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PC Sumpter, C Mutch Sample Presentation 3.6AS91578 Apply differentiation methods in solving problems
AS-3.6 Index 3.6-F : 2 17-May-14 ©PP Resources Graphical Calculator Hints Return Index of Sample Slides Find / check the value of the derivative Finding points of inflection. Find the maximum or minimum points. 2 – The Triangular Distribution 1 – Turning Points and Gradients 3 – Venn Diagrams 4 – Parabolas Title Page AS 3.6Conditions Each PowerPoint has an Index page. Conditions of Assessment 5 – Important information about these presentations. Achievement Standard
AS-3.6 Index 3.6-F : 3 17-May-14 ©PP Resources Return The GC can give find the value of the derivative for any given value of x. For example, to find the value of the gradient (or derivative) of y = 2x 3 -7 at (5, 1), enter: (At the very least you must show the derivative.) (To gain credit however you must show working) d/dx( 2X^3 - 7, 5 ) = 150
AS-3.6 Index 3.6-F : 4 17-May-14 ©PP Resources Decreasing f ’(x) < 0 “down as x increases” Increasing and Decreasing Functions Increasing f ’(x) > 0 “up as x increases” Decreasing f ’(x) < 0 “down as x increases”
AS-3.6 Index 3.6-F : 5 17-May-14 ©PP Resources e.g. Find the nature and position of the turning point of the function y = 3x² - 18x + 23 Turning point when 6x-18 = 0 So there is a turning point at (3,-4) x = 3,so y = 3.3² = -4 GC Link to method using the Graphics Calculator. Full control over steps to encourage student involvement.
AS-3.6 Index 3.6-F : 6 17-May-14 ©PP Resources e.g. Find the nature and position of the turning point of the function y = 3x² - 18x + 23 Check gradient on either side: turning point at (3, -4) Exercises D-15 f ’(2) = - 6, so the turning point is a minimum f ’(4) = + 6, y = 3x² - 18x + 23 has a minimum at (3, -4) Large print for clarity. Spreadsheet supplied to link this code to exercises in some commonly used texts and workbooks.
AS-3.6 Index 3.6-F : 7 17-May-14 ©PP Resources e.g. Find the nature and position of the turning point of the function y = 3x² - 18x + 23 Graph the function on the Calculator There is a turning point at (3,-4) Inspection shows the turning point is a minimum Solve for a minimum Solve without a GC Method using a GC Derivative must be shown.
AS-3.6 Index 3.6-F : 8 17-May-14 ©PP Resources Nina River Gorge In most units, sections are broken with photos of the New Zealand outdoors.
AS-3.6 Index 3.6-F : 9 17-May-14 ©PP Resources AS Algebra Identify discontinuities and limits of functions Choose and apply a variety of differentiation techniques to functions and relations using analytical methods CriteriaExplanatory Notes Apply differentiation methods in solving problems. Apply differentiation methods in solving problems involves: selecting and using methods demonstrating knowledge of concepts and terms communicating using appropriate representations. Apply differentiation methods, using relational thinking, in solving problems. Relational thinking involves one or more of: selecting and carrying out a logical sequence of steps connecting different concepts or representations demonstrating understanding of concepts forming and using a model; and also relating findings to a context, or communicating thinking using appropriate mathematical statements. Apply differentiation methods, using extended abstract thinking, in solving problems. Extended abstract thinking involves one or more of: devising a strategy to investigate or solve a problem identifying relevant concepts in context developing a chain of logical reasoning, or proof forming a generalisation; and also using correct mathematical statements, or communicating mathematical insight. Problems are situations that provide opportunities to apply knowledge or understanding of mathematical concepts and methods. Situations will be set in real-life or mathematical contexts. Methods are selected from those related to: ◙ derivatives of power, exponential, and logarithmic (base e only) functions ◙ derivatives of trigonometric (including reciprocal) functions ◙ optimisation ◙ equations of normals ◙ maxima and minima and points of inflection ◙ related rates of change ◙ derivatives of parametric functions ◙ chain, product, and quotient rules ◙ properties of graphs (limits, differentiability, continuity, concavity). Achievement Objectives Merit Excellence Achieved
AS-3.6 Index 3.6-F : May-14 ©PP Resources The triangular distribution is used where: 1.The most likely result is known or estimated 2.The probable upper and lower limits are known or estimated, and 3.The actual shape is unknown, but central values are more likely. The triangular distribution.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Finding maximum & minimum points on a Graphics Calculator Use the solve function to solve for a maximum or solve for a minimum. Graph the function –Ensure you have set the view window to show the stationery points. (To gain credit however you must show working) (At the very least you must show the derivative.) Answer: Maximum at (0, 36) and minimum at (4, 4) Example: Find the co-ordinates of the stationary points of the function y = x 3 – 6x Return
AS-3.6 Index 3.6-F : May-14 ©PP Resources Southern Alps - Culverden
