Marquette Area Public Schools

MARQUETTE AREA PUBLIC SCHOOLS 9-12 Math Curriculum Revised February 17, 2004

MARQUETTE AREA PUBLIC SCHOOLS HIGH SCHOOL MATHEMATICS CURRICULUM

The Marquette Senior High School Math Curriculum was developed utilizing current educational research, past practices, the Michigan Curriculum Framework, and the National Council of Teachers of Mathematics Principles and Standards for School Mathematics, 2000.

Strand I. Patterns, Relationships, and Functions Content Standard 1: Students recognize similarities and generalize patterns, use patterns to create models and make predictions, describe the nature of patterns and relationships, and construct representations of mathematical relationships. (Patterns)

PASSPORT MATH ALGEBRA GEOMETRY ALGEBRA II PRECALCULUS Aa =Algebra 1A IG = Informal Geometry A2 = Algebra II HP = Honors PreCalculus Ab =Algebra 1B G = Geometry HG = Honors Geometry + = Honors and Regular A1 = Algebra 1 HG = Honors Geometry + = All above + = All above + = All above 1. Describe, analyze and generalize patterns arising in a variety of contexts and express them in general terms.

+ Create tables from formulas + Use the properties of similar A2 Write numbers in scientific + Students graph functions Create tables from formulas. figures to find missing notation. (polynomial, rational, inverse, + Use exponents to express measures exponential, logarithmic, Write formulas from tables. numbers written as a product + Explore factoring patterns trigonometric, and polar) first of factors + Find the sum of the interior ax2+bx+c by making tables, and then by Test numbers for divisibility by angles of a convex polygon using symmetry and 2, 3, 4, 5, 6, 8, 9, 10. + Write expressions containing + Convert fractional exponents transformations based on the exponents in standard form G,HG Find the degree of a to radical form. parent function for each type. Write numbers in scientific node in a network notation in positive and + Write numbers in scientific + Students are shown how to negative exponents. notation using both positive IG,G Simplify radical use patterns to derive new and negative exponents expressions formulas and equations which, Use exponents to express in turn, are used to derive numbers written as a product + Test numbers for divisibility G Create truth tables from other formulas and equations. of factors. by 2,3,4,5,6,8,9,10 given statements HP Students use patterns of + Categorize numbers into counting to develop formulas

number systems for permutations and combinations.

+ Write a number as a product of its primes

Aa Identify similar figures

+ Recognize and apply algebraic properties in simplifying expressions and solving equations

+ Explore factoring patterns for various polynomial expressions

+ Recognize perfect square and perfect cube roots 2. Represent and record patterns in a variety of ways including; tables, charts and graphs, and translate between various representations.

Create read and interpret bar Aa, Ab Create, read and + Create a table for patterns A2 Create read and interpret + Students graph functions graphs, frequency tables, line interprets frequency tables, of convex polygons interior bar graphs, frequency tables, (polynomial, rational, inverse, plots, stem-and-leaf plots. bar graphs, line plots, line and exterior angles line plots, stem-and-leaf plots. exponential, logarithmic, graphs, stem and leaf plots, trigonometric, and polar) first and scatter plots. IG Make a table of the A2,HG Identify and distinguish by making tables, and then by dimensions of a rectangle to between arithmetic and using symmetry and + Solve and graph linear predict area. geometric sequences and transformations based on the equations/inequalities series. parent function for each type. G,HG Create, read, and Ab, A1 Solve and graph interpret polygons as networks HP Students use patterns of quadratic equations counting to develop formulas G Create truth tables from for permutations and + Solve and graph systems of given statements combinations. equations/inequalities

3. Use patterns and their generalizations to make and justify their inferences and predictions.

Calculate the mean, median, Aa Present data using mean, + Identify the quadrant that A2,HG Calculate the mean, + Students graph functions mode, and range. median, mode and range contains a given point median, mode, and range. (polynomial, rational, inverse, exponential, logarithmic, Present data using mean, Aa Calculate the mean, G,HG Recognize edges and + Solve and graph linear trigonometric, and polar) first median and mode. median, mode and range nodes as used in graph equations/inequalities. by making tables, and then by theory. using symmetry and Identify the quadrant that + Use Cartesian Coordinate transformations based on the contains the given point. Plane to justify solution sets G,HG Determine if a network parent function for each type. to various algebraic equations is traceable and/or complete. HP Students use patterns of Aa, Ab Create, read and G Create truth tables from counting to develop formulas interprets frequency tables, given statements to analyze for permutations and bar graphs, line plots, line possible outcomes combinations. graphs, stem and leaf plots, and scatter plots. HP Students use permutations and combinations to find + Solve and graph linear probabilities for use in equations/inequalities inferences and predictions.

Ab, A1 Solve and graph quadratic equations

+ Solve and graph systems of equations/inequalities

4. Explore and describe visual and numeric patterns, including linear expressions, near-linear patterns and symmetric and spatial patterns.

Write an algebraic equation or + Use Cartesian Coordinate + Find measure of interior HG,A2 Determine what comes + Students graph functions expression for a given Plane to justify solution sets angles of regular and non- next in a sequence or series. (polynomial, rational, inverse, situation. to various algebraic equations regular polygons exponential, logarithmic, + Write an algebraic equation trigonometric, and polar) first Match graphs to number Aa Determine what comes + Set up correspondence or expression for a given by making tables, and then by stories. next in a sequence or series between congruent figures situation using symmetry and based on reflections, transformations based on the translations, and rotations. parent function for each type. Replicate and continue a + Write an algebraic equation + Recognize the effect of + Solve and graph systems of visual pattern. or expression for a given changing the y-intercept in a equations/inequalities situation linear equation. + Solve and graph higher + Solve and graph linear G,HG Recognize and use the degree equations. equations/inequalities proportional relationships of corresponding perimeters, + Solve and graph systems of altitudes, angle bisectors, and equations/inequalities medians of similar triangles.

Aa Determine if a scatter plot suggests a trend

5. Use patterns and generalizations to solve problems and explore new content.

Use times tables to develop + Show functional + Generate equations by + Show functional + Students solve a variety of patterns. relationships such as: interpreting a pattern. relationships such as: equations (polynomial, -follow rules to complete table Show functional relationships -follow rules to complete table rational, exponential, Use common factors to learn -graph ordered pairs by: -graph ordered pairs logarithmic, and trigonometric) how to add fractions. -apply formulas -following rules to complete -apply formulas using a variety of techniques. tables Learn to recognize and + Create and solve equations -graphing ordered pairs + Create and solve equations + Students set up and solve represent a repeating decimal. based on various word -applying formulas to find the based on various word application problems related to problems area, surface area and volume problems the various functions covered of cylinders, pyramids, and (linear, quadratic, exponential, + Use factoring patterns to prisms + Use factoring patterns to logarithmic, and trigonometric) solve various polynomial solve various polynomial equations equations

A2,HG Use sequences and series to find new terms.

A2 Use patterns to convert repeating decimals to fractions.

