Lesson 12

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).

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. 

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. 

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.

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.

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.

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.

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.

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?”

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.