The document outlines the fundamentals and applications of instructional media and technology in mathematics education, focusing on various tools and software used for teaching. It discusses methods such as visualization, simulation, and interactive games, emphasizing their impact on learning processes. The paper further elaborates on the differences between manipulative and virtual manipulative materials, highlighting their educational significance.

1. M.Ed. First Semester(2080/081), ul0ft lzIff (Math Education) ;d"xsf] ul0ft lzIffsf cfwf/x? (Foundation of Mathematics
Education),Math. Ed. 515 sf] cf+lzs cfjZostf k"0f{ ug]{ k|of]hgsf nflu tof/ kfl/Psf] kl/of]hgf sfo{
Instructional Media and Technology for
Mathematics Teaching
-ul0ft lzIf0fsf z}Ifl0fs ;fwg tyf k|ljlwx?_
Submitted by
Tek Prasad Lingden
M.Ed., 1st Semester
Roll No.: 42
TU Registration no.: 9-2-214-111-2011
Campus Name: Sukuna Multiple Campus,
Sundarharaincha, Morang
Submitted to
Faculty of Education,
Master Level,
Sukuna Multiple Campus,Sundarharaincha , Morang

2. z}Ifl0fs ;fdu|Lsf] kl/ro
Pp6f lrlgofF pvfg 5
s'g} klg ljifosf] lzIf0f k|efjsf/L, /f]rs / Jojxf/d"ns agfO{ ;sf/fTds ¿kdf
lzIffsf] u'0f:t/ j[l4 ug{] dxTjk"0f{ ;fwg g} z}Ifl0fs ;fdu|L x'g . z}lIfs
lqmofsnfkdf k|of]u ul/g] ;fdu|L g} z}Ifl0fs ;fdu|L x'g . lzIf0f eg]sf] ljifoj:t',
z}Ifl0fs ;fdu|L, z}lIfs jftfj/0fsf ;fy ljleGg lcd"t{ ¿kdf /x]sf ljifoj:t'nO{ d"t{ ¿k
lbg z}Ifl0fs ;fdu|Ln] ;xhtf k|bfg ub{5 . ljBfy{L{sf] ;lqmotfdf j[l4 ug{], sl7g
kf7nfO{ ;/n agfpg], l;sfOnfO{ d"t{ ¿k lbO{ ljBfy{L{df 7f]; wf/0ff a;fng],
lzIf0f l;sfO k|lqmofnfO{ /f]rs / ?lrk"0f{ agfpg] sfo{df z}Ifl0fs ;fdu|Lsf]
dxTjk"0f{ e"ldsf /xG5 .
d}n] ;'g]+, d}n] la;{]+ (I hear, I forget)
d}n] b]v]+, d}n] ;Dem+] (I see, I
remember)
d}n] u/]F, d}n a'em]+ (I do, I
understand)

3. Ufl0ft lzIf0fdf z}Ifl0fs ;fduL|sf] cfjZstf /
dxTjM
• l;sf?df ul0ftk|lt pT;fx hufO{ l;Sgk|lt clek|]l/t ug{,
• l;sf?nfO{ ;lqmo agfpgsf nflu, ljifoj:t'sf cfwf/e"t s'/fx¿nfO{ ;/n agfO{
k|:t't ug{, cd"t{ s'/fx¿nfO{ d"t{¿kdf k|of]u u/L cy{k"0f{ lzIf0f ug{,
• cWoog ul//x]sf] ljifo j:t'nfO{ :yfgfGt/ / :yfoL :d[ltsf nflu tof/ u/fpg,
• ljifo j:t'k|lt ;s/fTds wf/0ffsf] ljsf; ug{,
• l;sfOaf6 ljBfy{LnfO{ pQd ;Gt'li6 k|bfg ug{ .

