Year 11 Maths B Differentiated Unit Plan Title: Unit 10 Rates of Change 1 Broad Objective: Time: Specific Objective: Students should be encouraged to develop an understanding of average and instantaneous rates of change and of the derivative as a function. This understanding should be developed using algebraic and graphical approaches. Students should be expected to apply the rules for differentiation and interpret the results. The use of technology should help students in these processes. Term 3 8 Hours By the end of these units, students will be familiar with: Rates of change Concept of rate of change calculation of average rates of change in both practical and purely mathematical situations And develop skills and understanding of: interpretation of the average rate of change as the gradient of the secant practical applications of instantaneous rates of change Senior Mathematics Learning Area PAJ- Edited 2016, 2017DSF Page 1
Chapter 12 Exercises 12A-12G Mathematics B Rates of change Syllabus Subject Matter - Concept of rate of change (SLEs 1, 2) 1 Sem 1 Sem 2 Sem 3 Sem 4 Exercises Covered this Unit 2 3 4 5 6 7 8 9 10 11 Calculation of average rates of change in both practical and purely mathematical situations (SLEs 1, 2) Interpretation of the average rate of change as the gradient of the secant (SLEs 1, 2) Understanding of a limit in simple situations (SLEs 3, 4) Definition of the derivative of a function at a point (SLEs 5, 6, 7) Derivative of simple algebraic functions from first principles (SLEs 3, 4, 5) of the derivative of a function at a point (SLEs 5, 6, 7) Interpretation of instantaneous rate of change at a point as the gradient of a tangent and as the derivative at that point (SLEs 1, 2, 5, 10) Rules for differentiation including, (Chain and Product Rule) Interpretation of the derivative as the gradient function (SLEs 1, 2, 5, 6, 7) Practical applications of instantaneous rates of change (SLEs 1, 2, 5 12). Senior Mathematics Learning Area PAJ- Edited 2016, 2017DSF Page 2
Assessment Date Use of ICT s Supporting and Learning Diagnostic (precedes instruction): Before Section 1 Casio Graphics Calculator Pretest on Rates of Change Graph 4.3 Formative (used during instruction to guide Geometers Sketchpad Ongoing teaching): Math Type quizzes, revisions sheets, Ongoing Spreadsheets Online /Quizzes Summative (used after instruction to measure achievement and allocate results): Logger Pro Term 3 Exam ONENOTE Internet Learn@SPS Senior Mathematics Learning Area PAJ- Edited 2016, 2017DSF Page 3
Section 1 CONSTANT RATES AND VARIABLE RATES Constant Rates Graphing Rates Calculating Gradients Questions : Ex 12A-B - All Diagnostic:A diagnostic session will be carried out in the form of a mini-quiz for the first 5 minutes. This session will be multimodal. Two teaching strategies will occur (Informal Formative Test and Brainstorming). Compare & Contrast: Discuss the difference between constant and variable rates of change. Students will then look at similarities and differences. Model:Definitions of gradient and how to calculate from given points or from graph. Represent rates on a graph showing correct conventions. Model:Referring to the first section, compare constant and variable. The goal for students is to be able to correctly identify variable vs. constant. Independent Practice. Students will complete Questions from Exercise 12A -B. The teacher provides feedback and assists evaluate their progress and achievements. Students will reflect on the learning of the lesson by summarizing the key points of the lesson. Students will continually review consolidating concepts such as Gradient and use strategies such as Word Maps to assist in their learning. Senior Mathematics Learning Area PAJ- Edited 2016, 2017DSF Page 4
Section 2 AVERAGE RATES OF CHANGE AND INSTANTANEOUS RATES Average Rates of Change and Applications Instantaneous Rates of Change and Applications Calculating AROC and IROC Questions : Ex 12C-D Activate Prior Knowledge: Refer to last lessons content and review differences of variable and constant rates Model: Define chord a straight line on the inside of a curve, joining two points on the curve. Define secant an extended chord (actually passes through the curve points). Average rate of change is the gradient of a chord (or secant line). Note that with average RoC, the actual path taken between the two points is ignored have students discuss the implications this has to the RoC. Practice calculating average rates of change. Venn Diagram: Compare and contrast the average rate and instantaneous rates of changer using this strategy. This will be teacher led. Model:Use an electronic spreadsheet to investigate the gradient of a secant as one of the points approaches the other (i.e. as the secant approaches the tangent). Practice drawing tangent lines. Use ClassPad to generate a table of instantaneous rates of change this can be used as a tool to check tangent values. Calculate the gradient of the tangent line - since point P lies on that tangent line, it must have the same gradient, therefore the instantaneous gradient at point P is the gradient of the tangent line. Independent Practice. Students will complete Questions from Ex 12C-D. The teacher provides feedback and assists evaluate their progress and achievements. Students will reflect on the learning of the lesson by summarizing the key points of the lesson. Students will continually review consolidating concepts such as gradient, secant, cord and rate of change and use strategies such as Venn Diagrams to assist in their learning. Senior Mathematics Learning Area PAJ- Edited 2016, 2017DSF Page 5
