TI-Time. In this issue. Contents

Transcription

1 TI-Time In this issue Developing the use of Graphics Calculators at Key Stage 3 Distance-Time graphs using the CBR The TI-83 and the Numeracy Strategy Probability activities at Key Stage 3 Contents Developing the use of Graphics Calculators at Key Stage p. 3 4 Distance-Time graphs using the CBR p. 5 6 The TI-83 and the Numeracy Strategy p. 7 Framework Examples p Rolling a die an awful lot of times! p T 3 Teachers Teaching with Technology p. 18 Workshop Loan Programme p. 19 FREE Workshops and Demonstrations p. 20 Instructional Dealer List p. 20 GENERAL INFORMATION: If you have general questions about using a product, to order products, or before returning a product for service: CSC Customer Service Centre ti-cares@ti.com Phone: Fax: Write to: Texas Instruments CSC, c/o SITEL Researchdrive 4 B-1070 Brussels Belgium TI-Time Summer 2001 EDUCATIONAL INFORMATION: For information on integrating hand-held technology and workshop ideas contact: Melanie Horsburgh, Educational Marketing Manager mhorsburgh@ti.com Phone: Write to: Texas Instruments 800 Pavilion Drive Northampton NN4 7YL

2 Introduction Dear Teacher Welcome to this edition of TI-Time! I d like to take this opportunity to introduce myself. My name is Melanie Horsburgh and I am the new Educational Marketing Manager for Texas Instruments in the UK and Ireland. Before joining TI I worked for the educational publisher Longman as Product Manager for their Maths and Science lists. My predecessor, Guy Harris, has now started his new job as Market Development Manager for TI in the United States and is preparing to make the move back across the Atlantic. This 17th edition of TI-Time focuses on the Numeracy Strategy. The National Numeracy Strategy, Framework for teaching mathematics: Years 7 to 9 was published in draft form at the end of last year and the final version will be arriving in schools at about the same time as this edition of TI-Time. It has been very pleasing to see that the framework recognises the importance of hand-held technology in the teaching of mathematics. In the framework document the graphics calculator is often referred to as the primary ICT resource for mathematical activities, in advance of spreadsheets and databases. This will be welcome in many mathematics departments where access to computing facilities is difficult. However, not all mathematics teachers are confident users of graphics calculators, so most of the articles in this edition of TI-Time will be useful for teachers who may be attempting to use the technology at Key Stage 3 for the first time. Although the references to the Numeracy Strategy won t be directly relevant to teachers in Scotland and Ireland, I hope that you will find the articles and examples of interest and use in your teaching. A big thank you to those of you who have contributed articles and ideas to this issue. I would also like to thank Alan Wiltshire, author of the program APE, which was featured in the last edition of TI-Time. The next issue of TI-Time (due out at the beginning of the autumn term) will concentrate on reviewing some of the powerful application programs which are coming out for the new generation of calculators with Flash ROM memory. Known as APPS, these include: Probability Simulator this application helps the user understand probability theory and provides animations to demonstrate examples, for instance: tossing coins and rolling dice. StudyCards allows teachers and students to create or download, review and study sets of electronic flashcards. CellSheet an extremely powerful application which allows students to have a portable spreadsheet in the palm of their hand. Compatible with Excel using the TI-GRAPH LINK. How would you like to test-drive one of these APPS and to write a review for TI-Time? We'd be pleased for you to have a TI-83 Plus on long-term loan. If you'd like to help, please contact me at the address below. Melanie Horsburgh Educational Marketing Manager Texas Instruments Limited 800 Pavilion Drive Northampton Business Park Northampton NN4 7YL Phone: mhorsburgh@ti.com 2

