Math 6A (Periods 1 & 2)

Multiplication & Division of Mixed Numbers  7-5
One of the best methods of finding the product of two mixed numbers is to first change them both to improper fractions and then multiply.
Multiply
6( 3 1/12)
First change 3 1/12 to 37/12
Then you have
(6/1) (37/12) =37/2 = 18  1/2

Estimates are a great way to check our computations.
5  3/4 X 4  2/3
First estimate 6 X 5 = 30
So our product should be a little less than 30
23/4 X 14/3 = 161/6 = 26 5/6
To divide one mixed number by another, we change the mixed numbers to improper fractions and use the method you learned in the previous lesson. Multiplying by the reciprocal
Divide 2 2/3 by  10 2/3
8/3 divided by 32/3
is really
8/3 X 3/32 = 1/4

Tuesday's lesson continued with
9/16  X  n = 21/20
We remembered to "squish the coefficient next to the variable..."
9n/16 = 21/20
Then we needed to isolate the variable by multiplying both sides by the reciprocal
(16/9)(9n/16) = 21/20(16/9)
We simplified as much as possible BEFORE we started multiplying.. and got
n = 28/15
n = 1  13/15

Another case was
4/7 X n =16/21
4n/7 = 16/21
Multiply both sides by the reciprocal of 4/7
(7/4)(4n/7) = 16/21 ( 7/4)
Simplify first
n = 4/3
n = 1  1/3

Math 6A ( Periods 1 & 2)

Division of Fractions 7-4

Certain numbers when multiplied together have the product 1
5 X 1/5 = 1
3/4 X 4/3 = 1

Two numbers whose product is 1 are called reciprocals of each other.
Thus 3/4 is the reciprocal of 4/3.
Zero does not have a reciprocal


Look at the following:
We know 18 = 3 X 6 and we know 18 ÷ 6 = 3 as well as 18 X 1/6 = 3
Dividing a number by a fraction is the same as multiplying the number by the RECIPROCAL of the fraction
a/b ÷ c/d = a/b ÷ d/c
Remember- you are using the reciprocal of the divisor... that is , as students want to say "You FLIP the 2nd number!!"

42/ 55 ÷ 36/11
you must rewrite the problem using the reciprocal of the 2nd number
42/55 X 11/36
Now using your skills of observing GCF simplify before you multiply ( MUCH EASIER and FASTER)
42/ 5 X 1/36 which becomes 7/5 X 1/ 6 = 7/30

Math 6A ( Periods 1 & 2)

Multiplication of Fractions 7-3

If a rectangle is divided into 4 equal parts, each part is ¼ of the whole. If each of these parts is then divided into 3 parts, that is into thirds, then there are 12 equal parts and each is 1/(3 ∙4) or 1/12 of the whole.

That is 1/3 of 1/4 is 1/(3 ∙4) or 1/12 and 1/3 ∙ 1/4 = 1/12 is

so another example 2/3 of 4/5 is 2∙4 /(3∙8) or 2/3 ∙4/5 = 8/15


Notice, that the numerator of the product, 8, is the product of the numerators 2 and 4. The denominator of the product, 15, is the product of the denominators 3 and 5

Rule
If a, b, c, and d are whole numbers with b ≠ 0 and d ≠ 0 , then

a/b(c/d) = a∙c/(b∙d)


When multiplying two fractions, you can simplify the multiplication by dividing either of the numerators and either of the denominators by common factors

6/35 ( 7/3) we can simplify first because both 6 and 3 are divisible by 3
2/35 (7/1) and then both 35 and 7 are divisible by 7 so 2/5 (1(1) = 2/5

Try the following

25/6 ( 42/5) What can we do there?

7/8(20/21) How about with these two sets of fractions?

19/20 ( 25/38) … and these fractions?

What happens when you have
15/2(7/8- 5/24)
What must we do first?

