MATH RHINEBECK PRIORITIZED CURRICULUM MATH B

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1 B 1.1.1A 1.1.2A 1.1.3A 3.1.1A 3.1.1B Mathematical Reasoning (Woven throughout the curriculum) How do you know? How did you arrive at that answer? Why did you choose that strategy? What other ways could the problem be solved? Can you restate this problem another way? What is a simpler way to say that? How could this problem or solution be graphically represented? Use abstraction and symbolic representation to communicate mathematically Use deductive and inductive reasoning to reach mathematical conclusions Use critical thinking skills to solve mathematical problem Use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence and construct an argument Construct valid arguments Follow and judge the validity of arguments These ideas should be woven throughout daily classroom instruction. Students use written language and/or graphic representations to describe/ illustrate problem solving methods Have students respond to assignments in journals Have students discuss solutions and strategies in small groups. Decide which strategies were most efficient Ask students to frequently restate problems in own language Introduce complex problems with simpler versions first Use graphic organizers designed to help students think about the steps involved (step ladder or sequence flow charts) - use a rubric for level of understanding, clarity of explanations, etc. Math B

2 B 3.1A 3.1B GUIDING QUESTIONS ESSENTIAL KNOWLDEGE SKILLS Euclidean Proofs Direct and indirect Congruence (ASA, SSS, SAS, H.L., CPCTC) Reflexive, transitive, symmetric, substitution property of equality Properties and definitions of quadrilaterals Geometric inequality theorem Triangle angle sum theorem Parallel and perpendicular theorems SUGGESTED CLASSROOM SUGGESTED ASSESSMENT Quizzes/tests (Should incorporate open-ended problemsolving situations requiring students to show work and/or explain the mathematics they used) Geometry Proofs What is the difference between direct and indirect proof? What are the ways of proving triangles congruent? What is the significance of CPCTC? What are the properties of equality and inequality? (equivalence relation?) How are the properties and definitions of quadulations applied? What is the relationship between angles and sides in a triangle? What is the sum of the angles of the triangle? Who can parallel and perpendicular concepts be used in geometric applications? What are the slope, distance and midpoint formulas and how are they used? Analytic Proofs (Coordinate Geometry) Slope Distance midpoint Discuss the general concept of a Two-Column Proof as a mathematical solution Construct with students a basic proof in logic Use the format of two-column proofs used in logic proofs to build on the format of two-column euclidean direct proofs in geometry Review if needed: - Congruent triangles and corresponding parts - Reflexive, symmetric, transitive properties - Properties of real numbers (addition, multiplication, substitution, etc.) - Ways to prove triangles congruent (SAS, SSS, ASA, HL, AAS) - Properties of geometric figures, parallel lines, perpendicular lines, etc.) Construct basic proofs in geometry like: Prove that the sum of the angles of a triangle is 180 Extend basic proofs into more lengthy proofs applying congruent triangles and CPCTC (prove diagonals of a parallelogram bisect each other) Discuss the format and logic of indirect proof. (Prove indirectly that if alternate interior angles are not equal, then lines are not parallel) Assign proofs, direct and indirect, to small groups in a class. Have students present solutions to the class Math B

3 B 3.2A 3.2B 3.2C 3.3A 3.7E 3.7H 3.7K Field of Real Numbers What are the properties of a field? How do field properties apply to subsets of the real numbers? How are operations in the reals used to simplify expressions and solve equations involving rational numbers, polynomials, and complex fractions? What is absolute value? (algebraic and geometric?) Field Properties Operations Rational numbers Polynomials Complex fractions Absolute Value Review if needed: - signed numbers - order of operations - what a variable, coefficient, exponent are - operations with fractions using real numbers Have the student simplify expressions and solve equations involving operations with rational numbers, polynomials and complex fractions Use real-life word problems and have students work in small groups to simplify expressions and solve equations that require operations with rational numbers, polynomials, and complex fractions. (work problems, mixture problems, conversions like Fahrenheit to centigrade, etc.) Have the students graph absolute value relations and functions by using a table of values. Students put solutions on the board. Use a graphing calculator, to graph absolute value, pointing out transformations that occur problem-solving Math B

