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Third Grade Mathematics

Number and Operations

NCTM FOCAL CONNECTION: Number and Operations: Building on their work in grade 2, students extend their understanding of place value to numbers up to 10,000 in various contexts. Students also apply this understanding to the task of representing numbers in different equivalent forms (e.g., expanded notation). They develop their understanding of numbers by building their facility with mental computation (addition and subtraction in special cases, such as 2,500 + 6,000 and 9,000 5,000), by using computational estimation, and by performing paper-and-pencil computations.

NCTM FOCAL PO INT: Nu mber a n d Operations and Algebra: Developing u nderstandings of m u ltiplication a nd division and strategies for basic m ultiplication facts and related division facts. Students understand the meanings of multiplication and division of whole numbers through the use of representations (e.g., equal-sized groups, arrays, area models, and equal "jumps" on number lines for multiplication, and successive subtraction, partitioning, and sharing for division). They use properties of addition and multiplication (e.g., commutativity, associativity, and the distributive property) to multiply whole numbers and apply increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving basic facts. By comparing a variety of solution strategies, students relate multiplication and division as inverse operations. NCTM FOCAL PO INT: Nu mber a n d Operations: Developing an u nderstanding of fractions and fraction equivalence. Students develop an understanding of the meanings and uses of fractions to represent parts of a whole, parts of a set, or points or distances on a number line. They understand that the size of a fractional part is relative to the size of the whole, and they use fractions to represent numbers that are equal to, less than, or greater than 1. They solve problems that involve comparing and ordering fractions by using models, benchmark fractions, or common numerators or denominators. They understand and use models, including the number line, to identify equivalent fractions.

-Copyright © 2006, National Council of Teachers of Mathematics.

Third Grade Mathematics

Number and Operations

Enduring Understandings

The number system is based on well-defined structure. Every numerical operation has an inverse. Rational numbers can be represented in multiple ways. Estimation serves as a tool for judging reasonableness of computations. Computational fluency requires efficient, accurate and flexible methods for computing.

Essential Questions

What does it mean to be "computationally fluent"? 1. How does understanding the structure of the number system help you solve problems? 2. How can you represent rational numbers in multiple ways? 3. Which mathematical skills are necessary to be a good "computer"? 4. Why is it important to estimate?

Third Grade Mathematics

Number and Operations

Outcomes

A. Students will understand relationships among numbers. 1. 2. 3. 4. 5. Represent decimals to the tenths place using parts of a whole and parts of a set Use place value concepts to order and compare money values to $100 Find 100 more or 100 less than any given four-digit number Find 1,000 more or 1,000 less than any given four-digit number Represent whole numbers to 10,000 in equivalent arithmetic expressions (in terms of groups of thousands, hundreds, tens, and ones) such as 3,108 (300 + 100 + 8 = 3,108 or 31 x 100 + 8 = 3,108) and non-numeric expressions such as words, pictures, and manipulatives 6. Read, write, compare, and order numbers to 10,000 in numerals and words 7. Identify, represent, read and write commonly used fractions of a whole and a set using physical models with numerators greater than one (equivalent parts, whole, fourths, thirds, halves, tenths) 8. Recognize that fractions also can be used to represent points on a number line or distances between points on a number line 9. Understand that the size of fractions part is relative to the size of the whole 10. Compare and order unit fractions and fractions with like denominators by using models and an understanding of the concept of numerator and denominator B. Students will understand meanings of operations and how they relate to one another. 1. Demonstrate the properties (commutative and associative) in computational situations 2. Understand the relationship between repeated addition and multiplication 3. Understand the relationship between equal sharing/repeated subtraction and division 4. Use the inverse relationship between addition and subtraction to compute and check results 5. Recognize the relationship between multiplication and division C. Understand how to compute fluently and make reasonable estimates 1. Demonstrate multiplication to 10 x 10 using words, manipulatives, diagrams, and rectangular arrays 2. Efficiently recalls multiplication and division facts through 5 x 9 and 1-digit numbers x 10 using concrete models 3. Develop and use strategies to estimate solutions to addition and subtraction problems (such as rounding up or down) 4. Round numbers to the nearest 1,000 , 100, and 10 5. Add and subtract multi-digit numbers using efficient and generalizable procedures based on knowledge of place value, including standard/traditional algorithms 6. Use addition and subtraction to solve real-world and mathematical problems including whole numbers 7. Assess the reasonableness of results for addition and subtraction problems based on context 8. Use various strategies, including the use of a calculator and the relationship between addition and subtraction, to check for accuracy 9. Represent multiplication facts by using a variety of approaches, such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line and skip counting

