# Math

Students learn mathematics is about: sense-making, flexible thinking, questioning, seeing and using patterns, problem solving, communicating, creative thinking about space, data, and numbers, making and analyzing connections and relationships.

We believe each student is capable of making sense of mathematical ideas and is able to use understanding and reasoning to solve authentic mathematical and real world problems. Mathematics learning involves students developing conceptual understanding and proficiency in applying mathematical procedures with understanding. Students learn through engagement in rich tasks and collaboration. Close attention is paid to students’ thinking, including emergent understanding and mistakes, in order to provide an appropriate level of feedback and support.

### Comprehensive K-12 Mathematics Framework

## Grade-Level Mathematics Learning

## Enduring Understandings

### En

### In Spring Lake Park Schools, Enduring Understandings are...

Statements that clearly articulate the big ideas that promote long term understanding of the discipline or subject area that have lasting value beyond the classroom. These are the important understandings that we want students to retain after they may have forgotten the details (Brown, 2004; Wiggins & McTighe, 1998).

#### K-12 Mathematics Enduring Understandings

- Students will create and use
**representations**to:- model mathematics and the world around them.
- organize and communicate their thinking, and solve problems.

- Students use
**reasoning**to:- develop and answer questions,
- use evidence to support or refute their own or other people’s thinking,
- think about their own thinking and reasoning (metacognition, and evidence they gather

- Students
**solve**problems to select strategies and explore new approaches to problems, think about, apply, monitor, and shift their strategies, build new knowledge and enhance their own thinking, and make inferences from data, formulate generalizations, and ask new questions. - Students at all levels should:
- demonstrate their understanding at increasing levels of abstraction,
- recognize and use
**connections.** - recognize and apply concepts within and outside of the discipline and, simplify challenges in our daily lives.

## Essential Questions

### In Spring Lake Park Schools, Essential Questions...

Focus our attention on what is important. They foster inquiry, understanding, and transfer of learning. They occur naturally and should be asked over and over (Brown, 2004; Wiggins & McTighe, 1998).

#### K-12 Language Arts Essential Questions include:

- How could I show my answer using the least amount of work?
- How can I express this in a different way?
- Why do we need so many different ways to represent our thinking?
- How can I argue that my answer is correct or the best answer possible?
- Why is it important to understand my own thinking?
- How do we decide when we have enough evidence or the right answer?
- What do I do if I'm stuck?
- What makes a problem interesting to me?
- How can I take something that's boring and make it interesting?
- When am I ever going to have to use this?
- How do I find out if I am right or wrong?
- What resources are available?
- Does my solution raise other questions?
- How do I know what approach to take when I don't know what to do?
- How can I use what I learned to solve new problems?
- Where does this come from?
- Where is this leading to?
- How is this useful to me?
- Have I done anything kind of like this before?
- How does this relate to what I learned in the past or what I will learn in the future?
- Why is this important to me?
- How do people use this outside the classroom?

## K-12 Learning Targets

### In Spring Lake Park Schools, Learning Targets...

Specify, in measurable terms, what all students should know and be able to do to achieve desired understandings and answer essential questions (Brown, 2004). These will be identified for each subject within each grade level.

## Grade-specific Mathematics Learning Targets

- Kindergarten
- First Grade
- Second Grade
- Third Grade
- Fourth Grade
- Fifth Grade
- Sixth Grade
- Seventh Grade
- Eighth Grade
- Quad Algebra
- Geometry
- Algebra II

## Kindergarten

- Recognize that a number can be used to represent how many objects are in a set or to represent the position of an object in a sequence.
- Read, write, and represent whole numbers from 0 to at least 31. Representations may include numerals, pictures, real objects and picture graphs, spoken words, and manipulatives such as connecting cubes.
- Count, with objects, forward to at least 31 and backward from 20.
- Find a number that is 1 more or 1 less than a given number up to 31.
- Compare and order whole numbers, with and without objects, from 0 to 31.
- Use objects and draw pictures to find the sums and differences of numbers between 0 and 10.
- Compose and decompose numbers up to 10 with objects and pictures.
- Identify, create, complete, and extend simple patterns using shape, color, size, number, sounds and movements. Patterns may be repeating, growing or shrinking such as ABB, ABB, ABB or ●,●●,●●●.
- Name basic two- and three-dimensional shapes such as squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, cones, cylinders and spheres. Describe basic 2D shapes: oval, square, circle, triangle, rhombus, rectangle, hexagon, trapezoid (sides and corners to describe 2D)
- Sort objects using characteristics such as shape, size, color and thickness.
- Use basic shapes and spatial reasoning to model objects in the real-world.
- Use words to compare objects according to length, size, weight and position.
- Order 2 or 3 objects using measurable attributes, such as length and weight.
- Count to 100 by tens.
- Count to 100 by ones.
- Identify the names and values of pennies, nickels, and dimes.
- Collect, organize, display, and analyze data and verbalize the meaning to others.

## First Grade

- Use place value to describe whole numbers between 10 and 120 in terms of tens and ones.
- Read, write and represent whole numbers up to 120. Representations may include numerals, addition and subtraction, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.
- Count, with and without objects, forward and backward from any given number up to 120.
- Find a number that is 10 more or 10 less than a given number.
- Compare and order whole numbers up to 120.
- Use words to describe the relative size of numbers.
- Collect, organize, display, and analyze data in tally charts and bar graphs. Verbalize the data's the meaning to others.
- Use words, pictures, objects, length-based models (connecting cubes), numerals and number lines to model and solve addition and subtraction problems in part-part-total, adding to, taking away from and comparing situations.
- Compose and decompose numbers up to 12 with an emphasis on making ten.
- Recognize the relationship between counting and addition and subtraction. Skip count by 2s, 5s, and 10s.
- Create simple patterns using objects, pictures, numbers and rules. Identify possible rules to complete or extend patterns. Patterns may be repeating, growing or shrinking. Calculators can be used to create and explore patterns.
- Represent real-world situations involving addition and subtraction basic facts, using objects and number sentences.
- Determine if equations involving addition and subtraction are true.
- Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as: 2 + 4 = , 3 + = 7, and 5 = – 3.
- Use addition or subtraction basic facts to represent a given problem situation using a number sentence.
- Describe characteristics of two- and three-dimensional objects, such as triangles, squares, rectangles, circles, rectangular prisms, cylinders, cones and spheres.
- Compose (combine) and decompose (take apart) two- and three-dimensional figures such as triangles, squares, rectangles, circles, rectangular prisms and cylinders.
- Measure the length of an object in terms of multiple copies of another object.
- Tell time to the hour and half-hour.
- Identify pennies, nickels and dimes; find the value of a group of these coins, up to one dollar.

