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 represent 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; determine 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 .
