Math 6A ( Periods 1 & 2)

Writing Mathematical Equations 2-2

The process of writing equations really is just writing two equal expressions and joining them by an equals sign. The words "is" "equals" or "equal to" all indicate that two phrases NAME the same number.
The equals sign is the VERB in a mathematical sentence-- without it you have a mathematical expression!!
Eight increased by a number x is equal to thirty-seven.
In translating this mathematical sentence, I always start but placing the equals sign directly under the words "is equal"
so my first step would be

Eight increased by a number x is equal to thirty-seven.
----------------------------------- = ------------------
Then I would translate each mathematical phrase separately.
Yes, thirty-seven is a mathematical phrase!!
8 + x = 37

Ten is two less than a number n
10 = n - 2

Twice a number w equals the sum of the number and four
2w = w + 4

Notice that when you are indicating multiplication the number (coefficient) ALWAYS is placed in front of the variable.

So three times a number b would be 3b
The only time you see the letter first-- is when you are looking for our ROOM--
which is K 101... a mathematician did not label the room numbers!! :)

Sometimes we need to write an equation for a word sentence that involves measurements. MAKE SURE that each side of the equation uses the SAME UNIT of Measurement!!
For example. Write an equation for: The value of d dimes is $27.50
WE know that the value of d dimes is 10d cents (from our previous lesson) .. but $27.50 is in terms of dollars so we need to change it to cents . $27.50 is 2750 cents ... So our equation becomes 10d = 2750.

Math 7 (Period 4)

Combining Like Terms 2.2

Lots of VOCAB today

Try to understand what each means—rather than just memorizing

terms- a number andor a variable added/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 a symbol used to represent a number

algebraic or variable: has at least one variable

simplify a numerical expression—do order of Operations O³ you will get a simpler expression

evaluate given the value of the variable you ‘plug and chug’ in an algebraic expression and simplify using O³

Coefficient: the number attached to a variable. The number is written BEFORE the variable

Constant: a number with no variable attached.

Combining Like Terms

Need the follow:

1. Same exact variable or variables (or NO variable- meaning that they are constants)

2. Same exact exponent

You can combine by adding or subtraction LIKE TERMS

You cannot combine UNLIKE TERMS

Example:

3a + 4a = 7a

but

3a + 4b  just stays the same it is 3a + 4b

3a + 4a² also cannot be combined it remains 3a + 4a²

You should always combine like terms BEFORE you evaluate.

It is so much simpler to do

25a + 5a – 10a  when a = 25

First combine like terms

25a + 5a – 10a = 20a

Then plug in for a = 25

20(25) = 500

You can always simplify unlike terms with multiplication or division

That is

(5x)(6y) = 30xy

WHY?

Our BFF’s the Commutative and Associative Properties of Multiplication JUSTIFY why you can multiply

(5)(x)(6)(y) Break up the pairs using Associative Prop of Mult

(5)(6)(x)(y) Switch the order using Commutative Prop of Mult

The numbers are together and the variables are in alphabetical order

(5∙ 6)xy  group the tw onumbers together using the Associative Prop of Mult

30xy Simplify using the Order of Operation (O³)

Our properties also Justify combining like terms

b + 3 + 2b

becomes

1b + 3 + 2b  That’s the Identity Prop of Multiplication (IDx) because it lets you include the 1 to b+3+ 2b-à 1b + 3 + 2b

1b + 2b + 3 Commutative Prop of Addition(C+) lets you change the order of the  terms

(1b +2b) + 3 Associative Property of Addition (A+) lets you GROUP the two b terms

b( 1+ 3) + 3 Distributive Property  DP+( backwards) lets you take out the common b that was districbuted to two of the terms

b(3) + 3 Simplify using Order of Operations ( O³)

3b + 3    Commutative Prop of Mult ( Cx) lets you switch the ORDER of the b and 3 to proper form

Algebra Honors ( Periods 6 & 7)

