FA Mathematics (Class IX)
Short Description
FA Mathematics (Class IX)...
Description
Formative Assessment Manual for Teachers Mathematics Class-IX PRICE : ` FIRST EDITION 2010 CBSE, India SECOND EDITION 2015 CBSE, India COPIES: 20,000
"This book or part there of may not be reproduced by any person or agency in any manner."
:
The Secretary, Central Board of Secondary Education, Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110092
DESIGN & LAYOUT :
Multi Graphics, 8A/101, WEA, Karol Bagh, New Delhi-110005, Phone: 011-25783846, 47503846
PUBLISHED BY
PRINTED BY
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B 6.
If the bisector of the exterior vertical angle of a triangle is parallel to the base, then prove that the triangle is isosceles. T
7.
Prove that the medians drawn on two equal sides of an isosceles triangle are equal.
8.
In the given fig. RT = TS, ∠ 1 = 2 ∠ 2 and ∠ 4 = 2 ∠ 3.
A 1
B 4
Prove that Δ RBT ≅ Δ SAT. 2
3
R
S C D
9.
In the given fig, prove that (i)
CD + DA + AB + BC > 2 AC
(ii)
CD + DA + AB > BC B
130
A
Task - 5: Assignment Topic
Triangles
Nature of task
Post-content
Content Coverage
Complete Chapter
Learning Outcomes
The students will be able to: l
get an idea of congruent figure
l
learn about congruent triangle
l
understand different criteria for congruence of triangle viz SSS, SAS, RHS, AAS, ASA
l
learn some properties of triangles
l
understand Inequalities in a triangle
Task
Assignment
Execution of task
Printed assignment may be given after completing the chapter
Duration
2 days
Assessment Criteria
Follow Rubric for C.W. Assignments
Follow-up
Reference material in the form of "Important points to Remember" can be given.
Assignment Questions 1.
In the figure if ∠x = ∠ y and AB = CB Prove that AE = CD. A D B
x y C
2.
In the fig PQRS is a quadrilateral and T and U are points on PS and RS such that PQ = RQ. ∠ PQT = ∠ RQU and ∠ TQS = ∠ UQS Prove that QT = QU. P T
S
Q
R
U
131
3.
In a right angle triangle, one acute angle is double the other. Prove that hypotenuse is double the smallest side.
4.
ABC is triangle in which ∠B = 2 ∠C. D is a point on BC such that AD bisects ∠ BAC and AB = CD. Prove that ∠ BAC = 72o.
5.
In fig if AD is the bisector of ∠ A, show that: (i)
AB > BD
(ii)
AC > CD
A
B
6.
C
D
In fig ABCD is a square. M is the mid point of AB and CM⊥PQ, Prove that CP = CQ. D
C
Q
A
M
B
P 7.
132
If two isosceles triangles have a common base, prove that the line joining the vertices bisects the base at right angle.
Task - 6 : Hands on Activity Topic
Triangles
Nature of task
During delivery of content
Learning Outcomes
The students will be able to verify that sum of two sides of a triangle is greater than the third side
Task
Maths Lab activity
Execution of task
An instruction sheet containing information about material required and steps involved to do the activity can be given to students for reference. They will be then asked to perform the activity and record the outcome in the recording sheet.
Duration
1 period
Assessment Criteria
This activity will be a part of Maths Lab activity, so it will be assessed according to the following parameters: l
Observation on thinking skills
l
Class Ethics
l
Performance of activity
l
File Record
(marks may be allotted by the teacher accordingly)
Instruction Sheet Learning Outcomes: To verify, using broom sticks, that a triangle is possible only if the sum of length of any two sides is greater than the third side. Previous Knowledge : Knowledge of triangle and its parts. Material Required : Sets of broom sticks of following lengths, scale, glue, paper sheet Step 1 : 5 cm, 7 cm, 11cm. Step 2 : 5 cm, 7 cm, 14 cm. Step 3 : 5 cm, 7 cm, 12 cm.
133
Procedure : (i)
Take Broom sticks of lengths 5 cm, 7 cm and 11 cm (Step 1). 5 cm 7 cm 11 cm
(ii)
Try to make a triangle using these broom sticks. Do you get a triangle?
5 cm
7 cm
11 cm (iii)
Now, take the other sets of broom sticks and repeat steps 1 and 2. Do you get a triangle ?
5 cm
7 cm
Observations:
5 cm
7 cm
14 cm
12 cm
For Set 1 : 5 + 7 > 11 5 + 11 > .......... 7 + 11 > .......... Triangle can ....................... (be formed/not be formed). For Set 2 : 14 + 7 > 5 5 + 14 > .......... 7 + 5 .......... 14 Triangle can .......................... (be formed/not be formed). For Set 3 : 12 + 7 > 5 5 + 12 > .......... 7 + 5 .......... 12 Triangle can .............................. (be formed/not be formed). Conclusions:
134
Sum of two sides of a triangle is always .............................. than the third side.
Chapter–8 Quadrilaterals Task : 1 Observe and Answer Topic
Quadrilaterals
Task
Observe and Answer
Nature of task
Pre-content
Learning Outcomes
Review the knowledge of various types of quadrilaterals
Content Coverage
Classification of quadrilaterals
Execution of task
Worksheet can be distributed to the class. Students can be given 15 minutes time to complete the task. Afterwards all questions can be discussed in class
Assessment Criteria
No assessment is required here. Worksheet can be used to test the previous knowledge and for diagnostic purpose.
Follow up
If children are not able to understand the concepts of a quadrilaterals then the pre–requisite knowledge of closed figures, to be given to students simple closed figures and polygons.
Suggestive questions for Worksheet-1 1.
Match the following: Parallelogram
Trapezium
Square
Rectangle
Rhombus
135
2.
Observe the pattern given below:
square ractangle parellelogram trapezium quadrilateral State whether the given statement is false or true:
136
1.
Every parallelogram is a square.
2.
Every trapezium is a parallelogram.
3.
Every square is a rectangle.
4.
Every rectangle is a quadrilateral.
5.
Every parallelogram is a trapezium.
Task - 2 : Hands on Activity Topic
Quadrilaterals
Nature of task
Content oriented
Content Coverage
Angle sum property of quadrilateral
Learning Outcomes
The students will be able to explore that sum of interior angles of a quadrilateral is 360o degrees
Execution of task
Cut-outs of various type of quadrilaterals can be distributed to students along with instruction sheet.
Duration
1 Period
Assessment Criteria
Assessment can be done according to the following parameters:
Follow up
l
Observation
l
Performance of activity independently
l
Response on follow up sheet
The teacher should help the students in arranging the angles to form a complete angle.
Instruction Sheet Learning Outcomes: To explore the sum of interior angles of quadrilateral. Material Required : Drawing sheet, scale, protractor, scissors, cut-outs of quadrilaterals. Instructions : 1.
Collect any cut out of quadrilateral from teacher.
2.
Cut the quadrilateral in four pieces so that each piece has one vertex angle.
3.
Arrange the angles in such a manner that can help to guess the sum of all angles of quadrilateral without measuring them.
4.
Verify result by drawing four types of quadrilaterals on prescribed space in response sheet, measuring them and completing the table in response sheet.
137
RESPONSE SHEET Date : Name :
138
Class :
1.
Trace the quadrilateral collected by you on the space given below:
2.
Paste the four cut outs of quadrilateral showing the sum of all four angles without the need of measuring them.
3.
Draw four different types of quadrilaterals in space prescribed below and complete the following table:
Name of quadrilateral
Angle 1
Angle 2
Angle 3
Angle 4
Sum of all angles
Trapezium Rectangle Parallelogram Rhombus CONCLUSION : _____________________________________________________________________
Task - 3 : Worksheet Topic
Quadrilaterals
Nature of task
Content oriented
Content Coverage
Properties of quadrilaterals
Learning Outcomes
The students will be able to understand the properties of all types of quadrilaterals with reference to its sides and angles
Execution of task
Worksheet can be distributed to the students with 25 minutes of time to complete it. Later on teacher can discuss all responses and students can do self evaluation
Duration
1 Period
Assessment Criteria
Correct Answers
Follow-up
A quiz may be organised where students may be asked to frame the relevant questions too.
139
Worksheet Complete the grid given below: Property
Square
Rhombus
Rectangle
Trapezium
Parallelogram
Opposite sides are equal
yes
yes
yes
No
No
Adjacent sides are equal All angles are of 900 Diagonals bisect each other Diagonals bisect at 900 Opposite angles are equal Diagonals divide it into two congruent triangles Diagonals are equal in length
140
Task - 4 : Topic
Quiz
Nature of task
Post-content oriented
Content Coverage
Properties of quadrilaterals
Learning Outcomes
The students will be able to review and revise all concepts learnt about quadrilaterals
Execution of task
1. Class can be divided into 4 groups as Team A, B, C, D. 2. There will be two rounds viz. Rapid fire round and problem solving round 3.. In rapid fire round each team would be given one minute to answer. They can answer as many questions as possible 4. Each correct answer will be awarded one point 5. In problem solving round the entire team will work together to solve question in prescribed time. There will be three question for each team 6. Each correct answer will be awarded 2 points
Duration
1 Period
Assessment Criteria
As per rules of quiz Score board ROUND
Follow up
TEAM A
TEAM B
TEAM C
TEAM D
The incorrect answers to be discussed in class after the quiz.
Suggested questions for RAPID FIRE ROUND. 1.
How many angles of a parallelogram are congruent?
2.