4. ul0ft sIffnfO{ ;';lHht ug{ cfjZos ljleGg k|sf/sf ;fdu|Lx¿ / ltgsf] Joj:yfkgsf] af/]df oxfF
rr{f ul/Psf] 5M
s_ kf7ok':tsx¿ / ;Gbe{ k':tsx¿M
k|fylds tx b]lv pRr :t/sf kf7ok':tsx¿M sIffsf] :t/cg';f/ ul0ftsf ljleGg ljifoj:t'x¿ ;dfj]z ePsf
k':tsx¿ .
;Gbe{ k':tsx¿M yk hfgsf/L / ulx/f] cWoogsf nflu .
v_ b[Zo ;fdu|Lx¿M u|flkms rf6{x¿: ljleGg u|fkmx¿, tflnsfx¿, / kmfd{'nfx¿ ;dfj]z ePsf
rf6{x¿ .
u_ lzIf0f pks/0fx¿M ;]tf] af]8{, / ltgsf nflu cfjZos dfs{/ / 8:6/x¿ .
k|f]h]S6/M l8lh6n ;fdu|Lx¿ k|bz{g ug{ .
3_ cGtlqm{ofTds pks/0fx¿M sDKo'6/ / 6ofAn]6x¿M z}lIfs ;km6j]o/ / cgnfOg
;|f]tx¿ kx'Fr ug{ / :df6{ af]8{M cGt/lqmofTds lzIf0fsf] nflu .
ª_ ul0ftLo pks/0fx¿M HofldtLo ;]6M sDkf;, k|f]6]S6/ h:tf pks/0fx¿ / Sofns'n]6/M
ljBfy{Lx¿sf] u0fgf ;xfotf ug{ .
r_ ;km6j]o/ / cgnfOg ;|f]tx¿M u|flkmË ;km6j]o/M GeoGebra, Desmos h:tf /
l;d'n]zg ;km6j]o/M MATLAB, Mathematica

5. The use of software packages in math teaching
ul0ft lzIf0fdf ;km6j]o/ Kofs]hx¿sf] k|of]u
ul0ft lzIf0fdf ;km6j]o/ Kofs]hx¿sf] k|of]un] lzIf0f / l;sfO k|lqmofnfO{ k|efjsf/L, cfsif{s, / cGtlqm{ofTds agfpF5 . oxfF ul0ft
lzIf0fdf ;km6j]o/ Kofs]hx¿sf] k|d'v k|of]u / ltgsf] dxTjsf] rr{f ul/Psf] 5gM
!= b[ZofÍg (Visualization):
u|flkm+u ;km6j]o/: ljBfy{Lx¿n] hl6n ul0ftLo sfo{x¿ / u|fkmx¿ ;lhn} a'emg ;S5G. pbfx/0fsf nflu, GeoGebra, Desmos h:tf
;km6j]o/x¿n] ul0ftLo u|fkmx¿nfO{ b[Zodfg / cGtlqm{ofTds agfpF5G.
y|L8L df]8nM y|L8L df]8n ;km6j]o/n] HofldtLo cfs[ltx¿sf] lqMcfofdLs b[Zo k|bfg ub{5, h;n] j:t'sf] jf:tljs hLjgdf kl/df0f /
cfsf/ a'emg d2t ub{5 .
@= cg's/0f (Simulation):
ul0ftLo l;d'n]zg: ;km6j]o/n] ljleGg ul0ftLo l;¢fGtx¿sf] cg's/0f u/]/ ljBfy{Lx¿nfO{ hl6n ;d:of ;dfwfg k|lqmofsf] cg'ej k|bfg ug{
;S5. pbfx/0fsf nflu, MATLAB, Mathematica h:tf ;km6j]o/x¿df ljleGg ul0ftLo df]8]nx¿ lgd{f0f ug{ ;lsG5.
#= cGtlqm{ofTds cEof;M ;'¿jftL / pGgt :t/sf cEof;x¿M ;km6j]o/ Kofs]hx¿n] ljleGg :t/sf cGtlqm{ofTds cEof;x¿ k|bfg u5{G.
of] cEof;n] ljBfy{L{x¿nfO{ pgLx¿sf] cfkmgf] ult / Ifdtfcg';f/ cEof; ug{ ;'ljwf lbG5.
$= lSjh / v]nx¿M ;km6j]o/n] lSjh / v]nx¿ k|bfg u/]/ ul0ft l;sfOnfO{ dgf]/~hgfTds agfpF5.
lzIfsx¿sf nflu ;xfotfM kf7 of]hgf / Joj:yfkg: ;km6j]o/ Kofs]hx¿n] lzIfsx¿nfO{ kf7 of]hgf agfpg], ljBfy{Lx¿sf] k|ult 6o{fs ug{],
/ JolQmut lzIf0f of]hgf lgd{f0f ug{ ;xof]u k'o{fpF5.
%= d"NofªsgM ;km6j]o/n] :jrflnt d"nofÍg / ljZn]if0f ug{] ;'ljwf lbG5, h;n] ub{f lzIfsx¿n] l56f] / k|efjsf/L ¿kdf ljBfy{Lx¿sf]
k|bz{g d"nofÍg ug{ ;S5g .
^= ;xof]ufTds l;sfOM cgnfOg Kn]6kmd{x¿: ;km6j]o/ Kofs]hx¿ h:t}M Google Classroom, Microsoft Teams n] ljBfy{Lx¿nfO{

