Year 2:

**Algebra/number/function:** Students will recognize equations that can be
considered identities.

Furthermore, they will use area models to illustrate the distributive property
and , in particular, justify the

identity that (x+y)^{2}=x^{2}+2xy+y^{2}. They will explore situations where one variable
is directly proportional

to another (y=kx) and situations where one variable varies as the square of
another (y=kx^{2}). Inverse

variation is also explored. They will find the constant of proportionality,
write an equation, and use it to

find additional information . Students will solve for a missing term in a
proportion using the technique of

cross-multiplying. They will be able to solve equations involving the
Pythagorean Theorem and use a

calculator to find the square root of a number, when needed. Students will
understand the meaning of

compound inequalities such as 2<x<5 and be able to separate them into two
inequalities and vice versa.

They will explore functions defined by multiple rules such as piecewise linear
and step functions . They

will have experience describing real situations with three variables and
identify independent and

dependent variables. In particular, they will explore real- valued functions of
two variables. With tables

and formulas. They will be familiar with notation such as T(x,y) to represent
the value of a function with

independent variables x and y. They will have a strategy
for envisioning such relationships involving

three variables using a 2-dimensional graph instead of a 3-dimensional graph,
especially when the

variables are discrete. The strategy involves plotting " contour lines" of the
graph determined by fixing

one of the variables at various points and graphing the resulting relation.
Using complete graphical

information obtained from this process, for example, students will be able to
determine whether or not

the function of two variables is linear. They will know what to look for
symbolically to determine

whether or not a function of any number of variables is linear. Also, here, a
matrix is defined as a

function of two variables where the independent variables represent values in
the set {1,2,..,n} for some

particular n. Students will be able to distinguish between categorical and
measurement variables. They

will have experience making some theoretical models and comparing the model to
actual data. They will

understand the term "residual" as the actual value minus the predicted value.
They will explore

"additive" models of functions versus "interactive" models of functions of two
variables (where there is

usually a term containing the product of the two independent variables.)

Students will be able to identify linear equations in two variables in several
symbolic forms such as

y=2x+1 and 2x+3y=6. They will be able to solve for one variable in terms of
another in these situations.

They will have experience solving systems in two unknowns using elimination. In
fact, they will be able

to work with the Gaussian Elimination algorithm and back substitution to solve
systems of liner equations

involving up to three variables. They will be able to represent such linear
systems using matrices and be

able to apply Gaussian elimination using row operations both by hand and using
technology. They will

be able to determine when this process reveals a system which has no solution ,
infinitely many solutions,

and a unique solution. They will be able to interpret the latter situation
geometrically with two and three

unknowns. Students will use these concepts in many applied settings. For
instance, within mathematics,

they will have experience "fitting" an equation of the form (x-a)^{2}+(y-b)^{2}=c to
three data points in the

plane by solving a system of linear equations in three unknowns using Gaussian
Elimination and back

substitution. By graphing the equation, they will see this as determining a
circle through the three points.

They will also have experience fitting a parabola to some sets of three points
using elimination. Students

will also be able to use matrices for organizing, sorting, and manipulating
information. They will have

contextual reasons for defining the addition, subtraction, and multiplication of
matrices, and know when

these operations can be performed. They will have seen and worked with some
abstract properties of

operations with matrices, e.g. commutativity of matrix addition, but not
multiplication; associativity of

addition and multiplication; and properties of the "zero" matrix (all entries
are zero) with addition and

identity matrices (ones along the diagonal and zeros otherwise) in matrix
multiplication. In addition, they

will understand and be able to use the concept of multiplying a matrix by a
scalar. Students will have

experience working with the least squares computations by hand and determining
the "coefficient of

determination" (usually called r^{2}). They will be introduced to ∑-notation. They
will know the basic

shapes of quadratic polynomials and at least the shape of the cubic y=x^{3}. They
will see what a, b, and c

do in each of the cases y=ax^{2} ; y=x^{2}+b ; y=(x+c)^{2} ; and y=x^{2} +cx, for at least
n=2.

