Enhanced Instructional Transition Guide
Grade 8/Mathematics
Unit 01:
Suggested Duration: 4 days
Unit 01: Numerical Understanding: Rational Numbers (10 days)
Possible Lesson 01 (6 days)
Possible Lesson 02 (4 days)
POSSIBLE LESSON 02 (4 days)
This lesson is one approach to teaching the State Standards associated with this unit. Districts are encouraged to customize this lesson by supplementing
with district-approved resources, materials, and activities to best meet the needs of learners. The duration for this lesson is only a recommendation, and
districts may modify the time frame to meet students’ needs. To better understand how your district is implementing CSCOPE lessons, please contact your
child’s teacher. (For your convenience, please find linked the TEA Commissioner’s List of State Board of Education Approved Instructional Resources and
Midcycle State Adopted Instructional Materials.)
Lesson Synopsis:
Students identify reasons for using scientific notation. Students convert numbers in standard notation into scientific notation and vice versa. Students apply scientific
notation in problem situations.
TEKS:
The Texas Essential Knowledge and Skills (TEKS) listed below are the standards adopted by the State Board of Education, which are required by Texas
law. Any standard that has a strike-through (e.g. sample phrase) indicates that portion of the standard is taught in a previous or subsequent unit.
The TEKS are available on the Texas Education Agency website at http://www.tea.state.tx.us/index2.aspx?id=6148
8.1
Number, operation, and quantitative reasoning.. The student understands that different forms of numbers are appropriate for different
situations. The student is expected to:
8.1D
Express numbers in scientific notation, including negative exponents, in appropriate problem situations. Supporting Standard
Underlying Processes and Mathematical Tools TEKS:
8.14
Underlying processes and mathematical tools.. The student applies Grade 8 mathematics to solve problems connected to everyday
experiences, investigations in other disciplines, and activities in and outside of school. The student is expected to:
8.14A
Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with
page 1 of 42 Enhanced Instructional Transition Guide
Grade 8/Mathematics
Unit 01:
Suggested Duration: 4 days
other mathematical topics.
8.15
Underlying processes and mathematical tools.. The student communicates about Grade 8 mathematics through informal and
mathematical language, representations, and models. The student is expected to:
8.15A
Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or
algebraic mathematical models.
8.15B
Evaluate the effectiveness of different representations to communicate ideas.
Performance Indicator(s):
Grade8 Mathematics Unit01 PI02
Express in a written form (e.g., web page, blog, etc.) examples of appropriate problem situations (e.g., speed of light, diameter of blood cell, etc.) with the use of both positive
and negative exponents in scientific notation and standard form
Sample Performance Indicator:
Record, in a blog, the appropriate scientific measurements that are best written in scientific notation (both positive and negative exponents). Include the standard form and
scientific notation for each problem situation.
Standard(s):8.1D ,8.14A , 8.15A ,8.15B
ELPS ELPS.c.1E , ELPS.c.5B , ELPS.c.5G
Key Understanding(s):
Scientific notation is an effective mathematical representation to communicate specific real-life problem situations involving very small or very large numbers using
powers of ten.
Misconception(s):
Some students may think that 10-2 = (-100), instead of 10-2 = 0.01.
Vocabulary of Instruction:
page 2 of 42 Enhanced Instructional Transition Guide
Grade 8/Mathematics
Unit 01:
Suggested Duration: 4 days
scientific notation
Materials List:
graphing calculator (1 per student)
graphing calculator with display (1 per teacher)
place value mat (large) (1 per teacher)
Powers of 10 by Charles and Ray Eames (1 per teacher)
projection device (film) (1 per teacher)
Attachments:
All attachments associated with this lesson are referenced in the body of the lesson. Due to considerations for grading or student assessment, attachments
that are connected with Performance Indicators or serve as answer keys are available in the district site and are not accessible on the public website.
The Powers of Ten KEY
The Powers of Ten
Investigating Scientific Notation Notes KEY
Investigating Scientific Notation Notes
Investigating Scientific Notation KEY
Investigating Scientific Notation
Applications of Scientific Notation KEY
Applications of Scientific Notation
Reflecting Scientifically KEY
page 3 of 42 Enhanced Instructional Transition Guide
Grade 8/Mathematics
Unit 01:
Suggested Duration: 4 days
Reflecting Scientifically
GETTING READY FOR INSTRUCTION
Teachers are encouraged to supplement and substitute resources, materials, and activities to meet the needs of learners. These lessons are one approach to
teaching the TEKS/Specificity as well as addressing the Performance Indicators associated with each unit. District personnel may create original lessons using
the Content Creator in the Tools Tab. All originally authored lessons can be saved in the “My CSCOPE” Tab within the “My Content” area. Suggested
Day
1
Suggested Instructional Procedures
Notes for Teacher
Topics:
Spiraling Review
Introduction to scientific notation
Engage 1
ATTACHMENTS
Students use reasoning skills, patterns, and a calculator to investigate the effects powers of 10
Teacher Resource: The Powers of Ten KEY
have when multiplied by a number. Students will view models of various powers of 10 that depict
(1 per teacher)
the relative scale of specific magnitudes, and discuss the powers of 10 relationships to their place
Handout: The Powers of Ten (1 per
value names.
student)
Instructional Procedures:
MATERIALS
1. Project the film clip Powers of 10 for students to visualize powers of 10.
Powers of 10 by Charles and Ray Eames (1
2. Display a large blank place value mat that includes millions to hundred-thousandths. Facilitate
per teacher)
a class discussion about the powers of 10 using the place value chart. Record student
projection device (film) (1 per teacher)
responses on the displayed place value chart.
place value mat (large) (1 per teacher)
Ask
graphing calculator (1 per student)
page 4 of 42 Grade 8/Mathematics
Unit 01:
Suggested Duration: 4 days
Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Notes for Teacher
graphing calculator with display (1 per
What are the place values to the left of the decimal point? (ones, tens, hundreds,
teacher)
thousands, etc.)
How do you represent 100 as a power of 10? 1000? 10? 1? (102) (103) (101) (100)
What are the place values to the right of the decimal point? (Tenths, hundredths,
thousandths, etc.)