PEMDAS... in my classroom...
15/2( 21/24 - 5/24)
= 15/2(16/24)
= 15/2(2/3)
then simplify to
15/1(1/3)
= 5

What about
8/9∗ 15/32∗ 9/10 = 3/8

or 16/11 × 33/20 × 5/3 = 4

Math 6A (Periods 1 & 2)

Addition & Subtraction of Mixed Numbers 7-2

To add or subtract mixed numbers we could first change the mixed numbers to improper fractions and then use the method from 7-1 .
1 4/9 + 3 1/9 = 13/9 + 28/9 = 41/9 = 4 5/9 but that was 5th grade….
In the second method, and the one I prefer, you work separately with the fractional and whole number parts of the given mixed numbers.


STACK THEM!!
3 4/9
1 7/9
4 11/9 = 5 2/9


If the fractional parts of the given mixed numbers have different denominators, we find equivalent mixed numbers whose fractional parts have the same denominator, usually the LCD.

5 3/10 + 7 7/15

Stack

5 3/10
+7 7/15

Draw a line separating the fractional part from the whole numbers Find the LCM of the denominators the LCD and add…

9 5/9 - 4 13/15

Math 6A ( Periods 1 & 2)

Addition and Subtraction of Fractions 7-1

Most of you already know how to add and subtract fractions, although some of you may need just a little review.

5/9 + 2/9 = 7/9
13/12 - 5/12 = 8/12 = 2/3
and that
7/9 – 2/9 = 5/9


13/12 - 5/12 = 8/12 = 2/3
a/c + b/c = (a +b)/c where c does not equal 0
a/c - b/c = (a -b)/c

The properties of addition and subtraction of whole numbers also apply to fractions.
If the denominators are the same— add or subtract the numerators AND use the numerator!!

In order to add two fractions with different denominators, we first find two fractions, with a common denominator, equivalent to the given fractions. Then add these two fractions.

The most convenient denominator to use as a common denominator is the least common denominator of LCD, of the two fractions. That is, the least common multiple of the two denominators.

LCD ( a/b, c/d) = LCM(b, d) where b and d both cannot be equal to 0

For example LCD ( 3/4, 5/6) = LCM(4,6) =12

3/4 = 9/12 and 5/6 = 10/12

Let’s do:
7/15 + 8/9

First find the LCD

LCM(15, 9) Do your factor trees or inverted division – or just by knowing!!

15 = 3• 5
9 = 3²

So LCM(15,9) = [every factor to its greatest power] 3²•5 = 45

Then find equivalent factions with a LCD of 45, and add

7/15 = 21/45

8/9 = 40/45

21/45 + 40/45 = 61/45 = 1 16/45

5/6- 11/24
Stack them and use the LCD
5/6 = 20/24
-11/24 = -11/24

9/24 = 3/8
7/12 + 4/9 + 3/4
several strategies ca be used. You can find the LCD for all three you can use the C+ and the A+
and change it to
(7/12 + 3/4) + 4/9
then add the first two factions
7/12 + 3/4 becomes 7/12 + 9/12 = 16/12 = 4/3
then add 4/3 + 4/9
change 4/3 to 12/9
12/9 + 4/9 = 16/9 = 1 7/9

What about 17/10 - ( 3/5 + 5/6)
You must do the parenthesis first
so 3/5 + 5/6
3/5 = 18/30
5/6 = 25/30
43/30
Now you have
17/10 - 43/30
stack those
17/10 = 51/30


51/30
^-43/30

8/30 = 4/15

Math 7 ( Period 4)

LEAST COMMON MULTIPLE  5.3

Mrs. Lovetoteach had her kindergarten class sorting a pile of buttons. When they separated the pile into groups of 5, there were 3 leftover. When they separated them into groups of 7, there were 3 leftover. When they separated the pile into groups of 9, there were NONE leftover. what is the LEAST number of buttons the kindergarteners could have had?  Hint; you may want to set up a guess and check chart or draw pictures of what’s happening, or use another strategy that you think might work?

