GLEN RIDGE PUBLIC SCHOOLS

Curriculum Guide

Course Title: MATHEMATICS

Subject: Mathematics

Grade Level: 4

Department/School: Mathematics/Ridgewood Avenue

Duration: Full Year

Number of Credits: N/A

Prerequisite: N/A

Elective or Required: N/A

Authors: Eileen Ippolito

Amanda Diglio

Date Submitted: Summer 2007

Course Description

The fourth grade mathematics curriculum prepares students to emphasize conceptual understanding while building a mastery of basic skills.  Throughout the course of the year, the students will explore many strands of mathematics.  These will include the areas of problem solving, numerical operations, measurement, geometry, place value, and analyzing data.  The students will become competent mathematicians through both classroom lessons, as well as investigations, games, construction, and other hands-on activities.

The fourth grade units will emphasize the concepts of multiplication and division as well as mastery of basic addition, subtraction, and multiplication facts.   The program will include the study of whole number and decimal models and their relationship to fractions.   In addition, the students will experience activities involving measurement and data using graphs and other models to read and analyze information.  Geometry is also highlighted as the students learn about linear measurement, area, and 2- and 3-dimensional shapes.   The year is concluded with an introduction to probability and chance.  The Everyday Mathematics program explores a broad mathematics spectrum preparing students to achieve their maximum potential in mathematics.

GLEN RIDGE PUBLIC SCHOOLS

MATHEMATICS MISSION STATEMENT

Mathematics and Computer Science are an integral part of our lives. Students must be actively involved in their mathematics education with problem solving being an essential part of the curriculum. The mathematics and computer science curricula should emphasize thinking skills through a balance of computation, intuition, common sense, logic, analysis and technology. Students will be engaged and challenged in a student-centered learning environment that is developmentally appropriate. Students will communicate mathematical ideas effectively by applying hands-on manipulatives, basic computational skills, mathematical models, and technology in order to solve practical problems.

Goals of the Ridgewood Avenue Mathematics Program

Provided with an environment that encourages problem solving as well as expression of mathematical concepts students will:

• become competent problem solvers who learn to analyze, evaluate, reflect upon, and respond to the ideas of others,
• approach math with an appreciation for various problem solving techniques,
• implement a sequential approach to problem solving including: planning, solving, and evaluating answers,
• apply appropriate problem solving techniques by applying the correct order of operation,
• incorporate mathematical vocabulary into everyday use,
• gather, evaluate, synthesize, and cite data from a variety of technological sources and print materials,
• connect mathematics to everyday experiences and,
• share, display, and/or publish individual and collaborative products.

New Jersey Core Curriculum Standards

The Mathematics Standards consist of five statements, which describe what is essential to excellent mathematics education, and present a view of mathematics teaching and learning that integrates the processes of mathematical activity, the content of mathematics, and the learning environment in the classroom. The following standards were adopted by the New Jersey State Board of Education.

All skills of the New Jersey Core Content Curriculum Standards for Mathematics and NJASK 4 are met or exceeded and referenced throughout the curriculum.

This course will cover the following Core Curriculum Standards:

Descriptive Statement:  Numbers and arithmetic operations are what most of the general public think about when they think of mathematics; and, even though other areas like geometry, algebra, and data analysis have become increasingly important in recent years, numbers and operations remain at the heart of mathematical teaching and learning.  Facility with numbers, the ability to choose the appropriate types of numbers and the appropriate operations for a given situation, and the ability to perform those operations as well as to estimate their results, are all skills that are essential for modern-day life.

Number Sense

Numerical Operations

Estimation

Descriptive Statement:  Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve describing the shapes we see all around us in art, nature, and the things we make.  Spatial sense, geometric modeling, and measurement can help us to describe and interpret our physical environment and to solve problems. 

Geometric Properties

Transforming Shapes

Coordinate Geometry

Units of Measurement

Measuring Geometric Objects

Descriptive Statement: Algebra is a symbolic language used to express mathematical relationships.  Students need to understand how quantities are related to one another, and how algebra can be used to concisely express and analyze those relationships.  Modern technology provides tools for supplementing the traditional focus on algebraic procedures, such as solving equations, with a more visual perspective, with graphs of equations displayed on a screen.  Students can then focus on understanding the relationship between the equation and the graph, and on what the graph represents in a real-life situation. 

