The goal of this Warm-up is for students to visualize 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, which 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 the number of segments of each length, they are observing and making use of the repeated structure of 10 segments joining at the different vertices (MP7, MP8).
Launch
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
Activity
Display the image.
“Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
1 minute: partner discussion
Record responses.
¿Cuántos ves? ¿Cómo lo sabes?, ¿qué ves?
Student Response
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Advancing Student Thinking
Activity Synthesis
Invite students to share their estimates for how many of the smallest line segments 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 1 long segment and then the number of tiny segments on 1 medium-size segment. Then I multiply those numbers.)
Invite students to count, and then display the expression: .
“¿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 es . 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 is . 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.”
Activity 1
Standards Alignment
Building On
Addressing
5.NBT.A.1
Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
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).
MLR2 Collect and Display. Circulate to listen for and collect the language students use as they work with exponential notation to represent large numbers. On a visible display, record words and phrases such as: million, billion, thousand, power of 10, exponential notation, represent, times, multiply by 10, number of zeros. Invite students to borrow language from the display as needed, and update it throughout the lesson. Advances: Conversing, Reading
Launch
Groups of 2
Activity
3–5 minutes: independent work time.
3–5 minutes: partner discussion
Monitor for students who:
Call 1,000,000,000 ”one thousand million” or “one million thousand.”
Make up names, such as “one zillion.”
Know that 1,000,000,000 is called “one billion.”
En Delaware viven aproximadamente 1,000,000 de personas.
¿Cómo se dice este número?
¿Cuántos miles es 1,000,000? Explica o muestra cómo razonaste.
Usa potencias de 10 para escribir el número.
Observa el diagrama del calentamiento. Si cada copo de nieve está formado por varios copos de nieve más pequeños, ¿cuántas veces tendrías que ampliar el diagrama para ver 1,000,000 de segmentos diminutos? Explica o muestra cómo razonaste.
En 2023, la población de los Estados Unidos era aproximadamente un tercio de 1,000,000,000.
¿Cómo dirías 1,000,000,000?
¿Cuántos millones es 1,000,000,000? ¿Cuántos miles es eso? Explica o muestra cómo razonaste.
Usa potencias de 10 para escribir el número.
Observa el diagrama del calentamiento. Si cada copo de nieve está formado por varios copos de nieve más pequeños, ¿cuántas veces tendrías que ampliar este diagrama para ver 1,000,000,000 de segmentos diminutos? Explica o muestra cómo razonaste.
Student Response
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Advancing Student Thinking
Activity Synthesis
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?” (1 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?” (, because there are 9 factors of 10.)
“Este número se llama ‘un billón’” // “This number is called ‘one 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 1 billón?” // “Using powers of 10, can you write a number that is bigger than 1 billion?” (Yes, , , .)
“¿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 also would have to count all those zeros to find out the number.)
Activity 2
Standards Alignment
Building On
Addressing
5.NBT.A.1
Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
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.
Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Record responses on a display, and keep it visible during the activity. Supports accessibility for: Organization, Social-Emotional Functioning
Launch
Groups of 2
Activity
1–2 minutes: quiet think time
6–8 minutes: partner work time
Monitor for students who:
Use multiplication equations, such as , , and , to help them find the solution to .
Name or write the number 10,000,000, in a way that makes sense to them, to represent the product of . For example, they may write “one hundred one hundred thousands,” “100 100,000,” or “.”
Write the number 10,000,000.
En cada caso, encuentra el número desconocido que hace que la ecuación sea verdadera. Explica cómo razonaste.
¿Cómo te ayudan los productos de 10 a resolver estos problemas?
Escribe cada potencia de 10 como un número.
Student Response
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Advancing Student Thinking
If students write too many zeros when they are representing a power of 10, consider asking:
“¿Cómo decidiste escribir ese número?” // “How did you decide to write that number?”
Display each power of 10 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?”
Activity Synthesis
Ask selected students to share how they found the unknown number that makes the equation true.
“¿Cómo les ayudó pensar en los factores de 10 a encontrar el número desconocido que hace que la ecuación sea verdadera?” // “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.
“¿Cómo encontraron el valor?” // “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.)
“¿Cómo se dice este número?” // “How do you say this number?” (10 million)
Display the equations:
“¿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.)
Activity 3
Optional
Standards Alignment
Building On
Addressing
5.NBT.A.2
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers 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).
Launch
Groups of 2
“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.
Activity
3 minutes: individual work time
3 minutes: partner discussion
Monitor for different ideas students have for what things in the world may number in the trillions.
¿Cómo dirías el número 1,000,000,000,000?
¿Cuántos billones es 1,000,000,000,000? ¿Cuántos millones? Explica o muestra cómo razonaste.
Usa potencias de 10 para escribir el número.
Escribe un ejemplo de algo de lo que haya 1,000,000,000,000 en el mundo.
Student Response
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Advancing Student Thinking
Activity Synthesis
Invite students to share things in the world that they think may number in the trillions.
“¿Cómo eligieron?” // “How did you choose?” (I picked something that there were too many to count.)
“¿Por qué es difícil adivinar si hay 1 trillón de algo, como, por ejemplo, de granos de arena?” // “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.)
Lesson Synthesis
“Hoy exploramos algunos números muy grandes que son potencias de 10 y los representamos usando exponentes” // “Today we looked at some really big numbers that are powers of 10 and represented them, using exponents.”
Display the image from the Warm-up.
“¿Cuántos segmentos de tamaño mediano hay?” // “How many of the medium-size segments are there?” (100) “¿Qué expresión pueden escribir para representar el número de estos segmentos?” // “What expression could you write to represent the number of these segments?” ()
“¿Cómo está representada la expresión en la imagen?” // “How does the image represent the expression ?” (There are 10 groups of these segments and 10 segments in each group.)
Display .
“También podemos representar esta expresión con una potencia de 10” // “We also can represent this expression with a power of 10.”
Refer to the smallest line segments in the Warm-up image.
“¿Qué expresión pueden escribir para representar el número de segmentos pequeños? ¿Cómo lo saben?” // “What expression could you write to represent the number of small segments? How do you know?” ( or because there are 10 more of these for each of the medium-size segments.)
Display .
“¿Qué potencia de 10 podemos escribir para que esta ecuación sea verdadera?” // “What power of 10 can we write to make this equation true?” ()
“Si continuaran y dibujaran 10 segmentos diminutos más, ¿cuántos habría?” // “If you kept going and drew 10 more tiny segments, how many of these would there be?” (10,000)
“¿Cómo podemos escribir el número de segmentos como una potencia de 10?” // “How can we write the number as a power of 10?” ( since there would be another factor of 10.)
“Si continuara el proceso y lo hiciera 6 veces en total, ¿cuántos segmentos de los más pequeños habría?” // “If I kept going a total of 6 times, how many of the smallest segments would there be?” (1 million or .)
Standards Alignment
Building On
Addressing
Building Toward
5.NBT.A.2
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
