Investigative Skills Toolkit (Numeric) Student Task Sheet TI-Nspire Numeric Version
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- Cornelius Stokes
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1 Introduction Investigative Skills Toolkit (Numeric) Student Task Sheet TI-Nspire Numeric Version This activity will develop your ability to use the TI-Nspire Numeric Handheld to investigate and generalise results from numerical patterns. The TI-Nspire and Mathematical skills that will be developed are: defining functions using sigma notation generating sequences using statistical regression models to reveal algebraic generalisations. Mathematical Aim As a context for developing the above 4 skills, you will use them to find the algebraic formula for the sum: n You will then find the algebraic formula for the sum: n 2 and finally find the formula for: n 3 Note: There are many ways of going about deriving the formula for the sequence sums that are listed above. Some of these ways are quicker than the approach you will follow here. However, an important purpose of this task is to have you experience how a wide variety of technological skills can be brought together to work in powerful ways. Such techniques are able to be used even when the formulae being sought are not as well known as the above three sequence sums. Structure of Tasks Tasks 1, 2, 3 and 4 introduce each of the 4 individual skills, listed above. Task 5 merges these skills together in order to acheive the Mathematical Aim given above.
2 Task 1 Defining Functions A function has three things: a name, an input and an output. Create a New Document and insert a Calculator Page. Type in g(x):=x+2 and press. as shown on the right. Notice the colon in front of the equals sign. You can quickly obtain := by pressing / then t. This line reads as g of x is defined as equal to x plus two Now type g, and you will see that it is already displayed in bold. This means that the Nspire system knows that something named as g already exists in the current problem. Investigative Skills Toolkit (Numeric) Now continue to type in g(4) and press. You should be given the answer of 6. The function g took the input of 4, and after adding 2 to it, presented the output of 6. Another way of accessing the function of g is to press the h key. Do that now. Nice that it includes the brackets for you, isn t it? Now type g(-7) and before you press, see if you can correctly predict what it will display. You will see that the function g takes whatever it is given as an input, and generates an output that is simply the input plus two. Add a new calculator page by pressing / then ~ then Add Calculator Define a new function called h(x,y):=x+y 2, as shown on the right. This function has two inputs, and one output. Use the h key to type h(2,3) and before you press, see if you can correctly predict what it will display. Repeat for h(5,-4). Watch carefully what happens when you type in h(g,g) and be sure that you know why it gives the error message that it does when you press. Ask your teacher if you are unsure.
3 Add a new calculator page by pressing / then ~ then Add Calculator Define a new function called discrim, as shown below discrim(tom,dick,harry):=dick 2-4 tom harry This function has three inputs, and one output, and the variable names are no longer single letters. Notice the multiplication dots between 4 and tom and harry. Use the h key to type discrim(1,2,3) and before you press, see if you can correctly predict what it will display. Repeat for discrim(5,-2,7).
4 Task 2a Sigma Notation Investigative Skills Toolkit (Numeric) Σ is the Greek capital letter called sigma. A lowercase letter sigma looks like this: σ σ is used in Statistics to represent a measure of how spread out a data set is. Σ is used to mean the adding up of several terms. It requires three inputs : a lower bound, an upper bound and an algebraic expression. Here are two examples, which you shall shortly check on the Nspire. Don t type anything in yet. Example (i). 5 r 1 ( r ), which reads as the sum from r equals 1 to 5 of r This is a short-hand way of writing , which is equal to 15. Example (ii). 9 c 6 ( c 2 ), which reads as the sum from c equals 6 to 9 of c squared This is a short-hand way of writing , which is equal to 230. In each example expression, the letter under the sigma sign increases by one each time from the lower bound until it reaches the upper bound. For each value it takes on this journey, the algebraic expression is evaluated. You will now check the two results for examples (i) and (ii): Insert a new Problem by pressing: ~... Insert... Problem... Add Calculator. To obtain the sigma notation template, press t and select the sigma notation icon, as shown on the right. You will then have an empty template as shown, and the cursor is flashing in the region where you need to type in the variable will appear in your expression. Set about entering in the letters and numbers as shown in example (i), above, and then press. Repeat for example (ii). continued.../
5 You should now have a screen display like this: Investigative Skills Toolkit (Numeric) Now edit one of these expressions to generate the sigma notation to find the following sum: You should obtain the answer of 139. Can you now predict the values of each of the following summations, before you press? So, to finish, sigma notation can be used very effectively to add up lots of terms in a sequence for which you already know an algebraic expression for each term. You can consider the sigma notation to be a function with 3 inputs: the lower bound, the upper bound and a formula. It generates a single output of a value, or an expression, for the sum of all the terms it represents.
