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

Math 6A ( Periods 1 & 2)

Decimals 3-3

Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.

Note that
1/10 = 1/10¹
1/100 = 1/10²
1/1000 = 1/10³

We also know that
1/10= 0/1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.


SO 1/10 = 1/10¹= 0.01 and it is equal to 10^-1
Notice that 10^-1 is NOT a negative number-- it is a small number
and 10^-21 is not a negative number it is a VERY TINY number

AS with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example

zero and six thousand three hundred ninety-four ten-thousandths.

Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?

Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.

So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."

Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so

zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400

Carefully see the distinction!!

Getting back to our 0.6394

to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004

0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)

0(10⁰) + 6(10^-1)+ 3(10^-2)+ 9(10^-3)+ 4(10^-4)

14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point