Mathematics - 3rd Grade Core

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Mathematics - 3rd Grade

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Course Description

Intended Learning Outcomes for Third Through Sixth Grade Mathematics

The main intent of mathematics instruction is for students to value and use mathematics and reasoning skills to investigate and understand the world.

The Intended Learning Outcomes (ILOs) describe the skills and attitudes students should acquire as a result of successful mathematics instruction. They are an essential part of the Mathematics Core Curriculum and provide teachers with a standard for student learning in mathematics.

ILOs for mathematics:

Develop a positive learning attitude toward mathematics.
Become effective problem solvers by selecting appropriate methods, employing a variety of strategies, and exploring alternative approaches to solve problems.
Reason logically, using inductive and deductive strategies and justify conclusions.
Communicate mathematical ideas and arguments coherently to peers, teachers, and others using the precise language and notation of mathematics.
Connect mathematical ideas within mathematics, to other disciplines, and to everyday experiences.
Represent mathematical ideas in a variety of ways.

Significant mathematics understanding occurs when teachers incorporate ILOs in planning mathematics instruction. The following are ideas to consider when planning instruction for students to acquire the ILOs:

Develop a positive learning attitude toward mathematics.
When students are confident in their mathematical abilities, they demonstrate persistence in completing tasks. They pose mathematical questions about objects, events, and processes while displaying a sense of curiosity about numbers and patterns. It is important to build on students' innate problem-solving inclinations and to preserve and encourage a disposition that values mathematics.
Become effective problem solvers by selecting appropriate methods, employing a variety of strategies, and exploring alternative approaches to solve problems.
Problem solving is the cornerstone of mathematics. Mathematical knowledge is generated through problem solving as students explore mathematics. To become effective problem solvers, students need many opportunities to formulate questions and model problem situations in a variety of ways. They should generalize mathematical relationships and solve problems in both mathematical and everyday contexts.
Reason logically, using inductive and deductive strategies and justify conclusions.
Mathematical reasoning develops in classrooms where students are encouraged to put forth their own ideas for examination. Students develop their reasoning skills by making and testing mathematical conjectures, drawing logical conclusions, and justifying their thinking in developmentally appropriate ways. Students use models, known facts, and relationships to explain reasoning. As they advance through the grades, students' arguments become more sophisticated.
Communicate mathematical ideas and arguments coherently to peers, teachers, and others using the precise language and notation of mathematics.
The ability to express mathematical ideas coherently to peers, teachers, and others through oral and written language is an important skill in mathematics. Students develop this skill and deepen their understanding of mathematics when they use accurate mathematical language to talk and write about what they are doing. When students talk and write about mathematics, they clarify their ideas and learn how to make convincing arguments and represent mathematical ideas verbally, pictorially, and symbolically.
Connect mathematical ideas within mathematics, to other disciplines, and to everyday experiences.
Students develop a perspective of the mathematics field as an integrated whole by understanding connections within mathematics. Students should be encouraged to explore the connections that exist with other disciplines and between mathematics and their own experiences.
Represent mathematical ideas in a variety of ways.
Mathematics involves using various types of representations, including concrete, pictorial, and symbolic models. Students use a variety of mathematical representations to expand their capacity to think logically about mathematics.

By the end of grade three, students develop understandings of multiplication and division of whole numbers. They use properties to develop increasingly more sophisticated strategies to solve problems involving basic multiplication and division facts. They relate division to multiplication. Students understand fraction equivalence for simple fractions; they recognize that the size of a fractional part is relative to the size of the whole. They understand meanings of fractions to represent parts of a whole, parts of a set, or distances on a number line. They compare and order simple fractions by using models, benchmark fractions, or common denominators.

Students investigate, analyze, and classify two-dimensional shapes by their sides and angles. They decompose, combine, and transform polygons to understand properties of two-dimensional space and use those properties to solve problems. Students construct and analyze frequency tables, bar graphs, picture graphs, and line plots and use them to solve problems.