6 Task 2b (Extension) - Sigma Notation Equations You can use sigma notation with other Nspire functions... Consider the problem of determining the value of n such that n = 153. You need to solve for n, which may at first glance seem only to be able to be done by a method of trial and improvement. As can be seen on the right, the nsolve(...) command can be used. Find this by pressing: b... Algebra... Numerical Solve Consider another problem of determining the value of p such that 5 p +6 p +7 p +8 p = Before you look at the screenshot solution on the right, can you set up an nsolve(...) statement to obtain the value of p? Consider a final problem of finding out the sum of which 40 odd numbers equals a total of Before you look at the screenshot solution on the right, can you set up an nsolve(...) statement to obtain the answer to this puzzle? Hint: the 40 th odd number after the start is 39 odd numbers after the first odd number!
7 Task 3 Generating Sequences Investigative Skills Toolkit (Numeric) The sequence command can generate a sequence of terms, based upon a formula, a start value and end value. The default setting is that it increases its variable by one each time. You can hopefully appreciate that it has very similar syntax to the sigma notation. However, the sigma notation sums all terms in a sequence without letting you see them. The sequence command will show you all the terms, but not sum them. Insert a new Problem by pressing: ~... Insert... Problem... Add Calculator. The first example (left) generates the sequence of squared numbers from 1 2 to 4 2. The second example generates the sequence of cubed numbers from 4 3 to 6 3. Notice that the output of this command is not a single value, but a list of values, as denoted by the curly brackets, {...}. The syntax of the command is: seq( formula, variable in formula, start value, end value ) Type in the following commands, and see if you can predict what they will display, before you press. seq(4-k,k,2,7) seq(5a+3,a,19,25) seq(3 a,a,2,6) Now try to type the sequence commands that will display each of the following outputs: 1. the first five odd numbers 2. the sequence {75, 69, 63, 57, 51} 3. the sequence {4 5, 5 6, 6 7, 7 8 }... which will be displayed as {1024,15625,279936, }
8 Task 4 Using Statistical Regression Models When you obtain a sequence of numbers and you are not sure what formula generated them, it is often helpful to plot them on a graph. Shown below are typical graphs of the most commonly occuring types of formulae formula with n 2 in it formula with n 3 in it formula with n 4 in it As you can see, there are only subtle differences between them, so a bit of intelligent decision making is required when trying to apply these formulae. For a worked example, you shall create a sequence using a formula that you already know, plot it, and then use Statistical Regression to provide you back with the formula you used. This will act as a self-check process, in readiness for when you are using the technique to find an unknown formula. Insert a new Problem by pressing: ~... Insert... Problem... Add Lists & Spreadsheet. Then type into the very top left cell on the screen, above the greyed-out row the variable name n.value and press. Then move to the very top of column B, and type in term.value and press. Notice that you can use the decimal point as a valid character in the names of our variables, and it helps to make them more readable. You shall now generate the first 10 terms in the sequence whose n th term is given by 4n 2-3n As ever, there are many ways to do this... First, you shall generate the n.values from 1 to 10. You can use the sequence command for this: seq(n,n,1,10) To enter this command, move the cursor to the grey row, second from the top, in column A, and press =. The screen will then look like that shown on the right: Type in seq(n,n,1,10) and press. You should then see column A fill up with the numbers 1 to 10.