## Second Grade

- Read, write and represent whole numbers up to 1000. Representations may include numerals, addition, subtraction, multiplication, words, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.
- Use place value to describe whole numbers between 10 and 1000 in terms of hundreds, tens and ones. Know that 100 is 10 tens, and 1000 is 10 hundreds.
- Find 10 more or 10 less than a given three-digit number. Find 100 more or 100 less than a given three-digit number.
- Round numbers up to the nearest 10 and 100 and round numbers down to the nearest 10 and 100.
- Compare and order whole numbers up to 1000.
- Use strategies to generate addition and subtraction facts including making tens, fact families, doubles plus or minus one, counting on, counting back, and the commutative and associative properties. Use the relationship between addition and subtraction to generate basic facts.
- Demonstrate fluency with basic addition facts and related subtraction facts up to 20.
- Estimate sums and differences up to 100.
- Use mental strategies and algorithms based on knowledge of place value and equality to add and subtract two-digit numbers. Strategies may include decomposition, expanded notation, and partial sums and differences.
- Solve real-world and mathematical addition and subtraction problems involving whole numbers with up to 2 digits.
- Use addition and subtraction to create and obtain information from tables, bar graphs and tally charts.
- Identify, create and describe simple number patterns involving repeated addition or subtraction, skip counting and arrays of objects such as counters or tiles. Use patterns to solve problems in various contexts.
- Understand how to interpret number sentences involving addition, subtraction and unknowns represented by letters. Use objects and number lines and create real-world situations to represent number sentences.
- Use number sentences involving addition, subtraction, and unknowns to represent given problem situations. Use number sense and properties of addition and subtraction to find values for the unknowns that make the number sentences true.
- Describe, compare, and classify two- and three-dimensional figures according to number and shape of faces, and the number of sides, edges and vertices (corners).
- Identify and name basic two- and three-dimensional shapes, such as squares, circles, triangles, rectangles, trapezoids, hexagons, cubes, rectangular prisms, cones, cylinders and spheres.
- Understand the relationship between the size of the unit of measurement and the number of units needed to measure the length of an object.
- Demonstrate an understanding of the relationship between length and the numbers on a ruler by using a ruler to measure lengths to the nearest centimeter or inch.
- Tell time to the five minutes and distinguish between a.m. and p.m.
- Identify pennies, nickels, dimes and quarters. Find the value of a group of coins and determine combinations of coins that equal a given amount.
- Recognize that fractions can be used to represent parts of a whole.

## Third Grade

- Read, write and represent whole numbers up to 100,000. Representations may include numerals, expressions with operations, words, pictures, number lines, and manipulatives such as bundles of sticks and base 10 blocks.
- Use place value to describe whole numbers between 1000 and 100,000 in terms of ten thousands, thousands, hundreds, tens and ones.
- Find 10,000 more or 10,000 less than a given five-digit number. Find 1000 more or 1000 less than a given four- or five-digit. Find 100 more or 100 less than a given four- or five-digit number.
- Round numbers to the nearest 10,000, 1000, 100 and 10. Round up and round down to estimate sums and differences.
- Compare and order whole numbers up to 100,000.
- Add and subtract multi-digit numbers, using efficient and generalizable procedures based on knowledge of place value, including standard algorithms.
- Use addition and subtraction to solve real-world and mathematical problems involving whole numbers. Use various strategies, including the relationship between addition and subtraction, the use of technology, and the context of the problem to assess the reasonableness of results.
- Represent multiplication facts by using a variety of approaches, such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line and skip counting. Represent division facts by using a variety of approaches, such as repeated subtraction, equal sharing and forming equal groups. Recognize the relationship between multiplication and division.
- Solve real-world and mathematical problems involving multiplication and division, including both "how many in each group" and "how many groups" division problems.
- Use strategies and algorithms based on knowledge of place value, equality and properties of addition and multiplication to multiply a two- or three-digit number by a one-digit number. Strategies may include mental strategies, partial products, the standard algorithm, and the commutative, associative, and distributive properties.
- Read and write fractions with words and symbols. Recognize that fractions can be used to represent parts of a whole, parts of a set, points on a number line, or distances on a number line.
- Understand that the size of a fractional part is relative to the size of the whole.
- Order and compare unit fractions and fractions with like denominators by using models and an understanding of the concept of numerator and denominator.
- Create, describe, and apply single-operation input-output rules involving addition, subtraction and multiplication to solve problems in various contexts.
- Understand how to interpret number sentences involving multiplication and division basic facts and unknowns. Create real-world situations to represent number sentences.
- Use multiplication and division basic facts to represent a given problem situation using a number sentence. Use number sense and multiplication and division basic facts to find values for the unknowns that make the number sentences true.
- Identify parallel and perpendicular lines in various contexts, and use them to describe and create geometric shapes, such as right triangles, rectangles, parallelograms and trapezoids.
- Sketch polygons with a given number of sides or vertices (corners), such as pentagons, hexagons and octagons.
- Use half units when measuring distances.
- Find the perimeter of a polygon by adding the lengths of the sides.
- Measure distances around objects using appropriate tools.
- Tell time to the minute, using digital and analog clocks. Determine elapsed time to the minute.
- Know relationships among units of time.
- Make change up to one dollar in several different ways, including with as few coins as possible.
- Use an analog thermometer to determine temperature to the nearest degree in Fahrenheit and Celsius.
- Collect, display and interpret data using frequency tables, bar graphs, picture graphs and number line plots having a variety of scales. Use appropriate titles, labels and units.