Core Standards of the Course

Standard 1
Students will understand the base-ten numeration system, place value concepts, simple fractions and perform operations with whole numbers.

Objective 1
Represent whole numbers up to 10,000, comprehend place value concepts, and identify relationships among whole numbers using base-ten models and symbolic notation.

Read, write, and represent whole numbers using standard and expanded form.
Demonstrate multiple ways to represent numbers using models and symbolic representations (e.g., fifty is the same as two groups of 25, the number of pennies in five dimes, or 75 - 25).
Identify the place and the value of a given digit in a four-digit numeral and round numbers to the nearest ten, hundred, and thousand.
Order and compare whole numbers on a number line and use the inequality symbols <, >, ≠, and = when comparing whole numbers.
Identify factors and multiples of whole numbers.

Objective 2
Use fractions to communicate and compare parts of the whole.

Identify the denominator of a fraction as the number of equal parts of the unit whole and the numerator of a fraction as the number of equal parts being considered.
Define regions and sets of objects as a whole and divide the whole into equal parts using a variety of objects, models, and illustrations.
Name and write a fraction to represent a portion of a unit whole for halves, thirds, fourths, sixths, and eighths.
Place fractions on the number line and compare and order fractions using models, pictures, the number line, and symbols.
Find equivalent fractions using concrete and pictorial representations.

Objective 3
Model problems involving addition, subtraction, multiplication, and division.

Demonstrate the meaning of multiplication and division of whole numbers through the use of a variety of representations (e.g., equal-sized groups, arrays, area models, and equal jumps on a number line for multiplication, partitioning and sharing for division).
Use a variety of strategies and tools, such as repeated addition or subtraction, equal jumps on the number line, and counters arranged in arrays to model multiplication and division problems.
Demonstrate, using objects, that multiplication and division by the same numbers are inverse operations (e.g., 3 x = 12 is the same as 12 ÷ 3 = and = 4).
Demonstrate the effect of place value when multiplying whole numbers by 10.
Write a story problem that relates to a given addition, subtraction, or multiplication equation, and write a number sentence to solve a problem related to the students' environment.

Objective 4
Compute and solve problems involving addition and subtraction of 3- and 4- digit numbers and basic facts of multiplication and division.

Find the sum or difference of numbers, including monetary amounts, using models and strategies such as expanded form, compensation, partial sums, and the standard algorithm.
Use a variety of methods to facilitate computation (e.g., estimation, mental math strategies, paper and pencil).
Compute basic multiplication facts (0-10) and related division facts using a variety of strategies based on properties of addition and multiplication (i.e., commutative, associative, identity, zero, and the distributive properties).

Mathematical Language and Symbols Students Should Use
sum, difference, expanded form, factor, product, array, multiple, numerator, denominator, halves, thirds, fourths, sixths, eighths, divisor, dividend, quotient, greater than, less than, equal to, <, >, =

Exploratory Concepts and Skills

Extend multiplication and division to larger-digit numbers.
Use concrete objects and visual models to add and subtract common decimals.
Investigate the distributive property of multiplication over addition for single-digit multipliers (e.g., 7 x 15 is equivalent to 7 x (10 + 5) is equivalent to (7 x 10) + (7 x 5).

Standard 2
Students will use patterns, symbols, operations, and properties of addition and multiplication to represent and describe simple number relationships.

Objective 1
Create, represent, and analyze growing patterns.

Create and extend growing patterns using objects, numbers, and tables.
Describe how patterns are extended using manipulatives, pictures, and numerical representations.

Objective 2
Recognize, represent, and simplify simple number relationships using symbols, operations, and properties.

Represent numerical relationships as expressions, equations, and inequalities.
b. Solve equations involving equivalent expressions (e.g., 6 + 4 = + 7).
Use the >, <, and = symbols to compare two expressions involving addition and subtraction (e.g., 4 + 6 3 + 2; 3 + 5 16 - 9).
Recognize and use the commutative, associative, distributive, and identity properties of addition and multiplication, and the zero property of multiplication.

