GLEN RIDGE PUBLIC SCHOOLS

Curriculum Guide

Course Title: MATHEMATICS

Subject: Mathematics

Grade Level: 3

Department/School: Mathematics/ Ridgewood Avenue School

Duration: Full year

Number of Credits: NA

Prerequisite: NA

Elective or Required: NA

Authors: Karen Bauer & Fiona Goldberg

Date Submitted: Summer 2007

Course Description

The third 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 third grade units will emphasize the concept 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 measures, 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 Glen Ridge Mathematics Program

Provided with an environment that encourages mathematic competence from the concrete to the abstract, students will:

• understand the meanings, uses, and representations of numbers,
• compute accurately while understanding the meanings of various operations,
• make reasonable estimates,
• select, create, analyze, and interpret appropriate graphical representations of collected data,
• understand and apply basic concepts of probability,
• investigate characteristics and properties of 2 and 3 dimensional geometric shapes,
• apply transformations and symmetry in geometric situations,
• understand the systems and processes of measurement using appropriate techniques, tools, and units in making measurements,
• extend, describe, and create numeric patterns,
• become competent problem solvers in everyday situations, applications, and purely mathematical contexts,

• convey methods and strategies when solving problems using writing, pictures, or graphs,
• approach mathematics with a variety of strategies whereby they analyze, evaluate, and reflect while applying their skills and understanding,
• understand and identify equivalent names for numbers,
• interpret numerical relations of whole numbers, decimals, and fractions.

New Jersey Core Curriculum Standards

All skills of the New Jersey Core Content Curriculum Standards for Mathematics and the NJASK 3 are met or exceeded and referenced throughout the curriculum.

STANDARD 4.1: ALL STUDENTS WILL DEVELOP NUMBER SENSE AND WILL PERFORM STANDARD NUMERICAL OPERATIONS AND ESTIMATIONS ON ALL TYPES OF NUMBERS.

Objectives:

· Use real-life experiences, physical materials, and technology to construct meanings for numbers; whole numbers through hundred thousands; commonly used fractions (denominators of 2,3,4,5,6,8,10) as part of a whole, as a subset of a set, and as a location on a number line.

· Demonstrate an understanding of whole number place value concepts.

· Identify whether any whole number is odd or even.

· Explore the extension of the place value system to decimals through hundredths.

· Understand the various uses of numbers: counting, measuring, and labeling.

· Compare and order numbers.

· 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").

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

· Use efficient and accurate pencil-and-paper procedures for computation with whole numbers: addition of 3-digit numbers; subtraction of 3-digit numbers; multiplication of 2-digit numbers by 1-digit numbers.

· Count and perform simple computations with money: 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.

· 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 result 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: ALL STUDENTS WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES, RELATIONSHIPS, AND MEASUREMENT TO MODEL, DESCRIBE, AND ANALYZE PHENOMENA.

Objectives:

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

· 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, pentagon, hexagon, octagon.

· Identify and describe relationships among two-dimensional shapes: same size, same shape; lines of symmetry.

· Understand and apply concepts involving lines, angles, and circles: line, line segment, endpoint.

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

· 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 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 (1/4, 1/2), mile, decimeter, kilometer; Area--square inch, square centimeter; Weight--ounce; Capacity--fluid ounce, cup, gallon, milliliter.

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

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

· Determine the perimeter of simple shapes by measuring all of the sides.

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

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

Objectives:

· Recognize, describe, extend, and create patterns: descriptions using words and number sentences/ expressions; 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,...)

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

· Recognize and describe change in quantities: graphs representing change over time (e.g., temperature, height).

· Construct and solve simple open sentences involving addition or subtraction (e.g., 3 + 6 = ___, n = 15 - 3, 3 + ___ = 3, 16 - c = 7) .

· Understand 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); any number multiplied by zero is zero.

STANDARD 4.4: 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 DRAW APPROPRIATE INFERENCES FROM DATA.

Objectives:

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

· Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data: pictograph, bar graph, table.

· Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability: likely, unlikely, certain, impossible; more likely, less likely, equally likely.

· 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).

· 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.

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

· Explore vertex-edge graphs: vertex, edge, path.

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

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

Objectives:

• 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.
• Analyze 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.
• 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: ROUTINES, REVIEW, AND ASSESSMENT

Objectives:

After completion of this unit, students will be able to:

1. Identify types and uses of numbers. (4.1)

2. Recognize and describe patterns on a number grid. (4.3)

3. Identify time to the nearest half hour, quarter hour, and five minutes. (4.2)

4. Measure to the nearest inch and centimeter. (4.2)

5. Identify and describe 2-dimensional shapes. (4.2)

6. Collect and organize data. (4.4)

7. Create a bar graph based on given data. (4.4)

8. Read, interpret, and draw inferences from displays of data. (4.4)

9. Calculate equivalent names for a number. (4.1)

10. Review the basic facts of addition and subtraction. (4.1)

11. Recognize and relate the vocabulary of chance. (4.4)

12. Compare whole numbers. (4.1)

13. Use a number grid to solve problems. (4.5)

14. Use a calculator to solve problems. (4.5)

15. Identify value of digits in decimal notation. (4.1)

16. Write money amounts in decimal notation. (4.1)

17. Calculate values of coin and bill combinations, order, and compare money amounts. (4.1)

18. Solve problems involving money. (4.5)

19. Practice estimation skills with money amounts. (4.1)

20. Compare and order numbers to solve number patterns. (4.3)

21. Practice and apply addition and subtraction facts. (4.1)

22. Identify and use math symbols + - = < > (4.3)

23. Communicate mathematical thinking. (4.5)

24. Use mathematic vocabulary to express mathematical ideas precisely. (4.5)

25. Evaluate the mathematical thinking and strategies of others. (4.5)

Approximate duration: 14 lessons, 4 weeks

Activities:

- Have students communicate their mathematical methods verbally and in written language. (Create problem of the week; pose open ended questions and solve.)

