Math 6 Honors ( Periods 1, 2, & 3)

Problem Solving 1-6 & 1-7

Follow the 5-step plan... handed out in class.. make sure to glue it into your Spiral Notebook,

MAKE SURE TO READ and RE READ the problem!!

When you are finished--> check your work to make sure your answers make sense .
LABEL, LABEL, LABEL

Sometimes you need to use more than one operation.

We turned to Page 19 and looked at problems 1, 2, 4 and 5 .. If you missed class on Friday, make sure to review those problems.
Then we looked at Example 2 on Page 21

Simon had $165 in his checking account. He wrote checks for $32, $19, and$47. How much did Simon have left in his account?

When you write checks, you will subtract them from your balance. First add all the checks that you wrote 32 + 19 + 47 = 98. So that's $98 to subtract from the beginning balance of $165.
165- 98 = 67
So Simon had $67 left in his checking account.

Then we looked at Page 22 the Class Exercises and discussed what operations had to be used for problems # 2 and # 4.

Make sure to check out those problems.
#2
1st OP: addition
2nd OP: division

#4
1st OP: multiplication
2nd OP: addition

Math 6 Honors ( Periods 1, 2, & 3)

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

Algebra Honors (Period 6 & 7)

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}

Math 6 Honors ( Periods 1, 2, & 3)

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!!

Algebra Honors (Period 6 & 7)

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}