Goal

The student will:

Content Knowledge and Skills:

Benchmarks:

1.1  Understand and use numbers.

1.  Apply properties of rational numbers.

1.  Use properties of the real number system to complete algebraic proofs.

2.  Understand properties of the real number system.

1.  Understand the real number system by examining relationships between its subsets.

3.  Apply properties of exponents

1.  Square and cube numerical expressions.

4.  Understand properties of roots

1.  Find the square roots and cube root of numerical expressions.

2.  Identify exact and approximate roots without simplification.

3.  Simplify radical expressions.

5.  Solve problems using number theory concepts such as factors, multiples, and primes

1.  Find the largest size square tile that could be used to tile two rooms of different dimensions using only whole tile.

2.  Use prime factorization to simplify radical expressions.

6.  Use appropriate vocabulary.

3.  See attached vocabulary.

1.2  Perform computations accurately.

1.  Use the proper order of operations. Perform operations with real  numbers.

1.  Evaluate and simplify expressions

2.  Use absolute value to find the distance between two points on the number line.

3.  Convert percent to decimal form and back

4.  Compute percents

1.3  Estimate and judge reasonableness of results.

1.  Apply number sense to every day situations.

1.  Estimate the amount of material needed to complete a building project.

2.  Identify that error acumulates in a computation when there is rounding.

1.  Compute exact and approximate answers involving irrational values such as pi and square roots.

Content Standard -

 The student will:

Content Knowledge and Skills:

Benchmarks:

2.1  Understand and use customary and metric measurements.

1.  Use geometric formulas to estimate and compute perimeter, circumference, area, surface area, and volume.

1.  Estimate and compute the perimeter of regular and irregular polygons.

2.  Estimate and compute the area of an irregularly shaped figure.

3.  Find the circumference of a circle given the radius, diameter, or the area of the circle.

4.  Estimate and calculate the area of polygons such as triangles, parallelograms, rhombi, squares, rectangles, trapezoids, regular pentagons, regular hexagons, and regular octagons.

5.  Estimate and calculate the area of circles, sectors and segments of circles.

6.  Estimate and calculate the lateral and total surface area of three-dimensional objects including cylinders, pyramids, prisms, cones, and spheres.

7.  Calculate the volume of three-dimensional objects including cylinders, pyramids, prisms, cones, and spheres.

8.  Use the relationship between the volume of a pyramid and the volume of a corresponding prism and the relationship between the volume of a cone and volume of a corresponding cylinder.

2.  Solve problems involving circumference, perimeter, or area of triangles, circles, and rectangles.

1.  Find the area of a circle inscribed in a square and given the coordinates for the 4 points of tangency.

2.  Determine volume and surface area in a problem requiring multiple calculations within the context of a real life scenario

3.  Find the area of the largest circle that can be inscribed in a square with a given area.

2.2  Apply the concepts of rates, ratios, and proportions.

1.  Use rates, ratios, proportions, map scales, and scale factors (one- and two- dimensional) in problem solving situations.

1.  Compute geometric mean.

2.  Describe and apply the similarity relationships for triangles.

Identify the scale factor of similar two-dimensional figures

3.  Solve ratio and proportion problems involving similar figures and their linear measurements, area, and volume.

4.  Use proportions to solve scale drawing problems.

2.  Apply concepts of rates and direct and indirect measurements.

1.  Find the height of an object using indirect methods such as Thales’ method of shadows, mirror reflection, and trigonometric ratios.

3.  Construct equivalent units, comparable units, and conversions.

1.  Create conversion factors to convert from one unit of measure to another.

2.  Convert between square units i.e. convert between square inches and square feet.

2.3  Apply dimensional analysis

1.  Use customary and metric units and their relationship to one another and to real world application involving length, area, capacity, weight, time, and temperature.

1.  Compute the cost, amount of materials needed, and time needed to complete a project such as building a shed, sewing a quilt, or constructing a model.

2.4.  Apply  appropriate techniques and tools to determine measurements.

1.  Determine and use appropriate units.

1.  Understand when to use linear, square, or cubic measures.

2.  Apply the Protractor Postulate.  Use protractors to measure and compare angles. 

3.  Estimate the measure of a given angle.

4.        Apply the Ruler Postulate.  Measure distance using customary units and metric units with appropriate unit labels

2.  Approximate error in mesurement situations.

1.  Understand tolerance, precision, and their applications.

2.  Understand that error accumulates in a computation when there is rounding at intermediate steps.

3.  Calculate percent error.

Content Standard -

The student will:

Content Knowledge and Skills:

Benchmarks

3.1  Use algebraic symbolism as a tool to represent mathematical relationships.

1.  Represent mathemetical relationships using variables, expressions, linear equations, and inequalities.

1.  Define a variable and write an equation to solve problems in geometric situations such find the value of x for which line l is parallel to line m.

3.2  Evaluate algebraic expressions.

1.  Use appropriate procedures for manipulating and simplifying algebraic expressions involving variables, integers, and rational numbers.

