Math 6A ( Periods 2 & 4)

Distributive Property 1-4
Distributive Property of Multiplication with Respect to Addition
DP+
a(b +c) =ab + ac
(b+c)a = ba +ca

Distributive Property of Multiplication with Respect to Subtraction
DP-
a(b -c) = ab -ac
(b -c)a = ba -ca
What's this property all about.. it is about making it easier for you to simplify!!

13(15) you could use your 5th grade skills and multiply it out or...
13(15) = 13(10 +5) = 130 + 65 = 195
OR
(10 +3)(15) = 10(15) = 3(15) = 130 + 45 = 195
SAME SOLUTIONS
What about
23(11) hmmm...
(20+3)11 = 20(11) + 3(11) = 220 + 33 = 253
OR
23(10 +1) = 23(10) + 23(1) = 230 + 23 = 253

we then took a little sidebar and discuss the trick of elevens.... If you missed that explanation.. ask a fellow student to explain multiply 2 digit numbers by 11... You will love it!!


(101)(34) = (100+1)(34) = 100(34) + 1(34)= 3400 + 34 = 3434

What about 54(29) What could you do to make that easier than just multiplying out???
54(29) = (50 +4)(29) = (50)(29) + 4(29) is that any easier? What about...
54(20 +9) = 54(20) + 54(9) ... hmmm... is that easier?? What about..
54(30-1) = 54(30) - 54(1) that looks easier...
We took the time to try all three methods and discovered that the results were all 1566!!
Thank goodness!!
9× 57 What could we do here?
9(50 +7) = 450 +63 =
OR
9(60 -3) = 540 -27
OR
(10-1)57 = 570 - 57
and we discovered again... they all equaled 513


If you have...
(7 × 9) + (13 ×9)
Look and see what they share...
They both have a nine
Pull that out and put one set of HUGS ( ) with the sign that is in the middle of the two
in this case it is +
so you have
( __+ ___)9
Now look at each term, what is the other number in each of the HUGS
7 is in the first one and 13 is in the second so
(7 +13)9    NOW in this case, you want to add them BEFORE you multiply
(7 +13)9 = (20)9= 180 That's TOO easy that way!!

(117 × 4) - (17 × 4)
Ask yourself.. what do they share? 4
Build your HUGS with the sign in the middle --> in this case it's subtraction
( __- ___ )4
Now put the other terms in place
(117-17)4 = (100)4 = 400 ....OMG that is way too easy now!!


(8× 6) + (2 × 6)
well we could just do it.. like 5th grade and get
48 + 12 = 60
or we could start the DP+
(8 + 2)6 and still get 48 + 12 = 60
OR
you could use the DP+ to make it even simpler
(8 +2)6 = (10)6 = 60 in 2 seconds flat!!

Math 6 H (Period 3)

Comparing and Ordering Decimals 1.5

It is really important to read  decimals correctly:
3.4 is Three and four tenths
3.04 is three and four hundredths
30.40 is thirty and forty hundredths

We graphed 2.6 on the numberline and noticed that we placed a dot directly on the numberline at 2.6
< means "is less than"
> means "Is greater than"
5<8 8="8" and="and"> 5  notice the distinctions 
Put the following in order from least to greatest
2.3,   2.09,  2.32, 2.27, 2.37
In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.
First stack them
To make it easier to compare, first express 2.3  to the same number of decimal places as the rest of the numbers.
2.30
2.09
2.32
2.27
2.37

All of them have a 2 in the units place so we look at the tenths place and notice  2.09 is the smallest
cross that number out and look again at the tenths place.
the next smallest is then 2.27
Now we are left with
2.30
2.32
2.37

The all have a 3 in the tenth's place so we look at the hundredths and notice that 2.30 would be the next in order. Then 2.32 and thus, 2.37 is the largest of the numbers so in order
2.09 < 2.27 < 2.3 <  2.32 <2 .37=".37" p="p">
How would you write One and thirty-five thousandths
1._ _ _   put the place value underlined so you know how far to start then look again at the decimal portion
that is thirty-five thousandths
so
1.035

