i HCF of Two Numbers
AIM/OBJECTIVE
ATERIALS REQUIRED
To obtain the HCF of two numbers using Euclid
Division Lemma. Plane sheet, coloured glazed
papers, scissor, glue, ruler, pencil!
KEY-CONCEPT pen
1. Lemma: A lemma is a proved statement which is
useful to prove other statements.
2. Euclid division lemma: Let a and b be any two positive integers. Then there exi
integers q and r such that. a = bg +r, 0.
2. Take green coloured glazed paper and cut out two strips each of length r units (r
3. Again take black coloured glazed paper and cut out two strips, each of length ¢ units (¢ < s).
tunits tunits sures sriven below
4 Paste the above given strips on sheet as shown 1? the fi
<—_—_—_— pits
runits
units s units
7
t units
‘s units units ¢ units
Fig. 1 Fig. 14
OBSERVATION
(A) From Buclid division lemma, we observe that
Figure-I represents, p= @* 147 i)
Figure-2 represents qarx2ts Ai)
Figure-3 represents pesxiltt .
and Figure-4 represents s = tx 2+0 “Ww
(B) Also, HCF ofp andg = HCF ofq andr
HCF ofr and s
HCF of s and ¢
as per assumption in E.D.A.
(from iv)
HCF ofs andt = ¢
HCF ofp andg = ¢
CONCLUSION
Ifa and b are positive integers such that
common divisor of b and r, and vice versa.
a= bg +r, then every common divisor of and b is @
APPLICATION
Pah process is an algorithm (series of well defined steps) whicl
‘ommon Factor (HCF) of two given positive integers.
ch is helpful to compute the Highest Quadratic Polynomial
ADM/OBJECTIVE
(i) Todraw the graph of a quadratic polynomial.
(ii) To recognize the shape of the curve based on
the sign of coefficient of =*.
(iii) To determine the number of zeroes (roots).
KEY-CONCEPT
Quadratic polynomial: An algebraic
real numbers anda = 0 i
. Zeroes of a quadratic polynomial: Let pir) =ax*
then a real number ‘a’ is called a zero if and only if pic
Ee
PROCEDURE
Take a chart paper and paste a graph paper on it with the help o!
1
Now, take a quadratic polynomial g(x) = ax® = 6x =,
2
3. Now, we will take different values of a, 6 and c by considering two
case (1): a>0(+ve)
case (2): a<0(-ve)
4, Fora>0:Leta=1,=-2,c=-8
So, glx) =x? 2-8
The following table gives the values of g(x) for various values of x
x -3 | -2 | -1 0 1 2 3 4 5
qa) | 7 [0 | -5 | -s | -9 | -s | -s 0 7 ah as when body
5. Now, plot these points on a graph pnper and draw ite eet
below:
jis graph as shown
nd draw its #
ph paper and drat
ther gral
. Igo on anotl se
Now, plot these points 50 yay
x
8. Now, by observing these figures we note down the number of times that these graphs
intersects with x-axis. OBSERVATION
1
2.
Graph-1 obtained in Step-5 represents the parabola which opens upward because of
sign of coefficient of x*.
sign of coefficient of x*.
Parabola given in the graph in step-5 cuts x-axis at two points. So.
Parabola given in the graph in step-7 does not cuts x-axis.
If coefficient of x? is positive, then parabola opens upward and coefficient o
is. So, number of real zeroes is 0
be a parabola.
Number of zeroes of quadratic polynomial = Number of times, points cuts x-axis
This activity will be helpful to understand the geometrical representatio:
It will be helpful to find the number of zeroes of quadratic polynomial using graph.
the shape of quadratic polynomial just by observing
3
q(x) =x? — 2x — Bis two.
4.
CONCLUSION
1. The shape of the quadratic polynomial will always
2.
then parabola opens downward.
3.
APPLICATION
a
polynomial.
2.
3. It will be helpful to les i
coefficient of x°.
