Operations with decimals

Addition
EX:    1
          .47
        +.36
          .83
Adding decimal game
http://www.math-play.com/baseball-math-adding-decimals/adding-decimals.html

Subtraction
EX:        51
              .625
            -.238
             .387

Multiplication

Multiplying by powers of 10
to multiply a decimal by a power of 10, move the decimal point one place to the right for each power of 10.
EX:  10x.165= 10 x (1/10+6/100+5/1000) = 10/10 + 60/100 + 50/1000
                                                                            1     +    6/10  +   5/100
                                                                                      =1.65

Division
Measurement concept:  repeatedly measuring off or subtracting one amount from another
 Sharing concept:  the shaded part of a decimal square is divided into four equal parts

.8/2=.4

there are .4 in each group









Dividing powers of 10
Move the decimal point one place to the left for each power of 10
EX:  .37/10= (3/10 + 7/100) x 1/10 = 3/100+7/1000 = .037

Decimals and rational numbers

Place Values

Reading and writing decimals

1208.0925

one thousand two hundred eight and nine hundred twenty-five ten-thousandths

Models for decimals

decimal square intereactive games (must download to play)

http://www.decimalsquares.com/dsGames/

Equality of Decimals

.4 = .40 = .400

Inequality of Decimals

.47 < .6

Even though 47 is greater than 6 more of the square is shaded for .6 than .47

Place value test for inequality of decimals

The greater of two positive decimals that are both less than on will be the decimal with the greater digit in the tenths place.  If the digits are equal you move to the digit to the right and so on.

Rational numbers

Any number that can be written in the form a/b where b does not = 0 and a and be are integers.

Dark blue shaded area

Terminating decimal

If a rational number is a/b is in simplest form it can be written as a terminating decimal if and only if b has obly 2s and or 5s in its prime factorization.

repeating decimals:

Rounding

Decimals can be rounded to the nearest whole number, tenth, hundreth, ect.

•  Find which place value the number is to be rounded to and check the digit to the right
• If the digit on the right is 5 or greater then all digits to the right are dropped and you round up
• If the difit on the right is less then 5 then all the digits to the right are dropped and the place value number stays the same

http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/roundoff.html

Number properties, mental calculations, and estimation

          Number Properties

Closure for addition of fractions
The sum of any two fractions is another unique fraction.
EX:  -2/3 + 4/5 = -2x5 + 4x3 = -10 + 12 = 2
                              3x5     5x3     15     15    15
Closure for multiplication of fractions
The product of any two fractions is another unique fraction
EX:  -2/3 x 4/5 = -2x4 = -8
                              3x5    15        
Identity for addition
The sum of any fraction and 0 is the given fraction.  Zero + any integer = the given integer
EX:5/6 + 0 =5/6 + 0/6 = 5+0 = 5
                                          6       6
Identity for multiplication
http://intermath.coe.uga.edu/dictnary/descript.asp?termID=172
EX:  1 x (-4/5) = 1x-4 = -4
                               5        5
Addition is commutative and associative
http://www.learningwave.com/lwonline/numbers/com_assoc_add.html
commutative:  two fractions that are being added can be interchanged without changing the sum
associative:  in a sum of three fractions the middle numer may be grouped with either of the other two numbers.

Multiplication is commutative and associative
http://www.learningwave.com/lwonline/numbers/com_assoc_mult.html
commutative: two fractions that are being multiplied can be interchanged without changing the product
associative:  In a product of three fractions the middle number may be grouped with either of the other two numbers.

Inverses for addition and multiplication
Addition:  for every fraction there is another fraction called its opposite or inverse for addition such that the sum of the two fractions is 0. 
EX:  3/4 and -3/4
Multiplication:  for every fraction not equal to zero there is a nonzere fraction called its reciprocal or inverse for multiplication such that the product of the two numbers is 1.
EX:  3/8 and 8/3
          Mental Calculations

Compatible numbers
Numbers that can be conveniently combined in a given computation

Substitutions
EX:  2  7 + 1 = 2  7 + 1 + 1 =   3 1
             8    4        8    8     8         8

           Estimation

Rounding
The sum or difference of mixed numbers and fractions can be estimated by rounding each number to the nearest whole number.

Practice:  http://www.ixl.com/math/practice/grade-5-round-mixed-numbers

Compatible numbers
Replacing a fraction with a reasonably close and compatible fraction can be useful when estimating

Operations with fractions

Addition
The concept of addition is the same for fractions as for whole numbers.  When adding whole numbers your combining two sets of objects.  When adding fractions your combining two amounts.