- Make a clock booklet – page 36 teacher’s manual/volume 1.

- See I Hate Mathematics! Book to explore probability, do the following: What’s the Chance of being invited to 2 birthday parties in one day, page 110; The Game of Pig, page 115.

- See A Million Fish … More or Less by Patricia C. McKissack to explore the language of chance, Alphabetical Probability, page 117.

- Length-of-Day Project – page 78 teacher’s manual.

- Create a chart with math vocabulary, add to it each unit.

- Find samples of menus, share them with the class. Have students create their own restaurant menu. Then have them write story problems involving money based on their menu.

- See Minute Math +, some experiences comparing money amounts: Measurement, pages 64, 65, 66, 67.

- See Minute Math+ , some experiences with time-telling skills: Measurement pages 71-73.

Games:

- Number-Line Squeeze – page 21, teacher’s manual.

- Less Than You – Student Reference Book, page 289.

- Addition Top-It – Student Reference Book, page 270.

- Name That Number- Student Reference Book, page 299 and 300.

- Number-Grid Difference Game – Student Reference Book, page 301.

- Beat the Calculator – Student Reference Book, page 278.

- Coin Top-It – Student Reference Book, page 270.

- Spinning for Money - Math Masters, pages 462 and 463.

UNIT 2: ADDING AND SUBTRACTING WHOLE NUMBERS

Objectives:

After completing this unit, students will be able to:

1. Use fact families to reinforce the basic addition and subtraction facts. (4.1)

2. Solve problems using mental math strategies. (4.1)

3. Use patterns and the commutative property for reinforcement of addition facts. (4.3)

4. Describe and extend patterns of addition and subtraction facts and their extensions. (4.3)

5. Relate basic facts in order to compute extended facts. (4.1)

6. Model rules for patterns and use them to solve problems. (4.3)

7. Use pictorial models to explore the concept of a function. (4.3)

8. Utilize diagrams and number models as strategies to solve number stories. (4.4)

9. Write number models to summarize addition and subtraction number stories. (4.3)

10. Solve multidigit number stories using diagrams which involve addition and subtraction. (4.4)

11. Practice making estimates as a check for reasonableness of answers. (4.1)

12. Model and practice the partial-sums algorithm. (4.1)

13. Use place-value concepts to apply the counting-up and trade-first algorithms. (4.1)

14. Solve addition number stories with three or more addends. (4.1)

15. Practice adding three or more addends in a convenient order. (4.3)

16. Use mathematic vocabulary to express mathematical ideas precisely. (4.5)

17. Evaluate the mathematical thinking and strategies of others. (4.5)

Approximate duration: 10 lessons, 3 weeks

Activities:

- Have students communicate their mathematical methods verbally and in written language. (Create problem of the week; pose open ended questions and solve.)

- Create addition and subtraction puzzles.- see math masters 58 and 59

- Practice basic facts and their extensions - see Minute Math+ book, Basic Routines pages 9 and 10.

- Practice comparison number stories. - see Number Stories in Minute Math + book pages 85, 157.

- Make subtraction posters.- see teacher’s guide/volume 1, page 146.

- Connect math and literacy with Math-terpieces and Grapes of Math by Gregory Tang.

Games:

- Target: 50 – Student Reference Book page 312

- Roll to 100 – Student Reference Book page 307 and 309

- Play Beat the Calculator Addition – Student Reference Book page 278

- Base- 10 Trading Game – see Math Masters page 411, teacher guide/volume one, page 146.

UNIT 3- LINEAR MEASURES AND AREA

Objectives:

After completing this unit, students will be able to:

1. Use nonstandard units to measure the lengths of objects. (4.2)

2. Estimate the length of objects. (4.2)

3. Practice measuring line segments to the nearest inch, half inch, quarter inch, half centimeter and decimeter. (4.2)

4. Estimate and measure using U.S. customary units and metric units of length. (4.2)

5. Apply basic facts to find perimeters. (4.2)

6. Identify, describe, and model polygons. (4.2)

7. Measure sides of polygons to the nearest inch. (4.2)

8. Collect, tabulate, and interpret experimental data. (4.4)

9. Use probability terms to describe the likelihood of an event. (4.4)

10. Create shapes with a given perimeter. (4.2)

11. Estimate and measure areas of surfaces and 2-dimensional shapes. (4.2)

12. Use multiplication facts, arrays, and counting to determine the area of rectangles. (4.1)

13. Use arrays to write number models to find the area of rectangles. (4.3)

14. Identify the circumference and the diameter of circular objects.

15. Measure the circumference and the diameter of circular objects to the nearest centimeter. (4.2)

16. Cite the maximum, minimum, range, and median of a set. (4.4)

17. Use mathematic vocabulary to express mathematical ideas precisely. (4.5)

18. Evaluate the mathematical thinking and strategies of others. (4.5)

Approximate duration: 10 lessons, 3 weeks

Activities:

- Have students communicate their mathematical methods verbally and in written language. (Create problem of the week; pose open ended questions and solve.)

- Read How Big Is a Foot.- see teacher’s guide/volume 1, page 174.

- Read Sir Cumference and the First Round Table: A Math Adventure to introduce circle measurements.

- Predict the results of rolling 2 die. - see Math Master 414 and teacher guide/volume 1.

- Find cans of various sizes, trace bottoms on paper and then find the circumference and diameter of each.

- Read Probably Pistachio by Stuart J. Murphy and review the probability terms with the story. Have students use probability terms to describe their day.

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