Patterns and functions recursive number patterns

Transcription

1 Ptterns nd functions recursive numer ptterns Look round you, cn you see pttern? A pttern is n rrngement of shpes, numers or ojects formed ccording to rule. Ptterns re everywhere, you cn find them in nture, rt, music nd even in dnce! In this topic, we re looking t numer ptterns. A numer pttern is sequence or list of numers tht is formed ccording to rule. Numer ptterns cn use ny of the four opertions (+,,, ) or even comintion. In the exmple elow, if we follow this instruction: strting t 1 dd 5 ech time we get this numer pttern: Write the next numers in ech sequence y following the rule: Rule: dd Rule: sutrct c Rule: multiply y Figure out the missing numers in ech pttern nd write the rule. Circle the scending ptterns c Rule + 7 Rule + 0 Rule 15 d e f Rule + 8 Rule 9 Rule 7 Complete these grid ptterns. Look closely t the numers in the grid nd follow the ptterns c F 1 1

2 Ptterns nd functions recursive numer ptterns Some numer ptterns cn e formed with opertions ech time. For exmple: The rule is multiply y nd dd ech time. With these numer ptterns, write the rule s opertions in the dimond shpes nd descrie it underneth The rule is The rule is Len nd Mx were sked to show numer pttern for different rules. Check ech sequence nd put circle round ny errors. You my use clcultor. Strt t, dd 1 nd multiply y No errors. Len Strt t, dd 1 nd multiply y Mx Look t ech pttern of shpes nd see if you cn predict the following: Wht will shpe numer 0 look like? Drw it here: Wht will shpe numer look like? Drw it here: Wht will shpe numer 15 look like? Drw it here: Wht will shpe numer look like? Drw it here: F 1

3 Ptterns nd functions function numer ptterns There re different types of rules tht numer pttern cn e sed upon: 1 A recursive rule used to continue the sequence y doing something to the numer efore it. A function rule used to predict ny numer y pplying the rule to the position of the numer. A function rule is rule sed on the position of numer. Consider this. Luci ws given this numer pttern: Her techer sked her to work out wht the 0th numer would e without continuing the sequence. Luci used tle to work out the rule etween the position of numer nd the numer in the pttern. She worked out the rule to e 5. Position of numer Function rule Numer pttern So, following the rule sed on the position of numer, the 0th numer is 100. This is function rule. 1 Use the function rule nd then pply the rule to position 0. Position of numer Function rule Numer pttern Position of numer Function rule Numer pttern HINT: In the lst pttern, the rule hs opertions. c Position of numer Function rule Numer pttern d Position of numer Function rule Numer pttern F 1

4 Ptterns nd functions function numer ptterns Function rules with opertions re esy to work out when we look t how they re linked to the multipliction tles. Position of numer 1 5 times tle Numer pttern Function rule Multiply y nd then dd This tle shows tht the numer pttern is the sme s the times tle with dded to ech nswer. Complete ech tle to show how function rules with opertions cn e linked to multipliction tles. Position of numer 1 5 times tle Numer pttern Function rule Multiply y nd then dd Position of numer 1 5 times tle Numer pttern Function rule Multiply y nd then dd c Position of numer times tle Numer pttern Function rule Multiply y 8 then dd Complete this tle to show the times tles with dded. Position of numer 1 5 times tle Numer pttern Function rule Multiply y then dd Wht would the numer in the 0th position e? 0 + = 8 F 1

5 Ptterns nd functions mtchstick ptterns Use the function rule to predict geometric ptterns with mtchsticks. Here is n exmple. Mi mde this sequence of shpes with mtchsticks: Shpe 1 Shpe Shpe Shpe If Mi followed this sequence, how mny mtchsticks will she need for shpe 0? Shpe numer Numer of mtchsticks Function rule Numer of mtchsticks = Shpe numer 1 Complete the tle for ech sequence of mtchstick shpes. Use the function rule for finding the numer of mtchsticks needed for the shpe in the 0th position. Shpe 1 Shpe Shpe Shpe numer Numer of mtchsticks Function rule Numer of mtchsticks = Shpe numer Shpe 1 Shpe Shpe Shpe numer Numer of mtchsticks Function rule Numer of mtchsticks = Shpe numer c Shpe 1 Shpe Shpe Shpe numer Numer of mtchsticks Function rule Numer of mtchsticks = Shpe numer 7 F 1 5

