revisiting numbers grade 8 Bộ Sách Toán THCS Của Mỹ

Revisiting
Numbers
Number

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Mathematics in Context is a comprehensive curriculum for the middle grades.
It was developed in 1991 through 1997 in collaboration with the Wisconsin Center
for Education Research, School of Education, University of Wisconsin-Madison and
the Freudenthal Institute at the University of Utrecht, The Netherlands, with the
support of the National Science Foundation Grant No. 9054928.
This unit is a new unit prepared as a part of the revision of the curriculum carried
out in 2003 through 2005, with the support of the National Science Foundation
Grant No. ESI 0137414.

National Science Foundation
Opinions expressed are those of the authors
and not necessarily those of the Foundation.

Abels, M., Wijers, M., and Pligge, M. (2006). Revisiting numbers. In Wisconsin
Center for Education Research & Freudenthal Institute (Eds.), Mathematics
in context. Chicago: Encyclopædia Britannica, Inc.

Copyright © 2006 Encyclopædia Britannica, Inc.
All rights reserved.
Printed in the United States of America.
This work is protected under current U.S. copyright laws, and the performance,
display, and other applicable uses of it are governed by those laws. Any uses not
in conformity with the U.S. copyright statute are prohibited without our express

written permission, including but not limited to duplication, adaptation, and
transmission by television or other devices or processes. For more information
regarding a license, write Encyclopædia Britannica, Inc., 331 North LaSalle Street,
Chicago, Illinois 60610.
ISBN 0-03-038568-7
1 2 3 4 5 6 073 09 08 07 06 05

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The Mathematics in Context Development Team
Development 2003–2005
Revisiting Numbers was developed by Mieke Abels and Monica Wijers.
It was adapted for use in American Schools by Margaret A. Pligge.

Wisconsin Center for Education

Freudenthal Institute Staff

Research Staff
Thomas A. Romberg

David C. Webb

Jan de Lange

Truus Dekker

Director

Coordinator

Director

Coordinator

Gail Burrill

Margaret A. Pligge

Mieke Abels

Monica Wijers

Editorial Coordinator

Editorial Coordinator

Content Coordinator

Content Coordinator

Margaret R. Meyer
Anne Park
Bryna Rappaport
Kathleen A. Steele
Ana C. Stephens
Candace Ulmer
Jill Vettrus

Arthur Bakker
Peter Boon
Els Feijs
Dédé de Haan
Martin Kindt

Nathalie Kuijpers
Huub Nilwik
Sonia Palha
Nanda Querelle
Martin van Reeuwijk

Project Staff
Sarah Ailts
Beth R. Cole
Erin Hazlett
Teri Hedges
Karen Hoiberg
Carrie Johnson
Jean Krusi
Elaine McGrath

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(c) 2006 Encyclopædia Britannica, Inc. Mathematics in Context and
the Mathematics in Context Logo are registered trademarks of
Encyclopædia Britannica, Inc.

Cover photo credits: (left to right) © William Whitehurst/Corbis;
© Getty Images; © Comstock Images
Illustrations
1, 2, 4, 6 Christine McCabe/© Encyclopædia Britannica, Inc.;
11 (top, bottom) Jerry Kraus/© Encyclopædia Britannica, Inc.; (middle)
Michael Nutter/© Encyclopædia Britannica, Inc.; 37, 54 Rich Stergulz
Photographs
1 © Tony Arruza/Corbis; 4 Victoria Smith/HRW; 5 (top) © Corbis;
(middle) © Tim Davis/ Corbis; (bottom) R. Clarke/Diomedea Images;
6 Sam Dudgeon/HRW; 8 (top) © Robert Galbraith/Reuters/Corbis;
(bottom) © David Madison/NewSport/Corbis; 10, 14 PhotoDisc/Getty
Images; 18 © Image 100; 20 Janice Carr/CDC; 23 Visuals Unlimited;
25, 36 Victoria Smith/HRW; 40 John Langfore/HRW

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Contents
Letter to the Student
Section A

Speed
The Wave
Make Some Waves
Rates and Units
Speed Records
Reaction Time
In a Jiffy
Speed of Light

