# Chapter. Contents: A Constructing decimal numbers

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting frtions to deimls

2 166 DECIMALS (Chpter 9) OPENING PROBLEM Arrnge these mounts of money in sending order: \$56:65, \$56:05, \$50:65, \$55:50, \$56:50 Whih is lrgest? DECIMAL NUMBERS ARE EVERYWHERE ALL THESE ITEMS AT ONE LOW PRICE.99 9 e Retil gints rep \$37.4n A CONSTRUCTING DECIMAL NUMBERS We hve seen previously how whole numers re onstruted y pling digits in different ple vlues. For exmple, 384 hs ple vlue tle sine 384= hundreds tens units Numers like 0:37 nd 4:569 re lled deiml numers. We use them to represent numers etween the whole numers. The deiml point or dot seprtes the whole numer prt to the left of the dot, from the frtionl prt to the right of the dot.

3 DECIMALS (Chpter 9) 167 If the whole numer prt is zero, we write zero in front of the deiml point. So, we write 0:37 insted of : :37 is short wy of writing , nd 4:569 is relly So, the ple vlue tle for 0: 37 nd 4: 569 is: Deiml numer units tenths hundredths thousndths Expnded form 0:37 0 : :569 4 : Exmple 1 Express in written or orl form: 0:9 3:06 11:407 0:9 is zero point nine. 3:06 is three point zero six. 11:407 is eleven point four zero seven. Orl form mens how you would sy it. Exmple 2 Write in ple vlue tle: 7 hundredths tens units tenths hundredths thousndths Numer Written Numerl 7 hundredths : 0 7 0: : :409 EXERCISE 9A 1 Express the following in written or orl form: 0:6 0:45 0:908 d 8:3 e 6:08 f 96:02 g 5:864 h 34:003 i 7:581 j 60:264 2 Convert into deiml form: eight point three seven twenty one point zero five nine point zero zero four d thirty eight point two zero six

4 168 DECIMALS (Chpter 9) 3 Write in ple vlue tle nd then s deiml numer: d + 6 e f Numer thousnds hundreds tens units 4 Write in ple vlue tle nd then s deiml numer: 8 tenths 3 thousndths 7 tens nd 8 tenths d 9 thousnds nd 2 thousndths e 2 hundreds, 9 units nd 4 hundredths f 8 thousnds, 4 tenths nd 2 thousndths g 5 thousnds, 20 units nd 3 tenths h 9 hundreds, 8 tens nd 34 thousndths i 6 tens, 8 tenths nd 9 hundredths j 36 units nd 42 hundredths : tenths hundredths thousndths PRINTABLE WORKSHEET Written Numerl If word for digit ends in ths then the numer follows the deiml point. Exmple 3 Express 5:706 in expnded form: 5:706 = = Express the following in expnded form: 5:4 14:9 2:03 d 32:86 e 2:264 f 1:308 g 3:002 h 0:952 i 4:024 j 2:973 k 20:816 l 7:777 m 9:008 n 154:451 o 808:808 p 0:064 6 Write the following in deiml form: d e 7 f g h i j k l m n o

5 Exmple 4 Stte the vlue of the digit 6 in the following: 0: :3964 So, the 6 stnds for. DECIMALS (Chpter 9) s 1 s 1 s 7 Stte the vlue of the digit 3 in the following: 4325:9 6:374 32:098 d 150:953 e 43:4444 f 82:7384 g 24:8403 h 3874:941 8 Stte the vlue of the digit 5 in the following: 18: :08 4:5972 d 94:8573 e :264 f 275:183 g :843 h 0:0005 Exmple 5 Write 39 in deiml form. 39 = = =0:03 + 0:009 =0:039 9 Write in deiml form: d 117 e 469 f 703 g 600 h 540 i j Convert the following to deiml form: d e f seventeen nd four hundred nd sixty five thousndths twelve nd ninety six thousndths three nd six hundred nd ninety four thousndths four nd twenty two hundredths 9 hundreds, 8 tens nd 34 thousndths 36 units nd 42 hundredths 11 Stte the vlue of the digit 2 in the following: d e f g h 5 652