6 Ptterns nd functions mtchstick ptterns This time the rule for this mtchstick pttern hs opertions. Cn you see why? Look for multipliction pttern nd how mny extr there re in ech shpe. Look for repeting element. Shpe 1 Shpe Shpe Then look to see wht is dded. These re circled in the sequence elow. Shpe 1 hs mtchsticks = Shpe hs 5 mtchsticks + 1 = 5 Shpe hs 7 mtchsticks + 1 = 7 Shpe numer Numer of mtchsticks Function rule Numer of mtchsticks = Shpe numer + 1 In ech of these ptterns, look for the repeting element nd then wht is dded ech time: Shpe 1 Shpe Shpe Shpe numer Numer of mtchsticks Function rule Numer of mtchsticks = Shpe numer + 1 Shpe 1 Shpe Shpe Shpe numer Numer of mtchsticks Function rule Numer of mtchsticks = Shpe numer c Shpe 1 Shpe Shpe Shpe numer Numer of mtchsticks Function rule Numer of mtchsticks = Shpe numer + 1 F 1

7 Ptterns nd functions function mchines This is function mchine. Numers go in, hve the rule pplied, nd come out gin. IN 8 10 RULE: OUT 0 1 Look crefully t the numers going in these function mchines nd the numers coming out. Wht rule re they following ech time? IN 8 RULE: OUT IN RULE: OUT Wht numers will come out of these function mchines? IN 11 9 RULE: OUT IN RULE: OUT Wht numers go in to these numer function mchines? IN 7 RULE: 1 OUT 15 IN 8 1 RULE: + OUT F 1 7

8 Ptterns nd functions function mchines Write the rule in ech doule function mchine. Ech rule is mde up of opertions ( then +). IN 8 RULE: + OUT 10 IN 10 RULE: OUT c d IN 0 RULE: + OUT 0 1 IN 5 RULE: OUT Which function mchine will win this gme of ingo? Write the numers tht come out nd colour ech mchine s numers in different colour. Check which mchine hs numers in line in ny direction. IN OUT This one does. 7 8 FREE 18 9 SPACE IN 1 OUT F 1

9 Ptterns nd functions function tles with ddition nd sutrction The function mchines showed us tht when numer goes in, it comes out chnged y the rule or the function. There re mny function ptterns in rel life. Look t this exmple: At their Christms fir, Middle Street Primry School chrges $1.50 for gift wrpping service. This tle shows the totl cost of ech wrpped gift nd shows the rule. Cost of unwrpped gift $7 $10 $15 $18 Cost of wrpped gift $8.50 $11.50 $1.50 $19.50 Rule Cost of unwrpped gift + $1.50 = Cost of wrpped gift 1 Complete the function tle for the totl cost of dy out t fun prk. You must py n entry fee of $1 nd purchse wrist nd for the mount of rides tht you wnt to go on. Wrist nd 5 rides for $0 rides for $5 7 rides for $0 8 rides for $5 Totl dmission $ $7 $ $7 Rule Wrist nd + $1 = Totl cost Complete the function tle for the totl cost of lunch t school cnteen. Students py $.0 for sndwich nd then choose wht else they would like. Work out the totl cost of lunch for ech option. Lunch option Drink: 80 Fruit: 95 Yoghurt: $1.10 Ice lock: $1.50 Totl cost of lunch $.0 $.5 $.50 $.90 Rule Lunch option + $.0 = Totl cost of lunch 5F hve fitness every Thursdy fternoon for 0 minutes. Ech week they complete fitness ctivity nd then ply running gmes. Work out how much time is left for gmes fter ech ctivity. Activity Skipping 10 minutes Str jumps 1 minutes Push ups 15 minutes Sit ups 1 minutes Time left for gmes 0 minutes 18 minutes 15 minutes 1 minutes Rule 0 minutes length of time of ctivity = Time left for gmes F 1 9