How do you represent 0.1 as a power of 10? 0.01? 0.001? (0.1 = one-tenth =
10-1) (0.01 = one-hundredth =
= 10-2) (0.001 = one- thousandth =
=
= 10-3)
If you view the place value chart as a number line of values, what relationship can
be used to represent the remaining powers of 10? Answers may vary. Each place value
position decreases by a power of 10 as you move from left to right on the place value chart;
Each place value position increases by a power of 10 as you move from right to left on the
place value chart; etc.
When considering powers of 10, what relationship do you see between the value of
the exponent and the place value position? Answers may vary. Powers of 10 with
positive exponent values are greater than or equal to 10; Powers of 10 with negative
exponent values are less than 1; etc.
page 5 of 42 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 01:
Suggested Duration: 4 days
Suggested Instructional Procedures
Notes for Teacher
3. Distribute a graphing calculator to each student.
4. Using a graphing calculator with a display, demonstrate how to enter exponents into the
calculator.
5. Place students in pairs and distribute handout: The Powers of Ten to each student.
6. Instruct students to complete handout: The Powers of Ten individually and then verify
solutions with a partner. Allow time for students to complete the activity. Monitor and assess
students to check for understanding. Invite students to share their conjectures to problem 24.
2
Topics:
Spiraling Review
Format for writing numbers in scientific notation
Explore/Explain 1
ATTACHMENTS
Students determine how to write numbers in scientific notation and compare calculator displays of
Teacher Resource: Investigating Scientific
numbers in scientific notation. Discuss the patterns for writing numbers in scientific notation and
Notation Notes KEY (1 per teacher)
practice converting between scientific notation, standard form, and vice versa. Use a calculator to
Handout: Investigating Scientific Notation
verify and validate solutions.
Notes (1 per student)
Teacher Resource: Investigating Scientific
Instructional Procedures:
Notation (1 per teacher)
Handout: Investigating Scientific Notation
1. Prior to instruction, set the mode for each graphing calculator to scientific notation so that
(1 per student)
when a number is entered, the number is then displayed in the calculator’s version of scientific
notation.
2. Distribute handout: Investigating Scientific Notation Notes and a graphing calculator to each
MATERIALS
page 6 of 42 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 01:
Suggested Duration: 4 days
Suggested Instructional Procedures
Notes for Teacher
student. Explain to students that the mode for the graphing calculator has been preset to
graphing calculator (1 per student)
scientific notation, therefore an adjustment will have to be made for proper scientific notation
graphing calculator with display (1 per
(e.g., The scientific mode of 735 on a typical display is 7.35 E 2, etc. ) and the “E” on the
teacher)
calculator will later need to be rewritten as x10. Remind students this “E” refers to exponents of
10. Facilitate a class discussion to complete the handout.
Ask:
TEACHER NOTE
Watch for common student errors such as counting
When is scientific notation used? (Scientific notation is used to write very large or very
small numbers.)
What is the format for writing numbers in scientific notation? (There can be only 1
non-zero digit to the left of the decimal AND the number must include multiplication by a
power of 10 held by the place value of the left-most, non-zero digit in the original number.
How do you know if the exponent should be positive or negative? (If the original
number is 10 or larger, the power of 10 is positive. If the original number is less than 1, the
place value movement from the end of the number
even when the original location of the decimal point
is elsewhere or counting zeros rather than place
value movement when determining the appropriate
power of 10. If students are struggling to complete
problems 1 – 2 on handout: Investigating Scientific
power of 10 is negative.)
Notation, work with the individual pairs and explain
How do you determine the value of the exponent when writing a number in
how to convert the data to actual scientific notation.
scientific notation? (You move the decimal to the right of the first non-zero digit.)
It is very important to emphasize the correct format
of scientific notation.
3. Place students in pairs and distribute handout: Investigating Scientific Notation to each
student.
4. Instruct student pairs to complete the Guided Practice problems 1 – 2 on handout:
Investigating Scientific Notation. Explain to students that on problem 1, they will identify
numbers in proper scientific notation while on problem 2, they will enter numbers in standard
form in their calculator; record the calculator’s display in E­form, then write the number in the
proper form of scientific notation. Allow time for students to complete the activity. Monitor and
page 7 of 42 Enhanced Instructional Transition Guide
Suggested
Day
Grade 8/Mathematics
Unit 01:
Suggested Duration: 4 days
Suggested Instructional Procedures
Notes for Teacher
assess student pairs to check for understanding.
5. Display teacher resource: Investigating Scientific Notation for the class to see. Invite
student volunteers to record their answers from handout: Investigating Scientific Notation.
Encourage students to correct their answers, as needed.
6. Instruct students to complete the remainder of handout: Investigating Scientific Notation
individually, without a calculator. Explain to students that they should use patterns for writing
numbers in scientific notation to complete the problems.
3
Topics:
Spiraling Review
Applications of scientific notation
Elaborate 1
ATTACHMENTS
Students apply numbers in scientific notation to problem situations and investigate real numbers
Teacher Resource: Applications of
on a number line.
Scientific Notation KEY (1 per teacher)
Handout: Applications of Scientific
Instructional Procedures:
1. Display the following scientific facts for the class to see:
Notation (1 per student)
Teacher Resource: Reflecting Scientifically
KEY (1 per teacher)
Atomic mass unit (amu) = 1.66 x 10−24 grams
Handout: Reflecting Scientifically (1 per
The diameter of a proton = 2 x 10−14 meters
student)
The diameter of an electron = 2.8 x 10−19 meters
The mass of a neutron = 1.6746286 x 10−27 kilograms
The mass of a proton = 1.67262158 x 10−27 kilograms
ADDITIONAL PRACTICE
page 8 of 42 Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
The mass of an electron = 9.10938188 x10−31 kilograms
Grade 8/Mathematics
Unit 01:
Suggested Duration: 4 days
Notes for Teacher
The handout: Reflecting Scientifically may be
used as additional practice if needed.
2. Instruct students to order the displayed scientific fact values written in scientific notation. Allow
time for students to complete the activity.
