The following ideas for lessons and activities are
provided to illustrate examples of each performance indicator. It is not
teachers use these specific ideas in their classrooms; rather, they should feel free to use them or adapt them if they so desire. Some of
the ideas incorporate topics in science and technology. In those instances the appropriate standard will be identified. Some classroom
ideas exemplify more than one performance indicator. Additional relevant performance indicators are given in brackets at the end of
the description of the classroom idea.
• Have students make multiplication and addition charts for a 12-hour clock, using only the numbers 1-12.
• Have students determine if the system is closed under addition and multiplication. If not, they should give a counterexample.
• Have students determine if multiplication and addition are commutative under the system, and if not, give a counterexample.
• Have students determine if there is an identity element for addition and multiplication, and if so, what are they?
• Have students determine if addition and multiplication are associative under the system, and if not, give a counterexample.
• Does each element have an additive and multiplicative inverse?
• Determine if multiplication is distributive over addition (if not give a counterexample) and if addition is distributive
over multiplication (if not, give a counterexample). [Also 3D.]
Identify the field properties used in solving the equation 2(x - 5) + 3 = x + 7.
Explain why the basic construction of bisecting a line segment is valid.
While watching a TV detective show, you see a crook running out of a bank carrying an attaché case. You deduce from
the conversation of the two stars in the show that the robber has stolen $1 million in small bills. Could this happen?
Why or why not?
Hints: 1. An average attaché case is a rectangular prism (18” x 5” x 13”).
2. You might want to decide the smallest denomination of bill that will work.
An odometer is a device that measures how far a bicycle (or a car) travels. Sometimes an odometer is not adjusted accurately
and gives readings which are consistently too high or too low.
Paul did an experiment to check his bicycle odometer. He cycled 10 laps around a
race track. One lap of the track is
0.4 kilometers long. When he started, his odometer read 1945.68 and after the 10 laps his odometer read 1949.88.
Compare how far Paul really traveled with what his odometer read.
Make a table that shows numbers of laps in multiples of 10 up to 60 laps, the distance Paul really travels, and the distance
the odometer would say he traveled.
Draw a graph to show how the distance shown by the odometer is related to the real distance traveled.
Find a rule or formula that Paul can use to change his incorrect odometer
readings into accurate distances he has gone
from the start of his ride.
An odometer measures how far a bicycle travels by counting the number of times
the wheel turns around. It then multiplies
this number by the circumference of the wheel. To do this right, the odometer has to be set for the right wheel circumference.
If it is set for the wrong circumference, its readings are consistently too high or too low. Before Paul’s experiment,
he estimated that his wheel circumference was 210 cm. Then he set his odometer for this circumference. Use the
results of his experiment to find a more accurate estimate for the circumference.
A box contains 20 slips of paper. Five of the slips are marked with a “X,” seven are marked with a “Y,” and the rest are
blank. The slips are well mixed. Determine the probability that a blank slip will be drawn without looking in the bag on
the first draw. Have students determine the probability theoretically and then have each conduct the experiment with
ten trials and see how close the empirical probability was to the theoretical probability. Combine data from all students
in the class to see if a larger number of trials will result in an empirical probability that more closely resembles the
theoretical probability. [Also 6B.]
Key Idea 1
Students use mathematical reasoning to analyze
make conjectures, gather evidence, and construct an argument.
|1A. Construct proofs based on deductive
|• Euclidean and analytic direct
|See Classroom Activity 1A.|
|1B. Construct indirect proofs.||• Euclidean indirect proofs.||See Classroom Activity 1B.|
Key Idea 2
Number and Numeration
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
|2A. Understand and use rational and irrational
|• Determine from the discriminant
of a quadratic equation whether
the roots are rational or irrational.
• Rationalize denominators.
• Simplifying of algebraic fractions
with polynomial denominators.
• Simplify complex fractions.
|See Classroom Activity 2A.|
|2B. Recognize the order of the real numbers.||• Give rational approximations of
irrational numbers to a specific
degree of accuracy.
|See Classroom Activity 2B.|
|2C. Apply the properties of the real numbers
to various subsets of numbers.
|• Use the properties of real numbers
in the development of algebraic
|See Classroom Activity 2C.|
|2D. Recognize the hierarchy of the complex
|• Subsets of complex numbers.||See Classroom Activity 2D.|
|2E. Model the structure of the complex number
|• Imaginary unit of complex
• Standard form of complex
|See Classroom Activity 2E.|
Key Idea 3
Students use mathematical operations and relationships among them to understand
|3A. Use addition, subtraction, multiplication,
division, and exponentiation with real
numbers and algebraic expressions.
|• Operations with fractions with
• Add and subtract rational fractions
with monomial and binomial
|See Classroom Activity 3A.|
|3B. Develop an understanding of and use the
composition of functions and transformations.
|• Understand the general concept
and symbolism of the composition
• Apply the composition of transformations
(line reflections, rotations,
translations, glide reflections).
• Identify graphs that are symmetric
with respect to the axes or origin.
• Isometries (direct, opposite).
• Applications to graphing (inverse
• Define and compute compositions
of functions and transformations.
|See Classroom Activity 3B.|
|3C. Use transformations on figures and functions
in the coordinate plane.
|• Apply transformations (line reflection,
point reflection, rotation,
translation, and dilation) on figures
and functions in the coordinate
• Use slope and midpoint to demonstrate
• Use the ideas of transformations to
investigate relationships of two circles.