6 170 DECIMALS (Chpter 9) B REPRESENTING DECIMAL NUMBERS DECIMAL GRIDS Deimls n lso e represented on 2-dimensionl grids. Suppose this grid represents one whole unit. This shded prt is or of the whole unit. This is 0:4 or 0:40. This shded prt is 27 or 0:27 of the whole unit. Exmple 6 Wht deiml numer is represented y: 0 : 3 6 units tenths hundredths MULTI ATTRIBUTE BLOCKS Multi Attriute Bloks or MABs re prtil 3-dimensionl wy to represent deimls. represents unit or whole mount. represents tenth or 0:1 of the whole. represents hundredth or 0:01 of the whole. nd represents thousndth of 0:001 of the whole. The smller the deiml numer, the more zeros there re fter the deiml point.

7 DECIMALS (Chpter 9) 171 Exmple 7 Write the deiml vlue represented y the MABs if the lrgest lok represents one unit. 1 : units tenths hundredths thousnds 2 : units tenths hundredths thousnds EXERCISE 9B 1 Write the deiml tht represents the shded re: d There re no hundredths shown. We must write tht with zero, 0. 2 Write the deiml vlue represented y the MABs if the lrgest lok represents one unit:

8 172 DECIMALS (Chpter 9) d e C DECIMAL CURRENCY Deiml urreny is one of the most prtil wys to ring mening to deimls. When tlking out nd using money we re lso using deiml numers. The deiml point seprtes whole numers from the frtions. For exmple: E27:35 is E27 plus 35 of one E. Suppose ountry hs the following oins nd nknotes: \$1 \$2 \$5 \$10 \$20 \$50 \$ The urreny is lled deiml euse it uses the se 10 system. 1 1 is or 0:01 of \$1 2 is 2 or 0:02 of \$1 5 5 is or 0:05 of \$1 10 is 10 or 0:10 of \$1 20 is 20 or 0:20 of \$1 50 is 50 or 0:50 of \$1

9 DECIMALS (Chpter 9) 173 Exmple 8 How muh money is shown? \$20 \$ \$2 5 2 We hve =27 whole dollrs nd = 77 ents So, we hve \$27:77 ltogether: Exmple 9 Using one euro (E) s the unit, hnge to deiml vlue: seven euros, 45 euro ents 275 euro ents E7: euro ents = 200 euro ents + 75 euro ents = E2 +E0:75 = E2:75 EXERCISE 9C 1 Chnge these urreny vlues to deimls of one dollr: \$5 5 \$20 50 \$2 20 \$2 \$2 \$50 \$1 d \$ 5 \$10 10 \$ \$5 e \$10 50 f \$ 50 \$2 20 \$

10 174 DECIMALS (Chpter 9) 2 Write eh mount s dollrs using deiml point: 4 dollrs 47 ents 15 dollrs 97 ents seven dollrs fifty five ents d 36 dollrs e 150 dollrs f thirty two dollrs eighty ents g 85 dollrs 5 ents h 30 dollrs 3 ents 3 Chnge these mounts to deimls using the euro s the unit: i 35 ents ii 5 ents iii 405 ents iv 3000 ents v 487 ents vi 295 ents vii 3875 ents viii ents D Strting with the top row, wht is the sum of eh row ove in euros? Wht is the sum of eh olumn ove in euros? USING A NUMBER LINE Just s whole numers n e mrked on numer line, we n do the sme with deiml numers. Consider the following numer line where eh whole numer shown hs ten equl divisions. Eh division on this numer line represents 1 10 or 0:1 Exmple 10 Mke sure the mounts hve their deiml point extly elow the other Find the deiml vlues of A, B, C nd D mrked on the numer line shown A 1 B C D Eh division on the numer line represents 0:1 So, A is 0:7, B is 1:3, Cis 2:1 nd D is 3:2 We n divide our numer line into smller prts thn tenths. 1 Suppose we divide eh of the prts whih represent 10 into 10 equl prts. Eh unit is now 1 divided into equl prts nd eh division is or 0:01 of the unit

