&&5BTLtExpressions and Equations
How Many Times in a Millennium?
Instruction
Common Core State Standard
Work with radicals and integer exponents.
8.EE.3.
Use numbers expressed in the form of a single digit times an integer power of 10 to
estimate very large or very small quantities, and to express how many times as much
one is than the other. For example, estimate the population of the United States as 3
times 108 and the population of the world as 7 times 109, and determine that the world
population is more than 20 times larger.
Task Overview
Background
This task is designed to illustrate the multiple ways a number can be expressed in written form.
Students will collect data based on repetition of a kinesthetic act. Students then use that value to
calculate, progressively, how many actions they could repeat in a millennium. Students will produce
a poster that shows their action, their count for 10 seconds and their count for 1 millennium, and will
express, mathematically, how many more times they can perform their action in a millennium than in
10 seconds.
The task also provides practice with:
•
unit conversion
•
multiplication of large numbers
•
terms used for the expression of time periods
•
powers of ten
•
noticing patterns
•
making predictions
Implementation Suggestions
•
Students may work individually, in pairs, or in small groups to complete one or both parts of
the task.
•
Students will need materials for poster production. Posters may be produced physically or digitally.
•
Allow 5–8 minutes for the introduction. Allow 10 minutes for data collection and number
calculation and 10 minutes for poster production. The poster walk and final debriefing can be
accomplished in 15 minutes.
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© Walch Education
Common Core State Standards Transition Tasks for Mathematics, Grade 8
&&5BTLtExpressions and Equations
How Many Times in a Millennium?
Instruction
Introduction
Introduce the task. Ask the students to suggest words used to express periods of time. Students
should respond with common time units (second, minute, hour, day, week, month, year, decade,
century). Remind students that these words indicate units of time and that time periods can be
converted between units. Students should know that multiplication is used to convert from a second
to a minute, a minute to a year, etc. List the following words and conversion factors on the board.
second
minute = 60 seconds
hour = 60 minutes
day = 24 hours
week = 7 days
year = 52 weeks
decade = 10 years
century = 100 years
millennium = 1,000 years
Have students observe that between decade and era the years expressed by the units given are
increasing by powers of ten. Students should be familiar with the use of exponents to express powers
of ten. You may want to compile a list on the board, making connections between the language
common to both (i.e., 100 years is a century, 1,000 years is a millennium).
100 = 1 (year)
101 = 10 (decade)
102 = 100 (century)
103 = 1,000 (millennium)
Each student will have to select a kinesthetic activity (blinking, snapping, clapping, jumping,
stamping their foot, etc.) that they can do multiple times in a 10-second period. Alternatively, you
may suggest a specific activity or present a list for students to select from.
Allow students a trial run to practice data collection. Keep time for the class, giving a verbal signal
to begin data collection at 0, then giving another verbal cue for them to stop after 10 seconds. Ask
for one or two students to volunteer to demonstrate their activity and talk through the process of
completing the statement: “I measured that I can (insert activity) (insert count) times in 10 seconds.”
(Example: I measured that I can blink 20 times in 10 seconds.)
Have students estimate how many times they think they could repeat the action in a millennium
(1,000 years) if it were somehow possible to keep going that long. For a more realistic scenario,
describe how pulsar stars emit regular radiation bursts. If a particular pulsar emits 3 bursts of
radiation every 10 seconds, how many bursts would it emit in a millennium?
You may wish to model the data collection and provide an example of value conversions on the
board before students begin.
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Common Core State Standards Transition Tasks for Mathematics, Grade 8
© Walch Education
&&5BTLtExpressions and Equations
How Many Times in a Millennium?
Instruction
Monitoring/Facilitating the Task
Ask questions and prompt student thinking so that they:
•
Focus on the multiple ways of writing each value. A sample response table is included to
illustrate the way students should be completing their work. Ask what has changed the
number from row to row in the table. Students should respond that the multiplier used for
unit conversion has made the number that many times larger. For example, the number of
repetitions per hour is 60 times larger than the number of repetitions per minute.
•
Realize the connection between the powers of ten and the exponents used. Ask what is
happening to their value as the exponent increases. Ask them to justify their response.
Students should observe that as their number grows larger, the exponents increase.
•
Observe that only the exponent is changing in the power of ten representations as they
calculate their repetitions for decade through millennium. Ask them to investigate the pattern
they are seeing. Students should observe that each increase in the exponent is a 10-time
increase in repetitions.
•
Compare decimal values and powers of ten.
As students fill in responses to Part 2, ensure that they properly use the terms measure, estimate, and
calculate as they discuss their work.
Allow students to complete their posters. Check their progress and remind them to include all the
required aspects on their poster. Display posters in the room for a poster walk during debriefing.
Debriefing the Task
•
Allow students time for a poster walk to observe the work done by their classmates. As they
review the posters, ask students to think about putting them in order. Ask them to decide
which activity could be done the most and the fewest times in a millennium (1,000 years).
•
During the poster walk, have students write their number per millennium on the board in
whole-number form.
•
Explore the ordering of larger numbers. Ask students to use the board to indicate the largest
and smallest repetition values expressed. Ask the students to find the largest and smallest
values from the board. Students should express difficulty in making the comparison because
of the number of integers written on the board.
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© Walch Education
Common Core State Standards Transition Tasks for Mathematics, Grade 8
&&5BTLtExpressions and Equations
How Many Times in a Millennium?