Graph-2 obtained in Step-7 represents the parabola which opens downward
number of zeroes 6
mn of quadratic Rana
System of Linear Equations
AIM/OBJECTIVE
To determine the conditions for con:
a system of linear equations in two variables by 3 graph papers, ruler, eraser,
graphical method. eon on
KEY-CONCEPT
L. Linear equation: An equation of the form ax + by +¢=0 or ax + by =e, where a 8 ¢ ag
real numbers, # 0,b #0 and, y are variables, is called a linear equation in two varia
2. Plotting of points on a graph paper.
PROCEDURE
1. Take the first pair of linear equations in two variables of the form
axtbyte, = 0
aye + by +e, = 0
eg: Gx—4y = 12
x-y=4
2,
Obtain a table of ordered pairs (x, y), which satisfy the given equation.
Find at least three such pairs for each equation.
For example:
6x —4y =12 bee 9 2
J -3 3
Forx-y=4 x 0 4 iz
ze 0
Now,
,
Matvemates Lab Manual x Observe if the lines are intersecting, parallel or coincident.
Also, note the following:
a
a
‘Take the second pair of linear equations in two variables, e.
o
e
Bx — By = 2, 6x — 10y =-14
Repeat the steps from (2) to (4).
Take the third pair of linear equations in two variables, e.g.: x — 4y = 5, 3x — 12y = 15
Repeat the steps from (2) to (4).
Fill in the following observation table:
aa i 2
Intersecting
Srr an
ole
Parallel
Coincident
a
10. Compare the values of 3, 5- a
2
parallel or coincident from the observation table.
Ces i :
ind = obtain the condition for two lines to be intersecting,
OBSERVATION
The student will observe that
(inerseastena-7 -
nee
SV | Ee
CONCLUSION |
The evant will learn that ions in two variables have a unique solution. Oe ee a
2 Some == have ian catans in bo vari (parallel lines)
8. Some Pairs have no solution,
APPLICATION
i ition.
This activity will be helpful to determine the nature of solutio
ae TRIANGLES
Basic Proportionality Theorem
(Thales Theorem)
AIM/OBJECTIVE
To verify the Basic Proportionality Theorem or
Thales Theorem
STATEMENT
Ifalineis drawn parallel to one side ofa triangle intersecting
the other two sides, then it divides the two sides in the same ratio.
KEY-CONCEPT
Parallel lines: Two lines are said to be parallel if they never meet.
PROCEDURE
1. Cut3 different triangles with 3 different sides from a coloured paper. Name them as AABC,
ADEF and APQR.
2, Take a board on which parallel lines are drawn as shown in Fig. 4.1 given below.
<<
<<
<>
Fig. 4.1
3. Place AABC cut-out on the board such that anyone side of the triangle is placed on one of
the parallel lines of the board as shown in /iy. 4.2 given below. permost line,
1 the lowermost or upperm
” r to place the triangles on ” e
pacartia : Q on AABC as shown in Mig, 4.2 given in step (3).
. Q and Q:
4. Mark the points P,, P
’, and Q,Q.. oa wy 80
Hate a pea rnoasiring the lengths of the respective line segments using a
List down the follo \s
ruler,
- Value
AP,
PB
AP;
Pc
AQ,
QB
AQ
QC
6. Repeat this activity for ADEF and APQR.
OBSERVATION
1. Students will observe that
aR,
In AABC, PB PC
a AQ _ AQ
QB ~ Qc
2. Similarly, students will observe the same for ADEF and APQR .
3. They will observe that the Basic Proportionality Theorem (Thales. theorem) is verified for
all the three triangles,
CONCLUSION
Students will be able to verify Basie Proportionality Theorem (BPT) through this activity.
Mathematics Lab Manual — x ATMOBJECTIVE : THATERIALS REQUIDED
To draw similar triangles, using Y-shaped strips 3 wooden strips of equal lengths
ea (approx. 10 cm long and 1 cm
KEY-CONCEPT
PROCEDURE
i
wide) fevicol, nails, hammer,
threads and cello-tape etc
Similar triangles: Two triangles are similar if
and only if
(a) their corresponding angles are equal and
(b) their corresponding sides are in the same ratio.
