Lesson 9

In this warm-up, 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 real-world contexts such as the number of cells in a human body (trillions), world population (billions), etc. 

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 whole-class discussion.

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.

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

In this activity, students use their understanding of decimal place value and base-ten 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.

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 whole-class discussion.

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 \(10^{\text-1}\) corresponds to the tenths place, \(10^{\text-2}\) corresponds to the hundredths place, etc.

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 base-ten numbers and place value system?” (In base-ten 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 base-ten 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 base-ten diagram to represent many different numbers?” (Because of the structure of our place value system—where every group of 10 of a base-ten unit composes 1 of the next higher unit—is consistent across all place values.)

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, ten-thousandth, 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.

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 whole-class discussion.

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 \(48 \boldcdot 10^{\text-13}\). We know that \(1 \boldcdot 10^{\text-13}\) is 1 multiplied by \(\frac{1}{10}\), 13 times, which is \(0.0000000000001\). So \(48 \boldcdot 10^{\text-13}\) 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.