11 DECIMALS (Chpter 9) 175 Exmple 11 Find the deiml vlues of A, B, C nd D mrked on the numer line shown. 2.4 A 2.5 B C 2.6 D Eh division on the numer line represents 0:01. So, A is 2:43, B is 2:51, C is 2:57 nd D is 2:62 EXERCISE 9D 1 Write down the vlue of the numer t N on the following numer lines. 0 N 1 2 N 3 6 N 7 d 21 N 22 e 11 N 12 f 8 N 9 2 Copy the numer lines given nd mrk the following numers on them. A =1:6, B =2:5, C =2:9, D =4: E =13:7, F =14:2, G =15:3, H =16: Red the temperture on the thermometer shown C 4 Red the length of Christin s skirt from the tpe mesure. 5 Wht weight is shown on the sles? How muh milk is in the jug? kg 1 litre

12 176 DECIMALS (Chpter 9) 6 Write down the vlue of the numer t N on the following numer lines. N N N d N e N f N Copy the numer lines given nd mrk the following numers on them. A =4:61, B =4:78, C =4:83, D =4: E =10:35, F =10:46, G =10:62, H =10: E ORDERING DECIMALS We n use numer line to help ompre the sizes of deiml numers. For exmple, onsider the following numer line: As we go from left to right, the numers re inresing. So, 1:08 < 1:25 < 1:7 < 1:89 To ompre deiml numers without hving to onstrut numer line, we ple zeros on the end so eh numer hs the sme numer of deiml ples. We n do this euse dding zeros on the end does not ffet the ple vlues of the other digits. Exmple 12 Put the orret sign >, < or =, in the ox to mke the sttement true: 0:305 0:35 0:88 0:808 We strt y writing the numers with the sme numer of deiml ples. 0:305 0:350 So, 0:305 < 0:350 0:880 0:808 So, 0:880 > 0:808

13 DECIMALS (Chpter 9) 177 EXERCISE 9E 1 Write down the vlues of the numers A nd B on the following numer line, nd determine whether A > BorA< B: B 6 7 A B A A B d A B e f A B B A Insert the orret sign >, < or = to mke the sttement true: 0:7 0:8 0:06 0:05 0:2 0:19 d 4:01 4:1 e 0:81 0:803 f 2:5 2:50 g 0:304 0:34 h 0:03 0:2 i 6:05 60:50 j 0:29 0:290 k 5:01 5:016 l 1:15 1:035 m 21:021 21:210 n 8:09 8:090 o 0:904 0:94 3 Arrnge in sending order (lowest to highest): 0:8, 0:4, 0:6 0:4, 0:1, 0:9 0:14, 0:09, 0:06 d 0:46, 0:5, 0:51 e 1:06, 1:59, 1:61 f 2:6, 2:06, 0:206 g 0:095, 0:905, 0:0905 h 15:5, 15:05, 15:55 4 Arrnge in desending order (highest to lowest): 0:9, 0:4, 0:3, 0:8 0:51, 0:49, 0:5, 0:47 0:6, 0:596, 0:61, 0:609 d 0:02, 0:04, 0:42, 0:24 e 6:27, 6:271, 6:027, 6:277 f 0:31, 0:031, 0:301, 0:311 g 8:088, 8:008, 8:080, 8:880 h 7:61, 7:061, 7:01, 7:06 5 Continue the numer ptterns y writing the next three terms: 0:1, 0:2, 0:3,... 0:9, 0:8, 0:7,... 0:2, 0:4, 0:6,... d 0:05, 0:07, 0:09,... e 0:7, 0:65, 0:6,... f 2:17, 2:13, 2:09,... g 7:2, 6:4, 5:6,... h 0:25, 0:50, 0:75,... i 1:111, 1:123, 1:135,... j 0, 0:125, 0:250,...

