Lesson 2

The goal of this warm-up is for students to visualize \(10^n\) for different exponents before they learn exponential notation in this lesson. Monitor for students who use the symmetry of the diagram to estimate how many line segments there are of each size. For example, the picture can be rotated 10 times around the center and each arm is the same so that means the number of each size segment has one factor of 10. This idea can be applied at a smaller scale to get a second and third factor of 10.

When students analyze the diagram and determine how many segments there are of each length, they are observing and making use of the repeated structure of ten segments joining at the different vertices (MP7,  MP8).

• Groups of 2
• “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you see them?”
• Display the image.
• 1 minute: quiet think time

• Display the image.
• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• Invite students to share their estimates for how many of the smallest line segments there are in the diagram.
• “¿Cómo pueden averiguar cuántos hay exactamente?” // “How can you find out exactly how many there are?” (I can count the number of long segments and then the number of medium size segments on one long segment and then the number of tiny segments on one medium size one. Then I multiply those numbers.)
• Invite students to count and then display the expression: \(10 \times 10 \times 10\).
• “¿Cómo se relaciona la expresión con el diagrama?” // “How does the expression relate to the diagram?” (It’s the total number of tiny segments.)
• “Otra forma de escribir \(10 \times 10 \times 10\) es \(10^3\). Esta se llama una potencia de diez. El número 3 nos dice cuántos factores de 10 hay o cuántas veces multiplicamos 10 para obtener el número” // “Another way to write \(10 \times 10 \times 10\) is \(10^3\). This is called a power of ten. The number 3 tells us how many factors of 10 there are, or how many times we multiply 10 to get the number.”

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 ten 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).

• Groups of 2

• 3–5 minutes: independent work time.
• 3–5 minutes: partner discussion
• Monitor for students who:
  • call 1,000,000,000 a thousand million or a million thousands
  • make up names such as a zillion
  • know that 1,000,000,000 is called a billion

• Ask previously selected students to share their names for 1,000,000,000 in the given order.
• “¿Cuántos millones hay en este número? ¿Cómo lo saben?” // “How many millions are in this number? How do you know?” (A thousand because there are 3 more zeros and that means multiplying by 10 three times.)
• “¿Cómo se escribe este número usando potencias de 10? ¿Por qué?” // “How do you write this number using powers of 10? Why?” (\(10^9\) because there are 9 factors of 10.)
• “Este número se llama un billón” // “This number is called a billion.” 
• If desired, display the numbers 1,000,000,000 and 1,000,000,000,000 and point to each number as you explain: “En los Estados Unidos e Inglaterra, ‘1 billón’ es mil millones. Sin embargo, en México y muchos otros países, ‘1 billón’ es 1 millón de millones. En este material seguiremos la convención de los Estados Unidos” // “In the United States and England, ‘1 billion’ is one thousand millions. However, in Mexico and many other countries, ‘1 billion’ is one million millions. In this material we will follow the convention of the United States.”
• “Usando potencias de 10, ¿pueden escribir un número que sea más grande que un billón?” // “Using powers of 10, can you write a number that is bigger than a billion?” (Yes, \(10^{10}\), \(10^{11}\), \(10^{100}\).)
• “¿Por qué las potencias de 10 son útiles para representar números muy grandes?” // “Why are powers of 10 useful for representing really big numbers?” (I would not want to write 100 zeros. I would also have to count all those zeros to see how big the number is.)

The purpose of this activity is for students to find the missing number that makes multiplication equations true where that value is a power of 10. The numbers in this activity were chosen to build toward numbers that are larger than what students have worked with before. 

The synthesis highlights that these powers of 10 are represented by a 1 followed by some zeros. The power of 10 tells us how many zeros the number has.

• Groups of 2

• 1–2 minutes: quiet think time
• 6–8 minutes: partner work time
• Monitor for students who:
  • use multiplication equations, such as \(10 \times 10 = 100\), \(10 \times 100 = 1,\!000\), \(10 \times 1,\!000 = 10,\!000\), to help them find the solution to  \(\underline{\hspace{2.5cm}} \,\times 10 = 100,\!000\)
  • name or write the number 10,000,000 in a way that makes sense to them to represent the product of \(1,\!000 \times 10,\!000\). For example, they may write “one hundred one hundred thousands, 100 100,000, or \(100 \times 100,\!000\).”
  • write the number 10,000,000

If students do not write a power of ten correctly, display each power of ten as a product of tens and ask, “¿En qué se parecen estas expresiones? ¿En qué son diferentes?” // “What is the same about these expressions? What is different?”

• Ask selected students to share how they solve the equation\(\underline{\hspace{2.5cm}} \,\times10 = 100,\!000\).
• “¿Pensar en los factores de 10 les ayudó a resolver la ecuación?” // “How did thinking about factors of 10 help to solve the equation?” (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 \(1,\!000 \times 10,\!000\) in the given order.
• “¿Cómo encontraron el valor?” // “How did you find the value?” (I know 1,000 is \(10 \times 10 \times 10\) 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.)
• “¿Cómo se dice este número?” // “How do you say this number?” (10 million)
• Display the equations:
  \(10^3 = 1,\!000\)
  \(10^4 = 10,\!000\)
  \(10^7 = 10,\!000,\!000\)
• “¿Qué observan acerca de las ecuaciones?” // “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 synthesis is to share and provide some ideas of things in the world of which there might be a trillion or more.

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

• “Vamos a investigar un número grande. ¿Cuál es el número más grande que se pueden imaginar? ¿Cómo se dice? ¿Cómo se escribe en su forma numérica?” // “We are going to investigate a big number. What is the biggest number you can think of? How do you say it? How do you write it in number form?”
• Consider asking students to write in their journal and then share with a partner.

• 3 minutes: individual work time
• 3 minutes: partner discussion
• Monitor for different ideas students have for what things there may be a trillion of in the world.

• Invite students to share things they think there may be a trillion of in the world.
• “¿Cómo eligieron?” // “How did you choose?” (I picked something that there were too many to count.)
• “¿Por qué es difícil adivinar si hay un trillón de algo, como, por ejemplo, de granos de arena?” // “Why is it hard to guess whether there are a trillion of something, like grains of sand?” (It’s too big a number to count or imagine.)
• Consider sharing some things that there are around a trillion of:
  • The total number of pennies made in U.S. is about 500,000,000,000.
  • The total number of fish in the oceans is about 3,500,000,000,000.
  • The total number of trees on earth is about 3,000,000,000,000.
  • The total number of insects on earth greatly exceeds a trillion: 10,000,000,000,000,000,000.
  • The total number of ants on earth also greatly exceeds a trillion: 10,000,000,000,000,000.