Note: Click any standard to move it to the center of the map.
[7.NS.A.3] -
Solve real-world and mathematical problems involving the four operations with integers and other rational numbers.
Mathematics | Grade : 8
Domain - Expressions and Equations
Cluster - Work with radicals and integer exponents.
[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.
- Irrational number
A number that cannot be expressed as a quotient of two integers, e.g.,√2 . It can be shown that a number is irrational if and only if it cannot be written as a repeating or terminating decimal. - Rational number
A number expressible in the form a∕b or – a∕b for some fraction a∕b. The rational numbers include the integers.
[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 better approximations.
[8.G.B.6] -
[See 8.G.B.6.a and 8.G.B.6.b.]
[8.G.C.9] -
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
[AI.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 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
[AI.N-RN.A.2] -
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
[AI.N-RN.B.3] -
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
[AI.A-CED.A.1] -
Create equations and inequalities in one variable and use them to solve problems. (Include equations arising from linear, quadratic, and exponential functions with integer exponents.)*
[AI.A-REI.B.4.a] -
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
[AI.A-REI.B.4.b] -
Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the solutions of a quadratic equation results in non-real solutions and write them as a ± bi for real numbers a and b.
[MI.A-CED.A.1] -
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and exponential functions with integer exponents.*
[MII.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 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
[MII.N-RN.A.2] -
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
[MII.N-RN.B.3] -
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
[MII.A-REI.B.4.a] -
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
[MII.A-REI.B.4.b] -
Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
[HS.PHY.3.1] -
Use algebraic expressions and the principle of energy conservation to calculate the change in energy of one component of a system when the change in energy of the other component(s) of the system, as well as the total energy of the system including any energy entering or leaving the system, is known. Identify any transformations from one form of energy to another, including thermal, kinetic, gravitational, magnetic, or electrical energy, in the system. Clarification Statement: Systems should be limited to two or three components and to thermal energy; kinetic energy; or the energies in gravitational, magnetic, or electric fields.