## Grade Eight

By the end of grade eight, students will understand
various numerical representations ,

including square roots, exponents and scientific notation; use and apply
geometric

properties of plane figures, including congruence and the Pythagorean theorem;
use

symbolic algebra to re present situations and solve problems, especially those
that

involve linear relationships ; solve linear equations, systems of linear
equations and

inequalities ; use equations, tables and graphs to analyze and interpret linear
functions;

use and understand set theory and simple counting techniques; de termine the
theoretical

probability of simple events; and make inferences from statistical data,
particularly

data that can be modeled by linear functions.

**C o n c e p t s / S k i l l t o M a i n t a i n**

Operations with rational numbers

Properties of equalities

Direct & inverse proportions

Solving multi-step equations

Properties of real numbers

Statistics

**Numbers and Operations**

Students will understand the numeric and geometric meaning of square root,

apply properties of integer exponents, and use scientific notation.

**M8N1 Students will understand different representations of numbers including**

square roots, exponents, and scientific notation.

a. Find square roots of perfect squares.

b. Recognize the (positive) square root of a number as a length of a side of a

square with a given area.

c. Recognize square roots as points and as lengths on a number line.

d. Understand that the square root of 0 is 0 and that every positive number has

two square roots that are opposite in sign.

e. Recognize and use the radical symbol to denote the positive square root of a

positive number.

f. Estimate square roots of positive numbers.

g. Simplify, add, subtract , multiply, and divide expressions containing square

roots.

h. Distinguish between rational and irrational numbers.

i. Simplify expressions containing integer exponents.

j. Express and use numbers in scientific notation.

k. Use appropriate technologies to solve problems involving square roots,
exponents,

and scientific notation.

**Geometry**

Students will use and apply geometric properties of plane figures, including
congruence

and the Pythagorean theorem.

**M8G1 Students will understand and apply the properties of parallel and**

perpendicular

lines and understand the meaning of congruence.

a. Investigate characteristics of parallel and perpendicular lines both
algebraically

and geometrically.

b. Apply properties of angle pairs formed by parallel lines cut by a
transversal.

c. Understand the properties of the ratio of segments of parallel lines cut by
one

or more transversals.

d. Understand the meaning of congruence: that all corresponding angles are
congruent

and all corresponding sides are congruent.

**M8G2 Students will understand and use the Pythagorean theorem.**

a. Apply properties of right triangles, including the Pythagorean theorem.

b. Recognize and interpret the Pythagorean theorem as a statement about areas

of squares on the sides of a right triangle.

**Algebra**

Students will use linear algebra to represent, analyze and solve problems.
They will

use equations, tables, and graphs to investigate linear relations and functions,
paying

particular attention to slope as a rate of change.

**M8A1 Students will use algebra to represent, analyze, and solve problems.**

a. Represent a given situation using algebraic expressions or equations in
one

variable .

b. Simplify and evaluate algebraic expressions.

c. Solve algebraic equations in one variable, including equations involving

absolute values.

d. Interpret solutions in problem contexts.

**M8A2 Students will understand and graph inequalities in one variable.**

a. Represent a given situation using an inequality in one variable.

b. Use the properties of inequality to solve inequalities.

c. Graph the solution of an inequality on a number line.

d. Interpret solutions in problem contexts.

**M8A3 Students will understand relations and linear functions.**

a. Recognize a relation as a correspondence between varying quantities.

b. Recognize a function as a correspondence between inputs and outputs where

the output for each input must be unique.

c. Distinguish between relations that are functions and those that are not
functions.

d. Recognize functions in a variety of representations and a variety of
contexts.

e. Use tables to describe sequences recursively and with a formula in closed
form.

f. Understand and recognize arithmetic sequences as linear functions with

whole-number input values.

h. Interpret the constant difference in an arithmetic sequence as the slope of
the

associated linear function.

i. Identify relations and functions as linear or nonlinear.

j. Translate among verbal, tabular, graphic, and algebraic representations of
functions.

**M8A4 Students will graph and analyze graphs of linear equations.**

a. Interpret slope as a rate of change.

b. Determine the meaning of the slope and y-intercept in a given situation.

c. Graph equations of the form y = mx + b.

d. Graph equations of the form ax + by = c.

e. Determine the equation of a line given a graph, numerical information that

defines the line, or a context involving a linear relationship.

f. Solve problems involving linear relationships.

**M8A5 Students will understand systems of linear equations and use them to
solve**

problems.

a. Given a problem context, write an appropriate system of linear equations.

b. Solve systems of equations graphically and algebraically, using technology as

appropriate.

c. Interpret solutions in problem contexts.

**Data Analysis and Probability**

Students will use and understand set theory and simple counting techniques;

determine the theoretical probability of simple events; and make inferences from

data, particularly data that can be modeled by linear functions.

**M8D1 Students will apply basic concepts of set theory.**

a. Demonstrate relationships among sets through use of Venn diagrams.

b. Determine subsets, complements, intersection, and union of sets.

c. Use set notation to denote elements of a set.

