Fifth Grade
Curriculum Unit Plan
Fifth Grade: Mathematics
Unit 2: Estimation and Computation
Lesson Plan
Overarching Question:
How does place value relate to multiplication of whole numbers and fractions in decimal form?
This Unit:
Previous Unit:
Next Unit:
Geometry
Estimation and Computation
Learning About Numbers
Estimation and
Computation
is about
by
understanding place
value & operations
of numbers
explaining place value
relationships from
billions to hundredths
to solve
problems: Use
contexts to apply
computations
to solve
including
including
computation using estimation:
Multiply a two-digit number,
reasonableness of computations
fluently multiplying
multi-digit numbers
using standard
algorithm
powers of ten (from
.01 to 1000)
Questions to Focus Assessment and Instruction:
1. How does working with the powers of ten help one when exploring the
concept of one million?
2. How does the estimate of a computation help one in solving word
problems?
3. How does one learn how to check for reasonableness in an answer?
4. How does understanding of the place-value structure of the base-ten
number system help one to represent and compare whole numbers
and decimals?
5. How do using problem solving strategies help one to estimate the
results of rational-number computations and judge the reasonableness
of the results?
Key Concepts
place value
powers of ten
digit
estimation
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Intellectual Processes (Standards for
Mathematical Practice):
• Make sense of multi-digit
problems and persevere in
solving them.
• Reason abstractly and
quantitatively when estimating
when multiplying two-digit
numbers.
• Look for and make use of
structure in arriving at solutions
involving large numbers.
• Model a larger number using a
million square millimeter grid.
millimeter
mathematical pattern
June 16, 2011
Fifth Grade
Lesson Abstract:
In this lesson, students will make predictions using estimation skills and problem solving strategies to decide how to best
find the solution. Additionally, they will use critical thinking to analyze situations and to identify mathematical patterns that
will enable them to develop the concept of very large numbers. Students will work with multiples of 10 to explore the
magnitude of 1,000,000.
Common Core Standards
Number and Operation in Base Ten (5.NBT)
Understand the place value system.
5. NBT. 1. Recognize that in a multi-digit number, a digit in the ones place represents 10 times as much as it
represents in the place to its right and 1/10 of what it represents in the place to its left.
5. NBT. 2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and
explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a
power of 10. Use whole-number exponents to denote powers of 10.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.5. Fluently multiply multi-digit whole numbers using the standard algorithm.
Instructional Resources
Sequence of Lesson Activities
Lesson Title: “Making Your First Million” http://illuminations.nctm.org/LessonDetail.aspx?ID=L367
(Control+Click to access the lesson.)
Two Class Periods
Materials Needed
Calculators
One roll of transparent tape
Scissors
One copy of “Making Your First Million” Activity Sheet for each student
http://illuminations.nctm.org/lessons/3-5/countonmath/CountOnMath-AS-Million.pdf (Control+Click to access the activity
sheet.)
A stopwatch, digital watch, or clock with a second hand
Selecting and Setting Up a Mathematical Task:
Advanced Preparation:
What are the mathematical
• The goal of this lesson is to engage students in using proportional reasoning in
objectives for the lesson?
problem solving when working with powers of ten to explore the concept of one
million.
• Fifth grade students have been working with place value since kindergarten. In
In what ways does the task
fourth grade they explored reading, writing, and modeling numbers to a million.
build on student’s previous
This lesson builds on that prior knowledge by deepening their understanding of
knowledge?
very large numbers.
•
The teacher should determine student understanding of place value by asking for
What questions will you ask
individual students to explain what they think place value is.
to access prior knowledge?
• Ask the students, "Have you been alive for 1 million days? Hours? Minutes?
Seconds?" Give the students an opportunity to explore these questions with their
calculators. Discuss their responses.
Launch
How would you introduce the
activity to the students?
•
•
•
•
Engage the students in discussing large numbers by recounting that some
scientists believe dinosaurs became extinct approximately 65 million years ago.
Consider a report of a certain athlete's salary reported as $3 million per year, or
that the sun is approximately 93 million miles from earth.
Ask the students, "How can we relate to such large numbers?" To help them
answer this question, focus the discourse on the magnitude of 1 million.
Ask the students to try to imagine the size of a tank that can hold 1 million
gallons of water, or a pile of garbage that weighs 1 million pounds, and discuss
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June 16, 2011
Fifth Grade
the notion that these images are difficult to visualize.
•
Explain that the focus of this investigation is to assist them in understanding and
appreciating the magnitude of large numbers.
•
Distribute a copy of Making Your First Million activity sheet to each student.
Explore
What questions will be asked
to focus students’ thinking on
the key ideas?
Students should record their answers to question 1 and write an explanation on the
Making Your First Million activity sheet before teacher asks them to share their responses
about how they arrived at their answers. A discussion about their thinking will help to
focus the students’ understanding. (1 million days is about 2700 years, 1 million hours is
just over 114 years, 1 million minutes is nearly 2 years, 1 million seconds is
approximately 11.5 days).
•
What will be asked to assess
understanding of key
mathematical ideas?
•
•
•
•
•
•
Discuss the conceptual difference and ability to visualize "1 million seconds"
versus the rough equivalent of "11.5 days". Often changing the way we look at a
subject helps us to see more detail.
Students should record their predictions to question number 2 on “Making Your
First Million” activity sheet.
Call the students' attention to the 100-mm by 100-mm grid on the activity sheet.
Ask the students to determine how many square mm are on each person's page.
Have them record their answers to question number 3 on the “Making Your First
Million” activity sheet.