Mathematical Language and Symbols Students Should Use
growing patterns, expressions, equations, <, >, =

Exploratory Concepts and Skills

Use concrete materials to build an understanding of equality and inequality.
Explore properties of equality in number sentences (e.g., when equals are added to equals, then the sums are equal; when equals are multiplied by equals, then the products are equal.

Standard 3
Students will describe and analyze attributes of two-dimensional shapes.

Objective 1
Describe and compare attributes of two-dimensional shapes.

Identify, describe, and classify polygons (e.g., pentagons, hexagons, octagons).
Identify attributes for classifying triangles (e.g., two equal sides for the isosceles triangle, three equal sides for the equilateral triangle, right angle for the right triangle).
Identify attributes for classifying quadrilaterals (e.g., parallel sides for the parallelogram, right angles for the rectangle, equal sides and right angles for the square).
Identify right angles in geometric figures, or in appropriate objects, and determine whether other angles are greater or less than a right angle.

Objective 2
Demonstrate the meaning of congruence through applying transformations.

Demonstrate the effect of reflection, translation, or rotation using objects.
Determine whether two polygons are congruent by reflecting, translating, or rotating one polygon to physically fit on top of the other.

Mathematical Language and Symbols Students Should Use
polygon, attribute, quadrilateral, equilateral triangle, isosceles triangle, right triangle, pentagon, hexagon, octagon, parallel, right angle, reflect, translate, rotate, slide, flip, turn, congruent

Exploratory Concepts and Skills

Explore line symmetry and rotational symmetry.
Investigate two-dimensional representations of three-dimensional objects.

Standard 4
Students will select and use appropriate units and measurement tools to solve problems.

Objective 1
Select and use appropriate tools and units to estimate and measure length, weight, capacity, time, and perimeter of two-dimensional figures.

Describe the part-whole relationships (e.g., 3 feet in a yard, a foot is 1/3 of a yard) between metric units of length (i.e., centimeter, meter), and among customary units of length (i.e., inch, foot, yard), capacity (i.e., cup, quart), and weight (i.e., pound, ounce).
Measure the length of objects to the nearest centimeter, meter, half- and quarter-inch, foot, and yard.
Measure capacity using cups and quarts, and measure weight using pounds and ounces.
Identify the number of minutes in an hour, the number of hours in a day, the number of days in a year, and the number of weeks in a year.
Describe perimeter as a measurable attribute of two-dimensional figures, and estimate and measure perimeter with metric and customary units.

Objective 2
Solve problems involving measurements.

Determine simple equivalences of measurements (e.g., 30 inches = 2 feet and 6 inches; 6 cups = 1 1/2 quarts; 90 min. = 1 hr. 30 min.).
Compare given objects according to measurable attributes (i.e., length, weight, capacity).
Solve problems involving perimeter.
Determine elapsed time in hours (e.g., 7:00 a.m. to 2:00 p.m.).

Standard 5
Students will collect and organize data to make predictions and identify basic concepts of probability.

Objective 1
Collect, organize, and display data to make predictions.

Collect, read, represent, and interpret data using tables, graphs, and charts, including keys (e.g., pictographs, bar graphs, frequency tables, line plots).
Make predictions based on a data display.

Objective 2
Identify basic concepts of probability.

Describe the results of events using the terms "certain," "likely," "unlikely," and "impossible."
Conduct simple probability experiments, record possible outcomes systematically, and display results in an organized way (e.g., chart, graph).
Use results of simple probability experiments to describe the likelihood of a specific outcome in the future.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Office of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Office of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.

For more information about this core curriculum, contact the USOE Specialist, DAVID SMITH or visit the Mathematics - Elementary Home Page. For general questions about Utah's Core Curriculum, contact the USOE Curriculum Director, LYNNE GREENWOOD . UEN Contact Info: 801-581-2999 | 800-866-5852 | Contact Us