• Number and Operations Standard
    Instructional programs from prekindergarten through grade 12
    should enable all students to--
     
    Understand numbers, ways of representing numbers, relationships among numbers, and number systems
     
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals; 
    * recognize equivalent representations for the same number and generate them by decomposing and composing numbers; 
    * develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers; 
    * use models, benchmarks, and equivalent forms to judge the size of fractions; 
    * recognize and generate equivalent forms of commonly used fractions, decimals, and percents; 
    * explore numbers less than 0 by extending the number line and through familiar applications; 
    * describe classes of numbers according to characteristics such as the nature of their factors.
     Understand meanings of operations and how they relate to one another
     
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * understand various meanings of multiplication and division; 
    * understand the effects of multiplying and dividing whole numbers; 
    * identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems; 
    * understand and use properties of operations, such as the distributivity of multiplication over addition. 

    Compute fluently and make reasonable estimates
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 3050; 
    * develop fluency in adding, subtracting, multiplying, and dividing whole numbers; 
    * develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results; 
    * develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience; 
    * use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals; 
    * select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools.

    Algebra
    Instructional programs from prekindergarten through grade 12
    should enable all students to--
     
    Understand patterns, relations, and functions
     
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * describe, extend, and make generalizations about geometric and numeric patterns; 
    * represent and analyze patterns and functions, using words, tables, and graphs. 

    Represent and analyze mathematical situations and structures using algebraic symbols
     
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers; 
    * represent the idea of a variable as an unknown quantity using a letter or a symbol; 
    * express mathematical relationships using equations.
     
    Use mathematical models to represent and understand quantitative relationships
     
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.
     Analyze change in various contexts
     
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * investigate how a change in one variable relates to a change in a second variable; 
    * identify and describe situations with constant or varying rates of change and compare them.
     
    Geometry Standard
    Instructional programs from prekindergarten through grade 12
    should enable all students to--
     
    Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
     
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes; 
    * classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids; 
    * investigate, describe, and reason about the results of subdividing, combining, and transforming shapes; 
    * explore congruence and similarity; 
    * make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions. 

    Specify locations and describe spatial relationships using coordinate geometry and other representational systems
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * describe location and movement using common language and geometric vocabulary; 
    * make and use coordinate systems to specify locations and to describe paths; 
    * find the distance between points along horizontal and vertical lines of a coordinate system. 
     Apply transformations and use symmetry to analyze mathematical situations
     Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * predict and describe the results of sliding, flipping, and turning two-dimensional shapes; 
    * describe a motion or a series of motions that will show that two shapes are congruent; 
    * identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs. 
    Use visualization, spatial reasoning, and geometric modeling to solve problems
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * build and draw geometric objects; 
    * create and describe mental images of objects, patterns, and paths; 
    * identify and build a three-dimensional object from two-dimensional representations of that object; 
    * identify and draw a two-dimensional representation of a three-dimensional object; 
    * use geometric models to solve problems in other areas of mathematics, such as number and measurement; 
    * recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.

    Geometry Standard
    Instructional programs from prekindergarten through grade 12
    should enable all students to--
     
    Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
    Understand measurable attributes of objects and the units, systems, and processes of measurement
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute; 
    * understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems; 
    * carry out simple unit conversions, such as from centimeters to meters, within a system of measurement; 
    * understand that measurements are approximations and how differences in units affect precision; 
    * explore what happens to measurements of a two-dimensional shape such as its perimeter and area when the shape is changed in some way. 
    Apply appropriate techniques, tools, and formulas to determine measurements.
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * develop strategies for estimating the perimeters, areas, and volumes of irregular shapes; 
    * select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles; 
    * select and use benchmarks to estimate measurements; 
    * develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms; 
    * develop strategies to 

    Data Analysis and Probability Standard
    Instructional programs from prekindergarten through grade 12
    should enable all students to--
     Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * design investigations to address a question and consider how data-collection methods affect the nature of the data set; 
    * collect data using observations, surveys, and experiments; 
    * represent data using tables and graphs such as line plots, bar graphs, and line graphs; 
    * recognize the differences in representing categorical and numerical data. 
    Select and use appropriate statistical methods to analyze data
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * describe the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed; 
    * use measures of center, focusing on the median, and understand what each does and does not indicate about the data set; 
    * compare different representations of the same data and evaluate how well each representation shows important aspects of the data. 
    Select and use appropriate statistical methods to analyze data
         
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * describe the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed; 
    * use measures of center, focusing on the median, and understand what each does and does not indicate about the data set; 
    * compare different representations of the same data and evaluate how well each representation shows important aspects of the data. 
     
    Develop and evaluate inferences and predictions that are based on data
     
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions or predictions. 
    Understand and apply basic concepts of probability
          
    Grades 3-5 Expectations:
    In grades 3-5 all students should-
    * describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, equally likely, and impossible;
    * predict the probability of outcomes of simple experiments and test the predictions; 
    * understand that the measure of the likelihood of an event can be represented by a number from 0 to 1. 
     
