Difference of Two Squares 6-2
Again, remember that FACTORING just UNDOES multiplication.
In this case, the multiplication that you'll be UNDOING is FOILING.
FOIL:
(a + b)(a - b)
You will get:
a2 - b2
This is the DIFFERENCE (subtraction) of TWO SQUARES.
Now FACTOR:
a2 - b2
You undo the FOILING and get:
(a + b)(a - b)
REMEMBER:
You must have two different signs because that's how the MIDDLE TERM disappears!
You will get ADDITIVE INVERSES which will become ZERO
HOW TO RECOGNIZE THE DIFFERENCE OF TWO SQUARES:
1) Is it a binomial?
2) Is it a difference?
3) Are both terms perfect squares?
IF YES TO ALL 3 QUESTIONS, THEN YOU HAVE A DIFFERENCE OF 2 SQUARES!!
HOW TO FACTOR THE DIFFERENCE OF 2 SQUARES:
1) Double hug ( )( )
2) Find square root of each term (sq rt sqrt)(sq rt sq rt)
3) Make one sign positive and one sign negative.
(sq rt + sqrt)(sq rt -sq rt)
Of course, they get more complicated!
We can combine pulling out the GCF with this!
ALWAYS LOOK FOR A GCF TO PULL OUT FIRST!!!!!!
EXAMPLE:
27y2 - 48y4
First, look for a GCF that can be pulled out.
The GCF = 3y2
Factor out the GCF (look at Chapter 6-1):
3y2(9 - 16y2 )
NOW YOU HAVE A DIFFERENCE OF TWO SQUARES TO FACTOR:
3y2(3 - 4y)(3 + 4y)
CALLED FACTORING COMPLETELY BECAUSE
YOU CANNOT FACTOR FURTHER!
Always check your factoring by distributing or FOILing back!
THERE IS NO SUCH THING AS THE SUM OF TWO SQUARES!
a2 + b2 CANNOT BE FACTORED!!!!!
BUT - b2 + a2
= + a2 -b2
= (a + b)(a - b)
BECAUSE IT'S JUST SWITCHED (COMMUTATIVE)
Tuesday, December 7, 2010
Monday, December 6, 2010
Algebra (Period 1)
Factoring Polynomials 6-1
Chapter 5 was a very important building block of Algebra but
CHAPTER 6 IS EVEN MORE IMPORTANT FOR HIGH SCHOOL!!!
REMEMBER THIS KEY CONCEPT: FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL …SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(You'll either distribute or FOIL.)
Factoring is a skill that you must understand to be successful in higher level math!!!
We did a simple version of this back in Chapter 1 and you had a RACE on it!
Factoring is simply UNDOING multiplying
Say you multiplied 5 by 10 and got 50
How would you undo it? DIVIDE by 5!
So FACTORING uses the concept of DIVIDING.
You're actually undoing the DISTRIBUTIVE PROPERTY.
How?
You look for the most of every common factor....the GCF!
Then you pull out the GCF (divide it out of) from each term,
placing the GCF in front of ( )
EXAMPLE:
FIRST, DISTRIBUTE: 2m2n (2n2 + n + 3) =
4m2n3 + 2m2n2 + 6m2n
Now, pretend you don't want the 2m2n to be distributed anymore...
What should you end up with once you UNDO the Distributive Property?
2m2n (2n2 + n + 3)
That's exactly what you started with!
So is it that easy?
Well yes... and no...
Yes because that is the answer …and …
No because it was only that easy because I gave you how it started!
You won't know how it started in a real problem!
THIS IS AN EXAMPLE OF THE WAY YOU WOULD USUALLY SEE IT.
The question would say:
FACTOR: 4m2n3 + 2m2n2 + 6m2n
Step 1: What does each term have in common (what is the GCF) ?
They each can be divided by 2m2n
Step 2: Put the GCF in front of a set of ( ) and divide each term by the GCF
2m2n ( 4m2n/2m2n + 2m2n2/2m2n + 6m2n/2m2n)
Step 3: SIMPLIFY and you'll get:
2m2n (2n2 + n + 3)
Step 4: Check your answer!!!!!
Always check your factoring of the GCF by distributing back!
(incognito, it should be the same thing)
Another check is to make sure you have factored out the entire GCF.
Look inside the parentheses and ask yourself if there is still any factors in common between the terms.
If there is, then you haven't factored out the GREATEST Common Factor.
For example, let's say in the prior example that you only factored out 2mn instead of 2m2n. You would have:
2mn ( 4m2n3/2mn + 2m2n2/2mn + 6m2n/2mn)
= 2mn(2mn2+ mn + 3m)
If you just check by distributing back, the problem will check.
BUT ...
Look inside the ( ) and notice that each term still has a common factor of m!
So this would not be the fully factored form!
So make sure you always look inside the ( )!!!
RELATIVELY PRIME TERMS -
TERMS WITH NO COMMON FACTORS
THAT MEANS THAT THEY CANNOT BE FACTORED
(GCF = 1)
We say they are "not factorable"
Chapter 5 was a very important building block of Algebra but
CHAPTER 6 IS EVEN MORE IMPORTANT FOR HIGH SCHOOL!!!
REMEMBER THIS KEY CONCEPT: FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL …SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(You'll either distribute or FOIL.)
Factoring is a skill that you must understand to be successful in higher level math!!!
We did a simple version of this back in Chapter 1 and you had a RACE on it!
Factoring is simply UNDOING multiplying
Say you multiplied 5 by 10 and got 50
How would you undo it? DIVIDE by 5!
So FACTORING uses the concept of DIVIDING.
You're actually undoing the DISTRIBUTIVE PROPERTY.
How?
You look for the most of every common factor....the GCF!
Then you pull out the GCF (divide it out of) from each term,
placing the GCF in front of ( )
EXAMPLE:
FIRST, DISTRIBUTE: 2m2n (2n2 + n + 3) =
4m2n3 + 2m2n2 + 6m2n
Now, pretend you don't want the 2m2n to be distributed anymore...