1.  Solve problems that use variables in expressions describing geometric quantities by solving for one variable.

3.3  Solve algebraic equations and  inequalities.

1.  Use appropriate procedures to solve multi-step, first-degree equations and inequalities such as

      3(2x - 5) = 5x + 7

       or 3(2x – 5)> 5x +7.

1.  Use inequalities to solve problems involving the triangle inequality theorem, hinge theorem, and exterior angle theorem.

2.       Differentiate between linear and non-linear equations and graphs.

1.  Use slope-intercept, point-slope, and standard form of linear equations.

2.  Identify linear equation for a straight line.

3.  Graph a circle and write the equation of a circle in standard form.

3.4  Solve simple linear systems of equations.

1.  Understand and use appropriate procedures to solve simple linear systems of equations.

x + y = 7

2x + 3y = 21

1.  Solve systems of linear equations in geometric context.

Content Standard -

The student will:

Content Knowledge and Skills:

Benchmarks

4.1   Apply concepts of size, shape, and spatial relationships.

1.  Understand congruence and similarity of two-dimensional figures including the concepts of reflection, rotation and translation.

1.   Identify congruent angles and segments.

2.  Identify congruent triangles and name corresponding parts of congruent triangles.

3.   Prove figures are congruent or similar.

4.   Identify a polygon according to its number of sides (e.g. quadrilateral, pentagon, and decagon) and describe polygons according to their characteristics (e.g. regular, similar).

5.   Comprehend congruence and similarity as it applies to transformations

6.   Rotate and flip figures on a coordinate graph, understanding the line of reflection and how a point transposed across that line is determined.

7.Perform transformations such as translations, rotations, reflections, dilations.

2.  Recognize and use similarity as it relates to size variations in two-and three-dimensional objects.

1.  Create and use models to represent two- and three- dimensional geometric objects as a means of problem solving

2.  Create a drawing to scale of a geometric figure.

3.  Use the relationships between the ratios of the areas and volumes of similar solids.

3.  Recognize and use properties of points, lines, and planes.

1.  Name and use relationships among lines, segments, and rays (e.g. parallel, perpendicular, intersecting).

2.  Name relationships among points (e.g. collinear, coplanar).

3.  Construct, using a straight edge and compass, congruent segments, congruent angles, perpendicular bisector, angle bisector, parallel lines.

4.  Recognize and use angle relationships.

1.  Classify /distinguish angles by their degree measure (e.g. acute, right, obtuse, and straight).

2.  Distinguish among supplementary, complementary, corresponding, interior, exterior, central, and inscribed angles.

3.  Find the complement and supplement of angles.

4.  Solve problems involving adjacent, vertical, exterior, complementary, and supplementary angles.

5.  Recognize and use properties of parallel lines.

1.  Solve for missing angle in problems involving parallel lines cut by a transversal.

2.  Find missing segment lengths for diagrams involving three or more parallel lines cut by two transversals.

3.  Distinguish between parallel lines and skew lines.

4.  Identify the special angle pairs formed by parallel lines and a transversal.

5.  Apply the properties of special angle pairs formed by parallel lines and the transversal/

6.  Formally or informally prove lines are parallel using special angle properties.

6.  Recognize and use properties of circles.

1.  Identify the parts of a circle (e.g. center, radius, circumference, diameter, chord, secant, tangent, major arc and minor arc) and show how these parts interrelate.

2.  Find the measure of angles and arcs related to circles (e.g. central, inscribed, angles formed by chords, secants, tangents) and lengths of segments formed inside and outside circles.

3.  Identify the measure of inscribed and central angles and intercepted arcs.

7.  Recognize and use properties of polygons.

1.  Represent problems involving triangles with an appropriate diagram and apply properties.

2.  Identify the parts of triangle (e.g. hypotenuse, altitude, angle bisector, perpendicular bisector)

3.  Identify/classify triangles by their angle measure and side lengths (e.g. acute, obtuse, right, scalene, isosceles, equilateral).

4.  Use and apply properties of isosceles triangles.

5.  Classify polygons (by sides/convex/concave)

6.  Find angles related to polygons (e.g. interior and exterior)

7.  Use the triangle inequality theorem to solve problems involving triangles

8.  Use properties of midsegments to solve problems.

9.  Find the length of altitudes, medians, and angle bisectors within triangles.

10.  Find missing angles and side lengths of quadrilateral for which diagonal length have been given.

11.  Identify parts of quadrilaterals (e.g. apothem, base, and diagonal)

12.  Classify special quadrilateral as parallelograms, rhombi, rectangles, squares, kites, trapezoids, and isosceles trapezoids.

13.  Recognize and use properties of parallelograms and other special quadrilaterals such as rectangles, squares, rhombi, and trapezoids.

