of Operations. Show each step by filling in the blanks.
12 • 3 – 6 • 3
___ – 6 • 3
36 – ___
18
4(7 – 6) + 3
4 ( __ ) + 3
4 + __
7
8 + 5 • (4 ÷ 2)
8 + 5 • ___
8 + ___
___
6 + 32 – 9 ÷ 3
___ + __ – 9 ÷ 3
___ + __ – ___
___ + ___
___
14 ÷ (5 + 2) x 3
___ ÷ ___ x 3
___ x ___
___
(4 − 2) 3 − 5
___ 3 − ___
___ – ___
___
3 • 12 – 6 ÷ 2
___ – ___ ÷ ___
___ – ___
___
5 2 −(32 − 1 2 ) + 2
5 2 −(__ − ___ ) + 2
5 2 − ___ + 2
___ − ___ + 2
___ + 2
___
48 + 7 2 ÷49 • 2
___ + ___ ÷ 49 • 2
___ + ___ • 2
___ + ___
___
7/7/2008
6th 1-9
100 ÷ 52 x (3 + 2)
100 ÷ ___ x ___
100 ÷ ___ x ___
___ x ____
___
152
DRAFT
Simplify each expression by applying the Order of Operations. Show each step.
Example:
10 x (9 – 4) ÷ 5
7 + 7 2 ÷ (3 + 4) – 6
10 x 5 ÷ 5
50 ÷ 5
10
7/7/2008
6th 1-9
35 ÷ 5 x 3 + 2
28 – 14 x (6 ÷ 3) + 5
8 x (42 + 6) ÷ (4 + 4) – 42
(49 + 6) ÷ 5 – 2 2
3 3 x (18 – 5)
(10 – 2) x 5 + 20 ÷ 5
153
DRAFT
Simplify each expression by applying the Order of Operations. Show each step.
(13 – 5) x (9 ÷ 32 ) + 21
11 + 4 x 32 – (6 + 5 x 2)
2 • 32 – 5 • 3 – 1
10 ÷ 5 – 2 2 ÷ 2
5 x (7 + 0 ÷ 4) – 24
7/7/2008
6th 1-9
5 2 + (30 + 24 ÷ 6 – 1) x 4
154
DRAFT
When more than one set of parentheses is needed, you can use nested parentheses (( )) or brackets [( )]
to group terms. The innermost grouped terms are solved first. Simplify from the inside out: first the
parentheses, then the square brackets.
Simplify each expression by applying the Order of Operations. Show each step by filling in the blanks.
Example
2 [6 – (3 + 2)]
12 • [(6 – 4) • 3]
2 [6 – 5]
___ • [2 • 3]
2 [1]
___ • ___
2
_____
[(8 + 5) • (4 ÷ 2)] + 10
[___ • ___ ) + 10
____ + ___
___
(5 + 2 ) [(6 + 3 2 ) – (9 ÷ 3)]
___ • [(6 + __ ) – __ ]
___ • [ ___ – __ ]
___ • ___
___
14 ÷ [(5 + 2) x 1 ]
___ ÷ [ ___ x 1]
___ ÷ ___
[ (4 − 2) 3 − 5 ] + 2
[ ___ 3 − ___ ] + 2
___ – ___
___
___
( 5 + 2) [(3 • 12) – (6 ÷ 2) ]
( 5 + 2) [ ___ – ___ ]
___ • ___
3 • [ 5 2 −(32 − 1 2 ) + 2 ]
3 • [ 5 2 −(__ − ___ ) + 2 ]
3 • [ 5 2 − ___ + 2 ]
3 • [ ___ − ___ + 2 ]
3 • [___ + 2]
3 • _____
___
_____
100 ÷ [(5 x 2) x (3 + 2)]
100 ÷ [ ___ x ___ ]
100 ÷ ___
( 3 + 4) [ 7 – (3 + 1)]
( 3 + 4) [ 7 - ___ ]
____ • _____
___
_______
7/7/2008
6th 1-9
154a
155
DRAFT
Simplify each expression by applying the Order of Operations. Show each step.
Example:
10 x [(9 – 4) ÷ 5 ]
(7 + 7 2 ) – [ (3 + 4) – 6 ]
10 x [5 ÷ 5]
10 x 1
10
7/7/2008
6th 1-9
(35 ÷ 5) x [ (3 + 3) ÷ 2 ]
( 28 ÷ 14 ) x [ (6 ÷ 3) + 5 ]
[ 8 x (42 ÷ 6) ] + [ (4 x 4) – 2 ]
( 2 2 + 3 ) [10 – (3 + 1) ]
( 3 + 2 )[ 3 3 x (18 – 5)]
[ (10 – 2) x 5 ] + 20 ÷ 5
154b
147
DRAFT
th
Investigating Student Learning: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard AF 1.2 Write and evaluate an algebraic expression for a given situation,
using up to three variables.
Lesson 1.10: Variables and Expressions
Concepts:
A variable is a letter used to represent a quantity.
A variable is the missing part of the expression.
Usually lower case letters such as n, x, etc. are used for the variables. When working with variable, it
can be helpful, but not necessary, to use a letter that will remind you of what the variable stand
for: let n be the number of students in 6th grade.
A numerical expression consists of numbers and operations. It has no equal sign.
An algebraic expression consists of one or more numbers and variables along with one or more
operations.
An expression represents one number.
When an expression has parts that are added or subtracted, each part is called a term.
When two or more numbers are multiplied, the numbers are called factors.
To evaluate (simplify) is to find the value of the expression.
To evaluate an expression, you substitute (replace the variable) a value (number) for the variable and
perform the operations (calculations).
Order of operations must be used when evaluating expressions.
Essential Question(s):
How do you evaluate expressions?
7/7/2008
6th 1-10
155
DRAFT
th
ISL Item Bank: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard AF 1.2 Write and evaluate an algebraic expression for a given situation,
using up to three variables.
