Teaching & Learning Plans Introducing e Leaving Certificate Syllabus Higher level The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve. Prior Knowledge points to relevant knowledge students may already have and also to knowledge which may be necessary in order to support them in accessing this new topic. Learning Outcomes outline what a student will be able to do, know and understand having completed the topic. Relationship to Syllabus refers to the relevant section of either the Junior and/or Leaving Certificate Syllabus. Resources Required lists the resources which will be needed in the teaching and learning of a particular topic. Introducing the topic (in some plans only) outlines an approach to introducing the topic. Lesson Interaction is set out under four sub-headings: i. Student Learning Tasks – Teacher Input: This section focuses on possible lines of inquiry and gives details of the key student tasks and teacher questions which move the lesson forward. ii. Student Activities – Possible Responses: Gives details of possible student reactions and responses and possible misconceptions students may have. iii. Teacher’s Support and Actions: Gives details of teacher actions designed to support and scaffold student learning. iv. Assessing the Learning: Suggests questions a teacher might ask to evaluate whether the goals/learning outcomes are being/have been achieved. This evaluation will inform and direct the teaching and learning activities of the next class(es). Student Activities linked to the lesson(s) are provided at the end of each plan. Teaching & Learning Plans: Introducing e Aims • To introduce the number e as the base rate of growth for all continually growing processes Prior Knowledge Prior knowledge and experience of handling fractions and percentages is required. Students have prior knowledge of • Patterns with numbers • Exponential functions such as y = a2x, y = a3x, where a ∈ N, x ∈ R • Indices • Compound interest and the compound interest formula • Logarithms Learning Outcomes As a result of studying this topic, students will be able to • Link continuously compounded interest and the number e • Understand the relationship between e and the natural logarithm (loge) Real Life Context The following examples could be used to explore real life contexts. • Continuously compounding interest • Bacterial growth • Radioactive decay • Rate of chemical reaction © Project Maths Development Team 2012 www.projectmaths.ie 1 Teaching & Learning Plan: Introducing e Relationship to Leaving Certificate Syllabus Sub-Topic Students learn about In addition students working at HL should be able to 5.1 Functions –– recognise surjective, injective and bijective functions –– find the inverse of a bijective function –– given a graph of a function sketch the graph of its inverse –– express quadratic functions in complete square form –– use the complete square form of a quadratic function to •find the roots and turning points •sketch the function –– graph functions of the form • ax2 + bx + c where a, b, c ∈ Q, x ∈ R • abx where a, b ∈ R •logarithmic •exponential •trigonometric –– interpret equations of the form f(x) and g(x) as a comparison of the above functions –– informally explore limits and continuity of functions © Project Maths Development Team 2012 www.projectmaths.ie 2 Teaching & Learning Plan: Introducing e Lesson Interaction Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning Teacher Reflections Review of exponential functions y = abx where a represents the starting value, b is the »» Remind students of the pocket money example and the story surrounding the payment by the Grand Vizier in grains of wheat for the invention of the game of chess. »» Are students able to recognise that in exponential functions the variable is in the exponent and not in the base? »» Do students know that when a function is exponential there is a constant called a growth factor and that during each time interval, the amount present is multiplied by this growth factor? »» Can they recall that exponential functions such as y = a2x, where a ∈ N, x ∈ R are always increasing? »» What are the variables and constants in this equation? • The starting value a and the rate of growth b, are constant and the variable x appears in the exponent. »» Write the equation y = abx on the board. »» Are students able to distinguish the variable from the constants in an exponential function? »» Plot a graph of y = 2x • Students draw up a table of »» With GeoGebra draw values and plot the graph y = ax and move the slider • Negative exponents lead to to change the value of a to values of y between 0 and highlight this.» 1 but never negative values (See page 16 on how to do of y. this using GeoGebra). • No. No value of x will give negative y. »» What exponential functions have you met to date? • constant rate of growth for a given time interval and x is the number of those time intervals. • In the pocket money example, if you started with €3, doubled your money each day you would have 3(2)4 euro after 4 days. »» What is the effect of negative exponents? What if we worked out 2-1000? »» Is it possible for this function to give negative y values? Note: Point out the similarity between this and geometric