# Mathematics

## Curriculum

Kindergarten through grade-twelve math instruction should emphasize practices and activities that promote and integrate the Eight Standards for Mathematical Practice in the Kentucky Academic Standards for Mathematics (KAS-M).  The practices should become the natural way in which students come to understand and engage in math.

## Mathematical Practices

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

### Elementary School

#### Kindergarten

In Kindergarten, instructional time should focus on two critical areas:

1. Representing, relating, and operating on whole numbers, initially with sets of objects
2. Describing shapes and space

More learning time in kindergarten should be devoted to numbers than to other topics.

In grade one, instructional time should focus on four critical areas:

1. Developing an understanding of addition, subtraction, and strategies for addition and subtraction within 20
2. Developing an understanding of whole number relationships and place value, including grouping tens and ones
3. Developing an understanding of linear measurement and measuring lengths as iterating length units
4. Reasoning about attributes of, and composing and decomposing, geometric shapes

In grade two, instructional time should focus on four critical areas:

1. Extending understanding of base-ten notation
2. Building fluency with addition and subtraction
3. Using standard units of measure
4. Describing and analyzing shapes

In grade three, instructional time should focus on four critical areas:

1. Developing understanding of multiplication and division and strategies for multiplication and division within 100
2. Developing understanding of fractions, especially unit fractions (fractions with numerator of 1)
3. Developing understanding of the structure of rectangular arrays and of area
4. Describing and analyzing two-dimensional shapes

In grade four, instructional time should focus on three critical areas:

1. Developing an understanding and fluency of multidigit multiplication and developing an understanding of dividing to find quotients involving multidigit dividends
2. Developing an understanding of fraction equivalence, addition, and subtraction of fractions with like denominators and multiplication of fractions by whole numbers
3. Understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry

In grade five, instructional time should focus on three critical areas:

1. Developing fluency with addition and subtraction of fractions and developing an understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions)
2. Extending division to two-digit divisors, integrating decimal fractions into the place value system, developing an understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations
3. Developing an understanding of volume

### Middle School

In grade six, instructional time should focus on four critical areas:

1. Connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems
2. Completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers
3. Writing, interpreting, and using expressions and equations
4. Developing an understanding of statistical thinking

In grade seven, instructional time should focus on four critical areas:

1. Developing an understanding of and applying proportional relationships
2. Developing an understanding of operations with rational numbers and working with expressions and linear equations
3. Solving problems involving scale drawings and informal geometric constructions and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume
4. Drawing inferences about populations based on samples

In grade eight, instructional time should focus on three critical areas:

1. Formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations
2. Grasping the concept of a function and using functions to describe quantitative relationships
3. Analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence and understanding and applying the Pythagorean Theorem

### High School

#### Algebra I

In Algebra I, instructional time should focus on five critical areas:

1. Students analyze and explain the process of solving an equation. Students fluently write, interpret, and translate between various forms of linear equations and inequalities, and use them to solve problems.
2. Students learn function notation and develop the concepts of domain and range. They explore many examples of functions, and they interpret multiple representations of functions. Students extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions. Students explore systems of equations and inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.
3. Students use regression techniques to describe approximately linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit.
4. Students apply understanding of number and strengthen their ability to see structure in and create quadratic and exponential expressions. They create and solve equations and inequalities involving quadratic expressions.
5. Students consider quadratic functions, comparing characteristics of quadratic functions to those of linear and exponential functions. Students anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. They identify the real solutions of a quadratic equation as the zeros of a related quadratic function.

#### Geometry

In geometry, instructional time should focus on six critical areas:

1. Students establish triangle congruence criteria. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems and solve problems about triangles, quadrilaterals, and other polygons.
2. Students identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry.
3. Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of area and volume formulas. Also, students use their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line.
4. Students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines.
5. Students prove basic theorems about circles, such as a tangent line is perpendicular to a radius; inscribed angle theorem; and theorems about chords, secants, and tangents dealing with segment lengths and angle measures. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students write the equation of a circle when given the radius and the coordinates of its center, and given an equation of a circle, they draw its graph.
6. Students expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible.

#### Algebra II

In Algebra II, instructional time should focus on three critical areas:

1. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multidigit integers and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. Students also connect the arithmetic of rational expressions to the same rules as the arithmetic of rational numbers.
2. Students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit.
3. Students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data and the role that randomness and careful design play in the conclusions that can be drawn.

#### Elective Courses

Pre-Algebra, Pre-Calculus, Calculus, Statistics, Math Concepts, College Algebra, College Readiness Mathematics, etc.