Patterns

Functions and Relationships

Modeling

Procedures

Descriptive Statement:  Data analysis, probability, and discrete mathematics are important interrelated areas of applied mathematics.  Each provides students with powerful mathematical perspectives on everyday phenomena and with important examples of how mathematics is used in the modern world.  Two important areas of discrete mathematics are addressed in this standard; a third area, iteration and recursion, is addressed in Standard 4.3 (Patterns and Algebra).

Data Analysis

Probability

Discrete Mathematics – Systematic Listing and Counting

Discrete Mathematics – Vertex-edge Graphs and Algorithms

Descriptive Statement:  The mathematical processes described here highlight ways of acquiring and using the content knowledge and skills delineated in the first four mathematics standards.

Problem Solving Reasoning

Communication Representations

Connections Technology

Scope and Sequence

Standard 4.1 (Number and Numerical Operations)

· Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 4 pertain to these sets of numbers as well): whole numbers through millions; commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10, 12 and 16) as part of a whole, as a subset of a set, and as a location on a number line; decimals through hundredths.

· Demonstrate an understanding of place value concepts.

· Demonstrate a sense of the relative magnitudes of numbers.

· Understand the various uses of numbers: counting, measuring, labeling (e.g., numbers on baseball uniforms), locating (e.g., Room 235 is on the second floor).

· Use concrete and pictorial models to relate whole numbers, commonly used fractions, and decimals to each other, and to represent equivalent forms of the same number.

· Compare and order numbers.

· Explore settings that give rise to negative numbers: temperatures below 0°, debts; extension of the number line.

· Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems: addition and subtraction--joining, separating, comparing; multiplication--repeated addition, area/array; division--repeated subtraction, sharing.

· Develop proficiency with basic multiplication and division number facts using a variety of fact strategies (such as "skip counting" and "repeated subtraction") and then commit them to memory.

· Construct, use, and explain procedures for performing whole number calculations with: pencil-and-paper, mental math, calculator.

· Construct and use procedures for performing decimal addition and subtraction.

· Count and perform simple computations with money: standard dollars and cents notation.

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

· Check the reasonableness of results of computations.

· Use concrete models to explore addition and subtraction with fractions.

· Understand and use the inverse relationships between addition and subtraction and between multiplication and division.

· Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.

· Construct and use a variety of estimation strategies (e.g., rounding and mental math) for estimating both quantities and the results of computations.

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

· Use estimation to determine whether the result of a computation (either by calculator or by hand) is reasonable.

Standard 4.2 (Geometry and Measurement)

· Identify and describe spatial relationships of two or more objects in space: direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?); relative shapes and sizes; shadows (projections) of everyday objects.

· Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them: vertex, edge, face, side, angle; 3D figures--cube, rectangular prism, sphere, cone, cylinder, and pyramid; 2D figures--square, rectangle, circle, triangle, quadrilateral, pentagon, hexagon, octagon; inclusive relationships--squares are rectangles, cubes are rectangular prisms.

· Identify and describe relationships among two-dimensional shapes: congruence; lines of symmetry.

· Understand and apply concepts involving lines, angles, and circles: point, line, line segment, endpoint; parallel, perpendicular; angles--acute, right, obtuse; circles--diameter, radius, center.

· Recognize, describe, extend, and create space-filling patterns.

· Use simple shapes to cover an area (tessellations).

· Describe and use geometric transformations (slide, flip, turn).

· Investigate the occurrence of geometry in nature and art.

· Locate and name points in the first quadrant on a coordinate gird.

· Use coordinates to give or follow directions from one point to another on a map or grid.

· Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.

· Select and use appropriate standard units of measure and measurement tools to solve real-life problems.: Length--fractions of an inch (1/8, 1/4, 1/2), mile, decimeter, kilometer; Area--square inch, square centimeter; Volume--cubic inch, cubic centimeter; Weight--ounce; Capacity--fluid ounce, cup, gallon, milliliter.

· Develop and use personal referents to approximate standard units of measure (e.g., a common paper clip is about an inch long).

· Incorporate estimation in measurement activities (e.g., estimate before measuring).

· Solve problems involving elapsed time.

· Determine the area of simple two-dimensional shapes on a square grid.