4 B Equation Solving 3.3D 3.4E 3.4H 3.5H 3.6A 3.7E 3.7G 3.7K GUIDING QUESTIONS ESSENTIAL KNOWLEDGE SKILLS SUGGESTED CLASSROOM SUGGESTED ASSESSMENT How are absolute value equations solved? What are the methods for solving quadratic equations? How is the solution expressed? How many possible solutions exist? What is equation for axis of symmetry for a parabola? What is the turning point? How do you graph absolute value? Linear (including absolute value) Quadratic Factoring Completing the square Solving for roots graphically Quadratic formula Systems Algebraically and graphically Linear Quadratic Linear/ quadratic Inequalities Quadratic- algebraically and graphically Absolute value- algebraically and graphically Use a number line to show the geometric reasoning involved in solving absolute value equations algebraically Have students solve quadratic equations by: - factoring - completing the square - quadratic formula - graphing Have students graph a parabola by: - a table of values - axis of symmetry and roots - 5 step solution of: 1. Equation of axis of symmetry 2. Turning point 3. Table of values 4. Plotting values from the table on graph 5. Roots- where does the parabola intersect the x axis? Graph parabolas on a graphing calculator Teacher observations Classroom assignment Quizzes/tests (Should incorporate open-ended problem-solving students to show work and/or explain the mathematics they used) Application Multi-step word problems solved by systems of equations Math B

5 B 3.2A 3.2D 3.2E 3.3D 3.4N 3.7I 3.7K Radicals and Complex Numbers When is a radical in simplest form? What is the conjugate of x+ y? of a+ bi? What steps must be taken to solve an equation with radicals? What are complex numbers and why are they needed? Simplifying Radical Expressions Operations with radicals Rationalizing the denominators Use of conjugates Solving radical equations Pythagorean theorem Isolating the radical Complex numbers Definition of imaginary numbers Simplification (I) Operations Discriminant (types of roots) Graphing parabolas Writing quadratic equations with complex conjugate roots (sum and product of roots) Review if needed: - simplifying radicals using perfect square factors - simplifying radicals using prime factors Have students simplify radicals: - that have perfect square factors other that 1 - that contain fractions - by rationalizing the denominator - by using conjugate of denominator Have students solve equations with radicals, then check for extraneous roots; 2x + 1=X problem-solving exercise Math B

6 B 3.3E 3.4D 3.4K 3.4J 3.7A 3.7C 3.7D 3.7E 3.7G 3.7J 3.7N 3.7O Relations and Functions What is a relation? What is a function? How do you determine domain and range? What is the difference between image and preimage? How can you tell by looking at a graph that it is a function? What is the horizontal line test used for? How do you compose function (composition?) How is the inverse determined (algebraically and graphically)? Definitions and Differences Relation Function Domain and range 1 1 Image and pre-image Identifying domains and ranges Algebraically Graphically Finding the image and pre-image Identifying functions Vertical line test Even and odd (symmetries) Horizontal line test (1 1) Composition of functions Notation Linear functions Inverses (algebraically and graphically) After defining Relation, Function, Domain, and Range, have students list and graph sets of ordered pairs that are functions and are not functions. Using graphs, have students draw conclusions about: - vertical line test (function) - horizontal line test (1 1) - domain (pre- image) and range (image) Use a graphing calculator to display graphs of absolute value and linear functions, pointing out the difference between 1 1 and not 1 1. Have students solve problem that combine functions using the basic operations and the composition of two functions as in the following cost analysis problem: - the cost c to produce x units of a given product per month is given by: C=F(X)=19, X - If the demand X each month at a selling price of $P per unit is given by : X= G(P) P/4. Find (F ο G)(P) and interpret What is the inverse of F(X) = 2X + 1? Have students find the solution algebraically and graphically (graphing calculator can be integrated) Teacher observation Class discussion problem solving Math B

