- Curriculum and SLO
- February 16, 2023
- 1:16 am

**The National Curriculum**

The National Curriculum lies at the heart of the national Department of Education’s policies to raise student performance standards. It sets out a clear, full, and statutory entitlement to learning for all students in FSM public schools. It determines the content of what will be taught, and sets attainment targets for learning. It also determines how student performance will be assessed and reported. The National Curriculum gives teachers, students, parents, and employers and the wider community a clear and shared understanding of the skills and knowledge that young people will gain at school. It provides a framework within which all partners in education can support young people on the road to further learning.

This document focuses on the Student Learning Outcomes (SLOs) for FSM National Curriculum Standard and Benchmarks in MATHEMATICS, from Grade 1 through High School. These SLOs have been developed by a team of curriculum developers, math content experts, and teachers representing the National and State Departments of Education. The SLOs are based upon existing National Curriculum Standards and Benchmarks and also draw upon revised versions of State Curriculum Standards and Benchmarks.

The SLOs have been approved by State Directors of Education and State Boards of Education. Furthermore, the SLOs have been endorsed by the Secretary of Education and are therefore mandatory for all schools within the FSM.

These SLOs are now part of the National Curriculum Minimum Standards, outlined under Section 110 of FSM Code Title 40. State Departments of Education must ensure that these minimum standards are implemented through their State Curriculum Frameworks.

All children of compulsory school age throughout the FSM, including children with special needs, have an entitlement to learning that allows them to meet or exceed the minimum standards contained within this curriculum framework. Public schools are legally obliged to follow the standards and guidelines laid out in this document.

**Coding**

The Coding for the current FSM National Math Standards, Benchmarks, and SLO document is as follows:

MTH.(Grade).(Standard).(Benchmark).(SLO letter)

For example, an SLO for fourth grade, Standard 1 Benchmark 3 is: is now coded “MTH.4.1.3.a”.

It is important to note that this is a change from the coding in the previous document of the FSM National Standards, Benchmarks and SLOs. Previously, the Standard number came before the Grade designation.

**Mathematics Standards**

**Numbers and Operations**– 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.

**Geometry**– Students understand geometry and spatial relationships, specifically, understanding the properties of objects and relationships among the properties and use transformations and symmetry to analyze mathematical situations.

**Measurement**– Students understand the purpose and application of various systems of measurement, including how to develop and use techniques, tools, and formulas for measuring the properties of objects.

**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.

**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.

**Student Learning Outcomes**

The Student Learning Outcomes (SLOs) describe what each student in the FSM should know and be able to do within each Benchmark at every grade level. The diagram below shows the relationship between Standards, Benchmarks, and SLOs.

Standard | In Mathematics, there are 5 standards. The standards describe, in general terms, what all students should study and they skills they should be able to demonstrate from Grade 1 to Grade 12.
| |||||

Benchmark | Benchmark | Benchmarks describe, in some detail, the skills students should be able to demonstrate at certain key stages, typically by the end of a particular grade. | ||||

SLO | SLO | SLO | SLO | SLO | SLO | Student Learning Outcomes (SLOs) help to “unpack” a benchmark by describing, in greater detail, the skills students should be able to demonstrate at each grade level for each Benchmark. |

SLOs act as a guideline that helps teachers think through what is required for successful attainment of Benchmarks and Standards. While SLOs contain information that can be used by teachers to plan their program of work, they should not be relied upon as lesson planning templates. Rather, they identify specific learning experiences that all students should enjoy as well as specific teaching strategies and methods.

SLOs also describe the scope and sequence of specific skills that students should master at each grade level in order to achieve a certain benchmark. The SLOs can be used to assess student progress at each grade level. Evidence of progress should be collected in the form of teacher observations and samples of student work.

**STANDARD 1. NUMBERS, OPERATIONS AND**

**COMPUTATIONS**

MTH.1.1.1 Use English and local systems to count, read, write and compare whole numbers up to 200.

**Student Learning Outcomes**

1.1.1.a. Count and identify numbers from 0 to 200 in the local language.

1.1.1.b. Count and identify numbers from 0 to 200 in English.

1.1.1.c. Write numbers from 0 to 200 in both the local language and English.

1.1.1.d. Compare whole numbers less than 200.

1.1.1.e. Count from 1-10 using local counting systems.

**STANDARD 1. NUMBERS, OPERATIONS AND**

**COMPUTATIONS**

MTH.1.1.2 Understanding base-ten by identifying the place value of numbers up to

hundreds.

**Student Learning Outcomes**

1.1.2.a. Name the number of tens or ones in a set and write the standard numeral.

1.1.2.b. Identify and name the place value of any digit in any given number up to hundreds.

**STANDARD 1. NUMBERS, OPERATIONS AND**

**COMPUTATIONS**

MTH.1.1.3 Demonstrate an understanding of the basic operations (+,-, x, ÷), and how they relate to each other.

**Student Learning Outcomes**

1.1.3.a. Identify and name the operation symbols.

1.1.3.b. Add any two 1-digit numbers with sums up to 18.

1.1.3.c. Solve subtraction problems with a minuend of 18 or less.

1.1.3.d. Skip-count by twos.

1.1.3.e. Group objects into equal sets of twos and threes.

1.1.3.f. Demonstrate understanding of the relationship between addition and subtraction.

1.1.3.g. Demonstrate an understanding for order in addition.

**STANDARD 1. NUMBERS, OPERATIONS AND**

**COMPUTATIONS**

MTH.1.1.4 Represent whole numbers and operations in a variety of ways using physical models, diagrams, and number expressions

**Student Learning Outcomes**

1.1.4.a. Draw pictures that represent number expressions or operations

1.1.4.b. Use physical models to show addition and subtraction of whole numbers.

1.1.4.c. Write numerals to show how many in all when two sets of objects are joined.

1.1.4.f. Write addition and subtraction sentences using facts up to 10.

**STANDARD 1. NUMBERS, OPERATIONS AND**

**COMPUTATIONS**

MTH.1.1.5 Use the basic operations to add and subtract 2- and 3-digit numbers.

**Student Learning Outcomes**

1.1.5.a. Demonstrate complete mastery of sums up to 18.

1.1.5.b. Demonstrate complete mastery of difference up to 9.

1.1.5.c. Add and subtract 2- to 3-digit numbers without renaming.

MTH.1.1.6 Use the basic operations to multiply and divide 1, 2 and 3 digit numbers by a single digit number.

1.1.6.a. Apply repeated subtraction to show division with whole numbers less than 20.

MTH.1.1.7 Use a variety of strategies including the understanding of number and operations to solve problems and explain the reasoning used to reach the solution.

1.1.7.a. Identify key words in a given problem to decide on the basic operation (+,-, x, ÷) needed to solve the problem.

1.1.7.b. Demonstrate at least 2 possible ways to solve a problem.

1.1.7.c. Use concrete objects or modeling to solve addition and subtraction problems.

**STANDARD 2. GEOMETRY, MEASUREMENT AND TRANSFORMATION**

MTH.1.2.1 Identify and classify two to three dimensional shapes

1.2.1.a. Identify and name the basic geometric figures (square, triangle and rectangles).

1.2.1.b. Draw two and three dimensional shapes (square, triangle and rectangle).

1.2.1.c. Identify points inside, outside or on a figure.

1.2.1.d. Draw a line of symmetry using basic geometric shapes.

MTH.1.2.2 Describe similarities and differences between common shapes.

1.2.2.a. Tell how two shapes are similar or different.

1.2.2.b. Classify shapes that are similar.

MTH.1.2.3 Demonstrate understanding of standards and non-standards units of measurement.

1.2.3.a. Find the next number in a sequence of numbers.

1.2.3.b. Identify standard units of measurement.

1.2.3.c. Identify non-standard units of measurement.

MTH.1.2.4 Use common instruments to measure and compare objects. (For example, students will be able to use a ruler, watch or balance with standard and non-standard units of measurement).

1.2.4.a. Use a ruler to measure and compare length of objects.

1.2.4.b. Use a watch/clock to tell and compare times.

1.2.4.c. Use a scale to weigh and compare objects.

1.2.4.d. Measure length with non-standard units of measurement.

1.2.4.e. Tell time using non-standard units of measurement.

1.2.4.f. Read a Celsius thermometer.

**STANDARD 3. PATTERNS AND ALGEBRA**

MTH.1.3.1 Describe and create patterns and find the next term using numbers, objects, and other materials.

1.3.1.a. Find the next term in a series of numbers.

1.3.1.b. Find the item in a series of patterns.

1.3.1.c. Create patterns.

MTH.1.3.2 Identify and use the inverse relationships between operations to solve problems.

1.3.2.a. Apply the horizontal form for addition and subtraction to find an unknown using single digit number.

By the end of GRADE 2, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.2.1.1 Use English and local systems to count, read, write and compare whole numbers up to 1,000.

2.1.1.a. Read and write numerals up to 1,000 in the local language.

2.1.1.b. Read and write numerals up to 1,000 in English.

2.1.1.c. Compare whole numbers less than 1,000.

2.1.1.d. Identify numbers as used in local counting systems from 1 to 100.

MTH.2.1.2 Understanding base-ten by identifying the place value of numbers up to 1,000.

2.1.2.a. Rename 10 ones as 1 ten or vice versa.

2.1.2.b. Identify the place value of any digit in any given whole number up to thousands.

MTH.2.1.3 Demonstrate an understanding of the basic operations (+, -, x, ÷), and how they relate to each other.