3 Jaiwant Timotheus, The Grange School, Stourbridge, West Midlands Developing the use of Graphics Calculators at Key Stage 3 Two years ago, our mathematics department invested in a set of TI-83 graphics calculators. Since then, we have been developing their use in our classrooms. Although we all see their potential, it has not always been easy for staff to develop the confidence to work with the calculators in the classroom. In this article I will give some practical ideas to simplify the management of the calculators in the classroom. Secondly, I will offer what I consider to be some key skills that we as teachers need in order to become confident users of graphics calculators in the teaching and learning that takes place in our own classrooms. Practical Ideas Introducing the graphics calculators for the first time at Key Stage 3 always seems to produce a positive response. Many of the children have never seen a calculator of this size before and one of the first questions I get asked is, How much do they cost? However, there are many other factors that the children need to take in. There are a lot of keys. Some pupils will wait until they are told what to press and follow the instructions rigidly. Some will press anything and then ask for immediate attention when they get an error message. As a teacher, there is a lot to co-ordinate. There may be some setting up to do at the start. There can be an element of uncertainty as children look for a particular key. There may be a number of error messages to sort out. Your attention is very much in demand. Another useful, but more expensive investment is a ViewScreen set. This allows you to display the contents of your calculator screen onto your whiteboard, or other screen using an overhead projector. This enables the pupils to get immediate reassurance when they press a key. They can see immediately when they are on the required menu. They will not need to ask you whether or not they are on the right screen, leaving you available to assist pupils who are not sure what to do. The ViewScreen also gives me the flexibility to work on particular points that need emphasis. I can produce a picture or a graph on my calculator and display it on the screen and then challenge the pupils to try and produce the same graph. I can flick quickly between the graph and the equations and demonstrate what happens when I change a particular variable. When projecting the display from my calculator onto a whiteboard, I am also able to use the whiteboard in conjunction with the projected calculator screen. I can ask questions, for example, What will the graph of y 5 3x22 look like? By asking the pupils to use their understanding of gradient and y intercept, they can make a prediction about the resulting line. A pupil could draw their estimate of where the line will be onto the axes that are projected onto the whiteboard. I then need only to enter the equation of the line and press s and the whole class can see how closely the line drawn by the pupil matches the line drawn by the calculator. Some of the activities I use also require the pupils to work as a pair, with one setting up a problem on the calculator for the other to solve. In general, I find that pupils work well with one graphics calculator per pair. This arrangement encourages them to talk about their mathematics and check each other s work. I have found it extremely helpful to display a large poster of the calculator at the front of the classroom, as close to my whiteboard as possible. These posters are available on request from Texas Instruments and they make the beginnings of lessons so much easier. As you give out instructions for children to find specific keys, you can point to the key on the poster. It saves a lot of time and confusion. Within our department, we have spent some INSET time looking at the calculators and exploring possible mathematical activities. We have also begun to develop a pool of ideas within our department. Some are our own, some we gleaned from a workshop session, and others are taken from publications such as Mathematics in School, Micromath and the series of books entitled Calculator Maths. We are still trying to develop ways of building the use of graphics calculators into our day to day teaching. We have found that some topics seem to be more suitable than others, and we make an effort to see how the calculator adds an extra dimension to the lesson and encourages mathematical thinking. It came as no surprise to us that the teaching of the mathematics involved in graphs of functions can be explored using graphics calculators. However, there is also a range of other topics that seem to work remarkably well. These vary from the solution of algebraic equations to an exploration of the angles at the centre of a regular polygon (using parametric graphing mode), to practising the use of A=πr 2 by trying to draw a circle of a specified area. 3

4 Jaiwant Timotheus, The Grange School, Stourbridge, West Midlands Developing the use of Graphics Calculators at Key Stage 3 continued Key Skills for Teachers Supporting one another within the classroom has also been a help to members of staff using the graphics calculators for the first time. The following are the key skills and common errors that are worth stressing to teachers to help them develop confidence with the graphics calculator. The Home Screen The graphics calculator has many screens. There is the home screen on which you enter calculations as with an ordinary scientific calculator; there is the mode screen, which is described below; there is the graphing screen on which the graphs are displayed, and many others. You may often find that a pupil is unsure what to do because they have accessed an unfamiliar screen or menu by mistake. It is useful to know that pressing y K on any screen will take you back to the home screen immediately. The MODE Menu There are some menus that are vital to the operation of the calculator. It is worth taking the time to investigate their options. The MODE menu, accessed by pressing the z key, is one such menu. The highlighted options can be changed by using the arrow keys and pressing on the required selection. It is worth finding out what each of these options do. For example, if you are drawing a trigonometric graph and are having problems, check to see whether you are in Degree or Radian mode. Other useful options let you choose the number of decimal places displayed, whether the points are connected or just dots, whether the graphs are plotted sequentially or simultaneously etc. Setting the axes Another important menu is the WINDOW menu. With any graph you need to make some decisions about the scale. The WINDOW menu allows you to tap in minimum and maximum values on the x and y axes, and then the graph is scaled to fit the screen. A problem that often occurs when pupils adjust the window settings, is that they press the subtract key instead of the negative key. Since the calculator screen is wider than it is tall, the above default settings will not result in a square graph. In other words, y5x will not result in a line at 45º to the horizontal. Fortunately, the calculator has some built-in window settings; pressing the q key can access them. One of these is ZSquare which adjusts the Xmin and Xmax settings to cause the graph to appear square. The ZOOM menu also includes a number of other useful settings. Try them! Resetting the calculators As there are so many possible settings, it can be helpful to reset the whole class set of calculators to the default settings before embarking on an exercise. This can be done using the RESET menu, accessed by pressing y Ø and choosing Reset. You then choose whether to reset All Memory or to reset Defaults. Resetting the defaults makes sure that all the menus are set back to their default values, whereas resetting all memory will erase stored data and programs as well. Resetting the contrast The display contrast is adjusted by first pressing 2 and then the up or down arrow. A number between 0 and 9 appears in the top right hand corner. As the batteries begin to wear out, you will need to increase the contrast setting in order to see the display. The brightness in the room may also require you to adjust the display contrast setting at times. Be aware that on the TI-83 (but not the TI-83 Plus) when you reset the calculator s memory, the display contrast is set to 0 and, unless the batteries are very new, the screen will go blank! You only need to press and hold 2 } in order to see the display again. References: 1. Micromath is published by the Association of Teachers of Mathematics 7 Shaftesbury Street, Derby, DE23 8YB Phone: Fax: Website: 2. Mathematics in School is published by The Mathematical Association 259 London Road, Leicester LE2 3BE Phone: Fax: Website: 3. The Calculator Maths series is published by Alan Graham and Barrie Galpin and is available from A+B Books 15 Top Lodge, Fineshade, Near Corby, NN17 3BB Phone: Website: 4

