Enter the Maths Zone: Algebra Net NotesPLUS

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1 Programme Worksheet : Palindromic Pursuits Take a number. Reverse it. Add the two numbers. Reverse the answer. Add the two numbers. Reverse the answer. Add the two numbers. Reverse the answer. Add the two numbers. The answer is a palindrome. It reads the same in both directions.. Investigate what happens when ou carr out the above process with the following starting numbers: 9. Some dates, like //, are palindromic. How man palindromic dates can ou find from the 990s?. Work out,,... What do ou notice? What about powers of 0,, 00,, 00,...? Challenge The answer to each clue in this crossword is a palindrome. A few digits have been placed to start ou off. Tr and complete the crossword. A calculator ma be useful. 0 9 Choose some numbers of our own; reverse the digits and add. Continue until a palindrome appears. 0. Record the number of steps that each number takes to produce a palindrome. Can ou describe what is happening?. If ou consider just two-digit numbers, how man would ou have to investigate altogether? Which numbers less than 00 give a palindrome in just one step? Can ou eplain our results? What about numbers with more than two digits? (9 + ) ( 0 ) 0 ( ) ( ) + (9 ) 9 Channel Four Television Corporation 000

2 Programme Worksheet : Cuts Me Up. If ou cut a piece of string once ou will have pieces. If ou cut the string again ou will have pieces.. Investigate and describe the connection between the number of cuts and the number of pieces for strings folded like these: What happens if ou continue cutting the string? Complete this table: number of cuts (n) 0 number of pieces (p) How man pieces would ou get if ou made 00 cuts? Can ou describe the connection between the number of cuts and the number of pieces of string?. Investigate what happens if ou fold the string once before ou cut. You can record our results in the table below. Use tables to record our results and to search for patterns. Tr to find the relation between n and p in each case.. You can also loop the string around one blade of the scissors before making a cut. Tr cutting with different numbers of loops and record our results in this table. number of loops (l) 0 number of pieces (p) Can ou describe the relationship between l and p? number of cuts (n) 0 number of pieces (p) What is the relation between n and p this time? What if ou made 00 cuts? Channel Four Television Corporation 000

3 Programme Worksheet : Chessboard Challenge How man squares are there on a chessboard? No not! You could start b looking at a smaller board. For eample, this board has squares. 9 squares squares square squares in total Check that ou can find these squares. Now investigate boards of other sizes. When ou have found how man squares there are on the board, tr to eplain how ou could work out the number of squares on a square board of an size. Channel Four Television Corporation 000

4 Programme Worksheet : Balancing Act The Balance of Nature must be maintained with vigilance. It is alwas recalibrated at New Year and at midsummer and the values of the elements are changed for securit purposes. The Wizard is responsible for divining the true values to be used and ensuring that the instructions for setting the balance are correct. Stabilit in the Garden of Reason depends on solving the balances to find the elements.. Last New Year the balances looked like this. One kilogram is shown b each circle. AIR A + = WATER AIR W + A + = 9 FIRE FIRE F = EARTH AIR AIR E + A = 0 Work out the value of each element.. The Wizard wrote a scroll with instructions for the Guard and Lisa to follow in case the balance was disturbed again and he wasn t around. It contained these four statements: Earth + ( Air) + Fire = Water = kg Water + Air + kg = kg ( Fire) + Water = kg Can ou show how to solve these to find the value the Wizard has chosen for each element?. The fifth element Custard is et to be set! Can ou write down some possible equations for Custard to go with the other values found in the programme? The were: Air =, Water = 9, Fire =, Earth =. Channel Four Television Corporation 000

5 Programme Worksheet : Amazing Balances START C A W+E W+C W E W F W F+ A F+E A W F+ A+W E+ A+C E F+C 0 C A W+E A+W F+E W C+A 0 E+F A A W E A+W+F E F+A+E W E W A The Wizard had to attend a Wizard s Convention at midsummer. and couldn t be in the Garden of Reason to recalibrate the Balance of Nature. He asked the Guard to do it, but (conscious as alwas of securit) he onl told the Guard the value of Custard. He hid information to work out the remaining values in a maze of balances. He also put some false balances in to add to the confusion! The Guard found it all too much and went to look for Lisa! The Wizard told the Guard that Custard was. Find the correct values for all the elements. To work through the maze: Move from the right-hand pan of the balance labelled START, b finding another balance that has the same epression written in bold on the left-hand pan. See what is written on the right of this balance and find the net balance with this epression on the left in bold. Continue until ou reach a dead end. The final balance should contain enough information for ou to find the value of an element. You ma find it useful to write down the equations represented b the balances as ou go along. When ou reach the last balance use the equations ou have found on our route to work out the values of all the elements and to check them. Channel Four Television Corporation 000