You’re taking care of your next door neighbor’s house for the next five week while they are away. These are your chores
1) Take the mail and newspaper in each day

2) Feed the fish every 3 days

3) Take out the garbage every 6 days

4) Feed the snake every 4 days

What is the first day that you will be doing all 4 chores on the same day? How often will you be doing all 4 chores on the same day during those 5 weeks?

This question CANNOT be answered using the GCF

You are NOT trying to find a factor of the days—but instead a Multiple

How to find the LCM  smallest number that your numbers can go into.

Just like the GCF let’s look at the letters backwards to understand it

MULTIPLE  each number given in the problem must go into this number multiples of 2: 2, 34, 6, 8,…

Multiples of 3: 3, 6, 9, 12…

Multiples of 5: 5, 10, 15, 20…

COMMON—must be a number that ALL of the numbers can go into

LEAST  must be the SMALLEST number that all of the numbers go into

There are the same ways to find it as the GCF

1) List all the multiples of each number and circle the smallest one that is common to all the numbers (Yuk- way too much work, if you ask me)

2) Circle every factor in the prime factorization of each number that is different and multiply

3) List the EXPANDED FORM prime factorization in a table and bring down ONE of EACH COLUMN. Then multiply (or you can do this with exponential form- you need the HIGHEST power of each column.

WINDOW BOX Method- My favorite

Create the window BOX method as if you are finding the GCF. The GCF will be on the left as usual.  But your LCM makes a capital L – find the product of the GCF and the last row of your window box ( the last row must be relatively prime) . You can also  multiply one of your numbers with the bottom relatively prime number of the opposite number.

The difference between GCF and LCM

For the GCF you need the LEAST POWER of only the COMMON FACTORS.

For the LCM you need the product of  GREATEST POWER of EVERY FACTOR

Why do we need EVERY FACTOR this time?

because it is a multiple of all your numbers

Mutiples start with each number so all the factors that make up each number have to be in the common multiple of the numbers.

Finding the LCM of 12 and 15. That multiple must be a multiple of 12

12: 12, 24, 36, …AS WELL AS

15: 15, 30, 45, ….

So the COMMON multiple just include

12: 2  x 2 x 3

and 15: 3 x 5

The LCM must have two 2’s and one 3 or 12 won’t do into it. It must also have that same 3 that 12 needs and one 5 or it won’t be a multiple of 15

Find the LCM 54 and 36

You could list all the multiples

This is more difficult than listing method for GCF for 2 reason

You don’t know where to stop as you list the first number—multiples go on forever…

the numbers get big very fast because they are multiples—not factors

Prime Factorization

54 = 2x3 x3 x3

36= 2 x 2 x 3 x 3

The LCM will be one of each factor of each number. (don’t double count a factor that is common to both numbers.

LCM = 2 x 2 x 3 x 3 x 3 = 108

What if by mistake you double up on a factor and use all of them? You will still get a common multiple—it won’t be the least common multiple. Generally you will get a really big number and the bigger the number is, the harder it is to use.

BOX Method

Checking the LCM to make sure it works

To show that the LCM works or each number, use the fraction simplifying concept. The LCM is on the top and the numbers are on the bottom each time

LCM =108

The numbers were 36 and 54

108/36 = 3

and

108/54 = 2

Use this same approach to prove the GCF works but this time you would put the GCF in the Denominator because it needs to go into the numbers

The numbers are still 36 and 54

We found that the GCF was 18

36/18 = 2

54/18 = 3

Math 7 ( Period 4)

GREATEST COMMON  FACTOR 
Factoring OUT the GCF

( the Distributive Property backwards—revisited. At the beginning of the year we reviewed the DP and talked about using it BACKWARDS to look at whether it was easier to simply use O3 to simplify—rathr than distribute. Today I will revisit this concept but we will call it FACTORING the GCF Factor the GCF means

1) find the GCF of 2 or more terms

2) set up a pair of (  )

3) Place the GCF in front of the ( )

4) divide the GCF OUT OF EACH term and place the quotients inside the (  ) with the applicable sign ( + or - )

Factor 36 + 45

the gcf is 9

9(4 + 5)

Notice when you look side the (    ) there shouldn’t be an common factors of the 2 terms—OR you did not USE the GCF.

