Elementary Curriculum for
Fifth Grade - Mathematics

I.  Patterns, Relationships and Functions

Content Standard 1: Students recognize similarities and generalize patterns, use patterns to create models and make predictions, describe the nature of patterns and relationships, and construct representations of mathematical relationships.  (Patterns)

• The learner will create, explain, extend and record complex number and geometric patterns.  (1.1)
• The learner will create, describe, record and interpret more complex patterned relationships using graphs, tables and charts.   (1.2)
• The learner will apply their experiences with patterns to help solve problems and explore new content.  (1.5)

Content Standard 2: Students describe the relationships among variables, predict what will happen to one variable as another variable is changed, analyze variation and sources of variability and compare patterns of change. (Variability and Change)

• The learner will recognize change and variability when it occurs in a variety of settings.  (2.1)
• The learner will recognize that change is often predictable, but variable, and that patterns emerge that help to describe the change.  (2.2)
• The learner will use tables, charts, open sentences and hands-on models to represent change and variability.  (2.4)
• The learner will explore variability and change in a variety of contexts, investigations and problems.  (2.6)

II.  Geometry and Measurement

Content Standard 1: Students develop spatial sense, use shape as an analytic and descriptive tool, identify characteristics and define shapes, identify properties and describe relationships among shapes.  (Shape and Shape Relationships)

• The learner will measure and describe attributes of 2D and 3D shapes and using appropriate tools.  (1.2)
• The learner will be given a set of attributes and will be able to construct a 3D shape.  (1.4)
• The learner will explore ways to combine, dissect and transform shapes.  (1.5)
• The learner will recognize parallel and perpendicular line segments and figures that has a similarity and/or congruence.  (1.6)
• The learner will use shape, shape properties and shape relationships to describe the physical world and to solve problems.  (1.7) 

Content Standard 2: Students identify locations of objects, identify location relative to other objects and describe the effects of transformations (e.g., sliding, flipping, turning, enlarging, reducing) on an object.  (Position)

• The learner will locate and describe objects in terms of their position including front, back, inside, outside, right, left, over, under, next to, between, locations on the number line and locate ordered pairs on a map.  (2.1)
• The learner will locate and describe objects in terms of their orientation, direction and relative position including up, down, front, back, N-S-E-W, flipped, turned, translated, recognize symmetrical objects and identify their lines of symmetry.  (2.2)
• The learner will use concepts of position, direction and orientation to describe the physical world and to solve problems.  (2.5)

Content Standard 3: Students compare attributes of two objects, or of one object with a standard (unit), and analyze situations to determine what measurements should be made and to what level of precision.  (Measurement)

• The learner will compare attributes of objects using nonstandard and standard (metric) forms of measurement (weight, capacity, temperature, length and volume).  (3.1)
• The learner will identify the attribute to be measured and select the appropriate unit of measurement (time, temperature, length, mass, money, perimeter, weight, area and volume).  (3.2)
• The learner will develop strategies for estimating measures and compare the estimates to the results of the measurement; decide if an estimate is “a good estimate.”  (3.3)
• The learner will explain the meaning of measurements and recognize that the number of units it takes to measure an object is related to the size of the unit.  (3.4)
• The learner will explore scale drawings, models and maps and relate them to measurements of real objects (rectangles, triangles, parallelograms).  (3.5)
• The learner will apply measurement to describe the real world and to solve problems.  (3.6)

III. Data Analysis and Statistics

Content Standard 1: Students collect and explore data, organize data into a useful form and develop skill in representing and reading data displayed in different formats.  (Collection, Organization and Presentation of Data)

• The learner will collect and explore data through counting, measuring and conducting surveys and experiments.  (1.1)
• The leaner will organize data using concrete objects, pictures, tallies, tables, charts, diagrams and graphs.  (1.2)
• The learner will present data using abstract representations and explain the meaning of the data.  (1.3)
• The learner will decide what data are important to answer the question or solve the problem and design and use strategies to obtain, organize, and present those data.  (1.4) 

Content Standard 2: Students examine data and describe characteristics of a distribution, relate data to the situation from which they arose, and use data to answer questions convincingly and persuasively.  (Description and Interpretation)

• The learner will read and explain data they have collected and organized themselves and progress to reading data from other sources.  (2.1)
• The learner will describe the shape of the graph.  (2.2)
• The learner will draw, explain and justify conclusions, such as trends based on data.  (2.3)
• The learner will raise and answer questions about the data collected.  Draw conclusions from the data.  (2.4)
• The learner will formulate questions and problems and gather and interpret data to answer those questions.  (2.5)

Content Standard 3: Students draw defensible inferences about unknown outcomes, make predictions and identify the degree of confidence they have in their predictions.  (Inference and Prediction)

• The learner will make and test hypotheses.  (3.1)
• The learner will conduct surveys, samplings and experiments to solve problems and answer questions of interest to them.  (3.2)
• The learner will formulate and communicate arguments and conclusions based on data and evaluate their arguments and those of others.  (3.3)
• The learner will make and explain predictions based on data.  (3.4)
• The learner will make predictions to answer questions and solve problems.  (3.5)

