Standard 4:Mathematics
STANDARD 4.1 (NUMBER AND NUMERICAL OPERATIONS) ALL STUDENTS WILL DEVELOP NUMBER SENSE AND WILL PERFORM STANDARD NUMERICAL OPERATIONS AND ESTIMATIONS ON ALL TYPES OF NUMBERS IN A VARIETY OF WAYS.
Descriptive Statement: Numbers and arithmetic operations are what most of the general public think about when they think of mathematics; and, even though other areas like geometry, algebra, and data analysis have become increasingly important in recent years, numbers and operations remain at the heart of mathematical teaching and learning. Facility with numbers, the ability to choose the appropriate types of numbers and the appropriate operations for a given situation, and the ability to perform those operations as well as to estimate their results, are all skills that are essential for modern day life.
Number Sense. Number sense is an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent that comes from investigating their characteristics and using them in diverse situations. It involves an understanding of how different types of numbers, such as fractions and decimals, are related to each other, and how each can best be used to describe a particular situation. It subsumes the more traditional category of school mathematics curriculum called numeration and thus includes the important concepts of place value, number base, magnitude, and approximation and estimation.
Numerical Operations. Numerical operations are an essential part of the mathematics curriculum, especially in the elementary grades. Students must be able to select and apply various computational methods, including mental math, pencil-and-paper techniques, and the use of calculators. Students must understand how to add, subtract, multiply, and divide whole numbers, fractions, decimals, and other kinds of numbers. With the availability of calculators that perform these operations quickly and accurately, the instructional emphasis now is on understanding the meanings and uses of these operations, and on estimation and mental skills, rather than solely on the development of paper-and-pencil proficiency.
Estimation. Estimation is a process that is used constantly by mathematically capable adults, and one that can be easily mastered by children. It involves an educated guess about a quantity or an intelligent prediction of the outcome of a computation. The growing use of calculators makes it more important than ever that students know when a computed answer is reasonable; the best way to make that determination is through the use of strong estimation skills. Equally important is an awareness of the many situations in which an approximate answer is as good as, or even preferable to, an exact one. Students can learn to make these judgments and use mathematics more powerfully as a result.
Number and operation skills continue to be a critical piece of the school mathematics curriculum and, indeed, a very important part of mathematics. But, there is perhaps a greater need for us to rethink our approach here than to do so for any other curriculum component. An enlightened mathematics program for today’s children will empower them to use all of today’s tools rather than require them to meet yesterday’s expectations.
Cumulative Progress Indicators
By the end of Grade 2, students will:
A. Number Sense
· Whole numbers through hundreds
· Ordinals
· Proper fractions (denominators of 2, 3, 4, 8, 10)
2. Demonstrate an understanding of whole number place value concepts.
3. Understand that numbers have a variety of uses.
4. Count and perform simple computations with coins.
· Amounts up to $1.00 (using cents notation)
5. Compare and order whole numbers.
B. Numerical Operations
· Joining, separating, and comparing
2. Explore the meanings of multiplication and division by modeling and discussing problems.
4. Construct, use, and explain procedures for performing addition and subtraction calculations with:
5. Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.
· Addition of 2-digit numbers
· Subtraction of 2-digit numbers
7. Check the reasonableness of results of computations.
8. Understand and use the inverse relationship between addition and subtraction.
C. Estimation
Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:
A. Number Sense
1. Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 3 pertain to these sets of numbers as well).
· 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
2. Demonstrate an understanding of whole number place value concepts.
3. Identify whether any whole number is odd or even.
4. Explore the extension of the place value system to decimals through hundredths.
5. Understand the various uses of numbers.
· Counting, measuring, labeling (e.g., numbers on baseball uniforms)
B. Numerical Operations
· Addition and subtraction: joining, separating, comparing
· Multiplication: repeated addition, area/array
· Division: repeated subtraction, sharing
2. Develop proficiency with basic multiplication and division number facts using a variety of fact strategies (such as “skip counting” and “repeated subtraction”).
3. Construct, use, and explain procedures for performing whole number calculations with:
· Pencil-and-paper
· Mental math
· Calculator
4. 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
5. Count and perform simple computations with money.
· Cents notation (¢)
6. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
7. Check the reasonableness of results of computations.
C. Estimation
1. Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.
4. Use estimation to determine whether the result of a computation (either by calculator or by hand) is reasonable.
Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:
A. Number Sense
1. Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 4 pertain to these sets of numbers as well).
· Whole numbers through millions
2. Demonstrate an understanding of place value concepts.
3. Demonstrate a sense of the relative magnitudes of numbers.
4. Understand the various uses of numbers.
6. Compare and order numbers.
7. Explore settings that give rise to negative numbers.
· Temperatures below 0o, debts
· Extension of the number line
B. Numerical Operations
· Addition and subtraction: joining, separating, comparing
· Multiplication: repeated addition, area/array
· Division: repeated subtraction, sharing
2. Develop proficiency with basic multiplication and division number facts using a variety of fact strategies (such as “skip counting” and “repeated subtraction”) and then commit them to memory.