9 You shall repeat this process for column B, only this time the sequence command will be: seq(4n 2-3n,n,1,10) When you have done this, your screen should look like this: Now insert a new page with Data & Statistics, and click on where it says Click to add variable at the foot of the screen. Select n.value Move to the left of the screen, where it (used to) say Click to add variable and click again. Select term.value You should then have a screen looking like the one shown on the right. You already know that the formula for this sequence is a quadratic expression, but you would be forgiven for thinking it might be cubic or quartic if you didn t already know. Look back and compare this graph with the 3 screenshots at the very start of Task 1d. You will now instruct the TI-Nspire to fit the best formula it can to these plotted points. Press b... Analyze... Regression... Show Quadratic The regression formula will then be displayed as shown on the left. You were expecting 4n 2-3n and it has displayed something very similar to this, only with different variable names and surplus characters. This is to be expected when using Regression techniques. Move the cursor over the displayed equation and press / then b then select Remove Regression If you didn t know it was a Quadratic formula that you were after, you could have done: b... Analyze... Regression... Show Cubic or even b... Analyze... Regression... Show Quartic Do each of these now.
10 You will notice that when selecting Show Quartic, the displayed regression equation seems overly complex. The full equation looks like this (split up so that you can see each term more clearly): y= E 13 x 4 or, in a shortened form, E 12 x x x -4. E 11 y= 1.464E 13 x E 12 x x x -4. E 11 You can see something like y=4 x 2-3 x lurking in the middle of this expression, albeit in a slight disguise due to rounding errors from the regression process! And you have surplus terms which are very small indeed. For example, look at the coefficient of the x 4 term. It is 1.464E 13 This is the TI-Nspire s way of displaying the number Which is the number You can therefore conclude that the x 4 term is not there as its coefficient is such a small number. Similarly for the coefficient of x 3 which is And for the constant on the end, which is This leaves you with y= x x You can take this to mean y=4 x 2-3 x after adjusting for rounding errors. It is very important that you understand the above logical process, else you may make errors when interpreting results from Statistical Regression calculations. You can, of course, have the TI-Nspire help you with this interpretation process! You can instruct the regression formula to be displayed with slightly reduced accuracy, thereby revealing the most important terms. To do this, insert a new Calculator Page. Press h and select stat.results Note: all the other variables that start with stat. were automatically generated by the Statistical Regression process that you commanded it to perform.
11 Then press: b... Number... Approximate Fraction Investigative Skills Toolkit (Numeric) You will have a command line that reads as: stat.results approxfraction(5.e 14) Don t press yet! Edit the last characters to change it from 14 to 10: stat.results approxfraction(5.e 10) This instructs it to start rounding at about the 10 th decimal place, rather than the 14 th decimal place. The full answer that you will obtain is shown above, but you will have to scroll up and down to read it on the handheld. This display lists the values of each of the coefficients of a, b, c, d and e in the regression equation of y= a x 4 +b x 3 +c x 2 +d x +e The values of a, b and e are fractions with such large denominators, that they can be ignored, as their contribution will be so small. This leaves the value of c = 4 and d = -3 And so, you can now see more clearly the desired formula of y=4 x 2-3 x Statistical Regression Conclusion Statistical Regression is very powerful, but it requires careful interpretation to draw the correct conclusions. It is also worth remembering that its output is not proof that the formula works. It is merely the best approximation given the data that was processed and the type of regression equation that you were trying to fit to that data.
12 Task 4b - Statistical Regression Practice The previous example was based around the formula for the n th term to be 4n 2-3n. You will now change the setup so that the TI-Nspire system tries to detect the formula 5n 3 -n/7 Return to the Lists & Spreadsheet page and edit the seq(4n 2-3n,n,1,10) command in column B to read seq(5n 3 -n/7,n,1,10) All the other dependant pages will automatically update, and the new regression equation will be displayed on the graph. On the Calculator page you should re-execute the command line: stat.results approxfraction(5.e 10) If this does not obviously reveal the expression 5x 3 -x/7, then you will have to adjust the tolerance of the approxfraction command from 5.E 10 to something like 5.E 8 or 5.E 6 This adjustment is because of the fraction 1/7 that is in the original formula. This is further evidence that you must remain flexible when using Statistical Regression, as the complexity of the formula that you are seeking often affects the accuracy of the calculations. Task 4c - Find the Formula Tasks 4a and 4b were built around already knowing the generating formula. This is not really realistic, as why would you try to find out something you already knew?!? Consider the following sequence: 2, 14, 50, 130, 280 Your task will be to find a formula for the n th term. To do this, return to the Lists & Spreadsheets page and do the following: 1. move to the grey formula row of column B and press. to delete the sequence statement. 2. with the selected cell still in the grey formula row of Column B, press / then b then select Clear Data You should then have a screen similar to that shown on the right. Now, type in the 5 sequence values into column B, to give a screen similar to that shown on the right: You may have noticed that column A still has a sequence running from 1 to 10, but you do not have 10 numbers in column B. You will need to edit the formula in column A to generate only the numbers from 1 to 5. Do that now.
