A. NUMBERS AND NUMBER SENSE
Students will understand and demonstrate a sense of what numbers mean and how they are used. Numbers are used to describe and interpret phenomena. Building a sense of number relationships is essential for developing the ability to deal with any set of numbers. Number sense involves understanding the meaning of numbers, relationships among numbers, relative number magnitudes, and the effects of operations on numbers. Skilled estimation is also an important component of number sense.
SECONDARY GRADES
1. Describe the structure of the real number system and identify its appropriate applications and limitations.
2. Explain what complex numbers (real and imaginary) mean and describe some of their many uses.
EXAMPLE
Given two numbers such as 1/2 and 1/3, describe the real numbers between them.
B. COMPUTATION
Students will understand and demonstrate computation skills. Understanding the fundamental operations of addition, subtraction, multiplication, and division is central to knowing mathematics. Proficiency in computational skills is essential to problem-solving and other mathematical activities. Estimating, evaluating reasonableness of answers, and obtaining accuracy in calculations are included in this proficiency. Understanding relationships in operations allows students greater facility with mental computation. Computational skill promotes efficient and confident learners.
SECONDARY GRADES
1. Use various techniques to approximate solutions, determine the reasonableness of answers, and justify the results.
2. Explain operations with number systems other than base ten.
EXAMPLE
If 10% of U.S. citizens have a certain trait, and four out of five with the trait are men, determine what proportion of men have the trait and what proportion of women have the trait. Explain whether the answer depends on the proportion of U.S. citizens who are women, and if so, how?
C. DATA ANALYSIS AND STATISTICS
Students will understand and apply concepts of data analysis. We are faced with massive quantities of information which must be selected, sorted, and analyzed to reach conclusions. Sound decision making requires the ability to collect data effectively, organize data, discover patterns, summarize trends, make inferences, draw conclusions, and make predictions. The ethical use of statistics is a paramount concern in the Information Age.
SECONDARY GRADES
1. Determine and evaluate the effect of variables on the results of data collection.
2. Predict and draw conclusions from charts, tables, and graphs that summarize data from practical situations.
3. Demonstrate an understanding of concepts of standard deviation and correlation and how they relate to data analysis.
4. Demonstrate an understanding of the idea of random sampling and recognition of its role in statistical claims and designs for data collection.
5. Revise studies to improve their validity (e.g., in terms of better sampling, better controls, or better data analysis techniques).
EXAMPLES
Draw a scatter plot of the height of each student in the class vs. their shoe length and find the line of best fit using a graphics calculator or computer software.
Design and conduct an experiment to estimate the population of clams in a given clam flat.
D. PROBABILITY
Students will understand and apply concepts of probability. Probability is the study of uncertainty. Informed consumers of information understand the basic principles of probability. People need to understand the uncertainties and limitations involved when drawing conclusions from data.
SECONDARY GRADES
1. Find the probability of compound events and make predictions by applying probability theory.
2. Create and interpret probability distributions.
EXAMPLE
Determine the probability that a 90% free throw shooter will make exactly one of his/her upcoming two free throws.
E. GEOMETRY
Students will understand and apply concepts from geometry. Geometry is the study of the spatial world and its symmetries. The ideas of geometry are used to describe, interpret, represent, and change the spatial world in which we live. The understanding and development of spatial and visual skills strengthens problem-solving abilities.
SECONDARY GRADES
1. Draw coordinate representations of geometric figures and their transformations.
2. Use inductive and deductive reasoning to explore and determine the properties of and relationships among geometric figures.
3. Apply trigonometry to problem situations involving triangles and periodic phenomena.
F. MEASUREMENT
Students will understand and demonstrate measurement skills. Measurement is valuable as an integrating skill throughout the curriculum and in everyday life. The use of estimation is vital in determining the reasonableness of measurement. Measurement attributes (e.g., length, volume, minutes), units, and tools enhance the ability to describe and understand the world.
SECONDARY GRADES
1. Use measurement tools and units appropriately and recognize limitations in the precision of the measurement tools.
2. Derive and use formulas for area, surface area, and volume of many types of figures.
EXAMPLES
Discover and explore the distance formula using the Pythagorean Theorem.
Using generalizations, compare the formula for the area of an n-sided, regular polygon to the formula for the area of a circle.
