Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.
Key ideas are
identified by numbers (1).
Performance indicators are identified by bullets.
Sample tasks are identified by triangles (s).
Elementary Mathematical Reasoning
1. Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.
Students:
This is evident,
for example, when students:
s build geometric figures out
of straws.
s find patterns in sequences
of numbers, such as the triangular numbers 1, 3, 6, 10, . . . .
s explore number
relationships with a calculator (e.g., 12 + 6 = 18, 11 + 7 = 18,
etc.) and draw conclusions.
Elementary Number and Numeration
2. Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.
Students:
This is evident,
for example, when students:
s count out 15 small cubes
and exchange ten of the cubes for a rod ten cubes long.
s use the number line to show
the position of 1/4.
s figure the tax on $4.00
knowing that taxes are 7 cents per $1.00.
3. Students use mathematical operations and relationships among them to understand mathematics.
Students:
This is evident,
for example, when students:
s use the fact that
multiplication is commutative (e.g., 2 x 7 = 7 x 2), to assist
them with their memorizing of the basic facts.
s solve multiple-step
problems that require at least two different operations.
s progress from base ten
blocks to concrete models and then to paper and pencil
algorithms.
Elementary Modeling/Multiple Representation
4. Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.
Students:
This is evident,
for example, when students:
s build a 3 x 3 x 3 cube out
of blocks.
s use square tiles to model
various rectangles with an area of 24 square units.
s read a bar graph of
population trends and write an explanation of the information it
contains.
5. Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.
Students:
This is evident,
for example, when students:
s measure with paper clips or
finger width.
s estimate, then calculate,
how much paint would be needed to cover one wall.
s create a chart to display
the results of a survey conducted among the classes in the
school, or graph the amounts of survey responses by grade level.
6. Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.
Students:
This is evident,
for example, when students:
s estimate the length of the
room before measuring.
s predict the average number
of red candies in a bag before opening a group of bags, counting
the candies, and then averaging the number that were red.
s determine the probability
of picking an even numbered slip from a hat containing slips of
paper numbered 1, 2, 3, 4, 5, and 6.
7. Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.
Students:
This is evident,
for example, when students:
s represent three more than a
number is equal to nine as n + 3 = 9.
s draw leaves, simple
wallpaper patterns, or write number sequences to illustrate
recurring patterns.
s write generalizations or
conclusions from display data in charts or graphs.
Intermediate Mathematical Reasoning
1. Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.
Students:
This is evident,
for example, when students:
s use trial and error and
work backwards to solve a problem.
s identify patterns in a
number sequence.
s are asked to find numbers
that satisfy two conditions, such as n > -4 and n < 6.
Intermediate Number and Numeration
2. Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.
Students:
This is evident,
for example, when students:
s use prime factors of a
group of denominators to determine the least common denominator.
s select two pairs from a
number of ratios and prove that they are in proportion.
s demonstrate the concept
that a number can be symbolized by many different numerals as in:
1 3 25 — = — = — = 0.25 = 25% 4 12 100
3. Students use mathematical operations and relationships among them to understand mathematics.
Students:
This is evident,
for example, when students:
s create area models to help
in understanding fractions, decimals, and percents.
s find the missing number in
a proportion in which three of the numbers are known, and letters
are used as place holders.
s arrange a set of fractions
in order, from the smallest to the largest: 3 1 2 1 1 —,
—, —, —, — 4 5 3 2 4
s illustrate the distributive
property for multiplication over addition, such as 2(a + 3) = 2a
+ 6.
Intermediate Modeling/Multiple Representation
4. Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.
Students:
This is evident,
for example, when students:
s build a city skyline to
demonstrate skill in linear measurements, scale drawing, ratio,
fractions, angles, and geometric shapes.
s bisect an angle using a
straight edge and compass.
s draw a complex of geometric
figures to illustrate that the intersection of a plane and a
sphere is a circle or point.
5. Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.
Students:
This is evident,
for example, when students:
s use box plots or stem and
leaf graphs to display a set of test scores.
s estimate and measure the
surface areas of a set of gift boxes in order to determine how
much wrapping paper will be required.
s explain when to use mean,
median, or mode for a group of data.
6. Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.
Students:
This is evident,
for example, when students:
s construct spinners to
represent random choice of four possible selections.
s perform probability
experiments with independent events (e.g., the probability that
the head of a coin will turn up, or that a 6 will appear on a die
toss).
s estimate the number of
students who might chose to eat hot dogs at a picnic.
Intermediate Patterns/Functions
7. Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.
Students:
This is evident,
for example, when students:
s find the height of a
building when a 20-foot ladder reaches the top of the building
when its base is 12 feet away from the structure.
s investigate number patterns
through palindromes (pick a 2-digit number, reverse it and add
the two—repeat the process until a palindrome appears) 42 86
+24 +68 palindrome 66 154 +451 605 +506 palindrome 1111
s solve linear equations,
such as 2(x + 3) = x + 5 by several methods.
