The purpose of this activity is for students to make sense of and then use exponential notation to represent large numbers, namely 1 million and 1 billion. Students should be encouraged to say the names of the numbers in a way that makes sense to them. Contexts, in the form of human populations, are provided for each of the large numbers to help students conceptualize the magnitude of the number. Students recognize that the purpose of exponential notation is to write large numbers efficiently and recognize how many factors of 10 are in a given number.

When students relate 1 million and 1 billion to products of 10 and powers of 10, they look for and make use of base-ten structure (MP7).

• Ask previously selected students to share their names for 1,000,000,000 in the given order.
• “How many millions are in this number? How do you know?” (1 thousand, because there are 3 more zeros and that means multiplying by 10 three times.)
• “How do you write this number, using powers of 10? Why?” (, because there are 9 factors of 10.)
• “This number is called ‘one billion.’”
• “Using powers of 10, can you write a number that is bigger than 1 billion?” (Yes, , , .)
• “Why are powers of 10 useful for representing really big numbers?” (I would not want to write 100 zeros. I also would have to count all those zeros to find out the number.)

The purpose of this activity is for students to find the unknown number that makes multiplication equations true when that value is a power of 10. The numbers in this activity were chosen to build toward numbers greater than those with which students have worked before.

The Activity Synthesis highlights that these powers of 10 are represented by a 1 followed by some zeros. The power of 10 tells us the number of zeros in the number.

• “How did you decide to write that number?”
• Display each power of 10 as a product of tens and ask, “What is the same about these expressions? What is different?”

• Ask selected students to share how they found the unknown number that makes the equation true.
• “How did thinking about factors of 10 help to find the missing number that makes the equation true?” (I started with 1,000, and 10 times that is 10,000, and then I knew I needed another factor of 10 to get to 100,000.)
• Ask selected students to share their responses for the value of in the given order.
• “How did you find the value?” (I know 1,000 is , so I started multiplying by 10. I got 100,000 and then 1,000,000, which is 1 million. I put one more zero in for the last 10.)
• “How do you say this number?” (10 million)
• Display the equations:
  
  
  
• “What do you notice about the equations?” (The number of zeros on each number is the same as the power of 10.)

The goal of this optional activity is to introduce one more number, a trillion, which is 1,000 billions. These large numbers become more and more difficult to conceptualize, and the goal of the Activity Synthesis is to introduce students to the idea that there are things in the world that number in the trillions.

Relating huge numbers to things in the world, as students do in the Activity Synthesis, is a part of modeling with mathematics (MP4).

• Invite students to share things in the world that they think may number in the trillions.
• “How did you choose?” (I picked something that there were too many to count.)
• “Why is it hard to guess whether there are 1 trillion of something, like grains of sand?” (It’s too big a number to count or imagine.)
• Consider sharing some quantities that number in the trillions:
  • The total number of seconds in 31,700 years is 1,000,000,000,000.
  • The total number of trees on Earth is about 3,000,000,000,000.
  • The total number of fish in the oceans is about 3,500,000,000,000.
  • The total number of ants on Earth greatly exceeds 1 trillion: 20,000,000,000,000,000 (20 quadrillion).
  • The total number of insects on Earth also greatly exceeds 1 trillion: 1,400,000,​0008,100,000,​000, which is greater than 11,000,000,000,000,000,000 (There are about 1.4 billion insects for every person on Earth, and there are about 8.1 billion people, so there are more than 11 quintillion insects.)