Proof in Algebra 3-8
Some of the properties discussed in the previous chapters are statements we assume to be true. Others are called theorems. A theorem is a statement that is shown to be true using a logically developed argument. Logical reasoning that uses given facts, definitions, properties, and other already proven theorems to show that a particular theorem is true is called a proof. Proofs are used extensively in Algebra as well as Geometry.
Prove: For all numbers a and b, (a + b) – b = a

Many times, only the KEY reasons are states—the substitute principle and the properties of equality are usually not stated. So, the above prove could be shortened to 4 steps:

Prove: For all real numbers a and b, such that a≠ 0 and b ≠0
1/ab = 1/a ⋅1/b

Since 1/ab is the unique reciprocal of ab, you can prove that

1/ab = 1/a⋅ 1/b by showing that the product of ab and 1/a ⋅ 1/b is 1

Once a theorem has been proved, you can use it as a reason in other proofs. Check the Chapter summary on page 88 of our textbook for the listing of properties and theorems that you can use as reasons in your proofs for our homework.

Math 6A ( Periods 1 & 2)

Writing Mathematical Expressions 2-1

Make sure to glue the 'pink 1/2 sheet' of math word phrases that we associate with each of the four basic operations -- into your spiral notebook (SN)

We can use the same mathematical expression to translate many different word phrases
Five less than a number n
The number n decreased by five
The difference when five is subtracted from a number n

All three of those phrases can be translated into the variable expression

n - 5

The quotient of a number y divided by ten becomes y/10. It may look like only a fraction to you-- but if you read y/10 as always " y divided by 10" you have used the proper math language.

Twelve more than three times a number m

Wait-- where are you starting from... in this case you are adding 12 to 3m so you must write

3m + 12


Not all word phrases translate directly into mathematical expressions. Sometimes we need to interpret a situation.. we might need to use relationships between to help create our word phrase.

In writing a variable expression for the number of hours in w workdays, if each workday consists of 8 hours...

8w would be our expression


Some everyday words we use to so relationships with numbers:
consecutive whole numbers are whole numbers that increase by 1 for example 4, 5, 6
So the next consecutive whole number after w is w + 1.

A preceding whole number is the whole number that is 1 less and the next whole number is the whole number that is 1 greater.

So the number which precedes x would be x - 1.
The next number after n is n + 1



What if I asked what is the next consecutive EVEN number after the even number "m"
It would be m + 2



What would it be if I asked what was the next consecutive odd number, after the odd number x?
x + 2


Wednesday September 18th  Lesson continues with Writing Expressions:


What if I asked what is the next consecutive EVEN number after the even number "m"
It would be m + 2





What would it be if I asked what was the next consecutive odd number, after the odd number x?
x + 2
Let's look at those relationships like the workdays from Monday's lesson... We are going to find another strategy to use for some of the more complicated expressions...

First set up a T chart- as discussed in class
put the unknown on the left side of the T chart... The unknown is always the one that reads like " w workdays"

so in this case

w workdays on the left side and under it you put
1
2
3

On the right side put the other variable-- in this case hours
under hours put the corresponding facts you know-- the relationship between workdays and hours as given in this case
hours
8
16
24

all of those would be on the right side of the T chart.

Now look at the relationships and ask yourself--
"What do you do to the left side to get the right side?"

and in this case, specifically...

What do you do to 1 to get 8?
What do you do to 2 to get 16?
What do you do to 3 to get 24?

Do you see the pattern?

For each of those the answer is "Multiply by 8" so
what do you do to w-- The answer is Multiply by 8
so the mathematical expression in this case is "8w."

What about writing an expression for
The number of feet in i inches

i inches is the unknown... so that goes on the left side of the T chart... with feet on the right

i inches ___feet
12...............1
24...............2
36...............3

I filled in three known relationships between inches and feet Now, ask your self those questions again...
"What do you do to the left side to get the right side?"

and in this case, specifically...