What about One and three hundred five ten-thousandths?
1._ _ _ _
so
1.0305

Algebra Honors (Periods 5 & 6)

Using Several Transformations 3-3

a+b - b = a
and ab ÷ b = a
Use inverse operations
We can use those to solve equations with more than one step
5n - 9 = 71
by adding 9 to both sides of the equation ( using the +prop=)
we get 5n = 80
By then dividing BOTH sides by 5 using the ÷prop=
we arrive at n = 16
Using set notation {16}

(½)x + 3 = 9
By subtracting 3 from both sides using -prop=
we get ½x = 6
By multiplying BOTH sides by the reciprocal of ½ which is 2 (using xprop=)
we get
x = 12

w-5 = 2
9

multiply BOTH sides by 9 using xprop=
we get
w-5 = 18
adding 5 to both sides using +prop=
w = 23
{23}

32 = 7a + 9a
Combine Like terms FIRST
32 = 16a
2 = a {2}

4(y +8) - 7 =15
You must use the Distributive Property first
4y + 32 - 7 = 15
combine like terms on the left side
4y + 25 = 15
subtract 25 from both sides
4y = -10
divide by 4
y = -10/4
simplify BUT leave as an improper fraction
y = -5/2
Using set notation {-5/2}


But what about
-4(m +12) = 36
Yes you could do the Distributive Property BUT... why not divide both sides by -4 FIRST
-4(m + 12)= 36/-4
-4

m + 12 = -9
subtract 12 from both sides using the -prop=
and get
m = -21
{-21}
(c + 3) -2c -(1-3c) = 2

BEFORE WE COMBINE terms let's look at
-(1-3c) = -1 + 3c that is the inverse of a sum!!

c + 3 -2c -1 + 3c = 2
combining like terms we get
2c + 2 = 2
2c = 0
c = 0
{0}

Algebra Honors (Periods 5 & 6)

Transforming Equations: All Four Op's 3-1 & 3-2

Addition Property of Equality
for all real numbers, a, b, and c
and given a= b
then
a + c = b + c
and c + a = c + b
we abbreviated it as
+prop=
Make sure to read this as the Addition Property of Equality
Now
a -c = b - c
is the Subtraction Property of Equality
-prop=
But realize with a tiny change
a + -c = b + -c you have the +prop=

x - 8 = 17
If we add 8 to both sides ---> That's using the +prop=
x -8 + (+8) = 17 + (+8)
x = 25


x -5 = 9
If we subtract 5 from both sides--> we are using the -prop=
x +5 (-5) = 9 + (-5)
x = 4

Multiplication Property of Equality
ac = bc
and ca = cb
and we write it
xprop=

Division Property of Equality
where c ≠ 0
a/c = b/c
and we write that
÷prop=

(-2/3)t = 8
If we use the Multiplication Property of Equality and
multiply BOTH SIDES by the reciprocal of -2/3
we have,
(-3/2)(2/3)t = 8(-3/2)
t = -12

⎮m⎮ =6
2
using the Multiplication Property of Equality (xprop=)

2⎮m⎮ =6(2)
2
⎮m⎮ =12
so m = -12 and 12
and using the set notation
we have the solution set as {-12, 12}

Math 6 High (Period 3)

Order of Operations 1.3

O³ because Order Of Operations

Grouping symbols ( ) hugs!! so important in middle school...and in math!!
[ ] brackets and { }


These Grouping symbols show which operations are performed first.
Even the Fraction bar is a grouping symbol!!

8 + 3 - 9⋅ 2 ÷ 3

Notice there are NO grouping symbols. WE then tried to come up with a number and in several of our classes we have over 6 different answers... we can't have that happening!!

So follow these rules:
When there are NO grouping symbols
1) Do ALL of the multiplication or division in ORDER from Left to Right
2) THEN do ALL of the addition or subtraction in ORDER from left to right!!