In groups of ten, ask students to cut out their grids and tape them together to
form 100-mm by 100-mm rectangles.
Then the class should determine how many of the square mm makes up their
group's rectangles. (Each group should have determined that the total is
100,000 square mm, and the students could conclude that it would require ten
group rectangles to piece together a square containing 1 million square mm.).
Optional: Consider cutting and pasting four copies of the grid found on the activity
sheet onto a blank sheet of paper so that twenty-five copies of this sheet would
allow students actually to see 1 million square mm.
Using counting and a calculator as another way to think about the magnitude of a
number, students should explore ways the calculator might be used to display a
count from 1 to 5. For example, students might press these keys:
1
2
3
4
5
or
1
•
+
1
•
+
1
=
+
1
=
+
1
=
If not already available, obtain a calculator that incorporates a repeat function so
that students will recognize that keying will allow the calculator to continue to add
"1" each time "=" is pressed, without having to repeatedly press "+1."
1
•
=
+
1
=
=
=
=
Ask the students, "How long will it take you to count from 1 to 1,000,000 using the
repeat function on your calculator?" (Field tests have shown that some studies
suggest using the calculator to count to 100 and then multiplying that amount of
time by 10,000, whereas others prefer to run calculator-counting trials for one or
more minutes and use a proportion to estimate the amount of time required to
count to 1,000,000.)
Ask students to implement their strategies, compare the times obtained through
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June 16, 2011
Fifth Grade
•
•
•
•
•
•
•
•
•
various methods, explore the range of estimates and the average time, and
determine which methods seemed to be the most efficient and reasonable.
Ask the students, "How long do you believe it would take to count out loud from 1
to 1 million?" They should record their answers for question 4 and explain the
strategy they used to arrive at their estimate.
Lead a discussion to allow students to suggest strategies to determine an
answer, such as timing a counting segment and extrapolating that result.
Have four or five students demonstrate the count somewhere in the hundred
thousands. Record the time required to count from 1 to 100, from 100 to 200,
from 1,000 to 1,100, from 10,000 to 10,100 from 100,000 to 100,100, and several
100 counts beginning with such large random numbers as 345,684.
Organize and use the collected data to form a hypothesis about how long it would
actually take to count out loud to 1 million.
Discuss factors that may not have been considered, such as the need for sleep
and food, and recognizing that most of the numbers counted will be timeconsuming six-place numbers. A "reasonable" work schedule could be devised to
complete the counting. Counting for only four hours a day, for example, multiplies
the number of days required to complete the task by six.
Discuss with students how long they believe it might take to write all the numerals
from 1 to 1 million. They should record their predictions to question 5 and explain
their strategy. Strategies similar to those used previously can be used here.
Have groups of students write out the numerals in order beginning at various
places in the count. Time trials will again yield data that can be analyzed to obtain
an estimate.
Ask the students how many notebook pages are needed to write all the numerals
from 1 to 1 million.
Students should then complete question 6 on the activity sheet to determine the
number of digits necessary to write these numerals.
To extend the lesson:
•
Encourage students to look for large numbers in the newspaper. Numbers in the
millions are frequently seen in the context of federal and state budgets, lottery
winnings, athletic salaries, corporate finances, astronomical measurements, etc.
This investigation can lead to discussions about the significance and appreciation
of large numbers in business, government, and science. Remind them that
estimating and understanding large numbers are useful mathematical skills.
•
The number pi, the constant ratio of the circumference to the diameter of a circle,
has been calculated by super computers to many millions of decimal places. How
many pages of printer paper would be needed to print out the first 1 million digits
of pi? In 1874, William Shanks calculated pi to 707 decimal places by hand. How
long might this task have taken? Discuss ways to find out. How long would it
take to write the 707 places? Have students discuss strategies that they would
use to find the solution. (Remember, Shanks was not only writing the digits but
calculating them as well. Estimates by historians suggest that he spent years
doing these calculations. Sadly, he made a mistake in the 527th place. A
contemporary computer takes only a few seconds to match Shanks' output,
without the mistake.
•
Have students discuss the difficulties in working with numbers into the millions
and billions.
Ask how understanding the powers of ten helped in devising a strategy for
counting to a million.
How would this knowledge help them to extrapolate to even larger numbers?
How will you extend the task
to provide additional
challenge?
Summary
What questions will be asked
so that students make
connections between the
different strategies?
•
•
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June 16, 2011
Fifth Grade
What will be seen or heard
that indicates all students
understand the mathematical
ideas you intended them to
learn?
Some questions to consider ask when wrapping up are:
•
•
•
What did you discover when you attempted to count to a million?
Were your predictions in question #2 reasonable? If not, what did you learn in this
activity that would help you make a better estimate in the future?
How did your knowledge of place value help in completing the chart in problem
number 6?
Teacher Reflections
The teacher will know if students understand the mathematical concepts being taught if
they can:
•
•
•
develop and use strategies to estimate the results of rational-number
computations and judge the reasonableness of the results.
understand the place-value structure of the base-ten number system and be able
to represent and compare whole numbers and decimals.
practice place value- skills and to read and write large numbers.
Extensions:
http://www.figurethis.org/challenges/c01/challenge.htm (Control+Click to access)
”Figure This! How long do you have to stand in Line?”
Students conduct an experiment to see how long they would have to stand in line if they
were number 300? They would need to begin by coming up with a reasonable estimate of
how long one person would take to purchase a ticket (e.g. a movie ticket) and then use
that calculation to find a reasonable answer.
Everyday Math’s “How Would You Spend $1,000,000? project would be a good
culminating activity.
This document is the property of MAISA.
June 16, 2011