· Distinguish between perimeter and area and use each appropriately in problem-solving situations.

· Measure and compare the volume of three-dimensional objects using materials such as rice or cubes.

Standard 4.3 (Patterns and Algebra)

· Recognize, describe, extend, and create patterns: descriptions using words, number sentences/ expressions, graphs, tables, variables (e.g., shape, blank, or letter); sequences that stop or that continue infinitely; whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11, ... or 800, 400, 200...); sequences that can often be extended in more than one way (e.g., the next term after 1, 2, 4, ... could be 8, or 7, or ...)

· Use concrete and pictorial models to explore the basic concept of a function: input/output tables, T-charts; combining two function machines; reversing a function machine.

· Recognize and describe change in quantities: graphs representing change over time (e.g., temperature, height); how change in one physical quantity can produce a corresponding change in another (e.g., pitch of a sound depends on the rate of vibration).

· Construct and solve simple open sentences involving any one operation (e.g., 3 x 6 = ___, n = 15 ÷ 3, 3 x ___ = 0, 16 - c = 7).

· Understand, name, and apply the properties of operations and numbers: commutative (e.g., 3 x 7 = 7 x 3); identity element for multiplication is 1 (e.g., 1 x 8 = 8); associative (e.g., 2 x 4 x 25 can be found by first multiplying ether 2 x 4 or 4 x 25); division by zero is undefined; any number multiplied by zero is zero.

· Understand and use the concepts of equals, less than, and greater than in simple number sentences: symbols ( = , <, >)

Standard 4.4 (Data Analysis, Probability, and Discrete Mathematics)

· Collect, generate, organize, and display data in response to questions, claims, or curiosity: data collected from the school environment.

· Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data: pictograph, bar graph, line plot, line graph, table; average (mean), most frequent (mode), middle term (median).

· Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability: likely, unlikely, certain, impossible, improbable, fair, unfair; more likely, less likely, equally likely; probability of tossing "heads" does not depend on outcomes of previous tosses.

· Determine probabilities of simple events based on equally likely outcomes and express them as fractions.

· Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color): what students think will happen (intuitive); collect data and use that data to predict the probability (experimental); analyze all possible outcomes to find the probability (theoretical).

· Represent and classify data according to attributes, such as shape or color, and relationships: Venn diagrams; numerical and alphabetical order.

· Represent all possibilities for a simple counting situation in an organized way and draw conclusions from this representation: organized lists, charts, tree diagrams; dividing into categories (e.g., to find the total number of rectangles in a grid, find the number of rectangles of each size and add the results).

· Follow, devise, and describe practical sets of directions (e.g., to add two 2-digit numbers).

· Play two-person games and devise strategies for winning the games (e.g., "make 5" where players alternately add 1 or 2 and the person who reaches 5, or another designated number, is the winner).

· Explore vertex-edge graphs and tree diagrams: vertex, edge, neighboring/adjacent, number of neighbors; path, circuit (i.e. path that ends at its starting point).

· Find the smallest number of colors needed to color a map or a graph.

Standard 4.5 (Mathematical Processes)

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

· Solve problems that arise in mathematics and in other contexts: open-ended problems; non-routine problems; problems with multiple solutions; problems that can be solved in several ways.

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

· Pose problems of various types and levels of difficulty.

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

· Use communication to organize and clarify mathematical thinking: reading and writing; discussion, listening, and questioning.

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

· Analyst and evaluate the mathematical thinking and strategies of others.

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

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

· 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.)

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

· Apply mathematics in practical situations and in other disciplines.

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

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

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

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

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

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

· Make and investigate mathematical conjectures: counterexamples as a means of disproving conjectures; verifying conjectures using informal reasoning or proofs.

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

· 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)

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

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

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

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

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

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

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

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

Curriculum Description

UNIT 1: NAMING AND CONSTRUCTING GEOMETRIC FIGURES

Objectives:

After completion of this unit students will be able to:

1. Use a compass and straightedge to construct geometric figures. (4.2)

2. Identify properties of polygons. (4.2)

3. Distinguish between convex and concave polygons. (4.2)

4. Explore geometric definitions and classification. (4.2)

5. Classify quadrangles according to side and angle properties. (4.2)

6. Define the properties of a circle. (4.2)

7. Name, draw, and label line segments, line and rays. (4.2)

8. Name, draw, and label angles, triangles, and quadrangles. (4.2)

9. Identify and describe right angles, parallel lines, and line segments. (4.2)

10. Demonstrate knowledge of addition and subtraction facts. (4.1, 4.5)

Approximate duration: 12 days

Activities:

- Use geometric manipulatives to construct figures to learn about the varying properties of shapes.