## Fourth Grade

- Demonstrate fluency with multiplication and division facts.
- Use an understanding of place value to multiply a number by 10, 100 and 1000.
- Multiply multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms.
- Estimate products and quotients of multi-digit whole numbers by using rounding, benchmarks and place value to assess the reasonableness of results.
- Solve multi-step real-world and mathematical problems requiring the use of addition, subtraction and multiplication of multi-digit whole numbers. Use various strategies, including the relationship between operations, the use of technology, and the context of the problem to assess the reasonableness of results.
- Use strategies and algorithms based on knowledge of place value, equality and properties of operations to divide multi-digit whole numbers by one- or two-digit numbers. Strategies may include mental strategies, partial quotients, the commutative, associative, and distributive properties and repeated subtraction.
- Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions.
- Locate fractions on a number line. Use models to order and compare whole numbers and fractions, including mixed numbers and improper fractions.
- Use fraction models to add and subtract fractions with like denominators in real-world and mathematical situations. Develop a rule for addition and subtraction of fractions with like denominators.
- Read and write decimals with words and symbols; use place value to describe decimals in terms of thousands, hundreds, tens, ones, tenths, hundredths and thousandths.
- Compare and order decimals and whole numbers using place value, a number line and models such as grids and base 10 blocks.
- Read and write tenths and hundredths in decimal and fraction notations using words and symbols; know the fraction and decimal equivalents for halves and fourths.
- Round decimals to the nearest tenth.
- Create and use input-output rules involving addition, subtraction, multiplication and division to solve problems in various contexts. Record the inputs and outputs in a chart or table.
- Understand how to interpret number sentences involving multiplication, division and unknowns. Use real-world situations involving multiplication or division to represent number sentences.
- Use multiplication, division and unknowns to represent a given problem situation using a number sentence. Use number sense, properties of multiplication, and the relationship between multiplication and division to find values for the unknowns that make the number sentences true.
- Describe, classify and sketch triangles, including equilateral, right, obtuse and acute triangles. Recognize triangles in various contexts.
- Describe, classify and draw quadrilaterals, including squares, rectangles, trapezoids, rhombuses, parallelograms and kites. Recognize quadrilaterals in various contexts.
- Measure angles in geometric figures and real-world objects with a protractor or angle ruler.
- Compare angles according to size. Classify angles as acute, right and obtuse.
- Understand that the area of a two-dimensional figure can be found by counting the total number of same size square units that cover a shape without gaps or overlaps. Justify why length and width are multiplied to find the area of a rectangle by breaking the rectangle into one unit by one unit squares and viewing these as grouped into rows and columns.
- Find the areas of geometric figures and real-world objects that can be divided into rectangular shapes. Use square units to label area measurements.
- Apply translations (slides) to figures.
- Apply reflections (flips) to figures by reflecting over vertical or horizontal lines and relate reflections to lines of symmetry.
- Apply rotations (turns) of 90˚ clockwise or counterclockwise.
- Recognize that translations, reflections and rotations preserve congruency and use them to show that two figures are congruent.
- Use tables, bar graphs, timelines and Venn diagrams to display data sets. The data may include fractions or decimals. Understand that spreadsheet tables and graphs can be used to display data.

## Fifth Grade

- Divide multi-digit whole and decimal numbers using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms. Recognize that quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction or mixed number, or a decimal.
- Consider the context in which a problem is situated to select the most useful form of the quotient for the solution and use the context to interpret the quotient appropriately.
- Estimate solutions to arithmetic problems in order to assess the reasonableness of results.
- Solve real-world and mathematical problems requiring addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies, including the inverse relationships between operations, the use of technology, and the context of the problem to assess the reasonableness of results.
- Read and write decimals using place value to describe decimals in terms of groups from millionths to millions.
- Find 0.1 more than a number and 0.1 less than a number. Find 0.01 more than a number and 0.01 less than a number. Find 0.001 more than a number and 0.001 less than a number.
- Order fractions and decimals, including mixed numbers and improper fractions, and locate on a number line.
- Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts.
- Round numbers to the nearest 0.1, 0.01 and 0.001.
- Add and subtract decimals and fractions, using efficient and generalizable procedures, including standard algorithms.
- Model addition and subtraction of fractions and decimals using a variety of representations.
- Estimate sums and differences of decimals and fractions to assess the reasonableness of results.
- Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed numbers, including those involving measurement, geometry and data.
- Create and use rules, tables, spreadsheets and graphs to describe patterns of change and solve problems.
- Use a rule or table to represent ordered pairs of positive integers and graph these ordered pairs on a coordinate system.
- Apply the commutative, associative and distributive properties and order of operations to generate equivalent numerical expressions and to solve problems involving whole numbers.
- Determine whether an equation or inequality involving a variable is true or false for a given value of the variable.
- Represent real-world situations using equations and inequalities involving variables. Create real-world situations corresponding to equations and inequalities.
- Evaluate expressions and solve equations involving variables when values for the variables are given.
- Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of edges, faces or vertices as well as the types of faces.
- Recognize and draw a net for a three-dimensional figure.
- Develop and use formulas to determine the area of triangles, parallelograms and figures that can be decomposed into triangles.
- Use various tools and strategies to measure the volume and surface area of objects that are shaped like rectangular prisms.
- Understand that the volume of a three-dimensional figure can be found by counting the total number of same-sized cubic units that fill a shape without gaps or overlaps. Use cubic units to label volume measurements.
- Develop and use the formulas
*V*=*ℓwh*and*V*=*Bh*to determine the volume of rectangular prisms. Justify why base area*B*and height*h*are multiplied to find the volume of a rectangular prism by breaking the prism into layers of unit cubes. - Know and use the definitions of the mean, median and range of a set of data. Know how to use a spreadsheet to find the mean, median and range of a data set. Understand that the mean is a "leveling out" of data.
- Create and analyze double-bar graphs and line graphs by applying understanding of whole numbers, fractions and decimals. Know how to create spreadsheet tables and graphs to display data.