2.1.3.a. Explore order in addition using the horizontal form.

2.1.3.b. Recognize that addition undoes subtraction and subtraction undoes addition.

2.1.3.c. Explore order in multiplication using the horizontal form.

2.1.3.d. Recognize that multiplication undoes division and division undoes multiplication.

MTH.2.1.4 Represent whole numbers and operations in a variety of ways using physical models, diagrams, and number expressions.

2.1.4.a. Compare numbers over 200 using >, < and =.

2.1.4.b. Write 2- and 3-digit numerals in expanded form.

2.1.4.c. Add and subtraction whole numbers in expanded form.

2.1.4.d. Add, subtract, multiply and divide whole numbers using a number line.(Use whole numbers less than 20.)

2.1.4.e. Draw pictures that represent number expressions or operations.

MTH.2.1.5 Use the basic operations to add and subtract 2- and 3-digit numbers.

2.1.5.a. Add and subtract tens and ones with and without renaming.

2.1.5.b. Add and subtract hundreds, tens, and ones without renaming.

MTH.2.1.6 Use the basic operations to multiply and divide 1, 2 and 3 digit numbers by a single digit number.

2.1.6.a. Skip count by 2’s, 3’s, 4’s and 5’s.

2.1.6.b. Multiply with products up to 25.

2.1.6.c. Break down even whole numbers less than 30 into equal sets of 2’s, 3’s, 4’s, and 5’s.

MTH.2.1.7 Use a variety of strategies including the understanding of number and operations to solve problems and explain the reasoning used to reach the solution.

2.1.7.a. Identify key words in a given problem to decide on the operation needed to solve the problem.

2.1.7.b. Demonstrate various ways to solve a problem.

**STANDARD 2. GEOMETRY, MEASUREMENT AND TRANSFORMATION**

MTH.2.2.1 Identify and classify two to three dimensional shapes.

2.2.1.a. Name the basic geometric figures (triangles, quadrilaterals).

2.2.1.b. Draw the basic shapes.

2.2.1.c. Classify basic (common) geometric shapes.

MTH.2.2.2 Describe similarities and differences between common shapes.

2.2.2.a. Tell how two shapes are alike.

2.2.2.b. Tell how two shapes are different.

MTH.2.2.3 Demonstrate understanding of standards and non-standards units of measurement.

2.2.3.a. Name appropriate standard units of measurement for length, weight, temperature and etc.

2.2.3.b. Estimate length, weight, temperature and etc using standard units of measurement.

2.2.3.c. Identify and make estimates using non-standard units of measurement.

MTH.2.2.4 Use common instruments to measure and compare objects. (For example, students will be able to use a ruler, watch or balance with standard and non-standard units of measurement).

2.2.4.a. Use a ruler to measure and compare objects.

2.2.4.b. Use a watch/clock to tell and compare times of day and night.

2.2.4.c. Use a scale to weigh and compare objects.

2.2.4.d. Measure length with non-standard units of measurement.

2.2.4.e. Tell time using non-standard units of measurement.

**STANDARD 3. PATTERNS AND ALGEBRA**

MTH.2.3.1 Describe and create patterns and find the next term using numbers, objects, and other materials.

2.3.1.a. Find the next term in a series of numbers.

2.3.1.b. Find the number that is one less or one more than a given number.

2.3.1.c. Find the item in a series of patterns.

2.3.1.d. Create patterns.

MTH.2.3.2 Identify and use the inverse relationships between operations to solve problems.

2.3.2.a. Apply the horizontal form for addition and subtraction to find the unknown.

2.3.2.b. Apply the horizontal form of multiplication to find the unknown.

By the end of GRADE 3, students will:

**STANDARD 1. NUMBERS, OPERATIONS, AND COMPUTATIONS**

MTH.3.1.1 Use English and local systems to count, read, write and compare whole numbers up to 1,000.

3.1.1.a. Identify numbers from 1 to 1,000 in English.

3.1.1.b. Identify numbers from 1 to 1,000 in the local language.

3.1.1.c. Write numbers from 1 to 1,000 in English.

3.1.1.d. Write numbers from 1 to 1,000 in the local language.

3.1.1.e. Compare whole numbers from 1 to 1,000.

3.1.1.f. Identify numbers as used in local counting systems from 1 to 1,000.

MTH.3.1.2 Understanding base-ten by identifying the place value of numbers up to 1,000.

3.1.2.a. Identify the place value of any digit in any given number from 10 to 1,000.

MTH.3.1.3 Demonstrate an understanding of the basic operations (+, -, x, ÷), and how they relate to each other.

3.1.3.a. Find sums of 2 digit by 2 digit addition problems with sums up to 198.

3.1.3.b. Solve subtraction facts with a minuend of 198 or less.

3.1.3.c. Find products where the multiplicands are nine or less.

3.1.3.d. Solve division facts whereas the dividends are 81 or less.

3.1.3.e. Write the inverse operation of a given problem.

MTH.3.1.4 Represent whole numbers and operations in a variety of ways using physical models, diagrams, and number expressions.

3.1.4.a. Write odd and even numbers less than 1,000.

3.1.4.b. Read and write Roman numerals to 30.

3.1.4.c. Draw pictures that represent multiple operations (addition and subtraction or multiplication and division).

MTH.3.1.5 Use the basic operations to add and subtract 2- and 3-digit numbers.

3.1.5.a. Add and subtract hundreds, tens and ones with and without renaming.

3.1.5.b. Add and subtract with amounts of money more than a dollar.

MTH.3.1.6 Use the basic operations to multiply and divide 1, 2 and 3 digit numbers by a single digit number.

3.1.6.a. Find the product of a 3 digit number by a single-digit multiplier with renaming.

3.1.6.b. Divide a 3 digit dividend with a single-digit divisor with no remainders.

MTH.3.1.7 Use a variety of strategies including the understanding of number and operations to solve problems and explain the reasoning used to reach the solution.

3.1.7.a. Identify key words in a given problem to decide on the operation needed to solve the problem.

3.1.7.b. Demonstrate various ways to solve a problem.

**STANDARD 2. GEOMETRY, MEASUREMENT AND TRANSFORMATION**

MTH.3.2.1 Identify and classify two to three dimensional shapes.

3.2.1.a. Identify the basic geometric figures (different types of polygons, cylinders, pyramids, cones and etc.)

3.2.1.b. Draw 2- and 3-dimensional geometric shapes.

3.2.1.c. Classify geometric figures based on shapes and number of sides.

MTH.3.2.2 Describe similarities and differences between common shapes.

3.2.2.a. Tell the difference between a closed curve and a simple closed curve.

3.2.2.b. Tell whether two geometric shapes are congruent.

MTH.3.2.3

Demonstrate understanding of standards and non-standards units of measurement.

3.2.3.a. Identify units of measurement used in standard form.

3.2.3.b. Identify units of measurement used in non-standard form.

MTH.3.2.4 Use common instruments to measure and compare objects. (For example, students will be able to use a ruler, watch or balance with standard and non-standard units of measurement.)

3.2.4.a. Use a ruler to measure and compare objects

3.2.4.b. Use a watch/clock to tell and compare times of day and night

3.2.4.c. Use a scale to weigh and compare objects

3.2.4.d. Measure length with non-standard units of measurement

3.2.4.e. Tell time using non-standard units of measurement

**STANDARD 3. PATTERNS AND ALGEBRA**

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

3.3.1.a. Find sums of 2 digit by 2 digit addition problems with sums up to 198.

3.3.1.b. Solve subtraction facts with a minuend of 198 or less.

3.3.1.c. Find the next term in a sequence of numbers.

3.3.1.d. Find the items in a series of patterns.

3.3.1.e. Create patterns.

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

3.3.2.a. Solve for the unknown when given simple algebraic addition problems (single-digit).

3.3.2.b. Solve for the unknown when given simple algebraic subtraction problems (single digit).

3.3.2.c. Explore order in addition and multiplication.

By the end of GRADE 4, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.4.1.1 Understand base-ten by identifying the place value of whole numbers up to 1,000 and decimal numbers down to 100th.

4.1.1.a. Identify numbers up to 1,000 in both local and English languages.

4.1.1.b. Write numbers up to 1,000 in both local and English languages.

4.1.1.c. Identify and name the place value of number down to 100th.

MTH.4.1.2 Demonstrate the ability to read, write and compare simple fractions and decimals in English and the local counting system.

4.1.2.a. Read and write fractions and decimals.

4.1.2.b. Compare fractions and decimals using a number line.

4.1.2.c. Apply cross multiplication to compare fractions.

4.1.2.d. Read decimal numbers down to 100th in the local language.

4.1.2.e. Compare decimal numbers down to 100th.

MTH.4.1.3 Represent whole numbers, fractions and decimals and operations involving them, in a variety of ways using physical models, diagrams and number expressions.

4.1.3.a. Draw pictures that represent fractions and decimal number expressions and operations.

4.1.3.b. Use a number line to show whole numbers, fractions and decimals.

4.1.3.c. Use physical models to show whole number, fractions and decimals.

MTH.4.1.4 Perform the basic operations to add, subtract, multiply and divide whole numbers and decimals and add and subtract fractions with like denominators.

4.1.4.a. Add, subtract, multiply and divide whole numbers and decimals.

4.1.4.b. Add and subtract like fractions.

MTH.4.1.5 Use a variety of methods and ways to round and estimate whole numbers, decimals and fractions.