Distance in Space
Summary
Check Your Work

Section B

1 ounce

25
27
29
32
33

Stir it up

9 ounces
10 ounces

Operations
Funny Zero
Negative Numbers
Number Properties
Summary
Check Your Work

Section E

16
17

20
22
23

Investigating Algorithms
Multiplication
Division
Fraction Operations
Summary
Check Your Work

Section D

1
2
2
5
6
8
8
10
12
13

Notation
Base Ten
Dilution
Small Numbers
Summary
Check Your Work

Section C

vi

36
37
40
42
43

Reflections on Numbers
Addition and Subtraction
Multiplication and Division
Random Number Activity
Powers and Roots
Summary
Check Your Work

45
46
47
50
52
53

Additional Practice

54

Answers to Check Your Work

60
Contents v

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Dear Student,
Have you ever been at a competitive event where
the race was too close to call? Electronic timers
today are very precise and can split a second into
a million parts. Precision is extremely important to
scientists as they map out unknown territories in
outer space and inside the human body.
In the unit, Revisiting Numbers, you use will learn to use numbers
more precisely. You will further investigate ways to represent very
large and very small numbers. You will reflect on all number
operations. You will improve your precision working with number
operations, by looking at related operations.
We hope you enjoy this unit.
Sincerely,

The Mathematics in Context Development Team

Neptune

Venus

vi Revisiting Numbers

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A

Speed

W

ES

T

RA

NO

MP

RT

HE

AS

T

R

A
P

N

R

TH

M

O

The Wave

NW

BROWNS

SW

695 f t

EAST SCOREBOAR D

BROWNS

WEST SCOREBOARD

NE

The Cleveland Browns Stadium can
seat over 73,200 people. Cleveland
fans often create a wave—people
stand up lifting their arms and then
quickly sit down. When one group
sits down, an adjacent group takes
over. The wave moves around the
entire stadium.

SE

SO

UT

HW

ES

T R
AM

T
SOU

P

HEA

ST

RA

M

P

933 ft

1. a. Estimate how many feet the wave travels one time around the
stadium. Describe how you made your estimate. The stadium
dimensions are 933 feet by 695 feet.
b. How much time will it take the wave to go once around the
stadium? Describe how you made your estimate.
c. Use your estimates to find the average distance the wave
travels in one second. You may want to use a ratio table for
your calculations.
Distance (in ft)
Time (in sec)

If a stadium wave travels 60 feet (ft) in 2 seconds (sec), it travels with
an average speed of 30 feet per second, or 30 ft/sec.

Section A: Speed 1

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Make Some Waves
For this activity, you will need:

•
•

stop watch
tape measure

Discuss with a classmate how to find the average speed of a wave.
Share your idea with the class.
Decide on a plan to find the average speed of a wave in your
classroom.
Record the time (in seconds) and distance (in feet) for a wave to
travel through your classroom.

2. a. Use the data from the activity to calculate the average
speed of the wave in your class.
b. Compare your class’s wave with the Cleveland Stadium
wave. Describe your findings.

Rates and Units
A rate is the ratio of two different measuring units. For example,
you can express the rate of speed in miles per hour (mi/h), or in feet
per second (ft/s). Using metric units, speed is usually expressed in
kilometers per hour (km/h), or meters per second (m/s).

2 Revisiting Numbers

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Speed A
3. What other rates do you know? Copy this table and complete it.
Units

Example

Heart Rate

Heartbeats/minute (bmp)

My heart beats at a rate of
65 beats per minute.

Data Transfer

Kilobytes/second (kB/s)

The speed of the download
stream is 1,024 kilobytes
per second.

Population
Density

You may remember from the unit Ratio and Rates how you used ratio

tables to express rates as a single number.
Stuart and Lexa are researchers who analyze average speeds of large
crowds. In a soccer stadium, they timed one wave taking 22 sec to
travel 440 seats. Each seat had a total width of 2 ft.
4. a. Calculate the average speed of the wave in seats per second.
b. Compare the average speed from this research to the speeds
you found in problem 2b. How do they compare?
Rita found 30 ft/s as the average speed of the wave in her class. She
wants to know how fast this is in miles per hour. Here is how she
started to solve the problem.
؋ 60

Distance (in ft)

30

Time (in sec)

1

؋ 60

Rita: “First, I multiplied by 60 to get the
number of feet per 60 seconds, or per one
minute.”
5. a. Explain Rita’s second step.

