maths revision.docx

8/10/2019 maths revision.docx

1/18

DIVISION is repeated subtraction :

20 4 = ??

20 4 4 4 4 4 = 0 .I subtracted 4 five times,so 20 4 = .

!4 2" = ??!4

2" "#$

2" "42

2" "2"

2" "0 4

I subtracted 2" four times, so !4 2" = 4

Ofte%, it is &a%dier to actua''( add i%stead of subtract:

Si%ce "$ ) "$ = 2#,"$ *oes to 2# t+o times.

So 2# "$ = 2

Si%ce 2" ) 2" ) 2" ) 2" = !4,2" *oes to !4 four times.

So !4 2" = 4

Example problems

". rite a mu'ti-'icatio% se%te%ce ND a divisio% se%te%ce t&at fits t&eadditio%/subtractio% facts.

) ) = "" = 0

=

=

"2 ) "2 ) "2 ) "2 = 4!4! "2 "2 "2 "2 = 0

=

=


2/18

2$ ) 2$ ) 2$ = 2$ 2$ 2$ = 0

=

=

40 ) 40 = 40 40 = 0

=

=

2. rite a subtractio% se%te%ce for eac& divisio% se%te%ce.

4 " =

4

$2 ! =

$2

Multiplication is re-eated additio%, a%d it is 'i1e um-s o% t&e %umber 'i%e.

3 4 = 20. ive um-s of 4 *ets (ou to 20.

Division is re-eated subtractio%. 5ou ma1e um-s of four backwards from 20 ti''(ou &it 0:

20 4 = . 20 4 4 4 4 4 = 0ive um-s of 4 *ets (ou from 20 ti'' 0.


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&at divisio% is i''ustrated &ere?

2. Dra+ um-s bac1+ards to i''ustrate t&e divisio% se%te%ces.

$0 =

2! 4 =

42 # =

4. So've usi%* re-eated subtractio% O6 addi%* u- to t&e %umber bei%* divided.

40 20 = 2 "$ = 4 " =


4/18

70 $0 =

$0 " =

$4 "8 =

#7 2$ =

70 " =

70 "! =

. If "2 3 2 = 24, t&e% "$ 3 2 is 9o+ about divisio%? se t&e -revious -rob'emto &e'- (ou so've t&e %e;t o%e.

a.24 2 =

2# 2 =

2! 2 =

$0 2 =

d.#0 2 =

## 2 =

80 2 =

8! 2 =

#.


5/18

A fraction describes a part of a whole when the whole is cut into intoequal parts!! Fractions can also be parts of a group. For example, ifthere is a group of fruit: 3 oranges and 4 apples, what fraction of the

group are apples? Four se enths. here are se en parts, andfour apples. "hat is the fraction for the oranges?hree se enths. he# are fractions that are not onewhole, it$s %ust a part of the whole.

&ere are some examples :

'ne fourth of the box is #ellow.hree fourths of the box is blue.

Examples of inductive reasoning;am-'es of i%ductive reaso%i%* are %umerous. >ots of I or i%te''i*e%ce tests are based o%i%ductive reaso%i%*. @atter%s a%d i%ductive reaso%i%* are c'ose'( re'ated.

i%d &ere a cou-'e of *ood e;am-'es of i%ductive reaso%i%* t&at +i'' rea''( &e'- (ou u%dersta%di%ductive reaso%i%*

Aut +&at is i%ductive reaso%i%*?

I%ductive reaso%i%* is ma1i%* co%c'usio%s based o% -atter%s (ou observe.


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If (ou &ave carefu''( observed t&e -atter%, ma( be (ou came u- +it& t&e fi*ure be'o+:

Example #2:

>oo1 at t&e -atter%s be'o+. Ba% (ou dra+ t&e %e;t fi*ure or %e;t set of dots usi%* i%ductivereaso%i%*?


7/18

4 3 C2 C2 ) C2 ) C2 ) C2 C !

$ 3 C8 C8 ) C8 ) C8 C 2"

3 C# C# ) C# ) C# ) C# ) C# C $0

&at do (ou %otice about t&e si*%s of t&e sums?

Si%ce t&e sum is a'+a(s %e*ative, t&e -atter% su**ests t&at t&e -roduct of a -ositive i%te*er a%da %e*ative i%te*er is %e*ative

Example #4:

>oo1 at t&e fo''o+i%* -atter%s:

$ 3 C4 = C"2

2 3 C4 = C!

" 3 C4 = C4

0 3 C4 = 0

C" 3 C4 = 4

C2 3 C4 = !

C$ 3 C4 = "2

ver( time t&e factor o% t&e 'eft is decreased b( ", t&e a%s+er is i%creased b( 4

9o+ever, t&e -atter% su**ests t&at a %e*ative times a %e*ative is a -ositive

I &o-e t&ese e;am-'es of i%ductive reaso%i%* +ere %ot com-'icated.