6 Programme Worksheet : Wizard Training Discover how to amaze and baffle our friends and famil. Follow the instructions in the eamples below to find techniques that will demonstrate our intellectual prowess and superior skills! Tr these puzzles:. Think of a number. Add. Take awa. Take awa the number ou first thought of. What happens? Tr another start number. Will the answer alwas be the same? Can ou prove it?. Think of a number. Add. Double it. Divide b. Take awa the starting number. What is the answer? Tr to eplain what is happening.. Think of a number. Add 0. Double it. Take awa. Multipl b. Add. Divide b. Subtract the number ou started with. Tr different starting numbers. Use algebra to show what is happening.. Think of a number. Add. Multipl b 0. Subtract 0. Divide b. Take awa twice the number ou first thought of. Can ou prove that the answer will alwas be the same?. Think of a number. Multipl b. Add. Multipl b. Subtract. Divide b 0. What is happening? Tr to eplain how this puzzle works. Can ou prove it? Make up our own think of a number puzzle and check using algebra that it will alwas work. Now memorise the steps and go and stun people with our mind-reading powers! Challenge Write down our house number. Multipl it b. Add. Multipl b 0. Add our age in ears. Add. Subtract. What is our answer? Tr to see how it works. Channel Four Television Corporation 000

7 Programme Worksheet : None Shall Pass! The Guard has more puzzles like the one he set Lisa, just in case there are an more visitors to the garden who want to pass through the doors. He is particularl keen to keep out the Ravening Hordes! See if ou can find the three numbers in each of the puzzles:. The number on the second door is more than the number on the first door.. The three numbers are consecutive whole numbers. The number on the third door is more than the number on the second door. The sum of the three numbers is 0.. The number on the second door is more than the number on the first door. The product of the three numbers is 0.. The number on the second door is twice the number on the first door. The number on the third door is twice the number on the second door. The number on the third door is more than the number on the second door. The sum of the three numbers is 0.. The number on the second door is double the number on the first door. The product of the three numbers is 000. Make up our own puzzle like these. Swap puzzles with a partner and tr to solve each other s. The number on the third door is times the number on the first door. The three numbers add up to 9. Channel Four Television Corporation 000

8 Programme Worksheet : Tr, Tr and Tr Again The number 9 can be epressed as the sum of a pair of numbers: 9 = + 9 = + 9 = If the two numbers are then multiplied together the give different products: = =. 0. = 9.0 Find the pair of numbers that gives the greatest product. Investigate the greatest product that can be found when other numbers are split into pairs. Can ou generalise our results? What if the number is split into parts? Can ou find the split that gives the greatest product? Etend our investigation to,... n parts. Tr to generalise our findings. Channel Four Television Corporation 000

9 Programme Worksheet : Pi in the Sk Find out more about the number π. What is it? How can ou find a value for it? _the_ages.html Background to the histor of calculating π, and links to other resources. How can it be used? What methods are used to calculate it? How accuratel has it been calculated? What did earlier civilisations know about this number? Wh is it so fascinating? You could create a set of references to information about π in books, newspapers or journals, or ou could use this list of web links: The website to accompan the book The Jo of Pi. An eample of mnemonic writing based on the digits of π, claiming to be the world s longest. Include some details about the content of the materials ou have chosen and, if appropriate, a description of how the could be used b other students. Make a displa of interesting, informative or amusing material and facts to do with π. A UK site aimed at GCSE students and featuring a π fan club. A list of π links. Channel Four Television Corporation 000

10 Programme 9 Worksheet : Restorative Vegetable Soup The Wizard is making his restorative soup. He claims it will increase intelligence, boost energ levels and act as a general tonic. He sends Lisa and the Guard to the vegetable plot to collect some unusual ingredients. As alwas nothing is straightforward. The have to identif the vegetables on the grid, using the Wizard s notes, given below. Use the graph to draw the lines that will help ou work out the coordinates of the points where the vegetables have been left. Here is the Wizard s list of ingredients: one artichoke at the origin curl kale at the point where = meets = a bunch of chervil at the point where = meets = an aubergine at the point where = crosses the -ais a marrow at the point where - = and = intersect - asparagus spears, one at each of the points on = where the value of is a positive whole number one head of chicor at the point where = crosses the -ais a root of celeriac at the point of intersection of = and = sorrel leaves where = crosses = zucchini at the vertices of the triangle formed b the intersection of the lines =, = and = + Channel Four Television Corporation 000