؋ 60

؋ 60

b. Copy Rita’s ratio table and
calculate the missing numbers.
Did you know? 5,280 ft are in 1 mi?
c. Use this information to find the
average speed of the wave into
miles per hour (mi/h).
Section A: Speed 3

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A Speed

You can use a similar technique to convert meters per second (m/s)
into kilometers per hour (km/h).
Kenny ran 60 m in 15 sec.
6. a. Copy this ratio table and use the arrows to
show the steps you have to make to find
Kenny’s distance in one hour.

Distance (in m)

60

Time (in sec)

15

b. What is Kenny’s average speed in meters
per hour? What is Kenny’s average speed
in km/h?
c. Will Kenny really be able to run that
distance in one hour? Explain.
d. Henri ran 50 m in 12 sec. Is Henri’s
average speed higher or lower than
Kenny’s? Show your work.

Maddie: “I did the 5K run in 25 minutes.
I wonder how fast I ran in kilometers per hour.”
7. Calculate Maddie’s average speed for the
5K run in km/h.

To compare speeds, you may have to change the
units. Changing kilometers per hour into miles
per hour is easy if you have an speedometer like
this one.
8. Explain why this speedometer is actually a
double number line.
4 Revisiting Numbers

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Speed A
In most European countries, the speed limit on highways is 120 km/h,
on county roads 80 km/h, and in towns 50 km/h.
9. How do these speed limits compare to those in the United States?

Speed Records
Here are some speed records for a variety of creatures.
a A cheetah, or hunting leopard, is
timed at 70 mi/h.

b The ostrich is a special sort of “sprinter.” It is a bird,
but it does not fly. It can run 20 mi in 40 min.
c American quarter horses are the
fastest horses in the world. They
can cover a quarter-mile in less
than 21 sec.
d In the water, the speed record is held by the sailfish, which
in a calm sea, can reach a speed of 100 m per 3.3 sec.
e The Indian spine-tailed swift bird
has repeatedly been clocked in level
flight over a carefully measured
two-mile track at 32.8 sec.

f On September 14, 2002, Tim Montgomery of the United States
set a world record in the 100 m, clocking 9.78 sec at the IAAF
Grand Prix Final.
10. Order these speed records on the odometer on Student Activity
Sheet 1. Show your work. Note that one of the speeds will not fit
on the odometer.
11. Reflect What is the difference between an average speed and a
maximum speed?
Can these two be the same? Explain your answers. Try to include
examples in your explanation.
Section A: Speed 5

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A Speed

Reaction Time
Digital timepieces improve accuracy.
12. a. What time is displayed on this timepiece?
b. What does the 04 mean?
c. How much more time needs to pass for the
timepiece to read 1 hour?
These hand-held timepieces are accurate; however, they
depend on the reaction time of the humans using the
timepiece. From the moment you see or hear a signal, it
takes time for the signal to go to your brain, time for
your brain to react, and finally time for your nerves and
the muscles in your fingers to react. You will determine
your personal reaction time using data collected from
the following activity.

For this activity, you need a centimeter ruler. You will
work with a classmate.
One student holds the ruler vertically, the other (the
catcher) holds his or her thumb and pointer finger
about 3 cm apart level with the 0 cm of the ruler.
Without any signal, the person holding the ruler lets go.
The catcher tries to react as quickly as possible and
catches the ruler. Record the number of centimeters

caught.
The number of centimeters caught is a distance that
will be used to calculate the catcher’s reaction time.
Do this experiment five times per person and record
the distances in centimeters.

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Speed A
When you know the distance in centimeters, you can use a formula to
calculate the reaction time in seconds.
tsec ‫؍‬

2؋d
980

d is the distance you caught, in centimeters.

Other formulas to find the reaction time are:

tsec ‫ ؍‬0.002 ؋ d

tsec ‫؍‬

d
490

or

tsec ‫ ؍‬0.045 ؋ d

13. a. Which of these three formulas gives the same result as the
first formula?
b. Use one of the correct formulas to calculate the five reaction
times you recorded in the previous activity. Calculate your
average reaction time.
Australia’s Cathy Freeman, a world-class athlete, had a reaction
time of 0.223 sec in the women’s 400 m final at the 1995 World
Championships. Her reaction time was measured with an electronic
device inside the starting block. This device recorded the interval
between the starting shot and the first athlete leaving the blocks.
14. a. On Student Activity Sheet 1 fill in the missing times indicated
by the blanks under the number line.

