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East Irondequoit Central School District Course: Math SL 12 IB Written: June 2010 Enduring Understandings: o o o o o There are a variety of mathematical strategies that can be used to identify, explore, analyze and solve problems. Mathematical reasoning and logic are necessary to develop, evaluate and defend (justify) mathematical ideas and conjectures. The ability to coherently communicate to others (both orally and in writing) mathematical ideas, processes and strategies is critical to demonstrating mathematical understanding. Mathematical concepts and ideas interconnect and have application to contexts outside of mathematics. Mathematical ideas and concepts can be represented in a variety of ways. Essential Questions Content/Skills Unit 1: Limits Review of Function What is a function? -Define relation/function -Identify a function in its standard form. -Graphing functions and listing its characteristics: domain/range, increasing/decreasing. What are the domain and range? Learning Experiences IB Topic Resource Assessments ch1 (Haese) 1 Family of Functions (graphic organizer) -Graphing rational functions by hand to understand the end behavior of a function (vertical and horizontal asymptotes) How are limits and asymptotes similar when discussing convergence? **< - Informal ideas of limit and convergence - Definition of a limit - Finding limits by direct substitution - Finding limits graphically - Finding limits using the table method - One-sided limits - Finding limits when x approaches infinity http://www.calculus-help.com/funstuff/phobe.html> # of Days 2 7.1 ch21 (Haese) Video** Mini-quiz on Limits (Formative Assessment) 3 East Irondequoit Central School District Course: Math SL 12 IB Essential Questions Written: June 2010 Content/Skills Unit 1 cont’d What does it mean for a function to be continuous? Learning Experiences IB Topic Resources - Horizontal asymptotes as limits at infinity - Vertical asymp. as limits that are infinite. 7.7 - Definition of continuity - Finding intervals of continuity using set and interval notation. 7.1 Assessments ch22 (Haese) Supplement: ch20 (Cirrito) # of Days 1 Unit 1 Test (Summative Assessment) 1 East Irondequoit Central School District Course: Math SL 12 IB Essential Questions Content/Skills Unit 2: Definition of a Derivative What is the derivative of a parabola? - Derivative/gradient as the slope and rate of change of the tangent line. Notations used for differentiation (students will discover finding the average velocity using slope). - Students will use the idea of a limit to get the distance between two points closer and closer to zero. - Graph the f ’(x) function given f (x) Why are limits used - Definition of derivative as in finding the derivative of a f ( x h) f ( x ) f ' ( x) lim function? How is it h 0 h used? (polynomial functions only) How can the definition of a derivative be used to help us find the equation of the tangent line? Written: June 2010 Learning Experiences IB Topic Resources 7.1 Assessments ch20 (Haese) # of Days 1 Website of secant line becoming tangent line. http://www.calculus help.com/funstuff/tu torials/limits/limit06. html 1 ch21 (Haese) 2 Supplement: ch19 (Cirrito) - Definition of derivative with fractions 1 - Find the equation of a tangent and normal line. – derivative as slope of tangent line at a point – normal line is perpendicular to tangent line (negative reciprocal slopes) 1 Unit 2 Review & Test (Summative Assessment) 2 East Irondequoit Central School District Course: Math SL 12 IB Essential Questions Written: June 2010 Content/Skills Unit 3: Rules of Differentiation - familiarity with both forms of notation, When taking the derivative of a dy polynomial, what is dx and f (x) the pattern between the original Power Rule: function and the - Derivatives of x n (n Q), sin x, cos x, tan derivative? x, ex, and ln x Learning Experiences IB Topics Resources 7.2 Assessments Supplement: ch19 (Cirrito) # of Days 1 1 Chain Rule: If y f (u ) where u u (x) then dy dy du dx du dx 1 Product Rule: d uv u dv v du dx dx dx 1 Quotient Rule: d u dx v v du dv u dx dx 2 v Review of rules all together Quiz on Rules of Differentiation (Formative Assessment) .5 East Irondequoit Central School District Course: Math SL 12 IB Essential Questions Written: June 2010 Content/Skills Unit 3 Rule of Differentiation cont’d Learning Experiences IB Topics Resources - Derivative of basic trigonometric quantities (composites with the linear function ax + b) 7.4 ch23-24 (Haese) Knowledge and use of the following formulas: 7.2 ch21 (Haese) Assessments # of Days 2.5 d u v du dv dx dx dx d u v du dv dx dx dx .5 d cu c du dx dx - Further extension