What do you do to 12 to get 1?
What do you do to 24 to get 2?
What do you do to 36 to get 3?

In each of these, the answer is divide by 12
so What do you do to i? the answer is divide by 12
i inches ___feet
12...............1
24...............2
36...............3
i................i/12



and it is written i/12

Math 7 (Period 4)

Translating Phrases into Expressions 2.1

Algebraic Expressions represent phrases that have an unknown or variable

Numeric expressions have only numbers

The main strategy to create an algebraic expression is called TRANSLATING. You translate each word by word to create the expression

That is when you need to know the typical operation words like sum, difference, product and quotient, You also need to know other operation words  such as  ‘more than’, times, minus, divided by… and so on….

For most translations you keep the same order of the  way the phrase is stated…
The sum of a number and 12

n + 12

The difference of 12 and a numbers

12- n

The product of 12 and a number

12n

The quotient of 12 and a number

12/n

[PLEASE DO NOT USE THE DIVISION SYMBOL—USE THE FRACTION BAR]

You NEVER use x for times because x is a variable!!
Use the dot in the middle or use  (  )

There are some “Algebra Style Notes” for writing expressions:

1.   You do not need any symbol if you are multiplying either 2 or more variables or a number with a variable because just putting them next to each other means multiplication.

3xy means triple the product of a number and another number.

6z means the product of 6 and a number

2.    For addition and multiplication with a number and a variable, generally the number is FIRST   
You would not see  (y)(3)  You would see 3y. Generally you change 3 + n to n + 3

3.    You generally use a fraction bar for division—especially when you have variables!

4.    You keep variables in alphabetical order and if there are exponents with the same variable, you ALWAYS list them with the HIGHEST power first.   If you have 3a+ 4a²  you would use the Commutative Property of Addition and switch it to 4a² + 3a

There are a couple of translations where the ORDER IS SWITCHED—from the way it is written.  They are both for subtraction. When in doubt of the order, always substitute any number  for the variable and you should be able to determine the order:

For example:

5 less than a number… Not sure?

Think: Change a number to let’s say 20

5 less than 20 is?     15

What did you do in your hear?    20 -5

Notice that the 5 is SECOND even though the word phrase has it FIRST

So now change the 20 back into the variable

n  - 5

Whenever you see LESS THAN… switch the order!!

The same is true for
5 subtracted from a number

You again have a number and then you subtract 5 from that number

n – 5

A more complicated translation is
when you have a number times a SUM or a DIFFERENCE

For example:

5 times the SUM of a number and 3

Thinkà change the number to any number of your choice

5 times the SUM of 4 and 3 is ?

Wait… that’s 5 times 7 or 35

How did you get 35?

First you found the sum (4 + 3) or 7

Then you found 5 time 7 or 35

Now substitute back the variable for the number of your choice

5(n+3)

So you need a set of ( ) to tell the reader to find the SUM first and then multiply.

The same is true of DIFFERENCE times a number
Translate:  Twice the DIFFERENCE of 5 and a number.

2( 5 -n)

When you just can’t translate word by word  you will need other strategies to help you

Stratgey # 2

DRAW a picture

When in doubt, Draw it out!!

I have 5 times the number of quarters as I have dimes.

I try translating it to 5q = d

I check If I assume that I have 20 quarters then 5(20) = 100 dimes.

Does that make sense? That would mean I have a lot more dimes than I have quarters and that isn’t what the words say that I have! The original problem says I have a lot more quarters. So my Algebra is WRONG

I need to switch the variables

5d = q

I check

If I assume that I have 20 quarters then 5d = 20

Does this make sense? YES

If I have 20 quarters and only 4 dimes

Sometimes it helps to make a quick pictures.

Imagine 2 piles of coins

The pile of quarters is 5 times as high as the pile of dimes

You can clearly see that you would need to multiple the number of  dimes to make that pile the same height as the number of quarters.