What should you end up with once you UNDO the Distributive Property?
2m2n (2n2 + n + 3)
That's exactly what you started with!
So is it that easy?
Well yes... and no...
Yes because that is the answer …and …
No because it was only that easy because I gave you how it started!
You won't know how it started in a real problem!
THIS IS AN EXAMPLE OF THE WAY YOU WOULD USUALLY SEE IT.
The question would say:
FACTOR: 4m2n3 + 2m2n2 + 6m2n
Step 1: What does each term have in common (what is the GCF) ?
They each can be divided by 2m2n
Step 2: Put the GCF in front of a set of ( ) and divide each term by the GCF
2m2n ( 4m2n/2m2n + 2m2n2/2m2n + 6m2n/2m2n)
Step 3: SIMPLIFY and you'll get:
2m2n (2n2 + n + 3)
Step 4: Check your answer!!!!!
Always check your factoring of the GCF by distributing back!
(incognito, it should be the same thing)
Another check is to make sure you have factored out the entire GCF.
Look inside the parentheses and ask yourself if there is still any factors in common between the terms.
If there is, then you haven't factored out the GREATEST Common Factor.
For example, let's say in the prior example that you only factored out 2mn instead of 2m2n. You would have:
2mn ( 4m2n3/2mn + 2m2n2/2mn + 6m2n/2mn)
= 2mn(2mn2+ mn + 3m)
If you just check by distributing back, the problem will check.
BUT ...
Look inside the ( ) and notice that each term still has a common factor of m!
So this would not be the fully factored form!
So make sure you always look inside the ( )!!!
RELATIVELY PRIME TERMS -
TERMS WITH NO COMMON FACTORS
THAT MEANS THAT THEY CANNOT BE FACTORED
(GCF = 1)
We say they are "not factorable"
Thursday, December 2, 2010
Math 6 Honors (Period 6 and 7)
Angles and Angle Measure 4-3
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol from the textbook.
To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.
To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.
We often label angels with their measures. When angles have equal measures we can write m angle A = m angle B
We say that angle A and angle B are congruent angles
If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”
Angles formed by perpendicular lines each have measure of 90° . A 90° angle is called a right angle. A small square is often used to indicate a right angle in a diagram
An acute angle is an angle with measure less than 90°. An obtuse angle has measure between 90° and 180°
Two angles are complementary if the sum of the measures is 90°
Two angles are supplementary if the sum of their measures is 180°
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol from the textbook.
To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.
To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.
We often label angels with their measures. When angles have equal measures we can write m angle A = m angle B
We say that angle A and angle B are congruent angles
If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”
Angles formed by perpendicular lines each have measure of 90° . A 90° angle is called a right angle. A small square is often used to indicate a right angle in a diagram
An acute angle is an angle with measure less than 90°. An obtuse angle has measure between 90° and 180°
Two angles are complementary if the sum of the measures is 90°
Two angles are supplementary if the sum of their measures is 180°
Tuesday, November 30, 2010
Algebra (Period 1)
Multiplying Polynomials 5-11
To multiply two polynomials, multiply each term of one polynomial by every term of the other. THEN ADD the results.
The textbook shows a column approach, please see page 249 for instructions.
In class we used the BOX method... and then combined terms.
When you multiply a trinomial by a binomial or two trinomials, it gets really tricky!
2 ways:
1) box method
2)column method (double or triple distributive with columns to combine like terms)
If you have a trinomial times a binomial, it's easier to use the Commutative Property
and make it a binomial times a trinomial:
(x2 + x - 1) (x - 1)
switch it to
(x - 1) (x2 + x - 1)
Remember the following rules
(A + B)(A + B) = (A + B) 2 = A2 + 2AB + B2
(A - B)(A - B) = (A - B)2 = A2 -2AB + B2
(A + B)(A - B)= A2- B2
You can use FOIL to multiply two binomials
remember FOIL is First Terms, Outside Terms, Inside Terms, Last Terms
You can always use FOIL-- or the BOX method but knowing these rules will make computation quicker if you know the above rules!!
To multiply two polynomials, multiply each term of one polynomial by every term of the other. THEN ADD the results.
The textbook shows a column approach, please see page 249 for instructions.
In class we used the BOX method... and then combined terms.
When you multiply a trinomial by a binomial or two trinomials, it gets really tricky!
2 ways:
1) box method
2)column method (double or triple distributive with columns to combine like terms)
If you have a trinomial times a binomial, it's easier to use the Commutative Property
and make it a binomial times a trinomial:
(x2 + x - 1) (x - 1)
switch it to
(x - 1) (x2 + x - 1)
Remember the following rules
(A + B)(A + B) = (A + B) 2 = A2 + 2AB + B2
(A - B)(A - B) = (A - B)2 = A2 -2AB + B2
(A + B)(A - B)= A2- B2
You can use FOIL to multiply two binomials
remember FOIL is First Terms, Outside Terms, Inside Terms, Last Terms
You can always use FOIL-- or the BOX method but knowing these rules will make computation quicker if you know the above rules!!
Math 6 Honors (Period 6 and 7)
Points, Lines, Planes 4-1
We can describe but CANNOT DEFINE point, line or plane in Geometry
We use a single small dot to represent a point and in class we labeled with a P and we called it Point P
A straight line in Geometry is usually just called a line...
Two points DETERMINE exactly ONE LINE
we connected Point P with Point Q and created Line PQ
We placed this type of arrow ↔ over PQ to show a line
↔
PQ
that would represent the line PQ but we found we could write
↔
QP
and mean the SAME line!!
You can name ANY line with ANY TWO points that fall on that line!! USE only TWO points to name a line!!
Three or more points on the same line are called collinear. Notice the word "line" in collinear.
collinear
Points NOT on a same line are called noncollinear.
We have a RAY if we have an endpoint and it extends through other points. We name the rame by Naming the endpoint FIRST
→
PQ is RAY PQ and it begins at P and goes through Q. It is NOT the SAME as
→
QP which is Ray QP, which begins at Q and goes through P
Segments are parts of lines with TWO ENDPOINTS.