14 178 DECIMALS (Chpter 9) F We re often given mesurements s deiml numers. For exmple, my throom sles tell me I weigh 59:4 kg. In relity I do not weigh extly 59:4 kg, ut this is n pproximtion of my tul weight. Mesuring my weight to greter ury is not importnt. We round off deiml numers in the sme wy we do whole numers. We look t vlues on the numer line either side of our numer, nd work out whih is loser. For exmple, onsider 1: :23 is loser to 1:2 thn it is to 1:3, so we round down. 1:23 is pproximtely 1:2. ROUNDING DECIMAL NUMBERS Consider 5716 ¼ 5720 (to the nerest 10) ¼ 5700 (to the nerest ) ¼ 6000 (to the nerest ) Likewise, 0:5716 ¼ 0:572 ¼ 0:57 ¼ 0:6 (to 3 deiml ples) (to 2 deiml ples) (to 1 deiml ple) RULES FOR ROUNDING OFF DECIMAL NUMBERS ² If the digit fter the one eing rounded is less thn 5, i.e., 0, 1, 2, 3 or 4, then we round down. ² If the digit fter the one eing rounded is 5 or more, i.e., 5, 6, 7, 8 or 9, then we round up. Exmple 13 Round: 3:26 to 1 deiml ple 5:273 to 2 deiml ples 4:985 to 2 deiml ples 3:26 is loser to 3:3 thn to 3:2, so we round up. So, 3:26 ¼ 3:3. 5:273 is loser to 5:27 thn to 5:28, so we round down. So, 5:273 ¼ 5:27. 4:985 lies hlfwy etween 4:98 nd 4:99, so we round up. So, 4:985 ¼ 4:99 : EXERCISE 9F 1 Write these numers orret to 1 deiml ple: 2:43 3:57 4:92 d 6:38 e 4:275 2 Write these numers orret to 2 deiml ples: 4:236 2:731 5:625 d 4:377 e 6:5237

15 DECIMALS (Chpter 9) Write 0:486 orret to: 1 deiml ple 2 deiml ples. 4 Write 3:789 orret to: 1 deiml ple 2 deiml ples. 5 Write 0: orret to: 1 deiml ple 2 deiml ples 3 deiml ples d 4 deiml ples. 6 Find deiml pproximtions for: 3:87 to the nerest tenth 4:3 to the nerest integer 6:09 to one deiml ple d 0:4617 to 3 deiml ples e 2:946 to 2 deiml ples f 0: to 4 deiml ples. G CONVERTING DECIMALS TO FRACTIONS Deiml numers n e esily written s frtions with powers of 10 s their denomintors. Exmple 14 Write s frtion or s mixed numer: 0:7 0:79 2:013 0:7 = :79 = 79 2:013 =2+ 13 =2 13 We hve seen previously how some frtions n e onverted to simplest form or lowest terms y dividing oth the numertor nd denomintor y their highest ommon ftor. Exmple 15 Write s frtion in simplest form: 0:4 0:72 0:275 0:4 0:72 0:275 = 4 10 = = 2 5 = 72 = = = 275 = = 11 40

16 180 DECIMALS (Chpter 9) EXERCISE 9G 1 Write the following s frtions in simplest form: 0:1 0:7 1:5 d 2:2 e 3:9 f 4:6 g 0:19 h 1:25 i 0:18 j 0:65 k 0:05 l 0:07 m 2:75 n 1:025 o 0:04 p 2:375 2 Write the following s frtions in simplest form: 0:8 0:88 0:888 d 3:5 e 0:49 f 0:25 g 5:06 h 3:32 i 0:085 j 3:72 k 1:096 l 4:56 m 0:064 n 0:625 o 0:115 p 2:22 Exmple 16 Write s frtion: 0:45 kg 3:40 m 0:45 kg 3:40 m = 45 kg = kg = 9 20 kg =3m m =3m m =3 2 5 m 3 Write these mounts s frtions or mixed numers in simplest form: 0:20 kg 0:25 hours 0:85 kg d 1:50 km e 1:75 g f 2:74 m g 4:88 tonnes h 6:28 L i E1:25 j E1:76 k E3:65 l E4:21 m E8:40 n E5:125 o \$3:08 p \$4:11 q \$18:88 r \$52:25 H CONVERTING FRACTIONS TO DECIMALS We hve lredy seen tht it is esy to onvert frtions with denomintors 10,,, nd so on into deiml numers. Sometimes we n mke the denomintor power of 10 y multiplying the numertor nd denomintor y the sme numers. 3 For exmple, 5 = = 6 10 =0: = = 28 =0:28 We need to multiply the numertor nd denomintor y the sme mount so we do not hnge the vlue of the frtion.