4.1.5.a. Round whole numbers greater than or equal to 1,000 to tens, hundreds and thousands.

4.1.5.b. Given decimals down to the 100th, round to the nearest tenth.

4.1.5.c. Given decimals down to the 100th, round to the nearest whole number.

4.1.5.d. Given mixed fractions, round to the nearest whole number.

MTH.4.1.6 Use a variety of strategies including the understanding of decimals and fractions to solve problems and explain the reasoning used to reach the solution.

4.1.6.a. Identify key words in a given problem to decide on the operation needed to solve problems involving fractions.

4.1.6.b. Identify key words in a given problem to decide on the operation needed to solve problems involving decimal numbers.

4.1.6.c. Explain the reasoning used to reach the solution.

MTH.4.1.7 Use a variety of strategies including the understanding of number and perations to solve problems and explain the reasoning used to reach the solution.

4.1.7.a. Demonstrate at least 2 possible ways to solve a problem.

4.1.7.b. Use problem solving processes to solve word problems and explain the reasoning used to reach the solution.

**STANDARD 2. GEOMETRY, MEASUREMENT AND TRANSFORMATION**

MTH.4.2.1 Identify and classify two and three dimensional shapes.

4.2.1.a. Identify the basic geometric figures (different types of polygons, cylinders,pyramids, cones and etc.).

4.2.1.b. Classify geometric figures based on shapes and number of sides.

MTH.4.2.2 Describe similarities and differences between one, two and three dimensional shapes.

4.2.2.a. Identify point, line, line segment and angles (right, acute and obtuse).

4.2.2.b. Compare and contrast one, two and three dimensional geometric figures.

4.2.2.c. Recognize and identify congruence and symmetry.

MTH.4.2.3 Demonstrate understanding of common units in English and metric systems by choosing appropriate units to measure common objects and quantities.

4.2.3.a. Convert units in one system into their equivalent measures in another system.

MTH.4.2.4 Use standard and non-standard units to determine length, volume and weight and describe the characteristics of each type of measurement.

4.2.4.a. Identify units of measurement used in standard form.

4.2.4.b. Identify units of measurement used in non-standard form.

MTH.4.2.5 Use the understanding of geometry, measurement and transformation to solve problems and explain the reasoning used to reach the solution.

4.2.5.a. Count to find perimeter of a triangle, square or rectangle.

4.2.5.b. Find area and volume by counting unit squares or cubes.

4.2.5.c. Make two shapes fit by sliding and flipping; by rotation and flipping.

**STANDARD 3. PATTERNS AND ALGEBRA**

MTH.4.3.1 Use patterns and functions to represent and solve real world situations and explain the reasoning used to reach the solution.

4.3.1.a. Add or subtract any two even numbers and tell whether the sum or difference is even or odd.

4.3.1.b. Add or subtract two odd numbers and tell whether the sum or difference is even or odd.

4.3.1.c. Find the next two terms in a series.

4.3.1.d. Find the number that is 2 more or 2 less than a given number, using simple word problems or patterns.

**STANDARD 4. STATISTICS AND PROBABILITY**

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

4.4.1.a. Collect and organize data using charts, tables, pictographs.

4.4.1.b. Describe in own words what the data show or mean.

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

4.4.2.a. Read tables, charts or pictographs to make meaning of data.

4.4.2.b. Make inferences or draw conclusions based on the data.

By the end of GRADE 5, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.5.1.1 Demonstrate the ability to read, write, and compare more complex decimals and fractions.

5.1.1.a. Read, write and compare complex decimals.

5.1.1.b. Read, write and compare complex fractions including the symbol for repeating decimals

MTH.5.1.2 Represent fractions as proper and improper fractions, mixed numbers and decimals.

5.1.2.a. Draw pictures for proper, improper and fractions.

5.1.2.b. Use a number line to show proper and improper fractions and mixed numerals.

MTH.5.1.3 Continue to develop fluency to do the basic operations to add, subtract, multiply and divide whole numbers, decimals, and simple fractions.

5.1.3.a. Add, subtract, multiply and divide whole numbers and decimals independently or with minimal assistance.

5.1.3.b. Add and subtract simple fractions independently or with minimal assistance. (Use the concept of tessellation to add unlike fractions.)

5.1.3.c. Use drawings to multiply fractions.

5.1.3.d. Multiply and divide fractions with minimal assistance.

MTH.5.1.4 Use rounding and estimation to solve problems.

5.1.4.a. Solves problems using rounding and estimation.

MTH.5.1.5 Choose and use appropriate computational procedures and tools (e.g. pencil and paper, mental computation, or calculators) to solve problems.

5.1.5.a. Use paper and pencil, mental computation and calculators to solve math problems and tell which one is better or faster.

MTH.5.1.6 Use variety of strategies including make a model, work backward, draw a diagram, guess and check, and etc., to solve problems and justify answers.

5.1.6.a. Solve problems using models or work backward and justify the answers.

5.1.6.b. Solve problems using diagrams, guess and check and etc. and justify the answers.

**STANDARD 2. GEOMETRY, MEASUREMENT AND TRANSFORMATION**

MTH.5.2.1 Recognize and classify triangles and quadrilaterals based on their properties (e.g. angles and sides).

5.2.1.a. Use angles and sides to identify and classify triangles as right, isosceles, scalene, equilateral and etc.

5.2.1.b. Use the number of sides of quadrilaterals to identify and classify the quadrilaterals.

MTH.5.2.2 Use the common units of the English and metric systems and carry out simple unit conversion within the systems (e.g. centimeters to meters, hours to minutes).

5.2.2.a. Convert units in the English system into their equivalent measures.

MTH.5.2.3 Develop and use formulas to determine perimeter and area.

5.2.3.a. Derive and use methods to find perimeter and area of triangles.

5.2.3.b. Use formulas for finding perimeter and area of triangles, squares, rectangles, rhombus and parallelograms.

MTH.5.2.4 Measure length, area, volume, and weight accurately using appropriate tools.

5.2.4.a. Use rulers, yard sticks, centimeter sticks, meter sticks and tape rules to measure lengths. (Make sure students can measure to fractions of an inch.)

5.2.4.b. Use scales to find the weights of objects.

5.2.4.c. Use graduated cylinders, flasks, syringes and etc. to find volume of liquids.

MTH.5.2.5 Use a variety of strategies (e.g. converting units and comparing, and comparing the capacities of a number of objects) by pouring including the understanding of measurement to solve problems and explain the reasoning used to reach the solution.

5.2.5.a. Demonstrate abilities to solve problems by making comparisons of converted units.

5.2.5.b. Be able to explain reasoning behind how solution was reached.

**STANDARD 3. PATTERNS AND ALGEBRA**

MTH.5.3.1 Represent and record patterns using tools such as charts, tables and graphs.

5.3.1.a. Use charts, tables and graphs to represent and record patterns.

5.3.1.b. Determine if charts, tables and graphs represent any pattern and specify what the pattern is.

MTH.5.3.2 Use words and simple algebraic expressions to describe quantities and situations.

5.3.2.a. Describe quantities and situations in words.

5.3.2.b. Describe quantities and situations using simple algebraic expressions.

MTH.5.3.3 Represent and investigate how a change in one variable relates to the change in the second variable (e.g. the height of a plant over time).

5.3.3.a. Design a simple investigation of one quantity to another.

MTH.5.3.4 Investigate and describe situations involving inverse relationships (e.g. the more friends, the fewer the cookies for each person; the larger the denominator in a unit fraction, the smaller the quantity).

5.3.4.a. Use addition and subtraction; multiplication and division to develop understanding of inverse relationship

5.3.4.b. Investigate real world situation and determine what the inverse relationships are

5.3.4.c. Investigate or survey to answer questions or problems.

**STANDARD 4. STATISTICS AND PROBABILITY**

MTH.5.4.1 Collect data using observations, measurements, surveys, or experiments.

5.4.1.a. Show understanding of data collection by collecting data from observations, measurements, surveys and experiments.

MTH.5.4.2 Organize data using tables and charts, and construct graphs (e.g. pictograph, bar graph, and line graph).

5.4.2.a. Use tables and charts to organize data.

5.4.2.b. Construct pictographs, bar graphs and line graphs based on data from tables and charts.

MTH.5.4.3 Discuss events as likely or unlikely and give a description of the degree of likelihood in informal terms (e.g. unlikely, very unlikely, certain, impossible).

5.4.3.a. Describe orally or in writing the degree of likelihood for something to happen using unlikely, very unlikely, likely, highly likely, certain, impossible and etc.

MTH.5.4.4 Estimate and describe probabilities in simple experiments involving coins, spinners, dice, or objects in a bag.

5.4.4.a. Use simple experiments to estimate and describe probabilities of events.(Examples: Throwing up a coin a certain number of times and record the number of times the head or tail lands; pulling out objects from a bag a certain number of times and record how many times each object is drawn.)

By the end of GRADE 6, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.6.1.1 Compare, order, round, and group rational numbers.

6.1.1.a. Define and compare rational numbers using the number line or cross multiplication.

6.1.1.b. Order rational numbers.

6.1.1.c. Round rational numbers to the nearest whole number.

6.1.1.d. Group rational numbers according to some specified criteria (rational numbers that are equal, having the same numerator or denominator, positive or negative, greater than or less than zero).

MTH.6.1.2 Demonstrate fluency in the basic operations to add, subtract, multiply and divide whole numbers, fractions, and decimals.

