STANDARD 4: Mathematics

Mathematics

Mission: Through mathematics, students communicate, make connections, reason, and represent the world quantitatively in order to pose and solve problems.

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.

Big Idea: Numeric reasoning involves fluency and facility with numbers.
4.1 A. Number Sense
Descriptive Statement: 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.
Essential Questions	Enduring Understandings
- How do mathematical ideas interconnect and build on one another to produce a coherent whole? (4.5C1; 4.5C6) - How can we compare and contrast numbers? (4.5A4) - How can counting, measuring, or labeling help to make sense of the world around us?	- One representation may sometimes be more helpful than another; and, used together, multiple representations give a fuller understanding of a problem. - A quantity can be represented numerically in various ways. Problem solving depends upon choosing wise ways. - Numeric fluency includes both the understanding of and the ability to appropriately use numbers.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
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.
4.1 B. Numerical Operations
Descriptive Statement: 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.
Essential Questions	Enduring Understandings
- What makes a computational strategy both effective and efficient? (4.5D1) - How do operations affect numbers? - How do mathematical representations reflect the needs of society across cultures? (An essential question with broad applicability across multiple standards) (4.5C5)	- Computational fluency includes understanding not only the meaning, but also the appropriate use of numerical operations. - The magnitude of numbers affects the outcome of operations on them. - In many cases, there are multiple algorithms for finding a mathematical solution, and those algorithms are frequently associated with different cultures.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
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. · Addition and subtraction · Scalar multiplication
4. Understand and apply the laws of exponents to simplify expressions involving numbers raised to powers.
4.1 C. Estimation
Descriptive Statement: 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.
Essential Questions	Enduring Understandings
- How can we decide when to use an exact answer and when to use an estimate?	- Context is critical when using estimation.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
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.

Big Idea Geometry: Spatial sense and geometric relationships are a means to solve problems and make sense of a variety of phenomena. Big Idea Measurement: Measurement is a tool to quantify a variety of phenomena.
4.2 A. Geometric Properties
Descriptive Statement: This includes identifying, describing and classifying standard geometric object, 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.
Essential Questions	Enduring Understandings
- How can spatial relationships be described by careful use of geometric language? - How do geometric relationships help in solving problems and/or make sense of phenomena?	- Geometric properties can be used to construct geometric figures. (4.5D1; 4.5D2; 4.5E3) - Geometric relationships provide a means to make sense of a variety of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
1. Use geometric models to represent real-world situations and objects and to solve problems using those models (e.g., use Pythagorean Theorem to decide whether an object can fit through a doorway).
2. Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views).
3. Apply the properties of geometric shapes. · Parallel lines – transversal, alternate interior angles, corresponding angles · Triangles a. Conditions for congruence b. Segment joining midpoints of two sides is parallel to and half the length of the third side c. Triangle Inequality · Minimal conditions for a shape to be a special quadrilateral · Circles – arcs, central and inscribed angles, chords, tangents · Self-similarity
4. Use reasoning and some form of proof to verify or refute conjectures and theorems. · Verification or refutation of proposed proofs · Simple proofs involving congruent triangles · Counterexamples to incorrect conjectures
4.2 B. Transforming Shapes
Descriptive Statement: This includes identifying, describing and classifying standard geometric object, 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.
Essential Questions	Enduring Understandings
- What situations can be analyzed using transformations and symmetries? (4.5E1; 4.5E2; 4.5E3)	- Shape and area can be conserved during mathematical transformations..
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
1. Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic object, and, conversely, determine whether and how one object can be transformed to another by a transformation or a sequence of transformations.
2. Recognize three-dimensional figures obtained through transformations of two-dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization.
3. Determine whether two or more given shapes can be used to generate a tessellation.
4. Generate and analyze iterative geometric patterns. · Fractals (e.g., Sierpinski’s Triangle) · Patterns in areas and perimeters of self-similar figures · Outcome of extending iterative process indefinitely
4.2 C. Coordinate Geometry
Descriptive Statement: 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.
Essential Questions	Enduring Understandings
- How can geometric/algebraic relationships best be represented and verified? (4.5C2; 4.5D2; 4.5E1; 4.5E2; 4.5F5)	- Reasoning and/or proof can be used to verify or refute conjectures or theorems in geometry (4.5D1; 4.5D3; 4.5D4; 4.5D5; 4.5F5) - Coordinate geometry can be used to represent and verify geometric/algebraic relationships.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
1. Use coordinate geometry to represent and verify properties of lines. · Distance between two points · Midpoint and slope of a line segment · Finding the intersection of two lines · Lines with the same slope are parallel · Lines that are perpendicular have slopes whose product is –1
2. Show position and represent motion in the coordinate plane using vectors. · Addition and subtraction of vectors
4.2 D. Units Of Measurement
Descriptive Statement: 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.
Essential Questions	Enduring Understandings
- How can measurements be used to solve problems? (4.5A6)	- Everyday objects have a variety of attributes, each of which can be measured in many ways. -What we measure affects how we measure it. (4.5A4; 4.5A6) - Measurements can be used to describe, compare, and make sense of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
1. Understand and use the concept of significant digits.
2. Choose appropriate tools and techniques to achieve the specified degree of precision and error needed in a situation. · Degree of accuracy of a given measurement tool · Finding the interval in which a computed measure (e.g., area or volume) lies, given the degree of precision of linear measurements
4.2 E. Measuring Geometric Objects
Descriptive Statement: 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 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.
Essential Questions	Enduring Understandings
- How can measurements be used to solve problems? (4.5A6)	- Everyday objects have a variety of attributes, each of which can be measured in many ways. - What we measure affects how we measure it. (4.5A4; 4.5A6)** - Measurements can be used to describe, compare, and make sense of phenomena.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
1. Use techniques of indirect measurement to represent and solve problems. · Similar triangles · Pythagorean theorem · Right triangle trigonometry (sine, cosine, tangent)
2. Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures. · Approximation of area using grids of different sizes · Finding which shape has minimal (or maximal) area, perimeter, volume, or surface area under given conditions using graphing calculators, dynamic geometric software, and/or spreadsheets · Estimation of area, perimeter, volume, and surface area

