Mathematics - 4th Grade Core

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Mathematics - 4th 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 four, students develop quick recall of the basic multiplication facts and related division facts. They develop fluency with efficient procedures for multiplying multidigit whole numbers, understand why the procedures work, and use them to solve problems. Students recognize decimal notation as an extension of the base-ten system. They relate their understanding of fractions to decimals. They generate equivalent fractions, simplify fractions, and identify equivalent fractions and decimals; compare and order whole numbers, simple fractions, and decimals to hundredths; and estimate decimal or fractional amounts in problem solving.

Students use transformations, including those that produce line and rotational symmetry. Students understand area as a measurable attribute of two-dimensional regions. They select appropriate units, strategies, and tools for solving problems that involve measuring area. They connect area measure to the area model for multiplication as a way to justify the formula for the area of a rectangle.

Core Standards of the Course

Standard 1
Students will acquire number sense and perform operations with whole numbers, simple fractions, and decimals.

Objective 1
Demonstrate multiple ways to represent whole numbers and decimals, from hundredths to one million, and fractions.

Read and write numbers in standard and expanded form.
Demonstrate multiple ways to represent whole numbers and decimals by using models and symbolic representations (e.g., 36 is the same as the square of six, three dozen, or 9 x 4).
Identify the place and the value of a given digit in a six-digit numeral, including decimals to hundredths, and round to the nearest tenth.
Divide regions, lengths, and sets of objects into equal parts using a variety of models and illustrations.
Name and write a fraction to represent a portion of a unit whole, length, or set for halves, thirds, fourths, fifths, sixths, eighths, and tenths.
Identify and represent square numbers using models and symbols.

Objective 2
Analyze relationships among whole numbers, commonly used fractions, and decimals to hundredths.

Compare the relative size of numbers (e.g., 475 is comparable to 500; 475 is small compared to 10,000 but large compared to 98).
Order whole numbers up to six digits, simple fractions, and decimals using a variety of methods (e.g., number line, fraction pieces) and the symbols <, >, and = to record the relationships.
Identify a number that is between two given numbers (e.g., 3.2 is between 3 and 4; find a number between 0.1 and 0.2).
Identify equivalences between fractions and decimals by connecting models to symbols.
Generate equivalent fractions and simplify fractions using models, pictures, and symbols.

Objective 3
Model and illustrate meanings of multiplication and division of whole numbers and the addition and subtraction of fractions.

Model multiplication (e.g., equal-sized groups, rectangular arrays, area models, equal intervals on the number line), place value, and properties of operations to represent multiplication of a one- or two-digit factor by a two-digit factor and connect the representation to an algorithm.
Use rectangular arrays to interpret factoring (e.g., find all rectangular arrays of 36 tiles and relate the dimensions of the arrays to factors of 36).
Demonstrate the mathematical relationship between multiplication and division (e.g. 3 x =12 is the same as 12 ÷ 3 = and = 4) and use that relationship to explain that division by zero is not possible.
Represent division of a three-digit dividend by a one-digit divisor, including whole number remainders, using a variety of methods (e.g., rectangular arrays, manipulatives, pictures), and connect the representation to an algorithm.
Use models to add and subtract simple fractions where one single-digit denominator is 1, 2, or 3 times the other (e.g., 2/4 + 1/4; 3/4 - 1/8).

Objective 4
Solve problems involving multiplication and division of whole numbers and addition and subtraction of simple fractions and decimals.

Use estimation, mental math, paper and pencil, and calculators to perform mathematical calculations and identify when to use each one appropriately.
Select appropriate methods to solve a single operation problem and estimate computational results or calculate them directly, depending on the context and numbers involved in a problem.
Write a story problem that relates to a given multiplication or division equation, and select and write a number sentence to solve a problem related to the environment.
Solve problems involving simple fractions and interpret the meaning of the solution (e.g., A pie has been divided into six pieces and one piece is already gone. How much of the whole pie is there when Mary comes in? If Mary takes two pieces, how much of the whole pie has she taken? How much of the pie is left?)

Objective 5
Compute problems involving multiplication and division of whole numbers and addition and subtraction of simple fractions and decimals.

Demonstrate quick recall of basic multiplication and division facts.
Multiply up to a three- digit factor by a two-digit factor with fluency, using efficient procedures.
Divide up to a three-digit dividend by a one-digit divisor with fluency, using efficient procedures.
Add and subtract decimals and simple fractions where one single-digit denominator is 1, 2, or 3 times the other (e.g., 2/4 + 1/4 = 3/4; 1/3 - 1/6 = 1/6).

Mathematical Language and Symbols Students Should Use
sum, difference, expanded form, standard form, square number, dividend, divisor, quotient, factor, product, array, multiple, numerator, denominator, sixths, eighths, tenths, equivalent, estimate, <, >, =, ≠

Exploratory Concepts and Skills

Use concrete objects and visual models to add and subtract common decimals.
Explore numbers less than zero by extending the number line and by using familiar applications such as temperature.
Investigate the concept of ratio (e.g., the number of students to the number of teachers).