Strand I. Patterns, Relationships and Functions Content Standard 2: Students describe the relationships among variables, predict what will happen to one variable as another variable is changed, analyze natural variation and sources of variability, and compare patterns of change. (Variability and Change)

PASSPORT MATH ALGEBRA GEOMETRY ALGEBRA II PRECALCULUS Aa =Algebra 1A IG = Informal Geometry A2 = Algebra II HP = Honors PreCalculus Ab =Algebra 1B G = Geometry HG = Honors Geometry + = Honors and Regular A1 = Algebra 1 HG = Honors Geometry + = All above + = All above + = All above

1. Identify and describe both complex and abstract situations, explore different kinds of change and patterns.

Give a range of reasonable + Give a range of reasonable A2,HG Explore relationship values for an equation. values for an equation between variables in cubic, exponential, absolute value, radical and rational equations. Describe a data set from its + Describe a data set from its + Determine the mathematical + Give a range of reasonable + Students identify the change graph. graph relationships between the values for an equation made to the parent function for formulas for volume and polynomial, rational, + Explore relationship surface area of 3 dimensional + Explore relationship exponential, logarithmic, and between variables in linear figures(ex: prisms, pyramids, between variables in linear trigonometric functions and and quadratic equations cones) and quadratic equations then use these changes to alter the graph of the parent A1 Explore relationship IG,G Discuss and work with function accordingly. between variables in cubic, the changes to a geometric exponential, absolute value, figure through + Students use past algebraic radical and rational equations Transformational Geometry concepts to explore and develop new formulas and Aa Determine the equations. mathematical relationships between the formulas for volume and surface area of 3 dimensional figures(ex: prisms, pyramids, cones)

2. Connect an initial state to a final state and generalize a rule that describes a pattern of change.

Follow order of operations to + Use roster, rule, graph to + Recognize the pattern A2 Examine relationship + Students solve problems by solve simple linear equations. specify number sets between the given information between input and output old techniques and are then and the prove statement in a values using function notation. shown more efficient ways of Aa Describe the relationship two-column proof. solving these problems by between experimental and analyzing the patterns used. theoretical probability IG,G Discuss and work with the changes to a geometric figure through Transformational Geometry

3. Changes are frequently interdependent (bivariate and linear relationships); exploring what will happen to one quantity if another variable is changed.

State the probability of a given + Describe how one variable + State the coordinate(s) that A2,HG Describe how the + Students identify the change situation with or without will change in relation to change when a reflection (flip), values for a, b and c in a conic made to the parent function for replacement. another variable rotation (turn) or translation section affect its graph. polynomial, rational, (slide) occurs. exponential, logarithmic, and Aa Predict and assess the trigonometric functions and effects of an outlier on a data + Identify which coordinate(s) then use these changes to set will change when a reflection alter the graph of the parent (flip), rotation (turn) or function accordingly. + Use words, tables and translation (slide) occurs. graphs to analyze and describe patterns and functions

Aa State the probability of a given situation with and without replacement

+ Explore relationship between variables in linear and quadratic equations A1 Explore relationship between variables in cubic, exponential, absolute value, radical and rational equations

4. Represent variability or change by ordered pairs, tables, graphs and equations.

Use equations and proportions + Recognize and apply + Identify the effect a + Recognize and apply + Students graph functions in to solve problems involving algebraic properties in transformation will have on an algebraic properties in the Cartesian coordinate plane percent. simplifying expressions and ordered pair. simplifying expressions and and the polar plane by making solving equations solving equations tables of ordered pairs and by Write equations and + Write equations and using equations in general expressions to represent + Use equations and expressions to represent + Represent relationship form. relationships involving time, proportions to solve problems relationships in similar figures. between variables in linear money, rate, and distance. involving percent and quadratic equations with + Students work with complex tables and graphs numbers graphed in the Transform equations using Aa Use formulas to solve for complex plane. multiplication, division, changes in Surface Area, + Represent relationship addition and subtraction. Volume, Perimeter and Area between variables in cubic, . when one dimension of a exponential, absolute value, given shape is changed radical and rational equations with tables and graphs. + Represent relationship between variables in linear and quadratic equations with tables and graphs A1 Represent relationship between variables in cubic, exponential, absolute value, radical and rational equations by solving techniques

5. Differentiate between functions and relationships such as linear vs. non linear and continuous vs. non continuous.

Describe a graph from its + Differentiate between linear IG Differentiate between a A2 Determine if a graph is a + Students graph, solve, and given rule or equation. and non linear relationships linear function and a quadratic function or relation. manipulate linear, polynomial, function. rational, inverse, exponential, + Describe a graph from its + Differentiate between linear logarithmic, trigonometric, and given rule or equation. and non linear relationships polar equations.

+ Describe a graph from its given rule or equation.

6. Explore relationships and use variables and relationships to solve mathematical problems.

Evaluate and simplify + Evaluate and simplify + Read, write and evaluate + Evaluate and simplify + Students solve linear, algebraic expressions. algebraic expressions expressions and equations. algebraic expressions polynomial, rational, inverse, exponential, logarithmic, Use a model and an equation + Write and solve algebraic + Use Pythagorean Theorem +Set up and solve various trigonometric, and polar involving positive and negative equations using a variety of to find the missing sides of a application problems equations. integers. formulas and techniques right triangle. + Students solve systems + Set up and solve various IG,G Use basic trigonometry equations of many types. application problems to find missing angles and sides of triangles. Use their knowledge of + Use the midpoint, distance + Students set up and solve a variability and change to solve formulas. variety of application problems and make problems. predictions. Use Pythagorean Theorem to find the missing sides of a right triangle.

Strand II. Geometry and Measurement Content Standard 1: Students develop spatial sense, use shape as an analytic and descriptive tool, identify characteristics and define shapes, identify properties and describe relationships among shapes. (Shape and Shape Relationships)

1. Spatial sense relies on the ability to recognize and describe shape.

Classify polygons by the + Recognize linear and + Classify polygons by the A2,HG Classify conic sections + Students utilize a graph to number of sides using the quadratic equations from their number of sides using the from their equations. find the equation of the prefixes of tri, quad, pent, hex, graphs prefixes of tri, quad, pent, hex, function it represents. oct, and deca. All others oct, and deca. All others classified as n-gons. Aa Classify polygons by the classified as n-gons. HP Students classify, graph, number of sides using the and manipulate the equation prefixes of tri, quad, pent, hex, + Classify quadrilaterals based of the conic sections. oct, and deca. All others on their properties of parallel classified as n-gons. and perpendicular sides. + Students use the basic geometry formulas for areas Aa Classify quadrilaterals + Classify triangles by their and lengths in many situations based on their properties of sides and angles (mostly the triangle and parallel and perpendicular quadrilateral formulas). sides + Classify 3-d objects as prisms, pyramids, or other. Aa Classify triangles by their sides and angles + Identify polygons as convex or concave. Aa Classify 3-d objects as prisms, pyramids, or other IG,G Discuss and work with the changes to a geometric figure through Transformational Geometry

2. Generalize the characteristics of shapes and apply their generalizations to classes of shapes.

Classify angles as acute, right, + Recognize linear and + Determine if a polygon is A2, HG Classify conic + Students work with the straight, or obtuse. quadratic equations from their regular sections from their equations. parent functions of many basic graphs functions and how these are + Find the area of irregular then transformed by changing Aa Determine if a polygon is figures the parent function. regular Aa Create nets to find surface + Identify the faces, edges, + Students do extensive work area of 3d figures and vertices of a polyhedron with polynomial equations and how the graphs relate to the Aa Identify the faces, edges, + Classify angles as acute, many characteristics of the and vertices of a polyhedron right, straight, obtuse equation itself.

