Topics
2. Exponents & Scientific Notation
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United States of America
Grade 8

Investigation: Patterns in powers of 10

Lesson
Practice
Overview
Activity 1
Activity 2
Reflection

Investigate patterns in powers of 10.

Objectives
  • To identify patterns in powers of 10.
  • To use the patterns found in powers of 10 to perform calculations more quickly.
Materials
  • Pen or pencil
  • Scientific calculator
  • Downloadable assets

Patterns in powers of 10

10 to the power of 3 is equals to 10 times 10 times 10. 10 is the base, 3 is the exponent or power. 10 is multiplying itself 3 times.
When you are discussing the patterns that you are noticing, be sure to use the vocabulary below:
  • Exponential form: 10^{3}
  • Expanded form: 10 \times 10 \times 10
  • Standard form: 1000
Procedure
  • Complete row 1, row 2, and row 3 in the first downloadable asset chart.
  • Turn and talk to a partner about what patterns you noticed while completing the table for row 1, row 2, and row 3.
  • Use the pattern that you established when considering row 1, row 2, and row 3 to determine the standard form of the number in row 4. Now use the scientific calculator to verify that your answer is correct.
  • Write a "rule" that you think might work to determine the value of any number raised to the zero power and test this rule be raising bases other than 10 to the zero power.
Investigate
Consider the following questions once you have completed the above procedure.
1.
Are there any patterns that you are noticing in the chart?
2.
What is happening to the standard form when the exponent of the number written in exponential form increases?
3.
What is happening to the standard form when the exponent of the number written in exponential form decreases?
4.
Now that we have a pattern for 10^{3} to 10^{0}, what do you think 10^{-1} will be? How about 10^{-2}?

Patterns in negative powers of 10

Follow the procedure below to discover patterns in negative powers of 10.
Procedure
  • Use the same pattern you used to establish standard form in row 4 of the first chart to determine standard form in row 5, row 6 and row 7 of the second downloadable asset chart. Use a calculator to verify that your responses are correct. (Be sure to write the numbers both in fraction and decimal form).
  • Write a general rule that you could use to describe the result of raising 10 to a negative exponent.
Investigate
Consider the following questions once you have completed the above procedure.
1.
Are there any patterns that you are noticing in the chart?
2.
Now that we have a pattern for 10^{-1} to 10^{-3}, what do you think 10^{-5} will be?

Consider the following questions once you have completed the above procedure.

Discussion
1.
Can you write a general rule that you can use to calculate the value of 10^{n}, for example 10^{7}?
2.
Can you write a general rule that you can use to calculate the value of a^{0}, for example 10^{0}?
3.
Can you write a general rule that you can use to calculate the value of 10^{-n}, for example 10^{-3}?
4.
Why do you think we use exponent notation if it is just repeated multiplication?

Outcomes

8.EE.A.1

Know and apply the properties of integer exponents to generate equivalent numerical expressions.

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