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In this activity for grades 4-6, students attempt to identify the concept of a million by working with smaller numerical units, such as blocks of 10 or 100, and then expanding the idea by multiplication or repeated addition until a million is reached. Additionally, they use critical thinking to analyze situations and to identify mathematical patterns that will enable them to develop the concept of very large numbers.

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, 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 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.

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.

Distribute a copy of Making Your First Million Activity Sheet to each student.

Students should determine the following:

• 1 million days is about 2700 years
• 1 million hours is just over 114 years
• 1 million minutes is nearly 2 years
• and 1 million seconds is approximately 11.5 days.

Discuss the conceptual difference and ability to visualize "1 million seconds" versus rough equivalent of "11.5 days." Often changing the way we look at a subject helps us to see more detail.

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.

In groups of ten, ask students to count 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 make 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.

Consider cutting and pasting four copies of the grid found on the act 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

 1 + 1 = = = =

will allow the calculator to continue to add "1" each time "=" is pressed, without having to repeatedly press "+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 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?" Encourage them to suggest strategies to determine an answer, such as timing a counting segment and extrapolating.

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 1100, 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 time-consuming 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. Strategies similar to those used previously can be used here.

Have groups of students write out the numerals in order beginning at various places inthe 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.

Ask students to "fill" the rectangle found in problem 7 on the activity sheet with as many legible digits as possible, which can be used to determine the approximate numbers of digits that can be legibly written on a standard 8 1/2-by-11-inch page and, consequently, how many pages would be required.

Discuss miscellaneous factors, such as handwriting styles and choice of digits to write in the box, which can influence the results.

Conclude this investigation by encouraging 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.

### References

• David M. Schwartz. (1985). How Much Is a Million? New York: Scholastic Books.
• David M. Schwartz. (1989). If You Made a Million New York: Scholastic Books.

Extensions

1. 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.
2. How long would it take to write the 707 places? 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.

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### Every Breath You Take

3-5
In the following open-ended exploration, students estimate, experiment, and display real-life data. Students use the number of breaths taken during a specified time period as the context for this exploration.

### Learning Objectives

Students will:
• Use proportional reasoning in problem solving.
• Use standard and nonstandard measurement.
• Work with 10s and 100s to explore the concept of 1,000,000.

### NCTM Standards and Expectations

• Understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals.
• Recognize equivalent representations for the same number and generate them by decomposing and composing numbers.
• Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.
• Design investigations to address a question and consider how data-collection methods affect the nature of the data set.
• Collect data using observations, surveys, and experiments.

### Common Core State Standards – Mathematics

Grade 4, Num & Ops Base Ten

• CCSS.Math.Content.4.NBT.A.1
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

Grade 5, Num & Ops Base Ten

• CCSS.Math.Content.5.NBT.A.1
Recognize that in a multi-digit number, a digit in one 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.

• CCSS.Math.Content.7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP6
Attend to precision.
• CCSS.Math.Practice.MP7
Look for and make use of structure.