5 Alison Clark-Jeavons Distance-Time graphs using the CBR NNS YEAR 7 OBJECTIVE: Construct functions arising from real-life problems and plot and interpret their corresponding graphs. Key Vocabulary x-axis y-axis distance time horizontal vertical steepness slope gradient speed Resources required A Calculator-Based Ranger (CBR), ViewScreen, OHP and single calculator. Photocopied resource sheets. An example is produced on the next page. You may need several of these. OHP transparency copy of the graphs, cut to fit the ViewScreen window. Be careful about the size of the graphs themselves the teacher needs a copy on OHP that exactly fits over the ViewScreen window. Mental/oral starter Pose some problems such as: If a car was travelling along a motorway at a steady speed of 70 mph, how far would you expect it to travel in 2 hours / 3 hours / half an hour Encourage visualisation using, for example, a blank number scale to allow pupils to relate the hourly jumps to represent increases of 70 miles on a linear scale. Working in small groups, the pupils are given a resource sheet and asked to describe the movements they would have to make to produce the graph. They are encouraged to consider where they would need to stand to start, which direction and roughly how far in how many seconds for each part of the graph. They record their instructions and elect a person who will later act out this graph for the group. The teacher then gets the class back together and, placing the OHP copy of the graph on the ViewScreen window, each group then has a go at matching the graph. Time allowing, the task can be repeated with the different graphs. Plenary Session The class need to be brought together to discuss what has been learned during the lesson: initial difficulties moving in front of the CBR forwards/backwards; the gradient of the lines representing the speed; is it possible to produce a vertical line? what would it mean? what do curved lines represent? Comment The success is due to the fact that all the groups work on the same graphs. This is an advantage over the random graph match facility built into the CBR program where the teacher is unable to control which graphs come up on the screen. hours 0 0 miles 1 70 Offer related problems such as: Estimate the time taken to travel 150 miles etc. Main Activity The CBR and ViewScreen will need to be set up in the classroom so that there is a clear tunnel within which the pupils can act out a distancetime graph. This may mean rearranging the desks to the perimeter of the room. It may be useful to chalk metre marks onto the classroom floor Use the CBR Ranger program and the default settings which give an elapsed time of 15 seconds. Pupils will need to be given a brief introduction to the CBR (used with the ViewScreen for whole-class teaching) and the opportunity be given for a few pupils to move in front of the CBR to produce a graph and stimulate the discussion as to what each part of the graph represents. Make sure pupils are happy with the initial concepts: standing still produces a horizontal line; moving very quickly produces a steep line; the differences between moving towards and away from the CBR. 5