Lesson 1.10: Variables and Expressions
How do you evaluate expressions?
Which examples represent multiplication?
5xn
Yes
No
n–6
Yes
(7)(n)
Yes
No
Yes
Yes
3(n)
No
10 • n
n
9
No
3n
Yes
No
Yes
n=n
No
–n
Yes
Yes
No
(3)n
No
Yes
No
No
Which of the following are examples of expressions?
2
Yes
x + 5 x (9 – 14)
No
– 67
Yes
7/7/2008
6th 1-10
No
5=5
No
x
5
Yes
Yes
Yes
No
x 3
=
5 8
No
Yes
No
3 • ( 4 + 2) = 12
Yes
No
7(x + 3)
Yes
No
3 2
1 ÷
5 3
Yes
156
No
8 + 112
Yes
No
3 – 12 = – 9
Yes
x=y
Yes
No
x+ 6 = 12
Yes
No
No
a2
Yes
No
y + 2b
Yes
No
DRAFT
Substitute n = 10 for the variable. Then evaluate the expression.
Example:
7+n
7 + 10
17
24n
15 – n
n
5
5•n
n + (–4)
(n + 9)
–3 – n
n2
n–n
(n)(n)
6n
Complete the table by evaluating the following expressions for x = 8 and list steps used.
Expression
Evaluation steps
Expression
Evaluation steps
Example:
Example:
x÷4x3
5x + 1
5(8) + 1
8÷4x3
Substitute 8 for x
Substitute 8 for x
Evaluate: multiply
Evaluate: divide
40 + 1
2x3
Evaluate: add
Evaluate: __________
41
____
29 – 22 • x
___________
Substitute__________
5 +(12 – x)
___________
_________________
___________
Evaluate: _________
____________
_________________
____________
_________________
____________
_________________
____________
_________________
_________________
48 – x ÷ 2
___________
_________________
2 + x2 – 1
______________
___________
_________________
_______________
_________________
____________
_________________
_______________
_________________
_______________
_________________
7/7/2008
6th 1-10
157
DRAFT
Evaluate the expression for m = 12. List each step.
22 + m
26 – m
4m
m
6
m + 13
m – 32
10m
96
m
(13 – 5) + m
127 – (mi3)
3 + 4m
m ÷ 3× 2 + 1
8 + (m − 3 2 ) − 6
4 + 3 ÷ 1 + m i 10
17 + 4m ÷ 2
(3m + 4) ÷ (10 – 2)
24 ÷ m + m ÷ 3
m − 42 ÷ 2 × 3
12 + 72 ÷ (m − 3) + m
123 – 7m
21 + m + (18 − m) × 3
m − 4(18 + m) ÷ 3
7/7/2008
6th 1-10
6×
158
m
×1 + 6
4
188 − m 2 ÷10 – 4
DRAFT
Evaluate the expression for a = 3 and b = 7.
a2 + b
b–a+4÷2
2b – 2a
3(a + b)
21 ÷b + 3 x a
(16 + b 2 ) ÷ ( a 2 )
(2a + 2) + (a + 6)
a x6÷2xb
(42 ÷ a) ÷ 7 + (4 – a)
(a • b) ÷ (b • a)
(a + 22 ) ÷ b
( 62 ÷ 13 ) x 2 – 6
7/7/2008
6th 1-10
159
DRAFT
Evaluate the expression for c = 6, d = 1, and c = 4
(c • 3) – (2d) x (e – 2)
(c + d) x (15 ÷ 3) – e
cxe–d
(2 x c) ÷ d x e
( e3 – c) ÷ c x d
⎛ 16 ⎞ ⎛ c ⎞ ⎛ d ⎞
⎜ ⎟ x⎜ ⎟ +⎜ ⎟
⎝ e ⎠ ⎝d ⎠ ⎝d ⎠
13 + d – c + c 2 – 3e
(15 x 2 ÷ c + d x e )
d + c2 - e
c+d+e÷2
Evaluate the expression for c = 6, d = 1, and c = 4
(5 + c – e) – ( d 3 + e ) ÷ d
7/7/2008
6th 1-10
3 • d + 4 + e •c
160
DRAFT
Evaluate each expression if x = 3, t =4, and y = 2
t2 + 3y
xit i y 3
6t + y
8(x – y) + 3t
Evaluate each expression if c = 3 and d = 7
6c + 4 – 3d
d 2 + 5d − 6
21 ÷d x c x d
c2 x d 2 + 1
Evaluate x 2 − ( y + 2) if x = 4
7/7/2008
6th 1-10
Evaluate
3(g – h) + 6 – (j + 1) for g = 4, h = 3, and j = 6
161
DRAFT
th
Investigating Student Learning: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard AF 1.2 Write and evaluate an algebraic expression for a given situation,
using up to three variables.
Lesson 1.11: Writing Expressions
Concepts:
A numerical expression consists of numbers and operations. It has no equal sign.
An algebraic expression consists of one or more numbers and variables along with one or more
operations.
Algebraic expressions represent relations expressed verbally.
When an expression has parts that are added or subtracted, each part is called a term.
A variable is a letter used to represent one or more numbers.
In mathematics there are expressions and equations or inequalities just as there are phrases and
sentences in English or other languages.
Word phrases can be translated into algebraic expressions.
Essential Question(s):
How do you write algebraic expressions?
7/7/2008
6th 1-11
162
DRAFT
th
ISL Item Bank: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard AF 1.2 Write and evaluate an algebraic expression for a given situation,
using up to three variables.
Lesson 1.11: Writing Expressions
How do you write and algebraic expression?
