Math 7 ( Period 4)

Multiplying and Dividing Integers 3.5 & 3.6

They have the same rules! (The rules are different from adding and subtracting integers… so keep these straight!)

Textbook Rules:
If you have 2 signs that are the SAME, the product or quotient is POSTIIVE

IF you have 2 signs that are DIFFERENT, the product or quotient is NEGATIVE

Good Guy/ Bad Guy Rules… or Life’s Lessons…

Good thing happens to Good person à that’s Good

Bad thing happens to a Good Personà that’s Bad (That’s not right! We hate when that happens)

Good thing happens to a Bad person à That’s Bad (We hate when good things happen to people who don’t deserve it.

Bad thing happens to a Bad person à that’s good (That’s Karma … they got what they deserved)

+ ∙ + = +

- ∙ + = -

+ ∙ - = -

-  ∙ - = +

It works with division as well!

Finger Rules:

If your index finger represents the negative sign then if  you have two negatives you have the index fingers of both your left hand and your right hand and they make a plus sign!

If you have just one negative it just stays negative because you don’t have another finger to cross it

If you have more than two negatives you just keep using your index fingers to determine the sign.

We did this in class. It’s fun… but once you get it.. you won’t need to keep doing it ( unless you want to keep having fun!)

What if there are more than 2 signs?

Use Aunt Sally’s  Rules (O³) and go left to right

or be a Sign Counter

…an ODD number of negatives = Negative

…an EVEN number of negatives = Positive

Only count the negative signs… you do not worry about how many numbers are involved nor the positives… only count the Negatives

(-2)(-5)(-3) = -30 (there are 3 negatives—and 3 is odd)

(2)(-5)(-3) = +30 (there are 2 negatives and 2 is an even number)

(2)(5)(-3) = -30 (there is only 1 negative and that is odd!)

Math 6A (Periods 1 & 2)

Multiplying or Dividing by a Power of Ten 3-7

We have learned that in a decimal or a whole number each place value is ten times the place value to its right.

10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000

10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01

Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.

Multiplying by ten moves the decimal point one place to the right

10 ∙ 762 = 7620

762 X 10 = 7620

4931 X 10 = 49,310


10⁴ = 10⋅10⋅10⋅10 = 10,000

2.63874 X 10⁴ = 26,387.4

To multiply a number by the nth power of ten--> move the decimal n places to the right.

0.0047 multiply by 100 = 0.47
0.0047 multiply by 1000 = 4.7

3.1 ÷ 10⁴ = 0.00031


10 ∙ 4.931 = 49.31

At the beginning of this chapter you learned about powers of ten

10⁴ = 10 ∙10 ∙ 10 ∙10 = 10,000

We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.

2.64874 ∙10⁴ = 26,387.4

Notice that we have moved the decimal point four places to the right.

Rule

To multiply a number by the n^th power of ten, move the decimal point n places to the right.



When we move a decimal point to the left, we are actually dividing by a power of ten.


Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros

Rule

To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.

2386 ÷ 10³ = 2.386

Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of two factors

(1) a number greater than or equal to 1, but less than 10,

AND

(2) a power of ten

are said to be written in scientific notation.

We can write 'a number greater than or equal to 1, but less than 10' as an mathematical inequality 1 ≤ n < 10 To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point. 4,592,000,000 in scientific notation First move the decimal point to the left to get a number between 1 and 10 4,592,000,000 the first factor in scientific notation becomes 4.592 Since the decimal point was moved 9 places, we multiply 4.592 by 10⁹ to express the number in scientific notation



4.592 x 10⁹ (Yes, you get to use the × symbol for multiplication .. but only for this!!



This is a way to write very large numbers AND very small numbers

Numbers expressed as products of a number greater than or equal to 1 BUT less than 10, AND a power of ten are called Scientific Notation.

Two Factors
(1) 1≤ n < 10
(2) Power of 10
4,592,000,000 becomes 4.592 X 10⁹
moved the decimal 9 places so we must multiply our number by a power of 10⁹

98,000,000 = 9.8 X 10⁷

320,000 = 3.2 X 10⁵

What if I give you 7.04 X 10⁸ and ask you to put it back into STANDARD NOTATION:

704,000,000.