⎯
PQ is a segment with endpoints Point P and POint Q
We then looked at the drawing from page 105 and the class named all of the names for the line in the drawing, all of the rays that existed in the figure as well as the segments. We found that there were just 3 collinear points: A, X, B but that we could name 3 sets of non collinear points
X, Y, B and A, X, Y, AND A, B, Y
Three non-collinear points determine a flat surface called a plane.
We name a Plane by using three of its non collinear points!! Plane ABC was out example.
Lines in the same plane that do not intersect are PARALLEL lines. Two segments or rays are parallel if they are parts of parallel lines.
↔
AB is parallel to
↔
CD
may be written
↔ ↔
ABllCD
using two straight lines to indicate parallel
Parallel lines DO NOT intersect.
Intersecting lines intersect in a single point!!
Planes that do not intersect are called parallel planes... we looked around the room and found examples of parts of planes.. noticing which ones were parallel!! (the floor and ceiling were a great example)
Then we drew the box from PAge 106 and identified parallel segments and lines from that box.
Two non parallel lines that do not intersect are called SKEW LINES.
We can describe but CANNOT DEFINE point, line or plane in Geometry
We use a single small dot to represent a point and in class we labeled with a P and we called it Point P
A straight line in Geometry is usually just called a line...
Two points DETERMINE exactly ONE LINE
we connected Point P with Point Q and created Line PQ
We placed this type of arrow ↔ over PQ to show a line
↔
PQ
that would represent the line PQ but we found we could write
↔
QP
and mean the SAME line!!
You can name ANY line with ANY TWO points that fall on that line!! USE only TWO points to name a line!!
Three or more points on the same line are called collinear. Notice the word "line" in collinear.
collinear
Points NOT on a same line are called noncollinear.
We have a RAY if we have an endpoint and it extends through other points. We name the rame by Naming the endpoint FIRST
→
PQ is RAY PQ and it begins at P and goes through Q. It is NOT the SAME as
→
QP which is Ray QP, which begins at Q and goes through P
Segments are parts of lines with TWO ENDPOINTS.
⎯
PQ is a segment with endpoints Point P and POint Q
We then looked at the drawing from page 105 and the class named all of the names for the line in the drawing, all of the rays that existed in the figure as well as the segments. We found that there were just 3 collinear points: A, X, B but that we could name 3 sets of non collinear points
X, Y, B and A, X, Y, AND A, B, Y
Three non-collinear points determine a flat surface called a plane.
We name a Plane by using three of its non collinear points!! Plane ABC was out example.
Lines in the same plane that do not intersect are PARALLEL lines. Two segments or rays are parallel if they are parts of parallel lines.
↔
AB is parallel to
↔
CD
may be written
↔ ↔
ABllCD
using two straight lines to indicate parallel
Parallel lines DO NOT intersect.
Intersecting lines intersect in a single point!!
Planes that do not intersect are called parallel planes... we looked around the room and found examples of parts of planes.. noticing which ones were parallel!! (the floor and ceiling were a great example)
Then we drew the box from PAge 106 and identified parallel segments and lines from that box.
Two non parallel lines that do not intersect are called SKEW LINES.
Pre Algebra (Period 2 & 4)
Graphing & Writing Inequalities Review
Some words to really know:
At least--> means greater than or EQUAL TO
I want at least $150 to go shopping for holiday gifts!!
m ≥ 150
AT MOST --> less than or EQuAL TO
You might say that you want at most 15 minutes of homework tonight. That means you will take 15 minutes but you sure would like less....
m ≤ 15
Review of some of the most missed problems from our recent races:
y is at most 5
y ≤ 5
x is no more than 7
x ≤ 7
y is at least 20
y ≥ 20
y is less than 8
y < 8. Now 5 < y reads " Five is less than than y" --> if that is true then y must be greater than 5 or
y > 5 so now you can graph easily because you know exactly which way the arrow ( solution set) must be pointing... in the same direction as the inequality symbol. THis ONLY works when you have the variable on the left!! as in y > 5.
ALWAYS CHANGE so the variable is on the left!!
Solve one step inequalities just like you do for equations.. just make sure to end with the variable on the left-- to be able to graph easily
y - 7 > -3
add 7 to BOTH Sides
y > 4
Now you can graph easily... It's an open dot on your graph Because it DOES NOT inlcude 4 as part of the solution
FORMAL CHECK
(1) REWRITE the inequality
(2) Substitute in for the variable--using ANY number that fits the solution (except the boundary number) In this example y > 4,
4 is the boundary number. We showed in class why you can't use the boundary number so if y > 4 then 5, 10, 100, etc would work...
In class we used 10
10 -7 > -3 (Remember to put a "?" over the inequality)
(3) REALLY DO THE MATH. that is, in this case,
do 10-7.
Well 10-7 = 3 so
3 > -3 That's true-- it works!!
If you get something that is not true, chances are you did your problem wrong. re work the problem and see if you can discover your error!!
WHen dividing or multiplying by a negative coefficient (that's the number attached to the variable) you must remember to
REWRITE the inequality AS YOU "FLIP THE SWITCH" (change the inequality symbol)
y/-5 ≥ -3
We need to multiply both sides by a -5 so we must REWRITE and FLIP
(-5)(y/-5) ≤ (-3)(-5)
y ≤ 15
FORMAL CHECK
(1) REWRITE THE INEQUALITY
(2) Substitute in one of the SOLUTIONS-- any one that really fits and is easy to work with!!
(3) DO THE MATH-- look at your results-- does it make sense?... don't just put a happy face!!
(1) y/-5 ≥ -3
well, we found out the y ≤ 15 that means any number less than 15 could be used to check I would pick something that was easy to work with, something that was divisible by 5... like 0, or 5 or even 10
we used 0 in class
(2) 0/-5 ≥ -3
(3) 0 ≥ -3 That's true!! IT WORKED
But what if I used 5?
well step 2 would be
(2) 5/-5 ≥ -3
(3) -1 ≥ -3 and that's still true !!
