This section looks at iterative or recursive formulae. Such formulae have two key characteristics:

Size: px

Start display at page:

Download "This section looks at iterative or recursive formulae. Such formulae have two key characteristics:"

Erin Atkinson
5 years ago
Views:

1 252 BLOCK 3 Task Quadratics and Sine Functions Now apply the same sort of thinking to answer the following two questions: (a) Are all quadratic curves essentially the same? (b) Are all sine function curves essentially the same? Note that there is an additional complication with quadratic and sine functions, namely that their shape is affected by the introduction of a scale factor. Learners often meet quadratic functions before sine functions. It is worth nothing that changes in sine functions are more easily interpreted than changes in quadratic functions. Pause for Reflection 12.2.R Task Reflection What is the difference between the variations considered in section 12.1 and the variations considered in this section: what is the same and what is different; what is invariant in each case, and what is permitted to change? 12.3 STEP BY STEP Quickie 12.3 Take a simple number, say 4 and apply the sequence divide by 2 and add 3. Keep applying the sequence to each new answer until it appears to settle down. Now take a different starting number and repeat the process. What do you notice? This section looks at iterative or recursive formulae. Such formulae have two key characteristics: As the names suggest, these involve repeating a procedure over and over. The formulae apply term-to-term (rather than position-to-term: see Chapter 4 on the use of this terminology). Most people are rather surprised with the result of the quickie investigation that you have just carried out. It may be surprising enough that the sequence seems to settle down to a stable value (a limit) but it is even more counterintuitive that the value of this limit is independent of the starting number. These sorts of investigations are much easier and more engaging when supported by ICT, allowing you to generate them quickly and easily, so enabling attention to be directed to the patterns and a search for an explanation of what lies behind it.

2 GRAPHS AND DIAGRAMS 253 Iterating with a Graphics Calculator Task Exploring Sequences with a Graphics Calculator Using the home screen of a graphics calculator, explore the sequence divide by 2 and add 3. s are given below. Here is a possible What if approach based on using a graphics calculator. Step 1 Try a couple of simple examples only a few iteration steps are shown. Step 2 What if I tried starting numbers bigger than 6? Step 3 What if I tried negative starting numbers? Step 4 What If I tried non-integer starting numbers? Letting these iterations run for a few hundred steps suggests that the endpoint really is 6, regardless of the starting point.

254 BLOCK 3 Using a Spreadsheet One key feature of using a spreadsheet is that, once the underlying formula is in place in a particular cell, it can quickly be filled down for 10, 30, or 100+ terms.

When you have generated some data, try graphing them so as to represent the settling down pattern visually. s s are given below. The spreadsheet will look something like the one on the left.

Once the spreadsheet has been created, it is simply a matter of entering some other starting value into cell B2 to see, directly, a new set of values.

3 254 BLOCK 3 Using a Spreadsheet One key feature of using a spreadsheet is that, once the underlying formula is in place in a particular cell, it can quickly be filled down for 10, 30, or 100+ terms.also, once the numbers are in place, they can easily be plotted graphically. Task Iterating with a Spreadsheet Using a spreadsheet, explore the sequence divide by 2 and add 3. When you have generated some data, try graphing them so as to represent the settling down pattern visually. s s are given below. The spreadsheet will look something like the one on the left. Note that the starting number, 4, is entered into cell B2.The divide by 2 and add 3 formula is entered into cell B3 and filled down column B. Once the spreadsheet has been created, it is simply a matter of entering some other starting value into cell B2 to see, directly, a new set of values.whatever number is entered into this cell, the result is always the same the sequence tends towards a limit of 6.Two more examples are shown above on starting values of 41 and 238: The benefit of including the step values in column A of the spreadsheet is that the results can now be plotted as a scatterplot. Note that these sequences are based on a series of discrete steps, so line graphs would not be appropriate.the scatter plot for the first ten terms of the first of the spreadsheets above will look like the following: Step

