Download Mathematics Summer Session: Transition Math Chapter 1 Notes

Mathematics Summer Session: Transition Math Chapter 1 Notes
Lesson 1-1 & 1-2:
Just as we started our in-class discussion, you should familiarize yourself with the following Roman numerals:
I: 1
V: 5
X: 10
L: 50
C; 100
D: 500
M; 1,000
Writing a regular number in Roman numeral form is a process of adding letters. Remember, you can never use
more than three of the same letter in a row.
See the examples below:
I: 1
II: 2
III: 3
IV: 4
V; 5
VI: 6
VII: 7
VIII: 8
IX: 9
X: 10
XX: 20
XXX: 30
XL: 40
L: 50
C: 100
CC: 200
CCC: 300
CD: 500
D: 500
M: 1000
MM: 2000
MMM:
3000
With those basics, you can build to more complex numbers.
For example, if we needed the number 3,763:
We would have: MMM + D + CC + L + X + III = MMMDCCLXIII
Our decimal system works much the same, but it’s centered on the number ten. The whole numbers, in
decimal notation, include: 0, 1, 2, 3, 4, and 5…all the way to infinity.
Each digit represents a certain place value. The number “1” appears in the spot of each written place value:
Millions: 1,000,000
Hundred Thousands: 100,000
Ten Thousands: 10,000
Thousands: 1,000
Hundreds: 100
Tens: 10
Ones: 1
Tenths: 0.1
Hundredths: 0.01
Thousandths: .001
Ten-Thousandths: .0001
Hundred-Thousandths: .00001
Millionths: .000 001
We can use decimals less than one to be even more accurate. The number line below is divided into intervals of
tenths, and each tick mark represents a fraction of the previous whole number.
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Tick Mark
0.1 interval
1.8
1.9
2.0
Lesson 1-3 & 1-4:
To save time mathematically, we often estimate by rounding to the nearest digit, rounding up, or rounding
down.
Let’s take the fraction 2/3 and round it …
a) up to the next hundredth: 0.67
b) down to the next hundredth: 0.66
c) to the nearest hundredth: .067
When rounding up or down, simply take the decimal place you are asked for, and to round up, make it one
bigger, and to round down, keep it the same.
To round to the nearest decimal place, you must look at the digit to the left of the decimal place you’re
rounding to:
If it is 0-4, round down by keeping the decimal place the same.
If it is 5-9, round up by increasing the decimal place by one.
Remember that money is usually rounded up to the nearest penny.
Lesson 1-6:
The fraction ½ can be written as 1/2 or as 1÷2.
The number on top of the fraction bar is the numerator and the number below it is the denominator. Simple
fractions have whole numbers in both the numerator and denominator.
To convert a fraction to a decimal, you can either use a calculator or divide by hand. 1÷2 = .5
You should memorize the following fractions and their decimal equivalents:
½ = .5
1/3 = .333…
2/3 = .667…
¼ = .25
¾ = .75
1/5 = .2
2/5 = .4
3/5 = .6
4/5 = .8
1/6 = .166…
5/6 = .833…
1/8 = .125
3/8 = .375
5/8 = .625
7/8 = .875
1/9 = .111…
2/9 = .222…
4/9 = .444…
5/9 = .555…
7/9 = .777
8/9 = .888
Lesson 1-7:
Mixed numbers consist of a whole number and a fraction. Basically, any decimal that is greater than one can be
represented as a mixed number. Look at a decimal like 2.5 and think of it as 2 + 1/2. So, as a mixed number, 2.5
can simply be written as 2 ½.
Mixed numbers can also be written as improper fractions. Multiply the denominator by the whole number and
then add the numerator. Keep the same denominator. So, 2 ½ would become (2 x 2 + 1)/2 = 5/2.
Lesson 1-8:
Throw a negative sign in front of any of the positive whole numbers, and you have negative numbers. Unlike
positive numbers, the closer a negative number is to zero, the bigger it is. So, -.0001 is bigger than -1,000,000.
Integers consist of the positive and negative whole numbers including zero. On a number line, to the left of
zero is negative and to the right is positive.
