Download Reading and Writing Small Numbers

Reading and Writing
Small Numbers
Objective To read and write small numbers in standard and
expanded notations.
e
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Apply place-value concepts to read, write,
and interpret numbers less than 1. [Number and Numeration Goal 1]
• Convert between standard and
expanded notations. [Number and Numeration Goal 1]
• Apply extended facts and order of
operations to express the value of digits
in a number. [Operations and Computation Goal 2]
Key Activities
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing High-Number Toss
(Decimal Version)
Student Reference Book, p. 324
Math Masters, p. 455
per partnership: 4 each of number
cards 0–9 (from the Everything Math
Deck, if available), calculator (optional)
Students practice reading, writing,
and comparing numbers through
thousandths.
Ongoing Assessment:
Recognizing Student Achievement
Curriculum
Focal Points
Differentiation Options
READINESS
Modeling and Comparing Decimals
Math Masters, pp. 46 and 411–413
scissors
Students use base-10 grids to model and
compare decimals through thousandths.
ENRICHMENT
Decimals between Decimals
Students explore the infinite number
of decimals between any two given
decimal numbers.
Use Math Masters, p. 455. Students read and write numbers to
thousandths in standard notation and
expanded notation. They also convert
between these notations.
[Number and Numeration Goal 6]
Math Boxes 2 2
Key Vocabulary
Math Journal 1, p. 51
Students practice and maintain skills
through Math Box problems.
standard notation expanded notation
Study Link 2 2
Materials
Math Masters, pp. 44 and 45
Students practice and maintain skills
through Study Link activities.
Math Journal 1, pp. 48–50
Study Link 21
Math Masters, p. 410
transparency of Math Masters, p. 410
Advance Preparation
Allow three days for Lessons 21 and 22. Make at least one copy of Math Masters, page 410 per student.
Teacher’s Reference Manual, Grades 4–6 pp. 94–98
108
Unit 2
Operations with Whole Numbers and Decimals
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP1, SMP3, SMP4, SMP6, SMP7
Getting Started
Content Standards
6.NS.6c, 6.NS.7a
Mental Math and Reflexes
Math Message
Students divide numbers by 10 and record their answers on slates or
dry-erase boards.
Complete the Math
Message on journal page 48.
$10.00 / 10 $1.00
$1.00 / 10 $0.10
$0.30 / 10 $0.03
0.01 / 10 0.001
0.001 / 10 0.0001
0.0001 / 10 0.00001
Study Link 2 1
Follow-Up
Refer to the above problems during discussions of place-value chart patterns.
Briefly review answers.
1 Teaching the Lesson
▶ Math Message Follow-Up
(Math Journal 1, pp. 48 and 49)
WHOLE-CLASS
DISCUSSION
Adjusting
the Activity
SOLVING
In the previous lesson, students used their knowledge of
place-value concepts to read and write whole numbers. In this
lesson, they will apply similar place-value concepts to read and
write fractional quantities.
ELL
Have students highlight each th ending in
the place-value chart on journal page 48.
Discuss the difference between tenth as an
ordinal number and tenth as a fractional part.
AUDITORY
The number 0.1016 that students recorded in the chart is written
in standard notation. Standard notation is a base-ten placevalue numeration.
KINESTHETIC
TACTILE
VISUAL
Ask students to look at 0.1016 and identify the digits in the
following places:
tenths 1
hundredths 0
thousandths 1
ten-thousandths 6
Time
LESSON
22
Reading and Writing Numbers between 0 and 1
Math Message
28
A grain of salt is about 0.1016 millimeter long.
Write the number 0.1016 in the place-value chart below.
th
s
Ask students to generate a few sentences using the different
meanings of tenth. For example, Mary finished in tenth place;
A dime is a tenth of a dollar.
Student Page
Date
10
100
hu
nd
r
te ed
ns s
on
1
es
an
d
te
0.1
nt
hs
hu
0.01
nd
r
th ed
0.001
ou th
s
te san
n- d
0.0001
th th
hu ou s
0.00001
nd san
r
m ed dth
0.000001 illi -th s
on ou
th
sa
s
nd
Have students work in pairs to look for patterns in the place-value
chart. Then ask them to share the patterns they found. Patterns
include:
1 the value of the place to its left and 10 times
Each place is _
10
0 . 1 0 1 6
the value of the place to its right.