SSS criterion of similarity: If the three sides of one triangle
are proportional to the three sides of another triangle, then the
two triangles are similar.
Take 3 wooden strips say, A, B and C and cut one end of each
strip. Now, join three ends of each strip using cello-tape or
fevicol and paste their ends such that they all lie in different
directions. (see Fig. 5.1)
Now, fix 5 nails at equal distances on each strip and name them
eeeey Cs OD
A,B and C respectively. (See Fig. Now, to create a triangle, we will wind the thread araund the nails A,, By, Con three
respective strips, (See Fix
Similarly, to find more triangles wind the threads around the r
ils of same subscript on th
respective strips
We will get triangles A,B\C,, A,B,C,, A,B,C), A,B,C, and AsB,C,. (See Fig. 5.1)
OBSERVATION
1. On each of strip:
s such that
B,B, on strip B and similarly,
re placed at equal distan
Band C, nails
B,Bs = BB
: A.A, on strip A and B,B;
C\C. = C.C, = CC, = C.C, on strip C.
‘Three wooden strips are fixed at some particular angles.
3. Measure the sides
A\B), ABs, AC). A.C, C\B,, CsBs in A,B,C, and AASBC;.
4. Find the ratios 4B! C1 ang S12! ana we will ob i
AB," A.C. eu ill observe that all ratios are equal.
Now, by actual measurement, we can find the following:
CA
B.C
BC. CA.
CONCLUSION
Since ratios are equal therefore, AA\B,C, ~ AAsB.C; (SSS similarity criterion)
1
2. Also, we can conclude that any two triangles formed on Y-shaped strips are similar.
APPLICATION
1. Concept of similarity is useful in reducing or enlarging images and in making photographs
of different sizes.
2, From this unknown dimensions of an object can be found using a similar object with known
dimensions.
‘Also, this concept is helpful to find out the height of the pole using the length of the shadow
of pole in sunlight.
QRS wn r
HATERIALS DEQuiDED
White chart paper, pencil erase,
tuler etc. igs
AIM/OBJECTIVE
To obtain a relation between the areas and sides of
similar triangles.
KEY-CONCEPT
1. Congruent triangles: Two triangles are congruent if and only if one of them can be
ae
to superpose on the other, so as to cover it exactly.
Or
‘Two triangles are congruent if and only if there exists a correspondence between thei,
vertices such that the corresponding sides and angles are equal (or congruent).
Similar triangles: Two triangles are said to be similar, if their
(a) corresponding angles are equal and
(b) corresponding sides are proportional
Thus, ALMN ~ APQR if
(a) 4L= ZP, 2M= ZQand ZN= ZR
LM _ MN _ LN
© pa ~ QR ~ PR
Theorem: Ratio of the areas of two similar triangles is equal to the ratio of the squares of
their corresponding sides.
PROCEDURE
1. Take a white chart paper and make a triangle PQR on it with the help of ruler of any length
of sides.
2. Divide the side PQ into six equal parts using ruler such that
PS = ST = TU = UV = VW=WQ (See Fig. 5.1)
3. Now, through the points of division S, T, U, V and W, we draw
line segments which are parallel to QR meet PR at the points
A, B, C, D and E respectively. (See Fig. 5.1)
4. Repeat the same procedure of step (3) in step (4).
ie., Draw line segments through the points of division on PR,
which are parallel to PQ meet PQ at points J, I, H, G and F.
