Foundations of Mathematical Reasoning
Student Pages 1.A, How big is a billion?
Lesson 1, Part A, How big is a billion?
Theme: Civic Life
A million, a billion, a trillion, a quadrillion—it is easy to lose a sense of the size of large
numbers. So just how big is a billion?
1)
Jot down anything you know about the
number one billion. Then share with at
least two neighbors.
2)
How many YouTube videos do you
think have over one billion views?
3)
If a billion people stood shoulder-toshoulder, how long would the line be?
Make a prediction.
Credit: iStockphoto
Objectives for the lesson
You will understand that:
o Large numbers can be represented in various ways.
o Collaborating with others can enhance learning.
You will be able to:
o Scale measurements of groups to represent individual elements
o Scale measurements to represent larger quantities of individual elements.
Stand in line shoulder-to-shoulder with your classmates.
4)
How many people are in the line?
5)
To the nearest inch, how long is the line?
6)
What is the average shoulder width of the people in your line? How do you know?
7)
How long would the line be if there were 1 thousand (1,000) people?
8)
How long would the line be if there were 1 million (1,000,000) people?
9)
How long would the line be if there were 1 billion (1,000,000,000) people?
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The University of Texas at Austin
Version 2.0 (2014)
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Foundations of Mathematical Reasoning
Student Pages 1.A, How big is a billion?
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The University of Texas at Austin
Version 2.0 (2014)
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 1.A, How big is a billion?
Lesson 1, Part A
How Big Is a Billion?
Overview and student objectives
Overview
Lesson Length: 25
minutes
The first lesson of the course addresses multiple goals:
Prior Lesson: None
•
Do mathematics.
•
Interact with each other and the instructor.
•
Begin to develop an understanding of the structure of the
course, which is designed around:
o Quantitative reasoning
o Active learning
o Productive struggle
o Embedded student success strategies
Next Lesson: Lesson 1,
Part B, “Building a
Learning Community” (25
minutes)
Constructive
Perseverance Level: 1
Theme: Civic Life
Outcomes: N1, N4, N5,
N6
Goal: Problem Solving
“How Big Is a Billion?” begins with Part A and is continued in Part C.
Objectives
Students will understand that:
•
Large numbers can be represented in various ways.
•
Collaborating with others can enhance learning.
Students will be able to:
•
Scale measurements of groups to represent individual elements.
•
Scale measurements to represent larger quantities of individual elements.
Suggested resources and preparation
Materials and technology
•
Computer, projector, document camera
•
Student Pages for Lesson 1, Part A
•
Name tags and markers
•
One or two measuring tapes
•
Calculator for each student
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The University of Texas at Austin
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 1.A, How big is a billion?
Prerequisite assumptions
Before beginning this lesson, students should be able to multiply and divide large
numbers with the use of a calculator.
Making connections
Many lessons in the course connect back to concepts that students typically will have
studied before; what is different is that students are now encountering these concepts in
an active-learning environment.
This lesson:
•
Connects back to measurement, estimation, and unit conversion.
•
Connects forward to scientific notation and population growth density.
Background context
None. In some future lessons, this section will provide information about the context—
including information that students encountered in preceding assignment—that may be
useful to you.
Suggested instructional plan
Frame the lesson
(8 minutes)
Classroom
Culture
•
Arrive a few minutes early to class, greet any students who are
already there, and ask students to fill out and put on name tags.
This action tells students right away that you want to get to know
them and learn their names and that you want all your students to
know one another’s names.
Student
Pages
•
Distribute or display the Student Pages for this lesson.
Think-PairShare
Questions 1–3
•
Ask students to read and think about questions 1–3, completing
them individually and then discussing with classmates. As
additional students enter the room, give them the same handout
and instructions. This strategy prevents dead time, sets the
expectation that class starts on time, and ensures that, from the
outset, students draw on their own prior knowledge.
•
Circulate the room and listen to student discussion.
•
Take student input.
•
Accept all appropriate responses and encourage thinking that
goes along with the philosophy of the course (e.g., students may
suggest what they know about world population or government
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 1.A, How big is a billion?
budget), but if necessary, prompt students to ensure that at least
the following representations are shared:
o One billion = 1,000,000,000 = 109 = 1,000 x 1,000 x 1,000.
o The representations 1 billion, 1,000,000,000, and 109 have the
same meaning.
Guiding
Question
•
Discuss the meaning of one billion. If some students think that one
billion is 1,000,000 (for example), don’t correct them; rather, ask,
“How could we check?” (Use resources such as smartphones; use
place value starting at ones and building up, etc.)
•
Interesting fact #1: Names of large numbers are different in
continental Europe. See, for example, the entry on “Names of
large numbers” in Wikipedia—
http://en.wikipedia.org/wiki/Names_of_large_numbers.
•
Interesting fact #2: There is actually only one video with over one
billion views, as of early 2014. See the entry on “List of most
viewed YouTube videos” in Wikipedia—
http://en.wikipedia.org/wiki/
List_of_most_viewed_YouTube_videos.
•
Accept several student predictions, such as “A line of one billion
people would stretch from here to the North Pole.”
•
Transition to the lesson activities by briefly discussing the
Objectives for the lesson.
•
Possible grounding statement: “One of the foundations of what it
means to reason mathematically is to have a sense of how
numbers of various sizes compare to each other.”