on finding equations of tangent and normal lines using differentiation rules. ch23 (Haese) .5 Unit 3 Review & Test (Summative Assessment) 2 East Irondequoit Central School District Course: Math SL 12 IB Written: June 2010 Essential Questions Content/Skills Unit 4: Applications of Differentiation Learning Experiences IB Topic Resources Assessments How can you use the derivative to determine if a function is increasing or decreasing? - Understand the meaning of a derivative as rate of change in order to determine whether a function is increasing or decreasing using the first derivative test. 7.1,7.3 ch22 (Haese) 1 If the first derivative represents the change in position (or velocity) then what would the second derivative represent? – Second derivative as the derivative of the d 2 y d dy first derivative dx 2 dx dx 7.2 Supplement: ch20 (Cirrito) 1 ch21 (Haese) – Familiarity with the following notation: d2y f (x) , y , dx 2 - Using the second derivative test to find concavity. How do the signs of the first and second derivative influence the shapes of the graphs? # of Days Fully Discussed Problem: – First Derivative Test (inc/dec test) – Second Derivative Test (concavity) – Points of inflection as changes in concavity with zero and non-zero gradients. – Inflection points (horizontal vs. nonhorizontal inflection points) – Combination of above knowledge and skills to sketch graphs 1 7.1,7.3 7.7 Quiz on one fully discussed question (Formative Assessment) 1 East Irondequoit Central School District Course: Math SL 12 IB Written: June 2010 Unit 4 Applications of Differentiation cont’d – Identification of local and global (absolute How can calculus extrema) max/min points. be used to find the max/min points of a function? – Use of first and second derivative in optimization problems such as profit, area, volume. 1 7.3 Optimization Packet 4 Unit 4 Review & Test (Summative Assessment) 2 East Irondequoit Central School District Course: Math SL 12 IB Written: June 2010 Essential Questions Content/Skills Unit 5: Kinematics What is the difference between distance and displacement? – Kinematics problems involving displacement, s, velocity, v, and acceleration, a. How would you find the rate of change in displacement? Velocity? Acceleration? How would you determine if velocity is increasing? Decreasing? Zero? v ds dv d 2 s , a dt dt dt 2 Learning Experiences IB Topic Resources 7.6 ch22 (Haese) Supplement: ch21 (Cirrito) Assessments Kinematics Quiz (Summative Assessment) # of Days 3-4 East Irondequoit Central School District Course: Math SL 12 IB Written: June 2010 Essential Questions Content/Skills Unit 6: Integration What if you were given the derivative of a function and you wanted to find the original function? Is there a way to work backwards? - Area estimation using rectangles - Left, right and midpoint - Introduction to integral as area under curve Learning Experiences Resources Strategies not a standard (further extension) ch25 (Haese) Supplement: ch22 (Cirrito) ch26 (Haese) Assessments # of Days 1 1 - Indefinite integration as antidifferentiation. 7.4 - Indefinite integral as a function of x with an arbitrary constant term. If F ( x) f ( x) then f ( x)dx F ( x) C ch26 (Haese) - Indefinite integral of the following basic 1 functions x n (n Q), sin x, cos x, and ex. x 1 Example: dx ln x C , x 0 x ch26 (Haese) 1 - Integrating: e ax + b and (ax + b)n ch27 (Haese) 2 - The composites ax + b of a linear function or trigonometric function. Example: f ' ( x) cos( 2 x 3) f ( x) 1 sin( 2 x 3) C. 2 East Irondequoit Central School District Course: Math SL 12 IB Written: June 2010 Unit 6 Integration cont’d What information is needed to determine the constant term of an indefinite integral? What’s the difference between definite and indefinite integral? - Integration with a boundary condition to determine the constant term. 7.5 Supplement: ch22 (Cirrito) Example: dy 3x 2 x and y=10 when If dx 1 x=0, then y x 3 x 2 10 2 - Finding definite integrals using the Fundamental Theorem of Calculus 1 Indefinite Integration Quiz (Formative Assessment) 1 7.4 b f ( x)dx F (b) F (a) a Pg 660 (Haese) Supplement: ch22 (Cirrito) - Indefinite integral as a function vs. definite integral as a number. - Method of Substitution (U-Substitution) 7.5 ch26-27 (Haese) 1 .5 - Basic integration properties (rules). Unit 6 Review & Test (Summative Assessment) 2 East Irondequoit Central School District Course: Math SL 12 IB Written: June 2010 Essential Questions Content/Skills Unit 7: Applications of Integration How can we use integration to find the exact area under a curve? What information is needed? – Using integration and differentiation to solve kinematics problems involving