**In this unit students will: **

• distinguish between rational and irrational numbers;

• find or estimate the square and cubed root of non-negative numbers,

including 0;

• interpret square and cubed roots as both points of a line segment and lengths on a number line;

• use the properties of real numbers (commutative, associative, distributive, inverse, and

identity) and the order of operations to simplify and evaluate numeric and algebraic expressions involving integer exponents, square and cubed roots;

• work with radical expressions and approximate them as rational numbers;

• solve problems involving the volume of a cylinder, cone, and sphere;

• determine the relationship between the hypotenuse and legs of a right triangle;

__ •__ use deductive reasoning to prove the Pythagorean Theorem and its converse;

• apply the Pythagorean Theorem to determine unknown side lengths in right triangles;

• determine if a triangle is a right triangle, Pythagorean triple;

• apply the Pythagorean Theorem to find the distance between two points in a coordinate system; and

__ •__ solve problems involving the Pythagorean Theorem.

__Key Standards__

**MCC.8.G.6 **

Explain a proof of the Pythagorean Theorem and its converse

** **

**MCC.8.G.7 **

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (Note Pythagorean Theorem in three-dimensions situations –slant height)

**MCC.8.G.8 A**pply the Pythagorean Theorem to find the distance between two points in a coordinate system

**Solve real-world and mathematical problems involving volume of cylinders, cones, and**

spheres.

**MCC.8.G.9 K**now the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems

**Work with radicals and integer exponents.**

**MCC.8.EE.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

• distinguish between rational and irrational numbers;

• find or estimate the square and cubed root of non-negative numbers,

including 0;

• interpret square and cubed roots as both points of a line segment and lengths on a number line;

• use the properties of real numbers (commutative, associative, distributive, inverse, and

identity) and the order of operations to simplify and evaluate numeric and algebraic expressions involving integer exponents, square and cubed roots;

• work with radical expressions and approximate them as rational numbers;

• solve problems involving the volume of a cylinder, cone, and sphere;

• determine the relationship between the hypotenuse and legs of a right triangle;

__•__use deductive reasoning to prove the Pythagorean Theorem and its converse;

• apply the Pythagorean Theorem to determine unknown side lengths in right triangles;

• determine if a triangle is a right triangle, Pythagorean triple;

• apply the Pythagorean Theorem to find the distance between two points in a coordinate system; and

__•__solve problems involving the Pythagorean Theorem.

__Key Standards__

**Solve real-world and mathematical problems involving volume of cylinders, cones, and**

spheres.

spheres.

**Work with radicals and integer exponents.**

**MCC.8.EE.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

**CONCEPTS & SKILLS TO MAINTAIN**

• properties of similarity, congruence, and right triangles

• understand the meaning of congruence: that all corresponding angles and sides are congruent

• two figures are congruent if they have the same shape and size

• represent radical expressions in radical form (irrational) or approximate these numbers as rational

• find square roots of perfect squares

• write a decimal approximation for an irrational number to a given decimal place

• measuring length and finding perimeter and area of quadrilaterals

• characteristics of 2-D and 3-D solids

• evaluating linear and literal equations in one variable with one solution

• properties of exponents and real numbers (commutative, associative, distributive, inverse and identity) and order of operations

• express solutions using the real number system

Key Vocabulary

Key Vocabulary

•

**Altitude of a Triangle:**The perpendicular distance between a vertex of a triangle and the side opposite that vertex. Sometimes called the height of a triangle. Also, sometimes the line segment itself is referred to as the altitude.

•

**Base (of a Polygon):**For two-dimensional figures, any side can be a base. Typically,

however, the bottom side, on which the polygon ‘sits,’ is called the base.

•

**Coordinate Plane**: A two-dimensional surface on which points are plotted and located by their

*x*and

*y*coordinates.

•

**Coordinate Point of a Plane:**

A pair of numbers defining the position of a point on a two-dimensional plane.

•

**Cone:**A three dimensional figure with a circular or elliptical base and one vertex.

•

**Converse of Pythagorean Theorem:**

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, the triangle is a right triangle.

•

**Cubed Root:**One of three identical factors of a number that is the product of those

factors.

•

**Cylinder:**A three dimensional object with two parallel, congruent, circular bases.

•

**Deductive Reasoning:**The process by which one makes conclusions using known facts,

definitions, rules, or properties.

•

**Diameter:**The distance across a circle through its center. The line segment that includes the center and whose endpoints lie on the circle.

•

**Distance Formula:**An application of the Pythagorean Theorem based on the distance between two points.

•

**Geometric Solid:**The collective term of all bounded three dimensional geometric figures.

•

**Height of Solids:**The vertical height (or altitude) which is the perpendicular distance

from the top down to the base.

•

**Hypotenuse:**The hypotenuse is the side of a right triangle that is directly across from the right angle.

•

**Irrational:**A real number whose decimal form is non-terminating and non-repeating that cannot be written as the ratio of two integers.

•

**Leg of a Triangle:**Either of the two shorter sides of a right triangle. These two sides

together form the right angle in the right triangle.

•

**Literal Equation:**An equation with more than one variable. Formulas are literal equations.

•

**Perfect Squares:**The product of a rational number multiplied by itself.

•

**Perfect Cubes:**In arithmetic and algebra, the cube of a number

*n*is its third power. It is the result of the number multiplying by itself three times.

•

**Pythagorean Theorem:**A theorem that states that in a right triangle, the square of the

length of the hypotenuse equals the sum of the squares of the lengths of the legs.

•

**Pythagorean Triples:**A set of positive integers,

*a*,

*b,*and

*c*that fit the rule

*a squared + b squared = c squared.*

•

**Radius:**The distance from the center of a circle to any point on the circle. Also, the line segment that has the center of the circle as one endpoint and a point on the circle as the other endpoint.

•

**Radical:**A symbol that is used to indicate square roots.

•

**Rational Number:**A number expressible in the form a/b or – a/b for some fraction a/b.

The rational numbers include the integers.

•

**Right Triangle:**A triangle with exactly one right angle.

•

**Sphere:**The set of all points in space that are equidistant from a fixed point, called the

center.

•

**Square Root:**The square root of a number is a special value that, when multiplied by itself, gives the number.

•

**Volume:**The amount of space occupied by an object.