The value of each place in the place-value chart is a power
of 10. The value of each place to the left of the decimal point
is a product of 10s, for example, 100 = 10 ∗ 10. The value of
1s
each place to the right of the decimal point is a product of _
10
1 ∗_
1 = 0.1 ∗ 0.1.
(or 0.1s), for example, 0.01 = _
10
10
1
NOTE Multiplying a number by _
10 is the same as dividing the number by 10.
Write each of the following numbers in standard form.
0.4
1.
four tenths
2.
twenty-three hundredths
3.
seventy-five thousandths
7.
0.23
0.075
one hundred nine ten-thousandths 0.0109
0.08
eight hundredths
1.54
one and fifty-four hundredths
twenty-four and fifty-six thousandths 24.056
8.
Write the word name for the following decimal numbers.
4.
5.
6.
a.
0.00016
b.
0.000001
Sixteen hundred-thousandths
One millionth
Math Journal 1, p. 48
EM3MJ1_G6_U02_45_81.indd 48
1/11/11 5:33 PM
Lesson 2 2
109
Student Page
Date
To read a small number such as 0.1016, underline the final digit
and identify its place value. 0.1016, ten-thousandths Then read
the number (ignoring the decimal point and any leading zeros)
followed by the place value. One thousand sixteen ten-thousandths
Time
LESSON
Reading and Writing Small Numbers
22
Complete the following sentences.
thousandths , of an inch long.
Example: A grain of salt is about 0.004, or four
1.
A penny weighs about 0.1, or
2.
A dollar bill weighs about 0.035, or
tenth
one
, of an ounce.
thousandths
thirty-five
On average, fingernails grow at a rate of about 0.0028, or
4.
Toenails, on average, grow at a rate of about 0.0007, or
5.
It takes about 0.005, or
,
of a second for a smell to transfer from the nose to the brain.
6.
A baseball thrown by a major-league pitcher takes about 0.01, or
7.
A flea weighs about 0.00017, or
hundred-thousandths, of an ounce.
8.
A snowflake weighs about 0.00000004, or
twenty-eight
Writing the number as a fraction or a mixed number can also help
738
students read it. For example, 6.738 as a mixed number is 6_
1,000
and is read six and seven hundred thirty-eight thousandths.
, of an ounce.
3.
ten-thousandths, of a centimeter per day.
To write a small number, such as three hundred fifty-nine
ten-thousandths, in standard notation, the last digit of the
decimal number should be in the place value named.
Ten-thousandths Draw a decimal point and mark spaces up to
and including the place identified. 0.
Write the number in
the spaces so the final digit is written in the last space at the
right. 0. 3 5 9 Fill any blank spaces with zeros. 0.0 3 5 9
ten-thousandths , of a centimeter per day.
five thousandths
seven
hundredth
one
four
28
, of a second to cross home plate.
seventeen
hundred-millionths , of an ounce.
Try This
About how many times heavier is a penny than a dollar bill?
About 3 times as heavy
NOTE In Everyday Mathematics, a zero appears to the left of the decimal point
About how many times faster do fingernails grow than toenails?
in any number greater than 0 and less than 1. This makes it easier to order
decimal numbers, to draw attention to the decimal point, and to correspond with
the display on most calculators.
About 4 times as fast
Math Journal 1, p. 49
Have students complete journal pages 48 and 49.
45_81_EMCS_S_G6_MJ1_U02_576388.indd 49
2/15/11 12:57 PM
6. 7 3 8
1
10
ones
Adjusting the Activity
1
1,000
1
100
Have students use the place-value template (Math Masters, p. 410) as
they work on the journal pages, Math Boxes, and Study Link for this lesson.
Six and seven hundred thirty-eight thousandths
A U D I T O R Y
s
s
sa
nd
th
th
hs
ou
nte
1.
