Lesson 9
Describing Large and Small Numbers Using Powers of 10
9.1: Thousand Million Billion Trillion (5 minutes)
Warmup
In this warmup, students connect thousand, million, billion, and trillion to their respective powers of ten—\(10^3, 10^6, 10^9\), and \(10^{12}\). Understanding powers of 10 associated with these denominations will help students reason about quantities in realworld contexts such as the number of cells in a human body (trillions), world population (billions), etc.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet work time and then 1 minute to compare their responses with their partner. Given the limited time, it may not be possible for students to create examples for each of the values in the second question. Tell students to try to at least find 1 or 2 examples and then to find others as time allows. Follow with a wholeclass discussion.
Student Facing
 Match each expression with its corresponding value and word.
expression \(10^{\text3}\) \(10^6\) \(10^9\) \(10^{\text2}\) \(10^{12}\) \(10^3\) value 1,000,000,000,000 \(\frac{1}{100}\) 1,000 1,000,000,000 1,000,000 \(\frac{1}{1,000}\) word billion milli million thousand centi trillion  For each of the numbers, think of something in the world that is described by that number.
Student Response
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Anticipated Misconceptions
Some students may think that, for example, \(1,\!000,\!000 = 10^7\) because the number 1,000,000 has 7 digits. Ask these students if it is true that \(10 = 10^2\) .
Some students may confuse the prefix “milli” with the word “million.” The word “million” literally means “a big thousand,” and so both “million” and “mille” are related to the Latin “mille,” meaning “thousand.” While “milli” is talking about a thousand parts (thousandths), “million” is talking about a thousand thousands.
Activity Synthesis
Ask students to share the corresponding expressions, words, and values. Record and display their responses for all to see. If time is limited, consider displaying the completed table for all to see and discussing any questions or disagreements. Then, invite students to share their examples for the final question. After each student shares, ask the class whether they agree that the given example could be described by that value.
If students struggle to find something that could be described by each value, consider sharing some of the following examples:
 Thousand (\(10^3\))
 Number of students in a school
 Population of an endangered species
 Cost of a car that barely runs
 Number of brain cells of a jellyfish
 Million (\(10^6\))
 Population of a state
 Cost of the most expensive car in the world
 Number of brain cells of a cockroach
 Billion (\(10^9\))
 Population of India
 Population of China
 Number of students in the world
 Number of brain cells of a monkey
 Number of trees in the U.S.
 Trillion (\(10^{12}\))
 Amount of wealth produced by a developed country in a year in U.S. dollars
 Number of stars in a large galaxy
 Number of brain cells of 10 students
9.2: Baseten Representations Matching (20 minutes)
Activity
In this activity, students use their understanding of decimal place value and baseten diagrams to practice working with the structure of scientific notation before it is formally introduced. They express numbers as sums of terms, each term being multiples of powers of 10. For example, 254 can be written as \(2 \boldcdot 10^2 + 5 \boldcdot 10^1 + 4 \boldcdot 10^0\).
Notice students who choose different, yet correct, diagrams for the first and last problems. Ask them to share their reasoning in the discussion later.
Launch
Display the following diagram for all to see. Pause for quiet think time after asking each question about the diagram. Ask students to explain their thinking. Here are some questions to consider:
 “If each small square represents 1 tree, what does the whole diagram represent?” (121 trees)
 “If each small square represents 10 books, what does the whole diagram represent?” (1,210 books)
 “If each small square represents 1,000,000 stars, what does the whole diagram represent?” (121,000,000 stars)
 “If each small square represents 0.1 seconds, what does the whole diagram represent?” (12.1 seconds)
 “If each small square represents \(10^3\) people, what does the whole diagram represent?” (121,000 people)
Give students 10 minutes to work on the task, followed by a brief wholeclass discussion.
Supports accessibility for: Conceptual processing; Organization
Student Facing
 Match each expression to one or more diagrams that could represent it. For each match, explain what the value of a single small square would have to be.