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Inductive Reasoning

Inductive reasoning is used when you need to draw a general conclusion from

specific instances.

For example, when a detective puts together specific clues to solve a mystery.

Looking for a Pattern

(Sequences)

In math, an example of inductive reasoning would be when you are given a patternand you need to come up with the rule for the pattern.

A lot of what we will be working with in this lesson are sequences. In general,a sequence is an ordered arrangement of numbers, figures, or objects.

pecifically, sequences of math are a string of numbers that are tied together withsome sort of consistent rule, or set of rules, that determines the next number in thesequence.

he following are some specific types of sequences of math!

"rithmetic sequence! a sequence such that each successive term is obtained fromthe previous term by addition or subtraction of a fixed number called a difference.!he sequence ", #, $%, $&, $', ... is an example of an arithmetic sequence. !he

pattern is that we are always adding a fixed number of three to the previous term toget to the next term. (e careful that you don)t think that every sequence that has a

pattern in addition is arithmetic. It is arithmetic if you are always adding theS"#$ num%er each time.

&eometric sequence! a sequence such that each successive term is obtained fromthe previous term by multiplying by a fixed number called a ratio. !he sequence *,$%, +%, "%, %, .... is an example of a geometric sequence. !he pattern is that we arealways multiplying by a fixed number of + to the previous term to get to the nextterm. (e careful that you don)t think that every sequence that has a pattern inmultiplication is geometric. It is geometric if you are always multiplying %y theS"#$ num%er each time.

'i%onacci sequence! a basic Fibonacci sequence is when two numbers are addedtogether to get the next number in the sequence. $, $, +, &, *, , $&, .... is anexample of a Fibonacci sequence where the starting numbers -or seeds are $ and $,and we add the two previous num%ers to get the ne t num%er in the sequence.


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ote that not all sequences fit into the specific patterns that are descri%eda%ove. !hose are just the more common ones. So as you look at patterns* look for

those as a possi%ility* %ut if it doesn+t fit one listed a%ove* don+t assume itdoesn+t have a pattern.

In general, when looking for a pattern start simple and then go from there. Forexample, see if there is some pattern in adding, subtracting, multiplying, ordividing. /aybe you are always adding the same number to the previous term toget the new term. 0r maybe you are subtracting the next multiple of three from the

previous number. 0r you are multiplying by a sequence of even numbers. 1erhaps,you are always adding or subtracting the two previous terms to get to the next one.2xponential growth is another good pattern to look for. /aybe you are alwayssquaring or cubing the term number to get your result. Also, don)t forget thatsometimes the pattern of a sequence is a combination of operations. /aybe youhave to multiply by + and then add * to get to the next number in a sequence or theoutput of a function. If a problem seems like it is taking forever to work, try adifferent approach 3 a different kind of sequence.

,nce you find your pattern* you can use it to find the ne t terms in thesequence.

$ ample - 4 5rite the next three numbers in the sequence *, #, $$,$#, +*, ...

/y first inclination is to see if there is some pattern in addition. 5ell, weare not adding the same number each time to get to the next number. (ut,it looks like we have * / , # 0 , $$ 1 , $# 2 , +*, .... I see a pattern in

addition 3 do you see it6 5e are always adding the next even number.

'inal "nswer! he pattern is to add the ne t even num%er. !he next three termswould have to be 34* 05* and 1-* since +* -6 7 &*, &* -/ 7 "#, "# -07 '$.


10/18

$ ample / 4 5rite the next three numbers in the sequence #, 3#, $",3"+, $' , ...

ince we are bouncing back and forth between positive and negativenumbers, a pattern in addition doesn)t look promising. 7et)s check outmultiplication. At first glance, I would say that a negative number is

probably what we are looking for here, since it does alternate signs. Itdoesn)t appear to be the same number each time, because # times 3$ is 3#,

but 3# times 3+ equals $". It looks like we have # (8-)* 3# (8/)* $" (83)* 3"+ (80)* $' , ... Aha, we have a pattern in multiplication 3 we aremultiplying by the next negative integer.

'inal "nswer! he pattern is multiplying %y the ne t negative integer. !he next threeterms are 8206* 4606* and 834/26 , since $' (84) 7 3 "%, 3 "% (81) 7 *%"%,*%"%(85) 7 3&*+ %.

$ ample 3 4 5rite the next three numbers in the sequence $%%, 8#,, '$, ...

ince the numbers are decreasing that should tell you that you are notadding a positive number or multiplying. o we want to check outsubtraction or division. At first glance it looks like it is some pattern insubtraction. 5e are not subtracting by the same number each time. 5ehave $%% 83, 8# 89, 8/5 , '$, .... 9ote how we are always subtractingthe next power of &. 5e have our pattern.

