Math 6A ( Periods 2 & 5)

Writing Expressions 3.2 

 Math Operations  vocab of what means addition, subtraction, multiplication, and division
Make sure to glue the '1/2 sheet' of math word phrases that we associate with each of the four basic operations -- into your spiral notebook (SN)

Use a "LET STATEMENT"
Let n = the number
That is the simplest "LET STATEMENT.  You are defining your variable!


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 product of a number and 3   3n
4 times a number  4n
The quotient of 3 and a number  3/n
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.
10 less than a number is n-10
a number increased by 9  n + 9
9 more than a number is also n + 9



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


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 how many dimes are in x dollars?

Use your T- chart
x dollars    dimes
1..................10
2..................20
3..................30
x...................10x

Put 3 known relationships and then ask yourself "What did I do to the left side to get the right side"
What did I do to 1 to get 10? ... answer multiply by 10
What did I do to 2 to get 20? ... answer multiply by 10
What did I do to 3 to get 30?  ... answer multiply by 10
So What must I do to x? it must be the same  ... answer multiply by 10  so 10x is what I am looking for!

How many dollars do we have in n nickles?
set up-- unknown on the left!
n nickles      dollars
20....................1
40....................2
60....................3
n......................?
Again, Ask yourself  "What did I do to the left side to get the right side"
What did I do to 20 to get 1? ... divide by 20
What did I do to 40 to get 2?  ... divide by 20
What did I do to 60 to get 3?  ... divide by 20
So what must I do to n? it must be the same  ... divide by 20 so n/20

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 6A ( Periods 2 & 5)

Algebraic Expressions 3.1 

An algebraic expression is an expression that may contain numbers, operations and one or more symbols (called variables) A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable.


Parts of an algebraic expression are called TERMS
5p + 4


When we write a product that involves a variable, we usually omit the multiplication symbol (whether that be written as x or as ∙ or even with parentheses). Thus, 3 x n is written as 3n
and 2 x a x b is written as 2ab

In numerical expressions for products a multiplication symbol must be used to avoid confusion.

9 x 7 may be written as 9 ∙ 7 or even 9(7)

5p + 4
Let's look at the term 5p
The number that modifies the variable is called a coefficient  Our book defines it as the numerical factor of a term that contains a variable
now let's look at the term +4 It is a term without a variable and it is called a constant.
We will
a)identify terms,
b)name the coefficients ,
c) name the constants

5x + 13                     a)5x , +13                b) 5              c) 13
2x²  + y + 3              a) 2x²  ,y ,  3            b) 2, 1           c) 3
4x²  -3y + z - 5      a) 4x²  ,-3y, z,  - 5     b) 4, -3, 1        c) -5


Using Exponents
d∙d∙d∙d = d⁴ 
1.5h∙h∙h = 1.5h³

h∙h∙h∙o∙o∙o= h³o³ = (ho)³ = hohoho!

When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equals sign is called the left side of the equation and the expression to the right of the equals sign is called the right side.
expression = expression

When a number is substituted for a variable in the variable expression and the indicated operation is carried out, we say that the variable expression has been evaluated. For example, if n has the value 6 in the variable expression 3n, then 3n has the value 3 (6), or 18
Example: Evaluate the expression 6a when the variable has the following values:
6a; 2, 4, 6, 8
You would substitute in each value for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48

148 ÷ 4 =
148/4
37


if m = 3 and n = 18
n ÷ m
substitute in
n/m or (18)/(3) = 6


If y = 18 and x = 8
4y ÷ 3x immediately set this up as
4y/3x

Now substitute in your values
4(18) / 3(8)
72/24 = 3

Evaluate k + 10 when k = 25
(25) + 10 = 35
When k = -25
(-25) + 10 = -15
Evaluate 4n when n = -12
4(-12) = -48
Now  a÷b   when a= 16 and b = 2/3
Yikes... think
16 ÷ 2/3  WOW-- instead of dividing we think multiply by its reciprocal.
The reciprocal of 2/3 is 3/2
so 16 ( 3/2) = 24
What happens with
3x-14 if x = 5
3(5) - 14 = 1
You must use the ORDER OF OPERATIONS
z² + 8.5 when z = 2
(2)² + 8.5
4 + 8.5
12.5
What about 30 - 24÷  y  when y = 6
THINK:  Order of Operations
30 - 24 ÷ 6
30 - 4 = 26
Saving for a Skateboard
It's on sale for $125. You have $45.00 and you are saving $3.00 each week.
How could we write an expression for how much you could save after w weeks?
We made a chart for weeks 1-4
and then realized we could write
45 + 3w
Why did we put the 3 in front of the w?
The three represents the $3 we save each week-- and you always put the coefficient first
How much did you save after
4 weeks? Substitute 4 for w--> 45 + 3(4) = 57  $57
10 weeks? 45 + 3(10) = 75  $75
20 weeks? 45 + 3(20) = 105  $ 105
What would be the expression if you could save $5 a week?
45 + 5w

