**MATHEMATICS**

**Standard 1: Number, Operations and Computation**

Students understand the number system, the meaning of operations and how they relate to each other and are able to use computational tools and strategies effectively.

**Standard 2: Geometry, Measurement and Transformation**

Students understand geometry, measurement and spatial relationships including units and systems of measurement and develop and use techniques, tools, and formulas for measuring the properties of objects and relationships among the properties and use transformations and symmetry to analyze mathematical situations.

**Standard 3: Patterns and Algebra**

Students understand various types of patterns and functional relationships, use symbolic forms to represent, model, and analyze mathematical situations and collect, organize, and represent data to answer questions.

**Standard 4: Statistics and Probability**

Students understand how to interpret data using methods of exploratory data analysis, develop and evaluate inferences, predictions and arguments that are based on data and understand and apply basic notions of chance and probability.

Grade 3

By the end of grade 3, students will:

**1. Number, Operations and Computation**

Mth.1.3.1 Use English and local systems to count, read, write and compare whole numbers up to 1,000.

Mth.1.3.2 Understand base-ten by identifying the place value of numbers up to 1,000.

Mth.1.3.3 Demonstrate an understanding of the basic operations (+, - , x, ^), and how they relate to each other.

Mth.1.3.4 Represent whole numbers and operations in a variety of ways using physical models, diagrams, and number expressions.

Mth.1.3.5 Use the basic operations to add and subtract 2- and 3-digit numbers.

Mth.1.3.6 Use the basic operations to multiply and divide 1, 2 and 3-digit numbers by a single digit number.

Mth.1.3.7 Use a variety of strategies including the understanding of number and operations to solve problems and explain the reasoning used to reach the solution.

**2. Geometry, Measurement and Transformation**

Mth.2.3.1 Recognize, classify, identify, and draw common shapes using English and local language terms.

Mth.2.3.2 Describe similarities and differences between common shapes.

Mth.2.3.3 Demonstrate the understanding of standard & non-standard units of measurement.

Mth.2.3.4 Use common instruments to measure & compare objects. (For example, students will be able to use a ruler, watch or balance with standard and non-standard units of measurement.)

**3. Patterns and Algebra**

Mth.3.3.1 Describe and create patterns and find the next term using numbers, objects and other materials.

Mth.3.3.2 Identify and use the inverse relationships between operations to solve problems.

**4. Statistics and Probability**

There are no benchmarks for this standard at this level

Grade 4

By the end of grade 4, students will:

**1. Number, Operations and Computation**

Mth.1.4.1 Understand base-ten by identifying the place value of whole numbers up to 1,000 and decimal numbers down to 100ths.

Mth.1.4.2 Demonstrate the ability to read, write and compare simple fractions and decimals in English and the local counting system.

Mth.1.4.3 Represent whole numbers, fractions and decimals and operations involving them, in a variety of ways using physical models, diagrams and number expressions.

Mth.1.4.4 Perform the basic operations to add, subtract, multiply and divide whole numbers and decimals and add and subtract fractions with like denominators.

Mth.1.4.5 Use a variety of methods and ways to round and estimate whole numbers, decimals and fractions.

Mth.1.4.6 Use a variety of strategies including the understanding of decimals and fractions to solve problems and explain the reasoning used to reach the solution.

**2. Geometry, Measurement and Transformation**

Mth.2.4.1 Identify and classify two- and three-dimensional shapes.

Mth.2.4.2 Describe similarities and differences between one, two- and three-dimensional geometric figures.

Mth.2.4.3 Demonstrate understanding of common units in the English and metric systems by choosing appropriate units to measure common objects and quantities.

Mth.2.4.4 Use standard and non-standard units to determine length, volume and weight and describe the characteristics of each type of measurement.

Mth.2.4.5 Use the understanding of geometry, measurement and transformation to solve problems and explain the reasoning used to reach the solution.

**3. Patterns and Algebra**

Mth.3.4.1 Use patterns and functions to represent and solve real world situations and explain the reasoning used to reach the solution.

**4. Statistics and Probability**

Mth.4.4.1 Collect, organize, display and describe data systematically.

Mth.4.4.2 Read and interpret data using pictographs, tables or charts.

Grade 5

By the end of grade 5, students will:

**1. Number, Operations and Computation**

Mth.1.5.1 Demonstrate the ability to read, write and compare more complex decimals and fractions.