Notice: You will get the same results either way because FACOTRING THE GCF DOESN’T change the value EVER!!!

36 + 45 = 81

9(4 + 5) = 9(4) + 9(5) = 81

The reason we need to know this is because of variables – so let’s look at a couple of algebraic terms

Factor 36a²b²c² – 45a²c⁵d

The GCF is 9a²c²

 9a²c² (4b²-5c³d)

If you now distribute back, you will get exactly what you started with.

If you look inside the (  )’s you will see that there are no longer any common factors between the 2 terms—they are now relatively prime. This example shows that you get c³ because you factored out 2 of the c’s leaving 3 more of them 
inside the (  )’s.

Math 7( Period 4)

Simplifying & Comparing Fractions 5.4

Equivalent Fractions- just multiply the numerator and the denominator by the same number and you will get an equivalent ( equal fraction to the one you started with.

GOLDEN RULE OF FRACTIONS à do unto the numerator as you do unto the denominator

Simplifying fractions
(your parents & MS Baril call this reducing)

Two great ways:

1) Just divide both the numerator and the denominator by the GCF

2) Rewrite the numerator and the denominator into prime factorization (use factor trees or inverted division) Then simply cross out (cross cancel) each common factor on the numerator and the denominator (they cross out because each becomes 1  such as 4/4 =1)  You are left with the simplified fraction every time!

THE GCF METHOD

One of the reasons we learn the GCF is because it is the FASTEST way to simplify fractions in one easy step!

Just divide both the numerator and the denominator by the GCF

(The problem with this method is if you are not comfortable finding the GCF, you really can’t do this method easily)

The best reason to use this method is because it is the fastest. So imagine you have a “GCF Magical Voice” in your hear.. the voice tells you the GCF of the numerator and the denominator.. you simply use that GCF to divide both the top and the bottom of your fraction and your done… It’s a “gut feeling” – combined with your knowledge of the divisibility rules.. and YOU CAN DO IT!

THE PRIME FACTORIZATION METHOD:

This is sort of the GCF “Incognito” ( In disguise)  Rewrite the numerator and the denominator in prime factorization form (Use Factor Trees or Inverted Division to find the prime factorization, if necessary)

Then simply cross out each common factor on the top with the bottom. You are actually using the ID Property of Multiplication because each “cross out_ is really a quotient of 1 – again 4/4 = 1 You will be left with the simplified fraction every time.

If you actually multiplied together all the “cross outs” you would get the GCF—so you are using the GCF without ever computing it.

THE CROSS OUT METHOD:

You simply think of the first number that comes to your mind that “GOZ-into” both the numerator and the denominator and keep going until its simplified. If the numbers are both even—many students start dividing it in half. and then half again—if both are still even… This probably takes the longest, but in practice, most people use this method!

The problem with this method—you may think that a fraction is simplified but you miss a factor—this especially happens when the number is odd and you are always using 2 to divide.

The best reason to use this method –no one ever forget how to do this method—it comes rather naturally and there are no “precise” steps to do.  

Comparing Fractions

I. Benchmarks

0, ¼1/4, ½ 1/2 , 3/4 ¾ , and 1 (using your gut feeling).

How do you figure out which benchmark to use?

When the numerator is close to the denominator , the fraction is approaching 1 ( 9/11 or  45/55)

When you double the numerator and that is close to the denominator the fraction is close to ½ ½1/2. ( 4/9)

When the numerator is very far from the denominator the fraction is approaching zero ( 1/9 )

Also if one number is improper or a mixed number and the other is a proper fraction, then obviously the number greater than 1 will be bigger.

