**CLASSROOM IDEAS
EXAMPLES FOR
Math A**

The following ideas for lessons and activities are
provided to illustrate examples of each performance indicator. It is not
intended that

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.

**2C.**

• 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.]

**3D.**

Identify the field properties used in solving the equation 2(x - 5) + 3 = x + 7.

**4B.**

Explain why the basic construction of bisecting a line segment is valid.

**5B.**

While watching a TV detective show, you see a crook running out of a bank
carrying an attachÃ© 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 attachÃ© case is a rectangular prism (18” x 5” x 13”).

2. You might want to decide the smallest denomination of bill that will work.

[Also 5A.]

**5H.**

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.

**6A.**

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

Mathematical Reasoning

Students use mathematical reasoning to analyze
mathematical situations,

make conjectures, gather evidence, and construct an argument.

PERFORMANCE INDICATORS |
INCLUDES |
EXAMPLES |

1A. Construct proofs based on deductive reasoning. |
• Euclidean and analytic direct proofs. |
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

ideas.

PERFORMANCE INDICATORS |
INCLUDES |
EXAMPLES |

2A. Understand and use rational and irrational numbers. |
• 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 skills. |
See Classroom Activity 2C. |

2D. Recognize the hierarchy of the complex number system. |
• Subsets of complex numbers. | See Classroom Activity 2D. |

2E. Model the structure of the complex number system. |
• Imaginary unit of complex numbers. • Standard form of complex numbers. |
See Classroom Activity 2E. |

**Key Idea 3
Operations**

Students use mathematical operations and relationships among them to understand

mathematics.

PERFORMANCE INDICATORS |
INCLUDES |
EXAMPLES |

3A. Use addition, subtraction, multiplication, division, and exponentiation with real numbers and algebraic expressions. |
• Operations with fractions with polynomial denominators. • Add and subtract rational fractions with monomial and binomial denominators. |
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 of transformations. • 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 functions, symmetry). • 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 plane. • Use slope and midpoint to demonstrate transformations. • 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 numbers. • Evaluate expressions with fractional exponents. • Basic arithmetic operations with complex numbers. • Simplify square roots with negative radicands. • 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
basic operations and the composition of two functions. |
• Determine the value of compound functions. • Pairs of equations. |
See Classroom Activity 3E. |

**Key Idea 4 Math B
Modeling/Multiple Representation**

Students use mathematical modeling/multiple representation to provide a means

of presenting, interpreting, communicating, and connecting mathematical
information

and relationships.

PERFORMANCE INDICATORS |
INCLUDES |
EXAMPLES |

4A. Represent problem situations symbolically by using algebraic expressions, sequences, tree diagrams, geometric figures, and graphs. |
• 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 variation (hyperbola). |
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 containing exponents. • 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 log _{b}a = c as a = b^{c}.• Solve equations, using logarithmic expressions. • 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. • Parabolas. • Using the intercepts, recognize the ellipse and non-rectangular hyperbola. |
See Classroom Activity 4D. |

4E. Model real-world problems with systems of equations and inequalities. |
• Solve systems of equations: linear, quadratic, and linear-quadratic systems. |
See Classroom Activity 4E. |

4F. Model vector quantities both algebraically and geometrically. |
• 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 cally. |
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 solutions. |
See Classroom Activity 4H. |

4I. Model the composition of transformations. | • The composition of two line reflections when the two lines are parallel. • 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 reflection). |
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 = a ^{x}. Characteristics to beemphasized are: -the domain of an exponential function is the set of real numbers -the range of an exponential function is the set of positive numbers -the graph of any exponential function will contain the point (0, 1) -the exponential function is one-to-one. • If a > 1, the graph rises, but if 0 < a < 1, the graph falls. • The graphs of y = a ^{x} 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 numbers -the range of the logarithmic function is the set of all real numbers -the graph of any logarithmic function will contain the point (1,0). • The graphs of y = a ^{x} and x = a^{y}, a>0, and a ≠ 1, are reflections of each other in the line y = x. |
See Classroom Activity 4J. |

4K. Use polynomial, trigonometric, and
exponential functions to model real-world relationships. |
• Recognize when a real-world relationship can be represented by a linear, quadratic, trigonometric, or exponential function. • Solve real-world problems by using linear, quadratic, trigonometric , and exponential functions. |
See Classroom Activity 4K. |

4L. Use algebraic relationships to analyze the conic sections. |
• 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} = r^{2}.• Recognize an equation in the form y = ax ^{2} + bx + c, a ≠0 as an equationof 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 parabola. • 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: -radian measure -sine -cosine -tangent -reciprocal trigonometric functions. • 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. |