13 Move back to the Data & Statistics graph to verify the plausibility of fitting a quartic, cubic or quadratic equation to the data. You can choose to fit whichever type of equation you think best suits. Remember that if you opt for, say, a quartic, and the data is only quadratic, then you should expect the coefficients of the x 4 and x 3 terms to be neglible. However if you opt for a quadratic and the data is quartic, then the regression line will not fit the data well and the coefficients in the equation will most likely be ugly Once you have completed this process, and chosen the required amount of accuracy for the approxfraction command, compare your result to the screen on the right. The display still looks unappealing with all the fractional coefficients, but it s really just saying: which can be tidied up by manual techniques as follows You should now check that this formula does indeed generate the first 5 terms that you were given - why not use the factorised form that is given above, and work through the evaluation process on paper, without using the TI-Nspire to help you! If you encounter difficulty, have your teacher show you how to process the numerical calculation with the most efficient ease. To practice the above process, edit the data on your TI-Nspire to help you identify the formulae for the n th term for each of the following two sequences: a) 0, 4, 22, 70, 170 b) 0, 3, 11, 26, 50
14 Task 5a Combining Formulae + Sigma Notation + Sequences + Statistical Regression Insert a new Problem by pressing: ~... Insert... Problem... Add Calculator. Enter both of the statements as shown on the right, which define a variable called power and a function called total. As the variable power equals 1, if you were to enter in total(5), it would calculate the sum Try it now. You will be able to change the value of power to other numbers later on, and therefore all the calculations that you will design in the next few steps will be automatically updated. Now define a variable called terms:=6 (see right) Entering the command total(terms) ought to calculate the sum Try it now. You are now going to use all the techniques learnt so far to find the formula for n 1. Insert a new Lists & Spreadsheet page. 2. Type into the very top left cell on the screen, above the greyed-out row the variable name n.value and press. 3. Move to the very top of column B, and type in term.value and press. 4. Move the cursor to the formula row in column A, and press =. 5. Type in seq(r,r,1,terms) and press. 6. Move the cursor to the formula row in column B, and press =. 7. Type in seq(total(r),r,1,terms) and press. You should have a screen that now looks like the one given below: The purpose of using the variable terms, makes sure that both lists have the number number of terms in them. You should now: 1. insert a Data & Statistics page 2. plot the data 3. fit an appropriate regression line 4. insert a Calculator page 5. if required, use the approxfraction command to extract the desired formula from the regression equation (stat.regeqn)
15 Compare your results to the screenshot on the right. Investigative Skills Toolkit (Numeric) Using the quartic regression approach, you can disregard the values for a, b and e as they are so small. This leaves you with: Task 5b Sum of the first n Square Numbers As declared at the outset of this whole activity, you will now attempt to find the formula for n 2 This requires very little extra work, as you have already constructed a fully dynamic system of pages in the current problem. 1. Return to the calculator page where you defined power to equal Now redefine power:=2 3. This will automatically update the sequences and data plots. 4. You may have to try a different type of regression function (quartic/cubic/quadratic), depending upon what you previously used. 5. Use the approxfraction command and some manual factorising to obtain the nicest form for the formula. 6. Check your factorised formula by evaluating it for the first 5 terms, or so. Task 5c Sum of the first n Cube Numbers Finally, you will find the formula for n 3 1. Repeat the above 6 step process, redefining power:=3 Task 5d (Extension) - Sum of the first n Quartic Numbers It is possible to attempt to find the formula for n 4 However, you need to try it yourself to discover what happens. Can you find a way of using Statistical Regression to obtain a working formula? Can you identify the feature of this last challenge that causes the whole statistical regression process on the TI-Nspire to falter?
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