\(2 \boldcdot 10^{\text 1} + 4 \boldcdot 10^{\text 2}\)

\(2 \boldcdot 10^{\text 1} + 4 \boldcdot 10^{\text 3}\)

\(2 \boldcdot 10^3 + 4 \boldcdot 10^1\)

\(2 \boldcdot 10^3 + 4 \boldcdot 10^2\)

\(2 \boldcdot 10^{\text 1} + 4 \boldcdot 10^{\text 2}\)

 Write an expression to describe the baseten diagram if each small square represents \(10^{\text 4}\). What is the value of this expression?
 How does changing the value of the small square change the value of the expression? Explain or show your thinking.
 Select at least two different powers of 10 for the small square, and write the corresponding expressions to describe the baseten diagram. What is the value of each of your expressions?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Some students may think that the small square must always represent one unit. Explain to these students that, as in the launch, the small square might represent 10 units, 0.1 units, or any other power of 10.
Some students may have trouble writing the value of expressions that involve powers of 10, especially if they involve negative exponents. As needed, suggest that they expand the expression into factors that are 10, and remind them that \(10^{\text1}\) corresponds to the tenths place, \(10^{\text2}\) corresponds to the hundredths place, etc.
Activity Synthesis
Select students to share their responses to the first and last problems. Bring attention to the fact that diagrams B or C could be used depending on the choice of the value of the small square.
The main goal of the discussion is to make sure students see the connection between decimal place value to sums of terms that are multiples of powers of 10. To highlight this connection explicitly, consider discussing the following questions:
 “How are the diagrams related to our baseten numbers and place value system?” (In baseten numbers, each place value is ten times larger than the one to its right; so every 1 unit of a place value can be composed of 10 units of the next place value to its right. The diagrams work the same way: each shape representing a baseten unit can be composed of 10 that represent another unit that is one tenth of its value.)
 “How are the diagrams related to numbers written using powers of 10?” (We can think of each place value as a power of ten. So a ten would be \(10^1\), a hundred would be \(10^2\), a tenth would be \(10^\text{1}\), and so on.)
 “If each large square represents \(10^2\), what do 2 large squares and 4 long rectangles represent?” (A long rectangle is a tenth of the large square, so we know the long rectangle represents \(10^1\). This means 2 large squares and 4 long rectangles represent \(2 \boldcdot 10^2 + 4 \boldcdot 10^1\).)
 “Why is it possible for one baseten diagram to represent many different numbers?” (Because of the structure of our place value system—where every group of 10 of a baseten unit composes 1 of the next higher unit—is consistent across all place values.)
Design Principle(s): Cultivate conversation; Maximize metaawareness
9.3: Using Powers of 10 to Describe Large and Small Numbers (15 minutes)
Activity
This activity motivates students to find easier ways to communicate about very large and very small numbers, using powers of 10 and working toward using scientific notation. Students take turns reading aloud and writing down quantities that involve long strings of digits, noticing the challenges of expressing such numbers.
As students work, monitor the different ways students communicate the number of zeros precisely to their partners. Some might use standard vocabulary (billion, tenthousandth, etc.), or some may communicate the number of zeros after the decimal point or after the significant digits. Select students using different strategies to share later.
Launch
Arrange students in groups of 2. Distribute a pair of cards (one for Partner A and one for Partner B) from the blackline master to each group. Ask partners not to show their card to each other.
Tell students that one partner should read an incomplete statement in the materials and the other partner should read aloud the missing information on the card. The goal is for each partner to write down the missing quantity correctly. Partners should take turns reading and writing until all four statements for each person are completed.
Consider explaining (either up front or as needed during work time) that students who have the numbers can describe or name them in any way that they think convey the quantities fully. Likewise, those writing the numbers can write in any way that captures the quantities accurately.
Give students 10 minutes to work. Leave a few minutes for a wholeclass discussion.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Your teacher will give you a card that tells you whether you are Partner A or B and gives you the information that is missing from your partner’s statements. Do not show your card to your partner.
Read each statement assigned to you, ask your partner for the missing information, and write the number your partner tells you.
Partner A’s statements:
 Around the world, about ______________________ pencils are made each year.
 The mass of a proton is ______________________ kilograms.
 The population of Russia is about ______________________ people.
 The diameter of a bacteria cell is about ______________________ meter.
Partner B’s statements:
 Light waves travel through space at a speed of ______________________ meters per second.
 The population of India is about ______________________ people.
 The wavelength of a gamma ray is _______________________ meters.
 The tardigrade (water bear) is ______________ meters long.
Student Response
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Student Facing
Are you ready for more?
A “googol” is a name for a really big number: a 1 followed by 100 zeros.
 If you square a googol, how many zeros will the answer have? Show your reasoning.
 If you raise a googol to the googol power, how many zeros will the answer have? Show your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Ask previously identified students to share how they described their partner's numbers or recorded those given to them. The purpose of the discussion is for students to hear different strategies for communicating very small and very large numbers. For example, 150,000,000,000 can be described as “one hundred fifty billion,” as “15 followed by 10 zeros,” as \(150 \boldcdot 10^9\), or as \((1.5) \boldcdot 10^{11}\).
For negative powers, discuss the idea that multiplying by \(10^\text{1}\) means multiplying by \(\frac{1}{10}\), which in turn increases the number of decimal places. For example, consider \(48 \boldcdot 10^{\text13}\). We know that \(1 \boldcdot 10^{\text13}\) is 1 multiplied by \(\frac{1}{10}\), 13 times, which is \(0.0000000000001\). So \(48 \boldcdot 10^{\text13}\) is \(48 \boldcdot (0.0000000000001)\), which is \(0.0000000000048\). Display the following table for all to see and briefly explain to students how to write each number using powers of 10.
quantities  using powers of 10 