4 GRAPHS AND DIAGRAMS 255 Task a More Sequences with a Spreadsheet (a) Using a spreadsheet, explore some more sequences like the sequence you have just investigated divide by 2 and add 3, by altering some dimensions of possible variation. (b) Try to create a sequence that settles down to the number 10. (c) Using an algebraic method or a logical argument, try to find a convincing explanation connecting the sequence rule to the final number that each sequence settles down to. s are given below. (a) The spreadsheet based on the sequence divide by 5 and add 2 settles down to a limiting value of 2.5. (b) There are many possible ways of setting up a sequence so that it settles down to a particular value. Here are three different sequences that all settle down to the same value, 10.They are, divide by 2 and add 5, divide by 5 and add 8 and divide by 4 and add 7.5.They are displayed in columns B, C and D respectively of this spreadsheet. (c) An argument or proof showing how the sequence links to the final number needs to be based on an important fact about such sequences, namely that, when it gets close to a stable value, it looks as though it is actually at that stable value.a stable value is unchanged by applying the iteration process.to take the first sequence (divide by 2 and add 3) as an example, you can let both the final number and its preceding value equal N. So, the following formula connects these two values: N = N/2 + 3 solving for N gives the answer 6, confirming the result from Task This method can be used to demonstrate that the other sequences proposed all have a stable value of 10. Task b Pythagorean Iteration Use the method for finding stable values to find the stable values for the Pythagorean iteration in Task 9.2.4, namely 2, 1+ 2, ,.... Graph the equations y = x and y = 1 + x and use the staircase construction to see the iteration process converging to one of the stable values. It may be worth contemplating the dimensions of possible variation in this iteration sequence. Sometimes they will settle down at a stable value, but sometimes they get larger and larger, or oscillate between two values.

5 256 BLOCK 3 Hero s Square Roots Hero was a celebrated mathematician and engineer who lived in Alexandria in the first century BCE. Amongst several inventions, he discovered the principle of feedback control devices. His self-filling wine bowl, for example, had a hidden float valve that automatically sensed the level of wine in a bowl.when guests ladled out wine, the bowl mysteriously refilled itself. Hero s method for estimating square roots was also based on the principle of a feedback loop. Suppose Hero wanted to find the square root of 10. His approach was to guess an answer (say, 3) and then apply the following sequence several times: Divide 10 by the guess, add the guess, then divide the answer by 2. This gave a sequence of ever-improving guesses. Task Be a Hero! (a) Using a calculator, run Hero s sequence four or five times and then compare your result with the calculator answer for the square root of 10. (b) Set up a graphics calculator or spreadsheet to run Hero s method for finding square roots, based on roughly 10 iterations. (c) Using an algebraic method or a logical argument, try to find a convincing explanation for Hero s method. s are given below. (a) On a graphics calculator, the initial value, 3, is entered.then, repeatedly apply the formula: (10/Ans + Ans)/2. Note that for each iteration the value of Ans is the most recent estimate. As you can see from the screenshots, Hero s square root algorithm settles down very quickly indeed usually only three or four iterations are needed to produce an estimate of a square root to ten figure accuracy. (b) A six-line graphics calculator such as the following could be used to provide ten iterations of Hero s method.the number whose square root you want to find is labelled N and the initial estimate of the square root is S.An explanation for each command is given here.

GRAPHS AND DIAGRAMS 257 Enter values for the number N and your initial estimate S. Set up a loop with ten iterations. Apply Hero s formula.

Press <ENTER> to move it on to the next iteration. In the spreadsheet set-up, notice that the value whose square root you wish to find is entered into cell A1.

Note the use of the dollar sign in the formula to refer to the value in cell A1 this ensures that the cell reference to A1 does not alter as the formula is filled down.

