3. How many gadgets must he make and sell to make a profit of R1000?

Transcription

1 The Entrepreneur An entrepreneur starts a new business. He manufactures gadgets at a cost of 4x rands and then sells them for 5,50x rands, where x is the number of gadgets produced. Profit = Income Expenditure 1. Calculate his profit for different numbers of gadgets by completing the table. Note: Assume that he sells all the gadgets he produces. # Gadgets (x) Profit (in rands) Discuss: How did you calculate the profit? 2. How many gadgets must he produce and sell to make a profit? 3. How many gadgets must he make and sell to make a profit of R1000? 4. Draw graphs of the Expenditure function E(x) = 4x and the Income function I(x) = 5,50x on the same system of axes. Explain how/where you can you see from the graph (a) when he is making a profit, and (b) what is the profit? 5. He can increase his profit by making and selling more gadgets explain why! But there are limits to the market and to his production capacity! Show how and why he can increase his profit by (a) reducing his production costs, and/or (b) increasing his selling price per gadget. 6. Draw the graph for the Profit function P(x) on the same system of axes. Explain how/where you can you see from this graph (a) when he is making a profit, and (b) what is the profit? Write down a formula for the Profit function P(x). How can you check that you are right? What is the situation if he does not sell all the gadgets that he makes?

2 Die Entrepreneur n Entrepreneur begin n nuwe besigheid. Hy vervaardig gadgets teen n koste van 4x rand en verkoop hulle dan vir 5,50x rand, waar x die getal gadgets is wat hy maak. Wins = Inkomste Uitgawes 1. Bereken sy wins vir verskillende hoeveelhede gadgets deur die tabel te voltooi. Let op: aanvaar dat hy al die gadgets wat hy maak verkoop. # Gadgets (x) Wins (in rand) Bespreek: Hoe het jy die wins bereken? 2. Hoeveel gadgets moet hy maak en verkoop om n wins te maak? 3. Hoeveel gadgets moet hy maak en verkoop om n wins van R1000 te maak? 4. Trek grafieke van die Uitgawe funksie U(x) = 4x en die Inkomste funksie I(x) = 5,50x op dieselfde assestelsel. Verduidelik hoe/waar jy in die grafiek kan sien (a) wanneer hy n wins maak, en (b) wat die wins is? 5. Hy kan sy wins verhoog deur meer gadgets te maak en te verkoop verduidelik hoekom! Maar daar is grense in die mark en in sy vervaardigingskapasiteit! Wys hoe en hoekom hy sy wins kan verhoog deur (a) sy produksiekoste te verlaag, en/of (b) die verkoopprys per gadget te verhoog. 6. Trek die grafiek van die Wins funksie P(x) op dieselfde assestelsel. Verduidelik hoe/waar jy in die grafiek kan sien (a) wanneer hy n wins maak, en (b) wat die wins is? 7. Skryf n formule vir die Wins funksie P(x). Hoe kan jy kontroleer dat jy korrek is? Wat is die situasie as hy nie al die gadgets wat hy vervaardig verkoop nie?