3. Select 1 student at random to identify the smallest value in the set of displayed scientific facts.
TEACHER NOTE
As a form of informal observation while students
Record the student’s response and encourage the class to agree or disagree with the selection
order the list of scientific facts, monitor and assess
and justify their reasoning. Select another student to identify the next smallest value. Record
student understanding of the value of the negative
the student’s response and encourage the class to agree or disagree with the selection and
exponent versus the smallness of the number.
justify their reasoning. Repeat this process until all the values are listed in order from least to
greatest. As students respond, facilitate a class discussion to lead to a greater understanding
of the emphasis on the exponent of 10 and a greater understanding of negative exponents
4. Place students in pairs and distribute handout: Applications of Scientific Notation to each
student.
5. Instruct student pairs to complete handout: Applications of Scientific Notation without
calculators.
4
Evaluate 1
Instructional Procedures:
1. Debrief handout: Applications of Scientific Notation. Instruct students to discuss their
responses to the handout with the class. Monitor and assess for student understanding and to
clarify any misconceptions.
2. Assess student understanding of related concepts and processes by using the Performance
page 9 of 42 Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Grade 8/Mathematics
Unit 01:
Suggested Duration: 4 days
Notes for Teacher
Indicator(s) aligned to this lesson.
Performance Indicator(s):
Grade8 Mathematics Unit01 PI02
Express in a written form (e.g., web page, blog, etc.) examples of appropriate problem situations
(e.g., speed of light, diameter of blood cell, etc.) with the use of both positive and negative exponents
in scientific notation and standard form
Sample Performance Indicator:
Record, in a blog, the appropriate scientific measurements that are best written in scientific notation
(both positive and negative exponents). Include the standard form and scientific notation for each
problem situation.
Standard(s):8.1D ,8.14A , 8.15A ,8.15B
ELPS ELPS.c.1E , ELPS.c.5B , ELPS.c.5G
04/04/13
page 10 of 42 Grade 8
Mathematics
Unit: 01 Lesson: 02
The Powers of Ten KEY
Use a graphing calculator to complete the following problems.
1. Multiply 523.473 by 10. What happened to the location of the decimal point when you
multiplied 523.473 by 10?
The decimal point moved one place to the right.
2. Multiply 523.473 by 100. What happened to the location of the decimal point when you
multiplied 523.473 by 100?
The decimal point moved two places to the right.
3. Multiply 523.473 by 1000. What happened to the location of the decimal point when you
multiplied 523.473 by 1000?
The decimal point moved three places to the right.
4. What direction did the decimal move when multiplying by tens?
It moved to the right.
5. How do the number of decimal places moved compare to the number of zeros? They are the
same.
6. Multiply 523.473 by 101. What happened to the location of the decimal point when you
multiplied 523.473 by 101?
The decimal point moved one place to the right.
7. Multiply 523.473 by 102. What happened to the location of the decimal point when you
multiplied 523.473 by 102?
The decimal point moved two places to the right.
8. Multiply 523.473 by 103. What happened to the location of the decimal point when you
multiplied 523.473 by 103?
The decimal point moved three places to the right.
9. What direction did the decimal move when the exponent on the ten is positive?
It moved to the right.
10. How do the number of decimal places moved compare to the exponent?
They are the same.
11. What is the numeric value of 101, 102, and 103?
10, 100, and 1000
12. Divide 523.473 by 10. What happened to the location of the decimal point when you divided
523.473 by 10?
The decimal point moved one place to the left.
©2012, TESCCC
04/26/13
page 1 of 2
Grade 8
Mathematics
Unit: 01 Lesson: 02
The Powers of Ten KEY
13. Divide 523.473 by 100. What happened to the location of the decimal point when you divided
523.473 by 100?
The decimal point moved two places to the left.
14. Divide 523.473 by 1000. What happened to the location of the decimal point when you divided
523.473 by 1000?
The decimal point moved three places to the left.
15. What direction did the decimal move when dividing by tens?
It moved to the left.
16. How do the number of decimal places moved compare to the number of zeros? They are the
same.
17. Multiply 523.473 by 10−1. What happened to the location of the decimal point when you
multiplied 523.473 by 10−1?
The decimal point moved one place to the left.
18. Multiply 523.473 by 10−2. What happened to the location of the decimal point when you
multiplied 523.473 by 10−2?
The decimal point moved two places to the left.
19. Multiply 523.473 by 10−3. What happened to the location of the decimal point when you
multiplied 523.473 by 10−3?
The decimal point moved three places to the left.
20. What direction did the decimal move when the exponent on the ten is negative?
It moved to the left.
21. How do the number of decimal places moved compare to the exponent?
They are the same.
22. What is the numeric value of 10−1, 10−2, and 10−3?
1
1
1
1
1
1
10-1 = 1 =
=0.1, 10-2 = 2 =
= 0.01, 10-3 = 3 =
= 0.001
10
10
10
100
1000
10
23. What patterns do you notice when you multiply by a ten to a negative power?
Answers may vary. Sample: Dividing by a multiple of 10 with the same number of zeros as the
exponent.
24. What conjectures can you make about multiplying by powers of ten?
Answers may vary. Sample: When the exponent is positive, move the decimal to the right the
same number of places as the exponent of 10. When the exponent is negative, move the
decimal to the left the same number of places as the exponent of 10.
©2012, TESCCC
04/26/13
page 2 of 2
Grade 8
Mathematics
Unit: 01 Lesson: 02
The Powers of Ten
Use a graphing calculator to complete the following problems.
1. Multiply 523.473 by 10. What happened to the location of the decimal point when you
multiplied 523.473 by 10?
2. Multiply 523.473 by 100. What happened to the location of the decimal point when you
multiplied 523.473 by 100?
3. Multiply 523.473 by 1000. What happened to the location of the decimal point when you
multiplied 523.473 by 1000?
4. What direction did the decimal move when multiplying by tens?
5. How do the number of decimal places moved compare to the number of zeros?
6. Multiply 523.473 by 101. What happened to the location of the decimal point when you
multiplied 523.473 by 101?
7. Multiply 523.473 by 102. What happened to the location of the decimal point when you
multiplied 523.473 by 102?