Mth.1.5.2 Represent fractions as proper and improper fractions, mixed numbers and decimals.

Mth.1.5.3 Continue to develop fluency to do the basic operations to add, subtract, multiply and divide whole numbers, decimals and simple fractions.

Mth.1.5.4 Use rounding and estimation to solve problems.

Mth.1.5.5 Choose and use appropriate computational procedures and tools to solve problems. (For example, students will be able to use pencil and paper, mental computation or calculators.)

Mth.1.5.6 Use a variety of strategies including making a model, working backwards, drawing a diagram or guessing and checking to solve problems and explain the reasoning used to reach the solution.

**2. Geometry, Measurement and Transformation**

Mth.2.5.1 Recognize and classify triangles and quadrilaterals based on their properties of angle and side.

Mth.2.5.2 Use the common units of the English and metric systems and carry out simple unit conversion within the systems. (For example, students will be able to convert centimeters to meters or inches to feet.)

Mth.2.5.3 Develop and use formulas to determine perimeter and area.

Mth.2.5.4 Measure length, area, volume and weight accurately using appropriate tools.

Mth.2.5.5 Use a variety of strategies to solve measurement problems and explain the reasoning used to reach the solution.

**3. Patterns and Algebra**

Mth.3.5.1 Represent and record patterns using tools such as charts, tables and graphs.

Mth.3.5.2 Use words and simple algebraic expressions to describe quantities and situations.

Mth.3.5.3 Represent and investigate how a change in one variable relates to the change in the second variable. (For example, students will be able to make a graph to show the height of a plant over time.)

Mth.3.5.4 Investigate and describe situations involving inverse relationships. (For example, students will be able to demonstrate that the larger the denominator in a fraction the smaller the quantity.)

**4. Statistics and Probability:**

Mth.4.5.1 Collect data using observations, measurements, surveys or experiments.

Mth.4.5.2 Organize data using tables and charts and construct pictographs, bar graphs and line graphs.

Mth.4.5.3 Identify events as likely or unlikely and give a description of the degree of likelihood in informal terms. (For example, students will be able to state whether an event is certain, unlikely, very unlikely or impossible.)

Mth.4.5.4 Estimate and describe probabilities in simple experiments involving coins, spinners, dice or objects in a bag.

Grade 6

By the end of grade 6, students will:

**1. Number, Operations and Computation**

Mth.1.6.1 Compare, order, round and group natural numbers.

Mth.1.6.2 Demonstrate fluency in the basic operations of adding, subtracting, multiplying and dividing whole numbers, fractions and decimals.

Mth.1.6.3 Identify the characteristics of prime numbers and composite numbers and decompose composite numbers into factor pairs and prime factors using exponents.

Mth.1.6.4 Use models and pictures to represent ratio and proportions and solve problems.

**2. Geometry, Measurement and Transformation**

Mth.2.6.1 Add and subtract customary units of length, mass, capacity and time.

Mth.2.6.2 Perform slides, flips, turns and rotations and indicate the motion, position and direction applied.

Mth.2.6.3 Use formulas to compute the perimeter and area of polygons.

Mth.2.6.4 Describe, compare and classify geometrical figures using mathematical terminology. (For example, students will be able to use terms such as number of edges and faces, number and size of angles and number of vertices.)

**3. Patterns and Algebra**

Mth.3.6.1 Represent patterns in a variety of ways, (For example, students will be able to represent patterns in numeric, algebraic, pictorial, oral and graphic form.)

Mth.3.6.2 Model and solve real world problems using various representations such as graphs and tables.

Mth.3.6.3 Locate whole numbers, fractions and decimals on a number line.

Mth.3.6.4 Solve simple algebraic expressions.

**4. Statistics and Probability**

Mth.4.6.1 Analyze and interpret data, including range, median, mode, mean and frequency and present information to an audience.

Mth.4.6.2 Make predictions that are based on experimental or theoretical probabilities and determine their reasonableness.

Mth.4.6.3 Formulate and solve problems that involve collecting and analyzing data to reach conclusions and make generalizations.

Grade 7

By the end of grade 7, students will:

**1. Number, Operations and Computation**

Mth.1.7.1 Understand and represent integers, adding and subtracting them in real world situations. (For example, students will be able to add and subtract negative and positive numbers to solve problems related to temperature, elevation and accounting.)