6 Resource Sheet (Distance-Time graphs using the CBR ) Group Name: Instructions: 6

7 Barrie Galpin and Alan Graham The TI-83 and the Numeracy Strategy The supplement of examples in the Framework for teaching mathematics; Years 7 9 provides ideas showing that a graphics calculator can be used to achieve many of the teaching objectives in this part of the Numeracy Strategy. The following examples develop eight of these ideas and provide teachers who may be new to using this technology with a little more support than is offered in the Framework document. Many of the examples chosen are numerical or algebraic ones, using the very basic facilities of the TI-83 or TI-83 Plus calculator and working on the home screen rather than the graphing screen. At the top of each page you will find a reference to the particular part of the Framework where the example may be found. For this selection we decided to keep to examples which can be found in the Year 7 teaching programme: there are many more examples in the Year 8 and 9 programme. Eg: Strand: Topic: Pupils should be taught to: Year: 7 Example: NUMBERS AND THE NUMBER SYSTEM Place value, ordering and rounding Compare and order decimals Use a graphics calculator to generate random numbers lying between 0 and 1, with a maximum of 2 decimal places. Arrange the numbers in order Several of the suggested activities are drawn from ideas in the series of five Calculator Maths books. For teachers wishing to develop their own expertise with the calculator, these books provide a wealth of advice and ideas. The first book in the series (Foundations Plus) deals with the basic skills of handling the technology, while the four other books (Number, Algebra, Shape and Handling Data) provide calculator activities that can be used alongside more traditional resources when teaching topics at Key Stages 3 and 4. Maths departments wishing to encourage teachers to use the calculator in their teaching will be well advised to provide a set of these books and, of course, a calculator to each of their members of staff. The Calculator Maths books are available from: A+B Books 15 Top Lodge Fineshade Corby, NN17 3BB Phone: a+b@fineshade.u-net.com. Website: Inspection copies are available for schools interested in bulk purchases. There are versions of the books written specifically for the TI-83 (and TI-83 Plus) and for the TI-80. 7

8 Barrie Galpin and Alan Graham The TI-83 and the Numeracy Strategy continued Strand: USING AND APPLYING MATHEMATICS TO SOLVE PROBLEMS Topic: Solving problems Pupils should be taught to: Solve word problems and investigate in a variety of contexts Year: 7 Example: Generate a sequence using the answer facility and the table facility ACTIVITY Generating Sequences 1) To generate, for example, the sequence of decimals starting with 0.3 and adding 0.3 using the answer facility: Start on the home screen and press: Before each press of, guess what the next answer will be. 2) To generate the same sequence using the Table facility: Press o. Clear any existing functions. Enter the function alongside Y1= by pressing 0.3 X,T,θ,n. 3) Use the two previous techniques to deal with questions such as: What is ? What is and ? Is 3 the same as 3.0? Is 0.3 the same as.3? What number is halfway between 0.3 and 0.6? Can you generate a sequence with the halfway values? Notes and Extensions Try pressing: 3 5. to produce a sequence on the home screen. Check that the Table is set to the default values by pressing 2 &and, if necessary, changing the values to those shown here. Investigate how to produce this sequence using the Table facility. Set challenges to produce particular sequences such as: the 37 times table, 64, 32, 16, , 3, 9, 27,... Another way of producing sequences is to use the seq command from the List menu. See Calculator Maths: Foundations Plus, page To see the values in the table press 2 [ TABLE]. You can scroll through the sequence using the blue cursor key,. }

9 Barrie Galpin and Alan Graham The TI-83 and the Numeracy Strategy continued Strand: USING AND APPLYING MATHEMATICS TO SOLVE PROBLEMS Topic: Solving problems Pupils should be taught to: Solve word problems and investigate in a variety of contexts Year: 7 Example: Use a graphics calculator to draw shapes ACTIVITY Drawing Shapes 1) Press p and create a friendly graphing window: the values shown here ensure that pixels have integer values. 3) Indirect drawing can be done by entering commands on the home screen. Press 2K to go to the home screen. Press y[draw] 2 to choose Line( from the menu and then complete the command by entering the coordinates of the end points, separated by commas. 2) Direct drawing can be done on the graphing screen. Press s and then y[draw] 2 to choose Line( from the menu. Use the blue cursor keys to move to the point where the line will start and press. Move to the point where the line will end and press again. Watch the coordinates at the bottom of the screen to make sure the end points are where you want them to be. Continue moving and pressing until the shape is complete. What shape is produced by the commands shown here? Enter a fourth line to complete the symmetrical shape. Notes and Extensions Points can be plotted directly or indirectly on the screen by pressing y[draw] ~ 1 to choose Pt-On( from the DRAW POINTS menu. Direct drawing is more fun than indirect the difficulty is then to ensure that students are using the coordinates at the bottom of the screen, rather than just producing the drawing by eye. There are lots of activities in Calculator Maths: Algebra (pages 16 19) which retain the fun aspect, while shifting attention to the coordinates. A third way of drawing shapes involves specifying coordinates as two lists and then drawing a scatterplot or line graph. (See the Activity on page 14). This is a particularly powerful idea and opens up the possibility of transforming the shape. 9 To clear the drawing press y[draw] 1 to choose ClrDraw from the menu. See Calculator Maths: Shape, pages Calculator Maths: Algebra, pages