Strategy #3

Make a  T-Chart

To translate known relationships to Algebra, it often helps to make a T-Chart
You ALWAYS put the unknown variable on the LEFT side AND  what you known on the right

Fill in the chart with 3 lines of numbers and look for the relationship between the 2 columns.
Then you use that mathematical relationship with a variable.

Ask yourself... What do I do to the left side to get the number on the right?

Example The number of hours in d days
Your unknown is d days so that does on the left side

d days----# of hours

1----------à 24

2---------à 48

3 --------à 72

Before you put the variable… look at the relationship between the left column and the right column. Ask yourself... What do I do to the left side to get the number on the right?

Specifically …

What do I do to 1 to get 24?
What do I do to 2 to get 48?
What do I do to 3 to get 72?

The answer to all of those questions is… Multiply it by 24

You must multiply the left column by 24 to get the right column.

So now back at your T chart

d days----# of hours

1----------à 24

2---------à 48

3 --------à  72

d---------à  24d

The last line of the column will then use your variable d
Notice your answer is 24d

What if the questions was to state the number of days in h hours ( the lip of the first example)
Your unknown is h hours this time so that does on the left side

h hours ---à   # of days

24  --------à  1

48  --------à  2

72 --------à  3

Why did I start with 24 not 1 hour this time?

Now look at the relationship between the left column and the right column…  Ask yourself... What do I do to the left side to get the number on the right?

Specifically

What do I do to 24 to get 1?
What do I do to 48 to get 2?
What do I do to 72 to get 3?

The answer to all those questions is…  DIVIDE by 24

You must divide the left column by 24 to get the right column.

The last line of the T-Chart will then use your variable h

h hours ---à   # of days

24  --------à  1

48  --------à  2

72 --------à  3

h  --------à h/24

Algebra Honors (Periods 6 & 7)

Cost,  Income, and Value Problems 3-7
Objective: To organize the facts of a problem in a chart & solve problems involving cost, income, and value
Using a chart to organize the facts of a problem can be a helpful problem solving strategy.
Cost = number of items X price per item
Income = hours worked X wage per hour
Total value = # of items X value per item

Example: Tickets for the senior class play cost $6 for adults and $3 for students. A total of 846 tickets worth $3846 were sold. How many student tickets were sold?
Let x = the number of student tickets sold
Then 846- x = the number of adult tickets sold

The only fact NOT recorded in this chart is that the total cost of the tickets is $3846.
The equation becomes
3x + 6(846 –x) = 3846.
3x + 5076 – 6x = 3826
-3x = -1230
x = 410
Check to make sure what x represented… in this case the number of student tickets so
410 student tickets were sold.


We then turned to Problem 4 on Page 128

A collection of 52 dimes and nickels is worth $4.50. How many nickels are there?
Let d = the number of dimes, so If there are 52 in the collection
52-d must = the number of nickels. So our Chart looks like:

This time we reread the given facts and with the second fact, we realize that we can add the two expressions and set them equal to $4.50
10d + 5(52 - d) = 450
solving this equation, we find that d = 38
Make sure to reread the question… it asked “how many nickels?”
So we need to use 52-d and substitute in 38… 52 -38 = 14
14 nickels is the correct answer


Here are two more good examples:
#6
Celia bought 12 apples, ate two and sold the rest at 20 cents more per apple than she paid. Her total profit was $1.00 How much did she sell each apple for?
Let b = the price she bought each of the apples for
We glued in the “yellow-colored” chart here and completed it to look like:

Now, her profit is the difference between what she sold them for and what she bought them for… Therefore the equation becomes
10(b + 20) – 12b = 100
10b + 200 -12b = 100
-2b = -100
b = 50
Re reading the question we realize we solved for what she bought the apples each for. We need to add 20cents to find out what she sold them for

She sold each apple for 70 cents.