Mth.1.7.2 Locate whole numbers, fractions, decimals and integers on a number line.

Mth.1.7.3 Convert between decimals, fractions and percentages.

Mth.1.7.4 Represent numbers in a variety of ways including expanded form and scientific notation.

Mth.1.7.5 Use the properties of numbers including zero, identity, commutative, associative and distributive, to solve problems.

Mth.1.7.6 Use the order of operations to evaluate expressions.

Mth.1.7.7 Round numbers to estimate solutions and check the reasonableness of results.

**2. Geometry, Measurement and Transformation**

Mth.2.7.1 Identify and draw points, lines, line segments, angles and rays.

Mth.2.7.2 Use pi (

**π**) , represented as both a decimal (3.14) and fraction (22/7), to find the circumference and area of circles. Mth.2.7.3 Use appropriate English and metric units to develop reasonable estimates of measures.

Mth.2.7.4 Describe symmetry, reflections and translations with appropriate notation.

**3. Patterns and Algebra**

Mth.3.7.1 Describe relationships and functions using word and symbols.

Mth.3.7.2 Write and solve one-step equations.

Mth.3.7.3 Locate points on the coordinate plane.

**4. Statistics and Probability**

Mth.4.7.1 Propose and support conclusions by summarizing data. (For example, after completing a survey, students will be able to propose and support the conclusion that over half the books students read in a year are read in April and May.)

Mth.4.7.2 Formulate questions or hypotheses based on initial data collection and describe further studies to explore them.

Grade 8

By the end of grade 8, students will:

**1. Number, Operations and Computation**

Mth.1.8.1 Represent, compare, order and use numbers in a variety of forms, including integer, fraction, decimal, percentage and exponents, in mathematical problem-solving situations.

Mth.1.8.2 Demonstrate fluency in computing with rational numbers, fractions, decimals, percentages and integers.

Mth.1.8.3 Square whole numbers, rational numbers and integers and find square roots of perfect squares. (For example, students will be able to find the square root of 1, 2, 4, 9, etc.)

Mth.1.8.4 Use ratio, proportion, and percentage in problem solving.

**2. Geometry, Measurement and Transformation**

Mth.2.8.1 Use a compass, protractor and straight edge to draw two-dimensional figures and do constructions (For example, students will be able to bisect an angle or line segment, create a right angle or draw a circle.)

Mth.2.8.2 Identify similar and congruent figures including lines of symmetry and diagonals.

Mth.2.8.3 Use formulas to find areas of quadrilaterals, triangles and circles, and the surface area and volume of cylinders and prisms using appropriate units of measure.

Mth.2.8.4 Use the Pythagorean Theorem to find lengths of sides of right triangles.

Mth.2.8.5 Solve simple problems involving rates and derived measures. (For example, students will be able to solve problems involving miles per hour or cost per yard.)

Mth.2.8.6 Use proportional reasoning and indirect measurements to draw inferences, such as measuring the thickness of a book and dividing by the number of pages to estimate the thickness of one page.

**3. Patterns and Algebra**

Mth.3.8.1 Write and solve two-step linear equations and one-step inequalities.

Mth.3.8.2 Graph linear functions in two variables using a table of ordered pairs.

Mth.3.8.3 Use symbolic algebra and additional techniques, such as tables, guess and check and diagrams, to represent situations and to solve problems, especially those that involve linear relationships.

Mth.3.8.4 Model and solve real-world problems using various representations, such as graphs and tables and understand the purpose and utility of each representation.

**4. Statistics and Probability**

Mth.4.8.1 Find, describe, and interpret mean, median, mode and range and determine which measure is best to use in a particular situation.

Mth.4.8.2 Read and interpret tables, charts and graphs and make inferences based on the data.

Mth.4.8.3 Use sampling and other data collection tools to gather and analyze data and make conclusions and predictions.

Mth.4.8.4 Compute simple probabilities using appropriate methods such as lists and tree diagrams or through experimental or simulation activities.

**Benchmarks for All High Schools**

All high school students will:

(Based on the shared requirement that all students complete at least Algebra I.)

**1. Number, Operations and Computation**

Mth.1.hs.1 Demonstrate the inverse relationship between square numbers and square roots.

