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Lesson
3 Working With Big Numbers
Problem Solving:
Chance Over Time
Working With Big Numbers
How do we work with big numbers?
Astronomers search the sky in an effort to understand the stars and
planets. They have spent thousands of years studying the motion of
planets. They have also calculated the distance of planets and stars
from our Sun. Their work helps us better understand the age of Earth,
the solar system, and our universe.
Astronomers have to work with large numbers because the distances
between planets, stars, and the Sun are vast. The numbers are hard
to read, and it is difficult to think about the actual distance of the
Sun to the Earth or how far light travels in one year. Here are some
measurements that astronomers have made.
Distance from the Sun
Location
Distance
Mercury
36,000,000 miles
Venus
67,000,000 miles
Earth
93,000,000 miles
Mars
142,000,000 miles
Jupiter
480,000,000 miles
Saturn
888,000,000 miles
Uranus
1,780,000,000 miles
Neptune
2,800,000,000 miles
Pluto
3,670,000,000 miles
Nearest star
25,260,788,000,000 miles
(Proxima Centauri)
Distance light
5,878,500,000,000 miles
travels in one year
492 Unit 7 • Lesson 3
Vocabulary
scientific notation
standard notation
Lesson 3
These distances are easier to describe using scientific notation .
Scientists use this kind of notation all the time as a shorthand for
describing large distances.
The method for writing numbers in scientific notation is not
complicated. We put a decimal point to the right of the number before
we begin. Then we count the number of places we are going to move the
decimal point.
Example 1 shows how we describe the distance from the Sun to the
Earth using scientific notation.
Example 1
Show the distance from the Sun to the Earth using scientific notation.
93,000,000 miles
9 3 0 0 0 0 0 0. = 9.3 · 107 miles
Move seven places to the left.
When we write the decimal number, we make sure we have one whole
number to the left of the decimal point.
In this example, we moved the decimal point seven places to the left.
This gives us a decimal number between 1 and 10 (including 1.0) that
is multiplied by 107. The key terms for our scientific notation are shown
below.
9.3 · 107
7th power of 10
Decimal number between
1 and 10
We always write scientific notation with a decimal number between 1
and 10 multiplied by a power with a base of 10.
Unit 7 • Lesson 3 493
Lesson 3
Often with scientific notation, we only include the tenths place in
the decimal number at the beginning. In the following example, this
means we need to round. The distance from the Earth to the Moon is
238,000 miles.
Example 2
Show the distance from the Earth to the Moon using scientific
notation.
238,000 miles
2 3 8 0 0 0. = 2.38 · 105 miles
Move five places to the left.
Rounding the number makes it easier to read. We round 2.38 up to the
nearest tenth, or 2.4.
The distance 2.38 · 105 miles can be rounded to 2.4 · 105 miles.
Another important term is standard notation . When we say a number
is written in standard notation, we mean it is written the “regular way.”
In Example 2, the number 238,000 is written in standard notation and
the number 2.4 · 105 is written in scientific notation.
Apply Skills
Turn to Interactive Text,
page 256.
494 Unit 7 • Lesson 3
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
Lesson 3
Problem Solving: Chance Over Time
How does probability change over time?
When we flip coins, roll dice, or draw from a deck of cards, we might get
results we do not expect. We have learned that the chance of rolling a
4 on a die is 1
6 . This means that after six rolls, we might expect to have
rolled a 4 at least one time. But what if we rolled no 1s, no 2s, and no
4s? That is not what we would expect. The bar graph shows the results
of six rolls.
Rolling a Die 6 Times
Roll of Die
5
4
3
2
1
0
one
two
three
four
five
six
Number on Die
This kind of outcome should be less surprising than we might think.
When we only try something a few times, we can get strange results.
Let’s say we rolled a die 20 times and got these results:
Rolling a Die 20 Times
7
Roll of Die
6
5
4
3
2
1
0
one
two
three
four
five
six
Number on Die
Unit 7 • Lesson 3 495
Lesson 3
Now each number has been rolled, but we have rolled more 5s than any
other number. There’s still a big difference between the number of 5s
and the number of 1s.
What happens after 100 rolls? Here is one result. The bars in the bar
graph are starting to even out. Each number on the die is starting to
appear about an equal number of times.
Rolling a Die 100 Times
Roll of Die
20
15
10
5
0
one
two
three
four
five
six
Number on Die
There are three facts about this kind of experiment that we need
to remember.
1. Every time we roll the die, each number has an equal chance
of appearing.
1
2. Each number on a side of the die has a 6 chance of landing
face up.
3. The more times we do an experiment, the closer we will get
to what we expect.
Problem-Solving Activity
Turn to Interactive Text,
page 257.
496 Unit 7 • Lesson 3
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
Lesson 3
Homework
Activity 1
Rewrite the numbers from standard notation to scientific notation. Remember
to round to the nearest tenth, if necessary.
Model 538,000 = 5.4 · 105
We rounded 5.38 to 5.4 in this problem.
1. 37,000 2. 4,000 3. 56,000 4. 120,000 5. 38,500 6. 5,600 Activity 2
Rewrite the problems as powers.
1. 2 · 2 · 2 · 2 · 2 · 2 2. 4 · 4 3. 5 · 5 · 5 4. 10 · 10 · 10 · 10 · 10 5. 3 · 3 · 3 · 3 6. 6 Activity 3
Write the probability for each situation. Show it in the form
(e.g., 3 out of 6) and as a fraction.
out of
1
13
1. What is the probability you will select an ace out of a regular deck of cards? 2. What is the probability you will turn up heads on a coin toss? 3. What is the probability you will roll a 6 on a six-sided die? 1
6
1
2
4. What is the probability you will select a red card out of a regular deck of cards? 5. If you roll a die 600 times, about how many times would you expect to roll a 4? 1
2
1
6
Activity 4 • Distributed Practice
Solve.
2
1. Write 5 as a percent. 2. Write 0.78 as a fraction. 3. Write 158% as a decimal number. 4.
5. 0.75 · 4 6. 1.54 ÷ 0.7 7. 3.7 + 2.9 + 8.8 + 4.02 8.
2
3
+ 19 10
4
7
9
− 13 78
100
1
6
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Unit 7 • Lesson 3 497