10 Ptterns nd functions function tles with multipliction Let s look t more rel life function tles, this time sed on multipliction. By working out the function, you cn extend the pttern to find out unknowns. For exmple: A kery mkes 10 cupckes n hour. The rule to work out the numer of cupckes this kery produces within certin mount of time is: Numer of hours 10 = Numer of cupckes Hours Cupckes How mny cupckes will it mke in 1 dy? This tle only goes up to 8 hours ut we cn use the function to nswer this question: hours 10 cupckes = 0 cupckes 1 Complete the function tles, write the rule nd nswer the question. A dry clener chrges $ to iron shirt. Numer of shirts Cost $ $ $ $8 $10 $1 $1 $1 Write the rule for finding out the cost of ironing shirts when you know how mny shirts: How much does it cost to hve 1 shirts ironed? Numer of shirts $ $ Monic nd Ann hve lemonde stnd outside their house. For every litre of lemonde they mke cups to sell. Litres Cups Write the rule for finding out how mny cups re needed when you know how mny litres hve een mde: How mny cups will e needed if they hve enough to mke 1 litres of lemonde? Numer of litres = Numer of cups 8 cups c At cinem, the lollies re sold y weight. 1 scoop costs 50. Scoops of lollies Cost 50 $1 $1.50 $ $.50 $ $.50 $ Write the rule to find out the cost of the lollies when you know how mny scoops: How mny scoops of lollies cn I get for $10? Numer of scoops 50 = Cost of lollies 0 scoops 10 F 1

11 Rows nd columns pply Getting redy This is gme for plyers. For this gme you will need dice, this pge nd 1 counters ech, in different colours. A clcultor is optionl. Wht to do Roll oth dice, dd them together nd put this vlue in the function rule. For exmple, if I roll nd 5, I dd these nd get 8. I put 8 into the first rule nd get 8 7 = 5. I plce one of my counters on 5. If the nswer is lredy tken, you lose turn. The winner is the plyer with the most counters in ny row or column fter rounds of ech function rule. (The numers do not hve to e next to ech other, lthough you could ply like tht if you wnted longer gme.) Function Rule 1 7 Function Rule Function Rule (8 ) Wht to do next Chnge the oject of the gme. For exmple, the winner might e the person who hs their counters on the most even numers. F 1 11

12 Pizz Pizzzz solve Getting redy Pizz Pizzzz is the nme of pizz delivery compny tht you work for on the weekends. You drive ll round town delivering hot nd tsty pizzs in record time. To encourge you to uphold the compny gurntee of delivering pizzs in record time, your oss hs given you choice of onus scheme. Wht to do Which scheme pys the est onus? Use the tles elow to work out which pyment system is est. Pyment System 1 For ech pizz tht you deliver, you will get $. Pyment System For ech pizz tht you deliver, your onus will doule, strting t 50. Numer of pizzs Bonus Numer of pizzs Bonus 1 $ $ $ $8 5 $10 $1 7 $1 8 $1 9 $18 10 $ $1 $ $ 5 $8 $1 7 $ 8 $ 9 $18 10 $5 Which onus scheme would you choose nd why? System is etter if I deliver more thn 5 pizzs. Wht to do next Cn you think of when the other onus scheme would e etter? If I delivered 5 or less pizzs. Which onus scheme do you think your oss would prefer you to choose? System 1. 1 F 1