11 Programme 9 Worksheet : Line Them Up You are going to pla noughts and crosses on the grids provided. Pla with a partner and take turns to place an O or an X on a grid point. Describe using coordinates the position where ou wish to place our mark. The winner is the first to get four points in a line. 0 Pla a few games and see what happens. What do ou notice about the coordinates of the points on horizontal winning lines? What about vertical or diagonal winning lines? Challenge What if ou add the rule that ou can onl claim a win if ou can correctl give the equation of our winning line? 0 What if ou continue until the grid is full? The winner is the plaer with the most winning lines. How man winning lines are possible on the grid? What are their equations? 0 Channel Four Television Corporation 000 page of

12 Channel Four Television Corporation 000 Enter the Maths Zone: Algebra Net NotesPLUS page of Programme 9 Worksheet : Line Them Up

13 Programme 9 Worksheet : Line Them Up Channel Four Television Corporation 000 page of

14 Programme 9 Worksheet : Plotting Patterns Here is a sequence of patterns made from matchsticks. Patterns Matches 9 Here is a table showing the pattern number () and the number of matches used to make it (). pattern number () 9 0 number of matches () 9 Complete the table. Tr to describe the number patterns that ou notice. Plot each pair of values as points on a graph. Describe the graph as full as ou can. Consider the difference between successive pairs of values of. How does this relate to the graph ou have plotted? Investigate other sets of matchstick patterns that grow according to a fied rule. Make tables and plot points as above. How are the graphs of these values related to the differences that can be found in the tables? How is the point where the graph meets the -ais related to our patterns in each case? Tr to find a general rule, for each set of patterns ou have made, that allows ou to calculate the number of matches () for a given pattern number (). How is our rule related to the graph ou have plotted? Channel Four Television Corporation 000

15 Programme 0 Worksheet : Plentiful Peppers A Lisa investigated what happened when she put just one pepper in the first machine and changed the number on the front. She worked out that the machine added to the number on the front and multiplied the result b the number of peppers. She could have fied the number on the front and put in different numbers of peppers instead. See what happens b completing these tables of values for the first machine.. The number on the front of the machine is. number of peppers put in number of peppers coming out. The number on the front of the machine is. number of peppers put in number of peppers coming out. The number on the front of the machine is 0. number of peppers put in number of peppers coming out. The number on the front of the machine is. number of peppers put in number of peppers coming out. The number on the front of the machine is. number of peppers put in number of peppers coming out Draw a graph of the values in each table. What do ou notice? What happens if the number on the front of the machine is? What about other values? B Lisa found that the second machine added the number of peppers she put in to the number on the front and then added. Draw and complete tables (like the ones above) that show what happens when ou fi the number on the front of the second machine and put in different numbers of peppers. Draw a graph of the values in our tables and describe what ou find out. What front numbers might give Lisa a strange result? Channel Four Television Corporation 000

16 Programme 0 Worksheet : Full Functional Here are some tables showing s and s from different function machines that the Wizard has ordered. These are a bit more sophisticated than the earl models that Lisa eperimented with. Some of them perform more than one operation to give the. The machines are still being tested. Unfortunatel, in each set of s there is one wrong value. Find a function that fits each pattern of s and s, and describe it. Find the wrong value and correct it Challenge Can ou use a spreadsheet to help ou discover the functions? Make up some functions of our own, and give our partner sets of s and s to for them to work out our function. Channel Four Television Corporation 000

17 Programme 0 Worksheet : The Ghost in the Machine Work in small groups of three to five. One person is going to act as a function machine and compute s from numbers b the rest of the group. Decide who is going to be the machine first. This person decides on a rule that will be applied to all s. The object of the game is to guess the rule b giving the machine s () and being told the resulting s (). Challenge What happens if ou have two s ( and ) that the machine combines in some wa to produce an (z)? What happens if two plaers are machines and one machine s result is fed into the other? (You could ask the first machine to give the second a piece of paper with their written on it read for processing). Before ou can win the game b stating the rule, ou must give a number () and correctl predict the result () that the machine would produce. Pla the game and see what happens. Pla again with a different person as the machine. Think about the best was to monitor and record our results. Can ou use graphs to help ou make our predictions? Can ou use algebra to describe what the machine is doing? Channel Four Television Corporation 000