0.847
(8 ∗ 0.1) + (4 ∗ 0.01) + (7 ∗ 0.001)
2.
3.093
(3 ∗ 1) + (9 ∗ 0.01) + (3 ∗ 0.001)
3.
25.3
(2 ∗ 10) + (5 ∗ 1) + (3 ∗ 0.1)
5.
15.9994
6.
23.62173
7.
387.29046
0.09, or 9 hundredths
0.00003, or 3 hundred-thousandths
0.2, or 2 tenths
41
3_
100
75
_
3
_
17 100
17.03
Example 2: 3.41
9.
235.075
235 1,000
10.
0.0543
10
b.
1 hundredth = 0.1 ÷
10
c.
1 thousandth = 0.01 ÷
10
50
Math Journal 1, p. 50
EM3MJ1_G6_U02_45_81.indd 50
Unit 2
io
nt
hs
us
th
o
dre
ill
m
nd
0.00001
0.000001
s
nd
th
sa
ou
nth
0.0001
te
hu
nd
dt
sa
re
nd
ou
th
0.001
hs
hu
Have students work independently to complete journal page 50.
543
_
10,000
Use extended facts to complete the following.
1 tenth = 1 ÷
an
dt
hs
0.05, or 5 hundredths
Write each number as a fraction or mixed number.
735
_
1,000
Example 1: 0.735
0.01
Display a transparency of Math Masters, page 410 and distribute
one copy of the same page to each student. Ask students to write
the number 0.495 in the place-value chart. Students will apply
skills they used in the previous lesson to write this decimal
number in expanded notation.
(4 ∗ 0.10) + (9 ∗ 0.01) + (5 ∗ 0.001)
0.0005, or 5 ten-thousandths
196.9665
es
0 . 4 9 5
Give the value of the underlined digit in each number below.
4.
te
nt
(5 ∗ 1) + (9 ∗ 0.1) + (6 ∗ 0.01)
Example: 2.3504
0.1
100
Write each of the following numbers in expanded notation.
Example: 5.96
d
nd
re
d
te
ns s
hu
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
10
100
.
hs
th
s
s
an
th
nd
hs sa
dt ou
s
th san -th s
s
t
d
h
d
d
d
u
n
t
re
s
re sa ho
re on
nd ns nes nd nth und ou n-t und illi
o a
hu te
te
th te
h
h
m
hs
on
110
WHOLE-CLASS
ACTIVITY
Expanded Notation for Small Numbers
22
a.
V I S U A L
Time
LESSON
11.
(Math Journal 1, p. 50; Math Masters, p. 410; Transparency of
Math Masters, p. 410)
Student Page
8.
T A C T I L E
Notation for Small Numbers
Three hundred fifty-nine ten thousandths
Date
K I N E S T H E T I C
▶ Interpreting Expanded
th
th
ou
sa
nd
dt
nt
hu
nd
re
te
on
es
hs
0.0 3 5 9
1
10.
10
9.
2/1/11 10:58 AM
Operations with Whole Numbers and Decimals
Student Page
Date
2 Ongoing Learning & Practice
Time
LESSON
Math Boxes
22
1.
2.
Write each of the following numbers
using digits.
Write each number in expanded form.
a.
PARTNER
ACTIVITY
five and fifty-five hundredths
b.
one hundred eight thousandths
5.55
c.
3.
26
27
This line graph shows the average monthly rainfall in Jacksonville, Florida.
B The average rainfall increases from
June through December.
C The average rainfall for May and
November is about the same.
Average Monthly Rainfall
in Jacksonville, Florida
8
7
6
5
4
3
2
1
0
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
D Jacksonville gets more rain on average than Tampa.
5.
Janessa is 3 years older than her
brother Lamont.
a.
Use Math Masters, page 455 to assess students’ ability to compare decimals
through thousandths. Students are making adequate progress if they are able to
identify the larger number. Some students may not need a calculator to find the
difference between scores.
(9 ∗ 1) + (4 ∗ 0.01)
two hundred six and nineteen
ten-thousandths
A At least 10 months of the year, the
average rainfall is less than 3.5 inches.