Name the quadrilateral with two pairs of equal adjacent sides.
3.
Name a quadrilateral with only one pair of parallel sides.
4.
Name a parallelogram where diagonals bisect each other at 90o.
True or False: 1.
A parallelogram is a trapezium.
2.
Rectangle is not a parallelogram.
3.
All squares are special case of rectangle.
4.
Diagonals of rhombus are equal.
141
Suggested questions for Problem solving round 1.
If the diagonals of a parallelogram are equal then show that it is a rectangle.
2.
ABCD is a quadrilateral. P, Q, R, S are midpoints of the sides AB, BC, CD and DA respectively. Show that PQRS is a parallelogram.
3.
Show that diagonals of a rhombus are perpendicular to each other.
Task - 5: Assignment Topic
Quadrilaterals
Nature of task
Post-content
Content Coverage
Complete Chapter
Learning Outcomes
To apply the knowledge attained about the quadrilaterals in solving the problems
Execution of task
For extra practise of content taught, assignment can be given after the completion of Chapter.
Duration
2 to 3 days
Assessment Criteria
Follow CW Assignment rubric
Follow-up
Class discussion. Answer to the questions may be discussed in class room and individual queries may be solved.
Assignment Questions 1.
In a quadrilateral ABCD, AO and BO are the bisectors of ∠A and ∠ B respectively. Prove that ∠AOB = ½ (∠ C + ∠ D)
2.
In figure, bisectors of ∠ B and ∠ D of quadrilateral ABCD meets CD and AB, produced at P and Q respectively. Prove that : ∠ P + ∠ Q = ½ (∠ ABC + ∠ ADC) A
P
142
D
B
C
Q
3.
In a parallelogram, the bisectors of any two consecutive angles intersect at right angle. Prove it. D P C
4.
In figure, ABCD is a parallelogram and ∠ DAB = 60o. If the bisector AP and BP of angles A and B respectively meet P on CD. Prove that P is the mid point of CD. 60o A
5.
B
A
In Δ ABC, AD is the Median through A and E is the mid point of AD. BE produced meets AC in F such that BF//DK. Prove that AF = 1/3 AC.
F E K
B
6.
7.
The angles bisectors of a parallelogram forms a D rectangle. Prove it.
C
In the given figure, ABCD is a parallelogram and E is the mid point of side BC, DE and AB when produced meet at F. Prove that AF = 2 AB.
E
A
C
D
F
B
8.
ABCD is a square E, F, G, H are points on AB, BC, CD and DA respectively such that AE = BF = CG = DH. Prove that EFGH is a square.
9.
ABCD is a parallelogram and line segments AX, CY bisect the angles A and C respectively. Show that AXCY is a parallelogram. D X C
A
Y
B
143
Task - 6 : MCQ Worksheet Topic
Quadrilaterals
Nature of task
Post-content
Content Coverage
Complete Chapter
Learning Outcomes
The students will be able to apply the knowledge attained about the quadrilaterals in solving the problems
Execution of task
Teacher may give printed worksheet to the students
Duration
1 Period
Assessment Criteria
l
For each correct answer marks may be alloted
l
In case MCQ is used as practise worksheet then, it is not necessary to assign marks
A Classroom discussion should follow. Answer to the questions and common errors may be discussed in the class.
Follow-up
MCQ Worksheet 1.
Given four points P, Q, R, S such that three points P, Q, R are collinear. By joining these points in order, we get (a) a straight line
2.
a triangle
(c)
a quadrilateral
(d)
a circle
(d)
Trapezium
In quadrilateral ABCD, AB = BC and CD = DA, then the quadrilateral is a (a) Parallelogram
3.
(b)
(b)
Rhombus
(c)
Kite
ABCD is a rectangle. The value of x: D
(a) cannot be found
C x
(b) 70o o
(c) 110 (d) 55 4.
6.
144
A
B
The bisectors of the angles of parallelogram enclose a (a) parallelogram
5.
55o
o
(b)
rhombus
(c)
rectangle
(d)
square
Which of the following quadrilaterals is a rhombus? (a) Diagonals bisects each other
(b)
All four sides are equal
(c) Diagonals bisect opposite angles
(d)
One angle between the diagonals is 60
(a) equal
(b)
supplementary
(c) complementary
(d)
equal to the straight angle
o
Consecutive angles of parallelogram are
7.
Given a rectangle ABCD and P, Q, R, S are mid points of AB, BC, CD and DA respectively. Length of diagonal of rectangle is 8 cm the quadrilateral PQRS is (a) parallelogram with adjacent sides 4 cm and 5 cm (b) rectangle with adjacent sides 4 cm and 5 cm (c) rhombus with side 4 cm (d square with side 4 cm
8.
In parallelogram ABCD, bisectors of angles A and B intersect each other at O. The value of AOB is: D C
O
A (a) 9.
10.
30o
(b)
B 60o
(c)
90o
(d)
120o
Which of the following is not true in a ||gm (a) opposite sides are equal
(b) opposite angles are equal
(c) diagonals bisect opposite angles
(d) diagonals bisect each other
In a parallelogram ABCD, ∠ A = 160o. Then ∠ B (a) 20o (b) 168o (c) 12o o
(d) 84
145
Chapter–9 Areas of Parallelograms and Triangles Task–1 : Worksheet Nature of task
Pre-Content
Learning Outcomes
The students will be able to revise formulae of area of all types of quadrilaterals
Execution of task
Worksheet can be distributed to the students.
Duration
10 minutes
Assessment Criteria
No assessment in terms of grading required here. Task can be used for diagnostic purpose only.
Follow-up
Discussion of questions in worksheet may be followed by some more oral question.
Worksheet - 1 FIGURE
h b
b l h a
a d2 d1
146
AREA
Q.2 Fill in the blanks : (a)
Congruent figures have equal __________________.
(b)
A diagonal divides a parallelogram in two ______________ triangles.
(c)
Diagonals of a parallelogram divide the parallelogram into four congruent triangles having ___________ areas
(d)
Median joins the vertex of the triangle to the _______________ of the opposite side
(e)
Area of a rhombus is equal to the sum of the areas of four ______________ triangles formed by the diagonals.
Task - 2 : Worksheet Nature of task
Content
Learning Outcomes
The students will be able to identify the figures on the same base and between same parallels and name them
Execution of task
Worksheet can be distributed to the students.
Duration
10 minutes
Assessment Criteria
No assessment in terms of grading required here. Task can be used for diagnostic purpose only.
Follow-up
Discussion of questions in worksheet may be initiated. Some problems may be solved on blackboard/geoboard.
Suggestive questions for Worksheet - 2 Q1. Which of the following figures lie on the same base and between the same parallels? In each case write the common base and the parallel lines. P A B C D A B (i) (iv)
D
E
C P
Q
D
(ii)
(v) A S
F
G
H
M
C
P
Q
B R
A
A
Q
(iii)
L
B
P
B
D
C
147
Task - 3 : Geoboard / Dotted Sheet Nature of task
Content
Learning Outcomes
The students will be able to understand that area of parallelograms on the same base and between the same parallels are equal
Execution of task
With the help of geo-board/dotted sheets students can explore the area of different parallelograms on the same base and between the same parallels. Teacher may distribute the photocopy of the following geo-sheet to all students. In each grid students can draw a line joining the four dots horizontally and a line joining five dots vertically. Complete the parallelogram, rectangle on the same base line in each grid and between same height. Count the number of dots in each parallelogram to find the area.
Duration
1 Period
Assessment Criteria
Students can be assessed on the parameter of class ethics, observation and response on follow-up sheet. No assessment in term of grading required here. Task can be used for diagnostic purpose only.
Follow-up
148
If some students are not able to do this task the teacher must explain in simplified manner.
5x5 Geoboard Dot Paper Name ______________________
Date ______________________
CONCLUSION :
Teacher's signature :
149
Task-4 : Multiple Choice Questions Nature of task
Post-content
Learning Outcomes
The students will be able to apply the concepts learnt about the area of parallelogram and triangle on the same base and between same parallels
Execution of task
Teacher can distribute the MCQ sheet to students. Students can exchange their sheets to check the answers explained by the teacher after the worksheet is over.
Duration
25 minutes
Assessment Criteria
Correct Answer
Follow-up
Discussion in the classroom about common errors.