Word Phrase
Operation
Algebraic Expression
Addition
8+h
Addition
6 + b or b + 6
Example:
8 plus h
l divided by 4
7 times m
f minus 32
15 divided d
x added to 45
7 subtracted from t
16 multiplied by w
5 more than k
y less than 9
Example:
the sum of 6 and a number b
the product of the number n and 5
the difference of a number k and 4
t + 23 or 23 + t
the quotient of a number x divided by 12
15 times a number g
twice a number k
7/7/2008
6th 1-11
163
DRAFT
Word Phrase
Operation
Algebraic Expression
w multiplied by 6
y
4
the quotient of 9 and a number z
the difference of 20 and a number t
add 25 to f
12 + c or c + 12
4 less than d
n and 4 more
9 squared plus and number s
b4 + 8
60
x
A number decreased by 4
n more than 15
the quotient of p divided by 3
the product of a and 2
a number w doubled
s squared plus 4
7/7/2008
6th 1-11
164
DRAFT
Word Phrase
Operation
Algebraic Expression
4 less n
x squared minus 3
5 increased by m
2z
half a number l
66 plus x
m3
the product of a number a and 2
25 + p
5 to the fourth power increased by 7
r–4
17 less than b
5 added to a number d squared
the product of the number x squared and 6
10 more than a number p
7/7/2008
6th 1-11
165
DRAFT
Word Phrase
3 more than the product of 5 and c
Operations
Algebraic Expression
Addition
Multiplication
3 + 5c or 5c + 3
6 less than the quotient of x and 12
twice the sum of 5 and y
h divided by 9, plus 6
8 times the difference of a and 2
3 times the square of a number b
7 subtracted from the quotient of n and 8
4 times the sum of 17 and a number j
the product of 4 and y, minus 25
12 plus the quotient of k and 3
the sum of 15 and m, divided by 2
10 more than twice g
4 less than the product of 2 and f
7/7/2008
6th 1-11
166
DRAFT
Write a word phrase for the algebraic expressions.
Example:
5 more than the quotient of 42 divided by s
42
+5
s
42 divided by s, plus 5
4i
3
z
6(h – 4)
5t + 3
3 – 2v
6+8÷k
5 x (b + 3)
a2 − 5
11 ÷ 2y
n
−3
6
a+b–c
16 + d 3
7/7/2008
6th 1-11
167
OR
DRAFT
Write a word phrase for the algebraic expressions.
7i
f
4
3m + 1
11 + 9x
64
+ 12
j
2–k+e
11 i
6
d
3(9 – p)
x (6 + 12)
n2 + 4
m–7+j
9 + 8r
84 ÷ 3k
7/7/2008
6th 1-11
168
DRAFT
Match the algebraic expression with the correct word phrase.
4x+7
7 less than the product of 4 and x
4(x – 7)
4 times the sum of x and 7
4(x + 7)
4 times the difference of x and 7
4x – 7
7 more than the product of 4 and x
5(6 – d)
5 times 6 less than d
5(d – 6)
the product of 5 times the sum of 6 and d
5(6 + d)
5 times the product of 6 and d
5 (6d)
5 times the difference of d and 6
24
+3
t
3 less than 24 divided by t
24
+t
3
t more than the quotient 24 divided by 3
24
−3
t
3 more than the quotient 24 divided by t
24
−t
3
the quotient 24 divided by 3, less t
32 ÷ 4k
k more than the quotient 32 divided by 4
32 +
7/7/2008
6th 1-11
4
k
32 times the sum of 4 and k
32 ÷ 4 + k
the sum of 32 and the quotient 4 divided by k
32(4 + k )
32 divided by the product of 4 and k
169
DRAFT
Tamiya bought x cherries and ate 5 of them. Write an expression that describes the number of cherries
Tamiya has left.
7 motorcycles were parked in the alley. Later in the afternoon, f more motorcycles were parked in the
alley. Write an expression that describes the number of motorcycles in the alley in the afternoon.
Jonas had p pairs of shoes in his closet. After one year, he had 3 times the amount of shoes. Write an
expression that describes the number of pairs of shoes Jonas had after one year.
There were s number of students going on a field trip. Each car could hold 5 students each. Write an
expression that describes the number of cars needed for the field trip.
Jeremy has 2 times more baseball cards than Fanny (F) and Ramone (R) put together. Write an
expression that describes the number of baseball cards Jeremy has.
The three Roniak Brothers are different heights. Jordon is 2 inches taller than little brother Noah.
Oldest brother George is double the size of Jordan. Write an expression that describes how
tall George is.
Luis started the day off with p number of pennies. Half-way through the day, he discovered that he had
a hole in his pocket and lost 9 of the pennies. In the afternoon, Luis decided to mow his neighbors lawn
and earned 5 times the amount he had in the afternoon. Write an expression that describes the number
of pennies Luis had after he mowed his neighbors lawn.
Janice started with l lollipops. She bought 4 more lollipops. She decided to divide her lollipops evenly
among 5 friends. Write an expression that describes the number of lollipops Janice took home.
Tomas had m number of silver dollars. Luces had six times the amount of silver dollars that Tomas had.
Reardon had 9 more than Luces. Write an expression that describes the number of silver dollars that
Reardon had.
7/7/2008
6th 1-11
170
DRAFT
th
Investigating Student Learning: 6 Grade
Chapter 1: Whole Numbers and Decimals
Standard AF 1.1*: Write and solve one-step linear equations in one variable.
Lesson 1.12: Solving Addition and Subtraction Equations
Concepts:
An equation is a mathematical sentence.
An equation says that two expressions are equal; the value on the left side of the equal
sign is always equal to the value on the right side.
An equation is solved by finding the number that makes each side the same value.
There are two types of equations: identical equations and conditional equations.
An identical equation is true for all values of the variable.
e.g. a + b = b + a (the Commutative Property) is and example of an identical equation
because it is always true regardless of the values for the variables.
A conditional equation requires certain values for the variables to make the equation true.
e.g. x + 4 = 10 is an example of a conditional equation because it requires certain values
(6,
12 18
,
, etc.) to make the equation true.
2 3
You can use the inverse properties and the properties of equality to get the variable alone to solve an
equation. Addition and subtraction are inverse operations; addition undoes subtraction and
subtraction undoes addition..
