Math 6A ( Periods 2 & 4)

Fractions & Mixed Numbers 6-3

1/2 + 1/2 + 1/2 = 3/2

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.
Every improper fractions is greater than 1
A proper fraction is a fraction whose numerator is less than its denominator.
Thus, a proper fraction is always between 0 and 1

1/4, 2/3, 5/9. 10/12 17/18 are all proper fractions

5/2, 8/3, 18/15, 12/5 are all improper fractions

You can express any improper fraction as the sum of a whole number and a fraction
a number such as 1 1/2 is called a mixed number

If the fractional part of a mixed number is a proper fraction in lowest terms, the mixed number is said to be in simple form.

To change an improper fraction into a mixed number in simple form, divide the numerator by the denominator and express the remainder as a fraction.
14/3 = 4 2/3
30/4 = 7 2/4 = 7 1/2

To change a mixed number to an improper fraction rewrite the whole number part as a fraction with the same denominator as the fraction part and add together.

or multiply the denominator by the whole number part and add the fractional part to that...
In class I showed the circle shortcut. If you were absent, check with a friend or ask me in class!!

2 5/6 =

(2 x 6) + 5
6
=17/6


Practice these:

785 ÷ 3

852÷ 5

3751÷ 16

98001÷231


post your answers below in the comments for extra credit !!

Math 6 High (Period 3)

Dividing Integers 4.6

According to your textbook, a useful way to divide integer  is to think of it as a multiplication problem. that is, when you consider   -15  ÷ 3  =?  rethink it as  3 · ? = -15.  

You know that  (3)(-5) = -15  so you can conclude that -15  ÷ 3  = -5   

Use similar reasoning for other division problems!

More generally, when dividing two integers, divide their absolute values and then use the same rules applied to multiplying integers:

Specifically:

The quotient of two positive integers is positive

The quotient of two negative integers is positive

The quotient of two integers with different signs is negative

The quotient of 0 and a non-zero integer is 0.

Note: an expression with 0 in the divisor  such as 4 ÷ 0 is undefined. You cannot divide  a number by 0.

Evaluating an expression such as   -18/x 

a) when x = -3

1.  First substitute in using parentheses for the number     -18/(-3)

2.  Then DO THE MATH  remembering that the quotient of two negative numbers is positive. 

3.  The solution is +6.

b) when x = 3

1. First substitute in using parentheses for the number   -18/(3)

2. Then DO THE MATH remembering that the quotient of two integers with different signs is always negative.   

3. The solution is -6.

Finding an Average With Integers

To find the average,   always add the numbers first.

The example in the book uses the following measurements (in feet) taken from the bottom of Lake Michigan and find the average lake bottom’s elevation to its surface….

-844,  -865, - 900,  -893,  -888

so to add do the following

-844 + (-865) + (-900) +(-893) + (-888)

Now, I recommend stacking them to add carefully.  

Remembering the rule for adding integers—when adding integers with the SAME sign, just add them and USE THEIR sign… so we know the total must still be negative since they are ALL negative… and we get – 4390

Now, there are 5 numbers so we divide by +5   

            -4390/5

We see that we have a negative divided by a positive.  

We know our quotient will be negative

-4390/5 = -878

So  the average of the lake’s bottom elevation is -878 feet or you could state it is 878 feet BELOW the lake surface.

Math 6 High (Period 3)

Multiplying Integers 4.5 

As we discovered any number multiplied by -1  is equal to its opposite.

-1 ∙ a = -a

We use this Multiplicative Property of -1  to develop general rules for multiplying integers.

 Specifically:

The product of two positive integers is positive

The product of two negative integers is positive

The product of two integers with different signs is negative

The product  of 0 and a non-zero integer is 0.

Examples

Multiplying Integers with the same sign:

(4)(7) = 28

(-11)(-4) = +44

Multiplying Integers with different  signs:

(5)(-5) = -25

(-8)(9) = -72

Evaluating an expression such as:

a)   x y     when x = -15 and y = -6

1.  First substitute in using parentheses for the numbers     (-15)(-6)

2.  Then DO THE MATH  remembering that the product of two negative numbers is positive. 

3.  The solution is +90.

b) ( -a)(b) when a = -4 and b = -8 

(Be careful with these types—they can be tricky)

1. First substitute in using parentheses for the number   -(-4)(-8)

2. Now look at –(-4) first  that really says to take the opposite of -4 and we know that the opposite of -4 is 4 so we really have (4)(-8)

3. Then DO THE MATH remembering that the product of two integers with different signs is always negative.   

4. The solution is -32.

Multiplying Three or MORE  Integers

The following is our textbooks instructions:

• To multiply three or more integers work from left to right, multiplying two numbers at a time.
•   Find the product of the first two numbers, the multiply that product by the next number to the right
•      Continue until all numbers have been multiplied to find the final product.

EXAMPLES:

(2)(-3)(4)  = (-6)(4) = -24

(-4)(5)(0) = (-20(0) = 0 

My Note:  But why even do the multiplication? Any number  multiplied by 0 is 0 so you know right away the solution is ZERO!!!

(-1)(-2)(-3)(-4) = (2)(-3)(-4) = (-6)(-4) = 24

We discussed in class to be a sign counter—If the number of negative signs (in your non-zero multiplication problem) is EVEN then your solution will be POSITIVE. If the number is ODD, then your solution will be NEGATIVE.  The sign of the product depends on the number of factors (assuming all are non-zero) that are negative.

The product of two integers with the same sign is POSITIVE

    The product of two integers with different signs is NEGATIVE