Aa Classify angles as acute, + Classify angle pairs as HP Students classify, graph, right, straight, obtuse adjacent, vertical, and manipulate the equations corresponding, linear pairs, of the conic sections. Aa Classify angle pairs as supplementary, adjacent, vertical, complementary, alternate supplementary, interior, alternate exterior, and complementary consecutive interior.

IG,G Discuss and work with the changes to a geometric figure through Transformational Geometry

3. Derive generalizations about shapes and apply those generalizations to develop classifications of familiar shapes.

Determine if angles are Aa Use classifications and + Use classifications and A2,HG Classify conic sections + Students work with the congruent. properties of triangles and properties of triangles and from their equations. parent functions of many basic quadrilaterals to find missing quadrilaterals to find missing functions and how these are measures measures then transformed by changing the parent function. Aa Determine if shapes are + Determine if shapes are congruent, similar or neither congruent, similar or neither + Students do extensive work with polynomial equations and + Determine if shapes are how the graphs relate to the convex or concave. many characteristics of the equation itself. IG,G Discuss and work with the changes to a geometric HP Students classify, graph, figure through and manipulate the equations Transformational Geometry of the conic sections.

4. Construct familiar shapes using coordinates, appropriate tools, sketching and drawing two and three-dimensional shapes.

Use a protractor to measure Aa Use a protractor to + Construct using a compass + Using the coordinates, we + Students are to come up angles within five degrees of measure angles within a 5 a figure based upon its construct graphs for conic with the equation of many of accuracy. degree accuracy description or classification. sections. the functions studied from the Example: Draw a 4 sided points given in the Cartesian Draw 2D and 3D shapes using Aa Draw 2D and 3D shapes figure with opposite sides Coordinate Plane. a straight edge. parallel and all angles 90o (square) + Students graph all the equations covered to develop + Use a protractor to measure a better understanding of how angles within a 5 degree it all relates to other concepts accuracy being covered.

+ Use a compass to bisect and construct an angle and a line segment IG Create 3d models of a cylinder, prism and pyramid from 2d nets

IG,G Discuss and work with the changes to a geometric figure through Transformational Geometry

5. Combine, dissect and transform shapes.

Combine 2 rays to make an Aa Use a scale factor to + Transform a given figure A2,HG Write an equation for a + Students work with the angle. create a new shape from a through reflection, rotation or given conic section that has parent functions of many basic given shape translation. undergone a transformation. functions and how these are Combine segments to make then transformed by changing convex polygons. + Determine whether a figure + Determine the axis of the parent function. has point, line or rotational symmetry for a parabola. Use a scale factor to create a symmetry + Students do extensive work new shape from a given shape with polynomial equations and + Subdivide, combine and how the graphs relate to the transform shapes to form new many characteristics of the shapes equation itself.

HP Students classify, graph, and manipulate the equations of the conic sections.

6. Informally verify generalizations about the common properties of similar, congruent, parallel and perpendicular shapes.

Identify lines as parallel, Aa Determine if given shapes + Determine if given shapes N/A N/A perpendicular, or neither are congruent, similar or are congruent, similar or neither neither Determine if given shapes are congruent, similar or neither + Identify lines as parallel, + Identify lines as parallel, perpendicular, or neither perpendicular, skew, transversal or neither.

+ Identify if a figure is a given shape based on its characteristics.

7. Use shape, shape properties, and shape relationships to describe the physical world and solve problems.

Find the circumference and + Use classifications of + Use classifications of + Use linear and quadratic + Students take the area of circles and polygons triangles and quadrilaterals to triangles and quadrilaterals to equations to simulate real- information they have learned find missing measures find missing measures. world situations. about the functions covered Classify triangles by angle and apply this knowledge to measures and by their sides. Aa Find the circumference + Find the circumference and application problems. and area of circles and area of circles and polygons. polygons + Find the area of irregular Aa Find the area of irregular figures. figures G Use angle of elevation and angle of depression to solve problems.

G Use Law of Sines, Law of Cosines to solve problems.

G Use vectors to represent magnitude and direction.

Strand II. Geometry and Measurement Content Standard 2: Students identify locations of objects, identify location relative to other objects, and describe the effects of transformations (e.g., sliding, flipping, turning, enlarging, reducing) on an object. (Position)

PASSPORT MATH ALGEBRA GEOMETRY ALGEBRA II PRECALCULUS Aa = Algebra 1A IG = Informal Geometry A2 = Algebra II HP = Honors PreCalculus Ab = Algebra 1B G = Geometry HG = Honors Geometry + = Honors and Regular A1 = Algebra 1 HG = Honors Geometry + = All above + = All above + = All above

1. Locate and describe objects in terms of their position, including compass directions, cartesian coordinates, latitude, longitude and midpoints.

Graph figures using Cartesian + Extensive use of the + Plot points on a Cartesian + Graph a variety of functions HP Students locate the Coordinates. Cartesian Coordinate Plane plane in the coordinate plane. various parts of each of the including formulas relating to conic sections (foci, vertices, Locate the latitude and this system + Identify the quadrant that directrix, etc.) both from the longitude of specific locations. contains a given point equations and the graphs.

+ Identify the midpoint of a line + Students work bearing segment, the number problems involving line(zero) and the coordinate trigonometry. plane (origin)

2. Locate and describe objects in terms of their orientation and relative position, including symmetry.

Label quadrants in the + Extensive use of the + Label quadrants in the + Explore symmetry relating to HP Students find and graph coordinate plane Cartesian Coordinate Plane coordinate plane parabolas lines of symmetry with all of including formulas relating to the conic sections. this system + Describe the coordinates of A2,HG Graph conic sections an object after a on the Cartesian coordinate HP Students graph the conic transformation plane. sections on a rotated grid based on equation G Describe an object’s vector manipulation position + Students work with graphs of equations and their symmetry to the x-axis, y-axis, and origin. This leads to the concepts of even vs. odd functions. 3. Describe translations, reflections, rotations and dilations using the language of transformations and employ transformations to verify congruence of figures.

N/A + Explore translations relating + Translate, reflect, or rotate A2,HG Write an equation for a + Students work with the to lines to determine the congruence given conic section that has parents of the functions of objects. undergone a transformation. covered and how the changes in these equations bring about the translations of the graphs.