4.
9.0402
Which conclusion can you draw from the graph?
Fill in the circle next to the best answer.
Divide the class into pairs and distribute four each of number
cards 0–9 to each pair, as well as a game record sheet (Math
Masters, p. 455). Students may need to play a practice game.
At the end of each round and when finding the total score,
allow students to use calculators, if needed.
(5 ∗ 10) + (3 ∗ 1) +
+ (2 ∗ 0.0001)
206.0019
(Student Reference Book, p. 324; Math Masters, p. 455)
Math Masters
Page 455
b.
0.108
(Decimal Version)
Ongoing Assessment:
Recognizing Student Achievement
53.078
(7 ∗ 0.01) + (8 ∗ 0.001)
Rainfall (in inches)
▶ Playing High-Number Toss
a.
If Janessa is 18 years old, how old
is Lamont?
15 years old
b.
Find the perimeter
of the square if
s = 4.3 cm. Use
the formula P = 4 ∗ s,
where s represents the
length of one side.
How old is Janessa when she is twice
as old as Lamont?
P=
6 years old
17.2
s
cm
212
51
Math Journal 1, p. 51
EM3MJ1_G6_U02_45_81.indd 51
1/11/11 5:33 PM
[Number and Numeration Goal 6]
▶ Math Boxes 2 2
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 51)
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 2-4. The skills in Problems 4 and 5
preview Unit 3 content.
Writing/Reasoning Have students write a response to the
following: Explain why each of the other three answers for
Problem 3 is not the best choice. Sample answer: A is not
correct because only 5 months have an average rainfall less than
3.5 inches. B is incorrect because the average rainfall is very low
from October to December. D is not correct because there is no
way to know what the average rainfall is for Tampa.
▶ Study Link 2 2
INDEPENDENT
ACTIVITY
(Math Masters, pp. 44 and 45)
Home Connection Students practice place-value
skills and write small numbers in standard and
expanded notations.
Study Link Master
Name
STUDY LINK
22
䉬
1.
Date
Time
Writing Decimals
Build a numeral. Write:
9 in the thousandths place,
4 in the tenths place,
8 in the ones place,
3 in the tens place, and
6 in the hundredths place.
2.
Build a numeral. Write:
3 in the tenths place,
6 in the ten-thousandths place,
4 in the hundredths place,
0 in the thousandths place, and
1 in the ones place.
Answer:
Answer:
3 8.4 6 9
1 .3 4 0 6
26–28
Write the following numbers in words.
eight-tenths
ninety-five hundredths
five-hundredths
0.05
sixty-seven thousandths
0.067
4.0802 four and eight hundred two ten-thousandths
3.
0.8
4.
0.95
5.
6.
7.
Write a decimal place value in each blank space.
hundred-thousandths ,
8.
Bamboo grows at a rate of about 0.00004, or four
kilometer per hour.
9.
The average speed that a certain brand of catsup pours from the mouth of the bottle is
about 0.003, or three
10.
thousandths
, mile per hour.
A three-toed sloth moves at a speed of about 0.068 to 0.098, or sixty-eight
thousandths to ninety-eight thousandths , mile per hour.
Math Masters, p. 44
Lesson 2 2
111
Study Link Master
Name
Date
3 Differentiation Options
continued
READINESS
▶ Modeling and
0.000001
0.00001
0.0001
0.001
0.01
on
es
an
d
.
1
10
100
hu
nd
r
ed
s
te
ns
䉬
0.1
22
Time
te
nt
hs
hu
nd
re
dt
th
ou hs
sa
n
te
n- dth
th
s
o
hu usa
nd
nd
th
re
s
dm
th
illi
on ous
an
th
s
dt
hs
Writing Decimals
STUDY LINK
Example: 2.756 ⫽ (2 º 1) ⫹ (7 º 0.1) ⫹ (5 º 0.01) ⫹ (6 º 0.001)
(1º 0.01) ⫹ (3 º 0.001)
(1 º 100) ⫹ (9 º 1) ⫹ (3 º 0.1) ⫹
(5 º 0.01) ⫹ (2 º 0.001) ⫹ (7 º 0.0001)
11.