The Addition Property of Equality states that the same value can be added to both sides of an equation
and maintain equality.
The Subtraction Property of Equality states that the same value can be subtracted to both sides of an
equation and maintain equality.
Finding a solution to an equation involves isolating the variable, or “getting the variable alone,” so you
can find a value for the variable that will make the equation true.
The Inverse Property and the Addition and Subtraction Properties of Equality can be used to help
isolate the variable.
e.g.
a
+
5 = 11
Subtracting 5 undoes adding 5
+ 5 - 5 = 11 - 5
The Subtraction Property allows 5 to be
+
0
=
6
subtracted from both sides of the equation
a
=
6
Using these two properties maintains equality
When you solve an equation, you find the value of the variable that makes the equat6ion true.
You can check answers to equations by substituting the answer for the variable in the original equation.
When both sides of the equation can be simplified to the same number, the value of the
variable is correct.
Equations can be written with the number on the left side and the expression on the right side.
E.g. y + 7 = 15 is equivalent to 15 = y + 7 or 15 = 7 + y
a
a
Essential Question(s):
How do you use the Inverse Property and the Properties of Equality to solve addition and subtraction
equations?
7/7/2008
6th 1-12
171
DRAFT
ISL Item Bank: 6th Grade
Chapter 1: Whole Numbers and Decimals
Standard AF 1.1: Write and solve one-step linear equations in one variable.
Lesson 1.12: Solving Addition and Subtraction Equations
How do you use the Inverse Property and the Properties of Equality to solve addition
and subtraction equations?
What operation is used in the expression x + 7 ? _____________________
What is the inverse operation of addition?
______________________
If you want to isolate the x on one side of the equation, then you must use the inverse
operation of “add 7.” Look at the blank.
x + 7 = 16
– 7
The Property of Equality says that if you subtract one number from one side of the
equation you must also subtract the same number from the other side of the equation.
Fill in the blanks.
x + 7 = 16
–
= –
x + 0
=
9
x
=
9
What operation is used in the expression n + 8 ? _____________________
What is the inverse operation of addition?
______________________
If you want to isolate n on one side of the equation, then you must use the inverse
operation of “add 8.” Fill in the blank.
n + 8 = 15
–
The Property of Equality says that if you subtract one number from one side of the
equation you must also subtract the same number from the other side of the equation.
Fill in the blanks.
n + 8 =
15
–
= –
n + 0
n
7/7/2008
6th 1-12
172
=
7
=
7
DRAFT
What operation is used in the expression m + 4 ? _____________________
What is the inverse operation of addition?
______________________
If you want to isolate m on one side of the equation, then you must use the inverse
operation of “add 4.” Fill in the blank.
m + 4 = 19
–
The Property of Equality says that if you subtract one number from one side of the
equation you must also subtract the same number from the other side of the equation.
Fill in the blanks.
m + 4 = 19
–
= –
m + 0
=
m
=
What operation is used in the expression z + 18 ? _____________________
What is the inverse operation of addition?
______________________
If you want to isolate z on one side of the equation, then you must use the inverse
operation of “add 18.” Fill in the blank.
z + 18 = 62
–
The Property of Equality says that if you subtract one number from one side of the
equation you must also subtract the same number from the other side of the equation.
Fill in the blanks.
z + 18 = 62
–
z +
= –
=
=
7/7/2008
6th 1-12
173
DRAFT
What operation is used in the expression x + 5 ? _____________________
What is the inverse operation of addition?
______________________
If you want to isolate x on one side of the equation, then you must use the inverse
operation of “add 5.” Fill in the blank.
25 = x + 5
–
The Property of Equality says that if you subtract one number from one side of the
equation you must also subtract the same number from the other side of the equation.
Fill in the blanks.
25 = x + 5
–
=
20
–
= x +
=
What operation is used in the expression p + 24 ? _____________________
What is the inverse operation of addition?
______________________
If you want to isolate p on one side of the equation, then you must use the inverse
operation of “add 24.” Fill in the blank.
27 = p + 24
–
The Property of Equality says that if you subtract one number from one side of the
equation you must also subtract the same number from the other side of the equation.
Fill in the blanks.
27 = p + 24
–
=
–
= p +
=
7/7/2008
6th 1-12
174
DRAFT
What operation is used in the expression 3 + y ? _____________________
What is the inverse operation of addition?
______________________
If you want to isolate y on one side of the equation, then you must use the inverse
operation of “add 3.” Fill in the blank.
13 = 3 + y
–
The Property of Equality says that if you subtract one number from one side of the
equation you must also subtract the same number from the other side of the equation.
Fill in the blanks.
13 = 3 +
y
–
= –
10
=
+ y
=
What operation is used in the expression 9 + z ? _____________________
What is the inverse operation of addition?
______________________
If you want to isolate z on one side of the equation, then you must use the inverse
operation of “add 9.” Fill in the blank.
10 = 9 + z
–
The Property of Equality says that if you subtract one number from one side of the
equation you must also subtract the same number from the other side of the equation.
Fill in the blanks.
10 = 9 +
z
–
= –
=
=
7/7/2008
6th 1-12
175
+ z
DRAFT
What operation is used in the expression 2 + n ? _____________________
What is the inverse operation of addition?
______________________
If you want to isolate n on one side of the equation, then you must use the inverse
operation of “add 2.” Fill in the blank.
2 + n = 14
–
The Property of Equality says that if you subtract one number from one side of the
equation you must also subtract the same number from the other side of the equation.
Fill in the blanks.
2 + n = 14
–
= –
0 +
n
=
n
=
What operation is used in the expression 6 + a ? _____________________
What is the inverse operation of addition?
______________________
If you want to isolate a on one side of the equation, then you must use the inverse
operation of “add 6.” Fill in the blank.
6 + a = 24
–
The Property of Equality says that if you subtract one number from one side of the
equation you must also subtract the same number from the other side of the equation.