8. Multiply 523.473 by 103. What happened to the location of the decimal point when you
multiplied 523.473 by 103?
9. What direction did the decimal move when the exponent on the ten is positive?
10. How do the number of decimal places moved compare to the exponent?
11. What is the numeric value of 101, 102, and 103?
12. Divide 523.473 by 10. What happened to the location of the decimal point when you divided
523.473 by 10?
©2012, TESCCC
04/26/13
page 1 of 2
Grade 8
Mathematics
Unit: 01 Lesson: 02
The Powers of Ten
13. Divide 523.473 by 100. What happened to the location of the decimal point when you divided
523.473 by 100?
14. Divide 523.473 by 1000. What happened to the location of the decimal point when you divided
523.473 by 1000?
15. What direction did the decimal move when dividing by tens?
16. How do the number of decimal places moved compare to the number of zeros?
17. Multiply 523.473 by 10−1. What happened to the location of the decimal point when you
multiplied 523.473 by 10−1?
18. Multiply 523.473 by 10−2. What happened to the location of the decimal point when you
multiplied 523.473 by 10−2?
19. Multiply 523.473 by 10−3. What happened to the location of the decimal point when you
multiplied 523.473 by 10−3?
20. What direction did the decimal move when the exponent on the ten is negative?
21. How do the number of decimal places moved compare to the exponent?
22. What is the numeric value of 10−1, 10−2, and 10−3?
23. What patterns do you notice when you multiply by a ten to a negative power?
24. What conjectures can you make about multiplying by powers of ten?
©2012, TESCCC
04/26/13
page 2 of 2
Grade 8
Mathematics
Unit: 01 Lesson: 02
Investigating Scientific Notation Notes KEY
Powers of 10 provide a way to write very large or very small numbers. Look at the two numbers
below. They are hard to read because it’s difficult for the brain to comprehend this many digits
without actually counting them. Label the numbers as very large and very small.
0.00000000000000453
Very small
324000000000000000
Very large
When written in scientific notation, these numbers can be made more compact and easier to
comprehend. Do the numbers have a positive or negative power of ten?
3.24 x 1017
Positive
4.53 x 10−15
Negative
What can you conclude from the examples above about the power of ten, when writing numbers in
scientific notation?
Large numbers will have a positive power of ten, and small numbers will have a negative power of
ten.
When writing numbers in scientific notation they must have the following:
• only 1 non-zero digit to the left of the decimal AND
• multiplication by a power of 10 held by the place value of the left-most, non-zero digit in the
original number. (This is also the number of places the decimal must be moved. If the
original number is 10 or larger, the power of ten is positive. If the original number is less
than 1, the power of ten is negative.)
Examples
1. 4,623.8 = 4.6238 x 103
103 = 10 x 10 x 10 = 1000, which is the place value of the 4 in 4,623.8.
2. 0.0236 = 2.36 x 10−2
1
1
1
10−2 =
•
=
, which is the place value of the 2 in 0.0236.
10 10
100
3. For other numbers, it might be easier to determine the power of ten by counting decimal
places. 4500000000000 is a very large number. To write this in scientific notation, move
the decimal so that it will be behind the first non-zero digit. Multiply by a power of ten that is
equal to the number of places the decimal moved to relocate behind the leading digit. Since
it is a very large number, the exponent will be positive.
4500000000000
00
4.5 x 10
©2012, TESCCC
12
Decimal point starts here
Decimal point moves
12 spaces
Larger original
number denotes
positive exponent.
05/15/12
page 1 of 2
Grade 8
Mathematics
Unit: 01 Lesson: 02
Investigating Scientific Notation Notes KEY
4. Numbers with negative exponents can be written in scientific notation using the same
method. To find out what the exponent should be, count how far the decimal moves.
Decimal point starts here
Decimal point moves
8 spaces
Smaller original
number denotes
negative exponent.
0.00000008673
8.673 x 10−8
Summarization of changing from standard form into scientific notation:
• What did we do to write the original numbers above in scientific notation?
We moved the decimal behind the first non-zero digit; then multiplied by a power of ten held
by the place value of the left-most, non-zero digit in the original number. This exponent also
indicates how many places the decimal point moved.
How does it work in reverse?
5. When converting 4.56 x 104 into standard form:
a. What does the positive power of ten indicate in terms of the numeric value?
It is a number 10 or larger.
b. How many places does the decimal need to be moved? Explain your reasoning.
The decimal must move four places because the power of ten is 4.
c. Will the decimal be moved right or left? Explain your reasoning.
The decimal will be moved to the right to make the number larger than 10, since the
power of ten was positive.
d. What is the standard form of the number?
45,600
6. When converting 8.9 x 10−5 into standard form:
a. What does the negative power of ten indicate in terms of the numeric value?
It is a number smaller than 1.
b. How many places does the decimal need to be moved? Explain your reasoning.
The decimal must move five places because the power of ten is −5.
c. Will the decimal be moved right or left? Explain your reasoning.
The decimal will be moved to the left to make the number less than 1, since the
power of ten was negative.
d. What is the standard form of the number?
0.000089
©2012, TESCCC
05/15/12
page 2 of 2
Grade 8
Mathematics
Unit: 01 Lesson: 02
Investigating Scientific Notation Notes
Powers of 10 provide a way to write very large or very small numbers. Look at the two numbers
below. They are hard to read because it’s difficult for the brain to comprehend this many digits
without actually counting them. Label the numbers as very large and very small.
324000000000000000
0.00000000000000453
When written in scientific notation, these numbers can be made more compact and easier to
comprehend. Do the numbers have a positive or negative power of ten?
3.24 x 1017
4.53 x 10−15
What can you conclude from the examples above about the power of ten, when writing numbers in
scientific notation?
When writing numbers in scientific notation they must have the following:
• only 1 non-zero digit to the left of the decimal AND
• multiplication by a power of 10 held by the place value of the left-most, non-zero digit in the
original number. (This is also the number of places the decimal must be moved. If the
original number is 10 or larger, the power of ten is positive. If the original number is less
than 1, the power of ten is negative.)