5y ≤ -35
I just divide by 5. I don't need to switch anything because I am NOT dividing by a negative number
5y/5 ≤ -35/5
y ≤ -7
But what about
-15 ≤ -3m
In this case I must rewrite as I "FLIP THE SWITCH" (Remember the poster from class)
-15/-3 ≥ -3m/-3
5 ≥ m
Before we continue we need to realize that if five is greater than or equal to m... that means m must be less than or equal to 5 or m ≤ 5. Now its easy to graph.
FORMAL CHECK
-15 ≤ -3m
-15 ≤ -3(4)
-15 ≤ -12 which is true!!
In class we used a number that was not correct to show what happens.
When working with Inequalities with 2 steps and variables on both sides:
(1) Do Distributive Property (DP) carefully-- if necessary
(2) combine like terms on each side
(3) "Jump" the variables to one side of the equation
(4) Add or subtract
(5) Multiply or Divide-- being careful if have negative coefficients to rewrite the inequality and "flip the Switch"
(6) MAKE SURE the variable is on the LEFT Side
Some words to really know:
At least--> means greater than or EQUAL TO
I want at least $150 to go shopping for holiday gifts!!
m ≥ 150
AT MOST --> less than or EQuAL TO
You might say that you want at most 15 minutes of homework tonight. That means you will take 15 minutes but you sure would like less....
m ≤ 15
Review of some of the most missed problems from our recent races:
y is at most 5
y ≤ 5
x is no more than 7
x ≤ 7
y is at least 20
y ≥ 20
y is less than 8
y < 8. Now 5 < y reads " Five is less than than y" --> if that is true then y must be greater than 5 or
y > 5 so now you can graph easily because you know exactly which way the arrow ( solution set) must be pointing... in the same direction as the inequality symbol. THis ONLY works when you have the variable on the left!! as in y > 5.
ALWAYS CHANGE so the variable is on the left!!
Solve one step inequalities just like you do for equations.. just make sure to end with the variable on the left-- to be able to graph easily
y - 7 > -3
add 7 to BOTH Sides
y > 4
Now you can graph easily... It's an open dot on your graph Because it DOES NOT inlcude 4 as part of the solution
FORMAL CHECK
(1) REWRITE the inequality
(2) Substitute in for the variable--using ANY number that fits the solution (except the boundary number) In this example y > 4,
4 is the boundary number. We showed in class why you can't use the boundary number so if y > 4 then 5, 10, 100, etc would work...
In class we used 10
10 -7 > -3 (Remember to put a "?" over the inequality)
(3) REALLY DO THE MATH. that is, in this case,
do 10-7.
Well 10-7 = 3 so
3 > -3 That's true-- it works!!
If you get something that is not true, chances are you did your problem wrong. re work the problem and see if you can discover your error!!
WHen dividing or multiplying by a negative coefficient (that's the number attached to the variable) you must remember to
REWRITE the inequality AS YOU "FLIP THE SWITCH" (change the inequality symbol)
y/-5 ≥ -3
We need to multiply both sides by a -5 so we must REWRITE and FLIP
(-5)(y/-5) ≤ (-3)(-5)
y ≤ 15
FORMAL CHECK
(1) REWRITE THE INEQUALITY
(2) Substitute in one of the SOLUTIONS-- any one that really fits and is easy to work with!!
(3) DO THE MATH-- look at your results-- does it make sense?... don't just put a happy face!!
(1) y/-5 ≥ -3
well, we found out the y ≤ 15 that means any number less than 15 could be used to check I would pick something that was easy to work with, something that was divisible by 5... like 0, or 5 or even 10
we used 0 in class
(2) 0/-5 ≥ -3
(3) 0 ≥ -3 That's true!! IT WORKED
But what if I used 5?
well step 2 would be
(2) 5/-5 ≥ -3
(3) -1 ≥ -3 and that's still true !!
5y ≤ -35
I just divide by 5. I don't need to switch anything because I am NOT dividing by a negative number
5y/5 ≤ -35/5
y ≤ -7
But what about
-15 ≤ -3m
In this case I must rewrite as I "FLIP THE SWITCH" (Remember the poster from class)
-15/-3 ≥ -3m/-3
5 ≥ m
Before we continue we need to realize that if five is greater than or equal to m... that means m must be less than or equal to 5 or m ≤ 5. Now its easy to graph.
FORMAL CHECK
-15 ≤ -3m
-15 ≤ -3(4)
-15 ≤ -12 which is true!!
In class we used a number that was not correct to show what happens.
When working with Inequalities with 2 steps and variables on both sides:
(1) Do Distributive Property (DP) carefully-- if necessary
(2) combine like terms on each side
(3) "Jump" the variables to one side of the equation
(4) Add or subtract
(5) Multiply or Divide-- being careful if have negative coefficients to rewrite the inequality and "flip the Switch"
(6) MAKE SURE the variable is on the LEFT Side
Algebra (Period 1)
Multiplying Binomials: Special Products 5-10
LEARN TO RECOGNIZE SOME SPECIAL PRODUCTS - IT MAKES IT EASIER!
Remember: You can FOIL these just like the other products until you remember these special patterns....but when we get to factoring next week, it will really help you to know these patterns by heart.
When you do, you actually don't need to show any work because you do it in your head!
(That should make a lot of you happy! :)
DIFFERENCE OF TWO SQUARES:
You will notice that the two factors are IDENTICAL except they have DIFFERENT SIGNS
(x + 6)(x - 6) =
x2 - 6x + 6x - 36 =
x2 - 36
This will happen every time!
The middle terms are additive inverses so they become zero.
You're left with a difference (subtraction) of two terms that are squared.
SQUARING A BINOMIAL:
When you multiply one binomial by itself (squaring it), you end up with:
First term squared + twice the product of both terms + last term squared
(x + 6)2 =
(x + 6)(x + 6) =
x2 + 2(6x) + 62 =
x2 + 12x + 36
If you foiled you would have:
x2 + 6x + 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
Another example with subtraction in the middle:
(x - 6)2 =
(x - 6)(x - 6) =
x2 + 2(-6x) + 62 =
x2 - 12x + 36
If you foiled you would have:
x2 - 6x - 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
PLEASE NOTE:
NOTICING THESE SPECIAL PRODUCTS HELPS YOU DO THESE MULTIPLICATIONS FASTER!
