Math 6A (Periods 1 & 2)

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.

Then we looked at Page 22 the Class Exercises and discussed what operations had to be used for problems # 1- 4

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

#4
1st OP: multiplication
2nd OP: addition
We also looked at some of the even numbered word problems on Page 22
2. 18 adults $72 total for admission
To find out how much each ticket was
72/18 = 4
Mae sure to label
$4.00
The cost for each is $4.00

4. The Library had 1843 books ... after a shipment of 7 cases the total number of books at the library was 1955 books. How many new books were added to the library? We talked about the unnecessary factof 7 cases... and realized we just needed to subtract.... 1955 - 1843 = 112
They library had 112 new books.

Algebra Honors (Periods 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 6A (periods 1 & 2)

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

Math 7 ( Period 4)

Justifying the properties

What are properties- They are characteristics of math operations that can be identified

Why are they your friends... your BFF  you can count on properties. They  always  work. there are no (0) COUNTEREXMPLES

COUNTEREXAMPLE- an example that shows that something does NOT work.

It counters what you said!

Because you can ALWAYS count on properties, you can use the to JUSTIFY what you do mathematically.

JUSTIFY = giving a reason for doing what you did and you’re reasons are your BFF’s the properties

There are 2 parts of justifying

1) First of all what did you change  or if you’re looking at what someone else did, what changed? (did the order change? did the (  ) change? Has anything been simplified?)

2) What allowed you (or them) to make the change?

(Commutative ?  Associative ? Distributive?)

Example  you’re given (5)(565)(2) but you change it to:

(5)(2)(565)

(10)(565)

5650

JUSTIFY what you did... What did you do to get the solution?

1) You changed the order

2) Commutative Property of Multiplication allows you to change the order

This time you are given

(565)(5)(2) but you change it to  (565)[(5)(2)] and quickly get 565(10) = 5650

Justify... ( What did you do to get your answer?)

1) you put in a set of [  ]

2) Associative Property of Multiplication allows you to either add or take away a set of ( ) and [  ] are part of the group in ( )  

Why do you think Aunt Sally would dislike these properties intensely?

Because properties are the EXCEPTIONS to her RULES (Order of Operations or PEMDAS)

She’s happy though that sometimes your justification can be ORDER OF OPERATIONS ( in other words you just simplified or did the math in the proper order of PEMDAS)

Again why do we love properties?? Why should you care?

Because they make the math easier—sometimes!

Aunt Sally hates them because they allow us to break her rules!

Algebra Honors (Periods 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

Math 7( Period 4)

Distributive Property 1.8

Another Friend
You can make equivalent (equal expressions) using the Distributive Property... this property allows you to multiply each term inside the (  ) instead of doing the (  ) first.
Distributing literally means giving... so think of DP as giving the multiplication to each term inside the (  )

It's Halloween and you are giving candy to each child who comes in the door.  Every ghost, vampire, and princess wants the candy... you can’t just stop somewhere in the middle

c( g + p + v) =  cg + cp + cv

a( b + c + d) = ab + ac + ad

a(b – c- d) = ab – ac – ad

How is this property different from Order of Operations?

Aunt Sally or PEMDAS) says Please ( ) first... do the operations inside the parenthesis first but the DP allows you to multiply the number/variable outside the parenthesis by each term inside to get the same results.

REMEMBER  you must have the following characteristics for the DP:

1.   a number of variable outside the (  ) that is multiplied

2.   terms inside that are separated by  +  or  -

2(3 + 5)

Aunt Sally  ( order of operations) says 2 (8) = 16

Distributive property says 2 (3) + 2(5) = 6 + 10 = 16

Both are correct—just different ways to arrive at the same answer. Now in this case—Aunt Sally was smarter—using her method we arrived at the solutions faster... right?

So when does the DP help the most?

When variables are present

2( y + 5)

Now Aunt Sally can’t do ANYTHING... She is stuck

But the DP can help you simplify to 2y + 2(5) which equals 2y + 10

The DP is also great for mental math

15(23)

with this you just need to multiply out but using the distributive property you can change 23 into 20 + 3

15( 20 + 3) you probably can multiply 15(20) in your head thinking well 15 (2) is 30 so 15(20) must be 300 and 15(3) is 45 so I add 300 + 45 ... and I can do that in my head  345

15(20+3) = 15(20) + 15(3) = 300 + 45 = 345

You can also use the DP Backwards  = FACTORING the GCF in Algebra

You are given

11(24) + 11 (76)

What do you normally do?

Instead look at this expression... you notice that 11 is a COMMON FACTOR in each term....