Unlike denominators
Finding the least common denominator
http://www.onlinemathlearning.com/adding-fractions-2.html

Addition of fractions for fractions a/b and c/d
a + c = ad + bc = ad + bc
b    d    bd    bd         bd

Mixed numbers
They are the combinations of wholes numbers and fractions.  The sum of two mixed numbers can be found by adding the whole umbers and the fractions speratly.  If the denominators are unequal you must find a common denominator

Exaples on adding fractions
http://www.themathpage.com/arith/add-fractions-subtract-fractions-1.htm

Subtraction
Make sure the denominators are the same.  Then subtract the numerators and put the answer over the same denominator.

Examples on subtracting fractions with unlike denominators
http://cuip.uchicago.edu/~jwoods/wit/fsubex2.htm

Examples on subtracting mixed numbers

http://www.youtube.com/watch?v=yXiG2iv1Vq8

Multiplication

 whole number times a fraction (repeated addition)

 

• First change 3 into a fraction (1/3)
• Multiply the numerateor (3x3)=9
• Multiply the denominator (1x4)=4

The answer is 9/4

Whole number times a fraction

k x a= ka

      b    b

Example

3 x 2 = 3(2) = 6 = 2

      9      9       9     3

Fraction times a fraction

2/3 x 1/7

• Multiply numerators (2x1) = 2
• Multiply denominators (3x7) =21
• The answer is 2/21

Division

a/b  and c/d when c/d does not = 0

a/b divided by c/d = a/b x d/c = ad/bc

Great video explaining how to divide fractions

http://www.youtube.com/watch?v=OGUaN-F80NA

Fractions

fraction definitions
http://www.math.com/school/subject1/lessons/S1U4L1GL.html

Fraction refers to both a number and to the numeral.  When talking about the top number being the numerator and the bottom number is called the denominator we are thinking of the fraction as a numeral, and when we add two fractions we think of them as numbers.

3 <--- numerator
4 <--- denominator

3 Concepts

1. part-to-whole concept
This concept is the most common used for fractions.  It uses fractions to denote part of a whole.
Denominator:  Tells us how many equal parts are in the whole.
Numerator:  Tells us how many parts we are considering.
Teaching to kids
http://www.teach-kids-math-by-model-method.com/part-whole-concept.html
Example
The fractional part of an iceberg that is under water is 8/9.

2. Division concept
For any number a and b, with b not = to 0 a/b = a divided by b.
Sharing (partitive) concept:  dividing by 50 means there will be 50 parts
Example
dividing 25 pieces of gum to 50 people.  Each person will get 1/2 of a piece of gum.  There are 50 parts and each part is 1/2 of a whole piece of gum.
Video
http://www.youtube.com/watch?v=zq6on5kah3Q
Example
If you have 4 loaves of bread and you want to share it with 10 people you would cut the loaves into 10 equal pieces and each person would get 1/10.

3. Ratio concept
Fractions are used to compare one amount to another.
Example
A boys height is 1/3 of his mothers height.
Explanation
http://www.emathzone.com/tutorials/everyday-math/concept-of-ratio.html

Integers

The Number Line

The number line is a line labeled with the integers in increasing order from left to right, that extends in both directions:

For any two different places on the number line, the integer on the right is greater than the integer on the left.

All integers on the right side of zero are positive and all integers to the left of zero are negative.

Uses of integers

Credits are represented by positive numbers and debts are represented by negative numbers.  This is used when you want to count on both sides of a fixed point of reference.

The graph below shows the difference between exports and imports from 1960 to 2010. 

   p 

Addition

rules:

1. negative plus negative equals negative
2. positive plus positive equals positive if a>b            a+ -b=a-b
3. positive plus negative equals negative if a<b          a+ -b= -(b-a)

examples:

1. -3+-7=-10
2. 13+-5=8
3. 6+-11=-5

Subtraction

for any two integers a and b, a-b is the sum of a plus the opposite of b     a-b=a+-b

example: 15-7= 15+-7=8 (the opposite of 7 is -7)

Multiplication

rules:

1. positive times negative equals negative         ax-b=-(axb)
2. negative times positive equals negative         -axb=-(axb)
3. negative time negative equals positive          -ax-b=axb

examples:

1. 5x-2=-10
2. -7x3=-21
3. -4x-5=20

Division

for any integers a and b, with b not equal to 0

a/b=k if and only if a=bxk

for some integer k in this case b and k are factors of a

rules:

1. positive divided by negative equals negative            a/-b=-(a/b)
2. negative divided by positive equals negative           -a/b=-(a/b)
3. negative divided by negative equals positive           -a/-b=a/b

examples:

1. 24/-6=-4
2. -14/2=-7
3. -30/-6=5