**M8D2 Students will determine the number of outcomes related to a given event.**

a. Use tree diagrams to find the number of outcomes.

b. Apply the addition and multiplication principles of counting.

**M8D3 Students will use the basic laws of probability .**

a. Find the probability of simple independent events.

b. Find the probability of compound independent events

**M8D4 Students will organize, interpret, and make inferences from statistical
data.**

a. Gather data that can be modeled with a linear function.

b. Estimate and determine a line of best fit from a scatter plot .

**Terms/Symbols**

square root

radical

√

rational

irrational

exponent

additive inverse

multiplicative inverse

scientific notation

significant digits

inequality

sequence

arithmetic sequence

recursive

linear function

function

relation

rate of change

slope

intercept

linear equation

linear inequality

like terms

system of linear

equations

transversal

vertical angles

complementary angles

supplementary angles

alternate interior

angles

alternate exterior

angles

corresponding angles

Pythagorean theorem

legs

hypotenuse

set

{}

element

∈

subset

complement of a set

intersection

∩

union

∪

Venn diagram

tree diagram

multiplication principle

addition principle

line of best fit

**Process Standards**

The following process standards are essential to mastering each of the
mathematics

content standards. They emphasize critical dimensions of the mathematical
proficiency

that all students need.

**M8P1 Students will solve problems (using appropriate technology).**

a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts.

c. Apply and adapt a variety of appropriate strategies to solve problems.

d. Monitor and reflect on the process of mathematical problem solving.

**M8P2 Students will reason and evaluate mathematical arguments.**

a. Recognize reasoning and proof as fundamental aspects of mathematics.

b. Make and investigate mathematical conjectures.

c. Develop and evaluate mathematical arguments and proofs.

d. Select and use various types of reasoning and methods of proof .

**M8P3 Students will communicate mathematically.**

a. Organize and consolidate their mathematical thinking through
communication.

b. Communicate their mathematical thinking coherently and clearly to peers,

teachers, and others.

c. Analyze and evaluate the mathematical thinking and strategies of others.

d. Use the language of mathematics to express mathematical ideas precisely.

**M8P4 Students will make connections among mathematical ideas and to other**

disciplines.

a. Recognize and use connections among mathematical ideas.

b. Understand how mathematical ideas interconnect and build on one another to

produce a coherent whole.

c. Recognize and apply mathematics in contexts outside of mathematics.

**M8P5 Students will represent mathematics in multiple ways.**

a. Create and use representations to organize, record, and communicate
mathematical

ideas.

b. Select, apply, and translate among mathematical representations to solve
problems.

c. Use representations to model and interpret physical, social, and mathematical

phenomena.

**Reading Across the Curriculum**

Reading Standard Comment

After the elementary years, students are seriously engaged in reading for
learning.

This process sweeps across all disciplinary domains, extending even to the area
of

personal learning. Students encounter a variety of informational as well as
fictional

texts, and they experience text in all genres and modes of discourse. In the
study

of various disciplines of learning (language arts, mathematics, science, social
studies),

students must learn through reading the communities of discourse of each of
those

disciplines. Each subject has its own specific vocabulary; and for students to
excel

in all subjects, they must learn the specific vocabulary of those subject areas
in context.

Beginning in the middle grades, students start to self-select reading materials

based on personal interests established through classroom learning. Students

become curious about science, mathematics, history, and literature as they form

contexts for those subjects related to their personal and classroom experiences.
As

students explore academic areas through reading, they develop favorite subjects

and become confident in their verbal discourse about those subjects.

Reading across curriculum content develops both academic and personal interests

in students. As students read, they develop both content and contextual
vocabulary.

They also build good habits for reading, researching, and learning. The Reading

Across the Curriculum standard focuses on the academic and personal skills
students

acquire as they read in all areas of learning.

**M8RC1 Students will enhance reading in all curriculum areas by:**

a. Reading in All Curriculum Areas

• Read a minimum of 25 grade-level appropriate books per year from a variety

of subject disciplines and participate in discussions related to curricular
learning

in all areas.

• Read both informational and fictional texts in a variety of genres and modes

of discourse.

• Read technical texts related to various subject areas.

b. Discussing books

• Discuss messages and themes from books in all subject areas.

• Respond to a variety of texts in multiple modes of discourse.

• Relate messages and themes from one subject area to messages and themes in

another area.

• Evaluate the merit of texts in every subject discipline.

• Examine author’s purpose in writing.

• Recognize the features of disciplinary texts.

c. Building vocabulary knowledge

• Demonstrate an understanding of contextual vocabulary in various subjects.

• Use content vocabulary in writing and speaking.

• Explore understanding of new words found in subject area texts.

d. Establishing context

• Explore life experiences related to subject area content.

• Discuss in both writing and speaking how certain words are subject area
related.

• Determine strategies for finding content and contextual meaning for

unknown words .