6.1.2.a. Add, subtract, multiply and divide whole numbers, fractions and decimals independently or with minimal assistance.

MTH.6.1.3 Identify the characteristics of prime and composite numbers and decompose composite numbers into factor pairs and prime factors using exponents.

6.1.3.a. Identify prime and composite numbers.

6.1.3.b. Find prime factorization of composite numbers using exponents.

MAT.6.1.4 Use models and pictures to represent ratio and proportions and solve problems.

6.1.4.a. Write ratios based on models and pictures.

6.1.4.b. Solve problems involving ratios and proportions.

**STANDARD 2. GEOMETRY, MEASUREMENT AND TRANSFORMATION**

MTH.6.2.1 Adds and subtracts customary units of length, mass, liquid, and time measures.

6.2.1.a. Add and subtract inches, feet, yards, miles and etc.

6.2.1.b. Add and subtract ounces, pounds, tons and etc.

6.2.1.c. Add and subtract pints, quarts, gallons and etc.

6.2.1.d. Add and subtract seconds, minutes, hours and etc.

MTH.6.2.2 Perform slides, flips, turns, and rotations and indicate the motion, position and direction applied.

6.2.2.a. To be able to identify and demonstrate geometrical transformations such as translation (slides), reflection (flips), turns and rotation.

6.2.2.b. Indicate motion, position and direction of each transformation performed.

MTH.6.2.3 Use formulas to compute perimeter and area of polygons.

6.2.3.a. Be able to use formulas to find perimeter and area of polygons. (Eg.triangles, rectangles, squares, rhombus, parallelograms, trapezoids and etc.)

MTH.6.2.4 Describe, compare, and classify geometrical figures using mathematical terminology (number of edge and faces, number and size of angles, and number of vertices).

6.2.4.a. Describe and compare simple closed curves using such prefixes like tri, quad, penta, hexa, hepta, octa, nano,deca, etc.

6.2.4.b. Demonstrate understanding of different angles (right angle, acute, obtuse).

6.2.4.c. Be able to classify and compare polygons based on number of sides, angles and vertices.

**STANDARD 3. PATTERNS AND ALGEBRA**

MTH.6.3.1 Represent patterns in a variety of ways (numeric, algebraic, pictorial, oral, and graphic).

6.3.1.a. Represent patterns based on data provided, orally, pictorially, numerically, algebraically and graphically.

MTH.6.3.2 Model and solve real world problems using various representations such as graphs and tables.

6.3.2.a. Construct models based on data from graphs and tables representing real world situations.

6.3.2.b. Solve problems based on the constructed models.

MTH.6.3.3 Locate whole numbers, fractions and decimals on a number line.

6.3.3.a. Plot whole numbers, fractions and decimals on a number line.

MTH.6.3.4 Use the guess and check method to solve simple algebraic expressions.

6.3.4.a. Define the term variable.

6.3.4.b. Estimate possible answers in an algebraic equation, then substitute it with the variables expression to verify the answer.

**STANDARD 4. STATISTICS AND PROBABILITY**

MTH.6.4.1 Analyze and interpret data, including range, median, mode, mean, and frequency and present information to an audience.

6.4.1.a. Analyze data and calculate mean, median, mode, range and frequency.

6.4.1.b. Orally present the analyses to classmates, schoolmates, school staff and others.

MTH.6.4.2 Make predictions that are based on experimental or theoretical probabilities and determine their reasonableness.

6.4.2.a. Use basic experimental and theoretical probabilities to determine reasonableness of predictions made.

MTH.6.4.3 Formulate and solve problems that involve collecting and analyzing data to reach conclusions and make generalizations.

6.4.3.a. Analyze data collected to prove or disprove conclusions made.

6.4.3.b. Generate further studies to collect more information then make generalizations based on the initial conclusions made.

By the end of GRADE 7, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.7.1.1 Understand and represent integers, adding and subtracting them in real world situations (e.g., change in temperature, elevation, debt).

7.1.1.a. Write integers on a number line.

7.1.1.b. Add and subtract integers on a number.

7.1.1.c. Add and subtract integers using rules for addition and subtraction in real world situations.

MTH.7.1.2 Locate whole numbers, fractions, decimals, and integers on a number line.

7.1.2.a. Plot whole numbers, fractions, decimals and integers on a number line.

7.1.2.b. Write fractions as decimals and vice versa.

7.1.2.c. Write fractions and whole numbers as percentages or vice versa.

MTH.7.1.3 Represent number in a variety of ways including expanded form and scientific notation.

7.1.3.a. Write the expanded forms of whole numbers.

7.1.3.b. Apply scientific notation to expand whole numbers.

MTH.7.1.4 Use the properties of numbers (zero, identity, commutative, associative, and distributive) to solve problems.

7.1.4.a. Solve problems using the zero and identity properties.

7.1.4.b. Apply the commutative, associative and distributive properties of whole numbers to solve problems.

MTH.7.1.5 Use the order of operations to evaluate expressions.

7.1.5.a. Evaluate expressions using the order of operations.

MTH.7.1.6 Round numbers to estimate solutions and check the reasonableness of results.

7.1.6.a. Round numbers to estimate solutions and check the reasonableness of results.

MTH.7.1.7 Round numbers to estimate solutions and check the reasonableness of results.

7.1.7.a. Estimate the solutions to problems using rounding of numbers.

**STANDARD 2. GEOMETRY, MEASUREMENT, AND TRANSFORMATION**

MTH.7.2.1 Identify and draw points, lines, line segments, angles and rays.

7.2.1.a. Define points, lines, line segments angles and rays.

7.2.1.b. Draw points, lines, line segments, angles and rays.

MTH.7.2.2 Use pi (π), represented as both a decimal (3.14) and fraction (22/7), to find circumference and area of circles.

7.2.2.a. Identify the different parts of a circle, i.e. diameter, radius and circumference.

7.2.2.b. Apply the decimal form and the fraction form of pi to find circumference and area of a circle.

MTH.7.2.3 Use appropriate English and metric units to develop reasonable estimates of measures.

7.2.3.a. Estimate length, mass, and volume of any given object using appropriate units of measurement

MTH.7.2.4 Describe symmetry, reflections, and translations with appropriate notation.

7.2.4.a. Demonstrate understanding of symmetry, reflection, and translation and describe using appropriate notation

**STANDARD 3. PATTERNS AND ALGEBRA**

MTH.7.3.1 Describe relationships and functions using word and symbols.

7.3.1.a. Identify the independent and dependent variable of a function using domain and range.

7.3.1.b. Use words and symbols to describe relationships and functions.

MTH.7.3.2 Write and solve one-step equations.

7.3.2.a. Write one-step equations.

7.3.2.b. Solve one-step equations.

MTH.7.3.3 Locate points on the coordinate plane.

7.3.3.a. Write the ordered pairs for points on a coordinate plane.

7.3.3.b. Plot given ordered pairs on a coordinate plane.

**STANDARD 4. STATISTICS AND PROBABILITY**

MTH.7.4.1 Propose and support conclusions by summarizing data (e.g., in a survey of how many books students read each month, over half the books students read in a year are read in April and May.).

7.4.1.a. Summarize surveys and other collected data to argue for or against a conclusion.

MTH.7.4.2 Formulate questions or hypotheses based on initial data collection, and describe further studies to explore them.

7.4.2.a. Interpret initial data collected to formulate questions or hypotheses.

7.4.2.b. Describe other means or studies to further explore questions or hypotheses.

By the end of GRADE 8, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.8.1.1 Represent, compare, order and use numbers in a variety of forms (integer, fraction, decimal, percentages, and exponents) in mathematical problem-solving situations.

8.1.1.a. Write numbers as integers, fractions, decimals, percentages and exponents.

8.1.1.b. Compare integers, fractions, decimals, percentages and exponents.

8.1.1.c. Order integers, fractions, decimals, percentages and exponents.

8.1.1.d. Find answers to word problems dealing with integers, fractions, decimals, percentages, and exponents.

MTH.8.1.2 Demonstrate fluency in computing with rational numbers (fractions, decimals, percentages and integers).

8.1.2.a. Solve problems on fractions, decimals, percentages and integers independently or with minimal help.

MTH.8.1.3 Square whole, rational, and integers and find square roots of perfect squares (e.g. 1, 4, 9, 16, etc.).

8.1.3.a. Find the results for squaring whole numbers, rational numbers and integers.

8.1.3.b. Find square roots of perfect squares.

MTH.8.1.4 Use ratio, proportion, and percentages in problem solving.

8.1.4.a. Solve problems involving ratio, proportion and percentages.

**STANDARD 2. GEOMETRY, MEASUREMENT AND TRANSFORMATION**

MTH.8.2.1 Use a compass, protractor, and straight edge to draw two-dimensional figures and do constructions (e.g. bisecting an angle or line segment, creating a right angle, drawing a circle).

8.2.1.a. Be able to draw two-dimensional figures using protractors and straight edge tools

8.2.1.b. Be able to draw circles using a compass.

8.2.1.c. Bisect a line segment to form right angle.

8.2.1.d. Bisect an angle.

MTH.8.2.2 Identify similar and congruent figures including lines of symmetry and diagonals.

8.2.2.a. Demonstrate understanding of the meaning of similar and congruent figures.

8.2.2.b. Demonstrate ability to identify similar and congruent figures.

8.2.2.c. Demonstrate ability to identify lines of symmetry and diagonals.