13 Equtions nd equivlence understnding equivlence An eqution is like set of lnced scles. Both sides re equl. Look t the scle on the right. On one side re lck tringles nd grey tringles. On the other side is the prolem +. Is this lnced eqution? Yes, ecuse they oth represent 7. Sometimes, we hven t een given ll the informtion nd we hve to work it out. This is wht lger is solving missing numer puzzles. + = Mke these scles lnce y dding the missing vlue: These scles hve numer prolems on ech side. One side hs complete prolem. On the other side, you need to work out the missing vlue. Write the vlue in the ox so tht the scles lnce: c d e It will help to write the nswers next to ech sum. f 11 F 1

14 Equtions nd equivlence understnding equivlence If the sides re not lnced, we sy the eqution is unequl. Look t these scles: 5 is greter thn 5 + So insted of n equls sign, we use the greter thn sign: 5 > 5 + Complete the following scles nd inequlities y dding greter thn (>) or less thn (<): > < 1 In these prolems, you hve to dd oth the symol nd vlue tht would mke the eqution true. Rememer, just like with ordinry scles, the igger vlue will e lower down ? ? HINT: there re mny vlues tht would work in the oxes! 1 > 17 + (0 0) 7 7 < 100 (0 50) c ? d 8 8 9? 9 9 > 10 (0 10) 8 < 9 or more 1 F

15 Equtions nd equivlence using symols Symols help us when we hve more thn one numer to find. A symol cn e ny shpe nd stnds for ny unknown numers. 1 Work out the vlue of the dimond in ech question. Notice the sme symol is dded times. Your times tles will help here = = c = 5 Find the vlue of the symols. Rememer tht if symol is used more thn once, it mens it is the sme vlue gin. Guess, check nd improve strtegy will help here. + + = 9 = = = c = 9 = 7 Find the vlue of the symols nd then check if you re right y using the sme vlue in the question longside it. = 81 = = 9 = + + = 9 = 0 = 5 = 1 F 15

16 Equtions nd equivlence using symols Known vlues cn help us work out the vlues of the secret symols. Your knowledge of inverse opertions will lso come in hndy. = 1 + = 0 + = 1 = = By knowing the vlue of we cn work out 1 + = 0, so = 8 By knowing the vlue of, we cn work out + 8 = 1, so = 5 Look crefully t the exmple ove nd follow the steps to find out the vlues of these secret symols: = 15 + = 0 + = 5 = 5 = 9 = = 5 = = 0 = 18 5 This time you must find the vlue of different symols using the clues in ech step: = 1 + = 100 = + = 50 = 5 + = c + = 0 = 7 1 = 5 = = = = 0 = 9 9 = 5 = 11 9 = 5 = 8 1 F

17 Equtions nd equivlence keeping lnce We cn work out how mny counters re in ech ox y keeping lnce. Here is our eqution. How do we work out how mny counters re in ech ox? We use symol to represent the unknown. + = 10 If we tke wy from ech side, we mintin the lnce nd mke the prolem esier. We now hve to work out the vlue of = 8 = 8 This works ecuse + = 10 1 Find out how mny counters re in ech of the oxes. Rememer to tke wy the sme mount on oth sides so the lnce is kept. I will tke wy from ech side. This leves me with: = 9 = This works ecuse + = 11 I will tke wy from ech side. This leves me with: = 1 = This works ecuse + = 15 c I will tke wy from ech side. This leves me with: = 10 = 5 This works ecuse 5 + = 1 F 17

18 Equtions nd equivlence keeping lnce In this ctivity you need to find out wht ech counter is worth. Step 1 Mke the numer stnd lone y keeping lnce. Step Write n eqution to solve. = = Look crefully t ech lnced scle nd work out wht the symols equl: = 9 = 7 = = 7 c 0 d 0 = = = 15 = This time use guess, check nd improve to work out wht the vlue of the symols could e. The symols hve the sme vlue on oth scles. 1 1 = = 10 or or or or 7 Answers will vry. = or 8 or F