6. The various ways of using computers in teaching and mathematics
-ul0ft lzIf0fdf ljleGg tl/sfx¿af6 sDKo'6/sf] k|of]u+_
!_ ul0ftLo ;km6j]o/ / PlKns];gx¿
Geometry software M h:t} GeoGebra, h;n] ljBfy{L{x¿nfO{ HofldtLo cfsf/x¿ lgdf{0f ug{, cGj]if0f ug{, /
ljZn]if0f ug{ d2t ub{5.
Calculus tools M mathematica jf Wolfram Alpha h:tf ;km6j]o/n] hl6n u0fgfx¿ ug{ / Calculus sf ;d:ofx¿
;dfwfg ug{ d2t ub{5.
@_ cgnfOg z}lIfs Kn]6kmd{x¿
ljBfy{L{x¿n] ljleGg ul0ftLo ljifox¿sf] lel8of] tutorial x]g{ / cEof; ;d:ofx¿ ug{ ;S5g . ljBfy{L{x¿n] :jMcWoog
ug{ / cgnfOg cEof;x¿ dfkm{t cfkmgf] ;Lkx¿ ;'wf/ ug{ ;S5g .
#_ Virtual manipulative ;fdu|Lx¿
cgnfOg sfpG6/x¿, u|fkm Knf]6;{, / gDa/ nfOg h:tf er{‘cn pks/0fx¿ k|of]u u/L ljBfy{Lx¿n] ul0ftLo
cjwf/0ffx¿ a'emg ;S5g .
Interactive games: ul0ft l;sfpg] v]nx¿ h;n] ul0ftLo ;d:ofx¿ ;dfwfg ug{] Ifdtfsf] ljsf; ub{5 .
$_cgnfOg sf];{x¿ / j]lagf/x¿
MOOCs: Massive Open Online Courses h:t} Coursera, edX, / Udacity dfkm{t ljBfy{L{x¿n] pRr:t/Lo ul0ft
sf];{x¿ lng ;S5g .
Live webinars: ljBfy{L{x¿n] nfOe j]lagf/x¿df efu lnP/ ljz]if1x¿af6 k|ToIf ¿kdf l;Sg ;S5g .
%_ ul0ft l;sfpg] Apps

7. ^_ ;xof]ufTds cgnfOg pks/0fx¿
Google Classroom / Microsoft TeamM lzIfsx¿n] ljBfy{L{x¿;Fu c;fOgd]G6x¿ ;femf ug{,
k[i7kf]if0f lbg, / ;xof]ufTds k|f]h]S6x¿ ug{ k|of]u ug{ ;S5g .
cgnfOg / kmf]/d ;d"xx¿M ljBfy{L{x¿n] cgnfOg kmf]/dx¿df k|Zgx¿ ;f]Wg /
5nkmndf efu lng ;S5g .
&_ Data analytics
l;sfO pknlAw dfkg lzIfsx¿n] ljBfy{L{x¿sf] k|ult dfkg ug{ / ljZn]if0f ug{ sDKo'6/
pks/0fx¿ k|of]u ug{ ;S5g . h;n] pgLx¿sf] sdhf]/Lx¿ klxrfg ug{ / ;'wf/sf] nflu
of]hgf agfpg d2t ub{5 .
(_ Multiledia package
lel8of] / Plgd];gx¿M ul0ftLo cjwf/0ffx¿ a'emfpg / ljBfy{L{x¿sf] Wofgfslif{t ug{
dlN6ldl8of ;fdu|Lx¿ k|of]u ug{ ;lsG5 .
!)_ Powerpoint PresentationM cfsif{s / hfgsf/Ld"ns k|h]G6]zgx¿ dfkm{t lzIfsx¿n]
ul0ftsf] kf7x¿ a'emfpg ;S5g .
sDKo'6/x¿sf] k|of]un] ul0ft lzIf0fnfO{ cfw'lgs, cGt/lqmofTds, / k|efjsf/L agfpF5

