

FOCUS Developing an intuitive, commonsense approach to number relationships and operations is of primary importance and should permeate every area of the mathematics curriculum. Number sense involves the use of "friendly easy numbers" and of actively seeking alternative ways of making computations. Number sense is not a topic to be taught as a unit, but is a prevailing theme throughout all mathematics. All students should develop a conceptual understanding of number magnitude and number operations through participation in handson investigative activities. These activities should provide many opportunities for students to discover and develop problemsolving strategies. Student involvement in these activities should assist in the development of estimation skills (particularly when an approximate answer is sufficient) and other mental arithmetic skills (when an exact answer is required). When the numbers are not manageable for mental arithmetic and an exact answer is required, calculators or paper and pencil should be used. Parallel with the need to develop an understanding of the methods and usage of various computational techniques is the students’ need for an informal development of mathematical language and symbolism. Inherent in our increasing dependence on technology is the danger of accepting machine answers at face value. A welldeveloped number sense can combat this danger. Furthermore, number sense leads naturally to the development of symbol sense necessary for use with technology, such as graphing calculators and symbolic manipulators. This developing mathematical power will allow the students to function and communicate more effectively and with greater confidence in reallife experiences. STANDARD In problemsolving investigations, students demonstrate an understanding of the real number system and communicate the relationships within that system using a variety of techniques and tools. Students in Grades K4 use estimation, mental arithmetic, number lines, graphs, appropriate models, manipulatives, calculators, and computers as they investigate problems involving whole numbers. As a result, what they know and are able to do includes:

N1E  constructing number meaning and demonstrating that a number can be expressed in many different forms (e.g., standard notation, number words, number lines, geometrical representation, fractions, and decimals)  1,2,4  
N2E  demonstrating number sense and estimation skills, giving particular attention to common equivalent reference points (i.e., 1/4 = 25% = .25; ½ = 50% = .5; $1 = 100%, etc.)  1  
N3E  reading, writing, representing, comparing, ordering, and using whole numbers in a variety of forms (e.g., standard notation, number line, and geometrical representation  1,4  
N4E  demonstrating a conceptual understanding of the meaning of the basic arithmetic operations (add, subtract, multiply, and divide) and their relationships to each other  1  
N5E  selecting appropriate operation(s) (add, subtract, multiply, and divide) for a given situation  2,3,4  
N6E  applying a knowledge of basic math facts and arithmetic operations to reallife situations  2,4,5  
N7E  constructing, using, and explaining procedures to compute and estimate with whole numbers (e.g., mental math strategies)  1,4  
N8E  selecting and using appropriate computational methods and tools for given situations involving whole numbers (e.g., estimation, mental arithmetic, calculator, or paper and pencil)  2,4  
N9E  demonstrating the connection of number and number relations to the other strands and to reallife situations  1,4,5 
FOCUS
Algebra is much more than the study of generalized forms of arithmetic. It is a powerful language used to interpret realworld experience. This language is a communication tool used to analyze and describe relationships and mathematical structures. Beginning at the elementary level, the school mathematics curriculum should integrate the use of the language of Algebra throughout all strands of the curriculum to enable students to shift progressively from informal to formal concepts and from concrete to symbolic representations. The middle school mathematics curriculum should integrate the use of this language throughout all strands of the curriculum to enable students to progressively shift from the concrete to the symbolic. At this level, algebra should be conceptual and intuitive, not formally computational. It should involve actively seeking easy and alternative ways of looking at problems. These transitions should be powered by investigations involving the use of appropriate manipulatives, models, and technology, and should encourage the development of communication, reasoning, and problemsolving skills. Algebra, in the K8 classrooms, refers to informal explorations and understandings of symbolism. It is beneficial to introduce the algebraic terminology (equation, inequality, variable, etc.) in the early grades. In this way high school students will be able to understand algebra as a natural outgrowth of their study of various number properties. The high school curriculum should continue the development of symbolic representatives. The use of modern technology frees teachers and students from the need to develop complicated pencil and paper manipulative skills in algebra. More classroom time is now available to apply algebra in solving challenging realworld problems. This will allow students to recognize the worth, importance, and power of the mathematics of abstraction and symbolism.
STANDARD
In problemsolving investigations students demonstrate an understanding of concepts and processes that allow them to analyze, represent, and describe relationships among variable quantities and to apply algebraic methods to realworld situations.