- Become familiar with using tools such as a straightedge and a compass.

- Use the book The Greedy Triangle to distinguish the differences between polygons.

- Play games such as Touch and Match It Quadrangles, Geometry 5 Questions, and Name that Polygon to enhance concepts.

UNIT 2: USING NUMBERS AND ORGANIZING DATA

Objectives:

After completion of this unit students will be able to:

1. Review and find examples of the various ways in which number are used. (4.1)

2. Find equivalent names for numbers. (4.1)

3. Name values of digits in the number up to hundred-millions. (4.1)

4. Read and write numbers up to hundred-millions. (4.1)

5. Practice place value skills. (4.1)

6. Read and write large numbers. (4.1)

7. Organize and display data with tally charts. (4.4)

8. Use the statistical landmarks mean, median, mode, maximum, minimum and range for a set of data. (4.4)

9. Organize and display a set of data with a line plot. (4.4)

10. Organize and display a set of data with a bar graph. (4.4)

11. Add multidigit numbers. (4.4)

12. Subtract multidigit numbers. (4.4)

13. Measure length to the nearest ½ centimeter. (4.2)

Approximate duration: 14 days

Activities:

- Examine different uses of numbers and review how we record numbers.

- Discuss the concepts of minimum, maximum, mean, median, mode, and range as they collect and organize data.

- Use books such as How Much Is a Million? and If You Made a Million can be used to aid in the discussion of place value.

- Use many games to enhance concepts in this unit including; Name That Number, Number Top-It, Addition Top-It, Subtraction Top-It, High Number Toss, and Subtraction Target Practice.

UNIT 3: MULTIPLICATION AND DIVISION; NUMBER SENTENCES AND ALGEBRA

Objectives:

After completion of this unit students will be able to:

1. Review strategies for multiplication facts working towards instant recall. (4.1)

2. Explore the relationship between multiplication and division. (4.1)

3. Explore the relationship between division and fractions. (4.1)

4. Understand the function and placement of parentheses in number sentences. (4.1, 4.5)

5. Review the meaning of a number sentence. (4.1, 4.5)

6. Determine whether number sentences are true or false. (4.1, 4.5)

7. Solve addition and subtraction number stories. (4.1, 4.5)

8. Use a map scale to estimate distances. (4.2)

9. Show mastery of multiplication facts. (4.1)

10. Show mastery of division facts. (4.1)

11. Familiarize themselves with vocabulary and notation of open sentences. (4.5)

12. Solve open sentences. (4.1, 4.5)

13. Develop reasoning skills. (4.1, 4.5)

Approximate duration: 15 days

Activities:

- Review and practice basic multiplication facts and explore the relationship between multiplication and division.

- Read the books Each Orange Had 8 Slices: A Counting Book and Sea Squares to explore the relationship between multiplication.

- Explore algebraic concepts as they solve equations missing integers.

- Additionally, review general problem solving techniques.

- Play the games Baseball Multiplication and Multiplication Top-It to enhance the students’ multiplication skills.

UNIT 4: DECIMALS AND THEIR USES

Objectives:

After completion of this unit students will be able to:

1. Review basic concepts and notations for decimals through the hundredths. (4.1)

2. Read and write decimals though the thousandths. (4.1)

3. Compare and order decimals in the tenths and hundredths. (4.1)

4. Explore the significance of decimals. (4.1)

5. Estimate sums of decimals. (4.1)

6. Estimate differences of decimals. (4.1)

7. Extend methods of whole number addition to decimals. (4.1)

8. Extend methods of whole number subtraction to decimals. (4.1)

9. Solve 1 and 2 place decimal addition and subtraction problem and number stories. (4.1, 4.5)

10. Use dollars-and-cents notations. (4.1)

11. Compute balances in a savings account. (4.1)

12. Extend basic concepts and notation for decimals to thousandths. (4.1)

13. Review the relationship among metric units of