8. Differences between manipulative and virtual manipulative materials and discuss their use in mathematics
teaching
Manipulative materials / virtual manipulative materials aLrsf] km/s
s_ k|of]ufTds ;fdu|Lx¿ (Manipulative Materials):
k|of]ufTds ;fdu|Lx¿ jf:tljs, ef}lts j:t'x¿ x'g h'g ljBfy{L{x¿n] xftn] ;dfpg / rnfpg ;S5g . oL ;fdu|Lx¿n]
ljBfy{L{x¿nfO{ ul0ftLo cjwf/0ffx¿sf] 7f]; cg'e"lt lbG5, h;n] cjwf/0ffx¿nfO{ /fd|f];Fu a'emg / ;Demgdf d2t
k'of{pF5 . Ufl0ft lzIf0fdf, k|of]ufTds ;fdu|Lx¿ k|f/lDes afnljsf; / k|fylds txsf ljBfy{L{x¿nfO{ ;+VofTds
cjwf/0ffx¿, hf]8, 36fp, u'0ff, efu, HofldtLo cfsf/x¿, / dfkg h:tf ljifoj:t'x¿ l;sfpg Jofks ¿kdf k|of]u ul/G5 .
v_ er{'cn k|of]ufTds ;fdu|Lx¿ (Virtual Manipulative Materials)
er{'cn k|of]ufTds ;fdu|Lx¿ l8lh6n ¿kn] l;h{gf ul/Psf j:t'x¿ x'g h'g sDKo'6/, 6ofAn]6, jf :df6{kmf]gdf
OG6/km]; dfkm{t x]l/g] / rnfpg ldN5 . oL ;fdu|Lx¿df online tools interactional tools / l8lh6n u]dx¿ ;dfj]z 5g
h;n] ljBfy{L{x¿nfO{ ul0ftLo cEof;x¿ u/fpF5 . er{'cn k|of]ufTds ;fdu|Lx¿n] ljBfy{L{x¿nfO{ cfsif{s / OG6]/]lS6e
jftfj/0fdf ul0ft l;Sg d2t k'of{pF5, h;n] ub{f l;sfO k|lqmofnfO{ /dfOnf] / k|]/0ffbfoL agfpF5 . oL ;fdu|Lx¿
ljleGg txsf ljBfyL{x¿nfO{ k|of]u ug{ ;lsG5, / lzIfsx¿n] cfkmgf kf7oqmdcg';f/ er{'cn cEof;x¿ ug{ ;S5g .