The last problem we did from our textbook was # 14 on Page 129

Jo has 37 coins ( nickels, dimes & quarters) for $5.50 She has 4 more quarters than nickels. How many dimes does Jo have?
Let n = the number of nickels
n + 4 = the number of quarters
So if there was a total of 37 the rest must be dimes
37 – [ n + (n+4)] or 37 –(2n +4)
Let’s set up the chart

Now we know that all of them combined equal $5.50
so
5n + 10[37-(2n+4) + 25(n +4) = 550
5n + 370 – 20n – 40 + 25n + 100 = 550
combining all the n’s

10n + 330 +100 = 550
10n = 120
n = 12
Re reading the question, we find we need to see how many dimes so substitute in
37 -[2(12) +4] = 37 – [24 +4] = 37 -28 = 9
Jo had 9 dimes.

Algebra Honors (Periods 6 & 7)

Problem Solving: Using Charts 3-6
Objective: To organize the facts of a problem in a chart.
Using a chart to organize the facts of a problem can be a helpful problem solving strategy.

Organize the given information in a chart
A swimming pool 25 m long is 13 m narrower than a pool 50 m long
There are two different charts you could create:

In the first chart we started with
Let w = the width of the 2nd pool; so to write the 1st pool in terms of the second we would place w-13 in place of the ?

In the 2nd chart we started with
Lt w = the width of the 1st pool; so we write the 2nd pool in terms of the first, so we would place w + 13 in place of that ?


We then solved the following:

Find the number of calories in an apple & a pear…

1) the pear contains 30 calories more than the apple.
2) Ten apples have as many calories as 7 pears.
Let a = the number of calories in an apple
Then a + 30 = the number of calories in a pear.
We glued in the “orange- colored” chart here and completed it to look like:

This time we reread the given facts and with the second fact, we realize that we can set
The last column expressions equal to each other
10a = 7(a + 30)
solving this equation, we find that a = 70
Therefore, an apple has 70 calories and a pear has 100 calories
Next we used the following two given facts to set up a chart and create an equation
1) An egg scrambled with butter & milk has 1 more gram of protein than an egg fried in butter.
2) Ten scrambled eggs have as much protein as a dozen fried eggs.
Let x = the number of protein in a fried egg.
Then x + 1 = the number of protein in a scrambled egg.
We glued in the “orange- colored” chart here and completed it to look like:

Our equation would be 10(x + 1) = 12x
We then turned to our textbook to Page 122-123 and completed problems 1 and 3 in our spiral notebook as follows:

Solve each problem using the two given facts. Complete a chart to help you solve each problem

1. Find the number of full 8 hour shifts that Maria worked last month
1) She worked twice as many 6 hour shifts as 8 hour shifts
2) She worked a total of 280 hours.

We also found that some problems, such as # 16 on page 124 could be easily worked out without a chart. However, the goal of this section was to attempt to use charts NOW so that in the future, using charts would help organize the information given in more complext word problems.

the Eiffe; Tower is 497 Ft taller than the Washington Monument. If each were 58 ft shorter, the Eiffel Tower would be twice as tall. How tall is each?

Let w = the height of the Washington Monument.
then Let w + 497= the height of the Eiffel Tower.
Taking the phrase " if each were 58 feet shorter" write
w -58 for the Washington Monument and (w + 497) - 58 for the Eiffel tower.
Now it states that the Riffel tower whould be twice as tall as the Washington Monument... so you would need two Washington Monuments to equal the Eiffel Tower... write an equation from that as follows
2(w - 58) = (w+497) -58

2w - 116= w + 439
w = 555
So the Washington Monument  would be 555 ft tall
and the Eiffel Tower would be 1052 ft tall

Algebra Honors ( Periods 6 & 7)

Equations w/ The Variable on Both Sides 3-5

Partial Notes

6x = 4x +18
You can show the steps or if it is easy you can do them in your head... However, if you make too many errors, I will suggest that you actually show the balancing steps!
x = 9
Use the set notation
{9}

3y = 15 -2y
5y = 15
{3}

(4/5)x + 3 = x
This really becomes
3 = x( 1-4/5)
or 3 = (1/5)x
12= x
{15}

How do we do a formal check:
Step 1: Rewrite the original problem
Step 2: Substitute in the solutions
Step 3: REALLY do the MATH

(4/5)x + 3 = x
(4/5)(15) + 3 = 15
12 + 3 ?=? 15
15 = 15
Hooray!