**Geometry:** Students will realize the advantages and disadvantages of modeling
real-world situations; in

particular, they will distinguish between shortest distances on a map vs. a
globe. They will be able to

identify units of measure for length and time, measure lengths using standard
and nonstandard units, and

recognize the importance of standard units of measure. Students will be able to
compute the perimeter

of equilateral polygons using the formula P =ns, as well as
perimeters of other regions (e.g. rectangles,

triangles). They will identify polygons and name their vertices, sides, and
angles. Using a compass and

straightedge, they will construct the perpendicular bisector of a line segment
and the perpendicular from

a line to a point not on the line. They will also be able to construct a
triangle, given its side

measurements, and they will be introduced to the triangle inequality when they
are given measurements

that cannot form a triangle. They will construct an isosceles triangle, as well
as the bisector of its vertex

angle. They will be able to determine the axes of symmetry for a figure,
including an equilateral triangle,

rhombus, rectangle, square, and other regular polygons. They will investigate
the property that the

diagonals of a rhombus are perpendicular bisectors of each other. They will
classify certain

quadrilaterals (parallelogram, rhombus, rectangle, square) according to sides
and angles. Students will

investigate strategies for finding the area of a region, including the use of
formulas for a rectangle, a

triangle and a parallelogram, "tiling" a region and counting unit squares, and
the "divide and conquer"

algorithm which divides a region into smaller areas that can be found and summed
to yield total area.

Moreover, they will realize that the area of a polygon can be determined by
triangulating it and adding

together the areas of the triangles. They will identify the Pythagorean Theorem,
use it to find the length of

a side of a right triangle when the other two sides are given, and use its
converse to ascertain whether or

not a triangle is a right triangle. They will convert some English units of
measure (e.g. square inches to

square feet) and some metric measures (e.g. square centimeters to square
decimeters). They will be

able to determine the volume of a rectangular prism.

Students will investigate the concept of proportionality. They will be able to
use a scaling factor to

determine measurements of similar figures and create a proportional drawing.
Given measurements of

two similar figures, they will find the scaling factor. They will conclude
whether or not two triangles are

similar by comparing lengths of corresponding sides and finding the ratio of
similarity, if possible.

Students will identify kinds of angles (acute, obtuse, right, straight, reflex)
and examine different ways to

measure them using slope, degrees, and "A-measure" (where the class decides on a
unit of angle

measure). They will convert slope measure of an angle into degree measure and
vice versa, using the

tan-1 and tan calculator keys. They will realize that angle measurement is
unaffected by scaling. They

will compute and compare perimeters and areas of scaled figures (i.e. if a
figure is scaled by a factor of

k, its area will change by a factor of k^{2}). They will also compute the volume of
a scaled rectangular

prism and realize that its volume changes by a factor of k^{3}. Students will know
that the measures of

vertical angles are equal and that supplementary angles can form a straight
angle. They will explore

corresponding angles and alternate interior angles formed by two lines crossed
by a transversal. In

particular, they will see that two lines cut by a transversal are parallel if
and only if the measures of a pair

of corresponding angles (or alternate interior angles) are equal. [Software such
as the Geometer's

Sketchpad is suggested in the Teacher Commentary as a means to explore angle
measures.] Students

will be guided through a series of steps leading to the fact that the sum of the
angles of any triangle is

180°. They will explore the sum of the measures of the interior angles of any
polygon to eventually

discover the generalization (n-2)180° for an n-gon. They will realize that the
sum of the measures of the

exterior angles of any polygon is 360°. Furthermore, using appropriate formulas,
they will compute the

measure of a single interior angle and exterior angle of a regular polygon.
Focusing on triangles, students

will investigate the congruence principles (SSS, SAS, ASA, AAS) and use them to
determine a triangle

from given angle or side measurements. They will recognize that SSA and AAA are
not valid principles

for determining a triangle.