Examples
1. 4,623.8 = 4.6238 x 103
2. 0.0236 = 2.36 x 10−2
3. For other numbers, it might be easier to determine the power of ten by counting decimal
places. 4500000000000 is a very large number. To write this in scientific notation, move
the decimal so that it will be behind the first non-zero digit. Multiply by a power of ten that is
equal to the number of places the decimal moved to relocate behind the leading digit. Since
it is a very large number, the exponent will be positive.
4500000000000
©2012, TESCCC
05/15/12
page 1 of 2
Grade 8
Mathematics
Unit: 01 Lesson: 02
Investigating Scientific Notation Notes
4. Numbers with negative exponents can be written in scientific notation using the same
method. To find out what the exponent should be, count how far the decimal moves.
0.00000008673
Summarization of changing from standard form into scientific notation:
• What did we do to write the original numbers above in scientific notation?
How does it work in reverse?
5. When converting 4.56 x 104 into standard form:
a. What does the positive power of ten indicate in terms of the numeric value?
b. How many places does the decimal need to be moved? Explain your reasoning.
c. Will the decimal be moved right or left? Explain your reasoning.
d. What is the standard form of the number?
6. When converting 8.9 x 10−5 into standard form:
a. What does the negative power of ten indicate in terms of the numeric value?
b. How many places does the decimal need to be moved? Explain your reasoning.
c. Will the decimal be moved right or left? Explain your reasoning.
d. What is the standard form of the number?
©2012, TESCCC
05/15/12
page 2 of 2
Grade 8
Mathematics
Unit: 01 Lesson: 02
Investigating Scientific Notation KEY
Guided Practice:
1. Study the following 8 numbers in the table below. Discuss the questions with your partner.
• Which numbers are in scientific notation? List the letters below.
• Which numbers are NOT in scientific notation? List the letters below.
• Justify why the numbers are or are not in scientific notation format.
A
B
345
E
C
5.62 x 10
5
F
62.109
D
28.17 x 10
G
3.901 x 108
IN scientific notation form:
B, D, F
2
8.2 x 10−3
H
0.0382
0.69 x 105
NOT in scientific notation form:
A, C, E, G, H
Justification for numbers in scientific notation:
Numbers have only 1 non-zero digit to the left of the decimal point AND the numbers
include multiplication by a power of 10 denoting the place value of the first digit in the
original number
Justification for numbers NOT in scientific notation:
Numbers have more than 1 non-zero digit to the left of the decimal point OR numbers have
no non-zero digits to the left of the decimal point OR numbers do not include multiplication
by a power of 10
©2012, TESCCC
04/04/13
page 1 of 3
Grade 8
Mathematics
Unit: 01 Lesson: 02
Investigating Scientific Notation KEY
2. Type the standard form of the number into the calculator and press enter. Record the
calculator’s display in the 2nd column. Finally, write the number in the proper form of
scientific notation. On some calculators you will need to change the mode to scientific in
order to see this version.
Standard Form of
Number
Calculator Version of
Scientific Notation
Proper Form of
Scientific Notation
1
735
7.35 E 2
7.35 x 102
2
4,293.8
4.2938 E 3
4.2938 x 103
3
0.0069201
6.9201 E −3
6.9201 x 10−3
4
0.058
5.8 E −2
5.8 x 10−2
5
473.92
4.7392 E 2
4.7392 x 102
6
54,703.92
5.470392 E 4
5.470392 x 104
7
0.000908
9.08 E −4
9.08 x 10−4
8
0.00004367
4.367 E −5
4.367 x 10−5
9
0.00901
9.01 E −3
9.01 x 10−3
10
57
5.7 E 1
5.7 x 10
What pattern do you see?
Answer: The integer that follows “E” on the graphing calculator is the exponent for 10 when the
standard form of the number is written in proper form of scientific notation!
©2012, TESCCC
04/04/13
page 2 of 3
Grade 8
Mathematics
Unit: 01 Lesson: 02
Investigating Scientific Notation KEY
Practice Problems
Write the numbers given in standard form in scientific notation, and the numbers given in
scientific notation in standard form.
©2012, TESCCC
Standard Form
Scientific Notation
763
7.63 x 102
538.09
5.3809 x 102
0.0059
5.9 x 10−3
0.0000217
2.17 x 10−5
74,956
7.4956 x 104
8300
8.3 x 103
36,100
3.61 x 104
7,891.43
7.89143 x 103
0.018
1.8 x 10−2
0.00435
4.35 x 10−3
04/04/13
page 3 of 3
Grade 8
Mathematics
Unit: 01 Lesson: 02
Investigating Scientific Notation
Guided Practice:
1. Study the following 8 numbers in the table below. Discuss the questions with your partner.
• Which numbers are in scientific notation? List the letters below.
• Which numbers are NOT in scientific notation? List the letters below.
• Justify why the numbers are or are not in scientific notation format.
A
B
345
E
C
5.62 x 10
5
F
62.109
D
28.17 x 10
G
3.901 x 108
IN scientific notation form:
2
8.2 x 10−3
H
0.0382
0.69 x 105
NOT in scientific notation form:
Justification for numbers in scientific notation:
Justification for numbers NOT in scientific notation:
©2012, TESCCC
04/04/13
page 1 of 3
Grade 8
Mathematics
Unit: 01 Lesson: 02
Investigating Scientific Notation
2. Type the standard form of the number into the calculator and press enter. Record the
calculator’s display in the 2nd column. Finally, write the number in the proper form of scientific
notation. On some calculators you will need to change the mode to scientific in order to see
this version.
Standard Form of
Number
1
735
2
4,293.8
3
0.0069201
4
0.058
5
473.92
6
54,703.92
7
0.000908
8
0.00004367
9
0.00901
10
57
Calculator Version of
Scientific Notation
Proper Form of
Scientific Notation
What patterns do you see?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
©2012, TESCCC
04/04/13
page 2 of 3
Grade 8
Mathematics
Unit: 01 Lesson: 02
Investigating Scientific Notation
Practice Problems
Write the numbers given in standard form in scientific notation, and the numbers given in
scientific notation in standard form.