We tried it again... with all sorts of beeps.... and discovered the funnel process

8 + 3 - 9⋅ 2 ÷ 3
8 + 3 - 18 ÷3
8 + 3 -6
11-6
5

WE then tried it on
72 -24 ÷ 3
72 - 8
64

and then

8 + 6 ⋅ 12 ÷ 4 + 6
8 + 72 ÷ 4 + 6
8 + 18 + 6
26 + 6
32

The last one we did in class was
126 - 9÷ 3 ⋅ 28
126 -3 ⋅ 28
126 - 84
42


if you have a series of grouping symbols... do the inside ones first.. called nesting

{ [ ( )]}

(14 + 77) ÷ 7
91 ÷ 7
13

Make sure to show your division as a side bar. Be proud of your work!!
[3 + (4 ×5)]×10
[3 + 20]×10
[23]10
230

P--> Please --> Parentheses ( )
E --> Excuse --> Exponents e²
M --> My ----> Multiplication
D --> Dear ---> division
A--> Aunt --> Addition
S-->Sally --> Subtraction

PEMDAS but we talked about how it should really be
P
E
MD
AS

or
P
E
DM
SA

Because it is multiplication or division, whichever comes first!!
and then addition or subtraction, whichever comes first


After we have checked grouping symbols...
do all the multiplication and division in order from left to right
then,
do all the addition and subtraction in order from left to right

97 -7 5÷ 5×3 + 68
as we scan... or used our typewriter skills...(where are your sound effects???)
we discover we must divide first
funneling down, we get
97 -15× 3 + 68
97 -45 +68
52 + 68
120

what if
t = 12
w = 10
x = 9
y = 4
z = 3

tx- wz
(12)(9) - (10(3)
108 - 30
78

x(y + w)
9(4 +10)
9(14)
126

Math 6A (Periods 2 & 4)

Order of Operations 1-5

O³ because Order Of Operations

Grouping symbols ( ) hugs!! so important in middle school...and in math!!
[ ] brackets and { }


These Grouping symbols show which operations are performed first.
Even the Fraction bar is a grouping symbol!!

8 + 3 - 9⋅ 2 ÷ 3

Notice there are NO grouping symbols. WE then tried to come up with a number and in several of our classes we have over 6 different answers... we can't have that happening!!

So follow these rules:
When there are NO grouping symbols
1) Do ALL of the multiplication or division in ORDER from Left to Right
2) THEN do ALL of the addition or subtraction in ORDER from left to right!!

We tried it again... with all sorts of beeps.... and discovered the funnel process

8 + 3 - 9⋅ 2 ÷ 3
8 + 3 - 18 ÷3
8 + 3 -6
11-6
5

WE then tried it on
72 -24 ÷ 3
72 - 8
64

and then

8 + 6 ⋅ 12 ÷ 4 + 6
8 + 72 ÷ 4 + 6
8 + 18 + 6
26 + 6
32

The last one we did in class was
126 - 9÷ 3 ⋅ 28
126 -3 ⋅ 28
126 - 84
42


if you have a series of grouping symbols... do the inside ones first.. called nesting

{ [ ( )]}

(14 + 77) ÷ 7
91 ÷ 7
13

Make sure to show your division as a side bar. Be proud of your work!!
[3 + (4 ×5)]×10
[3 + 20]×10
[23]10
230

P--> Please --> Parentheses ( )
E --> Excuse --> Exponents e²
M --> My ----> Multiplication
D --> Dear ---> division
A--> Aunt --> Addition
S-->Sally --> Subtraction

PEMDAS but we talked about how it should really be
P
E
MD
AS

or
P
E
DM
SA

Because it is multiplication or division, whichever comes first!!
and then addition or subtraction, whichever comes first


After we have checked grouping symbols...
do all the multiplication and division in order from left to right
then,
do all the addition and subtraction in order from left to right

97 -7 5÷ 5×3 + 68
as we scan... or used our typewriter skills...(where are your sound effects???)
we discover we must divide first
funneling down, we get
97 -15× 3 + 68
97 -45 +68
52 + 68
120

what if
t = 12
w = 10
x = 9
y = 4
z = 3

tx- wz
(12)(9) - (10(3)
108 - 30
78

x(y + w)
9(4 +10)
9(14)
126