Transcript Fractions, Decimals, Percents, & Scientific Notation

Fractions, Decimals, Percents, &
Scientific Notation
Converting, Comparing, & Ordering
SOL 7.1
Fractions
A fraction is a way of
representing division of a ‘whole’
into ‘parts’.
𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟
𝐷𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
Examples
𝟐
A graphical example of fraction 𝟓 is shown by the marching
men below. There are five marching men in the illustration.
𝟐
There are 2 men in orange uniform, thus 𝟓.
Is there another fraction that can be made from the picture below?
3
5
Create a fraction for the pictures below.
3
1
𝑜𝑟
4
4
4 2
1
= 𝑜𝑟
6 3
6
Ordering Fractions
There are three major ways to order
fractions:
1.Drawing a picture
2.Converting
3.Thinking about it!!!
Before we start ordering we need to learn/review …
Vocabulary:
•
•
•
•
•
Descending: Greatest to Least
Ascending: Least to Greatest
Numerator: the part, the top of the fraction
Denominator: the whole, the bottom of the fraction
Quotient: answer to a division problem
Ordering by Drawing a Picture
Suppose you have to place the following in
ascending order… Least to Greatest!
2 3 1
1. Draw a box.
2 1 3
, ,
, ,
4 2
2. Draw lines in your box to 5
5 2 4
create rows. You need as many
rows as you have fractions.
3. Now break
each row up
into the number
of whole parts
of each
fraction…the
denominator.
4. Now
shade each
part of the
whole…the
numerator.
5. The row with the least color is the smallest and so on.
You try it!
3 2 5
Place in descending order: , ,
8 4 12
2 5 3
, ,
4 12 8
Ordering by Converting
Fractions can be converted into fractions for easy
ordering.
To convert a fraction, like
2
into
5
a decimal you must…
Divide the numerator (2) by the denominator (5). The
quotient is your decimal.
0. 4
Do this for each
fraction you
5 20
have!
-0
20
-20
0
But first…
Common Fraction Conversions Chart
Copy these down and
learn them…it’ll save
you some time!
Once you have your fractions converted…
2 1 1
, ,
5 9 10
0.4, 0.111, 0.1
1. Line the decimals up by their decimal points.
0 .4
0 .1 1 1
0 .1 0
2. Look at the numbers in the first column, the
smallest number corresponds to the smallest
fraction.
3. If the numbers are the same, move on to the
next column… then the next, and the next.
2 1 1
, , They were already in descending order!
5 9 10
Try It!
3 1 3
, ,
4 6 8
Place in order from
least to greatest.
0.75
0.167
0.375
1 3 3
, ,
6 8 4
Ordering by Thinking!
Fractions can arranged in 4 categories
when it comes to size….
Placing your fractions in these categories
can help you order them quickly!
How do you know if a Fraction is…
Less Than Half?
3
𝐻𝑎𝑙𝑓 𝑜𝑓 8 𝑖𝑠 4. 3 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 4.
8
𝑆𝑜, 𝑡ℎ𝑖𝑠 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 ℎ𝑎𝑙𝑓.
Half?
5
𝐻𝑎𝑙𝑓 𝑜𝑓 10 𝑖𝑠 5.
10
𝑆𝑜, 𝑡ℎ𝑖𝑠 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 ℎ𝑎𝑙𝑓.
More Than Half?
A Whole?
7
𝐻𝑎𝑙𝑓 𝑜𝑓 9 𝑖𝑠 4.5. 7 𝑖𝑠 𝑚𝑜𝑟𝑒 𝑡ℎ𝑎𝑛 4.5.
9
𝑆𝑜, 𝑡ℎ𝑖𝑠 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑚𝑜𝑟𝑒 𝑡ℎ𝑎𝑛 ℎ𝑎𝑙𝑓.
3
𝑊ℎ𝑒𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑎𝑛𝑑 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
3
𝑚𝑎𝑡𝑐ℎ 𝑖𝑡 𝑖𝑠 𝑜𝑛𝑒 𝑤ℎ𝑜𝑙𝑒.
𝑆𝑖𝑛𝑐𝑒 3 𝑎𝑛𝑑 3 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙, 𝑡ℎ𝑖𝑠 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑎 𝑤ℎ𝑜𝑙𝑒.
Try It!
Place in ascending order!
5 3 1
, ,
8 6 9
5 is more than half of 8!
3 is half of 6!
1 is less than half of 9!
So…
1 3 5
, ,
9 6 8
Your Assignment…
1. Order the following fractions from greatest to least by drawing a
picture!
1 4 3
𝑎) , ,
2 6 8
5 1 3
b) , ,
8 3 5
2. Place the following fractions in ascending order by converting!
7 3 1
𝑎) , ,
9 10 4
2 4 1
b) , ,
3 4 5
3. Order the following fractions from least to greatest by thinking about
it!
2 1 6
𝑎) , ,
9 7 8
5 1 2
b) , ,
6 4 5
Decimals
A decimal is similar to a fraction in
that it is not a whole number. It is a
part of a number.