Standard 2
Students will use patterns and relations to represent mathematical problems and number relationships.

Objective 1
Identify, analyze, and determine rules for describing numerical patterns involving operations and nonnumerical growing patterns.

Analyze growing patterns using objects, pictures, numbers, and tables to determine a rule for the pattern.
Recognize, represent, and extend simple patterns involving multiples and other number patterns (e.g., square numbers) using objects, pictures, numbers, and tables.
Identify simple relationships in real-life contexts and use mathematical operations to describe the pattern (e.g., the number of legs on a given number of chairs may be determined by counting by fours or by multiplying the number of chairs by 4).

Objective 2
Use algebraic expressions, symbols, and properties of the operations to represent, simplify, and solve mathematical equations and inequalities.

Use the order of operations to evaluate, simplify, and compare mathematical expressions involving the four operations, parentheses, and the symbols <, >, and = (e.g., 2x (4 - 1) + 3; of the two quantities 7 - (3 - 2) or (7 - 3) - 2, which is greater?).
Express single-operation problem situations as equations and solve the equation.
Recognize that a symbol represents the same number throughout an equation or expression (e.g., + = 8; thus, = 4).
Describe 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 pattern, order of operations, parentheses, inequality, expression, equation, associative property, commutative property, distributive property, zero property of multiplication, >, <, =

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 understand attributes and properties of plane geometric objects and spatial relationships.

Objective 1
Identify and describe attributes of two-dimensional geometric shapes.

Name and describe lines that are parallel, perpendicular, and intersecting.
Identify and describe right, acute, obtuse, and straight angles.
Identify and describe the radius and diameter of a circle.
Identify and describe figures that have line symmetry and rotational symmetry.

Objective 2
Specify locations using grids and maps.

Locate coordinates in the first quadrant of a coordinate grid.
Give the coordinates in the first quadrant of a coordinate grid.
Locate regions on a map of Utah.
Give the regions on a map of Utah.

Objective 3
Visualize and identify geometric shapes after applying transformations.

Identify a translation, rotation, or a reflection of a geometric shape.
Recognize that 90°, 180°, 270°, and 360° are associated, respectively, with 1/4, 1/2, 3/4, and full turns.

Mathematical Language and Symbols Students Should Use
parallel, perpendicular, intersecting lines, right angle, acute angle, obtuse angle, straight angle, circle, radius, diameter, line symmetry, rotational symmetry, coordinate, first quadrant, degree, translate, rotate, reflect, transformation

Exploratory Concepts and Skills

Analyze results of transformations (e.g., translations, rotations, reflections) on two-dimensional shapes.
Investigate two-dimensional representations of three-dimensional objects.

Standard 4
Students will describe relationships among units of measure, use appropriate measurement tools, and use formulas to find area measurements.

Objective 1
Describe relationships among units of measure for length, capacity, and weight, and determine measurements of angles using appropriate tools.

Describe the relative size among metric units of length (i.e., millimeter, centimeter, meter), between metric units of capacity (i.e., milliliter, liter), and between metric units of weight (i.e., gram, kilogram).
Describe the relative size among customary units of capacity (i.e., cup, pint, quart, gallon).
Estimate and measure capacity using milliliters, liters, cups, pints, quarts, and gallons, and measure weight using grams and kilograms.
Recognize that angles are measured in degrees and develop benchmark angles (e.g., 45°, 60°, 120°) using 90° angles to estimate angle measurement.
Measure angles using a protractor or angle ruler.

Objective 2
Recognize and describe area as a measurable attribute of two-dimensional shapes and calculate area measurements.

Quantify area by finding the total number of same-sized units of area needed to fill the region without gaps of overlaps.
Recognize that a square that is 1 unit on a side is the standard unit for measuring area.
Develop the area formula for a rectangle as the number of unit squares that fit in the rectangle, and identify the unit of measure as square units.
Develop and use the area formula for a right triangle by comparing with the formula for a rectangle (e.g., two of the same right triangles makes a rectangle).
Develop the formulas and justify the relationships among area formulas of triangles and parallelograms by decomposing and comparing with areas of right triangles and rectangles.
Determine possible perimeters, in whole units, for a rectangle with a fixed area, and determine possible areas when given a rectangle with a fixed perimeter.

Standard 5
Students will interpret and organize collected data to make predictions, answer questions, and describe basic concepts of probability.

Objective 1
Collect, organize, and display data to answer questions.

Identify a question that can be answered by collecting data.
Collect, read, and interpret data from tables, graphs, charts, surveys, and observations.
Represent data using frequency tables, bar graphs, line plots, and stem and leaf plots.
Identify and distinguish between clusters and outliers of a data set.

Objective 2
Describe and predict simple random outcomes.

Describe the results of investigations involving random outcomes as simple ratios (e.g., 4 out of 9, 4/9).
Conduct simple probability experiments, with and without replacement, record possible outcomes systematically, and display results is an organized way.
Use the results of simple probability experiments, with and without replacement, 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, Lois Richins 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