    ALL GRADES
    Problem Solving
    Instructional programs from prekindergarten through grade 12
    should enable all students to--
     
    * Build new mathematical knowledge through problem solving  
    * Solve problems that arise in mathematics and in other contexts  
    * Apply and adapt a variety of appropriate strategies to solve problems  
    * Monitor and reflect on the process of mathematical problem solving  
      
     
     
    Reasoning and Proof
    Instructional programs from prekindergarten through grade 12 should enable
    all students to--
    * Recognize reasoning and proof as fundamental aspects of mathematics  
    * Make and investigate mathematical conjectures  
    * Develop and evaluate mathematical arguments and proofs  
    * Select and use various types of reasoning and methods of proof  
     

     
     
     
       Communication
    Instructional programs from prekindergarten through grade 12 should enable
    all students to--
    * Organize and consolidate their mathematical thinking through communication  
    * Communicate their mathematical thinking coherently and clearly to peers, teachers, and others  
    * Analyze and evaluate the mathematical thinking and strategies of others;  
    * Use the language of mathematics to express mathematical ideas precisely.   
     

     
     
     
       Connections
    Instructional programs from prekindergarten through grade 12 should enable
    all students to-- 
    * Recognize and use connections among mathematical ideas  
    * Understand how mathematical ideas interconnect and build on one another to produce a coherent whole   
    * Recognize and apply mathematics in contexts outside of mathematics  
     

     
     
     
       Representation
    Instructional programs from prekindergarten through grade 12 should enable
    all students to--
    * Create and use representations to organize, record, and communicate mathematical ideas  
    * Select, apply, and translate among mathematical representations to solve problems  
    * Use representations to model and interpret physical, social, and mathematical phenomena  
     Grades 3-5
     
     
      
    5.1 Communicating about Mathematics Using Games
    Mathematical games can foster mathematical communication as students explain and justify their moves to one another. In addition, games can motivate students and engage them in thinking about and applying concepts and skills. The first part of this example, Playing Fraction Tracks, contains an interactive version of a game (based on the work of Akers, Tierney, Evans, and Murray 1998) that can be used in the grades 3-5 classroom to support students' learning about fractions. By working on this activity, students have opportunities to think about how fractions are related to a unit whole, compare fractional parts of a whole, and find equivalent fractions, as discussed in the Number and Operations Standard. In the second part, The Role of the Teacher, two video clips illustrate communication about mathematics among a teacher and her students. The third part, Communication among Students, shows how activities like this allow students to use communication as a tool to deepen their understanding of mathematics, as described in the Communication Standard. In the fourth part, Reflecting on Practice, the teacher reflects on her own mathematical learning that occurs as a result of using activities like this game with her 5th-grade students.
     
     

     
      
    5.2 Understanding Distance, Speed, and Time Relationships Using Simulation Software
    This example includes a software simulation of two runners along a track. Students can control the speeds and starting points of the runners, watch the race, and examine a graph of the time-versus-distance relationship. The computer simulation uses a context familiar to students, and the technology allows them to analyze the relationships more deeply because of the ease of manipulating the environment and observing the changes that occur. Activities like these can help students in the upper elementary grades understand ideas about functions and about representing change over time, as described in the Algebra Standard.
     
     

     
       5.3 Exploring Properties of Rectangles and Parallelograms Using Dynamic Software
    Dynamic geometry software provides an environment in which students can explore geometric relationships and make and test conjectures. In this example, properties of rectangles and parallelograms are examined. The emphasis is on identifying what distinguishes a rectangle from a more general parallelogram. Such tasks and the software can help teachers address the Geometry Standard.
     
     

     
       5.4 Accessing and Investigating Data Using the World Wide Web
    Data sets available on the Internet are valuable resources for studying real data to address questions that interest students. Teachers and students can download data sets from the World Wide Web, collaborate in online data-collection projects, and search electronic libraries and data files. This example describes activities in which students can use census data available on the Web to examine questions about population. Working on such activities, students can also formulate their own questions and use the mathematics they are studying to address these questions. They can propose and justify conclusions that are based on data and design further studies on the basis of conclusions or predictions, as described in the Data Analysis and Probability Standard.
     
     

     
       5.5 Collecting, Representing, and Interpreting Data Using Spreadsheets and Graphing Software
    Spreadsheets and graphing software are tools for organizing, representing, and comparing data. This activity illustrates how weather data can be collected and examined using these tools. In the first part, Collecting and Examining Weather Data, students organize and then examine data that has been collected over a period of time in a spreadsheet. In the second part, Representing and Interpreting Data, students use the graphing functions of a spreadsheet to help them interpret data. Working on activities like these, students learn to set up a simple spreadsheet and use it in posing and solving problems, examining data, and investigating patterns, as described in the Representation Standard.
     
    Standards
    http://standards.nctm.org/document/eexamples/index.htm#3-5