0.0031 = 3.1 X 10^-3
It isn't a negative number its just a very tiny number

1≤ n < 10 0.16 becomes 1.6 x 10 ^-1

Math 7 (Period 4)

Chapter 3.4 Subtracting Integers
NEVER SUBTRACT—Add the Opposite

I call it "Double Check method" because you always change 2 signs

1) Change the subtraction sign to an addition sign (check)

2) Change the subtrahend’s sign (the 2^nd number’s sign) to its opposite
(if it was negative change it to positive,
if it had no sign then put a negative because no sign meant it was positive) (double check)

3) Follow the rules of integer addition from  Section 3.3

Example

5 –(-10) = 5 +(+10) =15

-5 –(-10) = -5 +(+10) = +5

-5- 10 = -5 +(-10) = -15

More than 1 subtrahend?  Double check each one!

By the way NEVER EVER EVER CHANGE THE FIRST NUMBER”S SIGN!!

Identifying terms:

Terms are separated by ADDITION
(Remember there is no such thing as subtraction)

2xy is only ONE term—but it has 3 factors

2xy + 3  is 2 two terms

2xy – 3z- (-10) is 3 terms made up of the following

2xy, -3z, and +10
(you must put the terms in Addition Format to determine their signs)

ALWAYS SIMPLIFY BEFORE EVALUTATING

11x + 14 – 21x + 6 + 12x  when x = 5

You will get the same answer if you plug and chug x = 5 in all of the terms as when you SIMPLIFY FIRST and then plug and chug just once.

11(5) + 14 – 21(5) + 6 12(5)

55 + 14 -105 + 6 + 60

135 + (-105)

30

VS

(11x -21x + 12x) + ( 14 + 6)

2x + 20

2(5) + 20

10 + 20

30

Which way do you want to do these types of problems?

Simplifying first is usually the least work!

Quick review of Coefficient and Constants

Remember that a coefficient goes along with a variable and EVERY VRIABLE MUST HAVE  COEFFICIENT

so

2a – 3b –(-c) –d – 12

has 5 terms, 4 variables, 4 coefficeints and 1 constant

The coefficients are 2, 3, 1, -1

The IDENTITY Property of Multiplication (IDx) says that you can sneak in the “1” by multiplication in front of any variable that has no other coefficient.  

Fractional Coefficients

2x

3

can be written 2/3(x)

2 x

3

So the Coefficient of this one term is 2/3

Review of Absolute Value with subtraction inside

Absolute value symbols are similar to parentheses  in that you must simplify inside using Order of Operations (O³) BEFORE applying the absolute value at the end

│2∙3²-30│

You need to do the power first, then multiply, then subtract and THEN absolute value at the very end.

│2∙9-30│= │18-30│=│-12│ = 12

Math 6 A (Periods 1 & 2)

Adding & Subtracting Decimals 3-6

Rules

1) Write the given numbers one above the other with the decimal points in line

I call that “Stack’ Em”

2) Add any zeros to get the same number of decimal places and then add or subtract ( +/-) as if the numbers were whole numbers.

We put the added zeros in colored pencil to distinguish them!

3) Place the decimal point in the number for the sum or difference in position directly under the decimal points in the given numbers.

add 6.47 + 3.40.8 + 73.523

Stack them   Lining up the decimals

420.793

Subtract 13.94 – 7.693

Again, stack and align the decimal points

6.247

Chelsea (Lindsey) had $ 317.58. She made three purchases of $19.95, $ 27.49, and $ 89.98

How much did she have left?

Add the purchase first… stack them

137.42

Take the total of her purchases and subtract that from the amount she started with $ 317.58

 

180.16

She had $180.16 left!

Estimations-  round to the highest place value of the SMALLEST number.

8.574 + 81.03 + 59.432

First figure out which is the smallest number?

In this case it is 8.574

Round that to its highest place value

8.574 rounded  to the ONES place is 9

so round all the other numbers to the ONES places

and add them

9 + 81 + 59

You can stack them and add normally

9 + 81 + 59 = 149

76.061 – 3.211

when you estimate you get 76  - 3= 73

When you actually add them you arrived at

76.061
-3.211

72.850

Which we write as 72.85

What happens with

(75.004 – 1.32) + ( 41.13 – 2.891)

First take the difference of each set of ( )

then add the differences

73.684 + 38.239 = 111.923

Math 7 (Period 4)

Using Rules to Add Integers 3.3

To add two integers with the same sign à just add them and use their sign

-          7 + (-5) = -12

To add two integers with different signs find the absolute values and take the difference. Use the sign of the number with the larger absolute value.

In 7^th grader terms:

Thinkà  Two teams:  the negatives and the positives

Ask yourself.. Who wins?  (that tells you who has the greatest absolute value)..