3 VERY VERY BRIEF QUICK NOTES: ENTREPRENEUR THE PROBLEM: Operation (calculation) vs structure In die vorige lesanalise het ons gesien dat selfs in so n maklike aktiwiteit soos Bouers, goeie Graad 8 leerders sukkel om n algebraise formule op te stel, omdat o hul rekenkunde agtergrond hulle dwing om alles in sig te bereken dan kan hulle nie veralgemeen nie, omdat hulle die situasie induktief deur patroonherkenning in getalle ( antwoorde ) benader, in plaas daarvan om die struktuur (die metode!) te veralgemeen! Dit was dus problematies vir kinders om hierdie tabel te veralgemeen: # dae n Hoogte 1,2 2,4 3,6 8,4 Dit sou teoreties vir kinders maklik wees om hierdie tabel te veralgemeen: # dae n Hoogte 1 1,2 2 1,2 3 1,2 7 1,2 o hulle (rekenkundige of algebraise) uitdrukkings beskou as rekenvoorskrifte (opdrag om te bereken), en dus ook die = -teken as n operator sien ( a do-something sign ), dus kan hulle nie n algebraise uitdukking as n antwoord n geheel, n ding, n naam, n objek sien nie (acceptance of lack of closure) hulle kan nie 7 1,2 of 1,2 n los nie, maar moet dit bereken! ) en dus het ons gesiendat hulle gedryf is om te skryf 1,2 n = x! Sulke leerders skryf dikwels 5 + x = 5x!. Kan jy dit verklaar? Leerlinge dink dus operasioneel aan rekenkundige en algebraise uitdrukkings in plaas van struktureel! Dit raak die wese en die aard van algebra presies wat doen ons, en waarom doen ons dit, wanneer ons sê 2x + 3x en 2(x + 3) = 2x + 6?? Algebra is glad nie bereken met letters nie, maar is n veralgemening van die metode (struktuur), en ekwivalente metodes! Bestudeer hierdie ou-ou opsomming van my Navorsing toon dat as leerders nie strukturele begryping ontwikkel nie, hulle basies gedoem is en algebra nie behoorlik kan hanteer nie. Die teorie sê dat ons strukturele begryping deur middel van die operasionele moet ontwikkel. Maar dit is juis wat nie op skool gebeur nie (in Graad 8 leer ons hulle van terme en om gelyksoortige terme op te tel, maar ons doen nie funksies nie, dus gebruik hulle omtrent nooit algebraise uitdrukkings vir berekening nie eenvoudige substitusie!). Die gevolg is wat Sfard en Linchevski noem pseudo-strukturalisme, dit is struktuur wat op sand gebou is, sonder die operasionele basis! Ons het dus aan die een kant die probleem dat leerlinge vasgevang is in die operasionele hantering van algebraise uitdrukkings en nie vorder na die strukturele nie, en aan die ander kant dat hulle die strukturele bloot instrumenteel, sonder verstaan, hanteer. Bestudeer asseblief Sfard en Linchevski as agtergrond! So wat kan goeie Graad 10 leerders doen???? Kom ons kyk of hulle die strukturele in Entrepreneur kan hanteer

4 Entrepreneur: Purpose and lenses The purpose of this activity is for us as student-teachers to analyse the activity and reflect on it: o mathematically (what are the mathematical ideas in the activity?), o didactically (1 - what are the cognitive requirements to solve the activity? 2 - from our observation: how much of that do these children have? 3 if there is a discrepancy between 1 and 2, can we explain it, and what are the implications for teaching?), and o to develop appreciation that such activities are worthwhile to teach and learn. Modeling This is an interesting situation, illustrating the power of mathematics to model a real-life situation. Mathematical modeling means that we replace the real-world objects with mathematical objects, and then manipulate the mathematical objects instead of the physical objects to gain more information about the physical situation. For example, working with an electric switch, we replace the on and off of the physical switch with the numbers 1 and 0 in the binary system, i.e. 1 and 0 are the only numbers, so a number can only be 1 or 0 this translates back to the physical world: if the result of a manipulation is 1, the switch is on; if the result is 0, the switch is off. Computers are built on this powerful idea... Here we simulate the whole business activity with a simple mathematical model! Functions This is a typical functional situation in which we study the relationship between two variables. Indeed we have here three functions: 1. The Expenditure function E(x) where the dependent variable expenditure (cost) depends on the independent variable number of gadgets (x) 2. The Income function I(x) where the dependent variable income depends on the independent variable number of gadgets (x) 3. The Profit function P(x) where the dependent variable profit depends on the independent variable number of gadgets (x) What are the relationships between these functions? Different representations of functions The first two function are here given as formulae in symbol form and can be transformed to words, a table, a graph or an equivalent formula, depending on which representation is more convenient for a particular purpose. These transformations are important skills in learning algebra, as shown below: PROBLEM SITUATION Interpret Verify Mathematise Symbolise MATHEMATICAL MODEL Transform EQUIVALENT MATH MODEL Analyse - find function values - find input values - behaviour of functions From...to Words Words Table Graph Table Graph Formula MORE INFORMATION OF MODEL Formula The questions can all be solved numerically, directly from the word formulation or the table representation. But the advantages of an algebraic formula should be clear!