10 Barrie Galpin and Alan Graham The TI-83 and the Numeracy Strategy continued Strand: NUMBERS AND THE NUMBER SYSTEM Topic: Place value, ordering and rounding Pupils should be taught to: Compare and order decimals Year: 7 Example: Use a graphics calculator to generate random numbers lying between 0 and 1, with a maximum of 2 decimal places. Arrange the numbers in order ACTIVITY Ordering Decimals 1) Set up the calculator Press 3 and fix the calculator to display 2 decimal places. If necessary clear lists L1 to L3. Various methods e.g.enter on the home screen ClrList L1, L2,L3 by pressing STAT 4 2 [L1] 2[L2] b2[l3] 4) Check your ordering by sorting L1 into ascending order. To enter SortA(L1) on the home screen press STAT 2 2 [L1] d To return to the list screen press STAT 1. 2) Produce 6 random numbers between 0 and 1 and store them in L1. Press ç 1 c 6 d ø 2 [L1] Note that rand is in the MATH PRB menu. Notes and Extensions If you want the whole class to get the same random numbers get them all to set the same random seed by entering e.g. 42 >rand. If you want them to get different numbers they should choose their own random seed. 3) Go to the list screen and enter the 6 numbers in L2 in ascending order. Press STAT 1 ~ etc. If necessary edit what you have entered in L2 by overtyping or using the { or 2 /keys. It is possible to put more numbers in list L1, but you then need to scroll up and down to see all the numbers. Try sorting into descending order and then using SortD (item 3 in the STAT menu) to check. There are various ways of changing the range of the random numbers: e.g. 2rand(6) produces six random numbers between 0 and 2. Set the Mode to show 3 decimal places and use rand(6)/10 for values between 0 and

11 Barrie Galpin and Alan Graham The TI-83 and the Numeracy Strategy continued Strand: NUMBERS AND THE NUMBER SYSTEM Topic: Place value, ordering and rounding Pupils should be taught to: Round numbers, including to a given number of decimal places Year: 7 Example: Round decimals to 0 or 1 decimal places ACTIVITY Rounding 1) Use the MODE setting to make the calculator display answers to just one decimal place. 4) Try counting on in eighths, still rounding to one decimal place. Guess the next answer before you press To do this press z ~ ~ ~ 2) Guess what the calculator will display when you enter numbers such as 0.21, 0.25, etc 5) Use the MODE setting to make the calculator display answers with no decimal places. (Choose 0 in the second line of the MODE menu). Repeat the above activities. What do you expect to see when you enter expressions such as 1/4, 1/2, 3/4, 1/3, 2/3 etc? 3) Can you explain the apparent inconsistency shown on this screen? Notes and Extensions It is important to be clear that using the second line of the MODE menu causes the calculator only to display the chosen number of decimal places. In the MATH NUM menu is a similar but subtly different facility. The command round(20/3, 1) for example, (with the MODE set back to Float) actually rounds the value to 1 decimal place in the calculator's memory, producing the value 6.7. You can see the difference if you now press p 3: the answer is not 20 but See Calculator Maths: Number, pages

12 Barrie Galpin and Alan Graham The TI-83 and the Numeracy Strategy continued Strand: Topic: Pupils should be taught to: ALGEBRA Equations, formulae and identities Know that algebraic operations follow the same conventions and order as arithmetic operation Year: 7 Example: Know that the commutative and associative laws apply to algebraic expressions as they do to arithmetic expressions ACTIVITY Commutative and Associative Laws 1) Begin by storing any values into A, B and C. You can either choose your own values or get the calculator to choose numbers at random. Notes and Extensions It is easy to try lots of different numbers stored in A, B and C using a semi-automated process as shown here. randint is in the MATH PRB menu. The three expressions can be re-entered very easily by pressing 2 three times. 2) Check the commutative law for addition and multiplication, firstly with arithmetic expressions and then with algebraic ones. Alternatively the expressions could be stored in a short program. Another approach is to use the calculator's TEST menu. If the expression entered is true 1 is generated and a 0 if the expression is false. 3) Check the associative law for addition and multiplication, firstly with arithmetic expressions and then with algebraic ones. See Calculator Maths: Algebra, page