displacement, s, velocity, v, and acceleration, a ds dv d 2 s v , a dt dt dt 2 What does the area under a timevelocity graph represent? How can we use integration to find the volume of revolution of a linear function? What shape does it make and what is the geometric formula to find the volume of this shape? -Areas under curves (between the curve and the x-axis), areas between curves. - Volumes of revolution about the x-axis and identifying the geometric shape. Learning Experiences IB Topics Resource 7.6 ch26 (Haese) Supplement: ch22 (Cirrito) 7.5 ch28 (Haese) Assessments # of Days Quiz on area under a curve & kinematic problems involving integration (Formative Assessment) 2 3 1 Supplement: ch22 (Cirrito) 1 - volume of revolution of one function b V f ( x) 2 dx a 1 - volumes for two defining functions b V f ( x) g ( x) dx 2 2 a Unit 7 Review & Test (Summative Assessment) 2 East Irondequoit Central School District Course: Math SL 12 IB Written: June 2010 Essential Questions Content/Skills Unit 8: Statistics What is the difference between discrete and continuous random variables? - Understand population, sample, and the process of statistics. - Discrete vs. continuous data 6.1 ch18 (Haese) 1 - Presentation of data using: frequency tables, frequency histogram, column graphs and stem-leaf-plot. - Grouped data (find the mean) - Mid-interval values - Interval width - Upper and lower interval boundaries 6.2 Supplement: ch13 (Cirrito) 2-3 How do organize data and make predictions using it? Learning Experiences IB Topics Resources Assessments # of Days - Calculate the range, mean, median and mode (or modal) of a table and/or diagram. - Construct and use a box and whisker plot to find the five-number summary of a data set. 1 - Determine if a graph is negatively or positively skewed. .5 - Construct a cumulative frequency table and curve. 6.4 1 - Use raw data, tables, and/or graphs to find the median, quartiles and percentiles. 6.3 1 - Understand the concept of standard deviation and variance and be able to calculate both 6.3 Unit 8 Test (Summative Assessment) .5 East Irondequoit Central School District Course: Math SL 12 IB Essential Questions Content/Skills Unit 9: Probability How do we mathematically notate sets of numbers? - Basic introduction of probability along with key vocabulary words. - Concepts of trial, outcome, equally likely outcomes, sample space (U) and event - The probability of an event A as n A P A nU - Theoretical vs. Experimental Probability - The complementary events A and A’ (not A) : P(A) + P(A’) = 1 Are non/mutually exclusive and de/independent events related to each other? What tool can you use to find the probability of non/mutually exclusive? What about dependent and independent events? Written: June 2010 Learning Experiences IB Topics Resources 6.5 Assessments ch19 (Haese) 1 Supplement: ch15(Cirrito) - Use of Venn Diagrams, tree diagrams and tables of outcomes to solve problems. 6.8 - Finding the probability of a combined event with and without replacement. 6.6 - Independent events - The definition: P A / B P A P A / B' 6.7 Venn & Tree Diagram Quiz (Formative Assessment) 2 2 1 - Conditional probability 1 P A B - The definition: P A / B P B -Mutually exclusive events: P A B 0 - Combined events using the formula: P A B P A PB P A B # of Days 6.6 Unit 9 Test (Summative Assessment) 1 East Irondequoit Central School District Course: Math SL 12 IB Essential Questions Content/Skills Unit 10: Statistical Distribution How can discrete random variables and their probability distributions be used in a game of chance? What are some examples of theses games? - Concept of discrete random variables and their probability distributions such as: 1 P X x 4 x for x {1,2,3} 18 - Applications of expectation, for example, game of chance. How are binomial distributions different than normal distributions? How are binomial distributions easily identified? - Calculate probabilities for the binomial distribution. - Find the mean of a binomial distribution. Written: June 2010 Learning Experiences IB Topics Resources 6.9 Assessments ch29 (Haese) 1 Supplement: ch16(Cirrito) 1 - Find the expected value (mean), E(x) for discrete data. - Knowledge and use of the formula: E X xP X x - Understand the properties of a normal distribution. -Calculate probabilities for a normal distribution and appreciate that the standardized value (z) gives the number of standard deviations from the mean. - Standardization of a normal variable. - Use of calculator or tables to find normal probabilities; the reverse process. # of Days 2 6.10 .5 6.11 Supplement: ch17(Cirrito) Normal & Binomial Distribution Quiz (Formative Assessment) Unit 10 Test (Summative Assessment) 1.5 2