# 6th Grade Study Guide for Decimals

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```					  6th Grade Study Guide for Chapter 2: Decimals
Games you will play for decimals:

 Doggone Decimal (Everyday Math) Students form two 2-digit numbers
with decimal points whose product is closest to the target number for each
round.

 Smallest to Greatest (Investigations) Students strategize through placing
decimal cards on a 3 by 3 grid such that they are able to play 9 decimal
numbers in order before their opponent.

Lessons 2-1+2-2: Reading and Writing Very Large and Very Small Numbers
 In standard notation digits take on the value of the place value they
occupy. For example: In 8,654.15 the 6 represents 6 hundred, while the 5
represents 5 tens or fifty.
 In expanded notation numbers are written as the sum of the values of its
digits. For example: 8,654.15 would be written as (8*1000) + (6*100) +
(5*10) + (4*1) + (1*1/10) + (5* 1/100).
 In number-and-word notation numbers are written as numerals and
words. For example: 8,654.15 would be written as 8 thousand, 6 hundred,
fifty-four, and fifteen hundredths.

Practice:
Standard notation         Expanded notation          Number-and-word notation
754

(3* 1/10) + (5* 1/100)

Seventy-five hundredths

(9*100) + (5* 10)

Sixty-six thousandths
Standard notation        Expanded notation           Number-and-word notation
754                       (7*100) + (5*10) + (4*1)    7 hundred, fifty-four
0.35                      (3* 1/10) + (5* 1/100)      Thirty-five hundredths
0.75                      (7* 1/10) + (5* 1/100)      Seventy-five hundredths

950                       (9*100) + (5* 10)           9 hundred, fifty
0.065                     (6* 1/100) + (5* 1/1000)    Sixty-six thousandths

Lessons 2-3 to 2-6: Addition, Subtraction, and Multiplication of Decimals
When adding or subtracting decimals, you have to line the decimals carefully
so that you are adding ones to ones, tenths to tenths, etc.
For example, 0.50 + 1.50 would be written as follows: 0.50
+ 1.50

When multiplying decimals, move the decimal to the right the same number of
places as the exponent when the exponent is positive. For example, \$0.50 * 100
students = \$50.00 because we moved the decimal two places to the right. Move
the decimal to the left the same number of places as the exponent when the
exponent is negative. For example, \$0.50 * 0.1 (or 1/10) = \$0.05.

Practice:
Player One: You’re playing Doggone Decimal. The target number is 100. You get
the digits 2, 5, 3, and 8. How will you build two decimal numbers whose product
is closest to the target number?

Player Two: Your opponent gets the digits 2, 1, 5, and 9. How should he/ she
build two decimal numbers?

Who wins? How much did they win by?
Player One: Possible Products:
23. * 5.8= 133.4         25. * 3.8= 95.0                28. * 3.5= 98.0
32. * 5.8= 185.6         35. * 2.8= 98.0                38. * 2.5= 95.0
52. * 3.8= 197.6         53. * 2.8= 148.4               58. * 2.3= 133.4
82. * 3.5= 287.0         83. * 2.5= 207.5               85. * 2.3= 195.5
Player Two: Possible Products:
21. * 5.9= 123.9         25. * 1.9= 47.5                29. * 5.1= 147.9
12. * 5.9= 70.8          15. * 9.2= 138.0               19. * 5.2= 98.8
52. * 1.9= 98.8          51. * 2.9= 147.9               59. * 2.1= 123.9
92. * 1.5= 138.0         91. * 2.9= 263.9               95. * 114.0

Player Two will win IF he/she does 52. * 1.9= 98.8 or 19. 8 5.2= 98.8, because
the best that player one can do is 35. * 2.8= 98.0 or 28. * 3.5= 98.0. Player Two
wins by 0.8 points. This was a close game!

*Play Doggone Decimal for more practice with multiplying decimals.

Lessons 2-8: Dividing Decimals and Interpreting Remainders
with the multiplication of decimals. Make sure that your answer is reasonable.

For example, 80 brownies divided by 100 people will result in less than one
whole brownie each, since 80 / 100 = 0.8000. 08.000 would not make sense,
since this is more than one.

Another way to think of dividing is multiplying by a decimal; so the problem above
could be represented as: 80 * .01= 0.80, which is less than one brownie each.

You can add as many zeroes as you’d like to whole numbers to the right of the
decimal point.

For example, if you wanted to find your average score for quizzes: 87, 63, 95,
and 68, you would compute the sum (313) then divide this by 4. You’ll find that 4
* 78 is 312, so you have a remainder of 1. If you add a decimal point and 2
zeroes, you can continue to divide until you get 78.25 with no remainder.
78.25
4/ 313.00
28
33
32
10
8
20
20
0
Practice:
You go out to eat with 3 friends. The bill comes to \$51. How much does each
person owe?