IF YOU EVER FORGET THEM, JUST FOIL!
(but you will need to recognize them for factoring in Chapter 6)
LEARN TO RECOGNIZE SOME SPECIAL PRODUCTS - IT MAKES IT EASIER!
Remember: You can FOIL these just like the other products until you remember these special patterns....but when we get to factoring next week, it will really help you to know these patterns by heart.
When you do, you actually don't need to show any work because you do it in your head!
(That should make a lot of you happy! :)
DIFFERENCE OF TWO SQUARES:
You will notice that the two factors are IDENTICAL except they have DIFFERENT SIGNS
(x + 6)(x - 6) =
x2 - 6x + 6x - 36 =
x2 - 36
This will happen every time!
The middle terms are additive inverses so they become zero.
You're left with a difference (subtraction) of two terms that are squared.
SQUARING A BINOMIAL:
When you multiply one binomial by itself (squaring it), you end up with:
First term squared + twice the product of both terms + last term squared
(x + 6)2 =
(x + 6)(x + 6) =
x2 + 2(6x) + 62 =
x2 + 12x + 36
If you foiled you would have:
x2 + 6x + 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
Another example with subtraction in the middle:
(x - 6)2 =
(x - 6)(x - 6) =
x2 + 2(-6x) + 62 =
x2 - 12x + 36
If you foiled you would have:
x2 - 6x - 6x + 36
CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP???
WHY???
PLEASE NOTE:
NOTICING THESE SPECIAL PRODUCTS HELPS YOU DO THESE MULTIPLICATIONS FASTER!
IF YOU EVER FORGET THEM, JUST FOIL!
(but you will need to recognize them for factoring in Chapter 6)
Wednesday, November 17, 2010
Algebra (Period 1)
Addition of Polynomials 5-7
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power
You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) + (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
Subtraction of Polynomials 5-8
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) - (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
5x4 - 3x2 - (-4x) + 3
- ( -10x4 + 3x3- 3x2 - x + 3)
-----------------------------------
For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x4 + 0x3 - 3x2 -(-4x) + 3
-(-10x4 +3x3- 3x2 - x + 3)
-----------------------------------
DISTRIBUTE THE NEGATIVE, THEN ADD:
5x4 + 0x3 - 3x2 - (-4x) + 3
+10x4 -3x3 +3x2 + x - 3
-----------------------------------
15x4 - 3x3 + 5 x
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power
You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) + (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
Subtraction of Polynomials 5-8
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) - (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
5x4 - 3x2 - (-4x) + 3
- ( -10x4 + 3x3- 3x2 - x + 3)
-----------------------------------
For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x4 + 0x3 - 3x2 -(-4x) + 3
-(-10x4 +3x3- 3x2 - x + 3)
-----------------------------------
DISTRIBUTE THE NEGATIVE, THEN ADD:
5x4 + 0x3 - 3x2 - (-4x) + 3
+10x4 -3x3 +3x2 + x - 3
-----------------------------------
15x4 - 3x3 + 5 x
Tuesday, November 16, 2010
Pre Algebra (Period 2 & 4)
Solving 1 Step Inequalities with Addition & Subtraction 2-9
These are solved just like equations except you have a greater than, less than, greater than or equal, less than or equal sign instead of the equal sign.
Your goal is still the same:
ISOLATE THE VARIABLE
You sitll balance with the INVERSE OPERATION
Your justifications are still almost the same: IDENTITY AND PROPERTIES OF INEQUALITY
Solving 1 Step Inequalities with Multiplication & Division 2-10
These are solved just like equations except you have a greater than, less than, greater than or equal, less than or equal sign instead of the equal sign.
Your goal is still the same: ISOLATE THE VARIABLE
You sitll balance with the INVERSE OPERATION
Your justifications are still almost the same: IDENTITY AND PROPERTIES OF INEQUALITY
THERE IS ONE MAJOR EXCEPTION!!!!
When you multiply or divide by a NEGATIVE,
the symbol CHANGES DIRECTION!
-3y > 9
You need to divide both sides by NEGATIVE 3 so the symbol will switch from > to < in the solution y < -3 is the answer If you want to understand why: 3 < 10 correct? Now multiply both sides by -1 (mult prop of equality) You get -3 < -10, but THAT'S NOT TRUE!!! You have to SWITCH THE SYMBOL to make the answer true: -3 > -10
These are solved just like equations except you have a greater than, less than, greater than or equal, less than or equal sign instead of the equal sign.
Your goal is still the same:
ISOLATE THE VARIABLE
You sitll balance with the INVERSE OPERATION
Your justifications are still almost the same: IDENTITY AND PROPERTIES OF INEQUALITY
Solving 1 Step Inequalities with Multiplication & Division 2-10
These are solved just like equations except you have a greater than, less than, greater than or equal, less than or equal sign instead of the equal sign.
Your goal is still the same: ISOLATE THE VARIABLE
You sitll balance with the INVERSE OPERATION
Your justifications are still almost the same: IDENTITY AND PROPERTIES OF INEQUALITY
THERE IS ONE MAJOR EXCEPTION!!!!
When you multiply or divide by a NEGATIVE,
the symbol CHANGES DIRECTION!
-3y > 9
You need to divide both sides by NEGATIVE 3 so the symbol will switch from > to < in the solution y < -3 is the answer If you want to understand why: 3 < 10 correct? Now multiply both sides by -1 (mult prop of equality) You get -3 < -10, but THAT'S NOT TRUE!!! You have to SWITCH THE SYMBOL to make the answer true: -3 > -10
Monday, November 15, 2010
Algebra (Period 1)
Polynomials 5-5
Polynomials = SUM of monomials
Monomials must have variables with whole number powers. Review section 5-3 for detail on monomials!!
no variables in the denominator, no roots of numbers!!
so 1/x is not a monomial
neither is x 1/2
constants have whole number power of zero..