150,000,000,000 meters  \(150 \boldcdot 10^9\) meters 
300,000,000 meters per second  \(300 \boldcdot 10^6\) meters per second 
0.0000000000048 meters  \(48 \boldcdot 10^{\text13}\) meters 
0.00000000000000000000000000167 kilogram  \(167 \boldcdot 10^{\text29}\) kilogram 
Design Principle(s): Maximize metaawareness; Support sensemaking
Lesson Synthesis
Lesson Synthesis
The focus of the discussion is the structure of our place value system and the rationale and usefulness of describing large and small numbers in different ways. Consider asking some of the following questions:
 “How do baseten diagrams help us make sense of (or explain) the exponents in powers of 10?” (When using diagrams, grouping 10 of the next smaller unit means multiplying by 10. When dealing with powers of 10, multiplying by 10 increases the exponent by 1. Likewise, decomposing a baseten unit into 10 of the next smaller unit means multiplying by \(\frac{1}{10}\), so the exponent in the power of 10 goes down by 1.)
 “How does using powers of 10 make it easier to communicate about very large or very small numbers?” (We can write in a smaller space. It’s also faster to read and easier to understand the size of a number and to compare numbers. Using powers of 10 helps us avoid errors of missing zeros or extra zeros.)
 “What are some different ways to describe a large number like 123 billion?” (\(123 \boldcdot (1,\!000,\!000,\!000\) or \(123 \boldcdot 10^9\).)
 “What are some different ways to describe a small number like 0.0000000789?” (789 tenbillionths, \(789 \boldcdot \frac{1}{10,000,000,000}\), or \(789 \boldcdot 10^\text{10}\).)
9.4: Cooldown  Better with Powers of 10 (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
Sometimes powers of 10 are helpful for expressing quantities, especially very large or very small quantities. For example, the United States Mint has made over
500,000,000,000
pennies. In order to understand this number, we have to count all the zeros. Since there are 11 of them, this means there are 500 billion pennies. Using powers of 10, we can write this as: \(\displaystyle 500 \boldcdot 10^9\) (five hundred times a billion), or even as: \(\displaystyle 5 \boldcdot 10^{11}\) The advantage to using powers of 10 to write a large number is that they help us see right away how large the number is by looking at the exponent.
The same is true for small quantities. For example, a single atom of carbon weighs about
0.0000000000000000000000199
grams. We can write this using powers of 10 as \(\displaystyle 199 \boldcdot 10^{\text25}\) or, equivalently, \(\displaystyle (1.99) \boldcdot 10^{\text23}\) Not only do powers of 10 make it easier to write this number, but they also help avoid errors since it would be very easy to write an extra zero or leave one out when writing out the decimal because there are so many to keep track of!