Standard Form
Scientific Notation
763
538.09
0.0059
0.0000217
74,956
8.3 x 103
3.61 x 104
7.89143 x 103
1.8 x 10−2
4.35 x 10−3
©2012, TESCCC
04/04/13
page 3 of 3
Grade 8
Mathematics
Unit: 01 Lesson: 02
Applications of Scientific Notation KEY
Select the correct answer choice for each question. Show all work and justify
your solution.
1. The diameter of Saturn at its equator is 74,977 miles. Scientists usually use
scientific notation to represent very large numbers and very small numbers. How
would Saturn’s diameter be written in scientific notation?
A.
74.977 × 103
B.
7.4977 × 101
C.
7.4977 x 10−4
D.
7.4977 x 104 Answer
2. The Earth’s average distance from the Sun is 9.3 × 107 miles. How would this
distance be represented in standard form?
A.
930,000,000
B.
0.00000093
C.
93,000,000 Answer
D.
9.30000000
3. The bulk density of helium is 0.1785 g / L at 00 C. How would this be written in
scientific notation?
A.
1.785 × 103
B.
1.785 × 101
C.
1.785 x 10−1 Answer
D.
1.785 x 10−3
4. An atomic mass unit is approximately 1.6604 × 10−24 gram. How would an atomic
mass unit be written in standard form?
©2012, TESCCC
A.
1,660,400,000,000,000,000,000,000
B.
16,604,000,000,000,000,000,000,000,000
C.
0.0000000000000000000000016604 Answer
D.
0.000000000000000000016604
04/04/13
page 1 of 5
Grade 8
Mathematics
Unit: 01 Lesson: 02
Applications of Scientific Notation KEY
5. The equatorial diameter of the Earth is 7,926.6 miles. How would this distance be
represented in scientific notation?
A.
7.9 × 103
B.
7.9266 × 103 Answer
C.
7.999999266 x 10−3
D.
7.9266 x 10−3
6. The surface gravity of Pluto is estimated to be 0.08. If the Earth’s surface gravity
is considered, how would this number be written in scientific notation?
A.
8.0 × 102
B.
8.0 × 101
C.
0.8 x 10−1
D.
8.0 x 10−2 Answer
7. The diameter of Mars at its equator is 4,221 miles. Scientists usually use
scientific notation to represent very large numbers and very small numbers. How
would Mars’ diameter be written in scientific notation?
A.
4.221 × 103 Answer
B.
4.221× 101
C.
4.221 x 10−3
D.
4.221 x 104
8. Mercury’s average distance from the sun is 3.6 × 107 miles. How would this
distance be represented in standard form?
©2012, TESCCC
A.
3.6000000
B.
0.00000036
C.
360,000,000
D.
36,000,000 Answer
04/04/13
page 2 of 5
Grade 8
Mathematics
Unit: 01 Lesson: 02
Applications of Scientific Notation KEY
9. The unit price of 1 ounce of Biff creamy peanut butter is $0.094. How would this
value be written in scientific notation?
A.
0.94 × 10−1
B.
9.4 × 102
C.
9.4 x 101
D.
9.4 x 10−2 Answer
10. The diameter of a proton is 2 x 10−14 meters. How would an atomic mass unit be
written in standard form?
A.
0.000000000002
B.
0.00000000000002 Answer
C.
200,000,000,000,000
D.
2,000,000,000,000,000
11. Neptune’s length of revolution in earth-years is 164.8 years. How would this
length of time be represented in scientific notation?
A.
1.648 × 103
B.
16.48 × 101
C.
1.648 x 102 Answer
D.
1.648 x 10−2
12. The surface gravity of Venus is estimated to be 0.91. If the Earth’s surface
gravity is considered, how would this number be written in scientific notation?
©2012, TESCCC
A.
9.1 × 102
B.
9.1 × 10−1 Answer
C.
9.1 × 101
D.
9.1 × 10−2
04/04/13
page 3 of 5
Grade 8
Mathematics
Unit: 01 Lesson: 02
Applications of Scientific Notation KEY
13. Explain the process for writing a number in scientific notation.
Place the decimal behind the first non-zero digit and multiply by the power of ten
that indicates the place value of the number in standard form (also the number of
places the decimal moved).
14. When a number is in scientific notation, what does the exponent indicate?
The exponent indicates the place value of the number in standard form or power
of ten used to convert the number into standard form (also the number of places
the decimal moved).
15. Why are some numbers written in scientific notation?
They are too large or too small to truly comprehend in standard form. They are
also too difficult to perform operations with (add, subtract, multiply, or divide).
16. Use a calculator and the given rational numbers:
9
1
5.0 x 10−1, −1.5, 2 , − , 125%, 100%, 2.5 x 100, 1.58
4
5
a. Graph each rational number on the number line.
−
9
4
−1.5
−2
5.0 x 10−1 100%
−1
0
1
1.58
125%
2.5 x 100
1
2 25
b. List the numbers in order from least to greatest.
1
9
−1
0
− , −1.5, 5.0 x 10 , 100%, 125%, 1.58, 2 , 2.5 x 10
5
4
©2012, TESCCC
04/04/13
page 4 of 5
8th Grade
Mathematics
Unit: 01 Lesson: 02
Applications of Scientific Notation KEY
17. Write the numbers given in standard form in scientific notation, and the numbers given in
scientific notation in standard form.
©2012, TESCCC
Standard Form
Scientific Notation
634
6.34 x 102
456.2
4.562 x 102
0.093
9.3 x 10−2
0.000714
7.14 x 10−4
34,923
3.4923 x 104
15
1.5 x 101
63,000
6.3 x 104
5,820
5.82 x 103
284.63
2.8463 x 102
0.00047
4.7 x 10−4
0.0192
1.92 x 10−2
3
3 x 100
04/04/13
page 5 of 5
Grade 8
Mathematics
Unit: 01 Lesson: 02
Applications of Scientific Notation
Select the correct answer choice for each question. Show all work and justify
your solution.
1. The diameter of Saturn at its equator is 74,977 miles. Scientists usually use
scientific notation to represent very large numbers and very small numbers. How
would Saturn’s diameter be written in scientific notation?
