Lesson 12

Applications of Arithmetic with Powers of 10

12.1: What Information Do You Need? (5 minutes)

Warm-up

The purpose of this warm-up is for students to reason about a real-world situation and consider the essential information required to solve problems (MP4).

Launch

Arrange students in groups of 2. Give students 1 minute of quiet think time, followed by 1 minute to share their responses with a partner. Follow with a whole-class discussion. 

Student Facing

What information would you need to answer these questions?

  1. How many meter sticks does it take to equal the mass of the Moon?
  2. If all of these meter sticks were lined up end to end, would they reach the Moon?

Student Response

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Activity Synthesis

Ask students to share their responses for each question. Record and display the responses for all to see.

Consider asking questions like these to encourage students to reason further about each question:

  • “Why do you need that piece of information?”
  • “How would you use that piece of information in finding the solution?”
  • “Where would you look to find that piece of information?”

If there is time, ask students for predictions for each of the questions. Record and display their responses for all to see. 

12.2: Meter Sticks to the Moon (20 minutes)

Activity

The large quantities involved in these questions lend themselves to arithmetic with powers of 10, giving students the opportunity to make use of scientific notation before it is formally introduced. This activity was designed so students could practice modeling skills such as identifying essential features of the problem and gathering the required information (MP4). Students use powers of 10 and the number line as tools to make it easier to calculate and interpret results.

Notice the ways in which students use relevant information to answer the questions. Identify students who can explain why they are calculating with one operation rather than another. Speed is not as important as carefully thinking through each problem.

Launch

From the warm-up, students have decided what information they need to solve the problem. Invite students to ask for the information they need. Provide students with only the information they request. Display the information for students to see throughout the activity. If students find they need more information later, provide it to the whole class then.

Here is information students might ask for in order to solve the problems:

  • The mass of an average classroom meter stick is roughly 0.2 kg.
  • The length of an average classroom meter stick is 1 meter.
  • The mass of the Moon is approximately \(7 \boldcdot 10^{22}\) kg.
  • The Moon is roughly \((3.8) \boldcdot 10^8\) meters away from Earth.
  • The distances to various astronomical bodies the students might recognize, in light years, as points of reference for their last answer. (Consider researching other distances in advance or, if desired, encouraging interested students to do so.)

Arrange students in groups of 2–4 so they can discuss how to use the information to solve the problem. Give students 15 minutes of work time.

Student Facing

  1. How many meter sticks does it take to equal the mass of the Moon? Explain or show your reasoning.
  2. Label the number line and plot your answer for the number of meter sticks.

    A number line with eleven evenly spaced tick marks. The first tick is labeled 0, the last tick is labeled 10 to the blank power, and the remaining tick marks are blank.
  3. If you took all the meter sticks from the last question and lined them up end to end, will they reach the Moon? Will they reach beyond the Moon? If yes, how many times farther will they reach? Explain your reasoning.
  4. One light year is approximately \(10^{16}\) meters. How many light years away would the meter sticks reach? Label the number line and plot your answer.

    A number line with eleven evenly spaced tick marks. The first tick is labeled 0, the last tick is labeled 10 to the blank power, and the remaining tick marks are blank.


     

Student Response

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Student Facing

Are you ready for more?

Here is a problem that will take multiple steps to solve. You may not know all the facts you need to solve the problem. That is okay. Take a guess at reasonable answers to anything you don’t know. Your final answer will be an estimate.

If everyone alive on Earth right now stood very close together, how much area would they take up?

Student Response

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Activity Synthesis

Select previously identified students to share how they organized their relevant information and how they planned to use the information to answer the questions. The important idea students should walk away with is that powers of 10 are a great tool to tackle challenging, real-world problems that involve very large numbers.

It might be illuminating to put 35 million light years into some context. It is over a thousand trillion times as far as the distance to the Moon, or about the size of a supercluster of galaxies. The Sun is less than \(1.6 \times 10^\text{-5}\) light year away from Earth.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . ..”
Supports accessibility for: Language; Social-emotional skills
Speaking: MLR8 Discussion Supports. Give students additional time to make sure everyone in their group can explain their response to the first question. Then, vary who is called on to represent the ideas of each group. This routine will prepare students for the role of group representative and to support each other to take on that role.
Design Principle(s): Optimize output (for explanation); Cultivate conversation

12.3: That’s a Tall Stack of Cash (20 minutes)

Optional activity

This activity also illustrates the utility of using powers of 10 to work with and interpret very large quantities. Students practice modeling skills, such as identifying essential features of a problem and gathering the required information (MP4). Students use numbers and exponents flexibly and interpret their results in context (MP2). 

As students work, look for students who use powers of 10 and the number line as tools to make it easier to calculate and interpret their results.