MCQ Worsheet Choose the correct answer: 1. If ar (A) = 20 sq. units and ar (B) = 20 sq. unit in the figures A and B, then: (a)
Figure A and B are congruent
(b)
Figure A and B are not congruent
(c)
Figure A and B may or may not be congruent
(d)
Data inadequate
2. A triangle and a parallelogram are on the same base and between the same parallels. The ratio of the areas of parallelogram and triangles is (a)
1:1
(b)
1:2
(iii)
2:1
(d)
3:1 A
3. In the given figure, BD = DE = EC. Mark the incorrect option (a)
ar (ΔABD = ar (ΔAEC)
(b)
ar (ΔDBA) = ar (ΔADC)
(c)
ar (ΔADE) = 1/3 ar (ΔABC)
(d)
ar (ΔABE) = 2/3 ar (ΔABC)
B D 4. ABCDE is a pentagon. A line through B parallel to AC meet DC produced at F
150
(a)
ar(ACB) = ar (ACF)
(b)
ar(ABF) = ar (CABF)
(c)
ar (ACF) = ar (CBF)
(d)
ar(ABF) = ar(ABC)
E
C
D C E A
B
5. If in the figure, ABCD is a parallelogram, then ar(ΔAFB) is D
F
C
4 cm
A (a)
16 cm2
(b)
B
4 cm
8 cm2
(c)
4 cm2
(d)
2 cm2
6. If the given figure, ABCD and ABFE are parallelograms and ar(quad. EABC) = 17 cm2 ar(parallelogram ABCD) = 25 cm2 then ar(Δ BCF) is D
E
C
A (a)
8 cm2
(b)
F
B
10 cm2
(c)
18 cm2
(d)
4 cm2
7. Given ar(ΔABC) = 32 cm2, AD is median of ABC, and BE is median of ΔABD. If BO is median of ΔABE, the ar(ΔBOE) is
A O E
B (a)
16 cm2
(b)
4 cm2
C
D (c)
2 cm2
(d) 1 cm2
151
2
8. In the given figure, find x, if ABCD is a rhombus and AC = 4 cm, ar(ABCD) = 20 cm D
4 cm
A
C x
B (a)
4 cm
(b)
5 cm
(c)
10 cm
(d) 2.5 cm
9. In a ΔABC, P, Q and R are the midpoints of the sides BC, CA and AB respectively. If ar(ΔABC) = 24 sq. units, then ar(ΔPQR) is A
R
B (a)
12 sq. units
(b)
Q
P
6 sq. units
C (c)
4 sq. units
(d)
3 sq. units
10. ABCD is a trapezium in which AB||DC. If ar(ΔABD) = 28 cm2 and AB = 7cm, then the distance between AB and CD is
D
A (a)
152
2 cm
(b)
C
B
7 cm 4 cm
(c)
7 cm
(d)
8 cm
Task - 5 : Assignment Topic
Area of Parallelogram and Triangles
Nature of task
Post-content
Learning Outcomes
The students will be able to apply the concepts learnt about the area of parallelogram and triangle on the same base and between same parallels
Execution of task
For extra practise of content taught, assignment can be given after the completion of Chapter.
Duration
2 to 3 days
Assessment Criteria
Follow CW assignment rubric
Follow-up
Class discussion. Answers to the questions may be discussed in class room and individual queries may be answered.
Assignment Questions 1. ABCD is a parallelogram X and Y are mid-points of BC and CD respectively. Prove that ar(ΔAXY) = 3/8 (parallelogram ABCD). 2. The medians BE and CF of a triangle ABC interest at G. Prove that ar(Δ GBC) = area of quadrilatral AFGE.
D
C
3. In the fig. O is any point on the diagonal BD of
O
the parallelogram ABCD. Prove that ar(ΔOAB) = ar(ΔOBC). A
A
B
B
4. In fig. ABCD is a parallelogram and BC is produced to a point Q such that AD = CQ. If AQ intersects DC at P. Show that ar(ΔBPC) = ar(ΔDPQ) P
D
C
Q
5. In a ΔABCD, M is the mid point of AC. Show that:
C
D M
ar(ΔABMD) = ar(□DMBC) A B
153
A 6. If the medians of a ΔABC intersect at G, Show that: ar(ΔAGB) = ar(ΔAGC) = ar(ΔBGC) =
1 ar(ΔABC) 3
F
G
B
E
D
C
7. The diagonals of a parallelogram ABCD intersect at O. A line through O meets AB at X and CD at Y. Show that D Y C (i)
ar(ΔAOX) = ar(ΔCOY)
(ii)
ar(ΔAXYD) = 1 ar(||gm ABCD)
2
O
A
B
X
8. In ΔABC, D is the mid point of AB. P is any point on BC. CQ||PD meets AB in Q.
A
Show that: ar(ΔBPQ) =
Q
1 ar(ABC) 2
D
[HINT : Join DC]
B
P
C P
9. ABCD is a quadrilateral. A line through D, parallel to AC meets BC produced in P. Prove that ar(ΔABP) = ar(□ABCD)
D
C
B
A 10. ABCD is a rectangles in which CD = 6cm,
D
6 cm
C
G
EB
F
AD = 8 cm. Find : (a)
ar(||gm CDEF)
(b)
ar(ΔGEF)
8 cm
A
154
Chapter–10 Circles Task : 1 Figures Speak Topic
Circles
Nature of task
Pre-content
Content Coverage
Definition and basic terms related to circle viz. interior, exterior, circular region, radius, diameter, arc, minor arc, major arc, segment, minor segment, major segment, sector
Learning Outcomes
Review definition and basic terms related to circle
Task
Figures Speak
Execution of task
1. Each student would be given the activity sheet. They would be then asked to write an appropriate word for the given picture. 2. Teacher may draw these figures on the chalk board also.
Duration
1 Period
Assessment Criteria
This is just a fun activity. Students are aware of these terms.
Follow-up
Teacher may use the given flash cards for review and recall.
Activity Sheet Write a suitable word corresponding to each figure.
Figure
1.
A
O
A
B
2.
155
Write a suitable word corresponding to each figure.
Figure
B 3.
A
4.
5.
6.
A
7.
A
O
B
O
8. P
156
B
O
Q
Flash Cards
Write a suitable word corresponding to each figure.
1.
O
radius
A
Circle P
A O
2.
Q
INTERIOR EXTERIOR
arc A
B
3.
B 4.
A
chord O
minor arc minor segment
5.
major segment
minor arc
SEMI CIRCLE
6.
A
O
B
DIAMETER
MAJOR SECTOR
7.
O MINOR
P SECTOR Q
157
Task-2 : Observe and answer Topic
Circles
Nature of task
Content
Content Coverage
Angles in the same segment
Learning Outcomes
The students will be able to apply the theorem, angles in the same segment of a circle are equal (Students would come to learn this relationship through given activity)
Task
Figures Speak
Execution of task
Each student would be given the activity sheet. They would be then asked to write to fill in the missing entries in the columns. Teacher may draw these figures on the chalk board also.
Duration
1 Period
Assessment Criteria
Teacher may make a note of record of students who are able to make the correct observation and write the result. It's a part of C.W. assessment.
Follow-up
Hands on activity (Given) Activity Sheet
1. Observe the following figures :
P 85
D
A O
500
500
B
Q C
What can you conclude? 2. Using the above conclusion :
D x A B
158
C
o
110
(i)
O
Find x?
S o
85o O
D R
P
S o
y
90
O
Q
R
(ii)
Find y?
(iii)
Find z?
(iv)
Find a?
(v)
Find b?
E z O F
o
50
H
G R
P
S
O
80o
a 40
o
Q
D R 35o O 70o A
b C
159
Follow up Hands on Activity Instruction sheet Aim : To verify by paper cutting and pasting "Angles in the same segment of a circle are equal" Observe the following pictures and perform hands on...
160
What do you observe? 161
Task - 3 : Figures Speak Topic
Circle
Nature of task
Content
Content Coverage
Angles subtended by an arc
Learning Outcomes
The students will be able to appreciate the theorem-angle subtended by an arc at the centre of a circle is twice the angle subtended by same arc at any other point on the remaining part of the circle (Students would come to learn this relationship through given activity)
Task
Figures Speak
Execution of task
Each student would be given the activity sheet. They would be then asked to write to fill in the missing entries in the columns. Teacher may draw these figures on the chalk board also.
Duration
1 Period
Assessment Criteria
Teacher may make a note of record of students who are able to make the correct observation and write the result. It's a part of C.W. assessment.
Follow up
Teacher may prepare such types of figures speak tasks for the other geometrical results. It helps in not only learning but also retaining the concept for longer.
Figures Speak 1. Observe the following figures :
A 70o
P O
40o
O 140o
80o
C
B
R Q
(i) What can you conclude ?
162
(ii)
2. Using the above conclusion :
M 60o O a
(i)
Find a?
(ii)
Find b ?
(iii)
Find x ?
(iv)
Find y and z ?
(v)
Find m ?
N
L
Y bo O 130o Z
X
Q
O 40o o x
o
30
R
P B y O 130o
C
A z D
O o
m
A
80 o 30 o
G 163
E A
D 19o
y
(vi)
x
x=? y=?
B
A y
C
D 35o
(vii)
y=?
C B Task - 4 : Crossword
164
Topic
Circle
Nature of task
Pre-content
Content Coverage
Basic term related to circles and statement of theorems
Learning Outcomes
l
To recall basic term related to circles
l
To revise statement of theorems related to circles
Execution of task
The teacher may provide printed Crossword Sheet to the students/ Each students would be given 10 minutes for solving the Crossword.
Duration
1 Period
Assessment Criteria
Correct Answer
Follow up
Answers the crossword to be discussed in class the next day.
Crossword Puzzle
1
B
I
E
S
2
C
T
S
Y
C
I R 4
5
R
D
L
I
C
L
A 6
C
I
A
I
M
E
T 7
U S
E
R 8
A
C
3
O
R 9
D
O
10 E
N
Q
G
U
R
A
U
E
A N
B
G
L
L
E
E
T
S
Across 1. perpendicular from centre of a circle to a chord ................................... the chord 5. sum of pair of opposite angles of ............................... quadrilateral is 180 degrees 6. longest chord 9. ................................... arcs of a circle subtend equal angles at the centre 10. Equal chords subtend ...................................... angles at the centre
Down 2. collection of points in a plane equidistant from a fixed point 3. ................................... is the same segment are equal 4. half of the diameter 7. part of a circle 8. angle subtended by an arc at centre of a circle is ............................ the angle subtended by it in remaining part of circle
165
Task - 5 : Oral Assessment Topic
Circle
Nature of task
Pre-content
Content Coverage
l
Basic term related to circles
l
Statement of Theorems
Learning Outcomes
The students will be able to enhance thinking skills, communication skills, understanding of concept
Execution of task
Oral Assessment can be an ongoing activity from beginning of Chapter till its completion.
