**Number and
Operations and Algebra: Developing quick recall of
multiplication facts and related division facts and fluency
with whole number multiplication.**

Students use understandings of multiplication to
develop quick recall of the basic multiplication facts and
related division facts. They apply their understanding of
models for multiplication (i.e., equal sized groups, arrays,
area models, equal intervals on the number line), place
value, and properties of operations (in particular, the
distributive property) as they develop, discuss, and use
efficient, accurate, and generalizable methods to multiply
multidigit whole numbers. They select appropriate methods
and apply them accurately to estimate products or calculate
them mentally, depending on the context and numbers
involved. They develop fluency with efficient procedures,
including the standard algorithm, for multiplying whole
numbers, understand why the procedures work (on the basis of
place value and properties of operations), and use them to
solve problems.

**Algebra:**
Students continue identifying, describing, and extending
numeric patterns involving all operations and nonnumeric
growing or repeating patterns. Through these experiences,
they develop an understanding of the use of a rule to
describe a sequence of numbers or objects.

- Grade 5 Curriculum Focal Points

**Number and
Operations and Algebra: Developing an understanding of and
fluency with division of whole numbers.**

Students apply their understanding of models for
division, place value, properties, and the relationship of
division to multiplication as they develop, discuss, and use
efficient, accurate, and generalizable procedures to find
quotients involving multidigit dividends. They select
appropriate methods and apply them accurately to estimate
quotients or calculate them mentally, depending on the
context and numbers involved. They develop fluency with
efficient procedures, including the standard algorithm, for
dividing whole numbers, understand why the procedures work
(on the basis of place value and properties of operations),
and use them to solve problems. They consider the context in
which a problem is situated to select the most useful form
of the quotient for the solution, and they interpret it
appropriately.

**Algebra:**
Students use patterns, models, and relationships as contexts
for writing and solving simple equations and inequalities.
They create graphs of simple equations. They explore prime
and composite numbers and discover concepts related to the
addition and subtraction of fractions as they use factors
and multiples, including applications of common factors and
common multiples. They develop an understanding of the order
of operations and use it for all operations.

- Grade 6 Curriculum Focal
Points

**Algebra: Writing,
interpreting, and using mathematical expressions and
equations.**

Students write mathematical expressions and
equations that correspond to given situations, they evaluate
expressions, and they use expressions and formulas to solve
problems. They understand that variables represent numbers
whose exact values are not yet specified, and they use
variables appropriately. Students understand that
expressions in different forms can be equivalent, and they
can rewrite an expression to represent a quantity in a
different way (e.g., to make it more compact or to feature
different information). Students know that the solutions of
an equation are the values of the variables that make the
equation true. They solve simple one-step equations by using
number sense, properties of operations, and the idea of
maintaining equality on both sides of an equation. They
construct and analyze tables (e.g., to show quantities that
are in equivalent ratios), and they use equations to
describe simple relationships (such as 3x = y) shown in a
table.

**Algebra:**
Students use the commutative, associative, and distributive
properties to show that two expressions are equivalent. They
also illustrate properties of operations by showing that two
expressions are equivalent in a given context (e.g.,
determining the area in two different ways for a rectangle
whose dimensions are x + 3 by 5). Sequences, including those
that arise in the context of finding possible rules for
patterns of figures or stacks of objects, provide
opportunities for students to develop formulas.

- Grade 7 Curriculum Focal Points

**Number and
Operations and Algebra and Geometry: Developing an
understanding of and applying proportionality, including
similarity.**

Students extend their work with ratios to develop
an understanding of proportionality that they apply to solve
single and multistep problems in numerous contexts. They use
ratio and proportionality to solve a wide variety of percent
problems, including problems involving discounts, interest,
taxes, tips, and percent increase or decrease. They also
solve problems about similar objects (including figures) by
using scale factors that relate corresponding lengths of the
objects or by using the fact that relationships of lengths
within an object are preserved in similar objects. Students
graph proportional relationships and identify the unit rate
as the slope of the related line. They distinguish
proportional relationships (y/x = k, or y = kx) from other
relationships, including inverse proportionality (xy = k, or
y = k/x).