0

_____

0.100

_____

_____

_____

seconds

b. How much longer does this number line need to be in order to
locate 1 sec?
c. Place Cathy Freeman’s reaction time on the number line. Use
an arrow to point to the location.
d. Tests have confirmed that nobody can react in less than 0.110 of
a second. Place this minimum reaction time and your reaction
time from problem 13b on the number line. If necessary, extend
the line.
e. Explain why a false start is declared if the interval between the
starting shot and the athlete leaving the block is less than
0.110 of a second.

Section A: Speed 7

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A Speed

In a Jiffy
In 2002, Tim Montgomery of the United States set a world
record in the 100 m, clocking 9.78 sec. His time was one
hundredth of a second faster than the previous record of
9.79 sec set by Maurice Green in 1999.
Suppose both athletes ran in the same 100-m race, at their
own world record pace. Would the race be too close to
call? To investigate, you will need to zoom in on the finish,

the moment when Tim reached the finish line.
15 a. How much time (in sec) elapses between Tim’s win
and Maurice’s finish?
b. Make a guess. What is Maurice’s distance from the
finish line when Tim wins the race?
Instead of guessing, you can calculate this distance in
centimeters. This ratio table will help you with your
calculations.
Tim Montgomery

Distance (in cm) 10,000
Time (in sec)

9.79

1

0.1

0.01

c. Explain the numbers 10,000 and 9.79 that are in the
ratio table.
d. Calculate the distance in the last column. What do
you know now?
e. If you were at the race, would you be able to tell
who finished first? Explain your answer.

Maurice Green

Speed of Light
The speed of light is 299,792,458 m/s.
16. a. How many kilometers does light travel per second?
b. What is the speed of light in kilometers per hour?

8 Revisiting Numbers

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Speed A
The speed of light is often rounded to 300,000,000 m/s.
This table shows the many different ways to write this number.
Speed of Light
300,000,000
30,000,000
3,000,000
300,000
...
...
...
...
3

؋
؋
؋
؋
؋

؋
؋
؋
؋

1
10
100
1,000
...
...
...
...
...

‫؍‬
‫؍‬
‫؍‬
‫؍‬
‫؍‬
‫؍‬
‫؍‬
‫؍‬
‫؍‬

300,000,000
30,000,000
3,000,000
300,000
...

...
...
...
...

؋
؋
؋
؋
؋
؋
؋
؋

101
102
...
...
...
...
...
...

17. Copy the last row of the table in your notebook and fill in the
missing numbers.
You may remember writing very large numbers in scientific notation
in the unit Facts and Factors.
A number written in scientific notation is the product of a number
between 1 and 10 and a power of 10. The first number is called the
mantissa.

1,680,900, written in scientific notation is 1.68 ؋ 106.
Notice that the mantissa is rounded to two decimal places.
A calculator may display this number as:

1.68

06

or as

1.68 ؋ E06

18. a. Write 43,986,000,000,000 in scientific notation. Round the
mantissa to one decimal place.
b. Write the speed of light in km/h from 16b in scientific notation.
Round the mantissa to two decimal places.

Section A: Speed 9

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A Speed

Distance in Space
The average distance from Earth to the sun
is about 1.5 ؋ 108 kilometers. Neptune is 30
times farther away from the sun.
19. Write the distance from Neptune to the

sun in scientific notation.

Neptune

Venus

The distance from the sun to Venus is 0.72
times the distance from the sun to Earth.
20. Write the distance from Venus to the
sun in scientific notation.

Samantha and Jennifer are checking over their homework answers.
They disagree on this problem: 102 ؋ 10 3 ‫________ ؍‬
Samantha claims the answer is 105, while Jennifer thinks that it has to
be 106.
21. a. Who is right, Samantha or Jennifer? How would you explain it
to the person who did the problem incorrectly?
b. Explain why 104 ؉ 104 ؉ 104 ؉ 104 ؉ 104 ؉ 104 ؉ 104 ؉ 104 ؉
104 ؉ 104 ‫ ؍‬105.
Samantha and Jennifer disagree on this problem:
106 ، 103 ‫________ ؍‬
Samantha claims the answer is 102, while Jennifer thinks it has
to be 103.
c. Who is right, Samantha or Jennifer? How would you explain it
to the person who did the problem incorrectly?
Calculate the following problems without the use of a calculator.
Write your answers in scientific notation.
22. a. Multiply 8 ؋ 103 by 4 ؋ 102.
b. Divide 8 ؋ 103 by 4 ؋ 102.
c. Add 8 ؋ 103 and 4 ؋ 102.