9. Implications of manipulative and virtual manipulative materials
• Doflgk'n]l6e ;fdu|Lx¿ k|of]u ubf{, lzIfsx¿n] ljBfy{L{x¿nfO{ j:t'x¿;Fu k|ToIf
cGt/lqmof u/fO{ ul0ftLo cjwf/0ffx¿ a'emg / cg';Gwfg ug{ ;xof]u ug{ ;S5g .
pbfx/0fsf nflu, ljBfy{L{x¿n] uGtL l;Sg Ansx¿ k|of]u ug{ ;S5g / cfsf/ / kl/df0fsf]
cjwf/0ffx¿ a'emg HofldtLo cfsf/x¿ k|of]u ug{ ;S5g .
• er{'cn k|of]ufTds ;fdu|Lx¿ k|of]u ub{f, lzIfsx¿n] Digital tools sf] k|of]udfkm{t
ljBfy{L{x¿nfO{ ;d:of ;dfwfg ug{, ul0ftLo cjwf/0ffx¿sf] cEof; ug{, / cfkm'nfO{
d'Nofªsg ug{ d2t k'o{fpg ;S5g . pbfx/0fsf nflu, Ps ljBfy{L{n] cgnfOg
Kn]6kmd{df ul0ftLo v]n v]Ng ;S5, h;n] ul0ftLo ;d:of ;dfwfg ug{sf] nflu ljleGg
pkfox¿sf] vf]hL ug{ k|]l/t ug{ ;S5 .
• b'j} k|sf/sf ;fdu|Lx¿, ul0ft lzIf0fdf dxTjk'0f{ 5g . b'j}sf] ldl>t k|of]un]
ljBfy{L{x¿sf] l;sfO cg'ejnfO{ dha't agfpg ;S5, pgLx¿sf] ljleGg l;sfO z}nLx¿nfO{
;Daf]wg ub{} hfg' cfhsf] cfjZostf xf] .

10. The Importance of Multimedia packages in teaching and learning math
ul0ft lzIf0f ug]{ / l;Sg] k|lqmofdf multimedia Kofs]hx¿sf] dxTjk"0f{ :yfg /x]sf] x'G5 . multimedia packages n]
lzIf0fnfO{ clws cfsif{s / k|efjsf/L agfpF5 . oxfF Multimedia packages sf] dxTj o; k|sf/ 5g M
!= b[Zo / >j0f ;fdu|Lsf] k|of]uM dlN6ldl8of Kofs]hn] tl:a/, lel8of], Plgd]zg, / cl8of] ;fdu|LnfO{ ;dfj]z
u5{, h;n] ljBfy{L{x¿nfO{ hl6n ul0ftLo cjwf/0ffx¿nfO{ ;lhn} a'emg d2t u5{ .
@= cGt/lqmofTdstfM dlN6ldl8of Kofs]hn] cGtlqm{ofTds ;fwgx¿ h:t} lSjh, v]n, / l;d'n]zgsf] k|of]u u5{ .
o;n] ljBfy{L{x¿nfO{ ;lqmo l;sfO{df ;+nUg u/fpF5 / pgLx¿sf] rf;f] a9fpF5 .
#= JolQmut ultM ljBfy{L{x¿n] dlN6ldl8of ;fdu|LnfO{ cfkmg} ultdf cWoog ug{ ;S5G. o;n] pgLx¿nfO{
sl7gfO ef]lu/x]sf ljifox¿df Wofg lbg / bf]xf]o{fP/ cWoog ug{ ;'ljwf lbG5 .
$= ljljw lzIf0f z}nLsf] ;dy{gM dlN6ldl8of Kofs]hx¿n] ljleGg lzIf0f z}nLx¿nfO{ ;dy{g u5{g. h:t} b[Zo, >j0f, /
cGtlqm{ofTds l;sfO . o;n] ljleGg k|sf/sf ljBfy{Lx¿sf] cfjZostf k"lt{ u5{ .
%= ;d:of ;dfwfgsf] ;LkM dlN6ldl8of ;fdu|Lx¿n] ljBfy{L{x¿nfO{ jf:tljs hLjgsf ;d:ofx¿ ;dfwfg ug{ cEof; u/fpF5, h;n]
pgLx¿sf] ;d:of ;dfwfg ug{] Ifdtfdf ;'wf/ nofpF5 .
^= pTk|]/0ff / ;xeflutfM dlN6ldl8of Kofs]hx¿ k|of]u ub{f ljBfyL{x¿nfO{ ul0ft l;Sg k|]l/t ug{ ;lsG5 . cfsif{s ;fdu|L
/ cGt/lqmofTds cEof;x¿n] pgLx¿nfO{ ;+nUg / pTk|]l/t agfpF5 .
o;/L, Multimedia packages n] ul0ft l;sfO{ k|lqmofnfO{ cToflws cfsif{s, cGtlqm{ofTds, / k|efjsf/L agfpF5 . o;n]