**Measurement and
Geometry and Algebra: Developing an understanding of and
using formulas to determine surface areas and volumes of
three-dimensional shapes.**

By decomposing two- and three-dimensional shapes
into smaller, component shapes, students find surface areas
and develop and justify formulas for the surface areas and
volumes of prisms and cylinders. As students decompose
prisms and cylinders by slicing them, they develop and
understand formulas for their volumes (Volume = Area of base
× Height). They apply these formulas in problem solving to
determine volumes of prisms and cylinders. Students see that
the formula for the area of a circle is plausible by
decomposing a circle into a number of wedges and rearranging
them into a shape that approximates a parallelogram. They
select appropriate two- and three dimensional shapes to
model real-world situations and solve a variety of problems
(including multistep problems) involving surface areas,
areas and circumferences of circles, and volumes of prisms
and cylinders.

**Number and
Operations and Algebra: Developing an understanding of
operations on all rational numbers and solving linear
equations.**

Students extend understandings of addition,
subtraction, multiplication, and division, together with
their properties, to all rational numbers, including
negative integers. By applying properties of arithmetic and
considering negative numbers in everyday contexts (e.g.,
situations of owing money or measuring elevations above and
below sea level), students explain why the rules for adding,
subtracting, multiplying, and dividing with negative numbers
make sense. They use the arithmetic of rational numbers as
they formulate and solve linear equations in one variable
and use these equations to solve problems. Students make
strategic choices of procedures to solve linear equations in
one variable and implement them efficiently, understanding
that when they use the properties of equality to express an
equation in a new way, solutions that they obtain for the
new equation also solve the original equation.

- Grade 8 Curriculum Focal Points

**Algebra: Analyzing and
representing linear functions and solving linear equations
and systems of linear equations.**

Students use linear functions, linear equations, and
systems of linear equations to represent, analyze, and solve
a variety of problems. They recognize a proportion (y/x = k,
or y = kx) as a special case of a linear equation of the
form y = mx + b, understanding that the constant of
proportionality (k) is the slope and the resulting graph is
a line through the origin. Students understand that the
slope (m) of a line is a constant rate of change, so if the
input, or x-coordinate, changes by a specific amount, a, the
output, or y-coordinate, changes by the amount ma. Students
translate among verbal, tabular, graphical, and algebraic
representations of functions (recognizing that tabular and
graphical representations are usually only partial
representations), and they describe how such aspects of a
function as slope and y-intercept appear in different
representations. Students solve systems of two linear
equations in two variables and relate the systems to pairs
of lines that intersect, are parallel, or are the same line,
in the plane. Students use linear equations, systems of
linear equations, linear functions, and their understanding
of the slope of a line to analyze situations and solve
problems.

**Geometry and Measurement:
Analyzing two- and three-dimensional space and figures by
using distance and angle.**

Students use fundamental facts about distance and angles
to describe and analyze figures and situations in two- and
three-dimensional space and to solve problems, including
those with multiple steps. They prove that particular
configurations of lines give rise to similar triangles
because of the congruent angles created when a transversal
cuts parallel lines. Students apply this reasoning about
similar triangles to solve a variety of problems, including
those that ask them to find heights and distances. They use
facts about the angles that are created when a transversal
cuts parallel lines to explain why the sum of the measures
of the angles in a triangle is 180 degrees, and they apply
this fact about triangles to find unknown measures of
angles. Students explain why the Pythagorean theorem is
valid by using a variety of methods—for example, by
decomposing a square in two different ways. They apply the
Pythagorean theorem to find distances between points in the
Cartesian coordinate plane to measure lengths and analyze
polygons and polyhedra.

**Data Analysis and Number and Operations and Algebra:
Analyzing and summarizing data sets.**

Students use descriptive statistics, including mean, median,
and range, to summarize and compare data sets, and they
organize and display data to pose and answer questions. They
compare the information provided by the mean and the median
and investigate the different effects that changes in data
values have on these measures of center. They understand
that a measure of center alone does not thoroughly describe
a data set because very different data sets can share the
same measure of center. Students select the mean or the
median as the appropriate measure of center for a given
purpose.

**Algebra:**
Students encounter some nonlinear functions (such as the
inverse proportions that they studied in grade 7 as well as
basic quadratic and exponential functions) whose rates of
change contrast with the constant rate of change of linear
functions. They view arithmetic sequences, including those
arising from patterns or problems, as linear functions whose
inputs are counting numbers. They apply ideas about linear
functions to solve problems involving rates such as motion
at a constant speed.