A. 74.977 × 103
B. 7.4977 × 101
C. 7.4977 x 10−4
D. 7.4977 x 104
2. The Earth’s average distance from the Sun is 9.3 × 107 miles. How would this
distance be represented in standard form?
A. 930,000,000
B. 0.00000093
C. 93,000,000
D. 9.30000000
3. The bulk density of helium is 0.1785 g / L at 00 C. How would this be written in
scientific notation?
A. 1.785 × 103
B. 1.785 × 101
C. 1.785 x 10−1
D. 1.785 x 10−3
4. An atomic mass unit is approximately 1.6604 × 10−24 gram. How would an atomic
mass unit be written in standard form?
A. 1,660,400,000,000,000,000,000,000
B. 16,604,000,000,000,000,000,000,000,000
C. 0.0000000000000000000000016604
D. 0.000000000000000000016604
©2012, TESCCC
04/04/13
page 1 of 5
Grade 8
Mathematics
Unit: 01 Lesson: 02
Applications of Scientific Notation
5. The equatorial diameter of the Earth is 7,926.6 miles. How would this distance be
represented in scientific notation?
A. 7.9 × 103
B. 7.9266 × 103
C. 7.999999266 x 10−3
D. 7.9266 x 10−3
6. The surface gravity of Pluto is estimated to be 0.08. If the Earth’s surface gravity
is considered, how would this number be written in scientific notation?
A. 8.0 × 102
B. 8.0 × 101
C. 0.8 x 10−1
D. 8.0 x 10−2
7. The diameter of Mars at its equator is 4,221 miles. Scientists usually use
scientific notation to represent very large numbers and very small numbers. How
would Mars’ diameter be written in scientific notation?
A. 4.221 × 103
B. 4.221× 101
C. 4.221 x 10−3
D. 4.221 x 104
8. Mercury’s average distance from the sun is 3.6 × 107 miles. How would this
distance be represented in standard form?
A. 3.6000000
B. 0.00000036
C. 360,000,000
D. 36,000,000
©2012, TESCCC
04/04/13
page 2 of 5
Grade 8
Mathematics
Unit: 01 Lesson: 02
Applications of Scientific Notation
9. The unit price of 1 ounce of Biff creamy peanut butter is $0.094. How would this
value be written in scientific notation?
A. 0.94 × 10−1
B. 9.4 × 102
C. 9.4 x 101
D. 9.4 x 10−2
10. The diameter of a proton is 2 x 10−14 meters. How would an atomic mass unit be
written in standard form?
A. 0.000000000002
B. 0.00000000000002
C. 200,000,000,000,000
D. 2,000,000,000,000,000
11. Neptune’s length of revolution in earth-years is 164.8 years. How would this
length of time be represented in scientific notation?
A. 1.648 × 103
B. 16.48 × 101
C. 1.648 x 102
D. 1.648 x 10−2
12. The surface gravity of Venus is estimated to be 0.91. If the Earth’s surface gravity
is considered, how would this number be written in scientific notation?
A. 9.1 × 102
B. 9.1 × 10−1
C. 9.1 × 101
D. 9.1 × 10−2
©2012, TESCCC
04/04/13
page 3 of 5
Grade 8
Mathematics
Unit: 01 Lesson: 02
Applications of Scientific Notation
13. Explain the process for writing a number in scientific notation.
14. When a number is in scientific notation, what does the exponent indicate?
15. Why are some numbers written in scientific notation?
16. Use a calculator and the given rational numbers:
9
1
5.0 x 10−1, −1.5, 2 , − , 125%, 100%, 2.5 x 100, 1.58
4
5
a. Graph each real rational on the number line.
−2
−1
0
1
2
b. List the numbers in order from least to greatest.
©2012, TESCCC
04/04/13
page 4 of 5
8th Grade
Mathematics
Unit: 01 Lesson: 02
Applications of Scientific Notation
17. Write the numbers given in standard form in scientific notation, and the numbers given in
scientific notation in standard form.
Standard Form
Scientific Notation
634
456.2
0.093
0.000714
34,923
15
6.3 x 104
5.82 x 103
2.8463 x 102
4.7 x 10−4
1.92 x 10−2
3 x 100
©2012, TESCCC
04/04/13
page 5 of 5
Grade 8
Mathematics
Unit: 01 Lesson: 02
Reflecting Scientifically KEY
1. Look at the following numbers. Write the letters in the correct box.
Numbers IN scientific notation form:
Numbers NOT in scientific notation form:
A, C, H
B, D, E, F, G
A
4.1 x 10
B
−5
C
72.69 x 10
E
3
F
38.572
D
8.081 x 10
6
G
98
0.32 x 102
H
0.6204
5.143 x 101
2. Write the numbers given in standard form in scientific notation, and the numbers given
in scientific notation in standard form.
©2012, TESCCC
Standard Form
Scientific Notation
715
7.15 x 102
4,205.79
4.20579 x 103
0.00083
8.3 x 10−4
8
8 x 100
5,200
5.2 x 103
0.0185
1.85 x 10−2
76,800
7.68 x 104
12.34
1.234 x 101
04/04/13
page 1 of 4
Grade 8
Mathematics
Unit: 01 Lesson: 02
Reflecting Scientifically KEY
Circle the correct answer choice for each question. Show all work.
3. The Nile River is the longest river in the world, although its end point remains
uncertain. Including its farthest mouth, the river is 4,195 miles in length. How
would the length of the Nile River be written in scientific notation?
A.
419.5 × 10−2 mi
B.
4.195 × 103 mi Answer
C.
41.95 × 102 mi
D.
4.195 × 10−3 mi
4. The most venomous land snake is the Taipan, a snake found in Australia, which
1
grows to approximately 5 feet. The average venom yield after milking is 1.55 x
2
10−3 oz.; however, 1 male snake produced 3.85 x 10 −3 oz. of venom during a
single milking. This is enough venom to kill 15 humans or 250,000 mice. How
would this particular male snake’s quantity of venom in a single milking be
represented in standard form?
A.
1,550 oz.
B.
3,850 oz.
C.
0.00155 oz.
D.