Assessment Criteria
Follow Oral Assessment rubric.
Follow up
If any student is not able to answer the question then he/she may be given another opportunity i.e. give a days or two to prepare again and appear for oral assessment sperately.
Suggested questions for oral assessment 1. Define circle. 2. Define circular region. 3. Differentiate between circumference and chord of a circle. 4. What does a theorem regarding angles subtended by equal chords in circle say? 5. If the angles subtended by an arc at centre of a circle measures 100o. What would be the measure of the angle subtended by the same arc at a point of major arc or minor arc? 6. When is a quadrilateral said to be cyclic? 7. What is the relationship between chord of a circle and a perpendicular drawn to it from the centre? 8. How would you find the measure of the angles of a cyclic Quadrilateral if only 2 angles of the quadrilateral are given, which are not opposite angles?
166
Task - 6 : Fill in the blanks Topic
Circle
Nature of task
Pre-content
Content Coverage
l
Basic term related to circles
l
Statement of Theorems
Learning Outcomes
The students will be able to enhance thinking skills, communication skills, understanding of concept of the student
Execution of task
The teacher may write the exercise on the board or dictate in class room.
Assessment Criteria
Teacher may give marks for each correct answer.
Follow up
Formative assessment tasks are meant for learning. It is not always necessary to assess all of them.
Suggested Fill up the blanks exercise (i)
Equal chords of a circle (or congruent circles) are ........................ from the centre.
(ii)
The line drawn through the centre of a circle to bisect a chord is ......................... to the chord.
(iii) There is one and only one circle passing through three given ................... points. (iv) Chords equidistant from the centre of a circle are .......................... in length. (v)
The .......................... from the centre of a circle to a chord bisects the chord.
(vi) The sum of either pair of opposite angles of a ............................. quadrilateral is 180o. (vii) Angle in a semicircle is a ......................... angle. (viii) Angles in the same segment of a circle are ............................... (ix) .................................... arcs of a circle subtend equal angles at the centre. (x)
Angle subtended by an arc at the centre of a circle is ............................. the angle subtended by the same arc at any other point on the remaining part of the circle.
167
Task - 7 : MCQ Worksheet Topic
Circle
Nature of task
Pre - content
Content Coverage
Complete Chapter
Learning Outcomes
The students will be able to revise basic terms related to circles
Execution of task
The teacher may give printed worksheet to the students.
Duration
1 Period
Assessment Criteria
l
For each correct answer mark may be alloted.
l
In case MCQ is used as practise worksheet then it is not necessary to assign marks.
Follow-up
Class room discussion. Answers to the questions and common errors may be discussed in the class.
Multiple Choice Questions 1. Distance of chord AB from the centre is 12 cm and length of the chord is 10 cm. Then diameter of the circle is (a)
26 cm
(b)
13 cm
(c)
244 cm
(d)
20 cm
2. Two circles are drawn with side AB and AC of a triangle ABC as diameters. Circles intersect at a point D, Then (a)
∠ADB and ∠ADC are equal
(b)
∠ADB and ∠ADC are complementary
(c)
Points B, D, C are collinear
(d)
none of these
3. The region between a chord and either of the arcs is called (a)
an arc
(b)
a sector
(c)
a segment
(d)
a semicircle
(d)
5 parts
4. A circle divides the plane in which it lies, including circle in (a)
2 parts
(b)
3 parts
(c)
4 parts
5. If diagonals of a cyclic quadrilateral are the diameters of a circle through the vertices of a quadrilateral, then quadrilateral is a (a)
parallelogram (b)
square
(c)
rectangle
(d)
trapezium
6. Given three non collinear points, then the number of circles which can be drawn through these three points are (a)
168
one
(b)
zero
(c)
two
(d)
infinite
7. In a circle with centre O, AB and CD are two diameters perpendicular to each other. The length of chord AC is 1
1
AB 2 2 8. If AB is a chord of a circle, P and Q are two points on the circle different from A and B, then
(a)
2 AB
(b)
(a)
∠APB = ∠AQB
(b)
∠APB = ∠AQB = 180o
(c)
∠APB + ∠AQB = 90o
(d)
∠APB + ∠AQB = 180o
2 AB
(c)
AB
(d)
9. If P is a point in the interior of a circle with centre O and radius r units, then (a)
OP = r
(b)
OP < r
(c)
OP > r
(d)
OP = 2r
(c)
acute
(d)
180o
10. An angle in a major segment is : (a)
right
(b)
obtuse
Task - 8 : Assignment Topic
Circle
Nature of task
Pre - content
Content Coverage
Complete Chapter
Learning Outcomes
The students will be able to apply knowledge gained on the topic circles to solve question
Execution of task
For extra practise of content taught, assignment can be given after the completion of Chapter
Duration
2 to 3 days
Assessment Criteria
Follow CW assignment rubric.
Follow-up
Class discussion. Answers to the questions may be discussed in class room and individual queries may be answered.
Assignment Questions 1. Two circles with centres A and B intersect at C and D. Prove that ∠ACB = ∠ADB 2. Bisector AD of ∠BAC of ΔABC passes through the centre of the circumcircle of ΔABC. Prove that AB = AC.
169
3. In fig. A, B, C are three points on a circle such that the angles subtended by the chords AB and AC at the centre O are 80 and 120 respectively. Determine ∠BAC. o
o
A B 80o O
120o C
4. In the fig. P is the centre of the circle. Prove that : ∠XPZ = 2 (∠XZY + ∠YXZ).
Y
X
Z P
5. Prove that the circle drawn with any side of a rhombus as a diameter, passes through the point of its diagonals. 6. Bisectors of angles A, B and C of a triangles ABC intersect its circumcircle at D, E and F A B C respectively. Prove that the angles of ΔDEF are 90o - , 90o - and 90o 2 2 A 2 HINT : ∠BED = ∠ BAD = ∠BEF = ∠BCF =
A
F
2 C
A
A
2
2
E
P C
B
2
2
2 B
C
2
2
B
∠ DEF = ∠BED + ∠BEF
C D
7. Prove that the mid-point of the hypotenuse of a right triangle is equidistant from its vertices. 8. In Fig. ABCD is a cyclic quadrilateral, O is the centre of the circle. If ∠BOD = 160o . Find ∠BPD.
A
O 160o
D P 170
B
C
9. O is the centre of the circle, Prove that : ∠a = ∠b +∠c
R c P
Q b O a B
A
10. O and P are the centres of two circles intersecting at B and C. ACD is a straight line, Find x ?
D C xo
O
P o
130 A
B
171
Chapter–11 Constructions Task - 1 : Task - 1
Oral Assessment
Topic
Constructions
Nature of task
Content
Content Coverage
Basic Constructions
Learning Outcomes
The students will be able to:
Execution of task
l
apply the knowledge of basic requisites to construct a triangle
l
construct a triangle with given conditions
Teacher may ask questions based on Learning Outcomes Note : Must provide an opportunity to every student to respond and to improve their response.
Duration
2 Periods
Assessment Criteria
No assessment in terms of grading required here. Task can be used for diagnostic purpose only.
Follow up
If children are not able to answer then the concept of triangles with their properties to be revised in class.
Suggested questions for oral assessment 1. Is it possible to construct a triangle with sides 3 cm, 4 cm and 8 cm ? How? 2. What are the instruments to be used in performing constructions ? 3. When do you say that a line is the perpendicular bisector of another line ? 4. What is perimeter of a figure? What is the perimeter of a given ΔABC ? 5. What is the sum of the angles of a Δ ? 6. The exterior angle of a Δ is equal to sum of the __________________________ .
172
Task - 2 : Topic
Constructions
Nature of task
Post - content
Content Coverage
Construction of angles
Learning Outcomes
The students will be able to test their knowledge of constructing angles using compasses
Task
To draw the protractor using ruler and compasses
Execution of task
Teacher can explain the construction of different angles which are multiples of 15o using ruler and compasses. Students can be then asked to do the same. 30 minutes time would be given.
Duration
1 Period
Assessment Criteria
Teacher may ask students to cross check each others work in pairs. Students can use a protractor to check the measure of different angles. It is not necessary to give marks for this assessment. It may be used for diagnostic purpose.
Follow up
Teacher to help students in construction of angles with the help of protractor and compass in the anti–clockwise direction.
o
o
0) o
o
o
o
90 (90 )
(12
)
o
(4
45
o
(1
35
5
o
)
o
0)
5 o) 3 o 0( 15 o 0)
o
0 (3 o
) (15
15 o(16
o
165
o
180 (0 )
o
0 o(180o)
5 o) 1 20 o (60 o ) 1 3
15
A
105 o(7
5
o
60
05 )
1 75 (
O
B
173
Task - 3 : Topic
Constructions
Nature of task
Post - content
Content Coverage
Complete Chapter
Learning Outcomes
The students will be able to revise terms related to construction of triangles
Task
MCQ Worksheet
Execution of task
MCQ worksheet can be distributed to the class. Students can be given 15 minutes time to complete the task. Afterwards all questions can be discussed in class.
Duration
1 Period
Assessment Criteria
Teacher may use the strategy of peers checking in pairs.
Follow up
All questions shall be discussed in class after the assessment.