IV.  Number Sense and Numeration

Content Standard 1: Students experience counting and measuring activities to develop intuitive sense about numbers, develop understanding about properties of numbers, understand the need for and existence of different sets of numbers, and investigate properties of special numbers.  (Concepts and Properties of Numbers)

• The learner will develop an understanding of whole numbers and read, write and count using whole numbers.  Investigate basic concept of fractions.  (1.1)
• The learner will investigate and develop an understanding of the base 10 place-value system.  (1.2)
• The learner will develop an understanding of the properties of numbers.  (1.3)
• The learner will apply their understanding of number systems to model and solve problems concretely, pictorially and symbolically.  (1.4) 

Content Standard 2: Students recognize that numbers are used in different ways such as counting, measuring, ordering and estimating, understand and produce multiple representations of a number and translate among equivalent representations.  (Representation and Uses of Numbers)

• The learner will represent whole numbers, decimals and fractions using concrete, pictorial and symbolic representations.  (2.1)
• The learner will explore and recognize different representations for the same number and explain why they are the same.  (2.2)
• The learner will investigate ways numbers are used.  (2.3)
• The learner will explore strategies for estimating quantity and evaluate the reasonableness of their estimates.  (2.4)
• The learner will select appropriate numbers and representations in order to solve real world problems.  (2.5) 

Content Standard 3: Students investigate relationships such as equality, inequality, inverses, factors and multiples and represent and compare very large and very small numbers.  (Number Relationships)

• The learner will compare and order numbers using “equal,” “less than,” or “greater than.”  (3.1)
• The learner will use part-whole relationships to explore numbers, develop number concepts and understand computation concretely, pictorially and symbolically.  (3.2)
• The learner will classify numbers as even or odd and explore concepts of factors and multiples.  (3.3)
• The learner will apply understanding of number relationships in solving problems  (3.5)

V.  Numerical and Algebraic Operations and Analytical Thinking

Content Standard 1: Students understand and use various types of operations (e.g. ,addition, subtraction, multiplication, division) to solve problems.  (Operations and their Properties)

• The learner will use manipulatives to model operations with numbers; develop their own methods of recording operations; and relate their models and recordings to standard symbolic expressions and algorithms.  (1.1)
• The learner will develop and apply the appropriate method of computation from among mental computation, estimation, paper-and-pencil or calculators; explain why they are choosing a method and how they know which operations to perform in a given situation.  (1.2)
• The learner will explore properties of operations (e.g., commutative and distributive properties) and give examples of how they use those properties).  (1.3)
• The learner will apply appropriate operations in solving problems.  (1.4) 

Content Standard 2: Students analyze problems to determine an appropriate process for solution, and use algebraic notations to model or represent problems.  (Algebraic and Analytic Thinking)

• The learner will write and solve open sentences and write stories to fit the open sentences.  (2.1)
• The learner will explore algebraic concepts with manipulatives such as balance scales, tables of input and output and pictorial representation of problems.  (2.2)
• The learner will find replacements for the variable(s) in open sentences.  (2.3)
• The learner will use analytic thinking to describe situations and solve problems.  (2.4)

VI.  Probability and Discrete Mathematics

Content Standard 1: Students develop an understanding of the notion of certainty and of probability as a measure of the degree of likelihood that can be assigned to a given event based on the knowledge available, and make critical judgments about claims that are made in probabilistic situations.  (Probability)

• The learner will explain the difference between chance and certainty and give examples to illustrate their understanding.  (1.1)
• The learner will compare events and describe them as “more likely” or “less likely” and use the language of fractions to describe simple probabilities.  (1.2)
• The learner will conduct experiments with concrete objects to explore concepts and develop an intuitive understanding of how the conditions of the experiment can affect the outcome.  (1.3)
• The learner will conduct the experiments, record the outcomes, examine those outcomes to determine if they make sense and search for explanations of the outcomes.  (1.4)
• The learner will conduct probability experiments and simulations to model and solve problems.  (1.5) 

Content Standard 2: Students investigate practical situations such as scheduling, routing, sequencing, networking, organizing and classifying, and analyze ideas like recurrence relations, induction, iteration and algorithm design.  (Discrete Mathematics)

• The learner will use manipulatives and diagrams to explore problems and involving counting and arranging objects.  (2.1)
• The learner will explore sets and set relationships by sorting and classifying objects.  (2.2)
• The learner will explore situations in which they model and trace paths using figures consisting of vertices connected by edges.  (2.3)
• The learner will explore now-next patterns.  (2.4)
• The learner will explore, develop and invent their own algorithms to accomplish a task or to solve numerical problems.  (2.5)
• The learner will use discrete mathematics concepts as described above to model situations and solve problems; and look for whether or not there is a solution (existence problems), determine how many solutions there are (counting problems) and decide upon a best solution (optimization problems).  (2.6)