sequences. © Project Maths Development Team 2012 www.projectmaths.ie KEY: » next step • student answer/response 3 Teaching & Learning Plan: Introducing e Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning »» What do you notice about • You need to compress the scale »» Remind students of »» Do students see that the scales you are using for on the y axis relative to the x the constant difference exponential functions the x and y axes for the axis scale. The y values start between successive outputs grow quickly because graph of y = 2x? very small and are very close to for linear functions of the of the multiplicative Are they the same? 0 initially but once the y values form y = a + bx and the nature of the start to grow they grow faster constant second differences relationship?» and faster. When x increases by for quadratic functions. E.g. 2x + 1 = 2.2x 1 more than its previous value, 2x + 2 = 22.2x y increases to twice more than it had been. There is a constant ratio between the output values for successive input values. »» What is meant by exponential growth when people speak about it in everyday terms? • Growing slowly initially then growing faster and faster. »» Can you think of any examples of exponential growth? • Bacterial growth 2x, population »» Tell students that population growth can growth approximate to, or be modelled by, exponential growth, but the growth factor in reality is not exactly constant. »» Exponentials functions always have a positive number other than 1 as a base. What does this mean? • © Project Maths Development Team 2012 Teacher Reflections »» Are students able to come up with examples of exponential growth? b in the formula y = abx is always positive www.projectmaths.ie KEY: » next step • student answer/response 4 Teaching & Learning Plan: Introducing e Student Learning Tasks: Teacher Input Student Activities: Possible Responses »» Recall the compound interest formula. What do the variables signify? • »» Compare this with the formula y = abx. What are the similarities? • It is the same type of formula where a = P, the growth factor b is represented by (1 + i) and the variable x is the number of periods of compounding. • Assessing the Learning »» Remind students to »» Are students F = P (1 + i)t use their Formulae familiar with the F = final value, and Tables book. compound interest P = principal (starting value), i = annual interest Teacher Reflections formula or do they need short revision on it? rate,» t = time in years. »» Let us have another look at our “doubling” function and relate it to the compound interest formula. Start with €1 at time t = 0, make a table to show the effect of doubling after each unit of time. »» What is i in the above “doubling” function? Use the C.I. formula to work out each value? Teacher’s Support and Actions Time t Final value F 0 1 1 2 2 4 3 8 4 16 »» Inform students that the banks use different kinds of compounding schemes – yearly, half yearly, quarterly, monthly, weekly and daily. »» Write on the board » »» Do students see i = 100% t t y = (1 + 1)t that the formula t = 1: F = P (1 + i) = 1 (1 + 100/100) = (1 + 1) = 2 t 2 y = 2t and the t = 2: F = P (1 + i) = 1 (1 + 1) = 4 compound interest t = 3: F = P (1 + i)t = 1 (1 + 1)3 = 8 t 4 formula with » t = 4: F = P (1 + i) = 1 (1 + 1) = 16 i = 100% and P = €1 give the same results? © Project Maths Development Team 2012 www.projectmaths.ie KEY: » next step • student answer/response 5 Teaching & Learning Plan: Introducing e Student Learning Tasks: Teacher Input Student Activities: Possible Responses »» The compound interest formula assumes growth occurs in discrete steps, not continuously. Let y = 2x represent the growth of 1 bacterium after x periods of time. A new bacterium does not suddenly appear after 1 unit of time. It is continuously growing from t = 0 to t = 1. Let us see what would happen if we reduced the intervals of time for which interest was compounded so that it simulated natural growth which occurs continuously and not in jumps. »» Let us first split up the year into 2 • periods so that interest is added every six months. We still start with €1 and • instead of 100% interest per year we get 50% every 6 months. Use the C.I. formula to compute the final value at the end of the year. »» Ask students where they've met “continuous” and “discrete” before? »» Are students associating the idea of continuous and discrete growth with continuous and discrete data which they encountered in statistics? Teacher Reflections F = 1 (1 + 1/2)2 = 2.25 • After 6 months the €1 earned 50% interest giving €1.50 which then earned 50% interest giving » €1.50 + €0.75 = €2.25. »» If we now have interest added after 4 equal intervals in the year with interest at (100/4)% after each interval what is the final value at the end of 1 year? • www.projectmaths.ie Assessing the Learning F = P (1 + i)t »» We now have €2.25 after 1 year instead of €2 after 1 year with 100% interest added at the end of the year. Explain. © Project Maths Development Team 2012 Teacher’s Support and Actions F = 1 (1 + 1/4)4 = €2.44 »» Can students see a pattern emerging i.e. the formula» F = 1 (1 + 1/n)n? KEY: » next step • student answer/response 6 Teaching & Learning Plan: Introducing e Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Assessing the Actions Learning »» What do you notice • The final value is greater the shorter the time about the final value period used for adding on interest. as you compound more often? »» Will this trend continue? Investigate in pairs using Student Activity 1. • Students fill in the table for Student Activity 1. »» Distribute Student Activity 1. Teacher Reflections »» Check the accuracy of students’ work.» »» Allow students, who reach the limit of their calculators’ computing ability, to use a spreadsheet. »» Did all students see their answers getting closer and closer to some “limit”? Did some students try to reduce the time further and reach the limit of their calculators’ computing ability? KEY: » next step • student answer/response »» Encourage students to look for patterns. »» What conclusion have you reached? • As the time interval for compounding decreases this becomes more like continuous growth.» • The more often we compound the greater the final value. However further increases in the number of compoundings per year seem only to cause changes in less and less significant digits.» • The rate of growth if we continually compound 100% on smaller and smaller time intervals seems to be about 2.7182 (to four decimal places) as those digits do not change as we reduce the time. © Project Maths Development Team 2012 www.projectmaths.ie 7 Teaching & Learning Plan: Introducing e Student Learning Tasks: Teacher Input Student Activities: Possible Responses »» This rate of growth is a number called » e = 2.71828........ and is the base rate of growth shared by all continuously growing processes. It is a fundamental constant like π and it is irrational like π. What does this mean? » » • A number is irrational when it cannot be written in the form a/b where a, b ∈ Z. »» What does π represent? Teacher’s Support Assessing the and Actions Learning »» Can students articulate what an irrational number is and what is meant by π? • Irrational numbers when expressed as decimals are nonterminating and non- repeating.» Teacher Reflections »» Can they recall any other irrational numbers? • π is the ratio of the circumference of any circle to its diameter. »» We have only approximated e here as we haven’t actually shown continuous growth – only for intervals of 1 second. »» The number e shows up in population growth and in radioactive decay – in systems which exhibit continuous growth or decay. »» Can you generalise the formula you were using where n represents the number of time intervals? • F = 1 (1 + 1/n)n »» If we find the limiting value of F = 1 (1 + 1/n)n as n • Using the calculator:» e1 ≈ 2.71828183 goes to infinity we have the number e. We write this as » » » »» The Swiss mathematician Euler was the first to use the notation e for this irrational number in 1731. »» Use your calculators to find an approximate value for e. © Project Maths Development Team 2012 www.projectmaths.ie KEY: » next step • student answer/response 8 Teaching & Learning Plan: Introducing e Student Learning Tasks: Teacher Input Student Activities: Possible Responses »» Have you any comments on the interest rate we used? • 100% interest – this is not realistic – no bank will ever give 100% interest. »» What if growth was 50% per year (i= 0.5) instead of 100% on a principal of €1 ,given n compounding periods per year and the interest rate for each compounding period = i/n, how would the final value relate to e? • A rate of 50% would give:» F = 1 (1 + 0.5/n)n Teacher’s Support Assessing the Learning and Actions Teacher Reflections »» Express F in terms of x by letting » and »» First write n in terms of x. » »» Assume that compounding is continuous – how will that affect n, the number of compounding periods?» • n will tend to infinity. • As n = 0.5x, as n tends to infinity, x will also tend to infinity. »» How will n tending to infinity affect x? »» What is the limiting value for F as x tends to infinity? © Project Maths Development Team 2012 www.projectmaths.ie »» Were students able to conclude that x would tend to infinity if n tends to infinity? »» Did students spot e in the formula for the limiting value for F? KEY: » next step • student answer/response 9 Teaching & Learning Plan: Introducing e Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning »» For any yearly interest rate i, and n compounding periods per year, where i/n is the compound interest rate per period, can you write the final value at the end of the year in terms of e, assuming continuous compounding i.e. as n tends to infinity? »» When the rate is 100% or 1, the final value after 1 year is e1. When the rate is 50%, the final value after 1 year is e0.5. Can you generalise what is happening? Teacher Reflections • Final value after 1 year for continuous compounding of €1 when the yearly interest rate is i is ei. »» If the time is t = 3 years for • F = (ei) (ei) (ei) = e3i = (ei)3 instance, and continuous F = erate x time compounding occurs yearly where r is the rate and t is the at interest rate of i what is time. the final value? »» The exponent of e in the above equation could be called x = it. »» If x is 0.20 what could the possible whole number values of i and t be? © Project Maths Development Team 