12.75
4l 51.00
4
11
8
30
28
20
20
0
Lessons 2-9: Scientific Notation for Very Large and Very Small Numbers
 102 has a positive exponent and means 10 * 10 or 100.
 10-2 has a negative exponent and means 1/10 * 1/10 or 1/100. This can
be written as decimals: 0.1 * 0.1 = 0.01.
 Zero exponents are always equal to one, while an exponent of one
means the number times one. For example, 100 = 1, while 101 = 10 * 1 or
10; likewise 20 = 1, while 21 = 2.
 We use powers of ten to write very large numbers or very small numbers
using scientific notation. Numbers written in scientific notation are
written as products of a factor more than one and less than 10 and a
power of ten.

For example, the width of a hair is about 2.5 * 10-7( or 0.0000025), while the
number of hairs on average will be about 5 * 106 (or 5,000,000).

Practice:
Standard notation            Scientific notation         Expanded notation
356,900,000
9.8 * 105
(3*0.01) + (5* 0.0001)
-3
7.45 * 10
(5* 1000)+(4* 100)+(3*1)

Standard notation      Scientific notation    Expanded notation
356,900,000            3.569 * 108            (3* 100,000,000)+ (5*10,000,000) +
(6*1,000,000)+(9*100,000)
980,000                9.8 * 105              (9*100,000)+(8*10,000)
0.0305                 3.05 * 10-2            (3*0.01) + (5* 0.0001)
0.0745                 7.45 * 10-3            (7*0.01)+(4*0.001)+(5*0.0001)
5,403                  5.403 * 103            (5* 1000)+(4* 100)+(3*1)
Expectations

Goals: Key Understandings: Students will understand that…
   Decimals have a distinct position on a number line and in an ordered list, which may
seem counter-intuitive at first, such that the longer numeral is not always the larger.
[Example: 0.099999 is less than 0.199.]
   There are equivalent forms for any real number, including standard, number-and-
word, expanded and scientific notations.
   The results of an operation depend on the types of numbers involved: multiplying
and dividing decimals might seem counter-intuitive as multiplying by a decimal gives
you a smaller value, while dividing by a decimal gives you a larger value.

Essential Questions:
   Where would decimals be positioned on a number on the number line and how can you
compare the values of various decimals?
   What is the relationship between place value and digits to the left and the right of the
decimal?
    What do you know about decimals as compared to whole #s and how does this help
   How can estimation be used to determine the reasonableness of an answer and when
and how should you round numbers to various place values?

Lesson by lesson expectations:

Students will know…                                Students will be able to…
    There are equivalent forms for any real           Read and write very large numbers.
number, including standard, number-               Interpret and translate between
and-word, and expanded notations.                  expanded, number-and-word, and
standard notations for very large
numbers. (2.1)
   Order very large numbers via the
__________________________________                          game Number Top-It (p 463+464 MM)
___________________________________________________
 Decimals represent specific points on
   Read and write very small numbers.
a number line and can be organized in
(2.2)
an ordered list.
   Interpret and translate between
standard, number-and-words, and
    You must line up the decimals when                 expanded, notations as well as
adding or subtracting so that you are              fractions, and decimals.
values (ones with ones and tenths with            Order decimal numbers via the game
tenths).                                           High-Number Toss (p455 MM).
_                                                          Add and subtract decimals. (2.3)
   Explore precision (Ex: 3.0) and
________________________________________________            rounding decimals (Ex: 2.9 to 3).
   We can use powers of 10 to facilitate        Multiply by powers of ten while
mental math.                                   exploring exponential notation, both
   Positive exponents involve multiplying         positive and negative. (2.4)
tens, while negative exponents involve       Play Doggone Decimal Game (p 310
multiplying tenths.                            SRB).
____________________________________           ______________________________________
 The relationship between the powers of          Estimate products of decimals. (2.5)
10 and place values.                          Practice decimal multiplication.
 Play Multiplication Bull’s Eye (p328
 Multiplying and dividing decimals might           SRB).
seem counter-intuitive as multiplying
by a decimal gives you a smaller            _______________________________
value, while dividing by a decimal            Locate decimal points in products.
gives you a larger value.                       (2.6)
 Multiply decimals using the lattice
method. *Only for those students who
already prefer to use the lattice
method.
__________________________________
 Divide whole numbers. (2.7)
 Partial quotients method reviewed.
_________________________________
___________________________________
 Divide decimals. (2.8)
 Remainders may be represented in a
number of ways depending upon the            Write remainders as decimals.
situation.                                   Interpret remainders.
____________________________________ _____________________________________
 A positive power of 10 is a number            Explore positive and negative powers
that can be represented as a product           of ten. (2.9)
whose only factors are 10’s.                 Translate between scientific and
 A negative power of 10 is a number              standard notations.
that can be represented as a product         Play Scientific Notation Toss (p 331
whose only factors are 0.1’s or 1/10’s.        SRB and p472 MM).
_________________________________              ________________________________
 Use the power key on a calculator.
(2.10)
 Review exponential notation.
 Play Exponent Ball (p 311 SRB and pp
404 + 436 MM)
 Multiply and divide decimals.
____________________________________
 Use scientific notation on a calculator.
(2.11)
 Explore patterns with powers of 10.

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