Students in Grades K4 use manipulatives, models, graphs, tables, technology, number sense, and estimation as they investigate problems involving the concepts and application of algebra. As a result, what they know and are able to do includes:
Benchmark 
Description 
Cluster 
Lesson Plans 
A1E  demonstrating a conceptual understanding of variables, expressions, equations, and inequalities (e.g., use letters or boxes to represent values; understand =, ¹, <, and > symbols  1,4  
A2E  modeling and developing strategies for solving equations and inequalities  1,2,3,4  
A3E  recognizing the connection of algebra to the other strands and to reallife situations (e.g., number sentences or formulas to represent realworld problems)  4,5 
FOCUS
Measurement is the connection between numbers and the real world and as such is a vital component of an attempt to organize the world. It allows one to communicate effectively and make decisions. It relates geometry and algebra, as well as geometry and numbers, in both intuitive and formal ways. It is also a connecting theme between such diverse fields as athletics, music, travel, astronomy, and engineering. The study of measurement should consist of active investigations based on realworld problems in both individual and group format. These explorations should include the appropriate use of manipulatives and technology and should encourage the development of communications, reasoning, and problemsolving skills. Students need to learn the effect of unit choice on mathematical entities, such as the shape of graphs and the magnitude of answers. Secondary students should become so adept with the use of units that they are comfortable with the use of compound units (footpounds, miles per second) and specialized units (atmospheres, millennia, gigabytes) as they occur in realworld problems.
STANDARD
In problemsolving investigations, students demonstrate an understanding of the concepts, processes, and reallife applications of measurement.
Students in Grades K4 use number sense, estimation, appropriate manipulatives, tools, and technology as they investigate problems involving measurement. As a result, what they know and are able to do includes:
Benchmark 
Description 
Cluster 
Lesson Plans 
M1E  applying (measure or solve measurement problem) the concepts of length (inches, feet, yards, miles, millimeters, centimeters, decimeters, meters, kilometers), area, volume, capacity (cups, liquid pints and quarts, gallons, milliliters, liters), weight (ounces, pounds, tons, grams, kilograms), mass, time (seconds, minutes, hours, days, weeks, months, years), money, and temperature (Celsius and Fahrenheit) to realworld experiences  1,2,3,4,5  
M2E  selecting and using appropriate standard and nonstandard units of measure (e.g., paper clips and Cuisenaire rods) and tools for measuring length, area, capacity, weight/mass, and time for a given situation by considering the purpose and precision required for the task  1,2,3,4  
M3E  using estimation skills to describe, order, and compare measures of length, capacity, weight/mass, time, and temperature  1,2,3,4  
M4E  converting from one unit of measurement to another within the same system (customary and metric); comparisons between systems should be based on intuitive reference points, not formal computations (e.g., a meter is a little longer than a yard)  2,3,4  
M5E  demonstrating the connection of measurement to the other strands and to reallife situations  2,4,5 
FOCUS
Geometry is the study of the physical shapes of the world in which we live. It provides a natural environment for the use of inductive and deductive reasoning. It is not only basic to design, construction, and engineering, but also to law, medicine, and other fields that depend on critical deductive thinking skills. It provides models for representing many numerical and algebraic concepts. In Grades K4, students must have opportunities to examine, manipulate, and construct geometric models using concrete materials. These activities should take place in a setting where students may freely explore and discuss ideas in order to develop and use appropriate vocabulary. After such firsthand experiences, many students should be able to progress to pictorial and abstract representations. The study of geometry should center around cooperative group investigations designed to promote the discovery of geometric concepts and principles and should encourage the development of communication, reasoning, and problemsolving skills. Secondary students should develop coordinate and transformational geometry as well as the usual axiomatic geometry. They should develop deductive reasoning skills by way of written proofs in a variety of formats. In the study of geometry, students should have access to appropriate manipulatives, technology, and construction materials to enhance their investigations.
STANDARD
In problemsolving investigations, students demonstrate an understanding of geometric concepts and applications involving one, two, and threedimensional geometry, and justify their findings.
