Exponents and Scientific Notation

Students learn to simplify complex-looking exponential expressions, and they learn efficient ways to describe, communicate, and operate with very large and very small numbers.

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Unit Summary

In Unit 1, eighth grade students learn how complex-looking expressions and very large or small numbers can be represented in simpler ways. Through investigation, students discover ways to write equivalent exponential expressions, and then formalize their understanding of these strategies into properties of exponents. Later in the unit, they learn efficient ways to describe, communicate, and operate with very large and very small numbers. Though there are many procedural elements in this unit, underneath these procedures are strong conceptual understandings. Throughout the unit, students look for structures and patterns that exist in exponential terms and powers of ten, and use those structures and patterns to make generalizations (MP.7 and MP.8).

In sixth grade, students wrote and evaluated expressions with exponents using the order of operations. They identified the parts of an expression, distinguishing a term from a factor from a coefficient. In eighth grade, students expand on these skills to go beyond just evaluation. They are presented with exponentials such as $$\frac{3^{16}}{3^4}$$ or $$(x^2y)^5$$ and are asked to simplify them or represent them in equivalent ways. In this way, students hone their abilities to manipulate algebraic expressions, which they will continue to do in future units in eighth grade. In fourth and fifth grades, students investigated patterns in powers of ten and how those patterns related to place value. In this unit, students will access these prior concepts and use them in representing and working with very large and small numbers.

In high school, students will need a strong understanding of exponents and exponent properties. They will apply the properties of exponents to exponential equations in order to reveal new understandings of the relationship. They will work with fractional exponents and discover the properties of rational exponents and rational numbers. In general, students’ ability to see the structure in an expression will support them in manipulating quadratic functions, operating with polynomials, and making connections between various relationships.

Pacing: 19 instructional days (15 lessons, 3 flex days, 1 assessment day)

For guidance on adjusting the pacing for the 2021-2022 school year, see our 8th Grade Scope and Sequence Recommended Adjustments.

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Assessment

This assessment accompanies Unit 1 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

Internalization of Standards via the Post-Unit Assessment

Take the Post-Unit Assessment. Annotate for:
- Standards that align to each question
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) to which the Assessment points

Internalization of Trajectory of Unit

Read and annotate the Unit Summary.
Notice the progression of concepts through the unit using the Lesson Map.
Do all Target Tasks. Annotate the Target Tasks for:
- Essential Understandings
- Connection to Post-Unit Assessment questions
Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

Unit-Specific Intellectual Prep

Read The Progressions for the Common Core State Standards in Mathematics, 6-8 Expressions and Equations, on the standards relevant to this unit

Essential Understandings

The properties of exponents summarize the structure and patterns that are inherent in exponential expressions. They are rooted in the conceptual meaning of exponents and provide a summary of how to efficiently work with exponential expressions.
Scientific notation is a useful tool to conceptualize, communicate about, and operate with very large and very small numbers.
The properties of exponents can be applied to numbers written in scientific notation to support efficient computation and comparison.
Looking for structure and patterns in expressions can provide insight into how to manipulate and work with complex expressions, as well as lead to generalizations and properties.

Vocabulary

exponential expression

base

exponent

power

scientific notation

properties of exponents

standard form/decimal form

To see all the vocabulary for this course, view our 8th Grade Vocabulary Glossary.

Materials

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

To see more information about the materials in this unit, view the Unit Materials Overview.

Lesson Map

Topic A: Review of Exponents

1

8.EE.A.1

Review exponent notation and identify equivalent exponential expressions.

2

8.EE.A.1

Evaluate numerical and algebraic expressions with exponents using the order of operations.

3

8.EE.A.1

Investigate patterns of exponents with positive/negative bases and even/odd bases.

Topic B: Properties of Exponents

4

8.EE.A.1

Investigate exponent patterns to write equivalent expressions.

5

8.EE.A.1

Apply the product of powers rule and the quotient of powers rule to write equivalent, simplified exponential expressions.

6

8.EE.A.1

Apply the power of powers rule and power of product rule to write equivalent, simplified exponential expressions.

7

8.EE.A.1

Reason with zero exponents to write equivalent, simplified exponential expressions.

8

8.EE.A.1

Reason with negative exponents to write equivalent, simplified exponential expressions.

9

8.EE.A.1

Simplify and write equivalent exponential expressions using all exponent rules.

Topic C: Scientific Notation

10

8.EE.A.3

8.EE.A.4

Write large and small numbers as powers of 10.

11

8.EE.A.3

Define and write numbers in scientific notation.

12

8.EE.A.3

8.EE.A.4

Compare numbers written in scientific notation.

13

8.EE.A.4

Multiply and divide with numbers in scientific notation. Interpret scientific notation on calculators.

14

8.EE.A.4

Add and subtract with numbers in scientific notation.

15

8.EE.A.1

8.EE.A.3

8.EE.A.4

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

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

Expressions and Equations

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.

Expressions and Equations

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.

Expressions and Equations

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.

Expressions and Equations

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.

Foundational Standards

Expressions and Equations

6.EE.A.1

Expressions and Equations

6.EE.A.1 — Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2

Expressions and Equations

6.EE.A.2 — Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.A.2.C

Expressions and Equations

6.EE.A.2.C — Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.

Number and Operations in Base Ten

4.NBT.A.1

Number and Operations in Base Ten

4.NBT.A.1 — Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
5.NBT.A.1

Number and Operations in Base Ten

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.
5.NBT.A.2

Number and Operations in Base Ten

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 Number System

7.NS.A.2

The Number System

7.NS.A.2 — Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Future Standards

Arithmetic with Polynomials and Rational Expressions

A.APR.D.6

Arithmetic with Polynomials and Rational Expressions

A.APR.D.6 — Rewrite simple rational expressions in different forms; write ^a(x/_b(x) in the form q(x) + ^r(x)/_b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
A.APR.D.7

Arithmetic with Polynomials and Rational Expressions

A.APR.D.7 — Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

High School — Number and Quantity

N.RN.A.1

High School — Number and Quantity

N.RN.A.1 — Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)³ = 5(^1/3)³ to hold, so (5^1/3)³ must equal 5.
N.RN.A.2

High School — Number and Quantity

N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Seeing Structure in Expressions

A.SSE.A.2

Seeing Structure in Expressions

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ — y⁴ as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
A.SSE.B.3.C

Seeing Structure in Expressions

A.SSE.B.3.C — Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.151/12)^12t 1.012^12t to reveal the approximate equivalent monhly interest rate if the annual rate is 15%.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Exponents and Scientific Notation

Unit Summary

Assessment

Unit Prep

Intellectual Prep

Internalization of Standards via the Post-Unit Assessment

Internalization of Trajectory of Unit

Unit-Specific Intellectual Prep

Essential Understandings

Vocabulary

Materials

Lesson Map

Topic A: Review of Exponents

1

2

3

Topic B: Properties of Exponents

4

5

6

7

8

9

Topic C: Scientific Notation

10

11

12

13

14

15

Common Core Standards

Core Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Foundational Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Number and Operations in Base Ten

Number and Operations in Base Ten

Number and Operations in Base Ten

Number and Operations in Base Ten

The Number System

The Number System

Future Standards

Arithmetic with Polynomials and Rational Expressions

Arithmetic with Polynomials and Rational Expressions

Arithmetic with Polynomials and Rational Expressions

High School — Number and Quantity

High School — Number and Quantity

High School — Number and Quantity

Seeing Structure in Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Standards for Mathematical Practice

Unit 2

Solving One-Variable Equations