7 B 3.4A 3.4B 3.4C 3.4J 3.4K 3.6G 3.7A 3.7B 3.7E 3.7G 3.7N Exponential and Logarithmic Functions What is an exponential function? What is the definition of logarithm? How are exponential and logarithmic functions related? Who do laws of exponents related to laws of logs? How do you find the logs? How do you find the log of a number and the ante-log of a number using a calculator? How are laws of logarithm used to solve various calculations? How can logarithmic and exponential functions be used to solve real life relationships? Exponential functions and graphs of y= a Inverse of exponential Logarithmic function Properties of logarithms Calculations using logarithms Solve equations using logarithmic expressions Problem solving using exponential and logarithmic functions Rewrite the equality logьa= c as Ьс = a Review if needed: - Laws of exponents (5) - Properties of zero, negatiate and fractional exponents After starting (3) laws of logarithms and relating them to laws of exponents, have students apply them through: - Simplifying expressions with logs - Graphing exponential functions (use a graphing calculator) - Solving equations with logs - Inverses of logs (algebraic and graphic) - Evaluating common logs (scientific calculator) - Performing calculations with logs like x(46.4)³ Formulas that give real world solutions to problems found in population growth and decline, radioactive decay, compound interest, inflation, etc. (A graphing calculator can be used to find algebraic and graphic solutions) Teacher observation Class discussion problem solving Math B

8 Circles 3.5C 3.5D 3.5H B How would you represent the following on a circle?, arc?, angle?, chord?, secant?, tangent? How do angles and arcs relate to each other in a circle? How would you apply segment length theorems for a circle? Basic definitions - Arcs - Angles - Chords - Secants - Tangents Find angle measure and arc degree measure in a circle Find lengths of segments in a circle Pythagorean Theorem Review if needed: - Perimeter of polygons - Circumference of circles - Area of polygons and circles - Volumes of solids Give students a diagram of a circle with related parts. Have them identify: - Chord - Secant - Tangent - Arc - Angles formed by Give students a diagram of a circle and of arc measures. Have them determine various types of angle measures from the arcs. Have students construct proofs of theorems: - In a circle, if central angles intercept congruent arcs, then the angles are congruent (use congruent triangles) - An inscribed angle is measured by one half the measure of its intercepted arc (use exterior angle theorem) If two chords intersect within a circle, then the product of the lengths of the segments of one chord equals the product of the length of the segments of the other chord (use similar triangles) Teacher observation Class discussion problem solving Math B

9 B 3.3B 3.3C 3.3E 3.4I 3.7F 3.7L 3.7M Transformational Geometry What is a transformation? Given the diagram of two images, how can you determine the type of transformation? What is an example for composition of transformations? What is an isometry? What is the difference between direct isometry and opposite isometry? What is the difference between point, line and rotational symmetry? How can slope and midpoint be used to demonstrate transformations? Reflections in a line and a point Rotations Translations Dilations Composition of transformations Isometries - Direct - Opposite Symmetry - Point - Rotational - Line Use slope and midpoints to demonstrate transformations Review if needed: - Reflections - Translations - Rotations - Dilations Give students the coordinates of the verticals of a triangle. Have them find the image triangle under a composition Rу=x RoT incorporated open-ended problem solving Math B