The DP allows you to go BACKWARDS BEFORE the 11 was distributed to both the 24 and 76.

Think: What would that look like????

Se up a pair of parentheses

(            )

Place the common factor in front

11(            )

Place the two other factors inside with the SAME operation given

11(  24 + 76 )

BUT WHY WOULD YOU WANT TO DO THIS????

Because it easier to do the math this way using Order of Operations

look

11(  24 + 76 )

 = 11(100) = 1100

So the DP can also be stated this way

ab + ac = a( b + c)

Algebra Honors (Periods 6 & 7)

More on Properties 
There are 2 types of Properties- Axioms and Theorems

Axioms = properties we accept as obvious and so we don’t need to prove them

Theorems =  properties that need to be proved USING the Axioms

Examples of Axioms 
Commutative, Associative, Identity, Distributive, Additive Inverse, Multiplicative Inverse

Example of Theorem:
The DP in reverse (a + b) c = ac + bc

Property of Negative 1 and the Inverse Property of a Sum

Both of these properties have to do with multiplying  -1 to another factor

Property of Negative 1

When you multiple any term by -1  you always get the opposite sign

-1(3a) = -3a  and   -1(-3a) = 3a

Both of these examples can be written with just the negative sign—without a 1

-(3a) = -3a  and –(-3a) = 3a

Inverse Property of a SUM

When you multiply by -1 by more than 1 term—you always get the opposite sign of each term

-1(3a + 5) = -3a – 5

-1(-3a – 5) = 3a + 5

Both of the above examples can be rewritten with just a negative sign ( without the 1)

Remember when you see a negative sign it is just a shortened for of -1.

-(3a + 5) = -3a – 5  and  -(-3a -5) = 3a + 5 

Properties of Equality (these are axioms)

Reflexive
a= a
 it looks the same – identical--on both sides !

Symmetric
a= b then b = a
3 + 5 = 8 the 8 = 3 + 5  You can switch the sides of an equation.

Transitive

a = b  b = c then a = c

3 + 5 = 8 and 2 + 6 = 8, then 3 + 5 = 2 + 6

If two things both equal a third, then we can say that they equal each other as well.

The Reflexive Property only has ONE equation

The Symmetric Property only has TWO equations

The Transitive Property only has THREE Equations

Algebra Honors (Periods 6 & 7)

Dividing Real Numbers 2-9
Quotient a divided by b or a ÷b should be written as a fraction
a/b
Definition of division --> Which we abbreviated as Def (÷)
a÷ b = a ⋅(1/b)
to divide by a non zero number, multiply by its reciprocal

a+b
c
= a/c + b/c
where c ≠ 0
and

a-b
c
= a/c - b/c

Justifying leads us to the following:
(a+b)/c
= (a+b)(1/c) --> def (÷)
=(a ⋅ 1/c) + (b ⋅ 1/c) --> DP
=a/c + b/c --> def (÷)

Math 6A( Periods 1 & 2)

 The Distributive Property 1-4

The Distributive Property of Multiplication with Respect to Addition 

DP+

For any whole numbers a, b, and c

 a(b + c) = ab + ac

(b+c)a = ba + ca

13(15) … well you could just multiply it out but using the distributive property you might be able to do this in your head…

13(15) = 13( 10+5) = 13(10) + 13(5) = 130 + 65 = 195

9(11)  now we know this but let’s see what happens if we try using the distributive property

9(11) = 9(10 + 1) = 90 + 9 = 99  YAY!!!

We could have completed this another way…

9(11) = (10-1)(11) = 10(11) – 1(11) = 110 – 11= 99

This last example is really  an example of the

Distributive Property of Multiplication with Respect to Subtraction or

DP-

For any whole numbers a, b, and c

 a(b - c) = ab- ac

(b - c)a = ba - ca

What should we do with the following:

(117∙ 4) – (17∙ 4)

Last year you would multiply  (117∙ 4) and then multiply (17∙ 4) and then take the difference… but that is a long process

What happens if we use the distributive property?

(117∙ 4) – (17∙ 4) = (117-17)∙4= (100)4= 400

WOW—that was so much easier

More Examples of the Distributive Property:
(7∙ 9) + (13 ∙9)

They both share a 9 so pull the 9 out

(7 + 13)9

(20)9 = 180

WOW.. look at that!

