OpjectvE
To find the HCF of two numbers
experimentally based on Euclid
Division Lemma.
Matera Required
Cardboard sheets, glazed papers of
different colours, scissors, ruler,
sketch pen, glue etc.
Memo oF ConstRucTION
1. Cut out one strip of length a units, one strip of length 6 units (6
—
Fig. 5
—— — a
Mahematies 188 Sager
units —> 7
Polynomial
» finding the number Of zeroes of a
quadratic polynomial,
Points on the graph Paper should
be joined by a free hand curve
only,
elOry. Manual OBJECTIVE
Mareriat Requirep
To verify the conditions of consistency/ Graph papers, pencil, eraser,
tency for a pair of linear cardboard, glue,
equations in two variables by graphical
method.
Metuop oF Construction
1, Take a pair of linear equations in two variables of the form
ax + by +0, =0
a)
ag + by +c, = 0, (2)
where a, b,, @,, b,c, and c, are all real numbers; 4y b,, a, and b, are not
simultaneously zero.
‘There may be three cases :
CaseT: £25
a,
Case Mi: &!
a.
Case III:
. Obtain the ordered pairs satisfying the pair of linear equ:
for each of the above cases.
ions (1) and (2)
3. Take a cardboard of a convenient size and paste a graph paper on it, Draw
two perpendicular lines X’OX and YOY’ on the graph paper (sce Fig. 1).
Plot the points obtained in Step 2 on different cartesian planes to obtain
different graphs [see Fig. 1, Fig. 2 and Fig3
Mathematics = < ¥ a
Fig.3
Demonstranion
Casel: We obtain tt
At One point P. Coxg;
Pair of linear equati
he graph as shown in Fig. S are imersecting
dinates of the point P (wy) give the unique solution for the
‘ons (1) and (2).
1. The wo line:
b
Therefore, the pair of linear equations with St. &
is consistent and h;
Ly
as the
%
unique solution,
We obtain the grap
graph as shown in Fig. 2. The {Wo lines are Coincident,
Thus, the pair of linear Cquations has infinitely Many solutions,
. . ith {4
Therefore, the Pair of linear equations With — = z =+ is also Consistent as
ay 2 ey
Well as dependent,
Mathematics ~ q two lines are Paraly
Gise HE We obtain the graph as shown in Fig. 3. The
each other,
: ir of equations W
f 7 ae solution, i.e., the pa
id ; This pair of quations has no solu
tek is inconsistent,
% be
Onstrvarion
1. a
Avruication
Conditions
Solution (5)
of consistency help to check whether a pair of line:
oF not.
In case, Solutions ‘XiSts, to find Whether the Solution js unique
OF the solutions are infinitely many,
7
‘Ar equations have
‘olution existe: in Sly
Ongecnve:
Marertar Reguirep
To identify Arithmetic Progressions Cardboard, white paper. pen/pencil,
in some given lists of numbers scissors, squared paper, glue.
(patterns).
Merton oF ConstRucTION
1. Take a cardboard of a convenient size and paste a white paper on it.
2. Take two squared papers (graph paper) of suitable size and paste them on
the cardboard.
Fig. 2
Marhematics 3. Let the lists of numbers be ‘|
(i) 1,2) 5,9) oe (ii) 1, 4, 7, 10, ...... 4
4. Make strips of lengths 1, 2, 5, 9 units and strips of lengths 1, 4, 7, 10 unit;
and breadth of each strip one unit.
5. Paste the strips of lengths 1, 2, 5, 9 units as shown in Fig. 1 and paste the
strips of lengths 1, 4, 7, 10 units as shown in Fig. 2.
DEMONSTRATION
1. In Fig. 1, the difference of heights (lengths) of two consecutive strips is not
same (uniform). So, it is not an AP.
2. In Fig, 2, the difference of heights of two consecutive Strips is the same
(uniform) throughout. So, it is an AP.
OBSERVATION
In Fig. 1, the difference of heights of first two strips =
the difference of heights of second and third strips =
the difference of heights of third and fourth strips =
Difference is (uniform/not uniform)
So, the list of numbers. 1, 2, 5, 9 form an AP. (does/does not)
Write the similar observations for strips of Fig.2.
Difference is (uniform/not uniform)
So, the list of the numbers 1, 4, 7, 10 form an AP. (does/does not)
APPLICATION -
This activity helps in understanding the
concept of-arithmetic progression.
Observe that if the left top
comers of the strips are joined,
they will be in a straight Tine in
case of an AP,
tI —— Laboratory Manual OBJECTIVE Materiat Requrep
To find the sum of first n natural Cardboard, coloured papers, while
numbers. paper, cutter, adhesive.
Metnon oF Construction
and paste a coloured paper
ind breadth 10 units.
1, Take a rectangular cardboard of a convenient si
on it. Draw a rectangle ABCD of length 1 un
v
. Divide this rectangle into unit squares as shown in Fig. 1.
w
. Starting from upper left-most corner, colour one square, 2 squares and so
on as shown in the figure.
DEMONSTRATION
1, The pink colour region looks like a stair case.
2. Length of Ist stair is 1 unit, length of 2nd stair is 2 units, length of 3rd stair
3 units, and so on, length of 10th stair is 10 units.