MTH.8.2.3 Use formulas to find areas of quadrilaterals, triangles, and circles, and the surface area and volume of cylinders and prisms including appropriate units of measure.

8.2.3.a. Be able to find area and surface are of geometric figures using appropriate formulas.

8.2.3.b. Be able to find volume of cylinders and prisms using appropriate formulas.

MTH.8.2.4 Use the Pythagorean theorem to find lengths of sides of right triangles.

8.2.4.a. Identify the legs and the hypotenuse of a right triangle.

8.2.4.b. Solve for lengths of sides of right triangles using the Pythagorean Theorem.

MTH.8.2.5 Solve simple problems involving rates and derived measures (e.g. miles per hour, cost per yard).

8.2.5.a. Apply rates and derived measures to find solutions to simple problems.

MTH.8.2.6 Use proportional reasoning and indirect measurements to draw inferences, such as measuring the thickness of a book to estimate the thickness of one page.

8.2.6.a. Draw inferences using proportional reasoning and indirect measurements.

**STANDARD 3. STATISTICS AND PROBABILITY**

MTH.8.3.1 Write and solve two-step linear equations and one-step inequalities.

8.3.1.a. Write a two -step linear equation.

8.3.1.b. Write a one-step for inequalities.

8.3.1.c. Solve two-step equations and one-step inequalities.

MTH.8.3.2 Graph linear functions in two variables using a table of ordered pairs.

8.3.2.a. Define a linear function.

8.3.2.b. Use a table of ordered pairs to graph linear functions.

8.3.2.c. Identify linear function.

MTH.8.3.3 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.

8.3.3.a. Represent given situations with symbolic algebra and use guess and check or diagrams to find answers to those given situations.

MTH.8.3.4 Model and solve real-world problems using various representations, such as graphs and tables, to understand the purpose and utility of each representation.

8.3.4.a. Use graphs and tables to represent data on real world issues.

**STANDARDS 4. STATISTICS AND PROBABILITY**

MTH.8.4.1 Find, describe, and interpret mean, median, mode, and range and determine which measure is best to use in a particular situation.

8.4.1.a. Find and describe mean, median, mode, and range.

8.4.1.b. Choose the best measure of central tendency or dispersion to use in a given situation.

MTH.8.4.2 Read and interpret tables, charts, and graphs, and make inferences based on the data.

8.4.2.a. Read and interpret tables, charts, and graphs.

8.4.2.b. Draw inferences from tables, charts, and graphs.

MTH.8.4.3 Use sampling and other data collection tools to gather and analyze data, and make conclusions and predictions.

8.4.3.a. Analyze data from sampling and other data collection tools.

8.4.3.b. Draw conclusions and make inferences based on analysis of data collection and sampling. (Draw conclusions and make inferences based on data analysis)

MTH.8.4.4 Compute simple probabilities using appropriate methods such as lists, tree diagrams or through experimental or simulation activities.

8.4.4.a. Find simple probabilities using lists and tree diagrams.

8.4.4.b. Find simple probabilities using experimental (e.g., rolling dice, picking a number out of a hat) or simulation (e.g., likely or seemingly) activities.

By the end of HIGH SCHOOL, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.HS.1.1 Demonstrate the inverse relationship between square numbers and square roots.

HS.1.1.a. Define inverse relationship and illustrate examples of such inverse relationship.

HS.1.1.b. Give examples of squared numbers and square roots.

HS.1.1.c. Express how the square root of a number can be represented as a positive rational exponent.

MTH.HS.1.2 Compare and order rational numbers and square roots using a number line.

HS.1.2.a. Define rational numbers.

HS.1.2.b. Compare rational numbers and square roots and illustrate their positions on a number line.

MTH.HS.1.3 Solve problems with squares and square roots, limited to square roots of square numbers.

HS.1.3.a. Define perfect squares and find square roots of perfect squares. Apply the algorithm for finding square roots.

MTH.HS.1.4 Represent numbers in a variety of forms including factors, multiples, exponents, primes, composites, fractions, decimals, and percentages and from one form to another.

HS.1.4.a. Define factors, multiples, exponents, prime, composite, fractions, decimals.

HS.1.4.b. Illustrate how each (factors, multiples, exponents, prime, composite, fractions, decimals) can be expressed in different forms.

MTH.HS.1.5 Apply an understanding of addition, subtraction, multiplication, division and the order of operations when calculating with rational numbers.

HS.1.5.a. Add, subtract, multiply and divide rational numbers.

HS.1.5.b. Apply the order of operations with rational numbers.

MTH.HS.1.6 Use ratios, proportions and percent to represent the relationship between two quantities and solve problems.

HS.1.6.a. Compare and contrast ratio, proportion and percent.

HS.1.6.b. Illustrate relationship between ratio and percent.

HS.1.6.c. Apply proportions to solve ration and percent problems.

MTH.HS.1.7 Add, subtract, multiply and divide numbers with positive and negative exponents.

HS.1.7.a. Apply each form below with real numbers to simplify.

an x a m ≡ an + m

an ÷ a m ≡ an – m

(an)m ≡anm

a¯n ≡1/a

an/m ≡ (√(m&a))n ≡√(m&a)n

MTH.HS.1.8 Estimate a reasonable solution to a problem.

HS.1.8.a. Find the best estimate to a given problem.

MTH.HS.1.9 Use rounding and estimation to solve real world situations and recognize the limitations.

HS.1.9.a. Solve real world problems using rounding and estimation.

HS.1.9.b. Recognize the limitations of solutions to real world problems when applying rounding and estimation.

**STANDARD 2. GEOMETRY, MEASUREMENT AND TRANSFORMATION**

MTH.HS.2.1 Apply an understanding of the English and metric systems of measurement to solve problems.

HS.2.1.a. Show relationship of the English System of measurement by converting one unit of measure to another (i.e. inches to feet, yard, miles; ounces to pounds, tons; pints to quarts, gallons and vice versa).

HS.2.1.b. Apply oral and written discourse on the history of the development of the metric system.

HS.2.1.c. Demonstrate understanding of the relationship between the metric units by converting one unit to another.

HS.2.1.d. Solve problems by converting units used in length, mass and capacity into each Measurement Systems

MTH.HS.2.2 Use formulas, including appropriate units of measure, to determine the surface area and volume of selected prisms, cylinders, and pyramids.

HS.2.2.a. Define surface area, volume, prism, cylinder and pyramid.

HS.2.2.b. Identify surface areas and volume of prism, cylinder and pyramid.

HS.2.2.c. Apply correct formulas to find surface areas and volume of prisms, cylinders, pyramids using different units of measure.

MTH.HS.2.3 Apply the Pythagorean Theorem to solve problems involving right triangles.

HS.2.3.a. Define the term theorem.

HS.2.3.b. Apply oral and written discourse in regards to the origin of the Pythagorean Theorem.

HS.2.3.c. Describe the characteristics of a right triangle.

HS.2.3.d. Compare the legs of a 45º, 45º, 90º and a 30º, 60º, 90º triangles.

HS.2.3.e. Apply an angle bisector to any equilateral triangle to find the length of the shorter leg.

HS.2.3.f. Apply the Pythagorean Theorem to find the missing length of any side of a right triangle.

MTH.HS.2.4 Perform transformations including reflection, rotation, and translation and describe the size, position and orientation of the resulting shapes.

HS.2.4.a. Define reflection, rotation and translation.

HS.2.4.b. Show the reflection, rotation, and translation of any geometric figure on a coordinate and describe its size, position and orientation.

**STANDARD 3. PATTERNS AND ALGEBRA**

MTH.HS.3.1 Represent a variety of patterns, including recursive patterns, with tables, graphs, words and symbolic rules.

HS.3.1.a. Use words and symbolic rule to represents patterns.

HS.3.1.b. Use tables and graphs to represent patterns and determine whether patterns are recursive.

MTH.HS.3.2 Represent mathematical situations as algebraic expressions and equations and describe algebraic expressions using words.

HS.3.2.a. Compare and contrast algebraic expressions and equations.

HS.3.2.b. Translate algebraic expressions and equations into words.

HS.3.2.c. Translate mathematical sentences into algebraic expressions and equations.

MTH.HS.3.3 Solve single-variable equations and inequalities using rational numbers.

HS.3.3.a. Differentiate rational numbers from other real numbers.

HS.3.3.b. Find solutions to equations and inequalities having a single variable.

MTH.HS.3.4 Use tables and graphs to represent linear relationships involving equalities and inequalities with two variables and solve problems.

HS.3.4.a. Represent linear relationship of two variable equations and inequalities.

HS.3.4.b. Solve two variable equations and inequalities and graph them.

MTH.HS.3.5 Justify the steps used in simplifying expressions and solving equations and inequalities.

HS.3.5.a. Solve equations and inequalities and write the reason(s) for each step used.

MTH.HS.3.6 Identify functions as linear or nonlinear and contrast their properties from tables, graphs and equations.

HS.3.6.a. Define linear and nonlinear function.

HS.3.6.b. Identify linear and nonlinear functions using graphs, tables and equations.

HS.3.6.c. Contrast the properties of linear and nonlinear functions from tables, graphs and equations.

MTH.HS.3.7 Represent data involving linear relationships from tables as graphs and equations.

HS.3.7.a. Draw graphs using data from data tables.

HS.3.7.b. Draw the graph of a linear equation.

MTH.HS.3.8 Solve linear equations and inequalities with two variables using algebraic methods, manipulatives or models.