Fill in the blanks.
6 + a = 24
–
= –
0 +
7/7/2008
6th 1-12
a
=
a
=
176
DRAFT
What operation is used in the expression x – 6 ? _____________________
What is the inverse operation of subtraction?
______________________
If you want to isolate x on one side of the equation, then you must use the inverse
operation of “subtract 6.” Look at the blank.
x – 6 = 4
+ 6
The Property of Equality says that if you add one number to one side of the equation
you must also add the same number to the other side of the equation. Fill in the blanks.
x – 6
=
4
+
= +
x – 0
x
=
10
=
10
What operation is used in the expression n – 7 ? _____________________
What is the inverse operation of subtraction?
______________________
If you want to isolate n on one side of the equation, then you must use the inverse
operation of “subtract 7.” Fill in the blank.
n – 7 = 9
+
The Property of Equality says that if you add one number to one side of the equation
you must also add the same number to the other side of the equation. Fill in the blanks.
n – 7 =
9
+
= +
n – 0
n
7/7/2008
6th 1-12
177
=
16
=
16
DRAFT
What operation is used in the expression m – 5 ? _____________________
What is the inverse operation of subtraction?
______________________
If you want to isolate m on one side of the equation, then you must use the inverse
operation of “subtract 5.” Fill in the blank.
m – 5 = 25
+
The Property of Equality says that if you add one number to one side of the equation
you must also add the same number to the other side of the equation. Fill in the blanks.
m – 5 =
25
+
= +
m – 0
m
=
=
What operation is used in the expression z – 1 ? _____________________
What is the inverse operation of subtraction?
______________________
If you want to isolate z on one side of the equation, then you must use the inverse
operation of “subtract 1.” Fill in the blank.
z – 1 = 78
+
The Property of Equality says that if you add one number to one side of the equation
you must also add the same number to the other side of the equation. Fill in the blanks.
z – 1 =
78
+
= +
z –
=
=
7/7/2008
6th 1-12
178
DRAFT
What operation is used in the expression x – 5 ? _____________________
What is the inverse operation of subtraction?
______________________
If you want to isolate x on one side of the equation, then you must use the inverse
operation of “subtract 5.” Fill in the blank.
30 = x – 5
+
The Property of Equality says that if you add one number to one side of the equation
you must also add the same number to the other side of the equation. Fill in the blanks.
30 = x – 5
+
=
35
+
= x –
=
What operation is used in the expression z – 6 ? _____________________
What is the inverse operation of subtraction?
______________________
If you want to isolate z on one side of the equation, then you must use the inverse
operation of “subtract 6.” Fill in the blank.
34 = z – 6
+
The Property of Equality says that if you add one number to one side of the equation
you must also add the same number to the other side of the equation. Fill in the blanks.
34 = z – 6
+
=
+
= z –
=
7/7/2008
6th 1-12
179
DRAFT
Solve each equation.
z + 7 =
8
–
m + 2
= –
z +
–
=
–
m +
=
=
=
2
11
= –
x +
44
–
–
–
–
=
=
m+
=
=
= t + 40
z + 18
=
–
–
=
t+
z +
=
y + 23
=
–
–
=
m+
y +
=
–
= –
7/7/2008
6th 1-12
26
=
= m + 2
y + 17
=
= –
= 125
= –
=
=
y +
= m + 5
=
=
38
11
= –
=
x + 1
=
=
43
37
–
= m + 9
=
–
=
=
m+
=
=
180
DRAFT
Solve each equation.
z – 9 =
6
+
x – 3
= +
z –
+
=
=
+
= +
n –
72
+
x –
=
=
4
15
+
+
+
=
=
m–
=
=
= n – 70
p – 32
=
+
+
=
n –
p –
+
t –
7/7/2008
6th 1-12
50
=
=
= x – 15
n – 6
=
+
+
=
x –
n –
=
=
= +
=
33
= +
=
=
t – 7
= m – 15
=
=
1
27
= +
=
n – 23
=
=
14
8
= +
+
= n – 8
=
+
=
=
n –
=
=
181
DRAFT
Problem
Inverse Operation
Property Used
To Solve
Solution
Show Work
Subtraction
Property of
Equality
x + 9 = 22
–9=–9
x – 0 = 13
x = 13
Example:
x + 9 = 22
Subtraction
a – 13 = 3
22 = n + 5
13 = 3 + c
17 = m – 12
b – 19 = 18
13 = x + 13
7/7/2008
6th 1-12
182
DRAFT
Problem
Inverse Operation
Property Used
To Solve
40 = m – 25
13 = 6 + n
s + 12 = 25
g–8=5
t – 45 = 72
129 = x + 63
297 = n + 154
7/7/2008
6th 1-12
183
Solution
Show Work
DRAFT
Solve the equations and check your solution.
Solve the Equation
Check Solution
Write the original equation
x + 15 = 129
– 15 – 15
x + 0 = 114
x = 114
x + 15 = 129
Substitute the value of x from your solution.
x = 114
114 + 15 = 129
Evaluate
129 = 129
Are both sides of the equation equal? If yes, the answer
is correct. If they are not equal the answer is incorrect.
Write the original equation
x + 48 = 397
Substitute
Evaluate
Are both sides are equal?
Yes
No
Yes
No
Yes
No
Yes
No
Write the original equation
x – 29 = 501
Substitute
Evaluate
Are both sides are equal?
Write the original equation
467 = x – 678
Substitute
Evaluate
Are both sides are equal?
Write the original equation
783 = x + 99
Substitute
Evaluate
Are both sides are equal?