7 is really 7x0
1 term = monomial
2 terms = binomial
3 terms = trinomial
TERMS are separated by addition
( if see subtraction-- THINK: add the opposite!!)
Coefficient - number attached to the variable ( it can be a fraction)
3x2 - 10x
the coefficients are 3 and -10. Make sure to attach the negative sign to the coefficient -- and ADD the OPPOSITE
y/6 is really (1/6)y so the coefficient is 1/6
if you have -x/3 that is really (-1/3)x so the coefficient is -1/3
Constant = the number that is not attached to ANY variable
Some TERMS YOU NEED TO KNOW
Degree of a term = SUM of the exponents of all its variables
-6x4 : the degree is 4
8x2 : the degree is 2
-2x : the degree is 1
9 : the degree is 0 ( think 9 is really 9x0
Degree of a polynomial - HIGHEST degree of any of its terms
so
-6x4 + 8x2 + -2x + 9
The degree of the polynomial is : 4
Leading term = term with the HIGHEST degree
Leading coefficient- the coefficient of the leading term
More on Polynomials 5-6
Descending order- write the variables with the highest power first ( This is the way it is usually written)
Ascending order- write the variables with the lowest powerr first ( actually NEVER used in practice)
Evaluating a polynomial- this is what we have been doing all year... plug it in, plug it in!!
Remember to ALWAYS put the number you substitute in parentheses!!
2x2y + 5xy - 4, where x = -4 and y = 5
Substitute carefully:
2(-4)2(5) + 5(-4)(5) - 4
= 2(16)(5) +(-20)(5) - 4
= 160 +(-100) -4
= 160 -104
= 56
Polynomials = SUM of monomials
Monomials must have variables with whole number powers. Review section 5-3 for detail on monomials!!
no variables in the denominator, no roots of numbers!!
so 1/x is not a monomial
neither is x 1/2
constants have whole number power of zero..
7 is really 7x0
1 term = monomial
2 terms = binomial
3 terms = trinomial
TERMS are separated by addition
( if see subtraction-- THINK: add the opposite!!)
Coefficient - number attached to the variable ( it can be a fraction)
3x2 - 10x
the coefficients are 3 and -10. Make sure to attach the negative sign to the coefficient -- and ADD the OPPOSITE
y/6 is really (1/6)y so the coefficient is 1/6
if you have -x/3 that is really (-1/3)x so the coefficient is -1/3
Constant = the number that is not attached to ANY variable
Some TERMS YOU NEED TO KNOW
Degree of a term = SUM of the exponents of all its variables
-6x4 : the degree is 4
8x2 : the degree is 2
-2x : the degree is 1
9 : the degree is 0 ( think 9 is really 9x0
Degree of a polynomial - HIGHEST degree of any of its terms
so
-6x4 + 8x2 + -2x + 9
The degree of the polynomial is : 4
Leading term = term with the HIGHEST degree
Leading coefficient- the coefficient of the leading term
More on Polynomials 5-6
Descending order- write the variables with the highest power first ( This is the way it is usually written)
Ascending order- write the variables with the lowest powerr first ( actually NEVER used in practice)
Evaluating a polynomial- this is what we have been doing all year... plug it in, plug it in!!
Remember to ALWAYS put the number you substitute in parentheses!!
2x2y + 5xy - 4, where x = -4 and y = 5
Substitute carefully:
2(-4)2(5) + 5(-4)(5) - 4
= 2(16)(5) +(-20)(5) - 4
= 160 +(-100) -4
= 160 -104
= 56
Pre Algebra (Period 2 & 4)
Inequalities & Their Graphs 2-8
What most real life situations call for
How often to you think...I need at least $20 (not exactly $20) ....
I want at most 20 minutes of homework (not exactly 20 minutes)?
Inequalities have many answers (most of the time an infinite number!)
Examples: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)
n ≥ 3 means still means that every real number greater than 3 is a solution,
but now 3 is also a solution
n < 3 means that every real number less than 3 is a solution! (but NOT 3) n ≤ 3 means still means that every real number less than 3 is a solution, but now 3 is also a solution GRAPHING INEQUALITIES: First, graphing an equation's solution is easy 1) Say you found out that y = 5, you would just put a dot on 5 on the number line 2) But now you have the y ≥ 5 You still put the dot but now also darken in an arrow going to the right showing all those numbers are also solutions I say: the equal sign part of it is a crayon...Pick up the crayon and color in the circle 3) Finally, you find in another example that y > 5
You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!
I say: there isn't a crayon so you can't color in the circle.
Therefore it stays open!
TRANSLATING WORDS:
Some key words to know:
AT LEAST means greater than or equal that is... ≥
AT MOST means less than or equal. THat is ≤
I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!
I want at most 15 minutes of homework means that I can have 15 minutes, but I'm hoping for even less!
What most real life situations call for
How often to you think...I need at least $20 (not exactly $20) ....
I want at most 20 minutes of homework (not exactly 20 minutes)?
Inequalities have many answers (most of the time an infinite number!)
Examples: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)
n ≥ 3 means still means that every real number greater than 3 is a solution,
but now 3 is also a solution
n < 3 means that every real number less than 3 is a solution! (but NOT 3) n ≤ 3 means still means that every real number less than 3 is a solution, but now 3 is also a solution GRAPHING INEQUALITIES: First, graphing an equation's solution is easy 1) Say you found out that y = 5, you would just put a dot on 5 on the number line 2) But now you have the y ≥ 5 You still put the dot but now also darken in an arrow going to the right showing all those numbers are also solutions I say: the equal sign part of it is a crayon...Pick up the crayon and color in the circle 3) Finally, you find in another example that y > 5
You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!
I say: there isn't a crayon so you can't color in the circle.
Therefore it stays open!
TRANSLATING WORDS:
Some key words to know:
AT LEAST means greater than or equal that is... ≥
AT MOST means less than or equal. THat is ≤
I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!