1. It is not possible to construct a triangle whose sides are : (i)
3 cm, 3 cm and 6 cm
(ii)
5 cm, 12 cm and 13 cm
(iii)
8 cm, 15 cm and 17 cm
(iv)
3 cm, 4 cm and 5 cm
2. The construction of a ΔABC, given that BC = 3 cm, ∠C = 60o is possible when difference of AB and AC is equal to (i)
3.3 cm
(ii)
3.2 cm
(iii)
3.4 cm
(iv)
2.8 cm
3. It is not possible to construct a Δ PQR with QR = 5 cm, ∠Q = 75o and PQ + PR equal to (i)
7 cm
(ii)
8 cm
(iii)
9 cm
4. Using ruler and compass, it is not possible to construct an angle of : o o ⎛ 1⎞ ⎛ 1⎞ (i) (ii) ⎛ 32 1 ⎞ (iii) 22 ⎜⎝ ⎟ ⎜⎝ ⎟ ⎜⎝ 52 ⎟⎠ 2⎠ 2⎠ 2 5. ∠BAC = 90o, If ∠BAD = ∠CAD, ∠DAE = ∠CAE, then ∠BAE = o ⎛ 1⎞ o 67 (i) 65 (ii) ⎜ ⎝ 2 ⎟⎠ (iii)
75o
(iv)
(iv)
3 cm
(iv)
⎛ 1⎞ ⎜⎝ 67 ⎟⎠ 2
? C
E
D
45o A
174
o
B
6. A unique triangle can not be constructed if its : (i)
three sides are given.
(ii)
two angles and included sides are given.
(iii)
a hypotenuse and one of the sides are given.
(iv)
perimeter is given.
7. (A) (B)
It is not possible to construct a ΔPQR with PQ = 8 cm, ∠P = 60o and PR + RQ = 12 cm Sum of any two sides of a triangles is always greater than the third side.
Which of the following is true : (i)
'A' is true ; 'B' is true and 'B' is a correct explanation for 'A'
(ii)
'A' is true ; 'B' is true but 'B' is not a correct explanation for 'A'
(iii) 'A' is true and 'B' is false (iv) 'A' is false and B is true 8. A unique triangle can be constructed if its
9.
(i)
base and one base angle are given
(ii)
two sides are given
(iii)
base and sum of the other two sides are given
(iv)
base, sum of the other two sides and one base angle are given.
An angle which can be constructed using a ruler and compass only is : (i)
5o
(ii)
10o
(iii)
15o
(iv)
25o
10. The perpendicular bisector of a line segment AB (i)
passes through point A
(ii)
passes through point B
(iii)
is perpendicular to AB at some point between A and B
(iv)
is perpendicular to AB and passes through the mid point of AB.
11. It is not possible to construct a triangle with : (i)
sides measuring 6 cm, 8 cm and 9 cm
(ii)
two base angles 60 , 70 and the included side of 8 cm.
(iii)
sides measuring 3 cm, 4 cm and 7 cm
(iv)
two sides 3 cm, 4 cm and their included angle of 90o.
o
o
175
Task - 4 : Task - 4
Class test worksheet
Topic
Constructions
Nature of task
Post - content
Content Coverage
Complete Chapter
Learning Outcomes
The students will be able to develop the skill of construction with given conditions.
Execution of task
The teacher may give printed worksheet (class test ) to the students or write questions on the board
Duration
1 Period
Assessment Criteria
Each question may be given marks. Students can be evaluated on neatness and accuracy in work.
⎛ 3⎞ 1. Draw a line segment AB = 8 cm. By ruler and compasses, obtain a line segment of length ⎜ ⎟ AB. ⎝ 4⎠ 2. Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to each other.
3. Construct a Δ ABC with base BC = 4.5 cm, ∠B = 60o, and AB + AC = 7.1 cm 4. Construct a Δ PQR with base PQ = 4.2 cm, ∠P = 45o and PR - QR = 1.4 cm 5. Construct a Δ XYZ with XY + YZ + ZX = 11 cm and ∠y = 450 and ∠Z = 60o.
Task - 5 : Task -5
Assignment
Topic
Constructions
Nature of task
Post - content
Content Coverage
Basic Constructions
Learning Outcomes
The students will be able to develop the skill of constructing a triangle with given conditions.
Execution of task
Teacher can give assignment containing questions on construction covering all types of conditions to draw a triangle.
Duration
Two days to complete the assignment.
Assessment Criteria
Students can be evaluated on neatness, accuracy in work and for timely submission of work.
Suggested Assignment Questions 1. Construct the following angles with the help of ruler and compass, if possible– 0
0
0
0
35 , 40 , 57 , 75 , 22
1° 2
0
0
, 15 , 135
2. Draw a ΔABC in which AB = 4cm, ∠A = 60 and BC = AC = 115cm. 0
3. Draw a ΔABC in which BC = 5cm. ∠B = 600 and AC + AB = 7.5cm. 4. Draw an equilateral Δ whose altitude is 6cm.
176
0
0
5. Draw a triangle ABC whose perimeter is 10.4cm and the base angles are 45 and 60 .
Chapter–12 Heron's Formula Task - 1 : Oral Questions Topic
Heron's Formula
Learning Outcomes
The students will be able to apply knowledge of basic concepts required for finding the area of triangle
Nature of task
Pre-content
Description of task
Students can be asked questions orally individually.
Execution of task
Students may be asked one by one. If the child is not able to respond, another chance can be give either by changing the question or by giving some hint.
Assessment Criteria
Students can be graded for number of correct responses.
Follow- up
If any student is not able to respond at first instance, he/she may be given another opportunity i.e. give a days or two time to prepare again and appear for oral assessment test separately.
Suggested Oral Questions 1. What is a scalene triangle ? 2. What is the name given to a Δ whose two sides are equal? Whose all the side are equal? 1 3. Area of a triangle = base x ________________. 2 4. When the sum of the squares of the lengths of two sides of a Δ is equal to the square of the length of the third side, it is called a ____________________ triangle. 5. State the Heron's formula for the area of a triangle? 6. What is the semi-perimeter of a triangle? 7. Area of a rectangle = length x ________________. 8. Perimeter of a rectangle = 2 (___________________ + ______________). 9. Area of rhombus =
1
(one diagonal) x (_____________). 2 10. The area of a parallelogram = (base) x (_____________). 1 11. Area of a trapezium = (_______________) x Altitude 2 12. Area of an equilateral triangle with sides of length x cm =
3 4
x ______________.
177
Task - 2: Figure Speak Topic
Heron's Formula
Nature of task
Post - content
Learning Outcomes
The students will be able to review the knowledge of basic concepts related to area of a triangle
Task
Figures speak
Execution of task
Each students would be given the activity sheet. They would be then asked to write an appropriate formula for the given figure. Teacher may draw these figures on the board also.
Duration
1 Period
Assessment Criteria
Teacher may make a note of number of correct answers given by the students. It's a part of C.W assessment.
Follow-up
Class room discussions answers to the questions and common errors may be discussed.
Figure
Write an appropriate formula corresponding to each figure
h
Area of triangle =
b
a
Area of right triangle =
b
a
b c
178
Semi perimeter (s) = Area of a triangle having sides of measure a , b and c and semi perimeter (s)
Figure
a
Write an appropriate formula corresponding to each figure
a
Area of an Equilateral Δ =
a b
a
e
c
Area of Quadrilateral where
s1 =
a+b+e 2
and s 2 =
c+d+e 2
is
d
Task - 3 Cross Word Topic
Heron's Formula
Nature of task
Post - content
Content Coverage
Basic terms related to triangle, quadrilateral and their areas.
Learning Outcomes
The students will be able to review basic terms related to triangle and Quadrilateral and their areas
Task
Crossword Puzzle
Execution of task
The teacher may provide printed crossword sheet to the students. Each student would be given 15 minutes for solving the crossword. Later on, teacher can discuss in class.
Duration
1 Period
Assessment Criteria
Correct Answer
Follow up
After the assessment, the questions should be discussed in class.
179
1. Solve the following crossword using clues given below:
1 2
3
5
6
4
7
8
9
Across 3. If length of each side of a square is 'a' then its ........................................ is of length √2 a. 6. A triangle with all the three sides of different lengths 8. A quadrilateral whose two sides are parallel (and the other two non-parallel) 9. If the sides of a triangles are of lengths a, b and c, then 1/2 (a+b+c) is called .................. perimeter.
Down 1. The area of a right angled triangle is equal to half the product of its ............................. and height. 2. The area of a parallelogram is equal to the product of its base and ..................................... 4. Area of a rectangle is the product of its ........................... and breadth. 5. Heron's formula is used to find the area of a ................................. 6. Area of a square is the square of its ........................................... 7. Area of a trapezium is the product of mean of its parallel sides and .........................................
180
Task - 4 : Assignment Topic
Heron's Formula
Learning Outcomes
The students will be able to: l
apply the Heron's formula to the problem of triangle and quadrilateral
l
develop the skill of finding areas
Nature of task
Post - content
Description of task
Students are required to complete Assignment in scheduled time.
Execution of task
Students may be given assignment sheet containing 8-10 questions. Some questions may be incorporated to help the students to follow the steps in systematic manner or with hints. These questions will benefit students while working independently and without any peer help or teacher's guidance (for example Q.9, 10, 11 suggested in worksheet).
Assessment Criteria
Students will be assessed for punctuality, presentation and accuracy. They shall be appreciated even if they come and discuss the problems with teachers before submission of assignment.
