Exponents and Scientific Notation

Lesson 15

Math

Unit 1

8th Grade

Lesson 15 of 15

Objective


Solve multi-step applications using scientific notation and properties of exponents.

Common Core Standards


Core Standards

  • 8.EE.A.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3-5 = 3-3 = 1/3³ = 1/27.
  • 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 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
  • 8.EE.A.4 — 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.

Criteria for Success


  1. Outline a solution pathway for a multi-step problem (MP.1).
  2. Understand the relationships between numbers in a problem in order to determine how you will use them (MP.1).
  3. Apply properties of exponents and use scientific notation to model and solve real-world problems (MP.4).
  4. Strategically use and interpret technology to support calculating with very large or very small numbers (MP.5).

Tips for Teachers


  • The Anchor Problems in this lesson are all lengthy problems which pull together concepts from the unit, specifically around scientific notation. Depending on time and class structure, there are a few different ways these problems could be used. One Anchor Problem could be selected for the whole class to engage with, or small groups could be assigned with each group getting a different problem. These problems could also be spread out over more than 1 day for students to engage with more problems, or for students to be able to share and discuss their work. As there are many different approaches students may take to solving each problem, it's worthwhile to showcase and discuss different strategies and solution pathways.
  • There is no Problem Set Guidance, as the Anchor Problem(s) is/are meant to take up the majority of the class. However, spiraled review could be prepared for students who finish early. 

Lesson Materials

  • Scientific calculator (1 per student) — These can be handheld or online.
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Anchor Problems

25-30 minutes


Problem 1

A penny is about 0.0625 of an inch thick.

a.   In 2021 there were approximately 8 billion pennies minted. If all these pennies were placed in a single stack, how many miles high would that stack be?

b.   In the past 100 years, nearly 550 billion pennies have been minted. If all these pennies were placed in a single stack, how many miles high would that stack be?

c.   The distance from the moon to the earth is about 239,000 miles. How many pennies would need to be in a stack in order to reach the moon?

Guiding Questions

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Student Response

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References

Illustrative Mathematics Pennies to heaven

Pennies to heaven, accessed on Aug. 4, 2017, 2:17 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Problem 2

This headline appeared in a newspaper.

Every day 7% of people in America eat at Giantburger restaurants.

 

Decide whether this headline is true using the following information:

  • There are about $${8\times10^3}$$ Giantburger restaurants in America.
  • Each restaurant serves on average $${2.5\times10^3}$$ people every day.
  • There are about $${3.2\times10^8}$$ people in America.

Explain your reasoning and show clearly how you figured it out.

Guiding Questions

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Student Response

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References

Illustrative Mathematics Giantburgers

Giantburgers, accessed on Aug. 4, 2017, 2:20 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 3

The average mass of an adult human is about 65 kilograms, while the average mass of an ant is approximately $${4\times10^{-3}}$$ grams. In 2010, the total human population in the world was approximately 6.84 billion, and it was estimated there were about 10,000 trillion ants alive.
 
Based on these values, how did the total mass of all living ants compare to the total mass of all living humans?

Guiding Questions

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Student Response

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References

Illustrative Mathematics Ants versus Humans

Ants versus Humans, accessed on Aug. 4, 2017, 2:22 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 4

There are approximately seven billion $${(7\times10^9)}$$ people in the world. 

In the 1990’s researchers calculated that if there were just 100 people in the world:

  • There would be 20 children
  • 25 people would not have food and shelter
  • 17 people would speak Chinese
  • 8 would speak English

a.   What fraction of people in the world did not have food and shelter? How many people in the world did not have food and shelter?

b.   How many more people in the world spoke Chinese than English? 

c.   Approximately $${3\times10^8}$$ people lived in the USA at the time of this comparison. In the world of 100 people, how many would live in the USA?

Guiding Questions

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Student Response

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References

MARS Summative Assessment Tasks for Middle School 100 People

100 People from the Summative Assessment Tasks for Middle School is made available through the Mathematics Assessment Project under the CC BY-NC-ND 3.0 license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed Aug. 4, 2017, 2:24 p.m..

Problem 5

a.   A computer has $$128$$ gigabytes of memory. One gigabyte is $${1\times10^9}$$ bytes. A floppy disk, used for storage by computers in the 1970's, holds about $$80$$ kilobytes. There are $$1,000$$ bytes in a kilobyte. How many kilobytes of memory does a modern computer have? How many gigabytes of memory does a floppy disk have? Express your answers both as decimals and using scientific notation.

b.   George told his teacher that he spent over 21,000 seconds working on his homework. Express this amount using scientific notation. What would be a more appropriate unit of time for George to use? Explain and convert to your new units.

c.   A certain swimming pool contains about $${3\times10^7}$$ teaspoons of water. Choose a more appropriate unit for reporting the volume of water in this swimming pool and convert from teaspoons to your chosen units.

d.   A helium atom has a diameter of about 62 picometers. There are one trillion picometers in a meter. The diameter of the sun is about 1,400,000 km. Express the diameter of a helium atom and of the sun in meters using scientific notation.  About many times larger is the diameter of the sun than the diameter of a helium atom?

Guiding Questions

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Student Response

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References

Illustrative Mathematics Choosing Appropriate Units

Choosing Appropriate Units, accessed on Aug. 4, 2017, 2:28 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Target Task

5-10 minutes


A new movie is being released and it is expected to be a blockbuster. Use the information below to predict how much money the movie will make in ticket prices over the opening weekend. 

  • The average movie ticket price in the country is $9.
  • There are approximately $${4\times10^4}$$ movie screens in the country.
  • The movie will be playing on 30% of the screens in the country on opening weekend.
  • For each screen, there is an average of 250 seats, and it is expected that each showing will be sold out. 
  • There are 8 showings per screen over the opening weekend.

Write your answer in scientific notation and standard form.

Student Response

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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Review of Exponents

Topic B: Properties of Exponents

Topic C: Scientific Notation

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