The mathematics program in Grades 3-5 focuses on building on students’ prior knowledge to allow them to progress from the foundation gained in early mathematics experiences to actively constructing new knowledge. In these grades, students develop mathematical skills and insights and use them in solving meaningful problems.
Students in Grades 3-5 are primarily concrete learners; however, they are developing skills to make the transition into abstract thinking through pictorial models and symbols. By nature they are inquisitive, respond well to genuine praise, and experience increased social and emotional development. They begin to make many of their own decisions and may progress from teacher dependency into a self-guided stage as they learn to evaluate their own thinking and the thinking of others. As students become empowered with the ability to interpret their world, they show enthusiasm and interest in mathematics.
The environment for students in Grades 3-5 encourages them to become independent thinkers as they relate mathematics to the real world. This environment is active, problem-rich, and stimulating. Students work together to build a community of mathematical learners as their ideas become a source of learning. A well-balanced mathematics curriculum provides materials for learning, technology for teaching, and opportunities for students to engage in cooperative learning. This environment includes on-going assessments with a focus on student understanding and procedural skills. Teaching practices reflect a commitment to both equity and excellence.
Students in Grades 3-5 encounter a range of representations and problem-solving situations that empower them to move from the concrete to the abstract. The curriculum emphasizes computational fluency in basic operations, problem solving, reasoning, and number sense. It also promotes student acquisition of the skills and strategies necessary to comprehend new and challenging mathematical concepts.
Students in third grade are active and inquisitive. They are primarily concrete learners, acquiring knowledge through hands-on experiences. Instructional tasks that relate to their personal lives stimulate their interest.
Third-grade students need a classroom environment that helps them learn to work together as a community of learners. This environment provides an atmosphere in which students are recognized as individuals whose ideas are valued, and one in which opportunities are provided for all individuals in the classroom to work together as members of a team. In such an environment, students feel less threatened about making mistakes and have a more positive attitude toward receiving ideas for improvement.
Third-grade students enjoy intellectually stimulating activities that promote enthusiasm and capture their interest. Such activities better enable students to make sense of mathematics. Students compare and order whole numbers, identify two-dimensional figures based on attributes, expand their knowledge of measurement and data analysis, and strengthen computational fluency by applying problem-solving strategies. The third-grade content enables students to use mathematics in other disciplines and to connect mathematics to the real world.
1. Demonstrate number sense by comparing, ordering, and expanding whole numbers through 9999.
· Comparing numbers using the symbols >, <, =, and
· Identifying the place value of any digit within a four-digit number
· Writing a four-digit number in words and locating it on a number line
· Determining the value of a number written in expanded notation to the ten-thousands place
Example: 3,000 + 400 + 20 + 1 = 3,421
· Rounding whole numbers to the nearest ten and hundred and money values to the nearest dollar
2. Solve addition and subtraction problems, including word problems, involving two- and three-digit numbers with and without regrouping.
· Estimating sums and differences by using compatible numbers, front-end estimation, and rounding
Examples: compatible numbers— 24 + 26 = 25 + 25
front-end estimation— 72 is approximately 70
rounding— 172 is approximately 200
· Demonstrating computational fluency in addition and subtraction
3. Multiply whole numbers with and without regrouping using single-digit multipliers.
· Applying concepts of multiplication through the use of manipulatives, number stories, arrays, repeated addition, or problem situations
· Applying basic multiplication facts through 9 x 9 by using manipulatives, solving problems, and writing number stories
· Recognizing properties of multiplication
4. Divide whole numbers using two-digit dividends and one-digit divisors.
· Recognizing division as repeated subtraction
5. Model equivalent fractions with concrete objects or pictorial representations.
6. Use coins to make change up to $1.00.
· Determining monetary values of sets of unlike coins and bills up to $5.00
7. Complete a given numeric or geometric pattern.
Examples: geometric pattern— ;