## Sixth Grade

### Operations and Numbers

- Locate positive rational numbers on a number line and plot pairs of positive rational numbers on a coordinate grid. (6.1.1.1)
- Compare positive rational numbers represented in various forms. Use the symbols <, = and >. (6.1.1.2)
- Understand that percent represents parts out of 100 and ratios to 100. (6.1.1.3)
- Determine equivalences among fractions, decimals and percents; select among these representations to solve problems. (6.1.1.4)
- Factor whole numbers; express a whole number as a product of prime factors with exponents. (6.1.1.5)
- Determine greatest common factors and least common multiples. Use common factors and common multiples to calculate with fractions and find equivalent fractions. (6.1.1.6)
- Convert between equivalent representations of positive rational numbers. (6.1.1.7)
- Identify and use ratios to compare quantities; understand that comparing quantities using ratios is not the same as comparing quantities using subtraction. (6.1.2.1)
- Apply the relationship between ratios, equivalent fractions and percents to solve problems in various contexts, including those involving mixtures and concentrations. (6.1.2.2)
- Determine the rate for ratios of quantities with different units. (6.1.2.3)
- Use reasoning about multiplication and division to solve ratio and rate problems. (6.1.2.4)
- Multiply and divide decimals and fractions, using efficient and generalizable procedures, including standard algorithms. (6.1.3.1)
- Use the meanings of fractions, multiplication, division and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions. (6.1.3.2)
- Calculate the percent of a number and determine what percent one number is of another number to solve problems in various contexts. (6.1.3.3)
- Solve real-world and mathematical problems requiring arithmetic with decimals, fractions and mixed numbers. (6.1.3.4)
- Estimate solutions to problems with whole numbers, fractions and decimals and use the estimates to assess the reasonableness of results in the context of the problem. (6.1.3.5)

### Algebra

- Understand that a variable can be used to represent a quantity that can change, often in relationship to another changing quantity. Use variables in various contexts. (6.2.1.1)
- Represent the relationship between two varying quantities with function rules, graphs and tables; translate between any two of these representations. (6.2.1.2)
- Apply the associative, commutative and distributive properties and order of operations to generate equivalent expressions and to solve problems involving positive rational numbers. (6.2.2.1)
- Represent real-world or mathematical situations using equations and inequalities involving variables and positive rational numbers. (6.2.3.1)
- Solve equations involving positive rational numbers using number sense, properties of arithmetic and the idea of maintaining equality on both sides of the equation. Interpret a solution in the original context and assess the reasonableness of results. (6.2.3.2)

### Geometry and Measurement

- Calculate the surface area and volume of prisms and use appropriate units, such as cm2 and cm3. Justify the formulas used. Justification may involve decomposition, nets or other models. (6.3.1.1)
- Calculate the area of quadrilaterals. Quadrilaterals include squares, rectangles, rhombuses, parallelograms, trapezoids and kites. When formulas are used, be able to explain why they are valid. (6.3.1.2)
- Estimate the perimeter and area of irregular figures on a grid when they cannot be decomposed into common figures and use correct units, such as cm and cm2. (6.3.1.3)
- Solve problems using the relationships between the angles formed by intersecting lines. (6.3.2.1)
- Determine missing angle measures in a triangle using the fact that the sum of the interior angles of a triangle is 180 degrees. Use models of triangles to illustrate this fact. (6.3.2.2)
- Develop and use formulas for the sums of the interior angles of polygons by decomposing them into triangles. (6.3.2.3)
- Solve problems in various contexts involving conversion of weights, capacities, geometric measurements and times within measurement systems using appropriate units. (6.3.3.1)
- Estimate weights, capacities and geometric measurements using benchmarks in measurement systems with appropriate units. (6.3.3.2)

### Data Analysis and Probability

- Determine the sample space (set of possible outcomes) for a given experiment and determine which members of the sample space are related to certain events. Sample space may be determined by the use of tree diagrams, tables or pictorial representations. (6.4.1.1)
- Determine the probability of an event using the ratio between the size of the event and the size of the sample space; represent probabilities as percents, fractions and decimals between 0 and 1 inclusive. Understand that probabilities measure likelihood. (6.4.1.2)
- Perform experiments for situations in which the probabilities are known, compare the resulting relative frequencies with the known probabilities; know that there may be differences. (6.4.1.3)
- Calculate experimental probabilities from experiments; represent them as percents, fractions and decimals between 0 and 1 inclusive. Use experimental probabilities to make predictions when actual probabilities are unknown. (6.4.1.4)