13.032
Place Value
Thousands
Hundreds
Tens
Ones
Decimal Point (.)
Tenths
Hundredths
Thousandths
Ten Thousandths
In our decimal number system, the value of a
digit depends on its place, or position, in the
number. Each place has a value of 10 times
the place to its right.
Decimals can be named and
ordered by their place
value!
Always line decimals up to
the chart and each other by
their decimal point.
The decimal with biggest number that is farthest left is the
largest!
Let’s try it!
Ordering Decimals
Remember
Descending- Greatest to Least
Ascending- Least to Greatest
Let’s place 0.45, 0.123, and 0.1045 in ascending order!
1. Line up your decimals by the decimal point.
0.45
0.123
0.1045
You cannot tell
which decimal is
biggest by which
is longest!
2. Since the first digits are all zeros, we can look to the right of the
decimal point
The largest decimal will have the largest number in the
tenths place!
3. The same rule applies for the hundredths place and so on.
0.1045, 0.123, 0.45
Let’s Try Some…
1) 0.42, 0.31, 0.402 Place in descending order.
0.42, 0.402, 0.31
2) 0.2, 0.11, 0.3 Place in ascending order.
0.11, 0.2, 0.3
3) 1.02, 0.57, 1.201 Order least to greatest.
0.57, 1.02, 1.201
4) -0.42, 0.01, -0.2 Place in descending order.
0.01, -0.2, -0.42
Percent
A ratio that compares a number to
100.
A ratio compares two numbers
using division!
15% = 15 = 3
100
20
Percents and Fractions
Percents and fractions are directly connected…with one
you can change to the other.
Percent to Fraction:
1. Drop the percent sign.
2. Make the number that was before the
percent sign your numerator.
3. Make 100 your denominator.
4. Simplify! Your answer is INCORRECT if your fraction is not in
simplest form!
25% = 25 = 1
100
4
How do you simplify?!
How to Find the Simplest
Form of a Fraction
1. Find the greatest common factor of the
numerator and the denominator.
2. Divide both the numerator and the
denominator by that number.
The greatest common factor is the largest
factor between two numbers.
A factor, in this case, is a number that divides
evenly into another.
6
2
12
÷
=
6
3
18
Let’s trying turning some
percents to fractions!
Remember:
1. Drop the percent sign.
2. Make the number that was in front of the percent
sign the numerator!
3. Make 100 your denominator.
4. Simplify.
45%
45 ÷ 5 = 9
100 5
20
To simplify or reduce:
1. Find the Greatest Common Factor.
2. Divide the numerator and denominator by that
GCF!
Let’s try a few more
examples!
15%
15 ÷ 5 = 3
100 5
20
75%
75 ÷ 25 = 3
100 25 4
12%
12 ÷ 4 = 3
100 4
25
Percents and Decimals
Percents and decimals are directly connected…with one
you can change to the other.
Percent to Decimal:
1. Drop the percent sign.
2. Move the decimal to the left twice…this is the
same as dividing by 100!
What about percents with decimals in
them, like 25.5%?
The percent sign supersedes the
decimal!
Even though, 25.5 is indeed a decimal it is more
importantly a percent. The same steps need to be
followed to make it a “real” decimal!
Let’s trying turning some
percents to decimals!
Remember:
1. Drop the percent sign.
2. Move the decimal to the left twice!
45% 45. 4.5 .45
one
5%
two
5. .5 .05
one
two
10% 10. 1.0 .10
one
two
Now that you can go from
percent to decimal…
Can you go from decimal to percent?
Decimal to percent:
1. Move the decimal to the right twice…this is the same as multiplying
by 100!
2. Tack on a percent sign.
0.76 7.6 76. 76%
one
1.05 10.5
one
two
105. 105%
two
Ordering Percents
Remember
Descending- Greatest to Least
Ascending- Least to Greatest
Let’s place 46%, 12.3%, and 26% in descending order!
1. Look at the numbers and ignore the percent sign.
46
12.3
26
2. The larger the number, the larger a percent it is.
46%, 26%, 12.3%
Percents can also be turned into decimals for
ordering...follow the decimal rules for ordering.
Your assignment…
1. Convert the following percents to fractions.
a) 24%
b) 4%
c) 19%
d) 98%
24
6
a) 100 = 25
4
1
b) 100 = 25
19
c) 100
98
49
d) 100 = 50
2. Convert the following percents to decimals and decimals to percents.
a) 2%
b) .14
c) 32.5%
d) .08
a) 0.02
b) 14%
c) .325
d) 8%
3. Place the following percents in descending order.
a) 2.4%
b) 48%
c) 99%
d) 9.8%
99%, 48%, 9.8%, 2.4%
4. Place the following percents in ascending order.
a) 1%
b) 100%
c) 10%
d) 10.1%
1%, 10%, 10.1%, 100%
Scientific Notation
A more convenient way to express
really large or small numbers.
How old is the Earth?
