STANDARD 4.1 (NUMBER AND NUMERICAL OPERATIONS) ALL STUDENTS WILL DEVELOP NUMBER SENSE AND WILL PERFORM STANDARD NUMERICAL OPERATIONS AND ESTIMATIONS ON ALL TYPES OF NUMBERS IN A VARIETY OF WAYS.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

A. Number Sense

 1. Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 5 pertain to these sets of numbers as well).

• All fractions as part of a whole, as subset of a set, as a location on a number line, and as divisions of whole numbers
• All decimals

 2. Recognize the decimal nature of United States currency and compute with money.

 3. Demonstrate a sense of the relative magnitudes of numbers.

 4. Use whole numbers, fractions, and decimals to represent equivalent forms of the same number.

 5. Develop and apply number theory concepts in problem solving situations.

• Primes, factors, multiples

 6. Compare and order numbers.

B. Numerical Operations

 1. Recognize the appropriate use of each arithmetic operation in problem situations.

 2. Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with:

• Pencil-and-paper
• Mental math
• Calculator

 3. Use an efficient and accurate pencil-and-paper procedure for division of a 3-digit number by a 2-digit number.

 4. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.

 5. Check the reasonableness of results of computations.

 6. Understand and use the various relationships among operations and properties of operations.

C. Estimation

 1. Use a variety of estimation strategies for both number and computation.

 2. Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.

 3. Determine the reasonableness of an answer by estimating the result of operations.

 4. Determine whether a given estimate is an overestimate or an underestimate.

STANDARD 4.2 (GEOMETRY AND MEASUREMENT) ALL STUDENTS WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES, RELATIONSHIPS, AND MEASUREMENT TO MODEL, DESCRIBE AND ANALYZE PHENOMENA.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

A. Geometric Properties

 1. Understand and apply concepts involving lines and angles.

       Notation for line, ray, angle, line segment

       Properties of parallel, perpendicular, and intersecting lines

       Sum of the measures of the interior angles of a triangle is 180°

 2. Identify, describe, compare, and classify polygons.

       Triangles by angles and sides

       Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi

       Polygons by number of sides

       Equilateral, equiangular, regular

       All points equidistant from a given point form a circle

 3. Identify similar figures.

 4. Understand and apply the concepts of congruence and symmetry (line and rotational).

B. Transforming Shapes

 1. Use a translation, a reflection, or a rotation to map one figure onto another congruent figure.

 2. Recognize, identify, and describe geometric relationships and properties as they exist in nature, art, and other real-world settings.

C. Coordinate Geometry

 1. Create geometric shapes with specified properties in the first quadrant on a coordinate grid.

D. Units of Measurement

 1. Select and use appropriate units to measure angles and area.

 2. Convert measurement units within a system (e.g., 3 feet = ___ inches).

 3. Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile).

 4. Use measurements and estimates to describe and compare phenomena.

E. Measuring Geometric Objects

 1. Use a protractor to measure angles.

 2. Develop and apply strategies and formulas for finding perimeter and area.

       Square

       Rectangle

 3. Recognize that rectangles with the same perimeter do not necessarily have the same area and vice versa.

 4. Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one’s foot).

STANDARD 4.3 (PATTERNS AND ALGEBRA) ALL STUDENTS WILL REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES AND SOLVE PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES.

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:

A. Patterns

 1. Recognize, describe, extend, and create patterns involving whole numbers.

 Descriptions using tables, verbal rules, simple equations, and graphs

B. Functions & Relationships

 1. Describe arithmetic operations as functions, including combining operations and reversing them.

 2. Graph points satisfying a function from T-charts, from verbal rules, and from simple equations.

C. Modeling

 1. Use number sentences to model situations.

 Using variables to represent unknown quantities

 Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations

 2. Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.