7/7/2008
6th 1-12
184
DRAFT
Solve each equation and check your answers.
x + 35 = 184
w + 56 = 905
436 = n + 77
506 = y + 506
7/7/2008
6th 1-12
Check:
Check:
Check:
Check:
185
DRAFT
Solve each equation and check your answers.
j + 673 = 2,390
d + 5 = 333
782 = 72 + n
469 = 123 + y
7/7/2008
6th 1-12
Check:
Check:
Check:
Check:
186
DRAFT
Solve each equation and check your answers
7/7/2008
6th 1-12
x – 606 = 154
w – 587 = 678
249 = n – 28
309 = y – 114
m – 349 = 28
z – 607 = 138
537 = t – 537
649 = p – 700
187
DRAFT
Solve each equation and check your answers.
7/7/2008
6th 1-12
x + 101 = 190
w – 174 = 377
397 = n – 196
491 = p + 18
x + 168 = 731
z – 285 = 582
900 = n + 709
1,704 = t – 78
188
DRAFT
th
6 Grade
Chapter 1 Whole Numbers and Decimals
Multiple Choice Math Test
1) What number is equal to three hundred
ten thousand, fifty-nine and forty-one
thousandths.
(1-1)
(NS 1.21 Gr 5)
A) 310,059.41
B) 310,000,059.041
C) 310,059.041
D) 300,010,059.041
Name: ________________________
Date: ____________
6)
A)
B)
C)
D)
600,428,000,264.0004
392.1
392.024
392.035
392.04
7) Use the break apart method to find the sum.
Show your work.
2) What short word form is equal to the number
(1-1)
(NS 1.21 Gr 5)
Which number is less than 392.03?
(1-2)
(NS 1.1*)
(1-4)
(AF 1.3)
837 + 65
A) 600 billion, 428 million, 264 thousand and 4
thousandths
B) 600 billion, 428 million, 264 and 4 thousandths
C) 600 billion, 428 million, 264 thousand and
4 ten-thousandths
D) 600 billion, 428 million, 264 and
4 ten-thousandths
A)
B)
C)
D)
3) Write the number 9,501.024 in expanded
form.
8) How could compensation best be used to
(1-4)
compute this problem mentally?
(1-1)
(NS 1.21 Gr 5)
(AF 1.3)
A)
B)
C)
D)
58 + 23
9,000 + 500 + 1 + 0.2 + 0.04
9,000 + 500 + 1 + 0.2 + 0.004
9,000 + 500 + 1 + 0.02 + 0.004
9,000 + 500 + 1 + 0.02 + 0.0004
4)
Compare.
(1-2)
37.583
(NS 1.1*)
A) >
C) =
B) <
D) +
5)
(1-2)
(NS 1.1*)
37.6
Write the numbers in the set from least to
greatest.
6.074; 6.3; 6.25; 6.1000
A) 6.3; 6.25; 6.074; 6.1000
B) 6.074; 6.1000; 6.3; 6.25
C) 6.074; 6.1000; 6.25; 6.3
D) 6.1000; 6.25; 6.3; 6.074
Revised 7/7/2008
895
897
902
912
A) add 2 to 23 = 25; add 2 to 58 = 60; 25 + 60 = 85
B) add 2 to 23 = 25; subtract 2 from 58 = 56; 25 + 56 = 81
C) add 2 to 58 = 60; 60 + 23 = 83
D) add 2 to 58 = 60; 60 + 23 = 83; then subtract 2 = 81
9) What two addends would you use as
compatible numbers to compute this problem
(AF 1.3) mentally?
(1-4)
53 + 24 + 17
A) 53 and 17
B) 24 and 17
C) 17 and 17
D) 24 and 53
10)
(1-5)
(NS 1.1 Gr 5)
A)
B)
C)
D)
1
Round 56,093,119.704 to the nearest
whole number.
56,093,120
56.093,119
56,093,119.700
56,000,000.000
Created collaboratively with grade 6 ISL teachers
DRAFT
6th Grade Chapter 1 Whole Numbers and Decimals Multiple Choice Math Test
11)
(1-5)
(NS 1.1 Gr 5)
A)
B)
C)
D)
Which place is the following number
rounded?
108,365,800,000
ten thousand
hundred thousand
one million
hundreds
(NS 2.0)
A) 71.239
12) Round 724.83 to the nearest tenth.
(1-5)
(NS 1.1 Gr 5)
A)
B)
C)
D)
16) Find 76.28 – 5.041.
(1-7)
724.7
724.8
724.82
724.92
13) Find 3.6 + 567 + 46.509.
(1-6)
(NS 2.0)
C) 71.241
B) 25.87
D) 71.238
17) Clarissa’s grandmother gave her $25 for her
(1-7) birthday. The next day, Clarissa spent $8.62
(NS 2.0)
on a new notebook for school. How much
of her birthday money does Clarissa have
left?
A) $17.62
C) $13.62
B) $16.48
D) $16.38
18) Find the difference.
(1-7)
4,500,321 – 605,720
(NS 2.0)
A) 471.12
C) 617.109
B) 506.76
D) 47,079.6
14) Find the best estimates for each addend.
3.2 + 579.7 + 18.287
(1-6)
(NS 2.0)
A)
B)
C)
D)
3 + 579 + 18
3 + 580 + 18
3 + 580 + 19
2 + 579 + 19
15) Find the sum.
(1-6)
346,259 + 4,280,367
(NS 2.0)
A) 7,742,957
C) 4,526,516
B) 7,642,857
D) 4,626,626
Revised 7/7/2008
A) 3,894,601
C) 3,994,601
B) 3,905,601
D) 4,105,401
19) Use the order of operations to evaluate the
expression.
(AF 1.4)
3 • 12 – 6 ÷ 2
(1-9)
A) 15
C) 27
B) 33
D) 9
20) Use the order of operations to evaluate the
(1-9)
expression.
(AF 1.4)
19 – 6 + 2 × 5
A) 3
C) 55
B) 23
D) 75
2
Created collaboratively with grade 6 ISL teachers
DRAFT
6th Grade Chapter 1 Whole Numbers and Decimals Multiple Choice Math Test
21) Use the order of operations to evaluate the
(1-9) expression.