4. Locate the position of points or objects described by two or more conditions; locate all of the points that satisfy a given condition.

Use latitude and longitude to + In depth study of linear + Determine if a point lies on a + Graph linear equations and HP Students find the find a given location. equations and their graphs in given plane inequalities. equations of the conic sections the Cartesian Coordinate based on given conditions. Find a value in a graph. Plane + Determine the signs of a + Graph systems of equations number given its quadrants and systems of inequalities. + Students find the equation of Identify all the values in a Aa Use a scatter plot to polynomial functions based on graph that meet a given predict information or + Use the distance and + Locate the foci, focus, given conditions. condition. determine if information if midpoint formulas to locate vertex, or vertices, or the points. center of a given conic Determine the signs of a section. number given its quadrant. G Use a scatter plot to predict information or determine if information fits

+ Locate all the points that satisfy a given condition

5. Concepts of position, direction and orientation to describe the physical world and to solve problems.

Recognize a proportion. + Use proportions to find IG Determine information A2,HG Apply quadratic + Students learn about vectors missing measures necessary to create a scale functions to real world to apply to physical situations. Determine needed or drawing. applications, such as necessary information. + Apply indirect measurement trajectory problems. HP Students use the to solve problems in a real information learned about the Use geometric vocabulary to world setting IG,G Use indirect conic sections to solve describe one object in relation measurement (proportions) to application problems. to another. Aa Determine information find missing measures. necessary to create a scale Apply indirect measurement to drawing. find missing measures. + Use geometric vocabulary Aa Determine appropriate (parallel, perpendicular, scale to fit an object within complementary, etc.) to given parameters describe one object in relation to another. + Use geometric vocabulary (parallel, perpendicular, complementary, etc.) to describe one object in relation to another Strand II. Geometry and Measurement Content Standard 3: Students compare attributes of two objects, or of one object with a standard (unit), and analyze situations to determine what measurement(s) should be made and to what level of precision. (Measurement)

1. Select and use appropriate tools; measure objects using standard units in both the metric and common systems and measure angles in degrees

Use non-standard, metric, and + Students compare attributes IG Use non-standard, metric, + Students compare attributes + Students learn to measure standard units to estimate, of two objects, or of one object and standard units to of two objects, or of one object angles in both degrees and measure, and compare: with a standard (unit), and estimate, measure, and with a standard (unit), and radians as it relates to -length analyze situations to compare: analyze situations to trigonometry. -weight determine what -length determine what -volume measurement(s) should be -weight measurement(s) should be -quantity made and to what level of -volume made and to what level of -area precision -quantity precision -perimeter -area -time -perimeter -degrees(angles) IG -time -degrees(angles)

+ Students compare attributes of two objects or of one object with a standard (unit), and analyze situations to determine what measurement(s) should be made and to what level of precision.

2. Identify the attribute to be measured and select the appropriate unit of measurement for length, mass, time, temperature, perimeter, area, volume and angle.

Select the most appropriate + Select the most appropriate + Select the most appropriate A2 Convert portions of + Students do various unit of measurement for a unit of measurement for a unit of measurement for a degrees into minutes. application problems in which given item given item given item units are used and answers must reflect the appropriate + Determine if a question calls + Determine if a question calls unit measure. Select the most appropriate for the object’s perimeter, for the objects perimeter or unit of measurement for a area, or volume area + Select the most given item appropriate unit of + Carry out unit conversions measurement for a given item Carry out unit conversions within a measurement system within a measurement system (ex. Millimeters to meters, IG Carry out unit conversions inches to feet, seconds to within a measurement system Use a proportion to convert minutes) (ex. Millimeters to meters, from one unit of measurement inches to feet, seconds to to another + Use a proportion to convert minutes) from one unit of measurement to another + Use a proportion to convert from one unit of measurement to another 3. Estimate measures with a specified degree of accuracy and decide if an estimate or a measurement is “close enough”.

Determine if estimates are + Determine if estimates are + Estimate an angles measure A2,HG Approximate radicals N/A reasonable reasonable within five degrees for graphing purposes. Estimate an angle measure within five degrees Aa Estimate an angle IG Use a grid sheet to measure within five degrees estimate the area of a polygon

4. Interpret measurements and recognize that two objects may have the same measurement on one attribute but not necessarily on the other.

Explore objects that have the N/A IG Explore objects that have A2,HG Two conic sections N/A same perimeter but not the the same perimeter but not the could have the same center, same area and reverse same area and reverse (The but have different shapes. Pig Pen problem)

IG Create 3d objects with the same surface area but different shape or volume and reverse.

G Interpret measurements and recognize that two objects may have the same measurement on one attribute but not necessarily on the other

5. Use proportional reasoning and indirect measurements to draw inferences.

Use proportions to solve real + Use proportions to solve real HG Compare two quantities A2 Use trig functions to solve + Students use proportional world problems world problems. with different units of measure real world problems. and inverse relationships to (example: comparing an arc’s set up and solve various Find a unit rate Aa Use maps and scale length in degrees and cm) + Use direct, indirect, and joint application problems. drawings to find actual variation to solve for a Recognize proportional measures + Recognize proportional specified variable. relationships in similar figures relationships in similar figures

Compare 2 quantities with G,IG Find a unit rate different units of measure.

6. Apply measurement to describe the real world and to solve problems.

Use an integer to represent a + Compare two quantities with IG,G Use proportions to solve A2,HG Apply quadratic + Students do various given situation. different units of measure real world problems. functions to real world application problems in which applications, such as units are used and answers Use maps and scale drawings + Recognize proportional + Find the scale factor given trajectory problems. must reflect the appropriate to find actual measures relationships the area and perimeter ratios unit measure.

Find the scale factor given the + Find a unit rate IG Use maps and scale area and perimeter ratios. drawings to find actual measures

Use proportions to solve real IG Discover how angle world problems measures can help you in billiards and golf. Strand III. Data Analysis and Statistics Content Standard 1: Students collect and explore data, organize data into a useful form, and develop skill in representing and reading data displayed in different formats. (Collection, Organization and Presentation of Data)

1. Collect and explore data through observation, measurement, surveys, sampling techniques and simulations.

Explore data through pie Aa, Ab Collect and record IG Collect and record data A2,HG Explore data through N/A charts, line plots, and bar data from an experiment or through surveys to be used in pie charts, line plots, box-and- graphs. survey the creation of a circle graph. whisker plots.

Collect and record data from + Recognize data through an experiment or survey observation of a figure for completing a proof.

2. Organize data using tables, charts, graphs, spreadsheets and data bases.

Organize data through pie Aa, Ab Independently IG Organize gathered A2,HG Organize data through N/A charts, line plots, and bar organize gathered information information and present data pie charts, line plots, box-and- graphs. and present data in a variety in a variety of forms such as whisker plots. of forms such as equations, equations, graphs, tables, Collect and record data from graphs, tables, charts, and charts, spreadsheets, and an experiment or survey. plots plots

+ Organize geometry definitions, postulates, theorems, and corollaries in a two column proof.