Follow up
Reference material on finding area of a triangle using Heron's formula can be given to students.
Assignment Questions 1. Find the area of a Δ whose sides are 35 cm, 45 cm and 50 cm 2. An isosceles triangles has perimeter 30 cm and each of its equal sides is 12 cm. Find its area (Use 15 = 3.88) 3. The measure of one side of a right triangular field is 48 m. If the difference of the lengths of hypotenuse and the other is 36 m. Find the sides of the triangle and its area. A
B
D 12 c m
C
6 cm
cm
15 cm
4.
7 cm 9
Find the area of the quadrilateral ABCD given in the figure alongside.
5. The perimeter of a rhombus is 40 cm. If one of its diagonal is 16 cm, find the area of the rhombus. 6. Two parallel sides of a trapezium are 60 cm and 77 cm and the other sides are 25 cm and 26 cm. Find the area of the trapezium.
181
7. Find the area of a quadrilateral ABCD in which AD = 24 cm, ∠BAD = 90o and B, C and D from an equilateral Δ of side 26 cm [use
3 = 1.73]
8. The height of an equilateral triangle measures 9 cm. Find its area, correct to two places of A decimals [Take 3 = 1.73] 9. Area of triangle - by Heron's formula. Step 1. Semiperimeter of Δ ABC, s =
2 Step 2. Area (ΔABC) = s(s - a)(s - b )(s - c)
1.
c
a+b+c B
b
a
C
Find the area of a triangle lengths of whose sides are 8 cm, 11 cm and 13 cm. a = 8 cm
Sol. Step . 1.
b = 11 cm c = 13 cm s=
a+b+c
= 2 s - a = ___________________
Step. 2.
s - b = ___________________ s - c = ___________________ s(s - a) (s - b) (s - c) = ___________________ area =
s(s - a)(s - b) (s - c)
= __________________ cm2 10. Find the area of a triangle length of whose two sides are 18 cm and 10 cm and the perimeter is 42cm. Sol. Hint : Length of third side = Perimeter - (sum of lengths of two given sides) Step. 1. Find the lengths of 3rd side Step. 2. s = Step. 3.
a+b+c
2 area (triangle) =
s(s - a)(s - b) (s - c)
sq units.
________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________
182
11. The length of sides of a triangular plot are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area. Sol. Hint : Let dimensions be a = 12 x b = 17 x c = 25 x Now find
s=
a+b+c 2
then area using the Heron's formula ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ Note : Area of a quadrilateral, lengths of whose sides and one diagonal are given, can be calculated by dividing the quadrilateral into two triangles and using the Heron's formula. 12. A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The non-parallel sides are 14 m and 13 m. Find the area of the field. D
A
C
B
Sol. _____________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________
183
Task - 5 : Multiple Choice Questions Topic
Heron's Formula
Learning Outcomes
The students will be able to: l
find the area of triangle when the sides of triangle are given
l
find the area of quadrilateral by dividing them into two triangle
Nature of task
Content Oriented
Content Coverage
–
Description of task
Multiple Choice Questions based on Heron's formula To find the area of triangle and the area of other figures like trapezium etc. (Where the Trapezium can be divided into two triangles).
Execution of task
Student can be given a 15 minutes Multiple Choice Question paper based on above learning objective.
Assessment Critera
Correct Answer
Follow-up
All questions shall be discussed in class after the assessment.
Multiple Choice Questions 1. The sides of a triangle are 3 cm, 4 cm and 5 cm. Its area is (a)
12 cm2
15 cm2
(b)
(c)
6 cm2
(c)
2
(d)
9 cm2
(d)
10 cm2
A 3cm
(a)
m
B
The area of Δ ABC is
3c
2.
4cm
C
20 cm2
(b)
4
5 cm 2
5 cm 2
3. The side of a triangle are in the ratio of 25 : 14 : 12 and its perimeter is 510 m. The greatest side of the triangle is (a)
120 m
(b)
170 m
(c) 250 m
(d)
270 m
4. The perimeter of a right triangle is 60 cm and its hypotenuse is 26 cm. The other two sides of the triangle are (a)
184
24 cm, 10 cm
(b)
25 cm, 9 cm
(c)
20 cm, 14 cm
(d)
26 cm, 8 cm
A
5. The area of quadrilateral ABCD in the adjoining figure is
(d)
16.4 cm
D
4 cm
C
2
6. The area of trapezium in the adjoining figure is
316 m2
(d)
296 m2
A 7. The area of quadrilateral ABCD in the adjoining figure is
A
57 cm
(b)
95 cm
(c)
102 cm2
(d)
114 cm2
cm
(a)
9c
m
2
17
2
B
40 m
B
8. Area of an isosceles right triangle is 8 cm2.
D
m
(c)
12 c
306 m
9m
(b)
C
m
286 m
28 m
D
2
15
(a)
2
15 cm
2
5 cm
15cm
14.8 cm
4 cm
(c)
2
(b)
m
15.2 cm
B
5c
2
(a)
3 cm
C
Its hypotenuse is (a)
32 cm
(b)
4 cm
(c)
4 3 cm
(d)
2 6 cm
9. The semiperimeter of a triangle is 30 cm. Its sides are in ratio of 1 : 3 : 2, then its smallest side is: (a)
1 cm
(b)
5 cm
(c)
10 cm
(d)
15 cm
185
Chapter–13 Surface Areas and Volumes Task - 1 : Worksheet Topic
Surface Areas and Volumes
Nature of task
Pre - content Oriented
Content Coverage
–
Learning Outcomes
The students will be able to: l
find out the total surface area of cube, cuboid, cylinder, cone, sphere.
l
differentiate the curved surface area from total surface area.
l
identify the kind of area required according to the problem.
Execution of task
Coloured papers can be distributed to each student. Class can be divided into groups each containing 4 to 5 students. Let them discuss, brain storm and solve the problems.
Duration
2 Consecutive Periods
Assessment Criteria
Teacher can observe each group to see that every child is taking interest and is participating. Every group is making effort to solve the problem. Students can be assessed on attributes of thinking, social and emotional skills.
Follow-up
Discuss response of each group and appreciate the students for their enthusiasm, involvement and other remarkable behaviours noticed.
Suggestive Problems 1. A conical tent of radius 3 cm and height 4 cm is to be made of canvas. Find the area of canvas required. 2. The walls of the room have to be painted. If the dimensions of the room are 5m x 4m x 3m. Find the area to be painted. 3. An open cylindrical tent is to be coated with rust resistant paint from inside. If the radius and the height of the tank are 7 m and 4 m respectively. Find the cost of painting it at a rate of ` 20 per m2. 4. A hemispherical bowl made up of a metal sheet of thickness 0.7 cm is to be painted completely. If the inner radius of the hemisphere is 7 cm, find the total area to be painted. 5. A roller 140 cm long has a diameter of 70 cm. To level the playground it makes 350 revolutions. Find the cost of leveling the playground at the rate of `10 per cm2.
186
Task - 2 : MCQ Worksheet Topic
Surface Area and Volumes
Nature of task
Content
Content Coverage
–
Learning Outcomes
The students will be able to:
Execution of task
l
find out the total surface area of cube, cuboid, cylinder, cone, sphere
l
differentiate the curved surface area from total surface area
l
identify the kind of area required according to the problem
l
be able to find out the volume of cube, cuboid, cylinder, cone, sphere
1. Teacher will distribute the worksheet. 2. Teacher can take round to see that every student is doing calculations in rough and not simply guessing.
Duration
2 Periods
Assessment Criteria
Correct Response
Follow-up
Discuss the answers after checking the MCQ and give assignment for more drill.
WORKSHEET Multiple Choice Questions Choose the correct answer: 1. If the dimensions of a cuboid are 3 cm, 4 cm, 10 cm then its surface area is (a)
2
82 cm
(b)
2
(c)
164 cm
2
(d)
216 cm
164 cm3
(c)
120 cm3
(d)
240 cm3
123 cm
2
2. The volume of the cuboid in Q.1 is (a)
17 cm3
(b)
3. The surface area of a cuboid is 1372 sq. cm. If its dimensions are in the ratio of 4 : 2 : 1, then its length is (a)
7 cm
(b)
14 cm
(c)
21 cm
(d)
28 cm
4. The base radius and height of a right circular cylinder are 7 cm and 13.5 cm. The volume of cylinder is (a)
1579 cm3
(b)
1897 cm3
(c)
2079 cm3
(d)
2197 cm3
187
5. The base radius of a cone is 5 cm and its height is 12 cm. Its slant height is (a)
13 cm
(b)
19.5 cm
(c)
26 cm
(d)
52 cm
6. The curved surface area of a cylinder of height 14 cm is 88 sq. cm. The diameter of the cylinder is (a)
0.5 cm
(b)
1.0 cm
(c)
1.5 cm
(d)
2.0 cm
7. The lateral surface area of a right circular cone of height 28 cm and base radius 21 cm is 2110 cm2 (d) 2310 cm2 264 2 m . The area of canvas required to 8. The circumference of the base of a 8 m high conical tent is 7 make the tent is 1360 1320 2 2 2 cm 2 (a) (b) (c) 286 cm (d) 98 cm m (a)
1155 cm2
(b)
1055 cm2
(c)
14
7
9. The area of metal sheet required to make a closed hollow cone of height 24 m and base radius 7 m is (a)
2
176 m
(b)
2
352 m
(c)
2
704 m
2
(d)
1408 m
(d)
10.5 cm
10. The diameter of a sphere whose surface area is 346. 5 cm2 is (a)
5.25 cm
(b)
5.75 cm
(c)
11.5 cm
11. The radius of a spherical baloon increases from 7 cm to 14 cm when air is pumped into it. The ratio of the surface area of original baloon to inflated one is (a)
1:2
(b)
1:3
(c)
1:4
(d)
4:3
12. The circumference of the base of a cylinderical vessel is 132 cm and its height is 25 cm. If 1000 cu.cm = 1 litre, the number of litres of water the vessel can hold is (a)
17. 325
(b)
34.65
(c)
34.5
(d)
69.30
13. The number of litres of milk, a hemispherical bowl of radius 10.5 cm can hold is (a)
2.47
(b)
2.426
(c)
2.376
(d)
3.476
14. The number of bricks, each measuring 18 cm x 12 cm x 10 cm are required to build a wall 12 m x 1 0.6 m x 4.5 m if of its volume is taken by mortar, is
10
(a)
15000
(b)
13500
(c)
12500
(d)
13900
15. The radius of a sphere is 10 cm. If its radius is increased by 1 cm, the volume of the sphere is increased by (a)
188
13.3%
(b)
21.1%
(c)
30%
(d)
33.1%
Task - 3 : Assignment Topic
Surface Areas and Volumes
Nature of task
Content
Content Coverage
Complete Chapter
Learning Outcomes
The students will be able to: l
apply the knowledge gained in 'Surface Area and Volumes' in solving the questions.
