integers

Integers- whole numbers such as 1,2,3,4,..... and the negatives of the whole numbers such as -1,-2,-3,-4,...

Rules for multipling and dividing integers
MULTIPLING-

1. Colored Chips- debts(-) credits (+)
1. When you have a positive number first [3*(-4)] then you think of the problem as 3 debts of the size 4. (-)(-)(-)  (-)(-)(-)  (-)(-)(-)  (-)(-)(-)= -12
2. When you have a negative number first [(-2)*3] then you think of the problem as take out 2 groups of 3. (+)(+)(+)(-)(-)(-) (+)(+)(+)(-)(-)(-)= -6, if you take away the 2 groups of positve 3 then it leaves you with 6 negatives.

  DIVIDING-

1. Sharing- divisor is positve and dividend is negative
1.  -10/2  - - - - -     - - - - - Think of the problem as you have 10 candy bars that you want to divide evenly between you and a friend. How many can each of you have? 5, but since the problem has a negative in it then you add the negative sign back onto the front of the number.
2. Measurement- both divisor and dividend are negative
1. -6/-3 think: 6 debts measured out 3 at a time, how many groups can there be?       (+)(+)(+)  (+)(+)(+)  there can be 2 groups.

Rules of signs

Addition:

- + - = -

+ + + = +

+ + - or - + + = find the difference and take the sign of the larger.

Subtraction:

A-B= A+(-B)

A-(-B)= A+B

Mulitplication:

positive x postive= positive

negative x negative= postive

postive x negative OR negative x positive= negative

Division:

positive / positive= positive

negative / negative= positive

positive / negative OR negative / positive= negative

http://www.loisterms.com/lois21.htm

Decimals to Fractions

As a review of decimals-
Powers of tens-
5.02/ 10^2= 0.0502
     4.07*10^-2=
     4.07*1/10^2= [4(1)+0(10^-1)+7(10^-2)]
     78.392/10^7= 0.0000078392
Converting decimals to fractions-
0.1111= 1/9
0.0101010=1/99
0.001001001=1/999
2/9=.2222...
3/9=1/3=.3333...
4/9=.4444...
5/9=.5555.
6/9=2/3=.6666...
7/9=.7777...
8/9=.8888...
9/9=1=.9999... In order to verify this: 2(1/9)=0.111(2)
48/99=.484848....
29/999=.029029029...

Operations with Decimals

Operations with Decimals-
Addition and Subtraction-
Many decimals numbers are written using fractions you can add and subtract them in a way similar to adding and subtracting fractions: you add and subtract "similar" fractions that have the same denominators and decimals that have the same decimal places.
Examples:   
     2.365                                21.05               15.02                     20.09
   +7.257                              +31.04             - 10.01                  - 10.02
----------------                       ---------------         -----------                -------------
     9.226                               52.09                  5.01                      10.07

Mulitplication-
Think of multiplication as repeated addition.
Examples:
  4*.06 So you have 4 groups of .06 added together  .06+.06+.06+.06=.24
  3*.2 So you have 3 groups of .2 added together    .2+.2+.2=.6
You can also set multiplication up vertically:
Examples:
 12.3
 * 1.7   within the 2 numbers that you are mulitplying you have 2 decimal places so that is  how many the answer will have
---------
     861
+1230
------------
20.91 which has the 2 decimal places from above.

Division-
Sharing-
Think of handing out the peices that you have using the divisor as the number of people to hand out to and the dividend as the amout of objects you have to hand out.
Examples:0.10/2 so think of the problem as having 2 friends and having 10 books to hand out to them. ~ ~ ~ ~ ~     ~ ~ ~ ~ ~ so if you hand the books out evenly to each friend they both get 5 but since you have a decimal you add the decimal back into the answer equalling .5

http://www.coolmath.com/prealgebra/02-decimals/06-decimals-adding-01.htm
This is a helpful website that give good and creative ideas of how to teach children the fundamentals of decimals.

Decimals

Expanded Notation-
623.124= 6(100) * 2(10)*3(1)*1(1/10)*2(1/100)*4(1/1000)
34.827=3(10)*4(1)*8(1/10)*2(1/100)*7(1/1000)
Writing out the names-
345.326- three hundred fourty-five and three hundred twenty-six thousandths
9.3425- nine and three thousand four hundred twenty-five ten thousandths
0.25- twenty five hundredths
Models-
 Decimal Squares
 Base Ten Blocks
 Number Line

Multiplication of fractions

A/B * C/D= A(C)/B(D)

Properties-

• Closure Property- if you multiply two fractions you get a fraction
• Commutative Property- order doesnt matter ( a/b * c/d = c/d * a/b)
• Associative Property- (a/b * c/d)e/f= a/b(c/d * e/f)
• Distributive Property- a/b (c/d +- e/f)= a/b * c/d +- a/b * e/f
• Inverse Property- for every fraction a/b, a &b do not = 0 there exists an inverse b/a so that a/b * b/a =1
• Identity Property- a/b *1= a/b * 1/1 = a/b

Decimals and Fractions-
Decimals are based on the prefix "deci" meaning 10.
examples:
0.4=4(10^-1)=4(1/10)=4/10
0.06=6(10^-2)=6(1/100)=6/100=3/50



http://www.youtube.com/watch?v=fCzBRpqvA6k
This shows a video of how to multiply the fractions.

Fractions

Fraction- a quantity that is less than a whole unit, an ordered pair of integers a&b, with b not equaling 0, written as a/b, the integer a is the numerator and b is the denominator.

Models-

• Part to Whole-
• you must agree on a unit (cups, inches, centimeters, ect.)
• you must understand that the unit is subdivided into b parts of a equal sizes
• and that you are only considering a certain number of parts
• Color Region (Area Model)-
• a shape chosen to represent unit which is visualized by coloring some of the regions.
• The Set Model-
• A finite set of objects(u) is chosen to represent the unit which is divided into n(u)= equal parts A fracction is visualized by circling the number of objects that represent numerator of fraction
• Number Line-
• portions of the number line is subdivided into equal size parts

Simplifying Fractions-

• when the numerator and the denominator of a fraction have a common factor that is greater than one they can both be divided by that common factor inorder to reduce the fraction into smallest terms.

Common Denominator-

Fraction Bars-

 This shows that 1/2 and 3/6 is equal and also that 1/3 and 2/6 are equal.

Rules of Signs for Fractions-

-7/9= -7/-9 FALSE

-5/-12= 5/-12 TRUE

3/-5=-3/5 TRUE

Mental Calculation and Estimation-

which is smaller? 1/7 or 1/10    1/10 is smaller

which is larger? 2/3 or 1/3     2/3 is larger

are they equal? 1/3 and 2/6   yes

Rounding-

9//12 almost 1

7 1/5 almost 7