d. Subtract 4 ؋ 102 from 8 ؋ 103.
23. If m and n are natural numbers, write this product, 10m ؋ 10n, as
a power of ten.
10 Revisiting Numbers

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Speed A

Math History
Different Number Systems

2

3

4

7

8

9

56

Our number system uses ten digits,
which is the same as the number

of fingers we have.

10

1

So we use a base-10 number system.

Diyala

Why do we have 24 hours in a day? Why are there 60 minutes in an
hour and 60 seconds in a minute?
ZA

Eshnunna

G

N

RO

S

W

M

O
z

S

Susa

ARABIAN

Eridu

Kark

heh

ru

ER

Ka

SUM

Larsa
Ur

ris

es

g
Ti

Uruk

at

n

Adab
Lagash

Eu
ph
r

IN

De

BABYLONIA

Babylon
Borsippa Nippur
Isin

TA

Kish

N

Sippar A K K A D

E
S

U

To explain this, we have to go back in history
4,000 years, to the land between the Tigris and
Euphrates Rivers, where the Sumerians lived.
They also used their fingers to count things, but
they did it differently.

HAWRS

The Sumerians counted finger joints instead of
fingers, and they used their thumb to do the
counting. Their number system was a base-12 number system, which
is why they divided a day into twelve parts.
DESERT

0

0

25

50

50

75 mi

100 km

Persian
Gulf

A thousand years later, the Babylonians, who lived in the same area as
the Sumerians, used a base-60 number
1 4
7
2 5
system. It is not sure why they chose
8 10
6
3 9
11
60. One reason might be because
12
base-60 makes divisional operations
Day
Night
easy since 60 is divisible by 2, 3, 4, 5,
6, 10, and 12.
They divided a day into two times 12 hours because twelve fits nicely
in their system. (12 ‫ ؍‬3 ؋ 4, and 60 ‫ ؍‬3 ؋ 4 ؋ 5). Hours were further
divided into 60 minutes, and the minutes were divided into 60
seconds. This system for time is still used today.
For other divisions, they used their base-60 system. In our decimal

system, the first decimal is tenths. In the base-60 system, the first fractional place is sixtieths. The first fractional place is called a minute,
the second place is called a second. So that is why an hour has 60
minutes, and a minute has 60 seconds.
The Babylonians not only measured time, they also studied astronomy
and measured angles. They divided the heavens into twelve sectors,
the time it takes the earth to complete one revolution around the sun.
Each sector was 30°, so a complete year took 360° (12 ؋ 30). Each
degree was further divided into 60 minutes, and each minute was
divided into 60 seconds.
Section A: Speed 11

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A Speed

Rates
A rate is the ratio of two different measuring units, written as a single
number. Examples of rates are:

•
•
•

Resting heart rate, such as 85 bpm
Speed of a car on a highway, such as 55 mi/h or 88 km/h
Data transfer rate, such as 1,024 kB/s

A ratio table is a helpful tool for finding a rate.

Suppose you traveled 150 km in 2.5 hours. A ratio table can help you
determine your average speed in km/h.
؋2

،5

Distance (in km)

150

300

60

Time (in hr)

2.5

5

1

؋2

Units

Traveling 150 km in
2.5 hours, what is the
average rate of speed?
Answer: 60 km/h

،5

Sometimes you have to convert the measuring units of a rate.

•

How fast is 5 m/s in kilometers per hour?
؋ 60

؋ 60

Distance (in m)

5

300

18,000

Time (in sec)

1

60

3,600

Now convert:

؋ 60

How fast is 5 m/s in
kilometers per hour?
Answer: 18 km/h

؋ 60

18,000 m ‫ ؍‬18 km, and 3,600 sec ‫ ؍‬1 hr.

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•

How fast is 60 km/h in meters per minute?
First convert:
60 km ‫ ؍‬60,000 m, and 1 hr ‫ ؍‬60 min.
، 60
Distance (in m)
Time (in min)

60,000

1,000

How fast is 60 km/h in

meters per minute?