But… what happens with

57 ∙ 9

There are a number of different ways we might set this up

(50 + 7)(9) = 50(9) + 7(9)

Or

57(10-1) = 57(10) – 57(1)

Or

(60-3)(9) = 60(9) – 3(9)

We could even set it up with the following but this would not help us to solve it easier…

(37 + 20)(9)… remember our discussion in class—it was to make the multiplication easier

The results for all of the above is 513

What about 25 ∙ 38

We could do 25(40 -2) = 25(40) – 25(2)

Now that’s easy…. 1000 – 50 = 950

25(30 + 8) also works but probably isn’t  quite as easy

25(30) + 25(8) = 750 + 200 = 950

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

(89∙ 5) – (19∙ 5)

(89-19)(5)

(70)(5) =350

So much easier than multiplying each separately!

BUT… what happens if you have

4(17 + 42)

Well we would first add the two numbers in the (  )

That gives us 4(59)… but that really did not make it any easier..

So now look at

4(59) and ask yourself what can you do with that number

4(60-1) = 4(60) – 4(1) = 240 -4 = 236!!

YAY!!

Math 7 ( Period 4)

 The Commutative & Associative Properties 1.7

What are properties... They are characteristic of math operations that can be identified

Why are they your friends?  (BFF) you can count on properties they always work they will NOT let you down... there are NO COUNTER EXAMPLES..

 Properties are the exceptions to Aunt Sally

These properties give you a choice when
it is ALL MULTIPLICATION or
it is ALL ADDITION

(they DO not work for subtraction or division)  lots of counterexamples for those...

 10 – 2 does not equal 2- 10

Commutative property (works for all Multiplication or All addition)

You can switch the order and still get the same sum or product

This is the property that you can hear because you switched the order

 a+ b = b + a  or ab= ba 

Commutative Big C looks like an ear... 

Why care?  it makes the math easier ... sometimes

 Which  would you rather multiply

(2)(543) (5)  or (2)(5)(543)

Associative Property    another friend

 This friend allows you to group all multiplication or all addition ANYWAY you choose

a + (b+ c)  = ( a + b)  + c

a(bc) = (ab)c

again.. to make the math easier

This is the property that you can SEE instead of hear because  you use (parentheses) but you don’t change the ORDER GIVEN

[(543)(5)](2)  what would Aunt Sally say you MUST do? 

but our friend the associative property allows use to move the brackets...

(543)[(5)(2)] which is so much easier to multiply—even in your head...

Algebra Honors (Periods 6 & 7)

Problem Solving Consecutive Integers 2-7


Consecutive- list in natural order from least to greatest
-2, -1, 0, 1 ..

an integer n
the next three numbers are n + 1, n+ 2, and n+ 3
the number that receded n is n-1

Let's create the following equation: The sum of four consecutive integers is 66
Let n be the first integer
n + (n + 1) + (n + 2) + (n + 3) = 66
4n + 6 = 66
n = 15
Now, if the question asked what was the first integer the solution would be {15}
but if the question asks you to state all the integers the solution would be
{15, 16, 17, 18}

If x = 15, how would you write 14 and 16 in terms of x?
14 --> x -1
16 --> x + 1

if m is odd, m + 1 is even

How do you write 4 consecutive even integers starting with x ?
x, x + 2, x + 4, x + 6

How do you write 4 consecutive odd integers, starting with x ?
x , x + 2, x + 4, x + 6


In the above cases, you defined what x was-- in the first case x was even... so you need to add multiples of 2 to continue getting an even number... AND in the 2nd case you defined that this time x was odd.. and again you need to add multiples of 2 to continue getting an odd number.


The sum of 2 consecutive integers is 43. Let x = 1st integer
x + ( x + 1) = 43

The sum of three consecutive odd integers is 40 more than the smallest.
What are the integers?

(Hint: Reread the word problem carefully!!)

Let x represent the 1st odd integer
x + (x + 2) + (x + 4) = x + 40
3x + 6 = x + 40
2x = 34
x = 17
so
{17, 19, 21} is the solution.. and if you substitute in to check, you will see you have solved the question!!

However, in our book, it asks this question a little differently, It says with the domain for the smallest given as {13, 17, 25}.
In this case you could work backwards and substitute each of the three integers to see if it works.