13 Barrie Galpin and Alan Graham The TI-83 and the Numeracy Strategy continued Strand: CALCULATIONS Topic: Number operations and relationships between them Pupils should be taught to: Know and use the order of operations, including brackets Year: 7 Example: Calculate with mixed operations, including with a calculator ACTIVITY Order of Operations 1) The graphics calculator is an excellent tool for investigating the effect of changing the order of operations and introducing brackets. 4) What mathematical rules does this screen display illustrate? For example, the screen display shown here, viewed by means of a ViewScreen on an OHP, can provide the teacher with the basis for a very useful full class discussion. Notes and Extensions 2) Alternatively pupils, working on individual calculators, may be asked to predict the result of keying in the following expression: Two...plus...three...times...four. When they use the calculator, many will be surprised to find that the answer is not 20. And the need for brackets and the BODMAS rule become immediately clear. It is an easy step on the calculator to move on to generalisations: ask pupils to store whatever numbers they like in A, B and C. Then investigate whether A 1 B 3 C is ever the same as (A 1 B) 3 C and always the same as A 1 (B 3 C). Similarly is A(B1C) always the same as A 3 B 1 A 3 C? See Calculator Maths: Number, page 10. 3) The following exercise is based on one from Calculator Maths: Number, page 10. Insert brackets in the following sequences in order to achieve the desired output. Sequence Desired Output Key sequence

14 Barrie Galpin and Alan Graham The TI-83 and the Numeracy Strategy continued Strand: SHAPE, SPACE AND MEASURES Topic: Co-ordinates Pupils should be taught to: Use co-ordinates in all four quadrants Year: 7 Example: Read and plot points using co-ordinates in all four quadrants ACTIVITY Plotting a Triangle (1) Clear all the lists by pressing 2Ø 4. Press STAT 1 to enter the list screen. Move the cursor to list L1; enter the values 2, 3, 5 and 2. Move the cursor to list L2; enter the values 1, 8, 3 and 1. (4) How do the numbers entered in L1 and L2 match the co-ordinates you found in part (3)? Why are there four pairs of values in L1 and L2 and only three pairs of co-ordinates? Experiment by entering different numbers in L1 and L2 and see the effect on the triangle. Notes and Extensions This activity will enable pupils to practise using co-ordinates with positive and negative numbers. (2) Press 2 [STAT PLOT] to see the StatPlot screen. If any of the plots are set to ON, press 4 to turn all plots off. Press 2 [STAT PLOT] again; press 1 to select Plot1 and choose the settings shown below. They can investigate questions like: how can I display a point in each quadrant of the screen? how can I create an equilateral triangle? how can I create a square or rectangle? how can I draw my initials on the screen? what will happen if I change the sequence of numbers in lists L1 and L2? It is possible and easy to transform the shape see the next activity. See Calculator Maths: Algebra, pages (3) Press q 6 to display the triangle with standard axes. Press r and use ~ and to move around the vertices (i.e. corners) of the triangle. Write down the co-ordinates of the three vertices. 14

15 Barrie Galpin and Alan Graham The TI-83 and the Numeracy Strategy continued Strand: Topic: Pupils should be taught to: Year: 7 Example: SHAPE, SPACE AND MEASURES Reflections and combinations of transformations Recognise and visualise transformations and symmetries Understand reflection as a transformation of a shape in which points are mapped to images in a mirror line or axis of reflection ACTIVITY Reflecting in the Y Axis (1) Set up the calculator to draw a triangle using the technique in the previous activity. (2) Now find the negative of the values in L1 and store them in L3. From the home screen press: 2 [L1] ø 2 [L3] Press STAT 1 to see the new values in L3. Spend a few moments understanding where the new values in L3 have come from. Notes and Extensions Pupils can use the Horizontal command from the DRAW menu to see that the line joining each point to its image is at right angles to the mirror line. They can confirm from the coordinates that the image is the same distance to the left of the mirror as the original is to the right of it. (3) Press 2 [STAT PLOT] and choose Plot2 by pressing 2. Choose the settings shown below. By changing one of the vertices of the original triangle to lie on the Y axis, they can confirm that points on the mirror line do not change their position after reflection. Reflect the image triangle in the Y axis (Use - L3! L4 and change the lists in Plot2). Now they can confirm that a reflection which maps A to A also maps A to A. They can investigate how to reflect in the X axis, using a similar approach (i.e. using - L2! L5). See Calculator Maths: Shape, pages Press s to see the triangle reflected in the Y axis. Press r, and write down the coordinates of the new triangle. } 15