## Seventh Grade

### Number and Operation

- Know that every rational number can be written as the ratio of two integers or as a terminating or repeating decimal. Recognize that is not rational, but it can be approximated by rational numbers such as 22/7 and 3.14. (7.1.1.1)
- Understand that division of two integers will always result in a rational number. Use this information to interpret the decimal result of a division problem when using a calculator. (7.1.1.2)
- Locate positive and negative rational numbers on a number line, understand the concept of opposites, and plot pairs of positive and negative rational numbers on a coordinate grid. (7.1.1.3)
- Compare positive and negative rational numbers expressed in various forms using the symbols < > = ≤ ≥ (7.1.1.4)
- Recognize and generate equivalent representations of positive and negative rational numbers, including equivalent fractions. (7.1.1.5)
- Add, subtract, multiply and divide positive and negative rational numbers that are integers, fractions and terminating decimals; use efficient and generalizable procedures, including standard algorithms; raise positive rational numbers to whole-number exponents. (7.1.2.1)
- Use real-world contexts and the inverse relationship between addition and subtraction to explain why the procedures of arithmetic with negative rational numbers make sense. (7.1.2.2)
- Understand that calculators and other computing technologies often truncate or round numbers. (7.1.2.3)
- Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest. (7.1.2.4)
- Use proportional reasoning to solve problems involving ratios in various contexts. (7.1.2.5)
- Demonstrate an understanding of the relationship between the absolute value of a rational number and distance on a number line. Use the symbol for absolute value. (7.1.2.6)

### Algebra

- Understand that a relationship between two variables,
*x*and*y,*is proportional if it can be expressed in the form*y/x**= k*or*y = kx.*Distinguish proportional relationships from other relationships, including inversely - proportional relationships (xy
*= k*or*y = k/x).* - (7.2.1.1)
- Understand that the graph of a proportional relationship is a line through the origin whose slope is the unit rate (constant of proportionality). Know how to use graphing technology to examine what happens to a line when the unit rate is changed. (7.2.1.2)
- Represent proportional relationships with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Determine the unit rate (constant of proportionality or slope) given any of these representations. (7.2.2.1)
- Solve multi-step problems involving proportional relationships in numerous contexts. (7.2.2.2)
- Use knowledge of proportions to assess the reasonableness of solutions. (7.2.2.3)
- Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers. (7.2.2.4)
- Use properties of algebra to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols and whole number exponents. Properties of algebra include associative, commutative and distributive laws. (7.2.3.1)
- Evaluate algebraic expressions containing rational numbers and whole number exponents at specified values of their variables. (7.2.3.2)
- Apply understanding of order of operations and grouping symbols when using calculators and other technologies. (7.2.3.3)
- Represent relationships in various contexts with equations involving variables and positive and negative rational numbers. Use the properties of equality to solve for the value of a variable. Interpret the solution in the original context. (7.2.4.1)
- Solve equations resulting from proportional relationships in various contexts. (7.2.4.2)

### Geometry and Measurement

- Demonstrate an understanding of the proportional relationship between the diameter and circumference of a circle and that the unit rate (constant of proportionality) is
*.*Calculate the circumference and area of circles and sectors of circles to solve problems in various contexts. (7.3.1.1) - Calculate the volume and surface area of cylinders and justify the formulas used. (7.3.1.2)
- Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors. (7.3.2.1)
- Apply scale factors, length ratios and area ratios to determine side lengths and areas of similar geometric figures. (7.3.2.2)
- Use proportions and ratios to solve problems involving scale drawings and conversions of measurement units. (7.3.2.3)
- Graph and describe translations and reflections of figures on a coordinate grid and determine the coordinates of the vertices of the figure after the transformation. (7.3.2.4)

### Data Analysis and Probability

- Design simple experiments and collect data. Determine mean, median and range for quantitative data and from data represented in a display. Use these quantities to draw conclusions about the data, compare different data sets, and make predictions. (7.4.1.1)
- Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Know how to create data displays using a spreadsheet to examine this impact. (7.4.1.2)
- Use reasoning with proportions to display and interpret data in circle graphs (pie charts) and histograms. Choose the appropriate data display and know how to create the display using a spreadsheet or other graphing technology. (7.4.2.1)
- Use random numbers generated by a calculator or a spreadsheet or taken from a table to simulate situations involving randomness, make a histogram to display the results, and compare the results to known probabilities. (7.4.3.1)
- Calculate probability as a fraction of sample space or as a fraction of area. Express probabilities as percents, decimals and fractions. (7.4.3.2)
- Use proportional reasoning to draw conclusions about and predict relative frequencies of outcomes based on probabilities. (7.4.3.3)

## Eighth Grade

### Number and Operation

- Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non- zero rational number and an irrational number is irrational. (8.1.1.1)
- Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers. (8.1.1.2)
- Determine rational approximations for solutions to problems involving real numbers. (8.1.1.3)
- Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions. (8.1.1.4)
- Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved. (8.1.1.5)

### Algebra

- Understand that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable. Use functional notation, such as
*f(x),*to represent such relationships. (8.2.1.1) - Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount. (8.2.1.2)
- Understand that a function is linear if it can be expressed in the form
*f*(x) =*mx*+*b*or if its graph is a straight line. (8.2.1.3) - Understand that an arithmetic sequence is a linear function that can be expressed in the form
*f*(x) =*mx*+*b,*where*x*= 0, 1, 2, 3,... (8.2.1.4) - Understand that a geometric sequence is a non-linear function that can be expressed in the form
*f*(x) =*ab x,*where*x*= 0, 1, 2, 3,.... (8.2.1.5) - Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. (8.2.2.1)
- Identify graphical properties of linear functions including slopes and intercepts. Know that the slope equals the rate of change, and that the
*y-intercept*is zero when the function represents a proportional relationship. (8.2.2.2) - Identify how coefficient changes in the equation
*f*(x) =*mx*+*b*affect the graphs of linear functions. Know how to use graphing technology to examine these effects. (8.2.2.3) - Represent arithmetic sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems. (8.2.2.4)
- Represent geometric sequences using equations, tables, graphs and verbal descriptions, and use them to solve problems. (8.2.2.5)
- Evaluate algebraic expressions, including expressions containing radicals and absolute values, at specified values of their variables. (8.2.3.1)
- Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols. (8.2.3.2)
- Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships. (8.2.4.1)
- Solve multi-step equations in one variable. Solve for one variable in a multi- variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used. (8.2.4.2)
- Express linear equations in slope-intercept, point-slope and standard forms, and convert between these forms. Given sufficient information, find an equation of a line. (8.2.4.3)
- Use linear inequalities to represent relationships in various contexts. (8.2.4.4)
- Solve linear inequalities using properties of inequalities. Graph the solutions on a number line. (8.2.4.5)
- Represent relationships in various contexts with equations and inequalities involving the absolute value of a linear expression. Solve such equations and inequalities and graph the solutions on a number line. (8.2.4.6)
- Represent relationships in various contexts using systems of linear equations. Solve systems of linear equations in two variables symbolically, graphically and numerically. (8.2.4.7)
- Understand that a system of linear equations may have no solution, one solution, or an infinite number of solutions. Relate the number of solutions to pairs of lines that are intersecting, parallel or identical. Check whether a pair of numbers satisfies a system of two linear equations in two unknowns by substituting the numbers into both equations. (8.2.4.8)
- Use the relationship between square roots and squares of a number to solve problems. (8.2.4.9)