numeric pattern—20, 27, 34, 41, ____
8. Identify geometric representations for points, lines, perpendicular lines, parallel lines, angles, and rays.
· Recognizing real-life examples of points, lines, perpendicular lines, and parallel lines
· Drawing points, lines, and perpendicular lines
9. Specify locations on a coordinate grid by using horizontal and vertical movements.
10. Measure length in metric units.
11. Determine elapsed time to the day with calendars and to the hour with a clock.
· Calculating elapsed time to the minute within the same hour
· Applying vocabulary associated with time using a.m., p.m., noon, or midnight
Data Analysis and Probability
12. Recognize data as either categorical or numerical.
race, languages spoken, genre;
numerical—age, height, weight
· Comparing related data sets
13. Determine the likelihood of different outcomes in a simple experiment.
Example: determining that the spinner is least likely to land on red in this diagram
Students in fourth grade are intrigued with mathematics. To nurture this interest, students at this grade level need to be involved in an active learning process rather than one that only builds on memorization of concepts and procedures. Concrete experiences are also important at this stage of development. Such experiences allow students to develop and strengthen the skills needed to communicate, reason, solve mathematical problems, and reach higher levels of cognitive reasoning.
An effective classroom environment provides intellectually stimulating instruction and developmentally appropriate opportunities for students to learn mathematical concepts. This classroom environment fosters an atmosphere in which students are encouraged to find solutions through a variety of methods and feel less threatened about making and correcting mistakes. Instruction includes opportunities for students to communicate their mathematical thinking by talking, writing, and sharing with each other.
Fourth-grade content builds a foundation of basic number sense, operations, quantitative reasoning, patterns, relationships, geometric and spatial reasoning, measurement, and probability and statistics. The content builds on and expands students’ conceptual understanding of mathematics. Through the interweaving of mathematical concepts and processes, students learn to value mathematics, display confidence in their mathematical ability, solve problems, and make connections between mathematics and other subjects.
Number and Operations
1. Demonstrate number sense by comparing and ordering decimals to hundredths and whole numbers to 999,999.
· Identifying a number when given a pictorial representation of tenths and hundredths or groups of ones, tens, hundreds, and thousands
· Writing a number in expanded notation through the hundred-thousands
Example: 914,682 = 900,000 + 10,000 + 4,000 + 600 + 80 + 2
· Determining the place value of a digit in a whole number through the hundred-thousands and in a decimal to the hundredths
2. Write money amounts in words and dollar-and-cent notation.
· Identifying equivalent units of money
3. Rename improper fractions as mixed numbers and mixed numbers as improper fractions.
· Using a number line to simplify, compare, and order fractions and mixed numbers
· Writing equivalent forms of fractions
4. Demonstrate addition and subtraction of fractions with common denominators.
5. Round whole numbers to the nearest ten, hundred, or thousand and decimals to the nearest tenth.
6. Solve problems, including word problems, that involve addition and subtraction of four-digit numbers with and without regrouping.
· Estimating sums and differences of whole numbers by using appropriate strategies such as rounding, front-end estimation, and compatible numbers
· Adding and subtracting decimals and money amounts
7. Solve problems, including word problems, involving the basic operations of multiplication and division on whole numbers through two-digit multipliers and one-digit divisors.
· Estimating products and quotients of whole numbers by using appropriate strategies such as rounding, front-end estimation, and compatible numbers
· Identifying information needed to determine the appropriate operation to solve a problem
· Demonstrating computational fluency in multiplication and division fact families through 12
8. Recognize equivalent forms of commonly used fractions and decimals.
Examples: = .25, of a dollar = $.25 (25 cents)
9. Write number sentences for word problems that involve multiplication or division.
10. Complete addition and subtraction number sentences with a missing addend or subtrahend.
11. Identify triangles, quadrilaterals, pentagons, hexagons, or octagons based on the number of sides, angles, and vertices.