AS-3.6 Index 3.6-F : May-14 ©PP Resources The GC can find the Point(s) of Inflection To gain credit however you must show working (At the very least you must show the derivative.) Use the solve (min or max) function to find the value of X for which the derivative is at a maximum or minimum. Answer: Point of Inflection at (2, 20) Example: Find the co-ordinates of the point of inflection of the function y = x 3 – 6x Differentiate and graph the derivative. Return Substitute this X value into the original function to get Y.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Venn Diagrams: When dealing with two or more events a Venn Diagram is often a useful tool. A picture card (Jack/Queen/King) is drawn at random from a pack of cards. Example: Two events are defined: C = the card is a Club ( ) K = the card is a King Navigation is a breeze. Every slide has buttons for next, previous, last slide visited and index.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Venn Diagram: The symbol means "and" or "intersection" Sample space: K is in C and K KK QQ KK KK K JJ JJ JJ J QQ QQ Q K is in the intersection of both sets. Teacher has full control over the pace to encourage student participation.
AS-3.6 Index 3.6-F : May-14 ©PP Resources General assessment specifications for Mathematics external standards (1) Format for the assessment All questions will provide opportunity for candidates to demonstrate all levels of performance: Achievement, Achievement with Merit, and Achievement with Excellence. This can be achieved in a variety of ways: The questions may have multiple parts. The parts of a question may be linked. There may be scaffolding within the question. Opportunities for Merit and Excellence will be spread through the paper. Question parts may not be arranged in order of increasing difficulty. Correct answers only may not be sufficient for showing evidence of the level of thinking required by the standard. Unless a method is specified within a question, candidates may choose their method when solving a problem although the grade awarded may be affected by the level of thinking applied in solving the problem. Guess and check methods are unlikely to show the required thinking. Candidates must show any working that is asked for in the assessment e.g. derivatives, anti- derivatives, and equations. Equipment to bring Candidates must bring an approved calculator (preferably a graphing calculator). Candidates who do not have access to graphing calculators will be disadvantaged.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Venn Diagram: The symbol means "or" or "union" Sample space: All Clubs and Kings are in C or K (or both) KK QQ KK KK K JJ JJ JJ J QQ QQ Q This is called the union of both sets.
AS-3.6 Index 3.6-F : May-14 ©PP Resources General assessment specifications for Mathematics external standards (2) A Formulae and Tables Booklet will be provided. Special notes Candidates will be required to answer questions that demonstrate an understanding of the mathematical concepts rather than directly transferring results from their graphing calculator. This may involve use of unknown constants. As good mathematical practice, candidates should be encouraged to show intermediate steps clearly and logically, communicating what is being calculated. Candidates who give the correct response only may lose the opportunity to provide evidence for other grades or to have minor errors ignored. Unless otherwise stated, rounding of any numerical answers to three significant figures will be of sufficient accuracy. Minor errors caused by rounding will not be penalised. Inappropriate use of units may count as a minor error and may not be penalised. It is expected that relevant working will be shown. When graphing calculators are used to solve a problem, candidates must provide evidence of their differentiation and integration skills. Content / context details Problems will be set in real life or in mathematical contexts. Solutions for problems providing opportunities for Achievement with Merit and Achievement with Excellence may incorporate content knowledge from other level 3 Calculus achievement standards. Problems that allow candidates to provide evidence for Achievement with Excellence may require candidates to devise their own model.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Remember: The symbol means "and". Think of the rowlock that holds the oar on a row boat! The symbol means "or". We write P(A and B both occur) as P(A B) We write P(A or B occur) as P(A B) (as in “fish ‘n’ chips”)
AS-3.6 Index 3.6-F : May-14 ©PP Resources Assessment Specifications for AS91578 Differentiation Further clarification of the achievement standard Candidates using graphing calculators will not receive credit for correct solutions to problems assessed against this standard where they have not provided the correct derived function. Problems assessing optimisation at Achievement and Achievement with Merit level will not require candidates to prove that a solution is a maximum or a minimum. This will be given by a statement such as: “You may assume that “ ” OR “You may assume that your solution is a minimum”. Problems will assess understanding of concepts of differentiation. Candidates must show any derivatives that are needed to solve the problems. Candidates may be required to form their own equations for problems that provide evidence for Achievement with Merit and Achievement with Excellence. Equations of tangents may be included.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Venn Diagrams: Little Mount Peel, Canterbury
AS-3.6 Index 3.6-F : May-14 ©PP Resources The standard y = x 2 parabola is translated a ← and b↑ e.g sketch the curve y = (x + 1 ) 2 – 4 Note: If a is positive, the curve moves left y = (x + a ) 2 + b Minimum point = (-1, -4) Parabolas Revision (Transformed) The curve moves 1← and 4↓ Exercise G-02 Animations are in small steps to encourage student participation.