The Reciprocal of Real Numbers 2-8



Two numbers whose product is 1 are called reciprocals or multiplicative inverses
5 and 1/5 are reciprocals because 5⋅1/5 = 1
4/5 and 5/4 are reciprocals 4/5⋅5/4 = 1
and so are (-1.25) and (-0.8) because (-1.25) ⋅ (-0.8) = 1
look closely at (-1.25) and (-0.8) and you realize that they are really -5/4 ⋅ -4/5 = 1

If a is any real number, its reciprocal is 1/a

Property of Reciprocals
for any non zero real number a,
a⋅1/a = 1 and 1/a⋅ a = 1
-a and -1/a are also reciprocals
Notice, we discussed in class that
1/-a = -1/a = -(1/a) those are equivalent

(ab)(1/a⋅1/b) = a (1/a)⋅b(1/b) = a/a⋅b/b= 1⋅1 = 1


so try the following
1/3(42m -3v)
1/3(42m) - (1/3)(3v)
14m - v

Make sure to distribute the negative to each term

How about
-1/20( 5z -4w) - 6(-1/30w - 1/24z)

Again make sure to distribute the negative to each term!!
(-1/4)z + (1/5)w + (1/5)w + (1/4)z
or
-z/4 + w/5 + w/5 + z/4
= 2w/5 also written as (2/5)w

Math 6A (Periods 1 & 2)

Inverse Operations 1-3

Inverse Operations undo each other
Addition & subtraction are inverse operations
Multiplication &division are inverse operations

We can use these relationships between inverse operations to simplify some numerical and variable expressions

17- 8 + 8 instead of doing 17 - 8 and then adding 8 again simply cancel the + and the -
That is, 17 - 8 + 8 = 17

9 - x + x = 9 because adding x and subtracting x are inverse operations and they undo each other...

108 ÷ 6 × 6 ... the long way would be to divide 108 by 6 first. That's 18 and then multiply 18 by 6 and get 108.. but why do that? Using the inverse operations you realize that multiplication and division undo each other so 108 ÷ 6 × 6 is simply 108

3n ÷ 3 = 3

Subtraction Property of Zero
For any whole number a
a-0 = a and a-a = 0

In Algebra
a- a = 0 is called the Property of Opposites ( The Prop of Opp's)

Division Property of One
For any whole number a, except 0
a ÷ 1 = a and a ÷ a = 1

Division Property of Zero
For any whole number a, except 0
0 ÷ a = 0


Remember we discussed that we can never divide any number by 0
We said that any number
a ÷ 0 was said to be 'undefined.'

Math 7 (Period 4)

 Powers & Exponents 1.3
A Power has two parts:

BASE= the large number on the bottom ( base)

EXPONENT = the tiny number raise up

EXPONENTIAL NOTATION = the ay you write the expression with the exponent

EXPANDED NOTATION = showing the multiplication the appropriate number of times reflected by the exponent

In your book this is called “ repeated multiplication”

STANDARD NOTATION = the actual value ( what you would expect) you find this by evaluating the power—doing the repeated multiplication

3⁴ is exponential notation where 3 is the base and 4 is the exponent

(3)(3)(3)(3)  is expanded notation

81 is standard notation

The entire expression is called     3 to the power of 4

(4y)³ = (4y)(4y)(4y) = 64y³

Evaluating a Power

Evaluating is plugging and chugging when a variable is raised to a power

x⁴ when x = 3   3⁴ = 81

Some special powers to know

There are two special powers that have ‘nicknames”

3² can be read  “ 3 to the second power” but its nickname is 3 squared

This is based on finding the area of a square

If one side of the square is 3 then the area would be ( 3)(3) which is 3²

Likewise 3³ can be read 3 to the third power.... but its nickname is “ 3 cubed”

This name is based on finding the volume of a cube.

(3)(3)(3) = 3³

Math 7 (Period 4)

Expressions & Variables 1-2

(we are skipping 1.1 for now—will pick it up  with a unit on data and stats)

LOTS of VOCAB

term: a number and or variable added or subtracted to another number and or variable

factors : a number and or variable multiplied by another number and or variable

addition words: sum, total plus, more than greater than addends  altogether both

subtraction words: difference   minus, less than, less minuend subtrahend

multiplication words: product, times, factors, each

division words: quotient, dividend divisor , shared equally

expression: at least one term ( with NO equal sign)

equation: equal expressions ( it must have an = sign)

numerical: all numbers ( called constants)

variable: a letter ( or symbol) used to represent a number

algebraic (or variable): has at least one variable

simplify a numerical expression: do order of operations you will get a simpler expression

evaluate- given the value of the variable you “plug and chug” in an algebraic expression and simplify using operations

You should be able to write a numerical expression in words:

 3 + 7 is The sum of three and seven

You should be able to write a numerical equation in words:

 3 + 7 = 10

The sum of three and seven is ten

Evaluating VARIABLE ( or algebraic) expressions

Evaluate 5x + y when x = 1.2 and y = 0.8

5(1.2) + 0.8 = 6.8