14.  Model problems with quadrilaterals and apply properties.

15.  Model problems with polygons and apply properties.

16.  Find the angles of a polygon (e.g. central, exterior, and interior) and the sum of interior and exterior angles.

8.  Recognize and use properties of space figures.

1.  Identify three-dimensional figures and their parts (e.g. cylinder, cone, edge, face, vertex, and base).

2.  Identify a polyhedron according to the number of its faces (e.g. tetrahedron, hexahedron, and octahedron) and describe the number of vertices and edges.

4.2  Apply the geometry of right triangles.

1.  Use the Pythagorean Theorem to solve problems.

1.  Find the center of a circle using inscribed right triangles.

2.  Use the Pythagorean Theorem and its converse to find missing side lengths and determine whether a triangle is acute, right, or obtuse.

3.  Determine and use Pythagorean triples.

2.  Use trigonometric ratio methods to solve problems

1.  Express the trigonometric functions as ratios.

2.  Define and discuss the relationships among the sine, cosine, and tangent ratios.

3.  Use sine, cosine, and tangent ratios to find missing angles and side lengths of triangles.

4.  Solve real-world trigonometry problems, and include the angle of depression or elevation in calculations.

5.  Use the Law of Sines to solve problems.

3.  Use the properties of special right triangles to solve problems.

1.  Find missing angle measures and side lengths for special right triangles.

2.  Use properties of special right triangles (30-60-90, 45-45-90) to find the length of apothem and radius of regular triangles, quadrilaterals, and hexagons.

4.3  Apply graphing in  two dimensions.

1.  Identify attributes of the Cartesian Coordinate System, such as quadrants, origin, and axis.

1.  Find numerical unknowns related to the coordinate plane (e.g. distance, midpoint, slope, coordinate)

2.  Use a transformation to map a preimage onto an image.

3.   Determine endpoints and midpoint of a line segment on a coordinate graph.

2.  Graph scatter plots and identify informal trend lines.

1.  Measure the circumference and diameter of several circular objects. Create a scatter plot of the data and graph a trend line.

2.  Represent experimental data with graphs.

3.  Identify positive and negative correlations.

1.  Investigate the correlation of the circumference to the diameter of a circle.

4.  Understand the characteristics and uses of vectors.

1.  Apply vectors to realistic situations.

2.  Determine the magnitude and direction of a vector given the coordinates of the initial and terminal points.

4.4  Represent and graph    linear relationships.

1.  Create graphs and equations for linear relationships.

1.  Write a linear equation based on the graph of the equation.

2.  Represent linear relationships using tables graphs, and mathematical symbols

1.  Graph linear equations with one and two variables.

3.  Interpret attributes of linear relationships such as slope, rate of change, and intercepts.

1.  Find the slope of a line given two points on        the line or the graph of the line.

2.  Understand the concepts of parallel and      perpendicular lines and their slopes are related.

3.  Write the equations of parallel and perpendicular lines.

4.  Graph parallel and perpendicular lines from their equations.

4.5  Use reasoning skills

1.       Use logic to make and evaluate mathematical arguments.

1.  Use a variety of problem solving skills such as draw a diagram, work backward, and make a systematic list write an equation, use formula to solve nonroutine problems.

2.  Construct logical arguments , form conjectures, judge their validity , and give counterexamples to disprove statements

3.    Construct an if-then statement and test its       validity under a variety of conditions.

4.     Write the converse, inverse, and contrapositive of a conditional statement and determine the truth value of each statement.

5.    Write biconditional statements and recognize good definitions

6.    Define and apply terms associated with mathematical logic and reasoning (e.g. postulate, theorem, hypothesis, conclusion, and converse).

7.    Use inductive reasoning when appropriate to solve a problem situation or write a proof.

8.    Use deductive reasoning when appropriate to solve a problem situation or write a proof.

9.    Use logic to evaluate the validity of a mathematical argument or conclusion.

Content Standard

The student will:

Content Knowledge and Skills:

Benchmarks

5.1  Represent data with a variety of formats.

1.  Analyze and interpret tables, charts, and graphs (e.g., scatter plots, line graphs, three-dimensional graphs, and pie charts).

1.  Interpret information presented in tables and charts.

2.  Apply graphs to problem solving.

5.2  Collect, organize, and display data.

1.  Collect and organize data, and display the data in tables, charts, and graphs

1.  Construct a circle graph in order to represent a set of data.

5.3  Apply simple statistical measurements.

1.  Interpret and use basic statistical concepts, including mean, median, mode, range, and distribution of data including outliers.

1.  Given a set of data, determine measures of central tendency including mean, median, and mode.

2.  Given a set of data, determine the range.

2.  Make predictions and draw conclusions based on statistical measures.

5.4  Understand basic concepts of probability.

1.  Find probabilities based on dependent, independent, and compound events.

1.  Use segment and area models to find the probabilities of events.

2.  Contrast experimental and theoretical probabilities.

5.5  Make predictions or decisions based on data.

1.  Make predictions based on randomness, chance, equally likely events, and probability.

2.  Use appropriate tools/technology to conduct simulations and employ graphical models to make predictions or decisions based on data.

3.  Design, conduct, and interpret results of statistical experiments.