3. Present data using a variety of appropriate representations and explain why one representation is preferred over another or how a particular representation may bias the presentation.

Compare different Aa, Ab Compare different G,HG Determine whether a A2,HG Determine which N/A representations of the same representations of the same given problem would be measure of central tendency data (i.e. data in chart form data (i.e. data in chart form better represented by a direct (mean, median, mode and versus graph form, bar graph versus graph form, bar graph or indirect proof. range) best represents the vs. line graph). vs. line graph). data

Determine which measure of Aa Determine which measure A2,HG Compare different central tendency (mean, of central tendency (mean, representations of the same median, mode and range) best median, mode and range) best data (i.e. data in chart form represents the data. represents the data. versus graph form, bar graph vs. line graph). Aa, Ab Determine which type of graph (line, circle, bar) best fits a given data set.

4. Identify what data are needed to answer a particular question or solve a given problem, and design and implement strategies to obtain, organize and present those data.

Choose the right Aa Design investigations to + To determine the measures + Choose the right N/A organizational tool to address a question and of angles formed by chords, organizational tool to represent the given data. consider how data collection tangents, and secants. represent the given data. methods affect the nature of Determine if a graph is the data set. + To apply theorems, misleading. postulates, or corollaries to + Determine a graph’s scale solve a given problem or apply Identify why a graph is and interval to a given proof. misleading. Aa, Ab Determine if a graph is Identify unnecessary and misleading, why it may be needed information in a misleading, and how to make problem. it more realistic

Strand III. Data Analysis and Statistics Content Standard 2: Students examine data and describe characteristics of the distribution, relate data to the situation from which they arose, and use data to answer questions convincingly and persuasively. (Description and Interpretation)

1. Critically read data from tables, charts or graphs and explain the source of the data and what the data represent.

Identify the highest and lowest Aa Compare information from G Determine if a scatter plot N/A A2,HG Identify the mode from pieces of information from bar, different graphs noting shows a positive, negative or reading a line or stem and leaf circle, scatter plots, variances in scale and interval no relationship(correlation) plot. pictographs and line graphs. Aa Identify the mode from + In a given proof, justify the A2,HG Identify outliers in a Compare information from reading a line or stem and leaf data provided by theorems, given set of data. different graphs noting plot postulates, corollaries, and variances in scale and interval definitions. Aa Determine if a scatter plot Identify the mode from reading shows a positive, negative or G Analyze truth tables a line or stem and leaf plot no relationship(correlation)

2. Describe the shape of a data distribution and identify the shape, the spread, correlations and any outliers.

Find the range of a data set Aa Determine if a scatter plot G Identify a prediction A2,HG Identify outliers in a N/A shows a positive, negative or equation for a given scatter given set of data. no relationship(correlation) plot. A2,HG Find the range and Aa Identify a trend line (if any) G Find any outliers in a data interquartile range of a data in a data set set. set

Aa Find any outliers in a data set Aa Find the range of a data set

3. Draw, explain and justify conclusions based on data.

Summarize results of a survey Aa, Ab Summarize results of IG Present survey results A2,HG Draw conclusions on N/A in written form a survey in written form verbally to the class date based on a box-and- whisker plot. Present survey results verbally Aa, Ab Present survey results + Write a formal proof, to the class verbally to the class paragraph, or two-column, based on observed data.

+ Determine the number of sides in a polygon given the sum of the interior angles.

4. Critically question the sources of data; the techniques used to collect, organize and present data; the inferences drawn from the data; and the possible sources of bias in the data or the presentation.

Identify questions and Aa Determine if a graph’s + Justify each statement in a N/A N/A samples containing biased or scale and interval are formal proof with a reason misleading characteristics. appropriate for the data based upon postulates, theorems, definitions, and corollaries.

5. Formulate questions and problems and gather and interpret data to answer those questions.

Formulate questions and Aa, Ab Formulate questions + Formulate questions and A2,HG Formulate questions N/A problems, gather and interpret and problems, gather and problems, gather and interpret and problems, gather and data to answer questions interpret data to answer these data to answer these interpret data to answer these questions questions. questions.

Strand III. Data Analysis and Statistics Content Standard 3: Students draw defensible inferences about unknown outcomes, make predictions, and identify the degree of confidence they have in their predictions. (Inference and Prediction)

1. Make and test hypotheses.

Make predictions based on Aa Make predictions based on + Make geometric conjectures A2,HG Students predict the N/A data data based on inductive reasoning. shape of a conic section.

Compare predictions with + Prove a geometric theorem. actual results. A2,HG Students determine if a conic section opens up or down, or if it goes left/right, or up/down.

A2,HG Students predict the outcome in a probability problems.

2. Design experiments to model and solve problems using sampling, simulations and controlled investigations.

Conduct experiments involving Aa Conduct experiments IG Hands on activities to N/A N/A probability and compare their involving probability and demonstrate triangle results to the theoretical compare their results to the congruence theorems, and probability.(ex. Dice theoretical probability.(ex. triangle inequality theorems. outcomes) Dice outcomes) IG Use the geoboard to show Survey a small group and Aa Survey a small group and the area of a triangle is half compare the results to a larger compare the results to a larger the base times the height. group group IG Use materials to develop area and volume formulas.

G Use networking techniques to solve transportation problems

3. Formulate, and communicate arguments and conclusions based on data and evaluate their arguments and those of others.

Determine the fairness of G,HG Communicate N/A N/A N/A games of chance. conclusion of indirect proofs .

+ Communicate conclusions of proofs through cooperative learning.

4. Make predictions and decisions based on data including interpolations and extrapolations.

Make predictions based on Aa Make predictions based on N/A + Make predictions based on N/A data. data. a few points as to what the shape of a graph will be (ie. Aa Make predictions about Rational functions) future values on a graph A2 Make predictions based on data.

5. Employ investigations, mathematical models and simulations to make inferences and predictions to answer questions and solve problems..

Make inferences from data. Aa Make predictions based on IG Use manipulatives to A2 Make predictions about + Students will predict the data. generate predictions of data based on a stem-and-leaf shape of a graph based only Predict what will come next surface area vs. volume of plot. on the equation. based on data. Aa Make predictions about polyhedra. future values on a graph A2,HG Predict the shape of a IG Predict cost of carpeting graph based on its from room dimensions equation(conics). Strand IV. Number Sense and Numeration Content Standard 1: Students experience counting and measuring activities to develop intuitive sense about numbers, develop understanding about properties of numbers, understand the need for and existence of different sets of numbers, and investigate properties of special numbers. (Concepts and Properties of Numbers)

1. Develop an understanding of integers and rational numbers and represent rational numbers in both fraction and decimal form.

Read, write, & understand + Students develop and use N/A A2 Classify numbers by sets. + Students need to recognize numerals from millions to the real numbers and all their what form of answer is thousandths. subsets throughout the year. + Represent rational numbers acceptable for a situation in in both fraction and decimal which it is being used. Read & write decimals. + Students write answers to form. various problems in both exact Classify a given number as and approximate form. whole, decimal, rational integers. Represent numbers as rational numbers in lowest terms.