Commencement Mathematical Reasoning
1. Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.
Students:
This is evident,
for example, when students:
s prove that an altitude of
an isosceles triangle, drawn to the base, is perpendicular to
that base.
s determine whether or not a
given logical sentence is a tautology.
s show that the triangle
having vertex coordinates of (0,6), (0,0), and (5,0) is a right
triangle.
Commencement Number and Numeration
2. Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.
Students:
This is evident,
for example, when students:
s determine from the
discriminate of a quadratic equation whether the roots are
rational or irrational.
s give rational
approximations of irrational numbers to a specific degree of
accuracy.
s determine for which value
of x the expression 2x + 6 is undefined. x - 7
3. Students use mathematical operations and relationships among them to understand mathematics.
Students:
This is evident,
for example, when students:
s determine the coordinates
of triangle A(2,5), B(9,8), and C(3,6) after a translation (x,y)
(x + 3, y - 1).
s evaluate the binary
operation defined as x * y = x -2 + (y + x) 2 for 3 * 4.
s identify the field
properties used in solving the equation 2(x - 5) + 3 = x + 7.
Commencement Modeling/Multiple Representation
4. Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.
Students:
This is evident,
for example, when students:
s determine the locus of
points equidistant from two parallel lines.
s explain why the basic
construction of bisecting a line is valid.
s describe the various conics
produced when the equation ax 2 + by 2 = c 2 is graphed for
various values of a, b, and c.
5. Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.
Students:
This is evident,
for example, when students:
s change mph to ft/sec.
s use the tangent ratio to
determine the height of a tree.
s determine the distance
between two points in the coordinate plane.
6. Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.
Students:
This is evident,
for example, when students:
s construct a tree diagram or
sample space for a compound event.
s calculate the probability
of winning the New York State Lottery.
s develop simulations for
probability problems for which they do not have theoretical
solutions.
Commencement Patterns/Functions
7. Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.
Students:
This is evident,
for example, when students:
s determine, in more than one
way, whether or not a specific relation is a function.
s explain the relationship
between the roots of a quadratic equation and the intercepts of
its corresponding graph.
s use transformations to
determine the inverse of a function.
Four Year Sequence in Mathematics Mathematical Reasoning
1. Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.
Students:
This is evident,
for example, when students:
s prove indirectly that: if
n2 is even, n is even.
s prove using mathematical
induction that: 1 + 3 + 5 + . . . + (2n - 1) = n 2.
s explain the axiomatic
differences between plane and spherical geometries.
Four Year Sequence in Mathematics Number and Numeration
2. Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.
Students:
This is evident,
for example, when students:
s relate the concept of
infinity when graphing the tangent function.
s show that the set of
complex numbers form a field under the operations of addition and
multiplication.
s show that the set of
complex numbers forms a field under the operations of addition
and multiplication.
s represent a complex number
in polar form.
Four Year Sequence in Mathematics Operations
3. Students use mathematical operations and relationships among them to understand mathematics.
Students:
This is evident,
for example, when students:
s relate specific matrices to
certain types of transformations of points on the coordinate
plane.
s evaluate expressions with
fractional exponents, such as 8 2/3 4 -1/2.
s determine the value of
compound functions such as (f o g) (x).
Four Year Sequence in Mathematics Modeling/Multiple Representation
4. Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.
Students:
This is evident,
for example, when students:
s determine coordinates which
lie in the solution of the quadriatic inequality, such as y <
x 2 + 4x + 2.
s find the distance between
two points in a three-dimension coordinate system.
s describe what happens to
the graph when b increases in the function y = x 2 + bx + c.
Four Year Sequence in Mathematics Measurement
5. Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.
Students:
This is evident,
for example, when students:
s use a chi-square test to
determine if one cola really tastes better than another cola.
s can illustrate the various
line segments which represent the sine, cosine, and tangent of a
given angle on the unit circle.
s calculate the first
derivative of a function using the limit definition.
Four Year Sequence in Mathematics Uncertainty
6. Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.
Students:
This is evident,
for example, when students:
s verify the probabilities
listed for the state lottery for second, third, and fourth prize.
s use graphing calculators to
generate a curve of best fit for an array of data using linear
regression.
s determine the probability
of getting at least 3 heads on 6 flips of a fair coin.
Four Year Sequence in Mathematics Patterns/Functions
7. Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.
Students:
This is evident,
for example, when students:
s transform polar coordinates
into rectangular forms.
s find the maximum height of
an object projects upward with a given initial velocity.
s find the limit of
expressions like n - 2 as n goes 3n + 5 to infinity.