0.013
12.
109.3527
13.
Using the digits 0, 3, 6, and 8, write the greatest decimal number possible.
(Math Masters, pp. 46, 411–413)
To provide experience comparing decimals, have students use
base-10 grids. Provide each pair with one copy of Math Masters,
page 46, two copies of page 411, one copy of pages 412 and 413,
and scissors. Review the worth of the flat, long, unit, and fractional
parts of the unit, as shown on Math Masters, page 46. After
students have prepared their base-10 grids, ask them to record
their models and compare the given decimal numbers using <, >,
or =.
8.6 3 0
Using the digits 0, 3, 6, and 8, write the least decimal number possible.
0.3 6 8
Try This
C
A
D
0.6
B
0.65
0.7
Name the point on the number line that represents each of the following numbers.
D
0.6299
A
Refer to the number line above. Round 0.6299 to the nearest hundredth.
18.
0.695
B
0.66
19.
17.
0.6
C
15.
16.
5–15 Min
Comparing Decimals
Write each of the following numbers in expanded notation.
14.
PARTNER
ACTIVITY
0.63
Practice
20.
0.01 ⫹ 0.006 ⫹ 0.0008 ⫽
45.009
22.
0.0168
⫽ 40 ⫹ 5 ⫹ 0.009
21.
23.
0.7 ⫹ 0.04 ⫹ 0.0002 ⫽
0.5801
0.7402
⫽ 0.50 ⫹ 0.080 ⫹ 0.00010
Math Masters, p. 45
ENRICHMENT
▶ Decimals between Decimals
PARTNER
ACTIVITY
5–15 Min
Between any two decimal numbers, there is always another
decimal number. Help students explore this concept by asking
them to list 20 or more decimal numbers between a given pair
of decimals. Suggested decimal pairs include:
NOTE Collect and store students’ base-10
grids for future use.
0.1 and 0.2; 0.33 and 0.34; 2.561 and 2.562.
Have students describe any patterns or strategies they used
to generate their lists.
Continue the discussion by asking students to comment about
the relative positions of their answers on the number line. Ask
questions such as the following:
Teaching Master
Name
LESSON
22
r
Date
Time
Modeling and Comparing Decimals
●
If a number is between 0.33 and 0.34, is it greater than or less
than 0.33? Greater than 0.33 Should the number be to the left
of 0.33 or to the right of 0.33 on the number line? To the right
of 0.33
●
If a number is between 0.33 and 0.34, is it greater than or less
than 0.34? Less than 0.34 Should the number be to the left of
0.34 or to the right of 0.34 on the number line? To the left
of 0.34
●
Choose two of your decimal numbers between 2.561 and 2.562.
Which number would be farther to the right on a number line?
Answers vary.
One way to compare decimals is to model them with base-10 grids.
The flat is the
whole, or 1.0.
The long is
worth 0.1.
The cube is
worth 0.01.
The fractional
part of the cube
is worth 0.001.
Another way to compare decimals is to draw pictures.
The flat is the
whole, or 1.0.
1.
The long is
worth 0.1.
The cube is
worth 0.01.
The fractional
part of the cube
is worth 0.001.
Use decimal models to complete the following.
1.0 = 0.10 ∗
10
0.10 = 0.01 ∗
10
0.01 = 0.001 ∗
10
Model the decimal numbers in each pair. Draw a picture to record each model.
Then compare the decimal numbers using <, >, or =.
2.
3.
0.3
>
0.14
4.
1.56
5.
0.2
>
<
Encourage students to sketch number lines and plot the decimal
numbers to verify their relative positions. Have students make a
general statement about the relative positions of two numbers on a
number line. Sample answer: A greater number will be farther to
the right on a number line.
1.562
Model and record a decimal number that is between
0.41 and 0.42. Sample answer:
0.025
0.41 <
46
0.418
< 0.42
Math Masters, p. 46
EM3cuG6MM_U02_041-070.indd 46
112
Unit 2
1/7/11 3:38 PM
Operations with Whole Numbers and Decimals