Convert between a decimal and a rational number.

Identify fractions as rational numbers.

Convert between a decimal and a fraction.

Simplify, multiply, and divide fractional equations.

2. Extend their understanding of numeration systems to include decimal numeration scientific numeration, and non-decimal numeration systems..

Determine the number of + Students develop and use + Students understand the A2 Identify any number raised + Students work with all the decimal places to move in a the real numbers and all their exact and approximate values to the zero power having the number systems up to and given scientific decimal subsets throughout the year. of pi. value of 1. including the complex notation. numbers. + Students write answers to A2 Multiply and divide in Write numbers from millions to various problems in exact scientific notation. thousandths in expanded form, approximate form, and form. scientific notation. Add, subtract, multiply, and divide real numbers.

3. Develop an understanding of the properties of the integer and rational number systems and of the properties of special numbers including 0, 1 and pi and the additive and multiplicative inverses.

Compare and order from least + Students understand and + Students understand the + Use orders of operations to + Students are expected to to greatest a given set of use all of the basic algebraic exact and approximate values solve problems. know all the basic number numbers in either fractional, properties. of pi. properties and apply them integer, or decimal form. A2 Study the properties of without much thought. A1 Students use the basic + Use pi to find the integers including but not Use orders of operations to algebraic properties to do circumference, area and limited to the additive and solve problems. algebraic proofs. volume of circles/spheres multiplicative inverse.

Recognize various forms of a + Students use pi in the G Identify pi as the ratio A2 Study and use the number’s additive and development and use of between any circles properties of complex multiplicative inverse. formulas. circumference and diameter. numbers.

Use pi to find the + Students are shown how pi + Understand the properties of A2 Study and use the circumference and areas of fits into the real number addition of radicals and pi. properties of radicals. circles. system.

Identify pi as the ratio of any circle’s circumference to diameter.

4. Apply their understanding of number systems to solve mathematical and applied problems.

Solve real world problems + Students must analyze + Solve real world problems A2 Solve real world problems + Students at this level apply involving divisibility, greatest problems to determine the involving area, volume, involving divisibility, greatest their number sense on a daily common factor, and least number system appropriate for surface area and angles. common factor, and least basis. common multiple. the solution set. common multiple.

Solve problems linking + Students understand and + Solve distance problems. + Use factoring and factoring fractions to music, stocks, and use all of the basic algebraic patterns to solve problems. other situations. properties for manipulating + Write addition, subtraction, expressions and solving multiplication, and division A2 Use irrational numbers to Solve cost, income and value equations. equations to represent solve elevation and problems. geometric situations involving depression problems. A1 Students use the basic fractions, integers, and algebraic properties to do rational numbers. algebraic proofs. Strand IV. Number Sense and Numeration Content Standard 2: Students recognize that numbers are used in different ways such as counting, measuring, ordering, and estimating, understand and produce multiple representations of a number, and translate among equivalent representations. (Representation and Uses of Numbers)

1. Give geometric representations of various numbers and number concepts; represent rational numbers and integers on the number line.

Represent fractions using + Students identify and locate G,HG Represent an arc length + Represent irrational + Students use number lines cuisinaire rods. all real numbers and their in degrees numbers as radicals. as tools in solving higher subsets on the number line. degree inequalities. Identify a decimal, integer, or + Represent rational numbers fraction on a number line. + Students explore and integers on a number line. relationships of absolute values of numbers as they relate to the number line.

A1 Students solve absolute value equations and inequalities and relate the solution sets to the number line.

+ Students solve inequalities and relate the solution sets to the number line.

2. Recognize equivalent representations of a number, especially fractions, decimals and percents and translate freely among representations..

Identify equivalent fractions. + Students write a number + Recognize the various A2 Recognize equivalent + Students at this level apply using multiple forms (decimal, representations of pi. matrices. their number sense on a daily Write a number using multiple fraction, integer) basis. forms, as a decimal, fraction, + Represent a radical as an A2 Represent irrational integer. + Students convert between approximate decimal and as a numbers with both decimal fraction, decimal and percent simplified radical. and exact values. Use a calculator to convert a forms. fraction to a decimal IG Convert between units of equivalent. A1 Students learn to convert measure (ex. 1 ft = 12 inches) repeating decimals to Match a fraction to its decimal fractional form. equivalent and percent. 3. Distinguish between the various uses of numbers.

Use numbers to represent + Students must analyze + Use numbers to represent A2 Recognize that numbers + Students are expected to positions, locations, prices, problems to determine the angles, chords, segments, can represent positions, know all the basic number times etc. number system appropriate for arcs, areas, volumes, locations, angles, lengths, etc. properties and apply them the solution set. perimeters, lengths. without much thought. Demonstrate that a numbers value can be determined by a situation.

4. Develop and refine strategies for estimating quantity and evaluate the reasonableness of their estimates.

Use front end , compatible A1 Students explore the IG Estimate angle type: N/A N/A numbers and traditional concept of estimating and use obtuse, straight, acute, right. rounding rules to estimate it to determine the sums, products, quotients, and reasonableness of an answer. IG Estimate the type of units differences for all forms of an answer will have. (Ex. Cm numbers. cubed)

Determine if a given estimate + Use knowledge of triangles, is reasonable. such as largest side, largest angle to determine reasonable Estimate within a given range lengths of sides, or size of of reasonableness. angles.

5. Select appropriate representations for numbers, including integers and rational numbers in order to simplify and solve problems. .

Determine the most + Students must analyze + Determine the most + Determine the most + Students are expected to appropriate problems to determine the appropriate representation appropriate representation know all the basic number representation(fraction, number system appropriate for (fraction, decimal, integer or (fraction, decimal, and integer) properties and apply them decimal, integer or percent)of the solution set. percent)of a number to solve a of a number to solve a given without much thought. a number to solve a given given problem. problem. problem IG When measuring, determine the appropriate unit to use. Strand IV. Number Sense and Numeration Content Standard 3: Students investigate relationships such as equality, inequality, inverses, factors and multiples, and represent and compare very large and very small numbers. (Number Relationships)

1. Compare and order integers and rational numbers using relations of equality and inequality.

Compare and order integers + Students compare and order + Use triangle inequality N/A N/A and rational numbers (both all real numbers and their theorems. like and unlike denominators) subsets in both equations and Largest side, largest angle using equal, less than, or inequalities. Hinge theorem, converse greater than. hinge theorem.