16 George Wickham Rolling a die an awful lot of times! One of the problems in getting the idea of equal probability over to Key Stage 3 pupils is that for small samples it is difficult to draw conclusions from practical work. Unless the experiments take an inordinate amount of time and risk the boredom factor, samples are inevitably small. I well remember my first pathetic worksheet Throw a die ten times. What do you notice? I m sure that, like me, you will have a pupil who gets 8 ones where do we go from there? The TI-83 has in the MATH PRB menu an item marked randint( which you can use to simulate dice rolls. For example, randint(1,6,50) produces 50 dice scores. If you do this with a whole class, make sure they all get a different set of random numbers by getting each of them to set a different seed number first e.g 234! rand. Results from each of the pupils can be collected and summed to gain a larger sample, but this can be tedious and the display is not easy to read for many dice. When you execute the program, the prompt NUMROLLS= appears so that you can put in the number of rolls that you require. Press. The running display in the window shows the progress of the count and the probabilities. When the program is complete you can also press STAT to see the final scores and probabilities. My results for N = 150, 1500, 15000, are shown below. I had done this activity in the past using the BBC computer lovely to program! I decided to write a program for the graphics calculator that would enable me to get large samples with the results being easy to read off in the List screen. I also wanted to show the probability value for each face of the die for the experimental data. I calculated the probability of each face from: no. of times face comes up total number of rolls This would enable me to compare these probabilities with theoretical values, assuming a fair die. Before running the program it is essential to discuss the pupils expectations of the results. For example, with N= 150, 1500, how many of each number 1 6 would they expect? The actual results may help to reinforce the idea of in the long run more clearly, especially if pupils share their results for each sample size. The program is shown here. Prog:ROLLDIE Action explained: :ClrHome Clear screen :Input "NUM ROLLS=",N Enter number :ClrList L 1,L 2, Clear lists :For(X,1,6) :X! L 1 (X):0! L 2 (X) Initialise List values :End :For(X,1,N) :randint(1,6)! D Generate a value 1 6 :L 2 (D)11! L 2 (D) and add one in appropriate row of L 2 :Output(D12,2,D) Display current count :Output(D12,5,L 2 (D)) :End : L 2 /N! L 3 Store probability in L 3 : Stop Of course these are my figures, a good teaching point in itself! I chose these sample sizes because they were exactly divisible by 6, so that it would be easy to calculate whole number values for the theoretical distribution and thus have an intuitive closeness of fit. WARNING The last one takes a very long time on my calculator over 3 hours! I got eight pupils to do each and then we added up to get results for a sample size of one million a rare experience of what a million is like try a square metre of mm graph paper for another! 16

17 George Wickham Rolling a die an awful lot of times! continued Having done that, I wanted to compare these results with those of an unfair die. Bits of blue-tack don t always work, so I modified the program shown previously. I found an article on the TI teachers website that discussed fair and unfair dice from the random number generator. The same article had an interesting algorithm for an unfair die. The idea is to choose values for R at random from the integers 1 to 11. Then assign two of the possible R values to each of the dice scores 2 to 6 and only one of the possible R values to the dice score 1. We again ran samples sizes of 150, 1500, and and my results were as follows: These commands will have this effect: :randint(1,11)! R :R-6int(R/6)11! D They produce the following results. R D This means that 1 has a probability of 1/11, whereas the rest each have a probability of 2/11. We would therefore expect 1 to come up less frequently, in the long run, than the other faces. I modified the program, replacing the command :randint(1,6)! D with the two commands :randint(1,11)! R :R-6int(R/6)+1! D These results sparked off a good deal of discussion. Visually it can be seen that the count for a score of 1 is roughly half of that of the others. Variations on a theme.. I adapted the program to look at spinning a coin a large number of times. For a fair coin the modifications are: In command 4: For(X,1,2) In command 8: randint(1,2)! D For a biased coin I tried: randint(1,3)! R R-2int(R/2)+1! D This gives p(head) =1/3, p(tail) =2/3, so the results are obvious. When I introduced this idea in my last school, for most pupils I simply got them to run the program, but the most able pupils wanted to go further and look at the program itself. This is an ideal what if situation what if we use randint(1,3)? What would randint(1,5) produce? (H=2/5,T=3/5) What would randint(1,4) produce? (a fair coin!) Can we generalise? Could we do a similar thing to change the odds on the unfair die program? (Harder! try 6a-1 where a={2,3,.} ). I hope to try these ideas with a year 7 class soon. I hope that I have shown that these simple programs are capable of being used in a variety of ways and at different ages. Reference: Does a TI-8X Cast a Fair Die? by Constance C. Edwards, Eightysomething! [Spring 1997] 17