19 Mgicin s ht trick solve Getting redy Mndn the mgicin is the mster of opticl illusions, mgic tricks nd disppering cts. One of his fvourite tricks, is the disppering ct where he wves his wnd nd things dispper or do they? Arkzm rkzoo look crefully t these clues! Work out wht he hs hidden under his top ht. Clue: It is only one thing either rit, ook or pinepple. CLUE 1 CLUE Wht to do Underneth Mndn the mgicin s ht is: A rit. F 19

20 Dhiffushi islnd currency solve Getting redy On the holidy islnd of Dhiffushi, insted of money, they use shells, eds nd peles. Insted of dollr sign they hve this: D D, which stnds for Dhiffushi Dollrs. Wht to do Work out wht this currency is equl to y looking t these clues: = = = D D 8 Key Shell = Bed = Pele = Using the symol D D, convert the price of ech of the following : 1 pele = D D 8 so peles = D D 1 ed = D D 1 so eds = D D 1 shell = D D 9 so shells = D D Using Dhiffushi currency, drw wht I could use to py for the following: Snorkeling = D D eds or shells Rinforest trip = D D 0 eds nd peles or 5 peles Turtle wtching = D D 5 shells Diving = D D 7 eds or 8 shells or 9 peles In Dhiffushi currency, how much ws my ccommodtion if I pid: My ccommodtion would e D D 8 0 F

21 Using equtions lnce strtegy using inverse opertions How cn we find out the vlue of the symol in this eqution? We need to mke it stnd on its own while keeping the eqution lnced. This is clled the lnce strtegy. We do this y performing the inverse opertion to oth sides. Cn you see why? 5 = = 0 5 = Doing the inverse cncels out numer nd helps get the unknown to stnd on its own. 1 Prctise performing inverse opertions y getting ck to the first numer. The first one hs een done for you: 0 5 = 5 = = 5 7 = 5 c 8 = 8 8 = d 7 9 = 8 9 = 7 e 5 = 9 = 5 f 18 = = 18 Find out the vlue of ech symol y performing inverse opertions: 8 = 7 = = 5 7 = 8 = 8 8 = Find out the vlue of ech symol gin. Perform the inverse opertion in fewer steps. 9 = 5 1 = 5 = 5 9 = 5 1 = 5 = 0 Find out the vlue of ech symol y following the sme steps s ove. Set your work out netly: = 5 5 = 15 = 5 = 15 5 = 9 = 5 F 1

22 Using equtions lnce strtegy using inverse opertions Sometimes the symol is not t the eginning so you hve to rerrnge the eqution y performing n inverse opertion. This is ecuse it is esier to solve when the symol is on the left hnd side of the equls sign. 1 = 78 Step 1 Move the symol to the left with n inverse opertion. The inverse of + is : 1 + = 78 Step Mke the symol stnd lone with n inverse opertion. To do this, sutrct 1 from oth sides: 1 + = 78 1 Step Now we cn perform simple sutrction to find out the vlue of the symol: = 78 1 = 5 Follow the steps outlined ove to find the vlue of the symol. = 5 = 78 + = 5 + = 78 = 5 = 78 = = c = 11 d 5 = = = 105 = 11 = = 7 = 5 e = 78 f 1 = 9 + = = 9 = 78 = 9 1 = 5 = 78 F

23 Using equtions word prolems If you cn solve equtions with one unknown numer using the lnce strtegy, you will e le to solve word prolems with ese! A lrge group of friends signed up to prticipte in fun run. 5 of them got food poisoning the dy efore so hd to pull out. How mny people signed up if totl of 8 people rn the rce? To get the str on its own we use the inverse opertion nd do the sme to the other side. 5 = 8 5 = = 10 1 Solve the following word prolems using inverse opertions. Strt y choosing the mtching eqution from the ox elow. $50 + = $10 70 m = 8 m $8 + $100 + = $00 Jck hd piece of rope nd cut off 70 metres. He ws left with 8 metres. How long ws the rope? 70 m = 8 m = 8 m + 70 m = 108 m Tom found $50 on the us on Mondy nd ws given irthdy money y his Grn on Wednesdy. How much did his Grn give him if he ended up with $10? $50 + = $10 = $10 $50 = $80 c Mtild sved $8 towrds trip to the snow nd her prents gve her $100. How much more money does she need if the trip costs $00? $8 + $100 + = $00 = $00 $8 $100 = $117 F