Stack them with the winner on top and take the difference ( subtract)

Use the sign of the winner!

The two questions to ask yourself ...Who wins? and By how much? 

43 + (-152)

The negatives win here

stack

152

-43

109

The solution must be -109

What if there are several addends? Here is a great strategy to follow:

1) See if there are any additive inverse first. Cross them out  using Inv+

2) Add the positives to the positives

AND

the negatives to the negatives

What properties allow us to do that? C+ and A+

3) Finally add the positive sum to the negative sum…
See “Who wins?” and “By how much?”

We tried this with a string of addends:

+ 3 + (-2) + 17 + 20 + (-3) + (-17)

We noticed that 3 and (-3) were additive inverses as well as

17 and (-17)

We crossed them out and were left with

(-2) + 20

That became easy—The signs were different so we asked ourselves.. “Who wins? and “By how much?”

the positive won.. by 18 so the answer was

+18

Then we tried:

+4 + (-5) + 18 + 3+ 25 + (-18) +(-4) + (-6)

we noticed we could cross out the +4 and the (-4)

as well as the +18 and the *-18)

Those were both additive inverses.

We were left with

+(-5) + 3+ 35 + (-6)

Add the positives 3 + 25 = 28

Add the negatives +(-5)+(-6) = -11

Then take the difference

28 -11 = 17

Much easier than working from left to right. You will make less “silly” mistakes using these strategies!

Adding Integers with variable expressions

Just substitute in for the variable, putting the substituted number into hugs (  )

Then evaluate using the integer rules… Plug & Chug

y + 5  where y = -12

(-12) + 5 = -7

Math 7 (Period 4)

Using a Number Line to Add Integers 3.2

Adding integers on a Number Line-

To find the sum of two integers a and b

1)  Start at ZERO… and move  │a│ units to the right if a is positive or toe the left if a is negative

2) then move  │b│ units to the right, if b is positive or to the left it b is negative. The sum is the  final position on the number line.

Now it becomes easier to see some of the Rules of Integers…

To find the sum of two negative integers, first add the absolute values of the integers . Then use a negative sign to show that the sum of the integers is negative.

What do we do if the integers are both positive? That’s easy—you do what you have done since… 2^nd grade…

So, can we write a general rule?

Yes,  

The rule for adding integers with the same sign—just add the integers and use their sign!

What happens, however, if we are adding integers with Different signs?

We drew a number line to represent -7 + 3 and found it was equal to -4

We noticed that the absolute value of the sum of the integers │-4│  is the difference between the length of the longer arrow and the length of the shorter arrow.  The sign of the SUM (negative) is the same as the sign of the integer with the longer arrow on our graph…

To find the sum of two integers with different signs, find the absolute value of the integers. Then take the difference between the two absolute values. Use the sign of the integer with the larger absolute value.

The sum of 0 and any integer is that integer.

Algebra Honors ( Periods 6 & 7)

Dividing Monomials 5-2

There are 3 basic rules used to simplify fractions made up of monomials.
Property of Quotients
if a, b, c, d are real numbers with b≠0 and d ≠0

ac/bd = a/b ⋅c/d
Our example was 15/21 = (3⋅5)/(3⋅7) = 5/7
The rule for simplifying fractions follows ( when a = b)
(bc)/(bd) = c/d
This rule lets you divide both the numerator and the denominator by the same NON ZERO number.

35/42 = 5/6
-4xy/10x = -2y/5 which can also be written (-2/5)x as well as with out the (((HUGS)))

c⁷/c⁴ = c⁴c³/c4 = c³
another way we proved this was to write out all the c's
c⋅c⋅c⋅c⋅c⋅c⋅c⋅/c⋅c⋅c⋅c = and we realized we were left with
c⋅c⋅c = c³
In addition, we noticed that
c⁷/c⁴ = = c^7-4 = c³
THen we considered
c⁴/c⁷ =
c⋅c⋅c⋅c/c⋅c⋅c⋅c⋅c⋅c⋅c = 1/c⋅c⋅c = 1/c³ = c^-3

Since we all agreed that any number divided by itself was = 1
(our example was b⁵/b⁵ ), we proved the following
1 = b⁵/b⁵ = b^5-5 = b⁰

We finally arrived at the Rule of Exponents for Division

if m > n
a^m/aⁿ = a ^m-n

If n > m
a^m/aⁿ = 1/a ^n-m
and if m = n
a^m/aⁿ = 1

A quotient of monomials is simplified when
1)each base appears only once in the fraction,
2) there are NO POWERS of POWERS and
3)when the numerator and denominator are relatively prime, that is, they have no common factor other than 1.