Execution of task
For extra practise of content taught, assignment can be given after the completion of Chapter.
Duration
2 to 3 days
Assessment Criteria
Follow CW assignment rubric
Follow - up
Class Discussion. Answers to the questions may be discussed in class room and individual queries may be answered.
Assignment 1. The dimensions of a Prayer Hall are 20 m x 15 m x 8 m. Find the cost of painting its walls @ Rs. 10 per sq. m. 2. Find the curved surface area of a right circular cylinder whose height is 13.5 cm and radius of its base is 7 cm. Find its total surface area also. 3. The exterior diameter of an iron pipe is 25 cm and it is one cm thick. Find the whole surface area of the pipe if it is 21 cm long. 4. A roller 150 cm long has a diameter of 70 cm. To level a playground it takes 750 complete revolutions. Determine the cost of levelling the playground at the rate of 75 paise per sq. metre. 5. Find the total surface area of a cone, if its slant height is 21 m and the diameter of its base is 24 m. 6. The volume of a sphere is 4851 cm3. How much should its radius be reduced so that it volume becomes 7. A river, 3 m deep and 40 m wide, is flowing at the rate of 2 km/hour. How much water will fall into the sea in a minute? 8. Find the capacity, in litres, of a conical vessel whose diameter is 14 cm and slant height is 25 cm. 9. What is the total surface area of a hemisphere of base radius 7 cm? 10. A village having a population of 4000, requires 150 litres of water per head per day. It has a tank measuring 20 m x 15 m x 6m. For how many days, the water of the tank will be sufficient for the village?
189
Task - 4 : Oral Assessment Topic
Surface Area and Volumes
Nature of task
Content Oriented
Learning Outcomes
The students will be able to: l
use the formulae, units for different types of solids
l
calculate the same mentally for simple problems
l
recognise the need of finding curved surface area, total surface area or volume after reading the word problems
Execution of task
Teacher can prepare the slips of questions based on above LO 's and put them in basket. Students can be called one by one and must read the question loudly and respond to it. If he/she is not able to respond next student can be called. These students may get the chance in the end or in somebody's turn, may complete the table for students requiring more practice.
Duration
1 Period
Assessment Criteria
Correct Response
Follow up
Reference material on formulae and their comparison volume and surface area of three dimensional figures can be given as points to remember.
Suggested Oral Questions - Volume and Surface Area Fill in the blanks. 1. The volume of a cuboid of dimensions l, l and h is ________________ unit3. 2. The surface area of a cube of side x is __________________. 3. The surface area of a cuboid of dimensions l, b, h is ____________________. 4. The volume of a right circular cylinder of base radius r and height h is ______________. 3
5. The radius of base of a cylinder is 7 cm and its volume is 770 cm . The height of the cylinder is _____________. 6. Total surface area of cylinder given in above questions is ___________________ cm2. 7. The height of a right circular cone is 3.5 cm and its base radius is 5 cm. Its volume is ____________________ cm3. 8. The formula for volume of a right circular cone is ____________________. 9. If the base radius of a cone is 8 cm and height is 6 cm, then its slant height is _____________ cm.
190
10. The formula for the total surface area of a right circular cone of base radius r height h and slant height l is _______________. 11. The volume of a sphere of radius r is _________________ and its surface area is ______________. 12. The total surface area of a hemisphere of radius r is _______________________.
Task - 5 : Formulae Testing Name of Student : _________________________
Task List of formulae
Class/Sec : _______________________________
Duration 10 minutes
Roll No. : _________________________________
Max - Marks
Date : ____________________________________
Marks obtained ____________________
Complete the Following Table Shape
Dimensions
Cube
Side a units
Cuboid
l length/breadth b
Right Circular Cylinder
radius of base = r
Right Circular Cone
radius of base = r
C.S.A
T.S.A
Volume a3
2(lb + bh + lh) 2πrh
πr2h
height = h πr(l + r)
height = h slant height l
Sphere
radius = r
Hemisphere (solid)
radius = r
2
4πr
4πr2 2
3
πr3
191
Chapter–14 Statistics Task - 1 : Newspaper Activity Topic
Statistics
Nature of task
Warm-up/ Pre-content
Content Coverage
Introduction of Statistics
Learning Outcomes
The students will be able to understand the meaning of the term statistics and the need of collection of data, survey and statistical analysis.
Task
Group Discussion
Execution of task
1. Teacher can discuss some situational examples where the survey and statistical analysis is required e.g. census, impact of its analysis on planning, market survey conducted by companies to enhance their sales or to improve the quality of products. 2. Teacher can distribute few magazines or newspapers to the students and ask them to identify atleast one situation by each student where the data collection is required.
Duration
1 Period
Assessment Criteria
No grading or marking is required in this case. The task is to gear up the students for study of statistics.
Follow-up
Teacher must motivate the students to identify such situations.
Task - 2 : Worksheet -1
192
Topic
Statistics
Nature of task
Content Oriented
Content Coverage
Collection of data, Presentation of data, Graphical Representation of Data
Learning Outcomes
Develop the skill of representing data graphically as bargraph histogram of uniform width, histogram of varying width, frequency polygon.
Execution of task
Teacher may give worksheet to assess the skills developed by the students to draw the appropriate graph of given data and the basic conceptual knowledge.
Duration
1 Period
Assessment Criteria
Teacher may decide the criteria.
Follow-up
Teacher must discuss the worksheet in the class specially, the incorrect responses in order to modify everyone's understanding. More problems of the same kind may be given as assignment.
Worksheet Time : 15 Minutes 1. The marks of 30 students in a Mathematics Test are given below: 62, 29, 36, 41, 52, 21, 50, 75, 78, 16, 20, 35, 46, 24, 57, 65, 82, 16, 25, 30, 42, 24, 18, 32, 36, 57, 75, 16, 30, 58 (i)
Arrange these marks in a grouped distribution, where one of the groups is 35-45.
(ii)
How many have scored above 75?
(iii)
How many have scored above 75?
(iv)
Write the class - size of the distribution.
(v)
What is the range of the above data?
Production of Rice (in lakh tonnes)
2.
70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 1950-51
1960-61
1970-71 1980-81 1990-91 Year
Bar graph of the production of rice crop in India in different years. Read the bar graph and answer the following questions: (i)
What was the crop-production in 1980-81?
(ii)
In which year, the crop-production was maximum?
(iii)
Write the difference between maximum and minimum production.
193
3. Draw a histogram to represent the following data C.I.
10-15 15-20 20-25 25-60 30-35
Frequency
5
6
9
8
2
Task - 2: Worksheet Topic
Statistics
Nature of task
Content Oriented
Content Coverage
Measure of Central Tendency
Learning Outcomes
The students will be able to: l
learn about measure of central tendency i.e. mean, median, mode for raw/discrete data and to
l
apply the knowledge in solving the problems.
Execution of task
Teacher may give a 30 minutes worksheet to assess the skills developed by the students to draw the appropriate graph of given data and the basic conceptual knowledge.
Duration
1 Period
Assessment Criteria
According to the weightage of the marks the assessment will be done.
Follow-up
After checking the sheets a formula reference sheet along with problems can be given to students, to give them more practice.
Worksheet 1. If mean of 8 observations is 25, find the sum of all observations. 2. Complete the table.
194
x
f
f xx
6
4
____
12
___
36
___
8
72
8
7
__
10
___
20
___
6 ___
66 ___
∑f=
∑fx=
Now, find the mean of this data.