11. z}lIfs v]nx¿ / kf/f8S;x¿sf] z}lIfs p2]Zox¿ w]/} dxQjk"0f{ 5G. oL ljlwx¿n] lzIf0f / l;sfO
k|lqmofnfO{ cfsif{s / k|efjsf/L agfpg d2t u5{G. oxfF z}lIfs v]nx¿ / kf/f8S;x¿sf] k|d'v p2]Zox¿
5gM
s_ cg's/0fLo l;sfOM v]nx¿df cGtlqm{ofTdstf / k|lt:kw{fTdstf x'G5, h;n] ljBfy{L{x¿nfO{
;lqmo
/ ;+ agfpF5 .
v_ ;d:of ;dfwfg ug{] Ifdtf ljsf;M v]nx¿df ljleGg ;d:of ;dfwfg ug{] r'gf}tLx¿ x'G5G, h;n]
ljBfy{Lx¿sf] tfls{s / cfnf]rgfTds ;f]rnfO{ k|f]T;flxt u5{ .
u_ ;d"xsfo{ / ;xsfo{M s]xL v]nx¿n] ;d"xdf sfd ug{ cfjZos k5{, h;n] ljBfy{Lx¿nfO{ ;xsfo{ /
;+rf/ sf}znsf] ljsf; ug{ d2t u5{ .
3_ pTk|]/0ffM v]nx¿sf] k|lt:kw{fTds :jefjn] ljBfy{L{x¿nfO{ a9L l;Sg / d]xgt ug{ k|]l/t u5{ .
ª_ dgf]an / cfTdljZjf; j[l4M v]nx¿df ;kmntf kfpFbf ljBfy{Lx¿sf] cfTdljZjf; a95 /
The Educational objectives of Games and Paradoxes in teaching

12. s_ tfls{s / cfnf]rgfTds ;f]rsf] k|f]T;fxgM kf/f8S;x¿n] ljBfy{L{x¿nfO{ cfkmgf] a'emfOsf] k'glj{rf/
ug{ / ulx/f] ;f]Rg afWo kf5{ .
v_ 1fgsf] ;Ldf k/LIf0fM kf/f8S;x¿n] k|foM ;fdfGo 1fg jf ;xh ljZjf;x¿nfO{ r'gf}tL lbG5g, h;n]
ljBfy{L{x¿nfO{ cfkmgf] 1fgsf] ;Ldf klxrfg ug{ / lj:tfl/t ug{ d2t u5{ .
u_ /rgfTds ;f]rsf] ljsf;M kf/f8S;x¿sf] ljZn]if0fn] ljBfy{L{x¿nfO{ c;fdfGo / /rgfTds tl/sfn] ;f]Rg k|]l/t
u5{ .
3_ bz{gLotf / lh1f;f pTkGgM kf/f8S;x¿ k|foM rlst kfg{] / lh1f;f hufpg] vfnsf x'G5g, h;n]
ljBfy{L{x¿nfO{ ljifoj:t'k|lt a9L pT;'s / ;+nUg agfpF5 .
ª_ l;4fGtsf] ulx/f] a'emfOM kf/f8S;x¿sf] cWoogn] ljBfy{Lx¿nfO{ ul0ftLo / j}1flgs l;4fGtx¿sf] ulx/f]
/ lj:t[t a'emfO k|fKt ug{ d2t u5{ .
o;/L, z}lIfs v]nx¿ / kf/f8S;x¿n] lzIf0f / l;sfO k|lqmofdf ljleGg dxTjk"0f{ p2]Zox¿ k"/f u5{G. oL
Paradoxes x¿sf] p2]Zox¿