Standard 4.3 Patterns and Algebra
All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.

Big Idea Algebra provides language through which we communicate the patterns in mathematics.
4.3 A. Patterns
Descriptive Statement: Algebra provides the language through which we communicate the patterns in mathematics. From the earliest age, students should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and by doing so, to make mathematical discoveries. They should have opportunities to analyze, extend, and create a variety of patterns and to use pattern-based thinking to understand and represent mathematical and other real-world phenomena.
Essential Questions	Enduring Understandings
- How can change be best represented mathematically? (4.5C1; 4.5F1; 4.5F2; 4.5F3; 4.5F4) - How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations? (4.5C1)	- The symbolic language of algebra is used to communicate and generalize the patterns in mathematics. - Algebraic representation can be used to generalize patterns and relationships.
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
1. Use models and algebraic formulas to represent and analyze sequences and series. · Explicit formulas for n^th terms · Sums of finite arithmetic series · Sums of finite and infinite geometric series
2. Develop an informal notion of limit.
3. Use inductive reasoning to form generalizations.
4.3 B. Functions and Relationships
Descriptive Statement: The function concept is one of the most fundamental unifying ideas of modern mathematics. Student begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Essential Questions	Enduring Understandings
- How are patterns of change related to the behavior of functions? (4.5F1; 4.5F2; 4.5F3; 4.5F4)	- Patterns and relationships can be represented graphically, numerically, symbolically, or verbally. (4.5E1)
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.
2. Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies. · Slope of a line or curve · Domain and range · Intercepts · Continuity · Maximum/minimum · Estimating roots of equations · Intersecting points as solutions of systems of equations · Rates of change
3. Understand and perform transformations on commonly-used functions. · Translations, reflections, dilations · Effects on linear and quadratic graphs of parameter changes in equations · Using graphing calculators or computers for more complex functions
4. Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions. · Linear vs. non-linear · Symmetry · Increasing/decreasing on an interval
4.3 C. Modeling
Descriptive Statement: The function concept is one of the most fundamental unifying ideas of modern mathematics. Student begin their study of functions in the primary grades, as they observe and study patterns. As students grow and their ability to abstract matures, students form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, they use the more formal language of algebra to describe these relationships.
Essential Questions	Enduring Understandings
- How are mathematical models used to describe physical relationships? (4.5E2) - How are physical models used to clarify mathematical relationships? (4.5E3)	- Mathematical models can be used to describe and quantify physical relationships. (4.5E2) - Physical models can be used to clarify mathematical relationships. (4.5E3)
Areas of Focus/Cumulative Progress Indicators	Comments and Examples
By the end of Grade 12:
1. Use functions to model real-world phenomena and solve problems that involve varying quantities. · Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years) · Direct and inverse variation · Absolute value · Expressions, equations and inequalities · Same function can model variety of phenomena · Growth/decay and change in the natural world · Applications in mathematics, biology, and economics (including compound interest)
2. Analyze and describe how a change in an independent variable leads to change in a dependent one.