The Number System
Grade 8 Math
Unit Length and Description:
The year begins with students extending their understanding of whole number exponents learned in previous grades to integer exponents in Unit 1. Students develop the properties of exponents. The unit progresses to include work with scientific notation which supports student understanding of positive and negative exponents. Students also work with radicals as they use square roots and cube roots to represent solutions to equations of the form x2=p and x3=p. They evaluate square roots and cube roots of small perfect squares and cubes. Later in the unit, in preparation for work with geometric contexts such as the Pythagorean Theorem in Unit 3 of this course, students extend their concept of numbers beyond the system of rationals to include irrational numbers as they estimate roots of non-perfect squares. They represent these numbers with radical expressions and approximate these numbers with rationals. Students use the number line model to support their understanding of the rational numbers, along with the inclusion of the irrational numbers, to develop the real number line.
8.NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
8.NS.A.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., 𝜋2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get a better approximation.
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.2 Use square root and cube root symbols to represent solutions to equations of the form 𝑥2 = 𝑝 and 𝑥3 = 𝑝, where 𝑝 is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
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.
Focus Standards of Mathematical Practice:
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:
I can define rational and irrational numbers.
I can show that the decimal expansion of rational numbers repeats eventually.
I can convert a decimal expansion which repeats eventually into a rational number.
I can show informally that every number has a decimal expansion.
I can approximate irrational numbers as rational numbers.
I can approximately locate irrational numbers on a number line.
I can estimate the value of expressions involving irrational numbers using rational numbers.
Examples: Being able to determine the value of the √2 on a number line lies between 1 and 2, more accurately, between 1.4 and 1.5, and more accurately, etc.
I can compare the size of irrational numbers using rational approximations.
I can explain why a zero exponent produces a value of one.
I can explain how a number raised to an exponent of -1 is the reciprocal of that number.
I can explain the properties of integer exponents to generate equivalent numerical expressions.
Exponents of the same base.
2. Dividing Exponents of the same base.
For example, 3²/ 3-1 = 33 = 27
3. Expand an exponent by another exponent.
For example (2²)³ = 64
I can use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 =p, where p is a positive rational number.
I can evaluate square roots of small perfect squares.
I can evaluate cube roots of small perfect cubes.
I can understand that the square root of 2 is irrational.
I can express numbers as a single digit times an integer power of 10 (scientific notation).
I can use scientific notation to estimate very large and/or very small quantities.
I can multiply and divide numbers in scientific notation in order to compare growth or decay.
I can compare quantities to express how much larger one is compared to the other.
I can perform operations using numbers expressed in scientific notation.
I can use scientific notation to express very large and very small quantities.
I can interpret scientific notation that has been generated by technology.
I can choose appropriate units of measure when using scientific notation.
· The number system consists of numbers that are rational and irrational.
· Irrational numbers can be represented on a real number line.
· Every number has a decimal expansion. The decimal expansion of a rational number either terminates or repeats. The decimal expansion of an irrational numbers does not terminate or repeat.
· The value of any real number can be represented in relation to other real numbers such as with decimals converted to fractions and numbers written as exponents or radical.
· Properties of operations with whole and rational numbers also apply to real numbers.
· The properties of integer exponents are used to simplify expressions containing integer exponents.
· Numbers can be expressed in scientific notation to compare very large and very small quantities and to perform computations with those numbers.
· Perfect square roots and perfect cube roots can be written as rational numbers.
· Non-perfect square roots and non-perfect cube roots are irrational, but can be approximated as rational numbers.
· Expressions are powerful tools for exploring, reasoning about, and representing situations.
· Both rational and irrational numbers can be represented on a real number line.
· Why is it helpful to write numbers in different ways?
· How can you determine the difference in a rational and an irrational number?
· How can you evaluate positive exponents?
· How can you evaluate negative exponents?
· How can you develop and use the properties of integer exponents?
· How can you use scientific notation to express very large and very small quantities?
· Why are quantities represented in multiple ways?
· How is the universal nature of properties applied to real numbers?
· What is a square root?
· What is a cube root?
· What is the difference in a finite decimal and an infinite decimal?
· Can a radical be used to represent a finite decimal?
· Can a radical be used to represent an infinite decimal?