Lesson activities
(12 minutes)
Large Group
Groups of
3 to 4
Questions 4 and 5
•
Provide students with a measuring tape or stick and encourage
them to complete the activity for questions 4 and 5. If necessary
due to space constraints, form groups and take student
suggestions on what to do with the outcomes. Groups should be
large enough to have a variety of shoulder widths but small enough
that in every group, all students participate.
•
Compile the results from the groups. You may wish to ask the
students what to do with the results. They may want to combine the
group results, or they may want each group to work with their own
measurements and compare at the end.
Questions 6–9
•
As students return to their seats, have them choose their own
groups to work on questions 6–9. Circulate the room to monitor
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 1.A, How big is a billion?
student progress.
Classroom
Culture
Guiding
Questions
•
Recognize that students may choose to use different units of
measure. Allow students to persist through the questions using
their chosen unit of measure. Guiding students to use their
calculator may assist them with persevering and arriving at an
answer.
•
Rather than answering student questions, redirect them to their
own thinking as well as to their group’s. For example, if students
ask:
o “How do I . . .?”: Be prepared with prompts for questions you
think students will ask. For example, if students ask how to find
an average in question 6, ask, “Suppose you had two people,
one with a shoulder width of 18 inches and one with a shoulder
width of 20 inches. How could you split up the length between
those two people? How could you use that technique with 10
people?”
o “Is this right?”: Ask how they would justify the answer if
everyone in the group agrees. Could they check their answer
or could they try it a different way? Ensure that students justify
their work mathematically and that they are also able to justify
it verbally. This practice begins to establish a culture of
students thinking independently.
•
As you circulate, make sure students are including units of
measure in their answers.
•
Debrief students on the strategies they used in completing
questions 6–9, paying particular attention to operations.
•
Guiding question to facilitate completion of questions 7–9: “What
did you do to get from question 6 to question 7? [Multiply by
1,000.] What if we repeat that process? Use that process to
complete the questions.”
•
Circulate through the room to monitor student progress as they
continue the process of multiplying.
•
Debrief. Be sure to draw out that 1,000,000 people came from
1,000 x 1,000 and that 1 billion came from 1,000 x 1,000 x 1,000.
Also note that 1,000,000 is 1,000 x 1,000.
Wrap-up/transition
(5 minutes)
Wrap-up
•
Ask the class to summarize the results and compare them to their
original prediction in question 3. Recall that the students may have
used a different unit system (miles, kilometers, etc.) to make their
prediction. Talk about how measuring things with small units
makes the final answer harder to understand. Use this discussion
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 1.A, How big is a billion?
on small units to motivate the activity in Lesson 1, Part C.
Classroom
Culture
Transition
•
Have students refer back to the Objectives for the lesson and
check the ones they recognize from the activity. Alternatively, they
may check objectives throughout the lesson.
•
Help students take notes on the objectives in the summary section
of their notes. Students could include notes on the different
representations of a billion and a million and how to convert feet to
miles.
•
Thank the class for contributing to whole class and/or group
discussions.
•
Transition to Lesson 1, Part B: Ask students what they noticed
about the way the first portion of the class was conducted: “What
did you like about the activity? What did you learn?”
Suggested assessment, assignments, and reflections
•
Give the Preview Assignments, if any, for the lesson activities you plan to
complete in the next class meeting.
•
Explain to students that in future assignments, they will see questions that are
designed to prepare them for future learning. Those questions will refer them to
pages in their Resource documents that will help them answer questions. You
could also have this discussion when reviewing the syllabus, if you have included
a binder requirement, as suggested in the Prep Week document.
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The University of Texas at Austin
Version 2.0 (2014)
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 1.A, How big is a billion?
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The University of Texas at Austin
Version 2.0 (2014)
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 1.A, How big is a billion?
Lesson 1, Part A,
How big is a billion?
Theme: Civic Life
ANSWERS
A million, a billion, a trillion, a quadrillion—it is easy to lose a sense of the size of large
numbers. So just how big is a billion?
Sample answer: One billion is 1,000 times larger than a million.
1)
Jot down anything you know about the
number one billion. Then share with at
least two neighbors.
Sample answer: Since 1 million has 6
zeros, 1 billion probably has 9 zeros.
Credit: iStockphoto
2)
How many YouTube videos do you think have over one billion views?
Answers will vary. Allow for a variety of answers to this question.
3)
If a billion people stood shoulder-to-shoulder, how long would the line be? Make a
prediction.
Answers will vary. Allow for a variety of answers to this question.
Objectives for the lesson
You will understand that:
o Large numbers can be represented in various ways.
o Collaborating with others can enhance learning.
You will be able to:
o Scale measurements of groups to represent individual elements.
o Scale measurements to represent larger quantities of individual elements.
Stand in line shoulder-to-shoulder with your classmates.
4)
How many people are in the line?
Sample answer: 5 people
The Charles A. Dana Center at
The University of Texas at Austin
Version 2.0 (2014)
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 1.A, How big is a billion?
5)
To the nearest inch, how long is the line?
Sample answer: 84 inches
6)
What is the average shoulder width of the people in your line? How do you know?
Sample answer: 16.8 inches
7)
How long would the line be if there were 1 thousand (1,000) people?
Sample answer: 16,800 inches
8)
How long would the line be if there were 1 million (1,000,000) people?
Sample answer: 16,800,000 inches
9)
How long would the line be if there were 1 billion (1,000,000,000) people?
Sample answer: 16,800,000,000 inches
The Charles A. Dana Center at
The University of Texas at Austin
Version 2.0 (2014)
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