I want at most 15 minutes of homework means that I can have 15 minutes, but I'm hoping for even less!
Tuesday, November 9, 2010
Pre Algebra (Period 2 & 4)
Adding & Subtracting Decimals Equations 3-5
Most important... LINE UP THE DECIMALS!!
We did a number of examples from the textbook... check out this section!!
Make sure you use a SIDE BAR for your calculations!!
Multiplying & Dividing Decimal Equations 3-6
For multiplication, count the decimals and place the decimal from the right!!
THese are difficult to show here as well...but
( m/7.2) = -12.5
multiply both sides by -7.2
(-7.2)(m/-7.2) = -12.5(-7.2)
m = 90
Make sure to use a side bar and multiply carefully!!
s/2.5 = 5
multiply both sides by 2.5
(2.5)(s/2.5) = 5(2.5)
3 = 12.5
4.5 = m/-3.3
multiply both sides by -3.3
Notice your answer will be a negative!!
(-3.3) (4.5) = [m/-3.3](-3.3)
-14.85 = m
... and I used a sidebar in class to show the multiplication!!
For division
0.9r = -5.4
Divide both sides by 0.9
.9r = -5.4
0.9 0.9
r = -6
use a sidebar to divide carefully
divide both sides by -0.9
-0.9n = -81.81
-0.9n/-0.9 = -81.81/-0.9
notice that your solution will be a positive!!
and use a side bar to divide carefully
n = + 90.9
Most important... LINE UP THE DECIMALS!!
We did a number of examples from the textbook... check out this section!!
Make sure you use a SIDE BAR for your calculations!!
Multiplying & Dividing Decimal Equations 3-6
For multiplication, count the decimals and place the decimal from the right!!
THese are difficult to show here as well...but
( m/7.2) = -12.5
multiply both sides by -7.2
(-7.2)(m/-7.2) = -12.5(-7.2)
m = 90
Make sure to use a side bar and multiply carefully!!
s/2.5 = 5
multiply both sides by 2.5
(2.5)(s/2.5) = 5(2.5)
3 = 12.5
4.5 = m/-3.3
multiply both sides by -3.3
Notice your answer will be a negative!!
(-3.3) (4.5) = [m/-3.3](-3.3)
-14.85 = m
... and I used a sidebar in class to show the multiplication!!
For division
0.9r = -5.4
Divide both sides by 0.9
.9r = -5.4
0.9 0.9
r = -6
use a sidebar to divide carefully
divide both sides by -0.9
-0.9n = -81.81
-0.9n/-0.9 = -81.81/-0.9
notice that your solution will be a positive!!
and use a side bar to divide carefully
n = + 90.9
Algebra (Period 1)
Scientific Notation 5-4
You've had this since 6th grade! It is just a bit more complicated
You restate very big or very small numbers using powers of 10 in exponential form
Move the decimal so the number fits in this range:
less than 10 and greater than or equal to 1
That is 1 ≤ n < 10 Scientific notation is the product of two factors: one of the factors is a number (1)1 ≤ n < 10 and the other factor (2) is a power of ten Count the number of places you moved the decimal and make that your exponent Very big numbers - exponent is positive Very small numbers (decimals) - exponent is negative (just like a fraction!) Remember that STANDARD notation is what you expect (the normal number) When you multiply or divide scientific notations, use the power rules! Just be careful that if your answer does not fit the scientific notation range, that you restate it. (2.2 x 10 -3)(3.0 X 10 5)
(2.2)(3.0) X 10 -310 5
6.6 X 10 2
(6.1 x 10 9)(2.5 x 10-4)
(6.1)(2.5) x 10 9+4
15.25 x 10 5
But that isn't in scientific notation so change just 15.25 into proper scientific notation first
that is
15.25 = 1.525 x 101
put that back in your work
1.525 x 101 ⋅ 105
1.525 X 106
(2.5 x 10 -3)(4.0 X 10 -8)
multiply the first factors or (2.5)(4.) and multiply the powers of ten
(10 -3)(10 -8)
(2.5)(4)= 10
so initially you have 10 X 10 -11
but 10 does not fall into the required 1 ≤ n < 10 so changed 10 into scientific notation or 1.0 X 101
so you have 1.0 X 101 X 10 -11
or 1 X 10 -10
(5.4 X 10-6)(5.1 X 10 -8)
(5.4)(5.1) = 27.54
so
27.54 X 10 -6+-8
27.54 X 10 -14
but again 27.54 isn't in scientific notation... changed just that number into scientific notation... FIRST
2.754 X 101
put it back into your work
2.754 x 101 ⋅ 10-14
2.754 X 10 1 + -14
2.754 X 10 -13
What about
9.0 X 10 8
3.0 X 10 2
do your factors separately just like you did with multiplication
That is,
9.0 = 3
3.0
and
108
102
= 10 8-2
=106
so the answer is
3 x 10 6
6.9 X 10 4
3.0 X 10 8
2.3 X 10 4-8
2.3 X 10 -4
2.7 X 10 12
9.0 X 10 12
Divide 2.7 by 9
you get
0.3
1012 -12 = 10 0
but wait 0.3 isn't in the correct for so change that..
0.3= 3.0 x 10-1
so the answer is
3.0 X 10 -1
6.0 x 10 7
3.0 X 102
First divide 6.0/3.0 = 2.0
and use the exponent rules of 10 7 -2 or 105
so 2.0 X 105
4.2 X 105
2.1 X 103
= 2 X 102
2.5 X 10 -7
5.0 X 10 6
first 2.5/5 = 0.5 and 10-7/106 = 10-7-6 = 10-13
so initially you have
0.5 x 10-13 but 0.5 is not within 1 ≤ n < 10 so change 0.5 to scientific notation 5.0 X 10 -1 and mult that by 10-13
5 X 10 -14
How about this harder one:
(3.6 X 106)(4 X 10 -3
(4.8 x 10 -2)(1.2 X 10 6)
put all your factors together and see if you can simplify before you begin...