HS.3.8.a. Find solutions of two variable linear equations and inequalities using algebraic methods.

HS.3.8.b. Find a solution to two variable equations and inequalities using manipulatives.

HS.3.8.c. Find solutions to two variable equations and inequalities using models.

MTH.HS.3.9 Determine the slope of a line when given the graph, two points on the line, or the equation of a line.

HS.3.9.a. Write a given equation into slope-intercept form.

HS.3.9.b. Find the slope of a linear equation.

HS.3.9.c. Graph a linear equation when only two points are given.

MTH.HS.3.10 Select and use a variety of strategies including concrete modeling, pictorial and representation, and algebraic manipulation to add, subtract, multiply, divide and factor first and second degree binomials, and trinomials in one variable.

HS.3.10.a. Use modeling and pictorial representation to add, subtract, multiply, and divide and factor first and second degree binomial and trinomial in one variable.

HS.3.10.b. Use algebraic manipulation to add, subtract, multiply, divide and factor binomials and trinomials in one variable.

**STANDARD 4. STATISTICS AND PROBABILITY**

MTH.HS.4.1 Analyze and interpret data using mean, median, mode, range and frequency.

HS.4.1.a. Define mean, median, mode, range and frequency.

HS.4.1.b. Find mean, median, mode, range and the frequency of a set of data.

HS.4.1.c. Explain the data using mean, median, mode, range and the frequency.

MTH.HS.4.2 Design a study, collect data, and select the appropriate representation to make conclusions and generalizations.

HS.4.2.a. Make decisions on how to collect and represent data.

HS.4.2.b. Draw conclusions and make generalizations based on the data.

MTH.HS.4.3 Judge the validity of reported data, conclusions and generalizations.

HS.4.3.a. Read and analyze data for validity and draw conclusions and make generalizations based on the validity of the data.

MTH.HS.4.4 Calculate probabilities for simple events under different relationships, including independent, dependent, with replacement and without replacement.

HS.4.4.a. Calculate probabilities for simple events.

HS.4.4.b. Calculate simple events on independent and dependent events with and without replacement.

By the end of GEOMETRY, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.GEO.1 Use right triangle trigonometric ratios to determine the unknown length of a side or the measure of an angle.

GEO.1.a. Identify sine, cosine, tangent, cotangent, secant and cosecant of a right triangle. (sin = a/h, cos = o/h, tan = a/o, cot = o/a, sec = h/o, cosec = h/a)

GEO.1.b. Generalize relationships between the legs; or the legs and the hypotenuse of a 60°-30°-90° degree triangle and a 45°-45°- 90° degree triangle.

GEO.1.c. Use sine, cosine, tangent, cotangent, secant and cosecant to find the length of an unknown side.

MTH.GEO.2 Solve problems using the formulas for perimeter, circumference, area and volume of two or three-dimensional figures and solids and determine the effect of dimension changes to perimeter, area and volume.

GEO.2.a. Define and illustrate two- or three -dimensional figures and solids.

GEO.2.b. Apply the correct formula for finding perimeter, circumference, area, volume of two or three dimensional figures and solids.

GEO.2.c. Determine what will happen when the dimensions are increased or decreased in terms of perimeter, circumference, area and volume.

MTH.GEO.3 Use reasoning to create and defend geometric conjectures.

GEO.3.a. Define geometric conjectures.

GEO.3.b. Apply geometric conjectures to prove or disprove given conditions.

MTH.GEO.4 Use the concept of corresponding parts to prove that triangles and other polygons are congruent.

GEO.4.a. Define congruency.

GEO.4.b. Illustrate and identify corresponding parts of geometric figures or polygons. (vertical angles, alternate interior of exterior angles, corresponding angles and etc.)

GEO.4.c. Apply the congruency symbol and prove congruency of corresponding parts.

MTH.GEO.5 Explain the properties and characteristics of angle bisectors, perpendicular bisectors and parallel lines.

GEO.5.a. Define and illustrate angle bisector, perpendicular bisector and parallel lines.

GEO.5.b. Find the slopes of two linear equations and determine whether the lines are perpendicular or parallel.GEO.5.c. Apply the slope intercept form of a linear equation to find the equation of a line that is perpendicular or parallel to a given linear equation.

MTH.GEO.6 Use relationship between pairs of angles (complementary, supplementary, vertical, exterior and interior) to determine unknown angle measures or definitions of properties.

GEO.6.a. Define and illustrate complementary, supplementary, vertical, exterior and interior angles.

GEO.6.b. Determine the relationship between the angles

GEO.6.c. Apply the meaning of each to determine an unknown measure.

MTH.GEO.7 Apply the concept of special right triangle to real world situations.

GEO.7.a. Illustrate the application of the 45-45-90 or the 30-60-90 degree triangle for the foundation of a house.

GEO.7.b. Apply the Pythagorean Theorem to determine the length of rafters for a certain house with its dimensions given.

GEO.7.c. Apply the Pythagorean Theorem to determine distance between two points (locations).

MTH.GEO.8 Use the relationship among properties of circles to solve problems. (chords, secants, tangents, minor and major arcs, central angles, circumference, radius, diameter and inscribed polygons)

GEO.8.a. Define and illustrate chords, secants, tangents, arcs, circumference, radius, diameter and inscribed polygons.

GEO.8.b. Illustrate their relationships.°

GEO.8.c. Apply their relationships to solve problems.

MTH.GEO.9 Use coordinate geometry to produce formulas and prove theorems for the midpoint of a line segment, the distance formula and the forms of equations of lines and circles.

GEO.9.a. Draw a coordinate plane and identify the four quadrants.

GEO.9.b. Illustrate the values of x and y in quadrant 1, 2, 3 and 4.

GEO.9.c. Given two pairs of endpoints, draw a line to connect the two endpoints then illustrate how the midpoint formula and the distance formula are derived.

GEO.9.d. Give the equation for a circle and illustrate where the circle lies in the coordinate plane.

GEO.9.e. Apply the formulas for midpoint, distance and circle to solve problems.

MTH.GEO.10 Describe the concept of rigid motion on figures in the coordinate plane, including rotation, translation and reflection.

GEO.10.a. Define rigid motion, rotation, translation and reflection.

GEO.10.b. Illustrate rotation, translation and reflection of geometric figures on a coordinate plane.

MTH.GEO.11 Use concrete objects, pictorial representations, computer software or graphing calculator to solve geometric problems.

GEO.11.a. Construct a protractor and use it to find the height of a coconut tree, building or etc.

GEO.11.b. Use computer software or graphing calculator to solve system of linear equations.

By the end of ALGEBRA 2, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.ALG2.1 Apply arithmetic properties to operate on and simplify expressions that include radicals and other real numbers.

ALG2.1.a. Demonstrate ability to simplify expressions that include radical and real number (e.g. 6+√4, 2+√9).

MTH.ALG2.2 Use the complex number system, the notation for complex numbers, and the definition of “I” to solve problems in standard form.

ALG2.2.a. Understand the meaning of the complex number system, standard form for complex numbers, and “I”.

ALG2.2.b. Write real numbers as complex numbers.

ALG2.2.c. Solve problems involving complex numbers.

MTH.ALG2.3 Add, subtract, multiply, and divide complex numbers.

ALG2.3.a. Add and subtract complex numbers.

ALG2.3.b. Multiply and divide complex numbers.

MTH.ALG2.4 Use the inverse relationship between exponents and logarithms to solve exponential and logarithmic problems.

ALG2.4.a. Understand the inverse relationship between exponents and logarithms.

ALG2.4.b. Convert expressions from exponential form to logarithmic form and vice versa.

ALG2.4.c. Solve problems involving exponents and logarithms.

MTH.ALG2.5 Use advanced formulas or functions to solve problems.

ALG2.5.a. Identify advanced formulas or functions.

ALG2.5.b. Solve problems that require the use of advanced formulas or functions.

MTH.ALG2.6 Apply the properties of geometric sequences and series to solve problems.

ALG2.6.a. Understand geometric sequence and geometric series.

ALG2.6.b. Solve problems using the properties of geometric sequence and geometric series.

MTH.ALG2.7 Use exponential functions to solve exponential growth and decrease.

ALG2.7.a. Understand exponential function, exponential growth, and exponential decrease.

ALG2.7.b. Solve problems involving exponential growth and exponential decrease using exponential functions (e.g. compounded interest, global population growth).

MTH.ALG2.8 Use the properties of many types of functions including polynomial, absolute value, exponential, and logarithmic, to identify the function’s graph.

ALG2.8.a. Identify graphs of different types of functions (polynomial, absolute value, exponential, logarithmic).

ALG2.8.b. Sketch graphs of different types of functions.

MTH.ALG2.9 Use the appropriate terminology and notation to define functions and their properties, including domain, range, function composition, inverses, zeros, and asymptotes.

ALG2.9.a. Demonstrate ability to define and write functions using function notation.

ALG2.9.b. Determine and identify domain, range, inverse, and zeros of a function.

ALG2.9.c. Define and find asymptotes in a given function.

ALG2.9.d. Use correct notation to write compositions of functions.

MTH.ALG2.10 Describe the relationship among relations and functions.

ALG2.10.a. Understand the meanings of relation and function.

MTH.ALG2.11 Solve equations and inequalities involving absolute values.

ALG2.11.a. Solve equations containing absolute values.

ALG2.11.b. Solve inequalities containing absolute values.