II. LCD—give them all the same denominator using the LCM as the LCD.

III. Use cross multiplication—sometimes several times to compare

IV. Change them to decimals ( my least favorite)

Comparing negative fractions

1) if one is positive and the other negative—the positive is always bigger… no matter what!

If both NEGATIVEà remember that the one closest to ZERO is bigger.

2) both proper fractions  if you have -3/4 and -1/4 then -1/4 is BIGGER because it is closer to ZERO ( It is to the right of -3/4 on the number line)

3) One is a proper fraction and the other is a improper (or mixed number) the proper negative fraction  will always be bigger because IT IS CLOSER to ZERO

-1/4 will always be bigger than – 5 1/4¼.¼

When cross multiplying with negative fractions—be careful Remember that these cross products are NEGATIVE so use integer concepts.

-3/4 vs – 5/6

Using cross products

(-3)(6) and (4)(-5)

-18 > -20 (integer concept)

-3/4 > -5/6

Math 7 ( Period 4)

GREATEST COMMON  FACTOR Review

You want to tile a 66 in by 72 in area with equal sized tile.
What is the largest square tile you could use?

You invite boys and girls to your birthday party ( 24 girls and 16 boys) the DJ decides to play some party games and each group needs to have the same number of girls and each group must be the same size. You’d like to keep the group size as small as possible because you don’t want to buy too many prizes for the winners. What is the largest number of groups possible and how many would be in each group?

At a music concert they need to set up lots of chairs. If they need 320   chairs for the special guests and 1000 chairs for the general audience AND they want ALL the rows to have the same number of chairs, what would be the GREATEST number of chairs they can put in each row?

Algebra Honors ( Periods 6 & 7)

The Pythagorean Theorem 11-6

Here is an interesting demonstration illustrating the Pythagorean Theorem.

We watched the following in class. Did you catch all the mistakes made?  

What about Homer Simpson?

Algebra Honors (Periods 6 & 7)

The Pythagorean Theorem 11-6

The Pythagorean Theorem can be used to find the lengths of a right triangle. The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side. The other two sides of a right triangle are called the legs of the triangle.

In any right triangle the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs.  a² + b² = c²

Example:

The length of one side of a right triangle I 28 cm the length of the hypotenuse is 53 cm. Write and solve an equation for the length of the unknown side.

a² + b² = c²       a² = c² –b²            

Pythagorean Triples

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

Constructions:

To draw a line segment with a length of √2 , draw a right triangle with legs of length 1 unit . Using that length You can construct a segment  √3 units long… and so on

Converse of the Pythagorean Theorem

If the sum of the squares of the lengths of the two shorter sides of a triangle is equal to the square of the length of the longest side of the triangle, then the triangle is a right triangle.

The Distance Formula

The distance between two pints on the x-axis 
(or a line parallel to that axis)  is the absolute value of the
difference between their x-coordinates.

The distance between two points on the y-axis
(or a line parallel to that axis) is the absolute value of the
difference between their y-coordinates.

To find the distance between two points NOT on an axis or a line parallel to either axis, use the Pythagorean Theorem.

This can be generalized as the Distance Formula

For any points P₁(x₁, x₂) and P₂ (x₂, y₂)

Math 6A (Periods 1 & 2)

Changing a Decimal to a Fraction 6-6

As we have seen, every fraction is equal to either a terminating decimal or a repeating decimal. It is also true that every terminating or repeating decimal is equal to a fraction.

To change a terminating decimal to a fraction in lowest terms, we write the decimal as a fraction whose denominator is a power of 10. We then write this fraction in lowest terms.

Change 0.385 to a fraction in lowest terms

0.385 = 385/1000 = 77/200

Change 3.64 to a mixed number in simple form

3.64 = 3+ 64/100 = 3 + 16/25 =  

We discovered that some fractions became repeating decimals.

For example

4/9 = 0.444444...

31/99= 0.31313131...