AS-3.6 Index 3.6-F : May-14 ©PP Resources The standard y = x 2 parabola can be plotted easily and simply: e.g sketch the curve y = (x – 2) 2 – 5 1. Locate the minimum point Minimum point = (2, -5) Parabolas – Quick Sketching(1) The curve moves 2→ and 5↓ 2. For each unit across, move up 1, then 3, then 5 etc.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Orari Gorge In most units, sections are broken with photos of the New Zealand outdoors.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Some points about these slides. These are a mixed random selection from over 500 slides at Level 2 and 1700 slides at Level 3. Every slide has these navigation buttons. Chapter PageAchievement Standard Index
AS-3.6 Index 3.6-F : May-14 ©PP Resources The slides are not intended to be a self- teaching resource. They are intended to supplement teaching in the class room. They are designed to replace hand written notes on the whiteboard, although the teacher will still use the white board to demonstrate more examples.
AS-3.6 Index 3.6-F : May-14 ©PP Resources The font size is large and clear. In our experience a small font size tends to make the content look more difficult. A font that is large and easily read from everywhere in the room will enhance the student’s understanding. Animation is used sparingly. Fancy “fly ins” and other effects (such as artistic backgrounds) have proved a distraction to the learners.
AS-3.6 Index 3.6-F : May-14 ©PP Resources These presentations are fully editable by the user. The notes are copyright protected. They are only for the use of teachers employed at the purchasing school. However, if a teacher employed for the full year in the purchasing school modifies the presentations for their own use, that teacher may continue to use them for their own teaching if they transfer to another school.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Students do not learn by staring at a screen! It is expected that a good teacher will coax each point from the students before clicking the mouse to display it on screen. This ensures student involvement in the explanations and examples. Points on the slides are revealed progressively by a mouse click.
AS-3.6 Index 3.6-F : May-14 ©PP Resources The authors believe that real learning takes place when the students are working from problems and exercises. Using PowerPoint to present notes significantly reduces the time that a teacher spends on whole class teaching. This results in more class time being available for individual support for the students by the teacher.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Some teachers will prefer to do all examples on the whiteboard. In this case simply click the “Right arrow” button below to by-pass the example. We expect that in some topics the slides will provide a framework, but much of your teaching will be done using either a spreadsheet or a graphing package on the data projector.
AS-3.6 Index 3.6-F : May-14 ©PP Resources The benefits of these presentations for the teacher: They reduce the pressure and workload for the teacher. They make teaching even more interactive and enjoyable! Find out more on They free the teacher from tedious whiteboard writing and allow more stimulating interaction with the class! (Especially if you use a remote mouse or presenter.)
AS-3.6 Index 3.6-F : May-14 ©PP Resources End of Presentation Right-Click to go back ESC to end Slideshow Hold Left and Right buttons down for 2 seconds to go to first slide.
AS-3.6 Index 3.6-F : May-14 ©PP Resources Through an extraordinary chance alignment, the Hubble telescope has captured a view of a face-on spiral galaxy lying precisely in front of another larger spiral. The unique pair is called NGC This line-up provides astronomers with the rare chance to see the dark material within the foreground galaxy, seen only because it is silhouetted against the light from the object behind it. NGC 3314 lies about 140 million light-years from Earth in the direction of the southern hemisphere constellation Hydra. Credit: NASA and The Hubble Heritage Team (STScI/AURA)NASASTScIAURA
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