(13 – 5) × (9 ÷ 32 ) + 21
(AF 1.4)
26) Write the phrase as an expression.
(1-11)
13 less than k squared
(AF 1.2)
A) 29
C) 176
A) k2 – 13
C) 13 – 2k
B) 93
D) 240
B) 13 – k2
D) 2k – 13
22) Evaluate the expression for m = 8
(1-10)
(AF 1.2)
A)
B)
C)
D)
m + 2(12 − m) ÷ 4
28
5
10
26
A) 4m + 5
B) 5m + 4
23) Evaluate the expression for a = 2 and b = 3.
(1-10)
(15 + b 2 ) ÷ ( a 2 )
(AF 1.2)
A)
B)
C)
D)
27) Tom had m number of silver dollars. Joe
(1-11)
had four times the amount of silver dollars
(AF 1.2) that Tom had. Maria had 5 more than
Joe. Write an expression that describes the
number of silver dollars that Maria had.
2
5
6
81
24) Evaluate the expression for a = 4, b = 3, and
(1-10)
c = 5.
2b + ( 3c − a )
(AF 1.2)
A) 54
B) 17
C) 12
D) 9
28) What inverse operation must you use to
(1-12) isolate z?
(AF 1.1*)
34 = z – 13
A) add 13
C) subtract 13
B) add 34
D) subtract 34
29) Solve the equation.
(1-12)
(AF 1.2)
A) n = 70.6
C) n = 46.3
B) n = 40.9
D) n = 16.6
A) (5 + p) × 2
C) 5 (p + 2)
B) (p × 2) + 5
D) 5 (2p)
Revised 7/7/2008
30) Solve the equation.
(1-12)
5 more than the product of p and 2
43.6 = n + 27
(AF 1.1*)
25) Write the phrase as an expression.
(1-11)
C) m + 4 + 5
D) m + 5
(AF 1.1*)
p − 47 = 129
A) 82
C) 154
B) 122
D) 176
3
Created collaboratively with grade 6 ISL teachers
DRAFT
th
6 Grade
Chapter 1 Whole Numbers and Decimals
Multiple Choice Math Test
Answer Key
1. C
2. D
3. C
4. B
5. C
6. B
7. C
8. D
9. A
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Revised 7/7/2008
A
B
B
C
B
D
A
D
A
B
B
A
C
C
B
B
A
A
A
D
D
4
Created collaboratively with grade 6 ISL teachers
DRAFT
th
6 Grade
Chapter 1 Whole Numbers and Decimals
Free Response Math Test
1) Write the standard form for eight hundred
sixty-one million, one hundred seven and
eight hundredths.
Name: ________________________
Date: ____________
6) Write a number greater than 267.47 but less
than 267.5
(1-2)
(NS 1.1*)
(1-1)
(NS 1.21 Gr 5)
2) Write the short word form for
(1-1)
(NS 1.21 Gr 5)
808,000,397,086.0035
3) Write the number 7,401.037 in expanded
form.
(1-1)
(NS 1.21 Gr 5)
7) Use the break apart method to find the sum.
(1-4)
Show your work.
(AF 1.3)
768 + 35
8) How could compensation best be used to
compute this problem mentally?
(1-4)
(AF 1.3)
28 + 43
4)
Compare.
(1-2)
843.3
(NS 1.1*)
843.189
9) What 2 addends would you use as compatible
(1-4)
numbers to compute this problem
(AF 1.3)
mentally?
28 + 61 + 42
5)
(1-2)
(NS 1.1*)
Write the numbers in the set from least to
greatest.
0.085; 0.12; 1.002; 0.3
Revised 7/7/2008
10)
(1-5)
(NS 1.1 Gr 5)
1
Round 64,253,008.381 to the nearest
tenth.
Created collaboratively with grade 6 ISL teachers
DRAFT
6th Grade Chapter 1 Whole Numbers and Decimals Free Response Math Test
11)
(1-5)
(NS 1.1 Gr 5)
Which place is the following number
rounded?
967,116,490,000
12) Round 854.52 to the nearest tenth.
(1-5)
(NS 1.1 Gr 5)
16) Find 93.15 – 7.089.
(1-7)
(NS 2.0)
17) Brody took $5.50 from his piggy bank to buy
candy for the movies. He started the day
(NS 2.0) with $24 in his piggy bank. How much
money does Brody have in his bank after he
buys the candy?
(1-7)
13) Find 302.7 + 45 + 0.86.
(1-6)
(NS 2.0)
18) Find the difference.
(1-7)
(NS 2.0)
14) What are the best estimates for each addend
before finding an estimated sum.
(1-6)
14.168 + 369.5 + 6.3
3,400,251 – 604,550
19) Use the order of operations to evaluate the
expression.
(AF 1.4)
3 • 8 – 6 ÷2
(1-9)
(NS 2.0)
15) Find the sum.
(1-6)
(NS 2.0)
456,374 + 2,390,245
20) Use the order of operations to evaluate the
(1-9)
expression.
(AF 1.4)
17 – 5 + 4 × 2
Revised 7/7/2008
2
Created collaboratively with grade 6 ISL teachers
DRAFT
6th Grade Chapter 1 Whole Numbers and Decimals Free Response Math Test
21) Use the order of operations to evaluate the
(1-9) expression.
(AF 1.4)
(12 ÷ 22 ) + (5•3) – 8
22) Evaluate the expression for m = 6
(1-10)
(AF 1.2)
m − 2(4 + m) ÷ 4
23) Evaluate the expression for a = 2 and b = 8.
(1-10)
(19 + b ) ÷ (1 + a 3 )
(AF 1.2)
24) Evaluate the expression for a = 5, b = 7, and
(1-10)
c = 3.
(AF 1.2)
26) Write the phrase as an expression.
(1-11)
(AF 1.2)
10c + ( 2a − b )
the quotient of a number t and 6 increased by 5
27) Gina had p number of pennies. Jorge had
nine times the amount of pennies that Gina
(AF 1.2)
had. Melinda had 3 less than Jorge.