6 GRAPHS AND DIAGRAMS 257 Enter values for the number N and your initial estimate S. Set up a loop with ten iterations. Apply Hero s formula. Display the value of R, the current estimate of the square root. Pause. End the loop. Execute the program. Each time the latest value of R is displayed, the program pauses. Press <ENTER> to move it on to the next iteration. In the spreadsheet set-up, notice that the value whose square root you wish to find is entered into cell A1.The first guess is entered into A2 and the initial formula is entered into cell A3; this is: =(A$1/A2+A2)/2.This is then filled down. Note the use of the dollar sign in the formula to refer to the value in cell A1 this ensures that the cell reference to A1 does not alter as the formula is filled down. To find the square root of a different number, simply enter it into cell A1. (c) An explanation for the solution is based on the same principle mentioned in the comments to Task part (c), namely that, when settling down has taken place, the final estimate is very nearly equal to the value preceding it, and at the stable value, the iteration makes no change. Let this estimate for the square root be labelled S. Then the formula for estimating the value of the square root of 10 can be written as: S = (10/S + S)/2 Multiplying out the brackets and rearranging, gives: S 2 = 10, and therefore S = (10). Replacing the particular value 10 with N provides the following general iterative formula for finding the square root, S, of any positive number, N: S new = (N/S old + S old )/2 Hero s method, although over 2000 years old, is extremely efficient and is essentially the same algorithm that is built into modern calculators to calculate square roots. Pause for Reflection Task 12.3.R Reflection Rehearse in your mind the technique for locating stable values of an iteration. Can you construct an iteration which does not settle down, even though there is a stable value? Just because you have a stable value for an iteration process, there is no guarantee that it will settle down at all, or it may only settle down for certain starting values. Deeper study of this topic belongs in an introduction to the calculus. New=Old 2 has 1 as a stable value but no static value settles down to 1.

7 258 BLOCK GRAPHING ITERATIONS Quickie 12.4 Iterative methods are designed to generate a sequence of numerical values that may or may not settle down to a particular value. Why might there be advantage in graphing the results of iteration? How might you get a picture of an iteration process? In this section you are asked to explore ways of representing iterations graphically. In the comments to Task you saw a graphical representation of the results of the iterative procedure divide by 2 and add 3. Here the values were plotted on the vertical scale and the step number on the horizontal. Now, in Task you are asked to take a different approach.the aim here is to lead gradually to the notions of staircase diagrams for plotting iterations. One-dimensional Plot Before developing alternative ways of graphing iterative values, it would be helpful to clarify the notation that will be used. Each term value will be labelled T 0, T 1, T 2, etc., where T 0 is the initial starting value (in this case, 2). The divide by 2 and add 3 function will be labelled F. Writing, say, F(4) means the value of the divide by 2 and add 3 with 4 as the input value. Using this notation, the first five iterations of this procedure can be written as follows: T 0 = 2 T 1 = F(2) = 2/2 + 3 = 4 T 3 = F(4) = 4/2 + 3 = 5 T 4 = F(5) = 5/2 + 3 = 5.5 T 5 = F(5.5) = 5.5/2 + 3 = 5.75 Task A One-dimensional Plot (a) Plot these values on a one-dimensional horizontal line. (b) What is stressed and what is ignored in this representation? (a) The one-dimensional representation should look something like this: (b) This representation stresses the particular values that are generated by the iteration formula but ignores the term values. In other words, the one-dimensional graph gives no clue as to the sequence of the values being depicted. It fails to signal an important feature of the iteration process, namely that the output value for one iteration becomes the input value for the next, and it fails to make use of the second dimension.

8 GRAPHS AND DIAGRAMS 259 Staircase Diagrams The staircase diagram is an attempt to solve the problem mentioned in the comments to the previous activity, in that it provides a clear and powerful pictorial image of how each new output value becomes the input value for the next iteration.to achieve this, you must return to a two-dimensional representation but this time with different variables depicted on the axes. But most students find staircase diagrams rather daunting when they first come across them. For this reason, they are presented here in two stages in Tasks and The first of these tasks sets out a simple principle of what is involved in plotting iterative values graphically and the second task looks at how this might be done more elegantly. But first, here is the rudimentary staircase diagram. Task A Rudimentary Staircase The graph shows input value on the horizontal scale and output value on the vertical scale. Try to follow the first few stages through now step by step. The straight line graph represents the divide by 2 and add 3 rule that you have been using. So, it takes the form of y = x/2+3. Step 1: (a) Input the starting value, 2. This can be represented as a vertical line from an input value 2 to the function line. (b) Read off the corresponding output value, 4. This can be represented as a horizontal line from the function line to the vertical axis. Step 2: The output value 4 now becomes the input value for the next iteration. (a) Input this value as a vertical line from 4 to the function line. (b) Read off the corresponding output value, 5. This can be represented as a horizontal line from the function line to the vertical axis. Continue subsequent steps in this manner until it settles down. (a) What are the coordinates of the point where the sequence settles down? What is the significance of this point? (b) Can you think of a more efficient way of creating this sequence graph? (a) The graph settles down at the point (6,6). There are two features worth noting about the point where x = 6, y = 6:

9 260 BLOCK 3 (i) (ii) This point lies on the line (check this by substituting the values x = 6, y = 6 into the equation y = x/2+3. The x and y values are equal; in other words, this is the only point on the line where the input value is equal to the output value (which is the condition of settling down). (b) Each new step begins by taking the output value from the previous step and making it the new input value. As carried out here, this is a rather cumbersome process as it involves taking particular values from the y-axis and measuring them out on the horizontal axis. It might be considerably speeded up if the line y = x were included on the graph.you will be asked to explore this simplification in the next task. Task A Fancy Staircase (a) Repeat the iterative graph that you have just drawn in Task but this time include the line y = x. Try to use this line to speed up your drawing. (b) Compare this version of the drawing with the one that you created in the previous task. What are the gains and the losses in including the y = x line? (a) As before, the first iteration starts at the value 2 on the Input axis and a vertical line is drawn to the line y = x/2+3. But now, instead of moving horizontally to the left to read off the corresponding output value, the movement is to the right, to meet the line y = x, ready for the next iteration.this effectively collapses the following two steps into one: Read the output value off the vertical axis. Turn this value into an input value by marking off the same interval on the horizontal axis. It is this stage that most students find difficult to grasp, for good reason, in that the thinking is fairly dense. Another version is to start on the x-axis at some point; go up to the graph, across to the line y = x, then up (or down as need be) to the graph again, then across to y = x, then up or down again, continuing in this fashion.the effect of going across to the line y = x is to transfer the current height into a distance along the x-axis. Continuing in this manner produces a series of steps that eventually end up at the point (6,6). It is not difficult to see from the final picture why this is often referred to as a staircase diagram. (b) The inclusion of the y = x. line has certainly streamlined the process and probably speeded up your drawing of the iterations, but there is certainly a cost in terms of clarity about what is going on. How comfortably might you have understood this more sophisticated

10 GRAPHS AND DIAGRAMS 261 procedure had you not already gone through the underlying thinking in Task ? This points to a wider in mathematical learning, namely that there is often a trade-off between algorithmic efficiency and transparency of method. It might be argued that, since machines are increasingly carrying out algorithms very quickly indeed (even for inefficient methods), inefficiency of method may be a price worth paying so that students can gain a clearer understanding of the big underlying ideas. Pause for Reflection The simple process of tracking the coordinates of each of the points on the staircase is a useful exercise in learning to interpret the graphs of equations: they are not simply end products, rather, they are intended to be informative, and so it is valuable to learn how to interpret them R Task Reflection What has using graphs to think about iterations highlighted for you about reading and interpreting graphs? 12.5 PEDAGOGIC ISSUES Algebraic versus Statistical Graphs Most teachers would probably agree that students find algebraic graphs difficult yet they seem to have a good intuitive understanding of statistical graphs based on everyday data. This finding was recorded by Daphne Kerslake in her chapter entitled Graphs in Hart et al., 1981.The research was carried out on roughly 1800 pupils aged They found that: (V)ery little difficulty was experienced by the pupils with elementary items on block graphs, and the use of rectangular coordinates to plot points, when the numbers involved were integers. About 90 per cent of the pupils were successful at such items. However, the movement from everyday to algebraic graphs, combined with the transition from discrete to continuous variables caused major problems. In one of the tasks set, the pupils were asked to plot the points (2,5), (3,7) and (5,11) and join them with a straight line.they were then further interrogated about what they had drawn: many pupils found difficulty with the idea that there are any more points on the line than those they had plotted. Several said that there were no points between (2,5) and (3,7), while others thought there was just one (presumably the mid-point). The pupils were also provided with paired data (height and waist measurements of five pupils) and asked to plot them as a scatter plot.they were then asked, Should we join up the points on the diagram? The commonest answers to this were based not on the nature of the data or the meaning of any lines used to join up the points but on the appearance of the graph.this was true both for many of pupils who thought the points should be joined up as well as for some of those who did not.

1

Zeros&asymptotes Example 1 In an early version of this activity I began with a sequence of simple examples (parabolas and cubics) working gradually up to the main idea. But now I think the best strategy