275/999= 0.275275275...
243/999 = 0.243243243...

and we found that you could write these with a vinculum.

so

We talked about the 9th's family.... and found the simple rule for changing a single digit decimal with the vinculum over it...

and then we saw that 

and

but we discovered that using our divisibility rules we could simplify 45/99  to 5/11

We then talked about the wonderful 11th's family and found the simple rule for changing those special fractions.

That that point we talked about

Again, using our divisibility rules we found we could simplify 243/999  because we saw that both the numerator and the denominator were divisible by 9. At that point we find that 243/999 = 27/111 But then we realize that both the numerator and the denominator are still divisible by 3. so we can simplify the fraction to 9/37.

This discussion continues with our next class! I can't wait! How about you?

Math 7 (Period 4)

Factoring Numbers & Expressions 5.1

A Prime number has exactly two factors—itself and 1

A whole number greater than 1 that has factors other than 1 and itself is called a composite number.

The number 1 is neither prime nor composite

Zero is neither prime nor composite.

You can use factor tress to find the prime factorization

and we found that there were many ways to use factor trees to arrive at the same prime factorization

So multiple Factor Trees could result in how you factor out number.

The product of prime numbers is called the prime factorization of that number.

You can also use something a called INVERTED DIVISION. It is similar to Factor Trees but you must begin with the smallest  prime factor that goes into the number and keep using it until it no longer works. Then go to the next higher prime factor… and keep going…

No matter which method you use,
you MUST write your PRIME FACTORIZATION from the smallest prime number to the biggest prime number.

54= 2∙3∙3∙3 That’s in expanded notation

54 =2∙3³ That’s in exponent notation

Negative numbers can be factored by using (-1) as a factor

What happens with variables?

-63a³ becomes (-1)∙3∙3∙7∙a³= (-1)∙3∙3∙7∙a∙a∙a

-27r3s = (-1)∙3∙3∙3∙r∙r∙r∙s

We could use exponents and when asked for prime factorization using exponential notation it would be:

(-1)∙3³∙r³∙s 

Math 6A (Periods 1 & 2)

Changing a Fraction to a Decimal 6-5
There are two methods that can be used to change a fraction to a decimal.

1) find an equivalent fraction whose denominator is a power of 10. (this method does not always work but when it does it becomes really easy to change to a decimal)

13/25 multiply the numerator and the denominator by 4 to get 52/100 and then just close your eyes and see Chapter 3... and 0.52


2) divide the numerator by the denominator. It's a great way to determine your score out of 100 and then figure out your percent.

If you got 67/75 on the last test

divide 67 by 75 carefully 0.8933333... you earned a B+

take 3/8 and divide 3 by 8 8 goes into 3 0.375 times

so 3/8 = 0.375

If the numerator is smaller than the denominator we know our number must be between 0 and 1--> it must be a decimal.


When the remainder is 0 as in the case of dividing 3 by 8, it is called a terminating decimal.


By examining a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal.

If the denominator has no prime factors other than 2 or 5, the decimal representation will terminate.

7/40

looking at 40 we notice the prime factorization ( oh no, it's Chapter 5)
40 = 2³·5 Since the only prime factors are 2 and 5

7/40 must terminate.

What about 5/12 ?

12 = 2²· 3 Since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal.


What about 9/12 ? At first it looks the same as the one above, but look carefully and realize 9/ 12 = 3/4

Since 4 = 2² , 4 has no other prime factors except 2, this can be expressed as a terminating decimal.


Let's look at 15/22
Since 22 has the prime factor of 11 we know that this will not terminate. In fact when you divide 15 by 22 you end up with 0.681818181...


We write this as ( Please check page 196) Notice that the bar is only over the 81 and represents a block  numbers that continues to repeat indefinitely and is called a repeating decimal.


EVERY FRACTION CAN BE EXPRESSED AS EITHER A TERMINATING DECIMAL OR A REPEATING DECIMAL.


Let's look at 4/9 = 0.4444444....

5/9 =

7/9 =


31/99 = 0.3131313131...

8/ 11 = 72/99 = .72727272...