35x³yz⁶/ 56x⁵yz
5z⁵/8x²

Finding the missing factor when you are given the following
48x³y²z⁴ = (3xy²z)⋅ (______)
we find that

48x³y²z⁴ = (3xy²z)⋅ (16x²z³)

Algebra Honors ( Periods 6 & 7)

Factoring Integers 5-1

When we write 56= 8⋅7 or 56 = 4⋅14 we have factored 56
to factor a number over a given set, you write it as a product of integers in that set ( the factor set).
When integers are factored over the set of integers, the factors are called integral factors.

We used the T- charts (students learned in 6th grade) to first find the positive integer factors
56 = 1, 2, 4, 7, 8, 14, 28, 56

A prime number is an integer greater than 1 that has no positive integral factors other than itself and 1.
The first ten prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29

To find prime factorization of a positive integer, you express it as a product of primes. We used inverted division (again taught in 6th grade)

504
Try to find the primes in order as divisors.
Divide each prime as many times as possible before going on to the next prime
we found 504 - 2⋅2⋅2⋅3⋅3⋅7
which we write as 2³⋅3²⋅7
Exponents are generally used for prime factors
The prime factorization is unique--> and the order should be from the smallest prime to the largest.
A factor of two or more integers is called a common factor of the integers.
The greatest common factor (GCF) of two or more integers is the greatest integer that is a factor of all the given integers.

Find the GCF(882, 945)
First find the prime factorization of each integer Then form product of the smaller powers of each common prime factor.
The GCF is only the primes (and the powers) that they SHARE!!
882 = 2⋅3²⋅7²
945 = 3³⋅5⋅7
The common factors are 3 and 7
The smaller powers of 3 and 7 are 3² and 7
You combine these as a PRODUCT and get

the GCF(882, 945) = 3²⋅7 = 63

We also talked about listing ALL pairs of factors--> thus including negative integers
For example:
List all the pairs of factors of 20
(1)(20) but also (-1)(-20)
(2)(10) and (-2)(-10)
(4)(5) and (-4)(-5)

Listing all the factors of -20, we discovered
(1)(-20) but also (-1)(20)
(2)(-10) and (-2)(10)
(4)(-5) and (-4)(5)

Math 6A (Periods 1 & 2)

Rounding 3-5 (cont'd)

Round the following number to the designated place value:
509.690285

tenths: 509.690285
You underline the place value you are rounding to and look directly to the right. If it is 0-4 you round down; if it is 5-9 you round up 1.
so here we round to
509.7

hundredths
509.690285
becomes 509.69

hundred-thousandths
509.690285
becomes
509.69029

tens
509.690285
becomes
510

(a) What is the least whole number that satisfies the following condition?

(b) What is the greatest whole number that satisfies the following condition?
A whole number rounded to the nearest ten is 520.
Well, 515, 516, 517, 58, 519, 520, 521, 522, 523, 524 all would round to 520
so

(a) 515
(b) 524

A whole number rounded to the nearest ten is 650
(a) 645
(b) 654

A whole number rounded to the nearest hundred is 1200
(a) 1150
(b) 1249
How about these...
(a) What is the least possible amount of money that satisfies the following condition?
(b) What is the greatest possible amount?

A sum of money, rounded to the nearest dollar is $57
(a) $56.50
9b) $57.49

A sum of money rounded to the nearest ten dollars $4980
(a) $4975
(b) $4984.99

Math 6A (Periods 1 & 2)

Rounding 3-5 



Our book talks about phonograph records... we needed to have a discussion about what a phonograph record is... Then we discussed what the cost of $7.98 would round to? We decided that $8 would be a good rounded number.

A general rule:
We underline the place we are rounding to and look to the number to its right. If the number is 5 or greater we round our underlined number up one and put zeros after it...
32, 567
rounded to the nearest ten thousand --> 30,000

rounded to the nearest thousand --> 33,000

rounded to the nearest hundred --> 32,600

rounded to the nearest ten --> 32,570

Rounding Decimals becomes a similar process BUT there are a few differences:

4.8637

rounded to the nearest thousandth --> 4.864

rounded to the nearest  hundredth --> 4.86

rounded to the nearest tenth  --> 4.9

rounded to the nearest unit  5

Math 6A (Periods 1 & 2)

Comparing Decimals 3-4



In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.