G. PATTERNS, RELATIONSHIPS, FUNCTIONS
Students will understand that mathematics is the science of patterns, relationships, and functions. Relationships are central to mathematical understanding. A study of patterns often reveals regularity, indicating the presence of a mathematical relationship. Studying relationships allows students to make generalizations and predictions about phenomena and occurrences.
SECONDARY GRADES
1. Create a graph to represent a real-life situation and draw inferences from it.
2. Translate and solve a real-life problem using symbolic language.
3. Model phenomena using a variety of functions (linear, quadratic, exponential, trigonometric, etc.).
4. Identify a variety of situations explained by the same type of function.
EXAMPLES
Express the diameter of a circle as a function of its area and sketch a graph.
Determine which of two ways of rolling a 8.5"x11" piece of paper into a cylinder gives the greater volume and whether there is a way to get even greater volume using a sheet of paper with the same area but different shape.
H. ALGEBRA CONCEPTS
Students will understand and apply algebraic concepts. Algebra and analytic thinking are fundamental tools for working in and thinking about mathematics. These tools provide ways to generalize and predict problem solutions when not all information is known. Taught within the context of mathematical and practical applications, the concept of functions is a unifying theme for algebraic concepts.
SECONDARY GRADES
1. Use tables, graphs, and spreadsheets to interpret expressions, equations, and inequalities.
2. Investigate concepts of variation by using equations, graphs, and data collection.
3. Formulate and solve equations and inequalities.
4. Analyze and explain situations using symbolic representations.
EXAMPLES
Use measurements from shopping carts which are nested together to find a formula for the number of carts that will fit in a given space and a formula for the amount of space needed for a given number of carts.
Solve the following problem: Given the formula for height of an object thrown upward with velocity v: h = ho + vt + (1/2) gt2, use quadratic functions and the quadratic formula to answer questions about the motion of projectiles and falling objects.
I. DISCRETE MATHEMATICS
Students will understand and apply concepts in discrete mathematics. Discrete mathematics studies discrete processes (e.g., all possible bus routes in a school district). This study includes the exploration of diagrams, networks, and flowcharts that students construct to model situations or use for planning, scheduling, and decision making. Three main concerns of discrete mathematics are: existence (Is there a solution?), counting (How many solutions are there?), and efficiency (What is the best solution?).
SECONDARY GRADES
1. Use linear programming to find optimal solutions to a system.
2. Use networks to find solutions to problems.
3. Apply strategies from game theory to problem-solving situations.
4. Use matrices as tools to interpret and solve problems.
EXAMPLE
Given a decreasing linear relationship between the selling price of a magazine and the number of people who will buy it, and given a fixed cost per copy that goes to production, analyze the profitability of the product and recommend a price range.
J. MATHEMATICAL REASONING
Students will understand and apply concepts of mathematical reasoning. Reasoning is fundamental to the knowing and doing of mathematics. To give more students access to mathematics as a powerful way of making sense of the world, it is essential that an emphasis on reasoning pervade all mathematics. Students need a great deal of time and many experiences to develop their ability to construct valid arguments in problem settings and to evaluate the arguments of others.
SECONDARY GRADES
1. Analyze situations where more than one logical conclusion can be drawn from data presented.
EXAMPLE
Given information about travel patterns in a local community, develop a convincing proposal for the logical placement of a bypass.
K. MATHEMATICAL COMMUNICATION
Students will reflect upon and clarify their understanding of mathematical ideas and relationships. Communication plays a key role in helping make important connections among physical, pictorial, graphic, symbolic, verbal, and mental representations of mathematical ideas. Providing individual and collaborative opportunities for discussions about issues, people, and the cultural implications of mathematics reinforce student understanding of the connection between mathematics and our society.
SECONDARY GRADES
1. Restate, create, and use definitions in mathematics to express understanding, classify figures, and determine the truth of a proposition or argument.
2. Read mathematical presentations of topics within the Learning Results with understanding.
EXAMPLES:
Having read the definition of "kite," a student analyzes a collection of figures to decide which are kites. The student then proceeds to apply the kite definition to the families of quadrilaterals to determine which are kites and why.
Student reads a manual or math text to successfully learn a new procedure.