2012 • i could be 20% for 1 year or 10% for 2 years or 5% for 4 years or 4 % for 5 years or 2% for 10. All will yield a final value of e0.2 = 1.22 (to 2 d.p.) if €1 is invested and interest compounded continuously. www.projectmaths.ie KEY: » next step • student answer/response 10 Teaching & Learning Plan: Introducing e Student Learning Tasks: Teacher Input »» When we invest €1 at 5% continuous compound interest for t years the final value is 1eit where i = 0.05. What is the final value if we invest €10 at 5% for t years? Student Activities: Possible Responses • F = eit • F = 10e0.05t Teacher’s Support and Actions Assessing the Learning Teacher Reflections »» We saw that a 100% increase became approximately a 171.8% increase after 1 year of continuous compounding. »» Can students distinguish between the final value of 2.1718 and the increase of 1.1718? »» Let’s see the difference continuous compounding and annual compounding makes for “normal” amounts of money, interest rates and times. »» Calculate the final value if €5,000 is invested for 5 years at 3% per annum or if it is invested at 3% continuous compounding. • F = P (1 + i)t • F = 5,000 (1 + 0.03)3 = 5,796.37 • F = Peit = 5,000e(0.03) (5) = 5,809.17 • Difference = €12.80 »» Wrap up: What have you discovered about e? • e is a fundamental constant like π. • It shows up as the base growth rate for continuously growing systems. © Project Maths Development Team 2012 www.projectmaths.ie KEY: » next step • student answer/response 11 Teaching & Learning Plan: Introducing e Student Learning Tasks: Teacher Input Student Activities: Possible Responses Teacher’s Support and Actions Assessing the Learning • If we calculate the final value when €1 is invested at 100% compound interest for 1 year where the interest is compounded for increasingly smaller intervals we arrive at an approximation of e. » Teacher Reflections • Homework »» Use a computer software package such as GeoGebra or Excel to plot graphs of y = a ex. »» How does the value of a affect the graph?» »» Plot a graph of » y = a e-x, a > 0. • a affects the rate of change »» Use GeoGebra here to of the function. When» a > 1, the graph passes through (a, 0) and the larger a is the greater the rate of change of the function.» • When a is negative the graph of y = a ex is a reflection of the graph of» y = a ex in the x-axis when a is positive.» see that the slope of the tangent is increasing and reinforce the concept:» y = aex dy/dx = aex » The effect of a is that the slope increases by a multiple of a. »» What do you notice about the y values now compared • The y values are decreasing, to the y values for very quickly initially and y = a ex, a > 0 then slowing down. © Project Maths Development Team 2012 www.projectmaths.ie KEY: » next step • student answer/response 12 Teaching & Learning Plan: Introducing e Student Activity 1 Invest €1 for 1 year at 100% compound interest. Investigate the change in the final value, if the annual interest rate of 100% is compounded over smaller and smaller time intervals. (The interest rate i per compounding period will be calculated by dividing the annual rate of 100% by the number of compounding periods per year.) Compounding period Final value F = P (1 + i)t where i is the interest rate for a given period and t is the number of compounding periods per year. Calculate F correct to eight decimal places. Yearly: i=1 Every 6 months: i=½ Every 3 months: i = _______ Every month: i = _______ Every week: i = _______ Every day: i = _______ Every hour: i = _______ Every minute: i = _______ Every second: i = _______ Conclusion: © Project Maths Development Team 2012 www.projectmaths.ie 13 Teaching & Learning Plan: Introducing e Student Activity 1 (Continued) Solutions How often interst Final Value is compounded Yearly Every 6 months Every 3 months Every month Every week: Every day Every hour Every minute Every second Conclusion: Conclusion: The final value gets bigger and bigger but the rate at which it is growing slows down and seems to be getting closer and closer to some fixed value close to 2.718. © Project Maths Development Team 2012 www.projectmaths.ie 14 Teaching & Learning Plan: Introducing e GeoGebra Tutorial GeoGebra File to illustrate the sequence with general term un = 2n 1. Plot the function f (x) = 2x in the Input Bar. 2. When you press Return the graph of the function is drawn. The diagram below shows the function drawn in Graphics View and the expression for the function in Algebra View. 3. Hide the graph of the function by clicking on the function button in Algebra View. 4. Create a slider by clicking on the Slider Button and then on Graphics View. In the dialogue box that then appears set the increments to 1 and click Apply. © Project Maths Development Team 2012 www.projectmaths.ie 15 Teaching & Learning Plan: Introducing e GeoGebra Tutorial (Continued) 5. In the Input Window type (a, f(a)) and press Return and the corresponding point appears in Graphics View. 6. Right click on the point (a, f(a)) and select "Show Trace" from the dialogue box which results. Move the slider to show the locus of the point (a, f(a)). © Project Maths Development Team 2012 www.projectmaths.ie 16

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