60

1

Answer: 1,000 m/min

، 60

Scientific Notation
Any positive number written in scientific notation is a product of
two factors: a number between 1 and 10 (the mantissa) and a
power of 10.
28,600,000 written in scientific notation is 2.86 ؋ 107.
A calculator may display this number as:

2.86

07

or as

2.86 ؋ E 07

You may round the mantissa to one decimal place: 2.9 ؋ 107.

1 hours. What was her average rate of speed
1. Helen hiked 18 km in 4 ᎑᎑᎑
2

in km/h?

On January 31, 2005, the International Programs Center of the U.S. Census
Bureau estimated the world population to be 6,415,905,543 people.
2. a. What is the meaning of the six in this number?
b. Write this number in scientific notation rounding the mantissa to
one decimal place.

Section A: Speed 13

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A Speed
The earth makes one complete revolution
on its axis in 24 hours. The rotational speed
begins at zero at either geographical pole
and increases as you head toward the
equator.

3. Calculate the earth’s rotational speed at the equator in miles per
hour. The circumference of the earth at the equator is about
2.5 ؋ 104 miles.
In one year, the earth travels about 5.8 ؋ 108 miles as it orbits around
the sun.
4. Calculate the earth’s average orbital speed around the sun in
miles per hour.
5. Mercury’s average orbital speed around the sun is about 48 km/sec.
Is Mercury’s orbital speed faster or slower than the orbital speed of

the earth? Explain your answer.
6. Calculate the following problems without the use of a calculator.
Write your final answer using scientific notation.
a. Multiply 3 ؋ 104 by 2 ؋ 102.
b. Divide 2 ؋ 1010 by 106.
c. Add 2 ؋ 103 and 4.5 ؋ 102.
d. Subtract 6 ؋ 102 from 2 ؋ 103.

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When timekeepers used hand-held stopwatches, it was very difficult
to rank evenly matched competitors. At the 1960 Olympic games in
Rome, Australia’s John Devitt and America’s Lance Larson finished
neck-and-neck in the final of the 100-m freestyle swimming events.
All three timekeepers for Devitt’s lane clocked him at 55.2 sec.
Larson was clocked at 55.0, 55.1, and 55.1 sec. The judges placed
Devitt as the winner. The official time for both swimmers was
recorded as 55.2 sec.
7. a. Is this fair? Explain your reasoning using your knowledge
about reaction time.
b. Suppose Larson swam 100 m in 55.2 sec and Devitt finished
0.1 sec before Larson. What is Larson’s distance (in cm) from
the wall when Devitt finished the race? Would this have been
visible?

Write 236.7 ؋ 104 as a number.

Why is 236.7 ؋ 104 not written in scientific notation?

Section A: Speed 15

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B

Notation

Base Ten
In the previous section, you worked with small numbers with one or
more decimals. You will now further investigate decimal numbers,
such as 9.78.
1. a. What is the value of each digit in the number 9.78?
b. What is the value in each digit in the number 97.8?
c. How does the number 9.78 compare to the number 97.8?
Describe how they are the same and how they are different.
In the unit Facts and Factors, you learned that when you multiply a
number by 10, you multiply the value of each digit by 10.
Consider 72.35 and 72.35 ؋ 10.
؉

3 ؋ 10

1

1

؉ 5 ؋ 100

72.35 ؋ 10 ‫ ؍‬7 ؋ 100 ؉ 2 ؋ 10 ؉

3؋ 1

1
؉ 5 ؋ 10

72.35

‫ ؍‬7 ؋ 10

؉ 2؋1

‫ ؍‬723.5
When you divide a number by 10, you divide the value of each
digit by 10.
72.35

‫ ؍‬7 ؋ 10

؉ 2؋1

؉

72.35 ، 10 ‫ ؍‬7 ؋ ___ ؉ 2 ؋ ___ ؉
‫؍‬

____

‫؍‬

____

؉

____

؉

1
3 ؋ 10

1
؉ 5 ؋ 100

3 ؋ ___ ؉ 5 ؋ ___
____

؉

____

2. Fill in each blank on Student Activity Sheet 2.

16 Revisiting Numbers

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7

2

3

5

؋ 10
7

2

3

Here are two schemas for
multiplying by ten and
dividing by ten.

thousandths

hundredths

5

tenths

hundredths

3

ones

tenths

2

tens

ones

7

hundreds

tens

hundreds

Notation B

، 10

5

3. On Student Activity
Sheet 2, complete the
schema that shows

dividing by ten.