MTH.ALG2.12 Solve systems of linear equations and inequalities in two or three variables using a variety of strategies, such as substitution, graphing or matrices.

ALG2.12.a. Use each strategy (substitution, elimination, graphing, matrices) to solve systems of linear equations and systems of inequalities in two or three variables.

MTH.ALG2.13 Solve equations containing radicals and exponents.

ALG2.13.a. Solve equations containing radicals.

ALG2.13.b. Solve equations involving exponents.

MTH.ALG2.14 Factor polynomials representing perfect squares, the difference of squares, perfect square trinomials, the sum and difference of cubes, and general trinomials.

ALG2.14.a. Use appropriate methods of factoring to factor perfect square polynomials.

ALG2.14.b. Use appropriate methods of factoring to factor sum and difference of cubes.

ALG2.14.c. Use appropriate methods of factoring to factor general polynomials.

MTH.ALG2.15 Apply quadratic equations to solve real-world situations and complex number problems.

ALG2.15.a. Demonstrate ability to solve real-world problem using quadratic equations (e.g., A pilot flies a distance of 600 miles. He could fly the same distance in 30 minutes less time by increasing his average speed by 40 miles/hour. Find his actual average speed.)

ALG2.15.c. Solve quadratic equations that involve complex roots.

MTH.ALG2.16 Use the binomial theorem to expand binomial expressions.

ALG2.16.a. Understand the binomial theorem.

ALG2.16.b. Apply the binomial theorem to expand expressions in the form: (a + b)ⁿ

ALG2.16.c. Apply the Pascal Triangle to expand binomial expressions.

MTH.ALG2.17 Use the fundamental counting principles for combinations and permutations to determine probability.

ALG2.17.a. Illustrate the fundamental counting principles for combination and permutations.

ALG2.17.b. Apply the counting principles to determine sample spaces and probability.

MTH.ALG2.18 Calculate probabilities of events under different relationships such as inclusion, disjoint, complementary, independent, and dependent, with and without replacement.

ALG2.18.a. Identify the different types of events.

ALG2.18.b. Apply the appropriate rules to find the probabilities (with and without replacement) of events.

MTH.ALG2.19 Use the right triangle relationships of trigonometric ratio, sine, cosine, and tangent to solve problems.

ALG2.19.a. Apply the definitions of the trigonometric ratios, sine, cosine, and tangent, to find solutions of right triangles (e.g., unknown lengths or angles).

By the end of TRIGONOMETRY, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.TRIG.1 Express complex numbers in standard and polar form and convert from one to the other.

TRIG.1.a. Understand complex number (every number in the form a + bi where a and b are real numbers and i = V-1. The expression, a + bi, is the standard form of a complex number and the two terms are called the real part and the imaginary part respectively.

TRIG.1.b. Explain modulus and amplitude of a complex number and demonstrate how to find them.

TRIG.1.c. Convert expressions in the form a + bi to r(cosθ + isinθ) and vice versa.

MTH.TRIG.2 Add, subtract, multiply, divide, and find powers of complex numbers.

TRIG.2.a. Distinguish between the real and imaginary parts of complex numbers.

TRIG.2.b. Understand the similarities between the multiplication of two binomials and that of complex numbers.

TRIG.2.c. Apply De Moivre’s Theorem to find powers of complex numbers.

MTH.TRIG.3 Use vector operations, including dot product and cross product, the Law of Sines and Law of Cosines to solve problems.

TRIG.3.a. Understand the meaning of: sum of vectors, dot product, and cross product.

TRIG.3.b. Show graphic representations of the sum and difference of two vectors.

TRIG.3.c. Apply the Law of Sines and the Law of Cosines to solve problems involving triangles.

MTH.TRIG.4 Calculate linear and angular velocity.

TRIG.4.a. Understand the meaning of linear and angular velocity.

TRIG.4.b. Explain the relationship between linear and angular velocity and use the correct unit of measures for each.

TRIG.4.c. Use the relationship between angular velocity and linear velocity to solve problems involving circular objects in motion.

MTH.TRIG.5 Find the sine, cosine, tangent, cotangent, secant, and cosecant of an angle is standard position.

TRIG.5.a. Understand the uses for each of the six trigonometric function and standard position of angles.

TRIG.5.b. Find values of each of the six trigonometric functions of a function.

MTH.TRIG.6 Use the relationship among the six trigonometric functions to translate among them.

TRIG.6.a. Express tangent, cotangent, secant, and cosecant in terms of sine and cosine.

TRIG.6.b. Simplify trigonometric expressions using trigonometric identities.

MTH.TRIG.7 Recognize the trigonometric functions of benchmark angles.

TRIG.7.a. Understand the meaning of benchmark angles.

TRIG.7.b. Solve problems involving benchmark angles.

MTH.TRIG.8 Translate between radians and degrees

TRIG.8.a. Define radian and degree

TRIG.8.b. Convert from radian measures to degree measures and vice versa

MTH.TRIG.9 Find the value of any trigonometric function and inverse trigonometric function and solve trigonometric equations.

TRIG.9.a. Understand the uses of trigonometric functions and inverse trigonometric functions and be able to distinguish between the two.

TRIG.9.b. Apply the trigonometric identities and algebraic techniques to solve trigonometric equations.

MTH.TRIG.10 Use the fundamental trigonometric identities, including the sum and difference formulas, double-angle formulas and half-angle formulas to solve problems.

TRIG.10.a. Understand the meaning of the fundamental trigonometric identities.

TRIG.10.b. Be able to explain the sum and difference formulas, double angle formulas, and half-angle formulas.

TRIG.10.c. Apply the identities and the formulas to solve problems involving trigonometric functions.

MTH.TRIG.11 Verify trigonometric identities.

TRIG.11.a. Understand the meaning of trigonometric identity.

TRIG.11.b. Convert trigonometric functions using the reciprocal, quotient, and the Pythagorean relationships to verify trigonometric identities.

MTH.TRIG.12 Solve trigonometric equations and inverse trigonometric equations that include all solutions or solutions with restricted domains.

TRIG.12.a. Understand the meaning and application of trigonometric equation and inverse trigonometric equation.

TRIG.12.b. Understand conditional equations and restricted domains.

TRIG.12.c. Apply algebraic techniques to solve trigonometric equations and write all solutions in general form.

MTH.TRIG.13 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 magnitude.

TRIG.13.a. Understand the meaning of: domain, range, period, phase shift, and magnitude as it relates to trigonometry.

TRIG.13.b. Analyze the graph of y = asin(bx + c) + d and identify its domain, range, period, phase shift, and magnitude.

MTH.TRIG.14 Identify real-world phenomena that can be represented by a trigonometric function in the form y = asin(bx + c) +d.

TRIG.14.a. Apply trigonometric function, y = asin(bx+c)+d, problems involving electricity and wave motion.

MTH.TRIG.15 Explain the relationship between trigonometric functions and their inverse.

TRIG.15.a. Understand the meaning of trigonometric function and inverse trigonometric function.

TRIG.15.b. Compare the domains of a trigonometric function and its inverse.

By the end of CALCULUS, students will:

**STANDARD 1. NUMBERS, OPERATIONS AND COMPUTATIONS**

MTH.CALC.1 Recognize limits from graphs and tables.

CALC.1.a. Understand limit and orally explain or in writing why limit is used in mathematics. (The limit is really the limit of a function and is the value that function approaches as the independent variable approaches a given value. For example, limit f(x) = 2x + 2 is read ‘the limit of f as x x1 approaches 1 is 2x + 2.) CALC.1.b. Use table of values of a function to determine the limit.

X | f(x) | X | f(x) |

0.5 | 3 | 1.5 | 5 |

0.8 | 3.6 | 1.2 | 4.4 |

0.9 | 3.8 | 1.1 | 4.2 |

0.95 | 3.9 | 1.05 | 4.1 |

0.99 | 3.98 | 1.01 | 4.02 |

0.999 | 3.998 | 1.001 | 4.002 |

0.9999 | 3.9998 | 1.0001 | 4.0002 |

0.99999 | 3.99998 | 1.00001 | 4.00002 |

CALC.1.c. Use the graph of a function to determine the limit.

MTH.CALC.2 Find limits of sums, differences, products, quotients and rational functions.

CALC.2.a. Demonstrate how to properly read and write a limit expression.

CALC.2.b. Be able to find limits of sums, differences, products, quotients and rational functions.

MTH.CALC.3 Understand continuity in terms of limits and functions.

CALC.3.a. Compare and contrast limits and functions. (It is important to point out to students that the limit of a function is not the value of the function. We can think of a function as a series of operations that

can be evaluated at certain points simply by substitution and this is not the same of limits.)

CALC.3.b. Be able to determine what a continuous function is.

CALC.3.c. Graph functions and identify where the line is not continuous.(Example: Given the function y=(x²-4) / (x+2), we can see that the function is undefined at x=-2. Why? If a line or a curve is continuous,

then the following conditions must exist: 1. Exist at every point in the defined interval.2. Have limits at every point equal to the value of the function at that point. Operationally, continuous functions are ones you can draw without lifting your pen.)

MTH.CALC.4 Find the derivatives of functions, including polynomial, rational, trigonometric, logarithmic, inverse, composite and exponential functions.

CALC.4.a. Understand derivative. Find derivative of a constant function, derivative of a power function and the derivative of a function multiplied by a constant.