Launch

Ask the class to predict which is taller, the Burj Khalifa or a stack of money it cost to build the Burj Khalifa. Push them further by asking them to predict how high they think the stack would go. Record some of these predictions. Students will ask for the information they need to solve these problems:

  • A 1-meter stack of 100-dollar bills is about 1,000,000 dollars. The Burj Khalifa is 830 meters tall and cost 1.5 billion dollars.
  • The Burj Khalifa weighs 450,000,000 kg. A penny weighs \((2.5) \boldcdot 10^\text{-3}\) kg. There are 100 pennies in a dollar.

Arrange students in groups of 2–4. Give students 10–15 minutes to work. As students work to finish the fourth problem (plotting the heights on a number line), tell the class that their next step is to read the fifth problem and think about what additional information they would need to know to solve the problem. When many students have finished problem 4, pause to allow the class to ask these questions before proceeding.

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts. Review an image or video of the Burj Khalifa to activate prior knowledge of the context of the problem.
Supports accessibility for: Conceptual processing; Memory
Writing, Speaking: MLR1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their response to the question: “Which is taller, the Burj Khalifa, or the stack of the money it cost to build the Burj Khalifa?” Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them to strengthen their ideas and clarify their language (e.g., “What did you do first?”, “How do you know…?”, “How did you compare the two values?”, etc.). Students can borrow ideas and language from each partner to strengthen their final version.
Design Principle(s): Optimize output (for explanation)

Student Facing

In 2016, the Burj Khalifa was the tallest building in the world. It was very expensive to build.

Consider the question: Which is taller, the Burj Khalifa or a stack of the money it cost to build the Burj Khalifa?

  1. What information would you need to be able to solve the problem?
  2. Record the information your teacher shares with the class.



     
  3. Answer the question “Which is taller, the Burj Khalifa or a stack of the money it cost to build the Burj Khalifa?” and explain or show your reasoning.



     
  4. Decide what power of 10 to use to label the rightmost tick mark of the number line, and plot the height of the stack of money and the height of the Burj Khalifa.

    A number line with eleven evenly spaced tick marks. The first tick is labeled 0, the last tick is labeled 10 to the blank power, and the remaining tick marks are blank.
  5. Which has more mass, the Burj Khalifa or the mass of the pennies it cost to build the Burj Khalifa? What information do you need to answer this?


     
  6. Decide what power of 10 to use to label the rightmost tick mark of the number line, and plot the mass of the Burj Khalifa and the mass of the pennies it cost to build the Burj Khalifa.

    A number line with eleven evenly spaced tick marks. The first tick is labeled 0, the last tick is labeled 10 to the blank power, and the remaining tick marks are blank.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Students may overlook the fact that there are 100 pennies in a dollar. Remind these students of this fact and ask, “If you use pennies instead of dollars, would there be more coins or fewer coins? How many times more? If 1.5 billion dollars is \((1.5) \boldcdot 10^9\), then how would you find the number of pennies?”

Activity Synthesis

If time allows, return to some of the recorded predictions. Acknowledge predictions that were accurate and discuss how powers of 10 made this problem much more approachable.

Lesson Synthesis

Lesson Synthesis

To prompt students to reflect on the modeling process and on using exponents to solve problems, consider asking some of these questions:

  • “To solve the problems in this lesson you had to determine what information was needed. Did you find that to be fairly straightforward or challenging? What made it straightforward or challenging?”
  • “Describe your thinking as you planned a solution path for the problems. For example, did you ask for information first and then decide what to do with it, or did you decide what needs to be done first before asking for certain information?”
  • “Once you had the information you needed, what were some difficulties you encountered? How did you work through them?”
  • “How did exponent rules and powers of 10 make the calculations easier?” (Powers of 10 make the numbers easier to express and interpret. The rules of exponents were handy for comparing how many times as large or as small one number is as another number.)
  • “Would an estimate be an acceptable answer for problems like these? Why or why not? When might we need more precise solutions?” (It depends on the questions and how the answers would be used. For example, if we were planning a space exploration, we would likely need a high level of precision to ensure that we hit our targets. But if the answers are for comparison or general information, estimates are likely adequate.)

12.4: Cool-down - Reflecting on Using Powers of 10 (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Powers of 10 can be helpful for making calculations with large or small numbers. For example, in 2014, the United States had

318,586,495 

people who used the equivalent of

2,203,799,778,107

kilograms of oil in energy. The amount of energy per person is the total energy divided by the total number of people. We can use powers of 10 to estimate the total energy as \(\displaystyle 2 \boldcdot 10^{12}\) and the population as \(\displaystyle 3 \boldcdot 10^8\) So the amount of energy per person in the U.S. is roughly \(\displaystyle (2 \boldcdot 10^{12}) \div (3 \boldcdot 10^8)\) That is the equivalent of \(\displaystyle \frac{2}{3} \boldcdot 10^4\) kilograms of oil in energy. That’s a lot of energy—the equivalent of almost 7,000 kilograms of oil per person!

In general, when we want to perform arithmetic with very large or small quantities, estimating with powers of 10 and using exponent rules can help simplify the process. If we wanted to find the exact quotient of 2,203,799,778,107 by 318,586,495, then using powers of 10 would not simplify the calculation.