Algebra ( Periods 1 & 4)

Chapter 4-3 Point Slope Form 

There is another way to solve for a line other than using slope-intercept form  Although most people use the slope- intercept form for all cases, the POINT-SLOPE Form  is actually easier—everything is built into the format!  You do not need to “put it altogether” at the end!

MY FAVORITE!!

It gives you exactly a point and the slope—just by looking at it!

You do not need to do anything BUT graph!!

Point- Slope Form of a line:

y-y₁ = m ( x- x₁)

Example: ( 3, 1) is a point on the line and m = 2

y – 1 = 2( x – 3)

What you have now is point-slope form of the line!

You can graph easily starting at (3, 1) and going up 2 and to the right 1

You can also simplify this and get the slope- intercept form of the line:

y - 1 = 2x – 6

y = 2x – 5

If you are trying to link the Slope-Intercept form to the Point-Slope form of the line:  The point slope version eliminates one step from using the slope intercept form

In the slope intercept form, you plug in the point and the slope and solve for b—
AND THEN rewrite the equation using the intercept that you found.

In point slope form, once you plug in the point and slope, you just simplify and the equation is already done!

IF you don’t have the slope, you will need to compute it with the formula—just like you did with Slope Intercept FORM

The biggest difference is that any point is plugged into this form, while  the Slope- Intercept Form focuses on the ONE specific point on the y axis 

Algebra Honors (Period 6)

Chapter 4-2 Writing Equations in Slope-Intercept Form 

We usually use the slope- intercept form of the line as our  ”template”

We know that y = mx + b so we can substitute that in what we know (what the problem gives us as information) and solve for whatever we are missing

It helps to memorize this little rhyme (Mrs Sobieraj made it up!)

Oh mystery line,
What could you be?
If I could just find you,
 y = mx + b
First I’ll find m,
Then I’ll find b
Then I’ll put it all together
And I will see:
y = mx + b

The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:

1) Do I have the slope (m)? If not find it by using the slope formula or counting it if you have the graph—(carefully pick two sets of integer points)

2) Do I have the y- intercept (b) ? If not, find it by plugging in a point and the slope and solving for b or if you have the graph, just read it on the y axis.

3) Remember: Put it all together in ONE equation at the end!

There are FIVE general cases of mystery lines

First Case:
You are given the slope and the y intercept (that is the easiest case)
For example:  you are given m = 3/2 and b = - 7/5

Just plug in to the generic slope intercept equation

y = (3/2)x – 7/5

Second Case:
You have a graph of a line and need to determine the equation
Look at the graph and find 2 easy points to use to find the slope ( make sure they are integers) If the y intercept is not an integer—then follow the FOURTH CASE (below) completely!

Put the information together in y = mx + b form

Third Case:
You are given a point and the slope and need to find the intercept ( b)
Example: ( 3, 1)is a point on the line and m = 2

Plug in the point and the slope and find b

That is, start with y = mx + b
You have a point ( 3, 1) plug it in to that equation:

1 = (2)(3) + b
1 = 6 + b

-5 = b or
b = -5

Now put it altogether with the given slope of m = 2 and the y intercept ( b) which you just found

y = 2x – 5

Fourth Case:
You are given a point and the y intercept and need to find the slope > Let’s use the point ( 3, 1) again but this time you are given b = 2

Again you can use y = mx + b . This time, however you are solving for m ( the slope)
1 = 3m + 2
-1 = 3m

-1/3 = m
m = -1/3

Again, NOW put it all together with the given intercept and the slope you just found
 y = (-1/3)x + 2

Fifth Case:
You are given 2 points on a line and need to find the slope and the y intercept
Example: ( 1, 3) and ( -2, -3) are 2 points on the line

You first need to find the slope using the formula
m = change in y/ change in x

m = (-3 -3)/(-2-1)  or  (3--3)/(1--2)  which really is (3+3)/(1+2) or
6/3 = 2

Now plug the slope in with one ( you get to pick—it will work with either) of the points and find the intercept, b

3 = 2(1) + b

3 = 2+b

b = 1

Finally put it all together

y = 2x +1

Chapter 4-3 Point Slope Form 

There is another way to solve for a line other than using slope-intercept form  Although most people use the slope- intercept form for all cases, the POINT-SLOPE Form  is actually easier—everything is built into the format!  You do not need to “put it altogether” at the end!