Lesson 1-9:
Numbers can be compared by using the < and > signs. We all know that 100 is greater than 25, and this can be
written mathematically as 100 > 25. This can be written as 25 is less than 100, which can be symbolized by 25 <
100. Think of the sign as an alligator, which always eats the bigger number.
What is this, second grade? Of
course you know I eat the bigger
number!
Lesson 1-10:
Equal fractions can all be reduced to the same simplest fraction in lowest terms. The fractions 6/9 and 20/30
can all be reduced to 2/3. So, they’re all equal.
The key to fractions is being able to reduce them into lowest terms. You must find the greatest common factor
of the numerator and denominator.
Take, for instance, the fraction 48/84. We must find the biggest number that goes into both 48 and 84. This
takes some practice, but try first writing down all the factors that go into each number.
The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
The factors of 84 are: 1, 2, 3, 4, 6, 14, 21, 28, 42, and 84.
What is the biggest factor they have in common? 12. So, the GCF is 12.
Now, divide the numerator and denominator by 12.
(48÷12)/(84÷12) = 4/7. You can also use any basic scientific calculator to reduce fractions or to change them
into decimals. I would suggest you become familiar using both your head and calculator to reduce fractions, or
to convert them.
Lesson 2-1:
To multiply numbers by 10, 100, 1000, etc., you move the decimal point of the original number to the right the
number of digits equal to the number of zeroes in the factor of ten.
To multiply by ten, move the decimal point one digit to the right.
21.3 x 10 = 213
To multiply by 1000, move the decimal place three spaces to the right.
21.3 x 1000 = 21300
We can simplify these “ten” numbers by using exponents.
Below I have taken “ten to the third power”:
Exponent
Base
103
What does this really mean though? You take the “base” times itself the number of times indicated by the
exponent.
103 = 10 x 10 x 10 = 1,000
It’s important to learn the powers of ten. They’ll become important later on.
Any number to the “0” power equals one. Any number to the first power is itself.
To multiply a number by a positive integer power of ten, move the decimal point to the right the same number
of places as the value of the exponent.
Lesson 2-3:
We can write very large numbers in a simple form, by using scientific notation.
In scientific notation, a number greater than or equal to one, but less than ten, is multiplied by a power of ten.
Let’s take the speed of light, which is 300,000,000 meters per second. How can we make this shorter.
300,000,000 is basically 3 times 100,000,000. And, we can easily write 100,000,000 as a power of ten. Just
count the number of zeroes, which is eight, and make that the exponent of the base ten.
So, in scientific notation, 3 x 108 = 300,000,000.
Lesson 2-4
To multiply by 1/10, 1/100, 1/1000…, you simply move the decimal point of the number to the left the same
number of places as there are zeros in the denominator of the fraction.
So, 21.3 x 1/100 = 0.213.
All I did was move the decimal two places to the left, because 100 has two zeroes.
To change any decimal to a percent, multiply it by 100, and add a % sign. Or, move the decimal two places to
the right.
To write .25 as a percent, you can multiply by 100 to get 25 (same as moving the decimal place two spaces to
the right), and it becomes 25%.
To change a percent back to a decimal, just move the decimal place two spaces to the left.
Lesson 2-5:
To take the percent of another number, it helps to first change the percent to a fraction or decimal.
30% of 100 is the same as .3 times 100 or 3/10 times 100.
Lesson 2-6:
We already learned how to change a fraction to a decimal. To change a fraction to a percent, you must first
change it to a decimal, and then use what you already know to change the decimal to a percent.
¼ = 1 ÷ 4 = .25 = 25%
To work backwards, move the decimal of the percent to places left, and change that decimal into a fraction.
We’ll add to this later in the summer.
Lesson 2-8:
Numbers can also have negative exponents. For now, we’ll only consider the negative powers of ten.
10-1 = .1
10-2 = .01
10-3 = .001
Lesson 2-9:
So, to put small numbers into scientific notation, again we start with a number between 1 and 10, and multiply
it by a negative integer power of ten.
Let’s take a very small decimal, such as .0000025.
This is like 2.5 times .000001 which is the same as
2.5 x 10-6