4. Calculate without using a calculator.
a. 4.8 ، 10

d. 9.8 ، 10 ، 10 ، 10

b. 4.8 ، 10 ، 10

e. 5 ، 1,000

c. 6.37 ، 10 ، 10

f. 1.25 ، 1,000

Dilution
You have to dilute many common household products before you use
them. You dilute dish soap with water before you wash dishes. Pool
water dilutes chemicals, such as chlorine, so the water can be clean.
You dilute condensed soup with milk or water before you cook it.
To investigate the dilution process,
Shelia dilutes food coloring with water.
She adds one ounce of food coloring
into a container that contains nine
ounces of water. Shelia has a 10-ounce
solution.

1 ounce

Stir it up

9 ounces
10 ounces

5. What portion of Shelia’s solution
is food coloring? What portion is
water?

Next, Shelia pours one ounce of the solution into an empty measuring
container. She adds enough water to make 10 ounces and stirs the
solution.
6. What portion of Shelia’s second solution is food coloring?

Section B: Notation 17

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B Notation

Shelia loves the new color. She decides to do it again and pours
one ounce of this second solution into another empty measuring
container. She adds enough water to make a 10-ounce solution.
7. What portion of Shelia’s third solution is food coloring?
8. Suppose Shelia repeats this process again. What portion of
Shelia’s fourth 10-ounce solution would be food coloring?
9. Copy and complete the table
for six dilutions.

Portion of the
10-oz Solution
That Is
Food Coloring

Number of
Dilutions

Suppose the table extended up
to 20 dilutions.
10. After 12 dilutions, what
portion of the 10-ounce
solution is food coloring?
What about after
20 dilutions?

1

1
0.1 or 10

2

1
0.01 or 100

3
4
5
6

You have probably found that it is just as tedious to write very small
numbers as it is to write very large numbers. An abbreviation system
like the one that you use for large numbers can be developed and
used for very small numbers as well.
When you were diluting food coloring, you divided the portion of
food coloring in each solution by 10.
Here is a way of describing this dilution process using arrow language.
0.1

، 10

0.01

، 10

0.001

، 10

_____

11. Copy and complete the arrow string until you reach the
sixth dilution.
Here is an arrow string starting at 10,000 and repeatedly dividing the
result by 10.
10,000

، 10

1,000

، 10

100

، 10

_____

12. Copy and continue the pattern until you have eight arrows.

18 Revisiting Numbers

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Notation B
10,000 ‫ ؍‬104

You can display this pattern of dividing by 10 vertically.
13. a. Copy the vertical display on the left in your
notebook. Write each result also as a power
of 10. Do this only with the numbers for which
you know the power of 10.

، 10
1,000 ‫؍‬
، 10

b. Describe any patterns you notice in the powers
of 10. How would you continue this pattern?

100 ‫؍‬
، 10

c. How would you explain to someone that 100 ‫ ؍‬1?

10 ‫؍‬

14. a. Look back at your chart of Shelia’s food coloring
dilutions. Suppose you have a solution that has a
10؊4 portion of food coloring. How many dilutions
have taken place? How do you know?

، 10
1‫؍‬
، 10

b. How many dilutions have taken place if the
solution is 10؊6 food coloring?

0.1 ‫؍‬
، 10

c. What portion of the solution is food coloring if
you have done five dilutions? Write this number
as a power of 10.

0.01 ‫؍‬
، 10

d. Describe the relationship between the number
of dilutions and the portion of the solution that
is food coloring.

0.001 ‫؍‬
، 10
0.0001 ‫؍‬

Think about diluting the seventh solution to make the eighth solution.
Remember that each dilution involves dividing by 10.
15. a. What is 10؊7 ، 10 written as a power of 10?
b. What is 10؊6 ، 100? (Hint: Dividing by 100 is two dilutions.)
16. Show that you can write the result of 100 ، 1,000 as 10؊1.
17. a. Calculate 106 ، 103.
b. Calculate 104 ، 107.

Section B: Notation 19

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