D
Mathematics — TOs
SS (1)
"Ajed of the shaded region = >
gion = > (area of rectangle ABCD)
=x10x je as obtained in (1) above Ss
11, which is sam btained in (1) above. This sh
1 1 is shows that
sum of the first 10 natural numbers is >x10x1]=—x +
fi mb
3 10(10+1)
This can be generalised to find the sum of first 1 natural numbers
Sas
1
S,==n(n+1)
(2)
OBSERVATION
E Forn=4,S,
Forn= 12,8, =
For 2 = 50, S,
For n = 100, 8, =~
ymbers:
APPLICATION
Result (2) may of first n terms of the list of nu
1. 17.2% By ae
be used to find the sum
ee
Tabaatey Mant! Opsective Material REQUIRED
To find the sum of the first n-even Cardboard, thermoco! balls, pins,
natural numbers. pencil, ruler, white paper, chart
: paper, adhesive.
Metuop of Consrruction
\, Take a piece of cardboard of a convenient size and paste a white paper on it.
mx ILem).
2. Draw a rectangle of suitable size on it (10 cI
3. Divide this rectangle into unit squares.
4, Fix a thermocol ball in each square using pin as show
5. Enclose the balls as shown in the figure.
nin the Fig. 1.
(ldIO
(olor)!
OlOIOlOlOlo!
DemonsTRATION
Starting from the uppermost left corner,
Mathematics Tae is
ny
the number of balls in first enclosure = 2.(= 1 x2)
the number of balls in first wo enclosures = 2 + 4 =6 (= 2 x3),
the number of balls in first three enclosures = 2 +4 + 6 = 12 (= 3 x4),
the number of balls in first six enclosures = 2444648 +104 12=42(=6%7)
the number of balls in first ten enclosures = 2-44 +648 + «.-+ 20110 (=10x 11)
This gives the sum of first ten even natural numbers.
This result can be generalised for the sum of first 7 even natural numbers us
S\=244464..4 eux (nd) (1)
ObseRvation
Forn=4in(1),S, =
For n =7 in QQ) S,
For 1 = 40 in (1), S,
For n = 70 in (1), §,
For n = 100 in (1), S,=
APPLICATION
The formula S, = n (n+1) is
numbers.
in finding out the sum of the first n even
Laboratory Manual OnJECTIVE
Mareriat Required
To establish a formula for the sum of Cardboard, coloured drawing
first n terms of an Arithmetic sheets, white paper, cutter,
Progression. adhesive.
Merion or ConstRUCTION
1, Take a rectangular cardboard of a convenient size and paste a white paper on
it. Draw a rectangle ABCD of length (2a+9¢) units and breadth 10 units.
2. Make some rectangular strips of equal length a units and breadth one unit and
some sips of length d units and breadth 1 unit, using coloured drawing sheets.
3. Arrange/paste these strips on the rectangle ABCD as shown in Fig. 1,
D
Teter
Ja +dfurea)
ja+ 2d .
at ad
_ jar ad
la+ sd
a+ 6d
ic
<< 9d >
Fig. 1
DEMONSTRATION
1. The strips so arranged look like a stair case,
2. The first stair is of length a units, the second stair is of length a+d (units),
third of «+2d units and so on and each is of breadth | unit, So, the areas (in
sq. units) of these strips are a, a +d, @ + 2c, ....., a+9d, respectively.
Mathemanes 3. This ar
3. This arrangement of strips gi
gement of strips gives pattem a, a + dat 2dat 3d,
which is al .
Alsthe aa a swith frst term «and the commman aiference
” u areas
¢ areas (in square units) of these striPS
sar
(at d) 4 (er 2d) + on 4 (a+ 9d) = LO + ASA ap)
§. Area of the desi,
1¢ designed formed by the stair case = 3 (area of rectangle ABCD)
1
= 3(10)(20+94)
=( ich i
(10a + 454), which is the same a5 ‘obtained in (1) above-
Thi
is shows thal the sum of first 10 terms of the AP= Zvoy(ea+oe)
(10) fea+(10-1)4]
=
2
This can be further generalised 10 ns of an AP AS
find the sum of first m term
n
Sz [2a+(n-)4]
OBSERVATION
On actual measurement:
d=
aeoowcrT
50, 53-5 [-+@e-)-1-
APPLICATION
ind the sum of first 7 terms of the list of numbers :
This result may pe used 10 fi
2. B23
died in Class XI.
= ee —
Taberatry Manu! OBJECTIVE MarteriAt REQUIRED
To verify the distance formula by Cardboard, chart paper, graph paper,
graphical method. glue, pen/pencil and ruler.
MetiHon oF Construction
1. Paste a chart paper on a cardboard of a convenient size.
2. Paste the graph paper on the chart paper.
3. Draw the axes X’OX and Y/OY on the graph paper [see Fig. 1].
joi ta
4. Take two points A(q, b) and B(c.d) on the graph paper and join them to get
line segment AB [see Fig. 2].
Fig. 1
Mathematics FIT fey) “ I.
Fig. 2
DEMONSTRATION
1. Calculate the distance AB using distance formula.
2. Measure the distance between the two points A and B using a ruler,
3. The distance calculated by distance formula and distance measured by the
ruler are the same.
OnSERVATION
1, Coordinates of the point A are
Coordinates of the point B are
2. Distance AB, using distance formula is
3. Actual distance AB measured by ruler is
4, The distance calculated in (2) and actual distance measured in (3) are .
APPLICATION
‘The distance formula
used in proving a number of results in geometry.
18 — 7 Laboratory Manual