10. Represent division facts by using a variety of approaches, such as repeated subtraction, equal sharing and forming equal groups 11. Solve real-world and mathematical problems involving multiplication and division, including both "how many in each group" and "how many groups" division problems 12. Use strategies and algorithms based on knowledge of place value and properties of addition and multiplication to multiply a two- and three-digit number by a one-digit number (strategies may include mental computation, partial products, the standard/traditional algorithms, and properties of operations) 13. Demonstrate mastery of basic addition and subtraction facts (0 through 9 without a calculator) 14. Use models to solve multiplication and division problems and use number sentences to record the solutions

Third Grade Mathematics

Algebra

NCTM FOCAL CONNECTION: Algebra: Understanding properties of multiplication and the relationship between multiplication and division is a part of algebra readiness that develops at grade 3. The creation and analysis of patterns and relationships involving multiplication and division should occur at this grade level. Students build a foundation for later understanding of functional relationships by describing relationships in context with such statements as, "The number of legs is 4 times the number of chairs." NCTM FOCAL PO INT: Nu mber a n d Operations and Algebra: Developing u nderstandings of m u ltiplication a nd division and strategies for basic m ultiplication facts and related division facts. Students understand the meanings of multiplication and division of whole numbers through the use of representations (e.g., equal-sized groups, arrays, area models, and equal "jumps" on number lines for multiplication, and successive subtraction, partitioning, and sharing for division). They use properties of addition and multiplication (e.g., commutativity, associativity, and the distributive property) to multiply whole numbers and apply increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving basic facts. By comparing a variety of solution strategies, students relate multiplication and division as inverse operations.

-Copyright © 2006, National Council of Teachers of Mathematics.

Enduring Understandings

A mathematical model has both descriptive and predictive power. Mathematical situations and structures can be represented and analyzed using algebraic symbols, numeric tables, graphs, and words.

Essential Questions

1. Why is it important to look for a pattern? 2. What can we learn from looking at patterns? 3. How do symbols, tables of numbers and graphs help us to understand mathematics?

Third Grade Mathematics

Algebra

Outcomes

The student will: A. Understand how to describe, extend, and create a wide variety of patterns and functional relationships. 1. Create, read, and interpret patterns represented on tables, charts, and graphs 2. Identify, describe, and examine the patterns in numbers when finding 10 more, 10 less, 100 more, 100 less of any given number 3. Identify, describe, and examine the patterns in skip counting on the 100s chart, number line, or diagrams (2's 12's) 4. Create, identify describe, extend and predict a given pattern both geometrically and numerically (addition, subtraction and multiplication patterns). 5. Create, describe and apply single-operation input-output rules involving addition, subtraction and multiplication to solve problems in various contexts B. Understand that mathematical situations can be represented and analyzed using algebraic symbols. 1. Create real-world situations to represent number sentences 2. Identify a missing number or operation in simple arithmetic equations which involve multiplication and division basic facts and unknowns (3 __ 4 = 21; 9__ 3= 3; x 8 = 40) 3. Apply commutative, associative, identity (adding zero) and multiplicative (multiplying by 1) properties with addition, subtraction and multiplication to solve problems

Third Grade Mathematics

Geometry

NCTM FOCAL PO INT: Geometry: Describing and analyzing properties of two-dimensional shapes. Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry.

-Copyright © 2006, National Council of Teachers of Mathematics.

Enduring Understandings

The study of geometry requires thinking and doing. Geometry requires visualization, spatial reasoning, and geometric modeling to solve problems. Two and three dimensional shapes can be described and classified by their properties.

Essential Questions

1. How do we use geometry to help us make sense of the world? 2. What is unique about each geometric shape? 3. How do we talk about and classify different shapes?