(example for constant function: f(x)= -10, then f’(x)= 0;

example for power function: f(x) = x-², then f’(x) = -2x-³= -2/x³;

example for a function multiplied by a constant: f(x)= 3x³ Let c=3 and g(x)=x³, then f’(x)= c g’(x)= 3(3x²)= 9x².)

Apply the following rules to find the derivative.

Derivative of the sum of functions (sum rule) -> derivative of f(x) = g(x) + h(x) is given by f’(x)=g’(x) +h’(x). (example: f(x) = x2² + 4 Let g(x) = x² and h(x)= 4, then f’(x)=g’(x)+h’(x) = 2x +0 = 2x)

Derivative of the difference of functions (difference rule) -> derivative of f(x) = g(x)- h(x) is given by f’(x) = g’(x) – h’(x).

(example: f(x)=x³-x-² Let g(x)= x³ and h(x)= x-², then f’(x)=g’(x)-h’(x)=x-²=3x²-(-2x-³)=3x²+2x-³)

Derivative of the product of two functions (product rule)-> the derivative of f(x)=g(x) h(x)is given by f’(x)= g(x) h’(x) + h(x) g’(x).

(example: f(x)=(x²-2x)(x-2) Let g(x)=(x²-2x) and h(x)= (x-2),then f’(x)=g(x) h’(x)+h(x) g ‘(x)=(x²-2x)(1)+(x-2)(2x-2)=x²-2x=2x²-6x+4=3x²-8x+4

Derivative of the quotient of two functions (quotient rule) -> the derivative of f(x) = ²g(x)/h(x) is given by f’(x) = (h(x))g’(x) – g(x) h’(x))/h(x)².

(example: f(x)= (x-2) / (x+1) Let g(x)=(x-2) and h(x)=(x+1), then f’(x)=(h(x) g’(x)-g(x)h’(x) /h(x)²=(x+1)(1)-(x-2)(1) /(x+1)²= 3 / (x+1)²

Derivative of a rational function-> a rational function is the ratio of two polynomials or the quotient of two polynomials. To find the derivative of a rational function, apply the quotient rule and the reciprocal rule. (Students need to be comfortable in using the quotient rule before they can differentiate rational functions.)

Understand the L’Hopital’s Rule and apply it to compute limit in the form lim f(x)X-c g(x)

CALC.4.b. Demonstrate how to take the derivative of a polynomial, rational, trigonometric, logarithmic, inverse, composite and exponential function.

MTH.CALC.5 Find the derivatives of implicitly-defined functions.

CALC.5.a. Understand derivative of an implicit function. (Implicit differentiation is nothing more than a special case of the chain rule for derivatives. (Chain Rule: D [f(g(x)]= f’(g(x)) g’(x))

CALC.5.b. Differentiate composite functions.

CALC.5.c. Find the derivatives of implicit functions.

MTH.CALC.6 Find points of inflection of functions.

CALC.6.a. Sketch a curve and show the maximum, minimum and the point of inflection.

CALC.6.b. Understand stationary or critical points. (The moment at which the derivative of position is exactly zero.)

CALC.6.c. Understand point of inflection and apply first and second derivative rules to determine the point of inflection of a function. (Inflection points are where the function changes concavity. Concave up

corresponds to a positive second derivative and concave down corresponds to a negative second derivative. This is not always the case and further illustrations are needed to determine why.)

MTH.CALC.7 Use implicit differentiation to find the derivative of an inverse function.

CALC.7.a. Understand the process of implicit differentiation.

CALC.7.b. Illustrate different types of functions and their inverses, then find the derivative of each using implicit differentiation.

MTH.CALC.8 Use integration by substitution or change of variable to evaluate integrals.

CALC.8.a. Apply and use the symbol ∫(Integral sign).

CALC.8.b. Understand antiderivative.

CALC.8.c. Compare and contrast an integral and an indefinite integral? (Only definite integral has limits above and below the integral sign.)

CALC.8.d. Apply integration by substitution using u = ax + b and u = g(x) for∫ f(g(x)g’(x) dx .

MTH.CALC.9 Use Riemann sums, the trapezoid rule and technology to evaluate definite integrals of functions represented algebraically, geometrically or by tables of values.

CALC.9.a. Understand Riemann sums and trapezoid or trapezium rule? (The Riemann sum is a method of approximating the total area underneath a curve on a graph, otherwise known as the integral.

The Trapezoidal or Trapezium rule is another approximate technique for calculating the definite integral. Riemann sum uses rectangles as partitions while the Trapezoidal rule uses trapezoids.)

CALC.9.b. Illustrate the Graph of a Riemann sum using the three basic types of Riemann sums, i.e. left sum, right sum and the middle sum.

n – 1

Riemann sum: Area ≈ ∑ f (xi)(xi + 1 – xi)

I = 0

n – 1

Left sum: Area ≈ ∑ f (xi)(xi + 1 – xi)

I = 0

n

Right sum Area ≈ ∑ f (xi)(xi – xi – 1)

I = 1

n

Middle sum Area ≈ ∑ f (xi + x I – 1) (xi – xi – 1)

I=1 2

CALC.9.c. Apply the three basic types of Riemann sums to approximate the integral. (Be mindful that the middle sum gives the best approximation for the integral or the area under the curve.

CALC.9.d. Apply the Trapezoidal rule :(|∫ f(x) dx – Atrap|≤ M₂(b – a)³(12n²) where M₂ is the maximum value of the absolute value of f’’(x).) to calculate the definite integral.

MTH.CALC.10 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.

CALC.10.a. Understand concept of initial condition. (Initial condition could be simply defined as when you are given the initial value of a quantity at some time.)

CALC.10.b. Position function is denoted by s(t), velocity function by v(t) and acceleration function by a(t). Demonstrate how to get the integral of dv = adt

CALC.10.c. Apply relevant formulas to find average velocity, average acceleration and instantaneous velocity.

CALC.10.d. Describe the difference between average velocity and instantaneous velocity. (Which one gives you the slope connecting the two points and which gives you the slope of the tangent line?

What is another name for the slope of the tangent line?)

MTH.CALC.11 Use definite integral 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.

CALC.11.a. Illustrate the area under a curve.

CALC.11.b. Define and illustrate a ‘closed interval’.

CALC.11.c. Review the meaning of definite integral and apply it to finding the area under a curve over a given closed interval.

MTH.CALC.12 Apply the intermediate value theorem and extreme value theorem on a function over a closed interval.

CALC.12.a. Understand the intermediate value theorem and the extreme value theorem.

CALC.12.b. Demonstrate how to apply the intermediate value theorem over a closed interval.

CALC.12.c. Demonstrate how to apply the extreme value theorem over a closed interval.

MTH.CALC.13 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.

CALC.13.a. Understand what a fundamental theorem of calculus is.

CALC.13.b. Apply the theorem to illustrate rate of change of a quantity over an interval as the change of the quantity over the interval.

MTH.CALC.14 Describe the concept of a derivative geometrically, analytically and verbally.

CALC.14.a. Graph any linear or a quadratic equation and find the slope. (What does a slope mean? How is a slope of a line or curve related to the derivative of that line or curve?

CALC.14.b. Find the limit of a tangent line. (What is a tangent line? What is a secant line? When we say that the tangent line is parallel to the curve, what does that mean? How do you tell if the function is

continuous or not?)

CALC.14.c. Differentiate any given function. (What does it mean by differentiating a function? If a function I is differentiable, is it continuous or not? Why?)

MTH.CALC.15 Find second derivatives and derivatives of higher order.

CALC.15.a. Understand higher order derivative or higher derivative. (Any derivative that is higher than the first derivative.)

CALC.15.b. Read and express higher order derivatives. (A derivative of a derivative is the second derivative and is expressed as f’’ or d²x / dx². The third derivative is the derivative of a derivative of a

derivative. How can the third, fourth and other higher derivatives be expressed?)

CALC.15.c. Find and explain the first, second, third or fourth derivative of a given function. Explain what these different derivatives tell us. (For example: The first derivative represents the rate of change of a

function with respect to the independent variable, or the slope. Thus the derivative of the position function for an object would be the object’s speed. The second derivative is the rate of change of

the first derivative. In the case where f(t) represents the position of an object, f”(t) is the objects acceleration. More generally, the second derivative of a function at a point tells you the concavity of

the function. A positive value of the second derivative indicates that function is concave up at that point, which means that it looks sort of like an upward opening parabola. A negative value for the

second derivative indicates that the function is concave down, and looks like an upside down parabola.)

MTH.CALC.16 Prove the mean value theorem.

CALC.16.a. Understand the mean value theorem. (The mean value theorem states, roughly, that given an arc of a smooth continuous (differentiable) curve, there is at least one point on that arc at which the derivative (slope) of the curve is equal (parallel) to the “average” derivative of the arc.)

CALC.16.b. Apply f’(c) = f(b) – f(a) to determine if a given point on the arc is p (b – a) parallel to the average derivative of the arc.

CALC.16.c. Verify that the equation proves the mean value theorem.

MTH.CALC.17 Find average and instantaneous rates of change.

CALC.17.a. Differentiate average rate of change and instantaneous rate of change.

CALC.17.b. Use correct formulas to find average rate of change and the instantaneous rate of change.

MTH.CALC.18 Use the first and second derivatives to describe the behavior of functions.

CALC.18.a. Find first and second derivative of a function.

CALC.18.b. State the conditions in which a function is continuous.

CALC.18.c. Determine whether the slope is positive or negative by the nature of the derivative found and also its concavity if applicable.