'inal "nswer! he pattern is we are su%tracting %y the ne t power of three. !he next three terms would be 8/6* 8/13* and 899/ , since '$ 8 2- 7 3+%,3+%8 /03 7 3+'&, 3+'& 8 5/9 7 388+.

Looking for a Pattern

Involving 'igures


11/18

:ere are some things to look for when trying to figure out a pattern involvingfigures!

7ook for counter clockwise and clockwise changes.

:ount sides of figures.

:ount lines in figures.

9ote changes in direction and figures.

As with the numeric patterns, this is not all the possible types of patterns involvingfigures. ;owever, it does give you a way to approach the problem.

$ ample 0 4 5rite the next three figures in the pattern

...

It looks like several things change throughout the pattern. 0ne thing isthat it alternates between a square with a line in it and a circle. Also theline in the square alternates from hori


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...

It looks like one row of asterisks is added at the bottom of each figure.!he row that is added contains the next counting number of asterisks.!here are + in the row added in the second term, there are & in the rowadded in the &rd term and " in the row added to the fourth term.

5ith all of that in mind, I believe the ne t two figures would %e


13/18

$ ample 1 4 >se the statements below to answer the question thatfollows4

$. All people wearing hats have blonde hair.+. ome of the people have red hair.&. All people who have blonde hair like hamburgers.". 1eople who have red hair like pi


14/18

=ou can use a process of elimination on this problem.

tatement A, Cerry and !odd eat lunch with the singer, doesnBt let usdefinitively eliminate anyone from being the manager.

;owever, statement (, @evin and /ark carpool with the manager,eliminates @evin and /ark from being the manager. And statement :,!odd watches : I with the manger and the singer, eliminates !odd.

he only one that could %e (-66A* without a dou%t) the manager isCerry.

Practice Pro%lems

!hese are practice problems to help bring you to the next level. It will allow you tocheck and see if you have an understanding of these types of problems. #athworks Dust like anything else* if you want to get good at it* then you need topractice it. $ven the %est athletes and musicians had help along the way andlots of practice* practice* practice* to get good at their sport or instrument. Infact there is no such thing as too much practice.

!o get the most out of these, you should work the pro%lem out on your own andthen check your answer %y clicking on the link for the answerEdiscussion forthat pro%lem . At the link you will find the answer as well as any steps that wentinto finding that answer.

Practice Pro%lems -a 8 -c! =rite the ne t three num%ers in thesequence.

$a. $, $, &, $*, $%*, ...-answerDdiscussion to $a

$b. $%%%, +%%, "%, , $.', ...-answerDdiscussion to $b
http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut35_reason_ans.htm#ad1ahttp://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut35_reason_ans.htm#ad1bhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut35_reason_ans.htm#ad1bhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut35_reason_ans.htm#ad1a


15/18

$c. *, *, $%, $*, +*, ...-answerDdiscussion to $c

Practice Pro%lem /a! =rite the ne t five figures in the pattern.

+a.-answerDdiscussion to +a

Practice Pro%lem 3a! 'our friends 8 Su;y* Cohn* Sally* and om 8

each has his or her own ho%%y. ,ne collect coins* one sews* one cooks* andone plays in a %and* not necessarily in that order.

>se the statements %elow to answer the question that follows.

&a.A. u


16/18

$, $, &, $*, $%*, ...

5hen you see a big jump in numbers all of the sudden 3 we start small with $, $, &,$*, and then all of the sudden we are at $%* 3 a good place to start is multiplication

or exponents. It is not a $%% rule, but it gives you a starting place. It looks likewe have $ (-) , $ (3) , & (4) , $* (5) , $%*,...!here is a pattern in multiplication, weare always multiplying the next odd integer.

'inal "nswer! !he pattern is to multiply the next odd number. !he next three terms would have to

be 904* -6394* -34-34* since $%*(9) 8"*, 8"* (--) $%&8*, and $%&8*(-3) $&*$&*.

-return to problem $a

"nswerE


17/18

*, *, $%, $*, +*, ...

!he numbers are going up again, so it is probably a sequence in addition,multiplication andDor exponents. ince it doesn)t go up high quickly, I)m thinking it

is addition. 7ooking at it closer, I see that we are always adding the two previousterms to get to the next term. !his is a Fibonnaci sequence 3 discussed in the lesson3 with starting values of * and *.

'inal "nswer! !he pattern is adding the two previous terms to get to the next term.!he next three terms would be 06* 14* and -64 since -4 /4 7 "%, /4 06 7 '*, 06 14 7 $%*.

-return to problem $c

"nswerE


18/18

"nswerE

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