13. ul0ftnfO{ cy{k"0f{ t/Lsfn]
lzIf0f ug{ v]nx? k|efjsf/L /
pkof]uL dfWodx? x'g . ulzt
lzIf0fdf v]n / Puzzles x?
k|of]udf Nofpg] ul0ft1
8fOg] x'g . pgsf cg';f/ ul0ft
lzIf0f l;sfOsf $ j6f v]nM
1. Preliminary games..
2. structural games
3. practice games
4. mental games
s_ cfwf/e't wf/0ff lzIf0fsf nflu v]n (Games for Teaching Basic Concepts)
ul0ft lzIf0fdf ljBfy{Lx?nfO{ gofF wf/0ff lzIf0f ug{ ljleGg v]nx?sf] k|of]u ug{
;lsG5 .
hfb'ju{ (Magic Square)
hfb'ju{sf] k|of]uaf6 afnaflnsfdf hf]8 ;DaGwL ul0ftLo wf/0ff ljsf; u/fpg ;lsG5 .
oxf lbOPsf] hfb'ju{ ;g !%!$ df Durer n] ljsf; u/]sf x’g . o;nfO{ Melancholia
elgG5 . oxfF ! b]lv !^ ;Ddsf ;+Vofx?nfO{ ju{fsf/ sf]7fx?df lgDgff';f/ n]lvPsf]
5M
oxfF lbOPsf] hfb'ju{df h'g;'s} nx/ / k+lQmnfO{ tyf 58s]nfO{ hf]8bf hf]8kmn
#$ x'G5 . o;sf/0f of] hfb'ju{ xf] .
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1

14. v_ pTk|]/0ff hufpg] v]nx? (Motivational Games)
i) Preserving Product (PP) Games: ljBfno txdf lgd{f0fafbsf] :ki6 wf/0ff a;fng of] v]n pkof]uL x'G5 . o; v]nsf]
lgod o;k|sf/ 5 :
a × b = (a × c) x (b × 1/c)
where a and b must be non-zerowhole numbers and c a real number, again but a x c and b x is C non zero whole number. Such a
numbers a and b called product preserving pair. For example, in this games the teacher would have to write two pair of PP
numbers.
Preserving product
e. g. 10×40= 20×20
And ask student to find the rule and find the others pair.
ii) Preserving Product and Altering the Place Value (PPAP) Games ljBfno txdf lgd{f0fjfbL ljlwjf6 ul0ft l;sfpg pkof]uL
x'G5 . o;sf] v]nsf] lgod o;k|sf/ 5 .
e. g. 12 x 84 = 21 × 48
And ask student to find the rule and find the other pairs.
The rule is: (10a + b) x (10 c + d) = (10b + a )x (10d + c) where a, b, c, d are non-zero one digit whole numbers and c is
obtained by multiplying b by some real number
and d is obtained by multiplying the same real number.

15. u_ ;+Vof 9fFrfx? (Number Patterns)
;+Vof 9fFrfx?sf] k|of]u u/]/ w]/} u0fLtLo v]nx? v]Ng ;lsG5 . o:tf a]nx? k|of]u u/L ul0ft lzIf0f ub{f
ljBfy{L{x? k|ToIf ?kdf ;xefuL x'G5g . ltgLx?sf] lg0f{o Ifdtfdf j[l4 x'G5 / lzIf0f k|efjsf/L x'G5 . ;+Vof
9fFrf;DaGwL s]xL pbfx/0fx? tn k|:t't ul/Psf 5g .
1. Number pattern in symmetrical
form
11 x 1= 1
11 x 11 = 121
...…......……...
2. Number pattern in one digit in
the product
(3 × 1) × 37 = 111
(3 × 2) × 37 = 222
....…..…..
3. Square number pattern whivh
ends in digit 5
(115)2 =11×12 = 132 (115)2 =13225
52 =25
(125)2 =12×13 = 156 (125)2 =15625
52 =25
3_ ul0ftLo uNtL (Mathematical Fallacy)
ul0ftLo v]n 1=2 lgsfNg]
dfgf}+ a = b
b'a}tkm{ a n] u'0fg u'0fg ubf{
ab = b2
b'a}tkm{ b 36fpFbf
ab - a2 = b2 - a2
or, a(b - a) = (b - a)(b + a)
or. a = a + b
since a = b ,then,
a = 2a
or, 1=2 (Find the mistake of this game)
oxfF ;fwf/0f tl/sfn] x]of}{+ eg] k|To]s r/0f Psbd l7s 5g t/ cln
ulxl/P/ ljrf/ ubf{ of] k|i6 x'G5 ls b2 - a2 ;dLs/0f sf] vl08s/0f ug{'
;/f;/ unt x'G5 lsgsL b - a = 0 x'G5 0 sf] vl08s/0f ug{' k"0f{?kdf unt