· Demonstrating slides (translations), flips (reflections), and turns (rotations) using triangles, quadrilaterals, pentagons, hexagons, or octagons
· Drawing lines of symmetry in triangles, quadrilaterals, pentagons, hexagons, or octagons
12. Find locations on a map or grid using ordered pairs.
13. Calculate elapsed time in hours and minutes.
14. Measure length, width, weight, and capacity, using metric and customary units, and temperature in degrees Fahrenheit and degrees Celsius.
· Estimating perimeter and area of irregular shapes using unit squares and grid paper
· Estimating area using unit squares
Data Analysis and Probability
15. Represent categorical data using tables and graphs, including bar graphs, line graphs, and line plots.
· Collecting data using observations, surveys, or experiments
· Creating tally charts to represent data collected from real-life situations
16. Determine if outcomes of simple events are likely, unlikely, certain, equally likely, or impossible.
17. Represent numerical data using tables and graphs, including bar graphs and line graphs.
Students in fifth grade experience increased social and emotional development. They become more aware of their independence, opinions, and level of thinking as compared to others. Students enjoy and benefit from content presented in a way that allows for in-depth understanding, heightens interest and enthusiasm, and provides relevance to real-world situations.
In fifth grade, students need a positive learning environment that encourages and challenges student effort and progress toward learning mathematics. This environment is supported through the use of active learning experiences and content-related questions that foster mathematical communication.
The mathematics curriculum in fifth grade emphasizes fluency in computing and problem solving with whole numbers, decimals, and fractions. Students apply basic operations to problem-solving situations with a greater understanding of the meanings of operations and how they relate to one another. By actively acquiring new knowledge of symbolic representation, fifth-grade students move toward an abstract level of thinking.
Number and Operations
1. Demonstrate number sense by comparing, ordering, rounding, and expanding whole numbers through millions and decimals to thousandths.
· Relating percents to parts out of 100 by using equivalent fractions and decimals
· Determining the value of a digit to thousandths
2. Solve problems involving basic operations on whole numbers, including addition and subtraction of seven-digit numbers, multiplication with two-digit multipliers, and division with two-digit divisors.
· Estimating products and quotients
· Determining divisibility by 2, 3, 4, 5, 6, 9, and 10
· Demonstrating computational fluency with addition, subtraction, multiplication, and division of whole numbers
3. Solve word problems that involve decimals, fractions, or money.
· Solving word problems involving elapsed time
4. Determine the sum and difference of fractions with common and uncommon denominators.
· Changing mixed numbers to improper fractions
· Solving problems involving addition and subtraction of fractions with common and uncommon denominators
· Using least common multiples
· Estimating sums and differences of fractions
5. Identify numbers less than zero by extending the number line.
Example: identifying negative temperatures (below 0o) on a thermometer
6. Demonstrate the commutative, associative, and identity properties of addition and multiplication of whole numbers.
7. Write a number sentence for a problem expressed in words.
8. Identify regular polygons and congruent polygons.
· Identifying angles as right, obtuse, acute, or straight
· Classifying triangles as equilateral, isosceles, or scalene
· Identifying figures that have rotational symmetry
· Predicting the results of a flip (reflection), turn (rotation), or slide (translation)
9. Identify components of the Cartesian plane, including the x-axis, y-axis, origin, and quadrants.
10. Identify the center, radius, and diameter of a circle.
11. Estimate perimeter and area of irregular shapes using unit squares and grid paper.
12. Calculate the perimeter of rectangles from measured dimensions.
13. Convert a larger unit of measurement to a smaller unit of measurement within the same system (customary or metric).
cups = 32 fluid ounces, 2 meters = 200 centimeters,
2 miles = 10,560 feet
14. Analyze data collected from a survey or experiment to distinguish between what the data show and what might account for the results.
· Evaluating different representations of the same data to determine how well each representation shows important aspects of the data
· Using given measures of central tendency (mean, median, and mode) to analyze data
15. Use common fractions to represent the probability of events that are neither certain nor impossible.
Example: finding the probability of stopping on a vowel when using a spinner with three vowels and five consonants