10 Conics 3.4A 3.4D 3.4L 3.4N B What are the four types of conic sections? What are general forms of the equations for the line, parabola, circle, ellipse, and hyperbola? How do you graph conic sections? Recognize the equation of: Straight line Parabola Circles Ellipses Hyperbolas Write the equation of a circle with a given center and radius in the form: (x-h)² + (y-k)² =r² Graph equations of all conics using intercepts, recognize the ellipse and non-rectangular hyperbola Use graphing calculator to demonstrate the graphs of conic sections: - Parabola - Circle - Ellipse - Hyperbola Given a circle with center C(5,-2) and radius of 4 units, write its equation and graph the circle. Check solution on a graphing calculator. Graph the parabolas (1)y=x²+6x-1 and (2) y = -x² = 4x + 1 by using a table of values and discussing: - axis of symmetry - turning point - roots (use graphing calculator for x- intercepts) Graph the ellipse and hyperbola using the x and y intercepts method: - ellipse x² - hyperbola Show students algebraic variations of conics like 4x¹ = 9y² Teacher observation Class discussion problem solving Math B

11 B 3.4M 3.5A 3.5C 3.5E Trigonometric Functions What does the word trigonometry mean? What special triangles do you know? What is the relationship between lengths of sides in and triangles? How do degree and radian measurements compare? What is a reference angle? In what quadrants are the circular functions positive and/or negative? What are reciprocal trigonometric functions? Use trigonometry to solve: - Right triangles - Word problems involving their applications Change degree measure to radian measure and vise versa Find reference angles - Special cases (exact values) - Non-special angles Define relationship between trig functions and ordered pairs on the unit circle - Two coordinates - Rectangular coordinates Reciprocal Function - csc - sec - cot Understand derivation and solve problems with Sine, Cosine, Tangent, and Reciprocal Functions Derive and apply for Sine and Cosine - Sum and difference of two angles - Double and half angles - Graphs of Sine and Cosine (with amplitude and frequency variations) - Reflections in line y=x(inverses) Review if needed: - Right triangle trig - Angle of elevation, depression and right angles Convert 150 to radians Convert 2π 3 radians to degrees Demonstrate an angle x of -210 in standard position on the unit circle: - Find sin x, cos x, tan x, cot x, sec x, csc x - How do we do this if reference angle is not a special case angle (30, 45,60 or quadrantal)? (use a scientific calculator to show how trig. Values are obtained) Identify positive and negative quadrants for six trig. Functions Find the trig coordinates for the point (5,-5 3) in the coordinate plane Use identity for sin (A+B) to derive identity for sin (A+A) Quizzes/tests should problem-solving situations requiring students to show work and /or explain the mathematics they used) Scientific calculator Math B

12 B 3.4I 3.4M 3.5E 3.6B 3.6G 3.7B 3.7D 3.7G GUIDING QUESTIONS ESSENTIAL KNOWLDEGE SKILLS Graph trigonometric functions and inverses and their transformations - apply them to the solutions of real-world problems Understand the concept that trig functions are cyclic in nature Use of graphing calculator to explore the effects of: - parameters (amplitude and frequency variations) - circular functions Domain and range for trig functions and their inverses Write the equation of a graph given the graph of a trig function SUGGESTED CLASSROOM SUGGESTED ASSESSMENT Quizzes/tests (Should incorporate open-ended problemsolving situations requiring students to show work and/or explain the mathematics they used) Trigonometric Graphs and Transformations How can you graph sine, cosine, tangent by using: - a table of values? - A graphing calculator? What do graphs of trig functions compare? What are definitions of amplitude, period, and frequency? What is the equation of a trig function given its graph? Graph by using a table of values beginning at 0 through 360 at intervals of 30 : - y = sin x - y = cos x - y = tan x Give domain and range for these graphs Sketch the six basic trig functions and their inverses on a graphing calculator by superimposing each function with its inverse Use a graphing calculator to demonstrate for different values of a and b : - y = a sin bx - y = a cos bx Give the domain and range for these graph Math B