Students will be able to construct regular polygons with a given number of sides
with geometry software

such as the Geometer's Sketchpad. They will see many properties of the circle.
They will know its

"center-radius" definition (i.e. the set of points in the plane a fixed distance
from a given point). They

will recognize that every diameter of a circle is an axis of symmetry. They will
know the definition of

rotational symmetry and be able to compare the rotational symmetry of regular
polygons to the

rotational symmetry of the circle. They will know that the set of circles that
pass through two given

points have centers on the perpendicular bisector of the segment joining the
points and conversely. They

will be able to construct a circle passing through three points with straight
edge and compass by locating

its center. They will be able to represent a circle parametrically using such
parametric representations as

(cosθ, sinθ) for the unit circle. They will investigate what a transformation (a+rcosθ,
b+rsinθ) does to

the unit circle. They will be able to determine parametric representations for
circles in the plane and be

able to utilize such representation along with technology that plots
parametrically (such as the TI-82) to

display designs created from circular arcs. Using results on changes in area and
perimeter of similar

figures, they will explore relationships between the area and perimeter of a
circle. For example, they will

compare the area and perimeter of a circle to the area and perimeter,
respectively, of the unit circle.

They will discover that the area of the unit circle is p (square units) and
develop several approximations

to its value using geometric means. They will see that the area of a circle can
be found by multiplying its

circumference by its radius and dividing by two. They will also express the area
and perimeter of a circle

as a function of its radius (i.e. A(r) = πr^{2} and C(r)=2πr). They will be able to
use the proportional

relationship between a central angle of a circle and the length of the
corresponding arc and the

proportional relationship between a central angle of a circle and the area of
the corresponding segment

of the circle. They will be able to supply justification for special cases. They
will also be able to use the

relationship between an inscribed angle and one of the central angles with the
same endpoints as the

inscribed angle and will have worked through a justification of this
relationship. They will know that

some properties don't characterize the circle. For example, they will know that
there are curves other

than the circle which have constant width (e.g. a Reuleaux Triangle). Students
will investigate three-dimensional

shapes. They will have two-dimensional strategies for describing these shapes
(such as a

cone). They will be able to use nets (two-dimensional patterns) that fold into
polyhedra and other solid

shapes. They will determine a method, using the " two-dimensional" Pythagorean
Theorem for

calculating the height of a pyramid with equilateral triangular sides and a
regular polygon as a base

knowing the length of an edge. They will realize that a pyramid with a regular
polygon as base and

equilateral triangular sides must have a base with fewer than six sides. They
will realize that a regular

polyhedron has the property that the sum of the angle measures at each vertex is
less than 360°. They

should have developed a justification for the fact that there are only five
regular polyhedra. They will be

able to use two-dimensional contour curves to describe irregular shapes such as
geological landscapes.

They will be able to describe a cone using a net formed by removing a sector of
a circle. They will be

able to calculate the radius of the base of a cone as a function of the central
angle measure of the sector

of the circle removed. They will be able to calculate the height of a cone as a
function of the radius of

the base of the cone. They will be able to compose these functions to represent
the height of a cone as a

function of the central angle of the central angle measure in the net. They will
be able to describe solids

such as prisms and cylinders using two-dimensional cross-sections. They will be
able to calculate the

surface area of a prism from information about its height,
the number of edges of its base, and the length

of each of those edges. They will provide some detail in the development of a
formula for the volume of

a cone using mostly geometric arguments. They will know Cavalieri's Principle
and have worked

through a plausibility argument by comparing cross-sectional slices of two
prisms of the same height

where one is a right prism and the other is oblique and letting the thickness of
these cross-sectional

slices shrink.

They will develop a very intuitive notion of this latter explanation as a
"limiting " procedure. They will

use Cavalieri's Principle to determine the volume of solids. For example, they
will have seen the

Principle used, with some algebraic manipulations , to develop a formula for the
volume of a sphere.