5 The activity involves the following function problem types can you identify which is which? Finding function values, i.e. given x, find f(x). Finding input values, i.e. solving equations, e.g. given f(x), find x. Working with the behaviour of function values (rate of change, domain and range). Finding an equivalent formula. Semantics and syntactics One aspect to understand is the syntactical meaning of the expressions: the expressions are shorthand notation telling us exactly which operations to carry out on which numbers in which order. So E(x) = 4x tells us that to calculate the value of E(x) for any value of x, we should multiply the value of x by 4 and then add the result to 200. The other aspect to understand is the semantic meaning of the formula, the meanings it derives from its referents, e.g. x is not just an abstract number, it is the number of gadgets, and 4x + 200, as a whole, is the production cost of x gadgets. Also, 4 is not just a constant, but the production cost per gadget, so if we analyse the situation, not just solve the particular given problem, the production cost can be changed that is why we prefer to call it a parameter, not a constant or a coefficient! Likewise, the 200 represent the fixed cost that is independent of the number of gadgets made, e.g. the hiring costs of the factory, telephone line rental, electricity, etc., so it can be manipulated (e.g. he can reduce his overall production costs by working from home ). Therefore, we can solve the initially stated questions using the formulae E(x) = 4x and I(x) = 5,5x, but to investigate the business situation we need to work with the mathematical objects E(x) = ax + b and I(x) = dx,, and study how changes in the parameters a, b and d influence the production costs, selling price and eventually the profit! And we need to understand this numerically, graphically and algebraically! The power of mathematics is immense, and at all stages symbols make a major contribution to this power. But without the ability of the mathematician to invest them with meaning, they are useless. Richard Skemp, 1971, p. 89 Break-even point In a way, the purpose of the activity is to introduce the concept of break-even point. The terminology probably derives from a graphical interpretation: the intersection of two straight lines. But breakeven also carries semantic meaning: the point where we change from a deficit (loss) to a profit The breakeven point can be found numerically, but in the end it is advantageous to understand it as the simultaneous solution of two linear functions, and in particular as the solution of the algebraic equality of the two function values, here 4x = 5,5x, which of course means that the profit = 5,5x (4x + 200) = 1,5x 200 = 0! Algebraic equivalence The other main mathematical and didactical issue here is whether learners follow a numerical approach or an algebraic approach to the problem. A numerical approach is to work out the profit for a given x by numerically calculating the cost, the income, and then the profit, e.g. for x = 300: E(300) = 5,5 300 = R1650 I(300) = = R1400 So P(300) = R1650 R1400 = R250

6 An algebraic approach is to first construct a general formula for the profit by being able to use the cost and selling functions as objects (that is being able to handle them as a whole, a thing, and not to think of them as calculating procedures), to manipulate them algebraically and use it to afterwards calculate the profit for a given x, e.g. for x = 300: First general, for any x: P(x) = I(x) E(x) = 5,5x (4x + 200) = 1,5x 200 Then, for x = 300, or for any x just as easily: P(x) = 1, = = R250. Note, this numerical vs algebraic approach is quite general and is supposed to be a leading idea in the development of algebraic understanding and skill, e.g.: If x= 7,64, evaluate 2x + 3x. 2 If x= 7,64, evaluate x 4. x 2 If you get 10% discount at Edgars on a R399 jersey, what do you pay? Here is a statement: An educated person understands the underlying algebraic concepts and understands and appreciates the advantage of the algebraic approach, and uses the algebraic approach as a habit of mind. You know and appreciate that e.g. it is easier, therefore more productive and empowering: for any calculation you do fewer operation (here 2 compared to the 4 operations of the numeric approach), and this is especially true if you have to repeat the job many times, e.g. when making a table of the profit for different x. And, even more importantly, while a numerical approach may offer a somewhat cumbersome solution for finding function values, i.e. given x, find f(x ), the algebraic approach is indispensable for solving equations, i.e. given f(x ), find x. For example: Solve for x if 2x + 3x = 48 2 Solve for x if x 4 17 x 2 Our entrepreneur will have difficulty calculating how many gadgets he should make for a profit of R1000 if he uses I(x) E(x) = 1000, but it is easy if he uses 1,5x 200 = 1000! Also see the interactive Excel activity! So, let s observe: do the learners use the algebraic approach as a habit of mind? Probably we will not get past question 3 but hopefully they can also tackle the graphs. Alwyn Olivier 17 April 2011