18 T 3 Teachers Teaching with Technology T 3 Scotland T 3 Scotland delivers Hand-held Technology Courses that are : Delivered in partnership with Education Authorities Delivered by current classroom teachers Designed to meet the needs of Scottish Teachers High quality content developed in Scotland FLEXIBLE OPTIONS half day, one day, two day, summer school Teachers: Become familiar and comfortable with Technology Get help with introducing new technology into mathematics teaching For more information about T 3 Scotland courses, please contact Ian Forbes: Phone: ian_forbes@education.ed.ac.uk Address: Faculty of Education Dept of Curriculum Studies: STMC Moray House Institute of Education The University of Edinburgh Holyrood Road Edinburgh EH8 8AQ T 3 England, Wales and Northern Ireland T 3 in England, Wales and Northern Ireland is offered in partnership with the Mathematical Association and co-ordinated by their Professional Development Officer, Rosalyn Hyde. The following courses are now offered: Mathematics at Key Stages 3 and 4 Data Handling in Maths and Science for Secondary Schools Numeracy and Transition at Key Stages 2 and 3 (TI-73) Data Handling in Science at Key Stage 3 The recently published draft National Numeracy Strategy for Key Stage 3 makes it clear that graphics calculators have a key role to play in teaching and learning at Key Stage 3. All T 3 materials are written by experienced educators in line with the National Numeracy Strategy, National Curriculum and G.C.S.E. criteria. Training sessions are hands-on workshops given by trainers experienced in using hand-held technology in the classroom. The courses cover basic operation of the calculator and data loggers, ideas and materials for use in the classroom across the attainment targets, practical help in managing the use of ICT in teaching and learning, and the impact of technology on teaching. We are also able to offer short introductory sessions for teachers, as well as longer courses in a variety of formats. A key part of our strategy for training and supporting teachers in using hand-held technology is to work in collaboration with Local Education Authorities. An important way forward for this partnership is for T 3 to train a team of local teachers to become trainers in their L.E.A. Those authorities already working in partnership with T 3 have been able to select an appropriate focus for their area from the list of available courses, to select teachers to act as trainers and to plan a strategy to develop the appropriate use of hand-held technology in their area. If you are interested in more details, please contact Rosalyn Hyde: Phone: hyde@tcp.co.uk Address: 158 Dale Valley Rd Southampton SO16 6QW 18

19 Workshop Loan Programme You may borrow calculators, at no charge, from TI for teacher workshops or in-service training. You may also borrow individual calculators so that you and other teachers in your department can review them. The following products are available through the Workshop Loan Programme: TI-73 TI-83 Plus TI-86 TI-89 TI-92/TI-92 Plus CBL /CBL 2 Probes for the CBL 2 CBR Cabri Géomètre II TI-Presenter At least six weeks before the calculators are needed, simply contact our Workshop Loan Co-ordinator. To make a request, please provide the following information: The objective of your teacher workshop The date and location of the workshop Quantity, type of calculator(s) and accessories required Delivery address and phone number Preferred delivery date To request a calculator loan or for information on this programme contact: CSC Customer Service Centre Phone: Fax: ti-loan@ti.com This could be your last issue of... We are updating our subscription list. 19

20 FREE Workshops and Demonstrations We can send an instructor to your school or education authority to help support the use of Texas Instruments technology in education. Training courses are available from an introductory two hour session to full-length in-depth courses in a variety of curriculum areas. In England, Wales or Northern Ireland: Call Rosalyn Hyde on COURSES INCLUDE: Mathematics at Key Stages 3 and 4 Data Handling in Maths and Science for Secondary Schools Numeracy and Transition at Key Stages 2 and 3 (TI-73) Data Handling in Science at Key Stage 3 In Scotland: Call Ian Forbes on T 3 SCOTLAND OFFERS HAND-HELD TECHNOLOGY COURSES: Delivered in partnership with Education Authorities High quality content developed in Scotland Taught by current classroom teachers FLEXIBLE OPTIONS half day, one day, two day, summer school ti-cares@ti.com CSC Customer Service Centre Phone: For Teacher Express Service, press 84 after calling the CSC Fax: ti-cares@ti.com Instructional Dealers Addex Limited (Ireland) Comcal George Waterstons & Sons Limited Jaytex Oxford Educational Supplies Science Studio Limited Shaw Scientific Limited (Ireland) All products available in Europe are manufactured to ISO 9000 certification. Cabri Géomètre II is a trademark of Université Joseph Fourier. All other trademarks are the property of their respective owners. Texas Instruments reserves the right to make changes to products, specifications, services and programs without notice. Printed on recyclable 100% chlorine-free paper by Thamesdown Colour Limited, UK. Desk Top Publishing Cloud 9 Publishing Limited, UK Texas Instruments CL2001NLM2/GB