They will investigate solids of revolution (without coordinates) of many bounded
plane regions around a

line. They will have seen evidence based on geometry and algebra to support the
Pappus-Guildin

Theorem which states that the volume of a solid of revolution is the product of
the area of the region and

the distance traveled by the center of gravity (centroid) of the region. They
will explore the center of

gravity of triangles, rectangles, circles, and semi-circles and, thus, be able
to determine the volume of the

solid obtained by rotating one of these regions about a line (which does not
intersect the region). They

will be introduced the a coordinatization of three dimensions using the three
standard axes in the

standard orientation. They will have seen the formula for the distance between
to points in the plane and

two points in three-space developed. They will be introduced to "set builder"
notation. They will be

able to describe many three-dimensional objects using sets of triples satisfying
algebraic equalities

and/or inequalities. They will know the equations for a (two-dimensional) circle
of radius r and a (three-dimensional)

sphere of radius r. They will see that the three axes do not need to represent
spacial

measures. They will see examples where solids can be used to convey information
when one of the

axes measures another quantity such as time.

**Trigonometry: **Building on the concept of similarity of right triangles, students
will be introduced to the

trigonometric functions (sinθ, cosθ, tanθ, cscθ, secθ, cotθ, where θ is an acute
angle) as ratios of the

sides of a right triangle. They will use the sine, cosine, and tangent functions
to find the length of a side

of a right triangle. They will find the measure of an acute angle of a right
triangle, using the inverse

trigonometric functions (sin^{-1}θ, cos^{-1}θ, tan^{-1}θ). They will generalize a formula
for the area of a

parallelogram in terms of its sides and an acute angle (either interior or
exterior): Area=absinA.

Students will follow developments of the Law of Sines and the Law of Cosines
(for acute angles only)

and use them to find the length of a side of a triangle. Through specific
questioning, they will explore a

proof that "the angle made by a tangent line and the radius at the point of
tangency is a right angle."

They will calculate (sinq)^{2}+(cosq)^{2} for several angles, make a conjecture, and
follow a generalization to

the conclusion that sin^{2}θ+cos^{2}θ=1. They will
also see that tan^{2}θ+1=sec^{2}θ. Given
tables of sine and

cosine values for 0°<θ≤50°, where θ is an integer, students will compare values.
Using a diagram of a

right triangle with sides a, b, c and angle θ, they will explain identities such
as sin(90°-θ)=cosθ and

tan(90°-θ)=cotθ. They will explain the connection between the slope of a line
and the tangent of an

angle. They will examine the graphs of y=sinθ and y=cosθ on a graphing
calculator and be introduced

to several real-world examples which could be modeled by the sine and/or cosine
curves. They will

relate the graph of the unit circle to the sine and cosine of an angle and,
hence, extend the domains of

the sine and cosine functions.

**Probability and statistics: **Students will perceive
differences between discrete and continuous data.

They will enhance their ability to create appropriate representations (e.g.
stem-and-leaf plot, boxplot,

scatterplot, dot plot (also known as line plot)) for sets of data and compare
and contrast these displays.

Using primarily a dot plot, they will examine the distribution of data in terms
of its symmetry, whether or

not it is bell-shaped or bimodal, and possible outliers. They will be introduced
to bias when they

encounter data that is not "distributed approximately symmetrically around the
true value."

**Logic and reasoning: **Students will write the converse of statements and decide
whether or not they

are true. If true, they will explain why; if not true, they will provide reasons
or counterexamples. For

example, they will state the converse of "If two polygons are similar, their
corresponding angles are

congruent"and give a counterexample to show that the converse of this statement
is false. Students will

be introduced to the idea of proof as "a logical argument that explains why a
statement must be true"

and they will be expected to provide logical arguments that justify facts such
as "angles of an equilateral

triangle must be equal" and "a quadrilateral must be a rhombus if its diagonals
are perpendicular

bisectors of each other."