4, 600, 000, 000 years old
This number is written in decimal
notation or standard form. When
numbers get this large, it is easier
to write them in scientific notation
What does Scientific Notation
look like?
# x 10
#
A number between 1 and 10
Times (multiplication)
BASE- In scientific notation there will always
be a base of 10!
EXPONENT- Power of ten is always an
integer.
How do you put a number
into Scientific Notation?
4, 600, 000, 000 years old
Where is the decimal point now?
After the last zero.
Where would you put the decimal to make
this number be between 1 and 10?
Between the 4 and the 6.
The Steps
4, 600, 000, 000 years old
1. Place a decimal point after the first natural number
(counting numbers).
4
. 600000000
2. Count the number of spaces you moved the
decimal point!
3. Drop all zeros AFTER the LAST natural number.
4
.6
4. Write your “x 10”.
.
4.6 x 10
4 6 x 10
5. Tack on your exponent (step 2). Remember,
numbers larger than 1 will have positive exponents!
9
The Steps
0. 0 0 0 0 0 0 0 2 0 4
1. Place a decimal point after the first natural number
(counting numbers).
.
0.00000002 04
2. Count the number of spaces you moved the
decimal point!
3. Drop all zeros BEFORE the FIRST natural number.
2
. 04
4. Write your “x 10”.
.
2.04 x 10
2 04 x 10
5. Tack on your exponent (step 2). Remember,
numbers smaller than 1 will have negative exponents!
-8
Remember!!!
Let’s try some!
2, 000, 000, 000
4.3 𝑥
0.0043
1, 203, 000, 000
0. 5
5𝑥
0.030201
2 𝑥 10
9
−3
10
1.203 𝑥
9
10
−1
10
3.0201 𝑥
−2
10
Ordering Scientific Notation
Order by the exponent!
The bigger the exponent, the bigger the
number and vice versus!
If the exponents are the same,
then order by the number in front of the
“x 10”!
Place in ascending order!
2.01 𝑥
1𝑥
2
10
8
10
2. 4 𝑥 10
4. 3 𝑥
1.1 𝑥
−1
2
10
−2
10
−2
1.1 𝑥 10
−1
2.4 𝑥 10
2
2.01 𝑥 10
4.3 𝑥
1𝑥
2
10
8
10
From Scientific Notation to
Standard Form!
1. Write down the number before the “x 10”.
2. Move the decimal point the number of
spaces that the exponent tells you.
Remember:
Positive moves Right
Negative moves Left
a) 2.1 𝑥 104
21,000
b) 4.06 𝑥 106
4,060,000
d) 5 𝑥 10−1
c) 1.1 𝑥 10−3
0.0011
0.5
Your Assignment
Course 2 Textbook
Chapter 1
1. Page 45 (10-34 EVEN)
2. Place the numbers written in
scientific notation in descending
order. Use the numbers from
problems 10-17 on page 45.
Ordering Fractions, Decimals,
Percents, and Scientific Notation!
Now that you know how to order these numbers
separately let’s order them all together!
1. Convert all the numbers to decimal or
standard form.
Type of Number
Convert to..
Fraction
Decimal
Percent
Decimal
Scientific Notation
Standard Form
Decimal
No Conversions Needed
2. Order them the way you order decimals.
3. Place all answers back in ORIGINAL
form.
Let’s Try Placing the Following
Numbers in Descending Order!
−2
3
3.6
×
10
37%
0. 345
8
1. Convert all the numbers to decimal or standard form.
−2
3
3.6
×
10
37%
0. 345
8
0. 37
0. 345 0. 375 0. 036
No need to
change
anything!
Numerator
divided by
the
denominator!
The
exponent is
negative, so
we moved
left.
Move the
decimal
point left
twice or
divide by
100!
Let’s Try Placing the Following
Numbers in Descending Order!
2. Order them the way you order decimals.
−2
3
3.6
×
10
0. 345
8
0. 37
0. 345 0. 375 0. 036
Start with
first column,
right now all
of the 3’s are
equal.
Looking at the
next column,
we can see 5 is
larger than 0,
making 0.375
the largest
decimal.
0. 3 4 5
0. 3 7 5
0. 0 3 6
0. 3 7
Let’s move
on to the
next column.
The two sevens
are tied for
largest!
Start from the
first column
again to find the
next largest!
Let’s Try Placing the Following
Numbers in Descending Order!
−2
3
3.6 × 10
37%
0. 345
8
0. 37
0. 345 0. 375 0. 036
3. Place all answers back in
ORIGINAL form.
3
8
37% 0. 345 3.6 ×
−2
10
Let’s try some…
−1
3.02 × 10
0.32
0.302
0.03215
32% 0.03215
0.03215
−1
3.02 × 10
2
5
4%
0.404
0.4
0.04
0.404
0.404
2
5
32%
1
1
4
1.25
1
1
4
4.11 × 10−2
4.11 × 10−2
0.0411
4%
Your Assignment…
Complete the
Ordering Practice
Sheet!