Third Grade Mathematics

Geometry

Outcomes

The student will: A. Understand characteristics and properties of two and three-dimensional geometric shapes. 1. Understand the differences between a 2D shape and a 3D shape using appropriate terminology 2. Recognize, name, build, draw, compare, classify, describe and sort 2D shapes in their world (circle, square, rectangle, triangle, rhombus, hexagon, and oval) according to their geometrical attributes (such as number sides/faces, corners/vertices, side lengths and edges). 3. Recognize, name, build, draw, compare, classify, describe and sort 3D shapes in their world (sphere, cube, rectangular prism and cylinder) according to their geometrical attributes (such as number sides/faces, corners/vertices, and edges). 4. Identify parallel and perpendicular lines in various contexts, and use them to describe and create geometric shapes, such as right triangles, rectangles, parallelograms and trapezoids 5. Sketch polygons with a given number of sides or vertices (corners), such as pentagons, hexagons and octagons B. Understand locations and describe spatial relationships using coordinate geometry and other representational systems. 1. Describe, name, and interpret direction and distance in navigating space and apply ideas about direction and distance 2. Find and name locations with simple relationships such as "near to" in coordinate systems such as maps C. Understand transformations and symmetry in mathematical situations. 1. Recognize, create, and identify shapes that have line of symmetry 2. Describe, recognize and predict what happens to a 2- or 3-dimensional shape as a result of slides (translation), flips (reflection), and turns (rotation) D. Understand the use of visualization, spatial reasoning, and geometric modeling for solving problems. 1. Identify common two- and three-dimensional shapes that are components of more complex shapes

Third Grade Mathematics

Measurement

NCTM FOCAL CONNECTION: Measurement: Students in grade 3 strengthen their understanding of fractions as they confront problems in linear measurement that call for more precision than the whole unit allowed them in their work in grade 2. They develop their facility in measuring with fractional parts of linear units. Students develop measurement concepts and skills through experiences in analyzing attributes and properties of two dimensional objects. They form an understanding of perimeter as a measurable attribute and select appropriate units, strategies, and tools to solve problems involving perimeter.

-Copyright © 2006, National Council of Teachers of Mathematics.

Enduring Understandings

Measurement processes are used in everyday life to describe and quantify the world. Measurements in the real world are approximate, in part because of the instruments used and because of human error in reading the scales of the instruments.

Essential Questions

1. How does measurement keep our world organized? 2. What is a precise measurement? 3. How might measurement errors occur?

Third Grade Mathematics

Measurement

Outcomes

The student will A. Understand measurable attributes of objects and the units, systems, and processes of measurement 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Measure length and weight using standard and non-standard units Select and use benchmarks to estimate length Measure linear objects to ½ and ¼ of a unit Use decimal point, $, and ¢ signs correctly Tell amount of change and denomination of coins from a $1.00 bill after purchasing items using multiple representations of coins Select and apply appropriate standard units and tools to measure length, weight, temperature, and time Use an analog thermometer to determine temperature to the nearest degree in Fahrenheit and Celsius Know relationships between units of length in systems of measurement (such as 12 inches = 1 foot, 100 cm = 1 meter) Tell and write time to the minute using a standard and digital clock also using a.m. and p.m. notation Determine elapsed time to the minute (less than 60 minutes, in multiples of 5) Know relationships among units of time (hours in a day, days in a week, etc) Make change up to one dollar in several different ways, including using as few coins as possible

B. Understand the connection between measurement concepts and geometry 1. Find the perimeter of a polygon by measuring and adding lengths of sides 2. Measure distance around objects 3. Use half units when measuring distances

Third Grade Mathematics

Data Analysis and Probability

NCTM FOCAL CONNECTION: Data Analysis: Addition, subtraction, multiplication, and division of whole numbers come into play as students construct and analyze frequency tables, bar graphs, picture graphs, and line plots and use them to solve problems.

-Copyright © 2006, National Council of Teachers of Mathematics.

Enduring Understandings

Probability is the measurement of the likelihood of events. Informed citizens and consumers use data to make inferences, sound predictions, and judgments. Data analysis is formulating questions that can be addressed, explored, and synthesized with relevant information.

Essential Questions

1. What are data and what can we learn from it? 2. How can we make predictions when we are faced with uncertainty?

Third Grade Mathematics

Data Analysis and Probability

Outcomes

The student will: A. Understand how to collect, organize, and display relevant data to answer questions 1. Collect data using observation or survey 2. Organize and represent data using pictographs, bar graphs, tally charts, line plots and circle graphs (using halves, thirds, and quarters) with appropriate key, title, labels and units B. Understand how to develop and evaluate inferences and predictions that are based on data 1. Summarize, in writing, conclusions based on data from tables, graphs, and diagrams 2. Interpret data from pictographs, bar graphs, tally charts, line plots and circle graphs (using halves, thirds, and quarters) C. Understand and apply basic concepts of probability 1. Use diagrams, lists, and/or simple charts to represent all possible outcomes for an event or experiment 2. Determine most likely or least likely outcomes (impossible, certain)

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