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=x2+2xy+y2. 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=kx2). 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 r2). They will be introduced to ∑-notation. They
will know the basic
shapes of quadratic polynomials and at least the shape of the cubic y=x3. They
will see what a, b, and c
do in each of the cases y=ax2 ; y=x2+b ; y=(x+c)2 ; and y=x2 +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 k2). They will also compute the volume of
a scaled rectangular
prism and realize that its volume changes by a factor of k3. 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) = πr2 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-1θ, cos-1θ, tan-1θ). 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 sin2θ+cos2θ=1. They will
also see that tan2θ+1=sec2θ. 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."