(8+x)/9 = x
We talked about the various ways to do this...
I suggested using cross products
9x = 8 + x
or 8x = 8
{1}

7(a -2) - 6 =2a + 8 + a
7a - 14 - 6 = 2a + 8 + a
combine like terms on each side FIRST
7a - 20 = 3a + 8
4a = 28
a = 7
{7}



2 unique types of equations:
Identity equation: You solve it and you get the same thing on both sides...if you solve until you cannot do anything more, you get 0 = 0.
What this means is that you can pick any number and the equation will work!
The Distributive Property is the simplest example of an Identity Equation:
3(x + 7) = 3x + 21
Distribute on the left side and you'll get:
3x + 21 = 3x + 21
At this point, you should already know this is an Identity!
If you keep solving, you would subtract 3x from each side and you'll get:
21 = 21
and you know again that this is an Identity.
If you now subtract 21 from each side:
0 = 0
BUT I WOULDN'T GO THIS FAR! AS SOON AS YOU HAVE THE SAME THING ON BOTH SIDES, YOU CAN STOP AND SAY IT'S AN IDENTITY EQUATION!!!

Null set equation:
You solve it and you get an impossible answer:
3(x + 7) = 3x + 10
3x + 21 = 3x + 10
21 = 10
WHEN WILL THAT HAPPEN??? NEVER!!! SO THERE IS NO POSSIBLE SOLUTION TO THIS! The answer is the null set.

Algebra Honors (Periods 6 & 7)

 Using Equations to Solve Problems 3-4

We looked at several problems in our textbook on Page 113-114
8. Find three consecutive ODD integers whose sum is 105
Let x = the first odd integer
let x + 2 = 2nd odd integer
Let x +4 = the 3rd odd integer

x + (x +2) + (x + 4) = 105
3x + 6 = 105
3x = 99

x = 33
{33, 35, 37}

10. Five consecutive integers whose sum is 195

let x = the 1st integer
x + 1 = the 2nd
x + 2 = 3rd
x + 3 = 4th
x + 4 = 5th
x + (x +1) + (x + 2) + (x + 3) + (x + 4)  =195
5x + 10 = 195
5x = 185
x = 37
{37, 38, 39, 40, 41}

12.  New tank holds 350 barrels of oil more than the old tank. Together they hold 3650 barrels of oil. How much does each hold?

Let x = the # of barrels in the old tank
let x + 350 = the # of barrels in the new tank
x + (x +350) = 3650

2x = 3300
x = 1650
The old tank holds 1650 barrels of oil
The new tank holds 2000 barrels of oil


23.  Angle A is 3 times as large as Angle B and Angle A is 16 degrees larger than Angle C

If you let x = the measure f Angel A
then
x/3 = measure of Angle B
x-16 = measure of Angle C
This became a bit complicated for some
But..
 x + (1/3)(x) + x - 16 = 180
(7/3)x - 16 = 180
(7/3)x = 196
x = 84
So Angle A is 84 degrees
Angle B is 24 Degrees
Angle C is  68 Degrees

But you could have set your variables as
Let x = the measure of Ange b
Then
3x = the measure of Angle A
3x -16 = the measure of Angle C
x + 3x + (3x - 16) = 180
x = 28
and you still get the same solutions
So Angle A is 84 degrees
Angle B is 24 Degrees
Angle C is  68 Degrees

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