Students in Grades K4 use number sense, estimation, models, drawings, manipulatives, and technology as they investigate problems involving geometric concepts. As a result, what they know and are able to do includes:
Benchmark 
Description 
Cluster 
Lesson Plans 
G1E  determining the relationships among shapes  1,2,3,4  
G2E  identifying, describing, comparing, constructing, and classifying twodimensional and threedimensional geometric shapes using a variety of materials  1,2,3,4  
G3E  making predictions regarding combinations, subdivisions, and transformations (slides, flips, turns) of simple plane geometric shapes  1,2,4  
G4E  drawing, constructing models, and comparing geometric shapes, with special attention to developing spatial sense  1,2,4  
G5E  identifying and drawing lines and angles and describing their relationships to each other and to the real world  1,4,5  
G6E  demonstrating the connection of geometry to the other strands and to reallife situations  1,2,3,4,5 
Data Analysis, Propability, and Discrete Math
FOCUS
Data analysis is the collecting, organizing, presenting, and analyzing of numerical information using appropriate statistical methods. Discrete mathematics is that branch of mathematics that involves finite sets and structured sets, including matrices and graph theory. Probability is that branch of mathematics that deals with uncertainty and the likelihood of events occurring or not occurring. These three subjects are closely interwoven. Concepts from these subjects should develop gradually through many varied experiences based on students’ natural interests. These concepts are essential to help students relate mathematical thinking to reallife situations, such as weather, games, sports, newspapers, and business. Classroom explorations involving these concepts should encourage the development of communication, connections, reasoning, and problemsolving skills and should effectively incorporate the use of appropriate models, manipulatives, and technology. Talking and writing should be of particular importance in this strand as students learn to analyze information and express similarities, differences, and patterns based on their investigations. The concepts studies will enable students to effectively communicate information in an organized and graphic manner that will enhance problemsolving skills.
STANDARD
In problemsolving investigations, students discover trends, formulate conjectures regarding causeandeffect relationships, and demonstrate critical thinking skills in order to make informed decisions.
Students in Grades K4 use collection and organizational techniques, number sense, estimation, manipulatives, and technology as they investigate problems involving data. As a result, what they know and are able to do includes:
Benchmark 
Description 
Cluster 
Lesson Plans 
D1E  collecting, organizing, and describing data based on reallife situations  1,3,4,5  
D2E  constructing, reading, and interpreting data in charts, graphs, tables, etc  1,2,3,4  
D3E  formulating and solving problems that involve the use of data  2,3,4  
D4E  exploring, formulating, and solving sequenceofpattern problems involving selection and arrangement of objects/numerals  2,3,4  
D5E  predicting outcomes based on probability (e.g., make predictions of same chance, more likely, or less likely; determine fair and unfair games)  1,2,4  
D6E  demonstrating the connection of data analysis, probability, and discrete math to other strands and reallife situations  1,2,3,4,5 
Patterns, Relations, and Functions
FOCUS
The concepts of patterns, relations, and functions play a central role in modern mathematics. These concepts arise naturally from observations of the world. Business people, social scientists, and physical scientists use mathematics to make predictions following their study of patterns and relationships found among the quantities measured in their respective fields. In Grades K8, students should use informal investigations to observe patterns created by nature and man (flowers, leaves, insects, music, predictable literature, wallpaper, fabric). Students should continue to use the study of patterns to explore mathematical relationships as they verbalize, complete, create, and analyze patterns. This gradual transition from the concrete to the symbolic provides a foundation for the study of functions. Not only does the high school curriculum contain the formal study of functions and inverse relations, it also uses functions and inverse relations as modeling tools for the study of relationships found in our world. This study of functions and how things change leads naturally to powerful analytic techniques, which are collectively called calculus.
STANDARD
In problemsolving investigations, students demonstrate an understanding of patterns, relations, and functions that represent and explain realworld situations.
Students in Grades K4 use number sense, estimation, manipulatives, drawings, tables, graphs, formulas, and technology as they investigate problems involving patterns, relations, and functions. As a result, what they know and are able to do includes:
Benchmark 
Description 
Cluster 
Lesson Plans 
P1E  recognizing, describing, extending, and creating a wide variety of numerical (e.g., skip counting of whole numbers), geometrical, and statistical patterns  1,2,3,4  
P2E  representing and describing mathematical relationships using tables, variables, open sentences, and graphs  1,2,4  
P3E  recognizing the use of patterns, relations, and functions in other strands and in reallife situations  2,3,4,5 