### Geometry and Measurement

- Use the Pythagorean Theorem to solve problems involving right triangles. (8.3.1.1)
- Determine the distance between two points on a horizontal or vertical line in a coordinate system. Use the Pythagorean Theorem to find the distance between any two points in a coordinate system. (8.3.1.2)
- Informally justify the Pythagorean Theorem by using measurements, diagrams and computer software. (8.3.1.3)
- Understand and apply the relationships between the slopes of parallel lines and between the slopes of perpendicular lines. Dynamic graphing software may be used to examine these relationships. (8.3.2.1)
- Analyze polygons on a coordinate system by determining the slopes of their sides. (8.3.2.2)
- Given a line on a coordinate system and the coordinates of a point not on the line, find lines through that point that are parallel and perpendicular to the given line, symbolically and graphically. (8.3.2.3)

### Data Analysis and Probability

- Collect, display and interpret data using scatterplots. Use the shape of the scatterplot to informally estimate a line of best fit and determine an equation for the line. Use appropriate titles, labels and units. Know how to use graphing technology to display scatterplots and corresponding lines of best fit. (8.4.1.1)
- Use a line of best fit to make statements about approximate rate of change and to make predictions about values not in the original data set. (8.4.1.2)
- Assess the reasonableness of predictions using scatterplots by interpreting them in the original context. (8.4.1.3)

## Quad Algebra

- 9.2.1.1 Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain.
- 9.2.1.2 Distinguish between functions and other relations defined symbolically, graphically or in tabular form.
- 9.2.1.3 Find the domain of a function defined symbolically, graphically or in a real-world context. For example: The formula f (x) = πx2 can represent a function whose domain is all real numbers, but in the context of the area of a circle, the domain would be restricted to positive x.
- 9.2.1.4 Obtain information and draw conclusions from graphs of functions and other relations. For example: If a graph shows the relationship between the elapsed flight time of a golf ball at a given moment and its height at that same moment, identify the time interval during which the ball is at least 100 feet above the ground.
- 9.2.1.5 Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f (x) = a(x – h)2 + k , or in factored form.
- 9.2.1.6 Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function.
- 9.2.1.7 Understand the concept of an asymptote and identify asymptotes for exponential functions and reciprocals of linear functions, using symbolic and graphical methods.
- 9.2.1.8 Make qualitative statements about the rate of change of a function, based on its graph or table of values. For example: The function f(x) = 3x increases for all x, but it increases faster.
- 9.2.1.9 Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. For example: Determine how the graph of f(x) = |x – h| + k changes as h and k change.
- 9.2.2.1 Represent and solve problems in various contexts using linear and quadratic functions. For example: Write a function that represents the area of a rectangular garden that can be surrounded with 32 feet of fencing, and use the function to determine the possible dimensions of such a garden if the area must be at least 50 square feet.
- 9.2.2.2 Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth.
- 9.2.2.3 Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions.
- 9.2.2.4 Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively. For example: A closed form formula for the terms tn in the geometric sequence 3, 6, 12, 24, ... is tn = 3(2)n-1, where n = 1, 2, 3, ... , and this sequence can be expressed recursively by writing t1 = 3 and tn = 2tn-1, for n ≥ 2. Another example: The partial sums sn of the series 3 + 6 + 12 + 24 + ... can be expressed recursively by writing s1 = 3 and sn = 3 + 2sn-1, for n ≥ 2.
- 9.2.2.5 Recognize and solve problems that can be modeled using finite geometric sequences and series, such as home mortgage and other compound interest examples. Know how to use spreadsheets and calculators to explore geometric sequences and series in various contexts.
- 9.2.2.6 Sketch the graphs of common non-linear functions such as f ( x)= x , f ( x)= x , ( ) 1x f x = , f (x) = x3, and translations of these functions, such as f (x)= x−2 + 4 Know how to use graphing technology to graph these functions.
- 9.2.3.1. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains.
- 9.2.3.2 Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree.
- 9.2.3.3 Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. For example: 9x6 – x4 = (3x3 – x2)(3x3 +x2).
- 9.2.3.4 Add, subtract, multiply, divide and simplify algebraic fractions.
- 9.2.3.6 Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. For example: 1 1 1 2 Å~ 7 = 22 Å~7 2 =142 = 14 . Rules for computing directly with radicals may also be used: 3 2 Å~ 3 x = 3 2x .
- 9.2.3.7 Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables.
- 9.2.4.1 Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. For example: A diver jumps from a 20 meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits the surface of the water, the resulting quadratic equation has a positive and a negative solution. The negative solution should be discarded because of the context.
- 9.2.4.2 Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations.
- 9.2.4.3 Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from rational numbers to real numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients.
- 9.2.4.4 Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which parts of the boundary are included in and excluded from the solution set using solid and dotted lines.
- 9.2.4.6 Represent relationships in various contexts using absolute value inequalities in two variables; solve them graphically. For example: If a pipe is to be cut to a length of 5 meters accurate to within a tenth of its diameter, the relationship between the length x of the pipe and its diameter y satisfies the inequality | x – 5| ≤ 0.1y.
- 9.2.4.7 Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. For example: The equation x −9 =9 x may be solved by squaring both sides to obtain x – 9 = 81x, which has the solution 980x = − . However, this is not a solution of the original equation, so it is an extraneous solution that should be discarded. The original equation has no solution in this case. Another example: Solve 3 −x+1 =−5 .
- 9.2.4.8 Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context.
- 9.4.1.1 Describe a data set using data displays, including box-and whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median, quartile and percentile. Measures of spread include standard deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics.
- 9.4.1.2 Analyze the effects on summary statistics of changes in data sets. For example: Understand how inserting or deleting a data point may affect the mean and standard deviation. Another example: Understand how the median and interquartile range are affected when the entire data set is transformed by adding a constant to each data value or multiplying each data value by a constant.
- 9.4.1.3 Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions.
- 9.4.1.4 Use the mean and standard deviation of a data set to fit it to a normal distribution (bell-shaped curve) and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve. For example: After performing several measurements of some attribute of an irregular physical object, it is appropriate to fit the data to a normal distribution and draw conclusions about measurement error. Another example: When data involving two very different populations is combined, the resulting histogram may show two distinct peaks, and fitting the data to a normal distribution is not appropriate.
- 9.4.2.2. Identify and explain misleading uses of data; recognize when arguments based on data confuse correlation and causation.

