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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? The Charles A. Dana Center at The University of Texas at Austin Version 2.0 (2014) 1 Foundations of Mathematical Reasoning Student Pages 1.A, How big is a billion? The Charles A. Dana Center at The University of Texas at Austin Version 2.0 (2014) 2 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 The Charles A. Dana Center at The University of Texas at Austin Version 2.0 (2014) 3 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 The Charles A. Dana Center at The University of Texas at Austin Version 2.0 (2014) 4 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 The Charles A. Dana Center at The University of Texas at Austin Version 2.0 (2014) 5 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 The Charles A. Dana Center at The University of Texas at Austin Version 2.0 (2014) 6 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. The Charles A. Dana Center at The University of Texas at Austin Version 2.0 (2014) 7 Foundations of Mathematical Reasoning Suggested Instructor Notes 1.A, How big is a billion? The Charles A. Dana Center at The University of Texas at Austin Version 2.0 (2014) 8 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) 9 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) 10