Instruction
•
Have the students move their focus to the posters. Ask them to order the posters by reading
the exponential notation representation. Students should observe their increased ability to
order values when powers of ten are used. Have them justify their responses using the powers
of ten and exponent values. Students should connect that the number with the largest power
of ten is the largest number. Ask for student input for ordering the posters in the classroom,
from smallest power of ten to largest.
•
In summary, make sure that students understand the purpose and value of scientific
notation/exponents in recording and working with very large numbers.
Answer Key
Answers will vary; all answers are sample answers.
1. I measured that I can snap my fingers three times in 10 seconds.
2. I estimate that I can snap my fingers three billion times in one millennium.
3. Sample table:
Multiplication
Integer value
Time span for
repetitions
Decimal
Power of ten
3•6
18
1 minute
1.8 • 10
1.8 • 101
18 • 60
1,080
1 hour
1.08 • 1,000
1.080 • 103
1,080 • 24
25,920
1 day
2.592 • 10,000
2.5920 • 104
25,920 • 7
181,440
1 week
1.8144 • 100,000
1.81440 • 105
181,440 • 52
9,434,880
1 year
9.43488 • 1,000,000
9.434880 • 106
9,434,880 • 10
94,348,800
1 decade
9.43488 • 10,000,000
9.434880 • 107
94,348,800 • 10
943,488,000
1 century
9.43488 • 100,000,000
9.434880 • 108
943,488,000 • 10
9,434,880,000
1 millennium
(1000 years)
9.43488 • 1,000,000,000
9.434880 • 109
4. I measured that I could snap my fingers three times in 10 seconds.
5. I estimated that I could snap my fingers three billion times in one millennium.
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6. I calculated that I could snap my fingers 9.43488 • 10 times in one millennium.
7. I can snap my fingers 9.4 billion times more in a millennium than I can in 10 seconds.
8. Check student posters for completion and accuracy. Make sure student posters include the
number of times they could complete their selected action in a millennium using the tens place
representation of the value.
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Common Core State Standards Transition Tasks for Mathematics, Grade 8
© Walch Education
&&5BTLtExpressions and Equations
How Many Times in a Millennium?
Instruction
Differentiation
•
Some students may benefit through the use of calculators during this task.
•
Interested students could collect data from multiple students prior to the activity.
•
Some students may wish to continue calculations beyond a millennium using higher powers
of ten.
•
Some students may benefit by further exploring the concept of ordering. Estimation values
could be used in an expansion of this task.
Technology Connection
•
Spreadsheet software could be used for generating multiples and recording data.
•
Graphic design software could be used for poster production.
Choices for Students
•
Interested students may choose to express large values (or smaller values) from a different
source (computing terms, scientific values), or order the estimations students give.
•
In place of the poster, students could produce a visual representation of the large value. For
examples of this, see the MegaPenny Web site (URL listed in Recommended Resources).
Meaningful Context
Students are routinely exposed to numbers in the millions, billions, and trillions in their world. It can
be difficult to visualize these numbers and put them in context with smaller values. This task offers
multiple representations for large numbers.
Recommended Resources
•
The MegaPenny Project
www.walch.com/rr/CCTTG8MegaPenny
The MegaPenny Project represents large numbers using pictures of pennies, with
excellent visual representations.
•
Wolfram MathWorld—Large Number
www.walch.com/rr/CCTTG8LargeNumbers
This site provides information on large numbers and words for various powers of ten.
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© Walch Education
Common Core State Standards Transition Tasks for Mathematics, Grade 8
NAME:
&&5BTLtExpressions and Equations
How Many Times in a Millennium?
Part 1
Use the following information to complete the task.
second
minute = 60 seconds
hour = 60 minutes
day = 24 hours
week = 7 days
year = 52 weeks
decade = 10 years
century = 100 years
millennium = 1,000 years
100 = 1 (year)
101 = 10 (decade)
102 = 100 (century)
103 = 1,000 (millennium)
Choose an action to perform, such as snapping your fingers. Count the number of times that you can
perform your action in 10 seconds. Use that piece of data to complete the records below, using the
same action for each.
1. I measured that I can ____________________________________________________
__________________ times in 10 seconds.
2. I estimate that I can _____________________________________________________
__________________ times in one millennium.
continued
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Common Core State Standards Transition Tasks for Mathematics, Grade 8
© Walch Education
NAME:
&&5BTLtExpressions and Equations
How Many Times in a Millennium?
3. Convert between different forms to fill out the information in the table. The first row is an
example, using 3 times per second as a sample answer. Look at it to see how each column is to
be filled in, using your own data.
Multiplication
Integer value
Time span for
repetitions
Decimal
Power of ten
3 • 6
18
1 minute
1.8 • 10
1.8 • 101
1 minute
1 hour
1 day
1 week
1 year
1 decade
1 century
1 millennium
(1,000 years)
continued
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© Walch Education
Common Core State Standards Transition Tasks for Mathematics, Grade 8
NAME:
&&5BTLtExpressions and Equations
How Many Times in a Millennium?
Part 2
Complete the following sentences using the power of ten representations from the table.
4. I measured that I could __________________________________________________
__________________ times in 10 seconds.
5. I estimated that I could __________________________________________________
__________________ times in one millennium.
6. I calculated that I could __________________________________________________
__________________ times in one millennium.
7. I can _______________________________________________________________
__________________ times more in a millennium than I can in 10 seconds.
8. Design and create a poster for display in your classroom. Make sure your poster includes the
number of times you can complete your action in a millennium using scientific notation.
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Common Core State Standards Transition Tasks for Mathematics, Grade 8
© Walch Education