2. Express numerical comparisons as ratios and rates.

Use ratios to compare + Students use ratios to IG Convert between different A2 Use long division to divide + Students are expected to quantities. compare quantities. units of measure. polynomials. know all the basic number properties and apply them Compare quantities involving + Students find equivalent + Use geometric means to without much thought. measures. ratios and rates. compare the altitude to the length of the hypotenuse. Find equivalent ratios and rates. G Use of trig ratios in triangles

Use tables to analyze ratios + Use of ratios with similar and rates. figures

3. Explore the relationships of primes, factors, multiples and divisibility.

Identify factors, multiples, + Students distinguish + Use of prime or square A2,HG Write a number or + Students are expected to common factors, and multiples between prime and composite factors to simplify radicals in algebraic expression as a know all the basic number of given numbers. numbers. the Pythagorean theorem or product of its prime factors properties and apply them special right triangles. using exponents. without much thought. Apply divisibility tests for + Students identify factors, 2,3,5,6,9,10. multiples, common factors, IG,G Use of scale factor in A2,HG Find the LCD in and relatively prime numbers. similar figures. rational expressions.

+ Students identify algebraic A2,HG Reduce rational expressions as prime or expressions. composite.

+ Students learn to factor various algebraic expressions. 4. Explain the meaning of powers and roots of numbers and use calculators to compute powers and square roots.

Use a calculator to find a + Students use powers and + Use powers and roots in the + Find monomial and binomial + Students are expected to number’s square and square roots to simplify expressions Pythagorean theorem, special factors. know all the basic number root. and solve equations. right triangles. properties and apply them + Find squares and square without much thought. Use a calculator to find the + Students use calculators to + Use powers and roots to find roots of binomials. standard form of a number simplify and approximate lengths of secant segments, raised to a given power. various roots and powers. tangent segments and chords + Find the value of a number in a circle. raised to a power. Determine if a number is a perfect square. + Find the value of a square root, using a calculator.

5. Apply their understanding of number relationships and solving problems.

Use understanding of number + Students apply their + Use understanding of + Use understanding of + Students are expected to relationships to solve knowledge of number number relationships to solve number relationships to solve know all the basic number problems involving real-world relationships by setting up and problems involving real-world problems involving real-world properties and apply them applications. solving various word applications. applications. without much thought. problems. Strand V. Numerical and Algebraic Operations and Analytical Thinking Content Standard 1: Students understand and use various types of operations (e.g., addition, subtraction, multiplication, division) to solve problems. (Operations and their Properties)

1. Use manipulatives and diagrams to model operations and their inverses with integers and rational numbers and relate the model to their symbolic expressions.

Use bingo games to N/A IG Use square tiles to + Represent the numbers in a N/A demonstrate and practice represent the Pythagorean given problem symbolically in addition, subtraction, theorem. order to solve for a missing multiplication and division of value. integers. IG Use square tiles to represent non-regular A2 Use Cramer’s Rule to Use cuisinaire rods to model polygons. solve systems of equations. fractions. + Represent the numbers in a Represent the numbers in a given problem symbolically in given problem symbolically in order to solve for a missing order to solve for a missing value. value.

2. Choose the appropriate algorithm and mathematical operations to manipulate numbers in a variety of forms.

Choose and apply the + Students learn many + Students will use the + Use the appropriate method + Students apply all the appropriate operations to fit a algorithms for solving a variety appropriate geometric formula to factor a problem (i.e. FOIL, algorithms they have learned given problem. of algebraic equations and to derive an answer. completing the square, in their previous algebra application problems. Examples include: quadratic formula) courses to solve a variety of One digit divisors should be Pythagorean theorem problems. done without a calculator. Distance formula A2,HG Determine whether Midpoint formula linear or circular permutations 2-D area are appropriate for a given surface area situation. volume geometric mean A2,HG Determine the special triangles appropriate algorithm to solve arc and chord formulas a logarithmic equation. etc. 3. Describe the properties of operations with rationals and integers and give examples of how they use those properties.

Describe and use the + Students identify the G Identify the operations used + Demonstrate the use of the + Students are expected to appropriate properties of properties of rational numbers in an algebra proof. commutative, distributive, know all the basic number addition and multiplication and use them to solve identity, associative, and zero properties and apply them when solving an equation. proportions. + Use the distributive property properties. without much thought. for the area of a trapezoid. + Students use the distributive A2,HG Demonstrate the use property to simplify of the properties of logarithms. expressions and solve equations. + Identify the closure property for real numbers. A1 Students use the properties of the real number system to do algebraic proofs.

4. Efficiently and accurately apply operations with integers, rational numbers and simple algebraic expressions involving problems.

Accurately solve addition, + Students accurately solve + Accurately solve addition, + Accurately solve addition, + Students are expected to subtraction, multiplication and addition, subtraction, subtraction, multiplication and subtraction, multiplication and know all the basic number division problems with multiplication and division division problems with division problems with properties and apply them integers, rational and real problems with integers, integers, rational and real integers, rational and real without much thought. numbers(both like and unlike rational and real numbers numbers(both like and unlike numbers(both like and unlike denominators) and one-step (both like and unlike denominators) and one-step denominators) and one-step and multiple step algebraic denominators) and one-step and multiple step algebraic and multiple step algebraic expressions both with and and multiple step algebraic expressions both with and expressions both with and without a calculator to solve expressions both with and without a calculator to solve without a calculator to solve geometry problem without a calculator. geometry problem. geometry problem. Strand V. Numerical and Algebraic Operations and Analytical Thinking Content Standard 2: Students analyze problems to determine an appropriate process for solution, and use algebraic notations to model or represent problems. (Algebraic and Analytic Thinking)

1. Read, write, create, and simplify algebraic expressions. Solve linear equations and inequalities.

Match a given expression to + Students accurately solve + Students write inequalities to + Write an equation for an + Students are expected to its word phrase. addition, subtraction, conclude if the triangle application problem. have the skills of simplifying multiplication and division inequality theorem, Hinge and solving linear equations Create an algebraic problems with integers, theorem, the converse hinge A2,HG Simplify rational and inequalities prior to this expression to match a given rational and real numbers theorem and right triangle expressions. course. They use these skills situation. (both like and unlike trigonometry hold for a given on almost a daily basis. denominators) and one-step problem. Students write Translate between verbal and and multiple-step algebraic equations to find lengths of algebraic expression. expressions both with and congruent sides and angles. without a calculator. Determine if a term will be a variable or a constant.

Use multiple grouping symbols.

2. Represent algebraic concepts with a variety of models and write corresponding algebraic expressions

Represent and model + Students translate from + Students write algebraic A2 Use matrices to solve + Students are expected to situations using a rule, table, words to expressions and model their systems of equations. have these skills prior to this or graph. expressions/equations and concept as to find the area of course. They will use it often. back. irregular polygons.

+ Students learn many strategies for solving linear equations and inequalities.

3. Solve linear equalities and inequalities using algebraic and geometric methods, and use the context of the problem to interpret and explain their solutions.

Solve problems using = and + Students use pictures to + Students write equations for A2 Use matrices to solve + Students are expected to not = using algebraic and represent problems to assist in angle of elevation and angle of systems of equations. have these skills prior to this geometric methods. setting algebraic equations. depression problems based course. They will use it often. on the situation given in the + Solve systems of linear Explain and justify a solution + Students examine solutions problem. inequalities and equations. to a given equation to determine if they are reasonable. + Students write algebraic + Interpret results by solution Order real numbers. expressions and model their set or graphing. concept as to find the area of irregular polygons. 4. Analyze, solve and evaluate strategies for solving problems using linear functions.