• Use translation and reflection to
investigate the parabola.
|See Classroom Activity 3C.|
|3D. Use rational exponents on real numbers
and all operations on complex numbers.
|• Absolute value of complex
• Evaluate expressions with fractional
• Basic arithmetic operations with
• Simplify square roots with negative
• Use the product of a complex number
and its conjugate to express the
quotient of two complex numbers.
• Cyclic nature of the powers of i.
• Solving quadratic equations.
• Laws of rational exponents.
|See Classroom Activity 3D.|
|3E. Combine functions, using the
and the composition of two
|• Determine the value of compound
• Pairs of equations.
|See Classroom Activity 3E.|
Key Idea 4 Math B
Students use mathematical modeling/multiple representation to provide a means
of presenting, interpreting, communicating, and connecting mathematical information
|4A. Represent problem situations symbolically
by using algebraic expressions,
sequences, tree diagrams, geometric figures,
|• Express quadratic, circular, exponential,
and logarithmic functions
in problem situations algebraically.
• Use symbolic form to represent an
explicit rule for a sequence.
• Definition and graph of an inverse
|See Classroom Activity 4A.|
|4B. Manipulate symbolic representations to
explore concepts at an abstract level.
|• Use positive, negative, and zero
exponents and be familiar with the
laws used in working with expressions
• In the development of the use of
exponents, the students should
review scientific notation and its
use in expressing very large or
very small numbers.
• Rewrite the equality
logba = c as a = bc.
• Solve equations, using logarithmic
• Rewrite expressions involving
exponents and logarithms.
• Compound functions.
|See Classroom Activity 4B.|
|4C. Choose appropriate representations to
facilitate the solving of a problem.
|• Select exponential or logarithmic
process to solve an equation.
• Recognize that a variety of phenomena
can be modeled by the
same type of function.
|See Classroom Activity 4C.|
|4D. Develop meaning for basic conic sections.||• Circles.
• Using the intercepts, recognize the
ellipse and non-rectangular
|See Classroom Activity 4D.|
|4E. Model real-world problems with systems
of equations and inequalities.
|• Solve systems of equations: linear,
quadratic, and linear-quadratic
|See Classroom Activity 4E.|
|4F. Model vector quantities both algebraically
|• The Law of Sines and the Law of
Cosines can be used with a wide
variety of problems involving triangles,
parallelograms and other
geometric figures in applications
involving the resolution of forces
both algebraically and geometric
|See Classroom Activity 4F.|
|4G. Represent graphically the sum and difference
of two complex numbers.
|• Represent the basic operations of
addition and subtraction.
|See Classroom Activity 4G.|
|4H.Model quadratic inequalities both
algebraically and graphically.
|• Use multiple representation to
show inequalities algebraically and
graphically to find the possible
|See Classroom Activity 4H.|
|4I. Model the composition of transformations.||• The composition of two line reflections
when the two lines are
• The composition of two rotations
about the same point.
• The composition of two translations.
• The composition of a line reflection
and a translation in a direction parallel
to the line of reflection (glide
|See Classroom Activity 4I.|
|4J. Determine the effects of changing parameters
of the graphs of functions.
|• Be able to sketch the effects of
changing the value of a in the function
y = ax. Characteristics to be
-the domain of an exponential
function is the set of real numbers
-the range of an exponential
function is the set of positive
-the graph of any exponential
function will contain the point (0, 1)
-the exponential function is
• If a > 1, the graph rises, but if 0 < a
< 1, the graph falls.
• The graphs of y = ax and y = a-x,
a > 0, and a ≠ 1, are reflections of
each other in the y-axis.
• The logarithmic function is the
inverse of the exponential function
with the following characteristics:
-since the exponential function
is one-to-one, its inverse, the
logarithmic function, exists
-the domain of the logarithmic
function is the set of positive real
-the range of the logarithmic
function is the set of all real numbers
-the graph of any logarithmic
function will contain the point
• The graphs of y = ax and x = ay, a
>0, and a ≠ 1, are reflections of
each other in the line y = x.
|See Classroom Activity 4J.|
|4K. Use polynomial, trigonometric, and
functions to model real-world relationships.
|• Recognize when a real-world relationship
can be represented by a
linear, quadratic, trigonometric, or
• Solve real-world problems by using
linear, quadratic, trigonometric ,
and exponential functions.
|See Classroom Activity 4K.|
|4L. Use algebraic relationships to analyze the
|• Write the equation of a circle with
a given center and radius and
determine the radius and center of
a circle whose equation is in the
form (x - h)2 + (y - k)2 = r2.
• Recognize an equation in the form
y = ax2 + bx + c, a ≠0 as an equation
of a parabola and
-be able to form a table of values
in order to sketch its graph
-find the axis of symmetry
-determine the abscissa of the
vertex to provide a point of reference
for choosing the x-coordinates
to be plotted
-find the y-intercept of the
• Turning point.
• Maximum or minimum.
|See Classroom Activity 4L.|
|4M.Use circular functions to study and model
periodic real-world phenomena.
|• Use the concept of the unit circle to
solve real-world problems involving:
• Relate reference angles, amplitude,
period, and translations to the
solution of real-world problems.
|See Classroom Activity 4M.|
|4N. Use graphing utilities to create and
explore geometric and algebraic models.
|• Graph quadratic equations and
observe where the graph crosses
the x-axis, or note that it does not.
|See Classroom Activity 4N.|