24 Using equtions word prolems Kte sved ech week for 5 weeks nd then spent $5. How much ws she sving ech week if she hd $100 left t the end of 5 weeks nd fter spending $5? Step 1 Set up the eqution. The tringle stnds for the mount Kte ws sving ech week. 5 5 = $100 Step Cncel out the 5 with the inverse opertion: = = 15 Step Cncel out 5 with the inverse opertion: 5 = 15 5 = $5 Kte ws sving $5 ech week. Mke the unknown numer stnd on its own while keeping the eqution lnced. We do this with inverse opertions. Solve the following word prolems using inverse opertions. The equtions re prtilly set up. You my like to use clcultor. For my school fete I ked tches of cookies, relised tht wsn t enough nd so I ought dozen more. How mny were in one tch if I hd 8 cookies ltogether? 8 sme sized Yer 5 clsses ssemled in the plyground for photo dy. There were 11 students sent. How mny students re there in ech clss if there were 1 there on the dy? + 1 = = 1 = = = 7 = 8 = = 8 There were cookies in ech tch. There were 8 students in ech clss. 10 = 7 0 = F

25 Using equtions think of numer Lim thinks of numer, dds to it nd then multiplies it y. The nswer is 0. Wht is Lim s numer? To nswer this, first we need to write n eqution with the unknown: Step 1 Set up the eqution. The hert shpe stnds for the unknown numer. + = 0 Step Cncel out the with the inverse opertion: + = 0 Step Cncel out the + with the inverse opertion: + = 5 = 5 = 1 Work out the numers these children re thinking of: Jmil sys: I m thinking of numer. I divide it y 7 nd then dd. My nswer is = 1 7 = 1 Plo sys: I m thinking of numer. I multiply it y nd then dd 7. My nswer is = 55 = = 7 = 8 = 7 7 = 8 = 9 = 8 c Mikel sys: I m thinking of numer. I multiply it y then sutrct 1. My nswer is 0. 1 = 0 = = = = 8 d Linh sys: I m thinking of numer. I divide it y 8 nd then dd 11. My nswer is = 19 8 = = 8 = 8 8 = F 5

26 Using equtions think of numer Follow the steps for different numers. Answers will vry. Think of numer 8 Doule it 1 1 Add Divide y 8 Sutrct 0 8 Wht hppens ech time? You end up with the sme numer you thought of. Follow the steps for different numers. Answers will vry. Think of numer 7 Add Doule it 1 0 Sutrct Hlve it 7 Wht hppens ech time? You end up with the sme numer you thought of. F

27 Numer tricks 1 solve Wht to do Try this numer puzzle y testing it out in the lnk oxes. Answers will vry. Think of numer ä Add ä + + Add first numer ä + ä Divide y ä Sutrct 1 ä Wht do you notice? You end up with the sme numer you thought of. Wht to do next This numer puzzle uses the sme trick. This time complete the column of oxes with the numer sentences using symols. Then test it in the lst column. Answers will vry. Think of numer ä Doule it ä + ä 8 Add ä + ä + 1 Divide y ä + 7 Sutrct ä Why does this work for ny numer? The opertions in the lst steps reverse the opertions in the first steps, which mens you lwys end up with the numer you strted with. F 7

28 Numer tricks solve Wht to do Write the symols for this puzzle in column nd test it out. Wht numer is left? Think of numer ä Add ä + + Doule it ä + ä Tke wy ä + ä + 10 Add the first numer ä + ä + ä + 1 Add ä + ä + ä Divide y ä + Sutrct the first numer You would e left with Answers will vry. 8 F