 Changes over time

 Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)

D. Procedures

 1. Solve simple linear equations with manipulatives and informally

 Whole-number coefficients only, answers also whole numbers

 Variables on one side of equation

STANDARD 4.4 (DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS) ALL STUDENTS WILL DEVELOP AN UNDERSTANDING OF THE CONCEPTS AND TECHNIQUES OF DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS, AND WILL USE THEM TO MODEL SITUATIONS, SOLVE PROBLEMS, AND ANALYZE AND DRAW APPROPRIATE INFERENCES FROM DATA.

Building upon knowledge and skills gained in preceding grades, by the end of

Grade 5, students will:

A. Data Analysis

 1. Collect, generate, organize, and display data.

• Data generated from surveys

 2. Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data.

• Bar graph, line graph, circle graph, table
• Range, median, and mean

 3. Respond to questions about data and generate their own questions and hypotheses.

B. Probability

 1. Determine probabilities of events.

• Event, probability of an event
• Probability of certain event is 1 and of impossible event is 0

 2. Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag).

• Given numbers of various types of items in a bag, what is the probability that an item of one type will be picked
• Given data obtained experimentally, what is the likely distribution of items in the bag

 3. Model situations involving probability using simulations (with spinners, dice) and theoretical models.

C. Discrete Mathematics—Systematic Listing and Counting

 1. Solve counting problems and justify that all possibilities have been enumerated without duplication.

• Organized lists, charts, tree diagrams, tables

 2. Explore the multiplication principle of counting in simple situations by representing all possibilities in an organized way (e.g., you can make 3 x 4 = 12 outfits using 3 shirts and 4 skirts).

D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms

 1. Devise strategies for winning simple games (e.g., start with two piles of objects, each of two players in turn removes any number of objects from a single pile, and the person to take the last group of objects wins) and express those strategies as sets of directions.

STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATION, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

At each grade level, with respect to content appropriate for that grade level, students will:

A. Problem Solving

 1. Learn mathematics through problem solving, inquiry, and discovery.

 2. Solve problems that arise in mathematics and in other contexts (cf. workplace readiness standard 8.3).

• Open-ended problems
• Non-routine problems
• Problems with multiple solutions
• Problems that can be solved in several ways

 3. Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.

 4. Pose problems of various types and levels of difficulty.

 5. Monitor their progress and reflect on the process of their problem solving activity.

B. Communication

 1. Use communication to organize and clarify their mathematical thinking.

• Reading and writing
• Discussion, listening, and questioning

 2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.

 3. Analyze and evaluate the mathematical thinking and strategies of others.

 4. Use the language of mathematics to express mathematical ideas precisely.

C. Connections

 1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).

 2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point).

 3. Recognize that mathematics is used in a variety of contexts outside of mathematics.

 4. Apply mathematics in practical situations and in other disciplines.

 5. Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).

 6. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

(Continued)

STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATION, CONNECTIONS, REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE MATHEMATICAL IDEAS.

At each grade level, with respect to content appropriate for that grade level, students will:

D. Reasoning

 1. Recognize that mathematical facts, procedures, and claims must be justified.

 2. Use reasoning to support their mathematical conclusions and problem solutions.

 3. Select and use various types of reasoning and methods of proof.

 4. Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.

 5. Make and investigate mathematical conjectures.

• Counterexamples as a means of disproving conjectures
• Verifying conjectures using informal reasoning or proofs.

 6. Evaluate examples of mathematical reasoning and determine whether they are valid.

E. Representations

 1. Create and use representations to organize, record, and communicate mathematical ideas.

• Concrete representations (e.g., base-ten blocks or algebra tiles)
• Pictorial representations (e.g., diagrams, charts, or tables)
• Symbolic representations (e.g., a formula)
• Graphical representations (e.g., a line graph)

 2. Select, apply, and translate among mathematical representations to solve problems.

 3. Use representations to model and interpret physical, social, and mathematical phenomena.

F. Technology

 1. Use technology to gather, analyze, and communicate mathematical information.

 2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.

 3. Use graphing calculators and computer software to investigate properties of functions and their graphs.

 4. Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).

 5. Use computer software to make and verify conjectures about geometric objects.

 6. Use computer-based laboratory technology for mathematical applications in the sciences.