5. APQR is divided into 36 congruent triangles, OBSERVATION
By observing the fig
W) APTBe
sf congrunent tr
sles with,
Gi) APUC contain:
congruent triangles with base UC 3s
Gi) APVD contains 16 congruent triangles with base VD = 4S 2mb= | uC
5 congruent triangles with base WE = 58A= TR
5 2ue ;
268
(iv) APWE contains 2
; vb
iangles, with base QR = 6SA
(v) APQR contains 36 conys
3 6
vp = Swe
Svp= 5 we
. (APSA) _ 1 ar(APSA) _1
r(APTB) 47 ar(APUC) 9
ar(QPSA) 1 ar(APSA)
ar(APVD) ~ 16° ar(APWE)
ar(APSA) 1
ca ar(APQR) 36
amo, (Se) =(a) = 9¢(ta) = (2) = 4
and (3) = (8) = an ame (Ge) = (a) * a
conciuston
We conclude from the above activity that the ratio of the areas of similar triangles is proportional
to the ratio of the squares of their corresponding sides.
APPLICATION
This activity will be helpful to find relationship between areas and sides of similar triangle
ratio of, their
ly. then what is the
1
Sol. a? 6"
5 AP 3 ar(SAPQ)
2. IfAABC - AAPQ and jp, = p+ then what is
SAPQand jyp = gto WHALE Gy SARC)
ar(QAPQ) AP?
Sol ess (6c AB=AP+BP=3+4=7)
© ar(QABC) ~ AB?
snd 16 em? respectively. What is the ratio of their as
Activity
Pythagoras Theorem
ise
AIMOBJECTIVE
To verify the Pythagoras theorem.
KEY-CONCEPT
1. Area of a square: If square has a side ‘a units
then the area of square is a
9. Now, from step (6) and (7) we get sec’a = 1 + tan*a, sec’ = 1 + tan’ and sec*y = 1 + tan
OBSERVATION
Students will i ivi
observe that this activity can be applied to verify other trigonometrical identities CONCLUSION
For an acute angle 0 (say) sec?@ = 1 + tan’0
APPLICATION
This activity is applicable for other identities also. Median
AIM/OBJECTIVE
To find median of given data using graph- ;
KEY.CONCEP ATERIALS PEQUIDS
“cons Graph paper, rut MRED
he value of the variable which eraser ete PPS bene
1. Median: It is #
divides it into two equal P
Cumulative frequency?
arts.
It is the total of the frequencies.
ative frequency: In this kind of cumulative fre
=
als are added to the frequencies ofthe 7
4
3. Less than type cumul
frequencies of all the proceeding class interv
PROCEDURE
1 Consider the data given below as follows:
CL (Marks) oo | 10-20 | 20-30 | 30-40 | 40-50 | soa
Frequency 7 10 23 51 6 3
> We first obtain the cumulative frequency distribution table by less than method.
CL Frequency Marks less than ‘Cumulative frequency _
0-10 7 10 7
10-20 10 20 7+10=17
20 23 30 17 +23=40
a 51 40 40 +51=91
cai 6 50 91+6=97
50-60 3 60 97 +3 = 100
Now, we mark the upper class limits along X-axis on a suitable scale and cums:
frequency along Y-axis.
20, 17), (30, 40), (40, 91), (50, 97) and (60, 100)
lative frequ
Thus, we plot the points (0, 0), (10, 7), (
Join the plotted points by a free hand curve and obtain an ogive (cum
curve) as shown as follows. | 1 2 wh no 7
| Mares ——>
6. Now, we calculate 72° = 50 and mark exrrespmding point on y-axis as chown in above
$item marked point on Y-axis, draw a line parallel to X-axis and meet the curve at point P
| Now, draw perpendicular PL from point Pon the Haris. The zonardinate of L gives the
median value.
OBSERVATION
Median obtained graphically is came as obteined algebraically except for error in plotting the scale
CONCLUSION
We can find median using graph.
APPLICATION
1. This activity will be helpful to find median graphically.
2. We can also find median using ‘more than type cumulative frequency!
1. Define median.
Sol. It is the value of the variable which divides it inw two equal parts.
2. What is ogive?
Sol. Ogive is a curve which represents cumulative frequency distribution graphically.
3. What is the use of ogive?
Sol. Ogive(s) can be used to find the median of a frequency distribution.
4
What is the formula for calculating the
Bel Medien