Mth.1.hs.2 Compare and order rational numbers and square roots using a number line.

Mth.1.hs.3 Solve problems with squares and square roots, limited to square roots of square numbers.

Mth.1.hs.4 Represent numbers in a variety of forms including factors, multiples, exponents, primes, composites, fractions, decimals and percentages and change from one form to another.

Mth.1.hs.5 Apply an understanding of addition, subtraction, multiplication, division and the order of operations when calculating with rational numbers.

Mth.1.hs.6 Use ratios, proportions and percent to represent the relationship between two quantities and solve problems.

Mth.1.hs.7 Add, subtract, multiply, and divide numbers with positive and negative exponents.

Mth.1.hs.8 Estimate a reasonable solution to a problem.

Mth.1.hs.9 Use rounding and estimation to solve real-world situations and recognize the limitations.

**2. Geometry, Measurement and Transformation**

Mth.2.hs.1 Apply an understanding of the English and metric systems of measurement to solve problems.

Mth.2.hs.2 Use formulas, including appropriate units of measure, to determine the surface area and volume of selected prisms, cylinders and pyramids.

Mth.2.hs.3 Apply the Pythagorean Theorem to solve problems involving right triangles.

Mth.2.hs.4 Perform transformations including reflection, rotation and translation and describe the size, position and orientation of the resulting shapes.

**3. Patterns and Algebra**

Mth.3.hs.1 Represent a variety of patterns, including recursive patterns, with tables, graphs, words and symbolic rules.

Mth.3.hs.2 Represent mathematical situations as algebraic expressions and equations and describe algebraic expressions using words.

Mth.3.hs.3 Solve single-variable equations and inequalities using rational numbers.

Mth.3.hs.4 Use tables and graphs to represent linear relationships involving equalities and inequalities with two variables and solve problems.

Mth.3.hs.5 Justify the steps used in simplifying expressions and solving equations and inequalities.

Mth.3.hs.6 Identify functions as linear or nonlinear and contrast their properties from tables, graphs or equations.

Mth.3.hs.7 Represent data involving linear relationships from tables as graphs and equations and vice versa.

Mth.3.hs.8 Solve linear equations and inequalities with two variables using algebraic methods, manipulatives or models.

Mth.3.hs.9 Determine the slope of a line when given the graph of a line, two points on the line, or the equation of the line.

Mth.3hs.10 Select and use a variety of strategies including concrete modeling, pictorial representation, and algebraic manipulation to add, subtract, multiply, divide and factor first and second-degree binomials and trinomials in one variable.

**4. Statistics and Probability**

Mth.4.hs.1 Analyze and interpret data using mean, median, mode, range and frequency.

Mth.4.hs.2 Design a study, collect data, and select the appropriate representation to make conclusions and generalizations.

Mth.4.hs.3 Judge the validity of reported data, conclusions and generalizations.

Mth.4.hs.4 Calculate probabilities for simple events under different relationships, including independent, dependent, with replacement and without replacement.

**Benchmarks for Students in College Preparatory Programs**

The earlier benchmarks define the important knowledge and skills in which all young people in FSM should demonstrate proficiency. The following outcomes reflect some of the additional knowledge and skills that students in high school college preparatory programs should demonstrate, as and if they complete the following mathematics courses.

**1. Algebra 2**

- Apply arithmetic properties to operate on and simplify expressions that include radicals and other real numbers.
- Use the complex number system, the notation for complex numbers, and the definition of “i” to solve problems in standard form.
- Add, subtract, multiply, and divide complex numbers.
- Use the inverse relationship between exponents and logarithms to solve exponential and logarithmic problems.
- Use advanced formulas or functions to solve problems.
- Apply the properties of arithmetic and geometric sequences and series to solve problems.
- Use exponential functions to solve problems involving exponential growth and decrease.
- Use the properties of many types of functions including polynomial, absolute value, exponential, and logarithmic, to identify the function’s graph.
- Use the appropriate terminology and notation to define functions and their properties, including domain, range, function composition, inverses, zeros and asymptotes.
- Describe the relationship among relations and functions.
- Solve equations and inequalities involving absolute values.
- Solve systems of linear equations and inequalities in two or three variables using a variety of strategies, such as substitution, graphing or matrices.
- Solve equations containing radicals and exponents.
- Factor polynomials representing perfect squares, the difference in squares, perfect square trinomials, the sum and difference of cubes and general trinomials.
- Apply quadratic equations to solve real-world situations and complex number problems.
- Use the binomial theorem to expand binomial expression.
- Use the fundamental counting principles for combinations and permutations to determine probability.
- Calculate probabilities of events under different relationships such as inclusion, disjoint, complementary, independent, and dependent, with replacement and without replacement.
- Use the right triangle relationships of trigonometric ratio, cosine, sine and tangent to solve problems.