Use working backwards, trying + Students solve linear + In right triangle trigonometry, + Use an equation and inverse + Students are expected to a smaller problem, guess-test equations using inverse students analyze a right operations to solve for a have these skills prior to this to solve a variety of problems operations. triangle and come up with the variable in a given equation. course. They will use it often. best strategy to solve Use the name, write, solve, + Students evaluate different problems. + Analyze graphs to determine check process to find the methods for solving linear the solutions for systems of solution to one-step algebraic functions. linear equations and equations. inequalities. + Students learn to solve Use an equation to solve a systems of linear equations by given story problem. graphing, substitution, and elimination.

5. Students employ algebraic and analytic thinking and the power of technology to explore problems in a modern society.

Select and use appropriate + Students learn when and + Using a compass, students + Select and use appropriate + Students are expected to technology (calculator, where calculator use is construct geometric figures. technology (calculator, have the algebraic skills prior compass, protractor, ruler, appropriate. Students use calculators to protractor, ruler, paper and to this course. They will use it paper and pencil) to enhance compute trigonometric pencil) to enhance often. mathematical understanding. Aa Students use appropriate functions in their equations. tools (compass, protractor, and ruler) to enhance understanding. Strand VI. Probability and Discrete Mathematics Content Standard 1: Students develop an understanding of the notion of certainty and of probability as a measure of the degree of likelihood that can be assigned to a given event based on the knowledge available, and make critical judgments about claims that are made in probabilistic situations. (Probability)

1. Describe events as likely or unlikely and give qualitative and quantitative descriptions of the degree of likelihood.

Use simple devices such as Aa Students use devices such + Students determine if 3 N/A HP Students use counting coins, spinners, dice in as coins, spinners, and dice to lengths can be the sides of principles, combinations, and activities to model probability. model probability. any triangle. Students apply permutations to determine the the converse Pythagorean probability of certain Describe a given event as Aa Students describe an theorem to determine if outcomes. likely or unlikely. event as likely or unlikely. lengths can be sides of a right triangle. Students qualify answers based upon its application within the context of the problem (negative numbers can’t be lengths or angles of triangles).

2. Describe probability as a measure of certainty ranging from 0 to 1 and conduct activities that allow them to express probabilities of simple events in mathematical terms.

Describe an events probability Aa Students use devices such G Students compute the A2 Model examples of HP Students use counting as a number between 0 and 1. as coins, spinners, and dice to geometric probability on a line independent and dependent principles, combinations, and model probability and express or on any 2 dimensional events. permutations to determine the Describe the likelihood of an their answers as a measure of figures. probability of certain event using fractions. certainty ranging from 0 to 1. outcomes.

Aa Students describe an event as likely or unlikely

3. Conduct experiments and give examples to illustrate the difference between dependent and independent events.

Conduct experiments using Aa Students conduct N/A N/A N/A cards, dice, spinners, coins, or experiments and record other simple devices that results illustrating the included events with and difference between dependent without replacement as well as and independent events. the use of events with the terms and / or. Aa Students solve various word problems involving Determine a given event as dependent and independent dependent or independent and events. find its probability. 4. Explain the difference between experimental and theoretical probability and explain how probability changes for large numbers of trials.

Conduct experiments and then Aa Students conduct N/A Use tree diagrams to N/A combine class data to experiments and then represent the counting highlight the difference combine class data to show principle HG, A2 between theoretical and how probability changes for experimental probability. large numbers of trils.

5. Conduct probability experiments and simulations to model and solve problems.

Conduct a probability Aa Students interpret and N/A A2 Use real world models to N/A experiment that simulates connect the uses of probability represent quadratic equations. winning in a game of chance. to the world around them. A2 Determine if a set is open Interpret and connect the uses or closed under an operation, of probability to the world such as the matrices. around them.

Strand VI. Probability and Discrete Mathematics Content Standard 2: Students investigate practical situations such as scheduling, routing, sequencing, networking, organizing and classifying, and analyze ideas like recurrence relations, induction, iteration, and algorithm design. (Discrete Mathematics)

1. Use manipulatives, diagrams, and the fundamental theorem of counting to count permutations and combinations.

Use a tree diagram to model Aa Students use the counting N/A N/A HP Students use counting the possible outcomes of a principle to find the number of principles, combinations, and given event. outcomes. permutations to determine the probability of certain List all the possible Aa Students use factorial outcomes. combinations of a given event. products to count arrangements.

Aa Students list all the possible combinations of a given event.

2. Use sets and set relationships to explore and solve simple algebraic and geometric problems.

Use Cuisinaire Rods to model + To solve geometric HG,A2 Study arithmetic and + Students will use set theory patterns using perimeter and problems, students confine geometric sequences and as applied to solution sets of area. answers to certain sets of series equations and inequalities. numbers, such as positive real Use real world models for numbers. + Develop binomial expansion. perimeter and area. + Classify figures as belonging + Use Pascal’s triangle to get to concave or convex coefficients for terms of polygons. polynomials. 3. Solve problems involving networks, for example planning delivery routes or counting paths between points.

Trace a figure without lifting N/A G,HG Students use mappings N/A N/A your pencil or finger. to determine if a network is complete and/or traceable. They determine how many nodes a network has.

4. Explore recurrence relations and iterations.

Determine if a decimal N/A + Students determine the HG,A2 Study arithmetic and + Students will use the idea of terminates or repeats. number of degrees of interior geometric sequences and recurrence to develop angles in a convex polygon series formulas. Express a common fraction as based on the number of either a terminating or degrees in a previous polygon + Develop binomial expansion. repeating number. (with one less side). + Use Pascal’s triangle to get Explore pi as a non- coefficients for terms of terminating or non-repeating polynomials. decimal.

5. Mathematical applications frequently require students to develop their own procedures for solving problems.

Provide problem solving + Students use algebraic + Students apply their own + Provide problem solving + Students solve a variety of opportunities that include but techniques to determine the strategy for doing geometric opportunities that include but application problems using a are not limited to oral and best procedure for solving a proofs. are not limited to oral and variety of strategies. written activities, given problem. written activities, manipulatives, group and manipulatives, group and individual work, pictures and individual work, pictures and diagrams to allow the students diagrams to allow the students to model mathematical to model mathematical situations and solutions. situations and solutions.

6. Use discrete mathematics concepts to model and solve problems with or without solutions.

Determine if a problem has a + Students determine if and + Students determine how + Determine the number of N/A solution. how many solutions exist for a many solutions exist for a zeros for a given polynomial given problem. given situation. equation. Be able to state what information is needed to find a + Students decide upon the + Use the discriminant to solution. best solution for a given classify the types of solutions problem. for a quadratic equation. Determine how many solutions exist for a given + Decide upon the best situation. solution for a given problem.

Decide upon the best solutions for a given problem.