3. Construct, use, and explain procedures for performing whole number calculations and with:
4. 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
· Division of 3-digit numbers by 1-digit numbers
5. Construct and use procedures for performing decimal addition and subtraction.
6. Count and perform simple computations with money.
· Standard dollars and cents notation
7. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
8. Check the reasonableness of results of computations.
9. Use concrete models to explore addition and subtraction with fractions.
C. Estimation
3. Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.
Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:
A. Number Sense
2. Recognize the decimal nature of United States currency and compute with money.
3. Demonstrate a sense of the relative magnitudes of numbers.
4. Use whole numbers, fractions, and decimals to represent equivalent forms of the same number.
5. Develop and apply number theory concepts in problem solving situations.
6. Compare and order numbers.
B. Numerical Operations
1. Recognize the appropriate use of each arithmetic operation in problem situations.
3. Use an efficient and accurate pencil-and-paper procedure for division of a 3-digit number by a 2-digit number.
4. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
5. Check the reasonableness of results of computations.
6. Understand and use the various relationships among operations and properties of operations.
C. Estimation
1. Use a variety of estimation strategies for both number and computation.
2. Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.
3. Determine the reasonableness of an answer by estimating the result of operations.
4. Determine whether a given estimate is an overestimate or an underestimate.
Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:
A. Number Sense
2. Recognize the decimal nature of United States currency and compute with money.
3. Demonstrate a sense of the relative magnitudes of numbers.
4. Explore the use of ratios and proportions in a variety of situations.
5. Understand and use whole-number percents between 1 and 100 in a variety of situations.
6. Use whole numbers, fractions, and decimals to represent equivalent forms of the same number.
7. Develop and apply number theory concepts in problem solving situations.
· Common multiples, common factors
B. Numerical Operations
1. Recognize the appropriate use of each arithmetic operation in problem situations.
2. Construct, use, and explain procedures for performing calculations with fractions and decimals with:
3. Use an efficient and accurate pencil-and-paper procedure for division of a 3-digit number by a 2-digit number.
4. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.
5. Find squares and cubes of whole numbers.
6. Check the reasonableness of results of computations.
7. Understand and use the various relationships among operations and properties of operations.
8. Understand and apply the standard algebraic order of operations for the four basic operations, including appropriate use of parentheses.
C. Estimation
1. Use a variety of strategies for estimating both quantities and the results of computations.
3. Determine the reasonableness of an answer by estimating the result of operations.
4. Determine whether a given estimate is an overestimate or an underestimate.
Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:
A. Number Sense
1. Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 7 pertain to these sets of numbers as well):
· Rational numbers
· Percents
· Whole numbers with exponents
2. Demonstrate a sense of the relative magnitudes of numbers.
4. Compare and order numbers of all named types.
5. Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number.
6. Understand that all fractions can be represented as repeating or terminating decimals.
B. Numerical Operations
2. Use exponentiation to find whole number powers of numbers.
C. Estimation
Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:
A. Number Sense
· Percents
· Roots
· Numbers represented in scientific notation
2. Demonstrate a sense of the relative magnitudes of numbers.
4. Compare and order numbers of all named types.
5. Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number.
7. Construct meanings for common irrational numbers, such as p (pi) and the square root of 2.
B. Numerical Operations
2. Use exponentiation to find whole number powers of numbers.
3. Find square and cube roots of numbers and understand the inverse nature of powers and roots.
4. Solve problems involving proportions and percents.
C. Estimation
1. Estimate square and cube roots of numbers.
3. Recognize the limitations of estimation and assess the amount of error resulting from estimation.
Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
A. Number Sense
1. Extend understanding of the number system to all real numbers.
2. Compare and order rational and irrational numbers.
3. Develop conjectures and informal proofs of properties of number systems and sets of numbers.
B. Numerical Operations
1. Extend understanding and use of operations to real numbers and algebraic procedures.
2. Develop, apply, and explain methods for solving problems involving rational and negative exponents.
3. Perform operations on matrices.
C. Estimation
1. Recognize the limitations of estimation, assess the amount of error resulting from estimation, and determine whether the error is within acceptable tolerance limits.
STANDARD 4.2 (GEOMETRY AND MEASUREMENT) ALL STUDENTS WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES, RELATIONSHIPS, AND MEASUREMENT TO MODEL, DESCRIBE AND ANALYZE PHENOMENA.
Descriptive Statement: Spatial sense is an intuitive feel for shape and space. Geometry and measurement both involve describing the shapes we see all around us in art, nature, and the things we make. Spatial sense, geometric modeling, and measurement can help us to describe and interpret our physical environment and to solve problems.