0.00385 oz. Answer
©2012, TESCCC
04/04/13
page 2 of 4
Grade 8
Mathematics
Unit: 01 Lesson: 02
Reflecting Scientifically KEY
5. The highest recorded jump by a dog is 146.5 inches. This record was set in 1993
by an 18-month-old lurcher dog named Stag who won the canine high-jump for a
leap-and-scramble over a smooth wooden wall without any climbing aids at the
annual Cotswold Country Fair in Cirencester, Glos, UK. How would this height be
represented in scientific notation?
A.
1.465 × 103 in.
B.
1.465 × 10−2 in.
C.
1.465 × 102 in. Answer
D.
14.65 × 101 in.
6. The world’s most venomous spiders are the Brazilian wandering spiders. The
Brazilian wandering spider has the most active neurotoxin venom of any living
spider. This spider’s venom is so strong that only 0.00000021 oz is needed to kill
a mouse. How would this quantity be written in scientific notation?
A.
2.1 × 106 oz.
B.
2.1 × 107 oz.
C.
2.1 × 10−7oz. Answer
D.
0.21 ×10−6 oz.
7. The record for the greatest distance walked in 24 hours is held by Jesse
Castaneda of the USA. In September of 1976 he walked 1.4225 × 102 miles in 24
hours. How would this distance be represented in standard form?
A.
14,225 mi
B.
1.4225 mi
C.
142.25 mi Answer
D.
1,422.5 mi
©2012, TESCCC
04/04/13
page 3 of 4
8th Grade
Mathematics
Unit: 01 Lesson: 02
Reflecting Scientifically KEY
Show all work.
8. Graph each real number on the number line.
3
9 3
{100%, −1.75, − , 2.5 x 10-1, −2, 2, , 1 }
4
4 4
−
−2 −1.75
−1
9.
3
4
2.5 x 10
0
-1
100%
1
3
4
9
2 4
1
Answer the following questions using examples when appropriate.
a. How do you write a number in scientific notation?
Answers may vary. Sample: Place the decimal so that only 1 non-zero digit is
to the left of the decimal point and multiply by the power of ten that states the
place value of the original leading digit.
b. When a number is in scientific notation, what does the exponent indicate?
Answers may vary. Sample: The exponent indicates the place value of the
original leading digit and the number of places that the decimal was moved to
write the number in scientific notation.
c. Why are some numbers written in scientific notation?
Very large and very small numbers are easier to interpret in scientific notation.
Calculations are also easier.
©2012, TESCCC
04/04/13
page 4 of 4
Grade 8
Mathematics
Unit: 01 Lesson: 02
Reflecting Scientifically
1. Look at the following numbers. Write the letters in the correct box.
Numbers NOT in scientific notation form:
Numbers IN scientific notation form:
A
4.1 x 10
B
−5
C
72.69 x 10
E
F
38.572
3
D
8.081 x 10
G
98
6
0.32 x 102
H
0.6204
5.143 x 101
2. Write the numbers given in standard form in scientific notation, and the numbers given in
scientific notation in standard form.
Standard Form
Scientific Notation
715
4,205.79
0.00083
8
5.2 x 103
1.85 x 10−2
7.68 x 104
1.234 x 101
©2012, TESCCC
04/04/13
page 1 of 4
Grade 8
Mathematics
Unit: 01 Lesson: 02
Reflecting Scientifically
Circle the correct answer choice for each question. Show all work.
3. The Nile River is the longest river in the world, although its end point remains uncertain.
Including its farthest mouth, the river is 4,195 miles in length. How would the length of the
Nile River be written in scientific notation?
A. 419.5 × 10−2 mi
B. 4.195 × 103 mi
C. 41.95 × 102 mi
D. 4.195 × 10−3 mi
4. The most venomous land snake is the Taipan, a snake found in Australia, which grows to
1
approximately 5 feet. The average venom yield after milking is 1.55 x 10−3 oz.; however, 1
2
male snake produced 3.85 x 10 −3 oz. of venom during a single milking. This is enough
venom to kill 15 humans or 250,000 mice. How would this particular male snake’s quantity of
venom in a single milking be represented in standard form?
A. 1,550 oz.
B. 3,850 oz.
C. 0.00155 oz.
D. 0.00385 oz.
©2012, TESCCC
04/04/13
page 2 of 4
Grade 8
Mathematics
Unit: 01 Lesson: 02
Reflecting Scientifically
5. The highest recorded jump by a dog is 146.5 inches. This record was set in 1993 by an 18month-old lurcher dog named Stag who won the canine high-jump for a leap-and-scramble
over a smooth wooden wall without any climbing aids at the annual Cotswold Country Fair in
Cirencester, Glos, UK. How would this height be represented in scientific notation?
A. 1.465 × 103 in.
B. 1.465 × 10−2 in.
C. 1.465 × 102 in.
D. 14.65 × 101 in.
6. The world’s most venomous spiders are the Brazilian wandering spiders. The Brazilian
wandering spider has the most active neurotoxin venom of any living spider. This spider’s
venom is so strong that only 0.00000021 oz. is needed to kill a mouse. How would this
quantity be written in scientific notation?
A. 2.1 × 106 oz.
B. 2.1 × 107 oz.
C. 2.1 × 10−7oz.
D. 0.21 ×10−6 oz.
7. The record for the greatest distance walked in 24 hours is held by Jesse Castaneda of the
USA. In September of 1976 he walked 1.4225 × 102 miles in 24 hours. How would this
distance be represented in standard form?
A. 14,225 mi
B. 1.4225 mi
C. 142.25 mi
D. 1,422.5 mi
©2012, TESCCC
04/04/13
page 3 of 4
8th Grade
Mathematics
Unit: 01 Lesson: 02
Reflecting Scientifically
Show all work.
8. Graph each real number on the number line.
3
9 3
{100%, −1.75, − , 2.5 x 10-1, −2, 2, , 1 }
4
4 4
−1
9.
0
1
Answer the following questions using examples when appropriate.
a. How do you write a number in scientific notation?
b. When a number is in scientific notation, what does the exponent indicate?
c. Why are some numbers written in scientific notation?
©2012, TESCCC
04/04/13
page 4 of 4