17 DECIMALS (Chpter 9) 181 Exmple 17 Convert to deiml numers: = = 75 =0: = = 35 =0: = = 184 =0:184 EXERCISE 9H 1 By wht whole numer would you multiply the following, to otin power of 10? d 8 e 20 f 25 g 50 h 125 i 40 j 250 k 500 l Convert to deiml numers: d e f g h i j k 1 4 l m n o p q 3 8 r Copy nd omplete these onversions to deimls: 1 2 = :::::: 1 5 = ::::::, 2 5 = ::::::, 3 5 = ::::::, 4 5 = ::::::, 1 4 = ::::::, 2 4 = ::::::, 3 4 = :::::: d 1 8 = ::::::, 2 8 = ::::::, 3 8 = ::::::, 4 8 = ::::::, 5 8 = ::::::, 6 8 = ::::::, 7 8 = ::::::: You should rememer the deiml vlues of these frtions. KEY WORDS USED IN THIS CHAPTER ² deiml ² deiml urreny ² deiml point ² frtion ² highest ommon ftor ² hundredth ² mixed numers ² ple vlue ² round off ² simplest form ² tenth ² thousndth

18 182 DECIMALS (Chpter 9) REVIEW SET 9A 1 If the dollr represents the unit, wht re the deiml vlues of the following? \$10 \$5 50 \$ \$ \$ \$ 20 \$ If eh grid represents one unit, wht deimls re represented y the following grids? 3 If represents one thousndth, write the deiml numers for: d 4 Write s deiml numer Write: 25 euros nd 35 euro ents s deiml numer \$107 nd 85 ents s deiml numer five nd twenty nine thousndths in deiml form d e f 436 : in two different frtionl forms 2:049 in expnded form the mening of the 2 digit in 51: Write down the vlue of the numer t B on these numer lines: B B

19 DECIMALS (Chpter 9) Whih numer is greter, 3:2 or 3:1978? 8 Round 3:8551 to: one deiml ple two deiml ples. 9 Convert to frtion in simplest form: 0:8 0:75 0:375 d 0:68 10 Convert to deiml numer: d Write in sending order: 0:216, 0:621, 0:062, 0:206, 0: Continue this numer pttern for 3 more terms: 0:81, 0:78, 0:75, 0:72,... REVIEW SET 9B 1 If represents one thousndth, write the deiml vlues of: 2 If eh grid represents one unit, wht deiml numers re represented y: 3 Write s deiml numer: d Convert sixteen point five seven four to deiml form. Write 0:921 in two different non-deiml forms. Stte the vlue of the digit 3 in 41:039. d Write \$12 nd 35 pene s deiml. 5 Write down the vlue of the numer t A on the following numer lines: A A Write the following deiml numers in desending order: 0:444, 4:04, 4:44, 4:044, 4:404 7 Continue the numer pttern y writing the next three terms: 0:3, 0:7, 1:1,...

20 184 DECIMALS (Chpter 9) 8 Whih numer is smller, 2:3275 or 2:3199? 9 Round 3:995 to: 1 deiml ple 2 deiml ples 10 Convert to frtion in simplest form: 0:62 0:45 0:875 d 10:4 11 Convert to deiml numer: d Copy nd omplete: 1 8 =0:125, 2 8 = ::::::, 3 8 = ::::::, 4 8 = ::::::, 5 8 = ::::::. ACTIVITY Wht to do: TANGRAMS 1 On piee of rd mrk out 20 m y 20 m squre. Then opy the following lines onto it nd ut long eh line. You should hve seven different piees. PRINTABLE TEMPLATE 2 Eh of the following shpes n e mde using ll seven piees of your tngrm. See how mny you n omplete. ridge puppy person running d t

### D e c i m a l s DECIMALS.

D e i m l s DECIMALS www.mthletis.om.u Deimls DECIMALS A deiml numer is sed on ple vlue. 214.84 hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed

### End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.

End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.