Compare the following:


1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68.


2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 .


3. 0.83 and 0.833

To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833

0.83 = 0.830 Then compare


0.830 and 0.833 since 0 <3 br="br">
Then 0.830 < 0.833.


Write in order from least to greatest


4.164, 4.16, 4.163, 4.1



First, express each number to the same number of decimal places

Then compare. 4.164, 4.160, 4.163, 4.100


The order of the numbers from least to greatest is


4.1, 4.16, 4.163, 4.164
4.1< 4.16 < 4.163 < 4.164

October 10, 2013 Powers of Ten

Powers of Ten Day
... and it is a Binary Day as well..
Check out this great Video on the Powers of Ten
POWERS OF TEN

Math 7 (Period 4)

Integers and Absolute Value 3.1

Why are operations with integers so important?

Integers, which include whole numbers and their opposites are used to describe signed quantities such as temperatures above and below 0 º, elevations above and below sea level, and gain and losses in value.

Operations with integers are used in many careers including sports writing and paleontology, for example. For example, sportswriters use integers as they calculate movement on the football field or scores over or under par in a golf game.

Sets of Numbers:
Counting or Natural Numbersà  1, 2, 3, 4, …

Whole Numbers à 0, 1, 2, 3, 4…

Integers à …-4, -3, -2, -1, 0, 1, 2, 3, 4….

Integers include the Natural Numbers and their Opposites AND Zero

The integer 0 is neither negative NOR positive.

Students normally read -3 as “negative 3”

We graphed the negative integers, ZERO, and the positive integers on a number line. When graphing specific integers on a number line, you must place a closed dot on the point that represents the integer. See Page 105 of your textbook looking at Example 1 as well as Example 2.

The Absolute Value of a number is the distance between the number and 0 on a number line. Absolute Value is a DISTANCE concept—and therefore the absolute value of any number CANNOT be negative. Distance cannot be negative!  

Absolute values are written with two vertical bars called absolute value signs.

│7│ = 7   You read this as… “The Absolute Value of 7 is 7”

│-7│ = 7  You read this as… “The Absolute Value of -7 is 7”

│0│ = 0  You read this as… “The Absolute Value of 0 is 0”

Two numbers that have the same absolute value BUT have different signs are called opposites. For example -7 and 7 are opposites.
The negative sign can also be thought of as “the opposite of ” 
Thus “the opposite of 7” and “negative 7” are the same number

We talked about the different temperatures around the country… Buffalo, NY can get really cold and in our book it has an example of -20º We compared that temperature with our 93º temperature that we had only a few days ago.

What is the difference between the two? 93 – (-20) = 113º WOW!

Algebra Honors ( Periods 6 & 7)

Problems Without Solutions 4-10

Not all word problems have solutions. We listed three of the reasons for this:
1) Not Enough Information ( NEI)
2) Unrealistic Results
3) Facts are contradictory

We used the following examples:
Jade/ Jamie drove at her normal speed for the first 2 hours of the trip--- but the road repairs slowed her down 10 mph slower than her normal speed. She made the trip in 3 hours. Find her normal speed.
Wait... just looking at this you realize you just don't have enough information.
We even made a chart with the information we had.. and it just was not enough

There is NO SOLUTION Not enough information or NEI

Shane/Brandon has a beautiful lawn that is 8 m longer than it is wide... and it is surrounded by a wonderful flower bed which he and his brother maintain for his mother The flower bed is 5m wide all around. Find the dimensions of the lawn if the area of the flower bed is 140m²

When you try to solve this problem by letting the dimensions of the lawn be w and w + 8 you find the following equation
(w + 10)(w + 18) - (w)(w +8) = 140
however that leads us to
w = -2
Since the width of the lawn cannot be negative,
There is No SOLUTION and the given facts are unrealistic.


Amit/Phillip says he has equal number of dimes and quarters but that he has 3 times as many nickels as he has dimes. He also tells us that the value of his nickels and dimes is 50 cents more than the value of his quarters. How many of each kind of coin does he have?

Let d = the number of dimes
well if he has the same number of quarters as he has dimes
then d also can equal the number of quarters
and with the other information
3d = the number of nickels.
Now looking at the information he gave us
10d + 5(3d) = 25d + 50
But that simplifies to
25d = 25d + 50
which is impossible.
There is NO SOLUTION
The given facts are contradictory