(3.6)(4)
(4.8)(1.2)
You can simplify
(3.6)
(4.8)
isn't that 3/4
and then your 4's simplify
left with
3.o
1.2
which simplifies to 2.5 ... so going back to the original problem you now have
2.5 x 106-3
10 -2 + 6
which
becomes
2.5 X 10 3-4 or
2.5 X 10 -1
You've had this since 6th grade! It is just a bit more complicated
You restate very big or very small numbers using powers of 10 in exponential form
Move the decimal so the number fits in this range:
less than 10 and greater than or equal to 1
That is 1 ≤ n < 10 Scientific notation is the product of two factors: one of the factors is a number (1)1 ≤ n < 10 and the other factor (2) is a power of ten Count the number of places you moved the decimal and make that your exponent Very big numbers - exponent is positive Very small numbers (decimals) - exponent is negative (just like a fraction!) Remember that STANDARD notation is what you expect (the normal number) When you multiply or divide scientific notations, use the power rules! Just be careful that if your answer does not fit the scientific notation range, that you restate it. (2.2 x 10 -3)(3.0 X 10 5)
(2.2)(3.0) X 10 -310 5
6.6 X 10 2
(6.1 x 10 9)(2.5 x 10-4)
(6.1)(2.5) x 10 9+4
15.25 x 10 5
But that isn't in scientific notation so change just 15.25 into proper scientific notation first
that is
15.25 = 1.525 x 101
put that back in your work
1.525 x 101 ⋅ 105
1.525 X 106
(2.5 x 10 -3)(4.0 X 10 -8)
multiply the first factors or (2.5)(4.) and multiply the powers of ten
(10 -3)(10 -8)
(2.5)(4)= 10
so initially you have 10 X 10 -11
but 10 does not fall into the required 1 ≤ n < 10 so changed 10 into scientific notation or 1.0 X 101
so you have 1.0 X 101 X 10 -11
or 1 X 10 -10
(5.4 X 10-6)(5.1 X 10 -8)
(5.4)(5.1) = 27.54
so
27.54 X 10 -6+-8
27.54 X 10 -14
but again 27.54 isn't in scientific notation... changed just that number into scientific notation... FIRST
2.754 X 101
put it back into your work
2.754 x 101 ⋅ 10-14
2.754 X 10 1 + -14
2.754 X 10 -13
What about
9.0 X 10 8
3.0 X 10 2
do your factors separately just like you did with multiplication
That is,
9.0 = 3
3.0
and
108
102
= 10 8-2
=106
so the answer is
3 x 10 6
6.9 X 10 4
3.0 X 10 8
2.3 X 10 4-8
2.3 X 10 -4
2.7 X 10 12
9.0 X 10 12
Divide 2.7 by 9
you get
0.3
1012 -12 = 10 0
but wait 0.3 isn't in the correct for so change that..
0.3= 3.0 x 10-1
so the answer is
3.0 X 10 -1
6.0 x 10 7
3.0 X 102
First divide 6.0/3.0 = 2.0
and use the exponent rules of 10 7 -2 or 105
so 2.0 X 105
4.2 X 105
2.1 X 103
= 2 X 102
2.5 X 10 -7
5.0 X 10 6
first 2.5/5 = 0.5 and 10-7/106 = 10-7-6 = 10-13
so initially you have
0.5 x 10-13 but 0.5 is not within 1 ≤ n < 10 so change 0.5 to scientific notation 5.0 X 10 -1 and mult that by 10-13
5 X 10 -14
How about this harder one:
(3.6 X 106)(4 X 10 -3
(4.8 x 10 -2)(1.2 X 10 6)
put all your factors together and see if you can simplify before you begin...
(3.6)(4)
(4.8)(1.2)
You can simplify
(3.6)
(4.8)
isn't that 3/4
and then your 4's simplify
left with
3.o
1.2
which simplifies to 2.5 ... so going back to the original problem you now have
2.5 x 106-3
10 -2 + 6
which
becomes
2.5 X 10 3-4 or
2.5 X 10 -1
Monday, November 8, 2010
Math 6 Honors (Period 6 and 7)
Multiplying Decimals 3-8
According to our textbook:
Place the decimal point in the product so that the number of places to the right of the decimal point in the product is the sum of the number of places to the right of the decimal point in the factors!!
You do NOT need to line up the decimal point when you are multiplying.
11.32 X 8.73
11.32
8.73
98.8236
estimate and you get 11 X 9 = 99
What would you do if you had
(19.81 x 5.1) + (19.81 X 4.9)
Wait... wait... remember the Distributive Property????
Look you can use it to make this problem soooooo much easier
19.81(5.1 _ 4.9)
19.81 (10) = 198.1
How about 50(.25) + 50(.75)
This is one you can even do in your head because
50(0.25 + 0.75) = 50 (1) = 50
Dividing Decimals 3-9
According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.
Check out our textbook for some examples!!
When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.
Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!
the quotient is 0.3431 which rounds to 0.343
To divide one decimal by another
Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number
Divide the new dividend by the new divisor
Check by multiplying the quotient and the divisor.
According to our textbook:
Place the decimal point in the product so that the number of places to the right of the decimal point in the product is the sum of the number of places to the right of the decimal point in the factors!!
You do NOT need to line up the decimal point when you are multiplying.
11.32 X 8.73
11.32
8.73
98.8236
estimate and you get 11 X 9 = 99
What would you do if you had
(19.81 x 5.1) + (19.81 X 4.9)
Wait... wait... remember the Distributive Property????
Look you can use it to make this problem soooooo much easier
19.81(5.1 _ 4.9)
19.81 (10) = 198.1
How about 50(.25) + 50(.75)
This is one you can even do in your head because
50(0.25 + 0.75) = 50 (1) = 50
Dividing Decimals 3-9
According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.
Check out our textbook for some examples!!
When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.
Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!
the quotient is 0.3431 which rounds to 0.343
To divide one decimal by another
Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number
Divide the new dividend by the new divisor
Check by multiplying the quotient and the divisor.
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