Geometric Properties. This includes identifying, describing and classifying standard geometric objects, describing and comparing properties of geometric objects, making conjectures concerning them, and using reasoning and proof to verify or refute conjectures and theorems. Also included here are such concepts as symmetry, congruence, and similarity.
Transforming Shapes. Analyzing how various transformations affect geometric objects allows students to enhance their spatial sense. This includes combining shapes to form new ones and decomposing complex shapes into simpler ones. It includes the standard geometric transformations of translation (slide), reflection (flip), rotation (turn), and dilation (scaling). It also includes using tessellations and fractals to create geometric patterns.
Coordinate Geometry. Coordinate geometry provides an important connection between geometry and algebra. It facilitates the visualization of algebraic relationships, as well as an analytical understanding of geometry.
Units of Measurement. Measurement helps describe our world using numbers. An understanding of how we attach numbers to real-world phenomena, familiarity with common measurement units (e.g., inches, liters, and miles per hour), and a practical knowledge of measurement tools and techniques are critical for students' understanding of the world around them.
Measuring Geometric Objects. This area focuses on applying the knowledge and understandings of units of measurement in order to actually perform measurement. While students will eventually apply formulas, it is important that they develop and apply strategies that derive from their understanding of the attributes. In addition to measuring objects directly, students apply indirect measurement skills, using, for example, similar triangles and trigonometry.
Students of all ages should realize that geometry and measurement are all around them. Through study of these areas and their applications, they should come to better understand and appreciate the role of mathematics in their lives.
Cumulative Progress Indicators
By the end of Grade 2, students will:
A. Geometric Properties
1. Identify and describe spatial relationships among objects in space and their relative shapes and sizes.
· Inside/outside, left/right, above/below, between
· Smaller/larger/same size, wider/ narrower, longer/shorter
· Congruence (i.e., same size and shape)
· 3D figures – cube, rectangular prism, sphere, cone, cylinder, and pyramid
· 2D figures – square, rectangle, circle, triangle
· Relationships between three- and two-dimensional shapes (i.e., the face of a 3D shape is a 2D shape)
3. Describe, identify and create instances of line symmetry.
B. Transforming Shapes
1. Use simple shapes to make designs, patterns, and pictures.
2. Combine and subdivide simple shapes to make other shapes.
C. Coordinate Geometry
1. Give and follow directions for getting from one point to another on a map or grid.
D. Units of Measurement
1. Directly compare and order objects according to measurable attributes.
· Attributes – length, weight, capacity, time, temperature
2. Recognize the need for a uniform unit of measure.
· Length – inch, foot, yard, centimeter, meter
· Weight – pound, gram, kilogram
· Capacity – pint, quart, liter
· Time – second, minute, hour, day, week, month, year
· Temperature – degrees Celsius, degrees Fahrenheit
E. Measuring Geometric Objects
1. Directly measure the perimeter of simple two-dimensional shapes.
2. Directly measure the area of simple two-dimensional shapes by covering them with squares.
Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:
A. Geometric Properties
1. Identify and describe spatial relationships of two or more objects in space.
· Vertex, edge, face, side, angle
· 3D figures – cube, rectangular prism, sphere, cone, cylinder, and pyramid
· 2D figures – square, rectangle, circle, triangle, pentagon, hexagon, octagon
3. Identify and describe relationships among two-dimensional shapes.
4. Understand and apply concepts involving lines, angles, and circles.
· Line, line segment, endpoint
5. Recognize, describe, extend, and create space-filling patterns.
B. Transforming Shapes
1. Describe and use geometric transformations (slide, flip, turn).
2. Investigate the occurrence of geometry in nature and art.
C. Coordinate Geometry
1. Locate and name points in the first quadrant on a coordinate grid.
D. Units of Measurement
2. 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
· Capacity – fluid ounce, cup, gallon, milliliter
3. Incorporate estimation in measurement activities (e.g., estimate before measuring).
E. Measuring Geometric Objects
1. Determine the area of simple two-dimensional shapes on a square grid.
2. Determine the perimeter of simple shapes by measuring all of the sides.
3. Measure and compare the volume of three–dimensional objects using materials such as rice or cubes.
Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:
A. Geometric Properties
1. Identify and describe spatial relationships of two or more objects in space.
· Shadows (projections) of everyday objects
· Vertex, edge, face, side, angle
· 3D figures – cube, rectangular prism, sphere, cone, cylinder, and pyramid
· 2D figures – square, rectangle, circle, triangle, quadrilateral, pentagon, hexagon, octagon
· Inclusive relationships – squares are rectangles, cubes are rectangular prisms
3. Identify and describe relationships among two-dimensional shapes.
4.