Write an expression that describes the
number of pennies that Melinda had.
(1-11)
28) What inverse operation must you use to
isolate z?
(AF 1.1*)
31 + z = 45
(1-12)
29) Solve the equation.
(1-12)
(AF 1.1*)
25) Write the phrase as an expression.
(1-11)
(AF 1.2)
Revised 7/7/2008
n – 32.7 = 91
30) Solve the equation.
(1-12)
8 less than the product of f and 4
(AF 1.1*)
n – 58 = 219
3
Created collaboratively with grade 6 ISL teachers
DRAFT
th
6 Grade
Chapter 1 Whole Numbers and Decimals
Free Response Math Test
Answer Key
1) Write the standard form for eight hundred
sixty-one million, one hundred seven and
eight hundredths.
(1-1)
(NS 1.21 Gr 5)
861,000,107.08
2) Write the short word form for
(1-1)
808,000,397,086.0035
(NS 1.21 Gr 5)
6)
(1-2)
(NS 1.1*)
Write a number greater than 267.47 but less
than 267.5
There are many possible answers.
(ie. 267.471 - 267.48 - 267.499)
7) Use the break apart method to find the sum.
Show your work.
(AF 1.3)
768 + 35
(1-4)
= 768 + (30 + 5)
= (768 + 30) + 5
= 798 + 5
= 803
808 billion, 397 thousand,
86 and 35 ten-thousandths
A student may
have also
chosen to
break apart the
768.
3) Write the number 7,401.037 in expanded
(1-1)
form.
8) How could compensation be used to
(1-4)
compute this problem simply?
(NS 1.21 Gr 5)
(AF 1.3)
28 + 43
7,000 + 400 + 1 + 0.03 + 0.007
4)
Compare.
(1-2)
843.3
(NS 1.1*)
843.189
Add 2 to 28 to get 30; 30 adds
easily to 43 = 73; then subtract 2
(compensate) to get 71.
9) What 2 addends would you use as compatible
numbers to compute this problem
(AF 1.3) mentally?
(1-4)
28 + 61 + 42
>
28 + 42
(28 + 42 = 70)
5)
(1-2)
(NS 1.1*)
Write the numbers in the set from least to
greatest.
0.085; 0.12; 1.002; 0.3
10)
(1-5)
(NS 1.1 Gr 5)
Round 64,253,008.381 to the nearest
tenth.
64,253,008.4
0.085; 0.12; 0.3; 1.002
Revised 7/7/2008
4
Created collaboratively with grade 6 ISL teachers
DRAFT
6th Grade Chapter 1 Whole Numbers and Decimals Free Response Math Test
11)
(1-5)
(NS 1.1 Gr 5)
Which place is the following number
rounded?
967,116,490,000
16) Find 93.15 – 7.089.
(1-7)
(NS 2.0)
86.061
ten thousands
12) Round 854.52 to the nearest tenth.
(1-5)
(NS 1.1 Gr 5)
17) Brody took $5.50 from his piggy bank to buy
candy for the movies. He started the day
(NS 2.0) with $24 in his piggy bank. How much
money does Brody have in his bank after he
buys the candy?
(1-7)
854.5
$18.50
13) Find 302.7 + 45 + 0.86.
(1-6)
(NS 2.0)
18) Find the difference.
(1-7)
(NS 2.0)
3,400,251 – 604,550
348.56
2,795,701
14) What are the best estimates for each addend
before finding an estimated sum.
(1-6)
14.168 + 369.5 + 6.3
19) Use the order of operations to evaluate the
expression.
(AF 1.4)
3 • 8 – 6 ÷2
(1-9)
(NS 2.0)
14 + 370 + 6
15) Find the sum.
(1-6)
(NS 2.0)
456,374 + 2,390,245
21
20) Use the order of operations to evaluate the
(1-9)
expression.
(AF 1.4)
17 – 5 + 4 × 2
2,846,619
Revised 7/7/2008
20
5
Created collaboratively with grade 6 ISL teachers
DRAFT
6th Grade Chapter 1 Whole Numbers and Decimals Free Response Math Test
26) Write the phrase as an expression.
21) Use the order of operations to evaluate the
(1-9) expression.
(AF 1.4)
(1-11)
(AF 1.2)
the quotient of a number t and 6 increased by 5
(12 ÷ 22 ) + (5•3) – 8
t + 5 or t ÷ 6 + 5
6
10
22) Evaluate the expression for m = 6
(1-10)
(AF 1.2)
m − 2(4 + m) ÷ 4
1
27) Gina had p number of pennies. Jorge had
(1-11)
nine times the amount of pennies that Gina
(AF 1.2)
had. Melinda had 3 less than Jorge.
Write an expression that describes the
number of pennies that Melinda had.
9p – 3
23) Evaluate the expression for a = 2 and b = 8.
(1-10)
(19 + b ) ÷ (1 + a 3 )
(AF 1.2)
28) What inverse operation must you use to
isolate z?
(AF 1.1*)
31 + z = 45
(1-12)
subtraction
3
24) Evaluate the expression for a = 5, b = 7, and
c = 3.
(1-10)
(AF 1.2)
10c + ( 2a − b )
29) Solve the equation.
(1-12)
(AF 1.1*)
33
n = 123.7
25) Write the phrase as an expression.
(1-11)
(AF 1.2)
n – 32.7 = 91
30) Solve the equation.
(1-12)
8 less than the product of f and 4
(AF 1.1*)
n – 58 = 219
(f x 4) – 8 or 4f – 8
n = 277
Revised 7/7/2008
6
Created collaboratively with grade 6 ISL teachers
Item Analysis for:
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Total Items
Correct
6th Grade Chapter 1 Tests
1-1
1-2
1-4
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2
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9
Teacher: ______________________
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Date Given: ________
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30
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