MY FAVORITE!!

It gives you exactly a point and the slope—just by looking at it!

You do not need to do anything BUT graph!!

Point- Slope Form of a line:

y-y₁ = m ( x- x₁)

Example: ( 3, 1) is a point on the line and m = 2

y – 1 = 2( x – 3)

What you have now is point-slope form of the line!

You can graph easily starting at (3, 1) and going up 2 and to the right 1

You can also simplify this and get the slope- intercept form of the line:

y - 1 = 2x – 6

y = 2x – 5

If you are trying to link the Slope-Intercept form to the Point-Slope form of the line:  The point slope version eliminates one step from using the slope intercept form

In the slope intercept form, you plug in the point and the slope and solve for b—
AND THEN rewrite the equation using the intercept that you found.

In point slope form, once you plug in the point and slope, you just simplify and the equation is already done!

IF you don’t have the slope, you will need to compute it with the formula—just like you did with Slope Intercept FORM

The biggest difference is that any point is plugged into this form, while  the Slope- Intercept Form focuses on the ONE specific point on the y axis 

Algebra ( Periods 1 & 4)

Chapter 4-2 Writing Equations in Slope-Intercept Form 

We usually use the slope- intercept form of the line as our  ”template”

We know that y = mx + b so we can substitute that in what we know (what the problem gives us as information) and solve for whatever we are missing

It helps to memorize this little rhyme (Mrs Sobieraj made it up!)

Oh mystery line,
What could you be?
If I could just find you,
 y = mx + b
First I’ll find m,
Then I’ll find b
Then I’ll put it all together
And I will see:
y = mx + b

The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:

1) Do I have the slope (m)? If not find it by using the slope formula or counting it if you have the graph—(carefully pick two sets of integer points)

2) Do I have the y- intercept (b) ? If not, find it by plugging in a point and the slope and solving for b or if you have the graph, just read it on the y axis.

3) Remember: Put it all together in ONE equation at the end!

There are FIVE general cases of mystery lines

First Case:
You are given the slope and the y intercept (that is the easiest case)
For example:  you are given m = 3/2 and b = - 7/5

Just plug in to the generic slope intercept equation

y = (3/2)x – 7/5

Second Case:
You have a graph of a line and need to determine the equation
Look at the graph and find 2 easy points to use to find the slope ( make sure they are integers) If the y intercept is not an integer—then follow the FOURTH CASE (below) completely!

Put the information together in y = mx + b form

Third Case:
You are given a point and the slope and need to find the intercept ( b)
Example: ( 3, 1)is a point on the line and m = 2

Plug in the point and the slope and find b

That is, start with y = mx + b
You have a point ( 3, 1) plug it in to that equation:

1 = (2)(3) + b
1 = 6 + b

-5 = b or
b = -5

Now put it altogether with the given slope of m = 2 and the y intercept ( b) which you just found

y = 2x – 5

Fourth Case:
You are given a point and the y intercept and need to find the slope > Let’s use the point ( 3, 1) again but this time you are given b = 2

Again you can use y = mx + b . This time, however you are solving for m ( the slope)
1 = 3m + 2
-1 = 3m

-1/3 = m
m = -1/3

Again, NOW put it all together with the given intercept and the slope you just found
 y = (-1/3)x + 2

Fifth Case:
You are given 2 points on a line and need to find the slope and the y intercept
Example: ( 1, 3) and ( -2, -3) are 2 points on the line

You first need to find the slope using the formula
m = change in y/ change in x

m = (-3 -3)/(-2-1)  or  (3--3)/(1--2)  which really is (3+3)/(1+2) or
6/3 = 2

Now plug the slope in with one ( you get to pick—it will work with either) of the points and find the intercept, b

3 = 2(1) + b

3 = 2+b

b = 1

Finally put it all together

y = 2x +1