13 B 3.4F 3.4K 3.5A 3.5B 3.5C 3.5E 3.5F 3.7Q Trigonometric Identities, Equations, and Problem- Solving What are vectors? What is resultant? What is Law of Sines? What is Law of Cosines? What does it mean to solve a triangle? What are alternate area formulas for area of a triangle and area of a parallelogram using trig? What is the ambiguous case? How do you use trig identities to solve trig equations? What techniques can be used to solve trig equations? Solve trig equations first and second degree - factoring - isolating the radical - quadratic formula Error analysis of solutions to trig equations Apply trig identities to simplify expressions Use vector to solve problems involving forces and resultant Apply Law of Sines and Laws of Cosines to solve a triangle Use trig area formulas to find area of: - triangle - parallelogram Trig Identities - basic - Pythagorean Use of real world situations in problems like the following: Have students draw a diagram and solve: - Two forces of 40lbs. and 62lbs. act on an object at an angle of 53. Find the magnitude of the resultant force and the angle between the resultant and the greater force - Solve the Δ ABC if m - Find the area of a triangle if a = 16.5, b = 9.3 and the measure of - How many triangles exit if the measure of - Find all values of x in interval 0 for which sin x cos 2x = 4 (give answers to nearest ten minutes or tenth of a degree) problem-solving Math B

14 B Statistic 3.5G 3.5I 3.5J 3.6A 3.6E 3.6F 3.6G 3.7P GUIDING QUESTIONS ESSENTIAL KNOWLEDGE SKILLS SUGGESTED CLASSROOM SUGGESTED ASSESSMENT What is statistics? What are measures of central tendency and how are they calculated? How do you choose appropriate statistical measures? What is most appropriate graph for a given set of data? What are measures of dispersion? What is sigma? Using a graphing calculator, how do you interpolate and extrapolate data to produce scatter plots, and lines of best fit? Find measures of central tendency and measures of dispersion for a given set of data and a grouped set of data Produce scatter plots and lines of best fit for a given set of data Calculate standard deviation Draw conclusions from a given set of data following a normal distribution Compare graphic and algebraic models for: - mean - median - variation - standard deviation - mode - range Use of Σ (Sigma) notation Use graphing calculator to interpolate and extrapolate data Use curve fitting to fit data: - linear - logarithm - power regressing from scatter plots - linear correction coefficient Calculate If 2400 students took a standardized test the results of which fall into a normal distribution, how many students scored within one standard deviation from the mean? Draw a bell curve that has a mean of 85 and a standard deviation of 3 Record in seconds, time for each student to run 100 meter dash. Also record their height in inches. Sketch a scatter plot of the data: - Can any conclusions be made concerning height and speed? - Using a graphing calculator, find the equation of the line of best fit - Does equation support conclusions? - Make predictions for other student s based on their height - Discuss the accuracy of these predictions Given the data for test scores: - Find the mean and standard deviation (use a graphing calculator) problem-solving Math B

15 B Probability 3.5I 3.5J 3.6C 3.6D 3.6F What is probability? How is Pascal s triangle formed? What is the binomal theorem? How do you find the 1 st, last, and n th term in a binomial expansion? How do you find exactly, at least, at most successes of an experiment? Know appropriate techniques for random sampling Make predictions and analyze for the possibilities of error in the predictions Pascal s triangle Use the binomial theorem in sample applications Find the nth term in a binomial expansion Find the probability of Bernoulli experiments Use normal distribution percentages to determine probability of results What is the probability of picking a king or a black card from a standard deck of cards? In how many ways can seven people be seated in three seats? How many sub-committees of four people can be chosen from a committee of 8? If a chorus has six altos and five sopranos, in how many ways can three altos and two sopranos represent the chorus? What is the probability of rolling exactly 2 fours in 5 tosses of a 6-sided die? If probability of Mudville winning is 3/5 when playing Beeville, what is the probability that Mudville wins at least one game out of five during the season? Create Pascal s Triangle using combinations Expand: - (x + Y) - (x 2y) What is the 6 th term in the expansion of (2x- 3y)9? problem-solving exercise Math B