## Geometry

- 9.3.1.1 Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. For example: Measure the height and radius of a cone and then use a formula to find its volume.
- 9.3.1.2 Compose and decompose two- and three- dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. For example: Find the volume of a regular hexagonal prism by decomposing it into six equal triangular prisms.
- 9.3.1.3 Understand that quantities associated with physical measurements must be assigned units; apply such units correctly in expressions, equations and problem solutions that involve measurements; and convert between measurement systems. For example: 60 miles/hour = 60 miles/hour °ø 5280 feet/mile °ø 1 hour/3600 seconds = 88 feet/second.
- 9.3.1.4 Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k 3 , respectively.
- 9.3.1.5 Make reasonable estimates and judgments about the accuracy of values resulting from calculations involving measurements. ACT (G 504) Recognize that real-world measurements are typically imprecise and that an appropriate level of precision is related to the measuring device and procedure For example: Suppose the sides of a rectangle are measured to the nearest tenth of a centimeter at 2.6 cm and 9.8 cm. Because of measurement errors, the width could be as small as 2.55 cm or as large as 2.65 cm, with similar errors for the height. These errors affect calculations. For instance, the actual area of the rectangle could be smaller than 25 cm2 or larger than 26 cm2, even though 2.6 °ø 9.8 = 25.48.
- 9.3.2.1 Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments.
- 9.3.2.2 Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. For example: The statement "If you don't do your homework, you can't go to the dance" is not logically equivalent to its inverse "If you do your homework, you can go to the dance."
- 9.3.2.3 Assess the validity of a logical argument and give counterexamples to disprove a statement.
- 9.3.2.4 Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. For example: Prove that the sum of the interior angles of a pentagon is 540˚ using the fact that the sum of the interior angles of a triangle is 180˚.
- 9.3.2.5 Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets.
- 9.3.3.1 Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. For example: Prove that the perpendicular bisector of a line segment is the set of all points equidistant from the two endpoints, and use this fact to solve problems and justify other results.
- 9.3.3.2 Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. For example: Prove that two triangles formed by a pair of intersecting lines and a pair of parallel lines (an "X" trapped between two parallel lines) are similar.
- 9.3.3.3 Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. For example: Use the triangle inequality to prove that the perimeter of a quadrilateral is larger than the sum of the lengths of its diagonals.
- 9.3.3.4 Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. For example: When building a wooden frame that is supposed to have a square corner, ensure that the corner is square by measuring lengths near the corner and applying the Pythagorean Theorem.
- 9.3.3.5 Know and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems and logically justify results. For example: Use 30-60-90 triangles to analyze geometric figures involving equilateral triangles and hexagons. Another example: Determine exact values of the trigonometric ratios in these special triangles using relationships among the side lengths.
- 9.3.3.6 Know and apply properties of congruent and similar figures to solve problems and logically justify results. For example: Analyze lengths and areas in a figure formed by drawing a line segment from one side of a triangle to a second side, parallel to the third side. Another example: Determine the height of a pine tree by comparing the length of its shadow to the length of the shadow of a person of known height. Another example: When attempting to build two identical 4-sided frames, a person measured the lengths of corresponding sides and found that they matched. Can the person conclude that the shapes of the frames are congruent?
- 9.3.3.7 Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. For example: Recognize that a rectangle is a special case of a trapezoid. Another example: Give a concise and clear definition of a kite.
- 9.3.3.8 Know and apply properties of a circle to solve problems and logically justify results. For example: Show that opposite angles of a quadrilateral inscribed in a circle are supplementary.
- 9.3.4.1 Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle.
- 9.3.4.2 Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. For example: Find the area of a triangle, given the measure of one of its acute angles and the lengths of the two sides that form that angle.
- 9.3.4.3 Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts.
- 9.3.4.4 Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments.
- 9.3.4.5 Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations.
- 9.3.4.6 Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90˚, to solve problems involving figures on a coordinate grid. For example: If the point (3,-2) is rotated 90˚ counterclockwise about the origin, it becomes the point (2, 3).
- 9.3.4.7 Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure.