### Know the sum of angles at a point, on a straight line and in a triangle

2.1 ngle sums Know the sum of ngles t point, on stright line n in tringle Key wors ngle egree ngle sum n ngle is mesure of turn. ngles re usully mesure in egrees, or for short. ngles tht meet t point mke

GRADE Frtions WORKSHEETS Types of frtions equivlent frtions This frtion wll shows frtions tht re equivlent. Equivlent frtions re frtions tht re the sme mount. How mny equivlent frtions n you fin? Lel eh

### Chapter. Fractions. Contents: A Representing fractions

Chpter Frtions Contents: A Representing rtions B Frtions o regulr shpes C Equl rtions D Simpliying rtions E Frtions o quntities F Compring rtion sizes G Improper rtions nd mixed numers 08 FRACTIONS (Chpter

### Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }

ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All

### State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127

ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not

### THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

### 11. PYTHAGORAS THEOREM

11. PYTHAGORAS THEOREM 11-1 Along the Nile 2 11-2 Proofs of Pythgors theorem 3 11-3 Finding sides nd ngles 5 11-4 Semiirles 7 11-5 Surds 8 11-6 Chlking hndll ourt 9 11-7 Pythgors prolems 10 11-8 Designing

### Proving the Pythagorean Theorem

Proving the Pythgoren Theorem Proposition 47 of Book I of Eulid s Elements is the most fmous of ll Eulid s propositions. Disovered long efore Eulid, the Pythgoren Theorem is known y every high shool geometry

### Maths Assessment Year 4: Number and Place Value

Nme: Mths Assessment Yer 4: Numer nd Plce Vlue 1. Count in multiples of 6, 7, 9, 25 nd 1 000; find 1 000 more or less thn given numer. 2. Find 1,000 more or less thn given numer. 3. Count ckwrds through

### The area of the larger square is: IF it s a right triangle, THEN + =

8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht

### Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

### The AVL Tree Rotations Tutorial

The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr-22-2007 Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk

### The remaining two sides of the right triangle are called the legs of the right triangle.

10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

### Fractions, Decimals and Percentages

G Teher Student Book SERIES Frtions, Deimls nd Perentges Nme Contents Series G Frtions, Deimls nd Perentges Topi Setion Frtions Answers (pp. (pp. ) ) equivlent frtions frtions _ mixed deiml numerls frtions

### Simple Electric Circuits

Simple Eletri Ciruits Gol: To uild nd oserve the opertion of simple eletri iruits nd to lern mesurement methods for eletri urrent nd voltge using mmeters nd voltmeters. L Preprtion Eletri hrges move through

### Chess and Mathematics

Chess nd Mthemtis in UK Seondry Shools Dr Neill Cooper Hed of Further Mthemtis t Wilson s Shool Mnger of Shool Chess for the English Chess Federtion Mths in UK Shools KS (up to 7 yers) Numers: 5 + 7; x

### Words Symbols Diagram. abcde. a + b + c + d + e

Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

### The Pythagorean Theorem Tile Set

The Pythgoren Theorem Tile Set Guide & Ativities Creted y Drin Beigie Didx Edution 395 Min Street Rowley, MA 01969 www.didx.om DIDAX 201 #211503 1. Introdution The Pythgoren Theorem sttes tht in right

### PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed

### Three squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.

1- The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor Self-Chek Quiz Three squres with

### Binary Representation of Numbers Autar Kaw

Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

### Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example

2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel

### Chapter 6 Solving equations

Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

### Chapter 2 Decimals. (A reminder) In the whole number chapter, we looked at ones, tens, hundreds, thousands and larger numbers. = 1

Chpter 2 Decimls Wht is Deciml? (A reminder) In the whole numer chpter, we looked t ones, tens, hundreds, thousnds nd lrger numers. When single unit is divided into 10 (or 100) its, we hve deciml frctions

### Right Triangle Trigonometry 8.7

304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R1 8.7 Right Tringle Trigonometry R1 8.7 Right Tringle Trigonometry T E G T I N G S T R T E D The origins of trigonometry, from the Greek trigonon (ngle) nd

### Quick Guide to Lisp Implementation

isp Implementtion Hndout Pge 1 o 10 Quik Guide to isp Implementtion Representtion o si dt strutures isp dt strutures re lled S-epressions. The representtion o n S-epression n e roken into two piees, the

### Simple Nonlinear Graphs

Simple Nonliner Grphs Curriulum Re www.mthletis.om Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle Liner euse their grphs re stright

### Square Roots Teacher Notes

Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

### 4.5 The Converse of the

Pge 1 of. The onverse of the Pythgoren Theorem Gol Use the onverse of Pythgoren Theorem. Use side lengths to lssify tringles. Key Words onverse p. 13 grdener n use the onverse of the Pythgoren Theorem

### PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

### Problem Set 2 Solutions

University of Cliforni, Berkeley Spring 2012 EE 42/100 Prof. A. Niknej Prolem Set 2 Solutions Plese note tht these re merely suggeste solutions. Mny of these prolems n e pprohe in ifferent wys. 1. In prolems

### MATH PLACEMENT REVIEW GUIDE

MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your

### Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

### Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

### excenters and excircles

21 onurrene IIi 2 lesson 21 exenters nd exirles In the first lesson on onurrene, we sw tht the isetors of the interior ngles of tringle onur t the inenter. If you did the exerise in the lst lesson deling

### Fractions to decimals

Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of

### 1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.

. Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry

Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

### 1 Fractions from an advanced point of view

1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning

### Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

### Ratio and Proportion

Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty

### Unit 6: Exponents and Radicals

Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

### Angles and Triangles

nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir

### Boğaziçi University Department of Economics Spring 2016 EC 102 PRINCIPLES of MACROECONOMICS Problem Set 6 Answer Key

Boğziçi University Deprtment of Eonomis Spring 2016 EC 102 PRINCIPLES of MACROECONOMICS Prolem Set 6 Answer Key 1. Any item tht people n use to trnsfer purhsing power from the present to the future is

### In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.

Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix

### P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

### WHAT HAPPENS WHEN YOU MIX COMPLEX NUMBERS WITH PRIME NUMBERS?

WHAT HAPPES WHE YOU MIX COMPLEX UMBERS WITH PRIME UMBERS? There s n ol syng, you n t pples n ornges. Mthemtns hte n t; they love to throw pples n ornges nto foo proessor n see wht hppens. Sometmes they

### Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

### Accuplacer Arithmetic Study Guide

Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how

### 8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

8-1 The Pythgoren Theorem nd Its Converse Voulry Review 1. Write the squre nd the positive squre root of eh numer. Numer Squre Positive Squre Root 9 81 3 1 4 1 16 1 2 Voulry Builder leg (noun) leg Relted

### Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

Alger Module A60 Qudrtic Equtions - 1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions - 1 Sttement of Prerequisite

### Reasoning to Solve Equations and Inequalities

Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

### Rational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and

Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions

### 4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A

Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter

### Lesson 18.2: Right Triangle Trigonometry

Lesson 8.: Right Tringle Trigonometry lthough Trigonometry is used to solve mny prolems, historilly it ws first pplied to prolems tht involve right tringle. This n e extended to non-right tringles (hpter

### NQF Level: 2 US No: 7480

NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................

### Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn

Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?

### Or more simply put, when adding or subtracting quantities, their uncertainties add.

Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

### 1. Area under a curve region bounded by the given function, vertical lines and the x axis.

Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given

### Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

### Place Value. Key Skills. Complete the daily exercises to focus on improving this skill.

Day 1 1 Write Four Million, Four Hundred and Twelve Thousand, Eight Hundred and Seventy One in digits 2 Write 6330148 in words 3 Write 12630 in words 4 Write 71643 in words 5 Write 28009 in words 6 Write

### . At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

### 2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

### c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00

Chter 19, exmle rolems: (19.06) A gs undergoes two roesses. First: onstnt volume @ 0.200 m 3, isohori. Pressure inreses from 2.00 10 5 P to 5.00 10 5 P. Seond: Constnt ressure @ 5.00 10 5 P, isori. olume

### Numbers Galore using. Print. font

Numbers Galore! Numbers Galore using Print font 0 zero Draw Tally 0 0 zero 1 one Draw Tally 1 1 1 1 1 one one 2 two Draw Tally 2 2 2 2 two two 3 three Draw Tally 3 3 3 3 three 4 four Draw Tally 4 4 4 4

### CONIC SECTIONS. Chapter 11

CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we

### Vectors 2. 1. Recap of vectors

Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

### Pythagoras theorem is one of the most popular theorems. Paper Folding And The Theorem of Pythagoras. Visual Connect in Teaching.

in the lssroom Visul Connet in Tehing Pper Folding And The Theorem of Pythgors Cn unfolding pper ot revel proof of Pythgors theorem? Does mking squre within squre e nything more thn n exerise in geometry

### not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions

POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the x-coordintes of the points where the grph of y = p(x) intersects the x-xis.

### Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

### CHAPTER 4: POLYGONS AND SOLIDS. 3 Which of the following are regular polygons? 4 Draw a pentagon with equal sides but with unequal angles.

Mthemtis for Austrli Yer 6 - Homework POLYGONS AND SOLIDS (Chpter 4) CHAPTER 4: POLYGONS AND SOLIDS 4A POLYGONS 3 Whih of the following re regulr polygons? A polygon is lose figure whih hs only stright

### The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center

Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,

### Quadratic Equations. Math 99 N1 Chapter 8

Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree

### In the following there are presented four different kinds of simulation games for a given Büchi automaton A = :

Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile

### Digital Electronics Basics: Combinational Logic

Digitl Eletronis Bsis: for Bsi Eletronis http://ktse.eie.polyu.edu.hk/eie29 by Prof. Mihel Tse Jnury 25 Digitl versus nlog So fr, our disussion bout eletronis hs been predominntly nlog, whih is onerned

### a 2 + b 2 = c 2. There are many proofs of this theorem. An elegant one only requires that we know that the area of a square of side L is L 2

Pythgors Pythgors A right tringle, suh s shown in the figure elow, hs one 90 ngle. The long side of length is the hypotenuse. The short leg (or thetus) hs length, nd the long leg hs length. The theorem

### SPECIAL PRODUCTS AND FACTORIZATION

MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

### Double Integrals over General Regions

Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

### Curve Sketching. 96 Chapter 5 Curve Sketching

96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

### 8.2 Trigonometric Ratios

8.2 Trigonometri Rtios Ojetives: G.SRT.6: Understnd tht y similrity, side rtios in right tringles re properties of the ngles in the tringle, leding to definitions of trigonometri rtios for ute ngles. For

### Example

6. SOLVING RIGHT TRINGLES In the right tringle B shwn in Figure 6.1, the ngles re dented y α t vertex, β t vertex B, nd t vertex. The lengths f the sides ppsite the ngles α, β, nd re dented y,, nd. Nte

### Right-angled triangles

13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Right-ngled tringles Syllus referene Mesurement 4 Right-ngled tringles

### Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle

: Using Trigonometry to Find Side Lengths of n Aute Tringle Clsswork Opening Exerise. Find the lengths of d nd e.. Find the lengths of x nd y. How is this different from prt ()? Exmple 1 A surveyor needs

### Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

### 50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS

0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS BASIC KNOWLEDGE () RATIOS: Rtios re use to ompre two or more numers For n two numers n ( 0), the rtio is written s : = / Emple : If 4 stuents in

### The theorem of. Pythagoras. Opening problem

The theorem of 8 Pythgors ontents: Pythgors theorem [4.6] The onverse of Pythgors theorem [4.6] Prolem solving [4.6] D irle prolems [4.6, 4.7] E Three-dimensionl prolems [4.6] Opening prolem The Louvre

### Algebra Review. How well do you remember your algebra?

Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

### Seeking Equilibrium: Demand and Supply

SECTION 1 Seeking Equilirium: Demnd nd Supply OBJECTIVES KEY TERMS TAKING NOTES In Setion 1, you will explore mrket equilirium nd see how it is rehed explin how demnd nd supply intert to determine equilirium

### Whole Number and Decimal Place Values

Whole Number and Decimal Place Values We will begin our review of place values with a look at whole numbers. When writing large numbers it is common practice to separate them into groups of three using

### Lesson 2.1 Inductive Reasoning

Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,

### Graphs on Logarithmic and Semilogarithmic Paper

0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

### Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

### ISTM206: Lecture 3 Class Notes

IST06: Leture 3 Clss otes ikhil Bo nd John Frik 9-9-05 Simple ethod. Outline Liner Progrmming so fr Stndrd Form Equlity Constrints Solutions, Etreme Points, nd Bses The Representtion Theorem Proof of the

### EduSahara Learning Center Assignment

06/05/2014 11:23 PM http://www.edusahara.com 1 of 5 EduSahara Learning Center Assignment Grade : Class VI, CBSE Chapter : Knowing Our Numbers Name : Words and Figures Licensed To : Teachers and Students