Section 3 MOTION GRAPHS Understand the difference between vector and scalar quantities: o Displacement cf. distance o Velocity cf. speed Be able to generate the gradient function from the original function using tangents Understand that the gradient of a displacement-time graph will give the velocity Understand that the gradient of a velocity-time graph will give acceleration. Activate Prior Knowledge: Calculating AROC and IROC using formulae etc. Be able to compare displacement-time graphs and draw contextual meaning. Be able to compare velocity-time graphs and draw contextual meaning. Questions : Ex 12E Model: Define the following terms, giving examples of each. Position describes where the object is or was ( x ), Distance how far an object has travelled ( d ), Displacement the change in an object s position ( s ), Speed = the average distance covered per unit of time ( sp d t ), Velocity = the rate of change of position with respect to time ( v s ), Relate velocity to the gradient of t the displacement graph this can be done through the formula or using units. Practice describing the gradient of position-time graphs continuously relate to velocity. Enquiry Learning:Show a video clip of person in motion running at variable velocities. In groups students will attempt to graph and describe these in graphical form. Guided Practice: Complete questions calculating gradients on displacement and velocity y curves and evaluate the results. Independent Practice. Students will complete Questions from Exercise 12E. The teacher provides feedback and assists evaluate their progress and achievements. Students will reflect on the learning of the lesson by summarizing the key points of the lesson. Students will continually review consolidating concepts such as Gradient of a Line and use strategies such as Word Maps to assist in their learning. Senior Mathematics Learning Area PAJ- Edited 2016, 2017DSF Page 6
Section 4 RELATING THE GRADIENT FUNCTION TO THE ORIGINAL FUNCTION Use the ClassPad to generate table of values for the tangents of polynomials. Generate a gradient function and compare the shape to the original function. Be given a gradient function and generate an original function. Questions : Ex 12F Activate Prior Knowledge: Calculating tangents from motion graphs to find velocity or acceleration. Think Pair Share: The objective for students is to be able to accurately select what displacement curve will match to the corresponding velocity curve. Model: Generate a table of instantaneous rates of change for: Graphs, Rules/formulae. Graph the gradient function stress that the x-axis values do not change. Have the students make links between the gradient function and the original function. Look at specific points, e.g. Maximums, minimums and points of inflection on the original function (stationary points) gradient is zero at these points and so will cross the x-axis on the gradient function. Have students link the direction Points half way between two turning points on the original function will give a maximum or minimum gradient (either positive or negative). Have students relate the direction to the sign of the gradient function. Give cubics or trig functions as examples. Asymptotes on the original function have students think about what is happening to the gradient at these points. Give log or exponential functions as examples. Guided Practice: Graphing gradient functions and comparing them to the original. Independent Practice. Students will complete Questions Exercise 12F. The teacher provides feedback and assists students to evaluate their progress and achievements. Students will reflect on the learning of the lesson by summarizing the key points of the lesson. Students will continually review consolidating concepts such as generating gradient values from the CASIO and use strategies such as Graphic Organisers to assist in their learning. Senior Mathematics Learning Area PAJ- Edited 2016, 2017DSF Page 7
Section 5 RELATING VELOCITY-TIME GRAPHS TO POSITION-TIME GRAPHS Construct a velocity-time graph from a displacement-time graph Construct a displacement-time graph from a velocity-time graph. Questions : Ex 12G Activate Prior Knowledge: Refer to relating the gradient function to the original. Talk with class about the methods needed to solve such problems. Model: Practice sketching the velocity-time graph from position-time graphs. Use the skills from the previous lesson (these may need to be re-covered briefly). Guided Practice: Complete sketching problems from Exercise 12G to scaffold the students learning. Self :Reflect on the content studied this unit. Highlight strengths and weaknesses and focus on these during revision for the exam. Independent Practice. Students will complete Questions 12G. The teacher provides feedback and assists evaluate their progress and achievements. Students will reflect on the learning of the lesson by summarizing the key points of the lesson. Students will continually review consolidating concepts such as Gradient graphs of functions and use strategies such as Retrieval Charts to assist in their learning. Senior Mathematics Learning Area PAJ- Edited 2016, 2017DSF Page 8