This activity introduces students to scientific notation that has been generated by technology. Technology may include physical and online calculators, apps, spreadsheet programs, and other digital tools that may display very large or very small numbers.
Launch
Give students 2 minutes of quiet think time followed by a whole-class discussion.
Activity
None
Diego and Priya were calculating .
Diego used a calculator and his display read .
Priya used her knowledge of exponent rules and got .
Clare used an online calculator and the screen showed .
What do you think these different results mean?
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is for students to understand the different ways scientific notation might be expressed by technology. Invite students to share their reasoning about Diego’s result. Here are some questions for discussion:
“How are Diego’s and Priya’s answers the same?” (They both have the same value.)
“How are Diego’s and Priya’s answers different?” (They have different factors. Diego’s result has a letter in it.)
If time allows, perform the calculation from the Task Statement on a calculator for all to see. Note that different calculators and programs may display scientific notation slightly differently. For example, some technology may use E, EE, or e.
12.2
Activity
Optional
Standards Alignment
Building On
Addressing
8.EE.A.3
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than times larger.
In this optional activity, students answer questions about small and large quantities in context. They use scientific notation as a tool to describe quantities, make estimates, and make comparisons. Use this activity if students would benefit from additional practice multiplying numbers expressed in scientific notation.
Launch
Arrange students in groups of 2 to allow for partner discussions as they work. Give groups 8–10 minutes to work, followed by a brief whole-class discussion.
Activity
None
Use the table to answer questions about different creatures on the planet. Be prepared to explain your reasoning.
creature
number on planet
mass of one individual (kg)
humans
cows
sheep
chickens
ants
blue whales
Antarctic krill
zooplankton
bacteria
Which creature is least numerous? Estimate how many times more ants there are than this creature.
Which creature is the least massive? Estimate how many times more massive a human is than this creature.
Which is more massive, the total mass of all the humans or the total mass of all the ants? About how many times more massive is it?
Which is more massive, the total mass of all the krill or the total mass of all the blue whales? About how many times more massive is it?
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for students to share their reasoning and strategies for comparing numbers using scientific notation. Begin by inviting students to share their responses to each question. To involve more students in the conversation, consider asking:
“Do you agree or disagree? Why?”
“Who can restate ’s reasoning in a different way?”
“Does anyone want to add on to ’s reasoning?”
12.3
Activity
Standards Alignment
Building On
Addressing
8.EE.A.3
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than times larger.
This activity gives students an opportunity to determine and request the information needed to describe quantities, make estimates, and make comparisons.
The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.
Launch
Ask students, “How many humans do you think there are for each cat in the world?” Ask for estimates that are too high, too low, and as reasonable as possible.
Explain that large numbers like populations are often estimated using scientific notation. There are an estimated humans and an estimated cats in the world. Guide students through the example: . So there are roughly 18 humans for each cat.
Tell students that making reasonable estimates will help to answer the question in the Task Statement. For example, estimating the number of humans for each cat could have looked like . In this case, the final estimate of 20 is not far from the original estimate of 18.
Tell students that they will be making estimates like this one about humans and cats based on very large numbers written in scientific notation. Display the Info Gap graphic that illustrates a framework for the routine for all to see.
Remind students of the structure of the Info Gap routine, and consider demonstrating the protocol if students are unfamiliar with it.
Arrange students in groups of 2. In each group, give a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give students the cards for a second problem, and instruct them to switch roles.
Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Info Gap graphic visible throughout the activity or provide students with a physical copy. Supports accessibility for: Memory, Organization
Activity
None
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:
Silently read your card, and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because . . . .”
Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.
If your teacher gives you the data card:
Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!
These steps may be repeated.
Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:
“Why are some of the estimates in the class different?” (Different people used different numbers to estimate. For example some people may have estimated 1.392 as 1.4 while another person may have used 1.5.)
“How did you come up with your estimates?” (I rounded the factor between 1 and 10 to the nearest whole number or half number.)
“How did you compare powers of 10?” (Using exponent rules, I knew that to divide two powers of 10, I could find the difference between their exponents.)
“Are the estimates still useful even if they are all different?” (Yes, the estimates may not be exactly the same, but they are close, especially relative to the quantities that they are comparing.)
Lesson Synthesis
The purpose of this discussion is for students to describe a process for multiplying and dividing numbers written in scientific notation. Begin by displaying this equation for all to see: .
Ask students if they think the statement is true or false and to explain their reasoning. (The statement is true.) Next display this equation for all to see: .
Ask students if they think this statement is true or false and to explain their reasoning. (The statement is false.) Ask students how to make the statement true and record responses for all to see. (The result should be because and .)
Ask students to rewrite the result in scientific notation (.)
Ask students to describe a strategy for multiplying or dividing numbers written in scientific notation and record responses for all to see. Here are some strategies students may describe:
To find the product of two numbers written in scientific notation, multiply the two factors that are a decimal number and multiply the two powers of 10 by adding their exponents. Write the result in scientific notation by making sure the first factor is greater than or equal to 1 but less than 10 and adjusting the power of 10 accordingly.
To find the quotient of two numbers written in scientific notation, divide the two factors that are a decimal number and divide the two powers of 10 by subtracting their exponents. Write the result in scientific notation by making sure the first factor is greater than or equal to 1 but less than 10 and adjusting the power of 10 accordingly.
Student Lesson Summary
Multiplying numbers in scientific notation extends what we do when we multiply regular decimal numbers. For example, one way to find is to view 80 as 8 tens and to view 60 as 6 tens. The product is 48 hundreds or 4,800. Using scientific notation, we can write this calculation as
To express the product in scientific notation, we would rewrite it as .
Calculating using scientific notation is especially useful when dealing with very large or very small numbers. For example, there are about 39 million, or residents in California. The state has a water consumption goal of 42 gallons of water per person each day. To find how many gallons of water California would need each day if they met their goal, we can find the product , which is equal to . That’s more than 1 billion gallons of water each day.
Comparing very large or very small numbers by estimation also becomes easier with scientific notation. For example, how many ants are there for every human? There are ants and humans. To find the number of ants per human, look at . Rewriting the numerator to have the number 50 instead of 5, we get . This gives us . Since is roughly equal to 6, there are about or 6 million ants per person!
Standards Alignment
Building On
Addressing
8.EE.A.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, .
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