## Algebra II

- 9.2.1.1 Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain.
- 9.2.1.2 Distinguish between functions and other relations defined symbolically, graphically or in tabular form.
- 9.2.1.3 Find the domain of a function defined symbolically, graphically or in a real-world context. For example: The formula f (x) = πx2 can represent a function whose domain is all real numbers, but in the context of the area of a circle, the domain would be restricted to positive x.
- 9.2.1.5 Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form f (x) = a(x – h)2 + k , or in factored form.
- 9.2.1.6 Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function.
- 9.2.1.7 Understand the concept of an asymptote and identify asymptotes for exponential functions and reciprocals of linear functions, using symbolic and graphical methods.
- 9.2.1.8 Make qualitative statements about the rate of change of a function, based on its graph or table of values. For example: The function f(x) = 3x increases for all x, but it increases faster.
- 9.2.1.9 Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. For example: Determine how the graph of f(x) = |x – h| + k changes as h and k change.
- 9.2.2.1 Represent and solve problems in various contexts using linear and quadratic functions. For example: Write a function that represents the area of a rectangular garden that can be surrounded with 32 feet of fencing, and use the function to determine the possible dimensions of such a garden if the area must be at least 50 square feet.
- 9.2.2.2 Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth.
- 9.2.2.3 Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions.
- 9.2.2.4 Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively. For example: A closed form formula for the terms tn in the geometric sequence 3, 6, 12, 24, ... is tn = 3(2)n-1, where n = 1, 2, 3, ... , and this sequence can be expressed recursively by writing t1 = 3 and tn = 2tn-1, for n ≥ 2. Another example: The partial sums sn of the series 3 + 6 + 12 + 24 + ... can be expressed recursively by writing s1 = 3 and sn = 3 + 2sn-1, for n ≥ 2.
- 9.2.2.5 Recognize and solve problems that can be modeled using finite geometric sequences and series, such as home mortgage and other compound interest examples. Know how to use spreadsheets and calculators to explore geometric sequences and series in various contexts.
- 9.2.2.6 Sketch the graphs of common non-linear functions such as f ( x)= x , f ( x)= x , ( ) 1x f x = , f (x) = x3, and translations of these functions, such as f (x)= x−2 + 4 Know how to use graphing technology to graph these functions.
- 9.2.3.1. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains.
- 9.2.3.2 Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree.
- 9.2.3.3 Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. For example: 9x6 – x4 = (3x3 – x2)(3x3 + x2).
- 9.2.3.4 Add, subtract, multiply, divide and simplify algebraic fractions.
- 9.2.3.5 Check whether a given complex number is a solution of a quadratic equation by substituting it for the variable and evaluating the expression, using arithmetic with complex numbers.
- 9.2.3.6 Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions, including those involving nth roots. For example: 1 1 1 2 Å~ 7 = 22 Å~7 2 =142 = 14 . Rules for computing directly with radicals may also be used: 3 2 Å~ 3 x = 3 2x .
- 9.2.3.7 Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables.
- 9.2.4.1 Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. For example: A diver jumps from a 20 meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits the surface of the water, the resulting quadratic equation has a positive and a negative solution. The negative solution should be discarded because of the context
- 9.2.4.2 Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations.
- 9.2.4.3 Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from rational numbers to real numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients.
- 9.2.4.4 Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which parts of the boundary are included in and excluded from the solution set using solid and dotted lines.
- 9.2.4.5 Solve linear programming problems in two variables using graphical methods.
- 9.2.4.6 Represent relationships in various contexts using absolute value inequalities in two variables; solve them graphically. For example: If a pipe is to be cut to a length of 5 meters accurate to within a tenth of its diameter, the relationship between the length x of the pipe and its diameter y satisfies the inequality | x – 5| ≤ 0.1y.
- 9.2.4.7 Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods. For example: The equation x −9 =9 x may be solved by squaring both sides to obtain x – 9 = 81x, which has the solution 980x = − . However, this is not a solution of the original equation, so it is an extraneous solution that should be discarded. The original equation has no solution in this case. Another example: Solve 3 −x+1 =−5.
- 9.2.4.8 Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context.
- 9.4.1.1 Describe a data set using data displays, including box-and whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median.
- 9.4.1.4 Use the mean and standard deviation of a data set to fit it to a normal distribution (bell-shaped curve) and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve. For example: After performing several measurements of some attribute of an irregular physical object, it is appropriate to fit the data to a normal distribution and draw conclusions about measurement error. Another example: When data involving two very different populations is combined, the resulting histogram may show two distinct peaks, and fitting the data to a normal distribution is not appropriate.
- 9.4.2.1 Evaluate reports based on data published in the media by identifying the source of the data, the design of the study, and the way the data are analyzed and displayed. Show how graphs and data can be distorted to support different points of view. Know how to use spreadsheet tables and graphs or graphing technology to recognize and analyze distortions in data displays. For example: Displaying only part of a vertical axis can make differences in data appear deceptively large.
- 9.4.3.1 Select and apply counting procedures, such as the multiplication and addition principles and tree diagrams, to determine the size of a sample space (the number of possible outcomes) and to calculate probabilities. For example: If one girl and one boy are picked at random from a class with 20 girls and 15 boys, there are 20 °ø 15 = 300 different possibilities, so the probability that a particular girl is chosen together with a particular boy is 1 300.
- 9.4.3.5 Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. For example: The probability of tossing at least one head when flipping a fair coin three times can be calculated by looking at the complement of this event (flipping three tails in a row).
- 9.4.3.7 Understand and use simple probability formulas involving intersections, unions and complements of events. For example: If the probability of an event is p, then the probability of the complement of an event is 1 – p; the probability of the intersection of two independent events is the product of their probabilities. Another example: The probability of the union of two events equals the sum of the probabilities of the two individual events minus the probability of the intersection of the events.
- Solve exponential equations using logarithms.
- Solve triangles and trigonometric equations.
- Solve triangles using SOCAHTOA, law of sines, law of cosines.