**2. Geometry**

- Use right triangle trigonometric ratios to determine an unknown length of a side or the measure or an angle.
- Solve problems using the formulas for perimeter, circumference, area and volume of two- and three- dimensional figures and solids and determine the effect of dimension changes to perimeter, area and volume.
- Use reasoning to create and defend geometric conjectures.
- Use the concept of corresponding parts to prove that triangles and other polygons are congruent or similar.
- Explain properties and characteristics of angle bisectors, perpendicular bisectors and parallel lines.
- Use the relationship between pairs of angles (complementary, supplementary, vertical, exterior, interior) to determine unknown angle measures or definitions of properties.
- Apply the concepts of special right triangles to real-world situations.
- Use the relationships among properties of circles (chords, secants, tangents, arcs, circumference, radius, diameter, inscribed polygons) to solve problems.
- Use coordinate geometry to produce formulas and prove theorems for the midpoint of a line segment, the distance formula and forms of equations of lines and circles.
- Describe the concept of rigid motion on figures in the coordinate plane, including rotation, translation and reflection.
- Use concrete objects, pictorial representations, computer software or graphing calculators to solve geometric problems.

**3. Trigonometry**

- Express complex numbers in standard and polar form and convert from one to another.
- Add, subtract, multiply, divide, and find powers of complex numbers in polar form.
- Use vector operations, including dot product and cross product, the law of sines and the law of cosines to solve problems.
- Calculate linear and angular velocity.
- Find the sine, cosine, tangent, cotangent, secant and cosecant of an angle in standard position.
- Use the relationship among the six trigonometric functions to translate among them.
- Recognize the trigonometric functions of benchmark angles.
- Translate between radians and degrees.
- Find the value of any trigonometric function and inverse trigonometric function and solve trigonometric equations
- Use the fundamental trigonometric identities, including the sum and difference formulas, double-angle formulas and half-angle formulas to solve problems
- Verify trigonometric identities.
- Solve trigonometric equations and inverse trigonometric equations that include all solutions or solutions with restricted domains
- Use the trigonometric functions in the form y = ASin (Bx+C) + D to determine various properties of the function including domain, range, period, phase shift and amplitude.
- Identify real-world phenomena that can be represented by a trigonometric function in the form y = ASin (Bx + C) + D.
- Explain the relationship between trigonometric functions and their inverse.

**4. Calculus**

- Recognize limits from graphs and tables.
- Find limits of sums, differences, products, quotients and rational functions.
- Understand continuity in terms of limits and functions.
- Find the derivatives of functions, including polynomial, rational, trigonometric, logarithmic, inverse, composite and exponential functions.
- Find the derivatives of implicitly-defined functions.
- Find points of inflection of functions.
- Use implicit differentiation to find the derivative of an inverse function.
- Use integration by substitution or change of variable to evaluate integrals.
- Use Riemann sums, the trapezoidal rule and technology to approximate definite integrals of functions represented algebraically, geometrically or by tables of values.
- Find specific antiderivatives using initial conditions, including finding velocity functions from acceleration functions, finding position functions from velocity functions and applications to motion along a line.
- Use definite integrals to find the area between a curve and the x-axis, the average value of a function over a closed interval and the volume of a solid with known cross-sectional area.
- Apply the intermediate value theorem and extreme value theorem on a function over a closed interval.
- Apply the fundamental theorem of calculus, that is, interpret a definite integral of the rate of change of a quantity over an interval as the change of the quantity over the interval.
- Describe the concept of a derivative geometrically, numerically, analytically and verbally.
- Find second derivatives and derivatives of higher order.
- Prove the mean value theorem.
- Find average and instantaneous rates of change.
- Use first and second derivatives to describe the behavior of functions.