3. Find the median of following observations 7, 4, 2, 5, 1, 4, 0, 10, 3, 8. 4. Find the mode of following observations 1, 2, 3, 5, 2, 1, 3, 2, 5, 2 5. Find the mean of following observations 5, 8, 2, 7, 8, 5, 3, 5, 6, 8 Follow up : Practice Worksheet 1. Find the true class limits of the first two classes of the distibution 1-9, 10-19, 20-29, ......... 2. The following are the marks obtained by 20 students in a class-test : 40, 22, 36, 27, 30, 12, 15, 20, 25, 31, 34, 36, 39, 41, 43, 48, 46, 36, 37, 40 Arrange the above data in frequency distribution with equal classes, one of them being (0-10), 10 not included. 3. The electricity bills of twenty house holds in a locality are as follows: 370, 410, 520, 270, 810, 715, 1080, 715, 1080, 712, 802, 775, 310, 375, 412, 420, 370, 218, 240, 250, 610, 570,. Construct a frequency distribution table with class size 100. 4. The enrolment in classes VI to X of a school is given below: Class :
VI
VII
VIII
IX
X
Enrolment :
70
65
60
45
35
Draw a bar chart to depict the data. 5. Draw a histogram and a frequency polygon for the following data: Marks No. of Students
10-20
20-30
30-40
40-50
50-60
8
12
15
9
6
6. Draw a histogram for the following data: Classess Frequency
10-15
15-20
20-30
30-50
50-80
6
10
10
8
18
7. Find the mean of the following data: 153, 140, 148, 150, 154, 142, 146, 147 8. The mean of the following data is 37. Find x 28, 35, 25, 32, x, 40, 45, 50
195
n
9. If the mean of 20 observations 2x1, 2x2, ............, 2xn is 2⎯x , show that
∑ (x
i
- x)= 0
i =1
10. The mean of 20 observations is 25. If each observation is multiplied by 2, then find the mean of new observations. 11. The mean of two groups of 15 and 20 observations are 20 and 25 respectively. Find the mean of all the 35 observations. 12. If the mode of the following data is 14, find the value of x 10, 12, 14, 15, 16, 14, 15, 14, 15, x, 16, 14, 16 13. The median of the observations, arranged in increasing order is 26. Find the value of x. 10, 17, 22, x + 2, x + 4, 30, 36, 40 14. If the mean of observations x, 15, x+7, x+9 and 20 is 15, what is the mean of third and fourth observation. 15. The mean of 16 number is 8. If 2 is subtracted from every number, what will be new mean.
Task - 3 : Remedial Worksheet Topic
Statistics
Nature of Task
Post–content
Content Coverage
Complete chapter
Learning Outcomes
The students will be able to revise the contents especially problem areas
Task
Step by step explanation of method
Execution of Task
Sheets can be distributed to the students with blanks for steps. They can by given 15 minutes to complete the task.
Duration
15 minutes
Assessment Criteria
Teacher may use the strategy of peer checking in pairs.
Follow-up
All questions shall be discussed in the class after the assessment.
I. Formulae Reference Sheet
Sum of all observations
1.
For Raw Data Mean, x =
2.
For Ungrouped Frequency Distribution
Total number of observations
n
∑fx i
mean =
i
i =1 n
∑f
i
i =1
3.
196
Median : (The value of the middle-most observations)
Two Cases If n is odd
If n is even
Median = value of
Median = mean of values of th
4.
th
⎛ n⎞ ⎛n ⎞ ⎜⎝ ⎟⎠ & ⎜⎝ + 1⎟⎠ observations 2 2
th
⎛ n + 1⎞ ⎜⎝ ⎟ observation 2 ⎠
Mode : It is the most frequently occurring observation.
II. Practice Questions 1.
The heights (in cm) of 9 students of a class are as follows. 155
160
145
149
150
147
150
144
148
Find the median of the data. Sol. Step 1. Arrange the given observations in an ascending order. 144 ∴
145
147
148
149
Median = Value of ⎛ n + 1 ⎞
th
⎛ 9 + 1⎞ = Value of ⎜ ⎝ 2 ⎟⎠
th
⎜⎝ ⎟ 2 ⎠
150
152
155
160
observation
observation
= Value of 5th observation = 149 Median height = 149 cm. 2. In a mathematics test 15 students appeared. Their marks (out of 100) are recorded as under: 41, 39, 48, 52, 46, 62, 54, 40, 96, 52, 98, 40, 42, 52, 60 Find the median marks. Sol.
Step 1. Arrange the given data in ascending order Step 2. n = __________________ (odd) Step 3. Median = Value of ⎛ n + 1 ⎞
⎜⎝ ⎟ 2 ⎠
th
observation
= ___________________________________________________________ = ___________________________________________________________ = ___________________________________________________________
197
3. The following observations have been arranged in ascending order. If the median of the data is 63, find the value of x. 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95 Sol.
Step 1. Observations are already arranged in ascending order Step 2. Note that the number of observations is even (n=10) Step 3. Use formula for even case th
th
Step 3. Median = Mean of ⎛ n ⎞ & ⎛ n + 1⎞ observations. ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠
2
2
th
th
⇒
63 = 5 observations + 6 observations
⇒
63 = ________________________________________________________________
2
________________________________________________________________ ________________________________________________________________
Task - 4 : MCQ Worksheet
198
Topic
Statistics
Nature of task
Post-content
Content Coverage
All concepts learnt in Statistics
Learning Outcomes
The students will be able to evaluate the understanding of all the concepts learnt in the chapter and the skill to apply them
Execution of task
Teacher may give a 15 minutes MCQ worksheet to the students.
Duration
1 Period
Assessment Criteria
For each correct response mark to be awarded
Follow-up
After checking the sheets questions can be discussed and another opportunity in the form of oral assessment can be given in order to improve the response. Oral assessment should be done individually.
MCQ Worksheet 1. The range of the data 14, 27, 29, 61, 45, 15, 9, 18, is (a)
61
(b)
52
(c)
47
(d)
53
(c)
135
(d)
150
2. The class mark of the class 120-150 is (a)
120
(b)
130
3. The class mark of a class is 10 and its class width is 6. The lower limit of the class is (a)
5
(b)
7
(c)
8
(d)
10
4. In a frequency distribution, the class-width is 4 and the lower limit of first class is 10. If there are six classes, the upper limit of last class is (a)
22
(b)
26
(c)
30
(d)
34
5. The class marks of a distribution are 15, 20, 25, ........, 45. The class corresponding to 45 is (a)
12.5 -- 17.5
(b)
22.5 -- 27.5
(c)
42.5 -- 47.5
(d)
None of these
No. of students
6. The number of students in which two classes are equal. 50 40 30 20 10 0 VI VII VII (a)
VI and VIII
(b)
VI and VII
(c)
VII and VIII
(d)
None
(c)
5.6
(d)
6.5
(c)
38.5
(d)
39.2
(c)
x+3
(d)
x + 17
zero
(d)
can not be found
7. The mean of first five prime numbers is (a)
5.0
(b)
4.5
8. The mean of first ten multiples of 7 is (a)
35.0
(b)
36.5
9. The mean of x + 3, x - 2, x + 5, x + 7 and x + 72 is (a)
x+5
(b)
x+2
10. If the mean of n observations x , x , x , ............, x is (a)
1
(b)
-1
(c)
11. The mean of 10 observations is 42. If each observation in the data is decreased by 12, the new mean of the data is (a)
12
(b)
15
(c)
30
(d)
54
199
12. The mean of 10 numbers is 15 and that of another 20 numbers is 24 then the mean of all 30 observations is (a)
20
(b)
15
(c)
21
(d)
24
(c)
15
(d)
16
13. The median of 10, 12, 14, 16, 18, 20 is (a)
12
(b)
14
14. If the median of 12, 13, 16, x + 2, x + 4, 28, 30, 32 is 23, when x + 2, x + 4 lie between 16 and 30, then the value of x is (a)
18
(b)
19
(c)
20
(d)
22
(d)
18
(d)
13.6
15. If the mode of 12, 16, 19, 16, x, 12, 16, 19, 12 is 16, then the value of x is (a)
12
(b)
16
(c)
19
16. The mean of the following data is xi
5
10
15
20
25
fi
3
5
8
3
1
(a)
12
(b)
13
(c)
13.5
Follow up : Oral Assessment Sheet
200
1.
The mid-point of a class is called _______________________.
2.
Data collected by the experimenter himself is called __________________ data.
3.
The difference between maximum and minimum observations in the data is called ________.
4.
Cumulative frequency of a class is the sum total of all frequencies _____________ that class.
5.
Are the class-limits and true class limits different? If yes, explain the difference.
6.
The sum total of all observations divided by their number is called _____________ of the data.
7.
The mode of a group of observations is that value of the variable which has ______________ frequency.
8.
The _____________ is the middle most observation in the data, when they are arranged in increasing/decreasing order.
9.
x , the mean of n observation x1, x2, ..............., xn, is given by _____________
10.
The mean of first ten natural numbers is ________________.
11.
The median of first 9 natural number is _________________
12.
If each observation in the data is increased by 'a', then their __________ is also increased by 'a'.
13.
The sum of deviations of the data (observations) from the mean is ____________.
Chapter–15 Probability Task - 1 : Worksheet Topic
Probability
Nature of task
Content Oriented
Learning Outcomes
The students will be able to: l
understand the commonly used word 'probability' in Mathematical sense
l
explain the key terms like random experiment, sample, space, events, outcomes, favourable and unfavourable outcomes
l
calculate probability using formula
Execution of task
Teacher may write these questions on board and discuss the answers after 10 minutes. After the completion on worksheet students can exchange them and check each other's worksheet with the inputs given by teacher.
Duration
1 Period
Assessment Criteria
Students can exchange papers and award marks to each other.
Follow-up
Discussion on incorrect response can be followed by assignment carrying similar questions or the questions based on the same concepts.
201
Worksheet 1. Write all possible outcomes when (i)
one coin is tossed
(ii)
two coins are tossed
(iii)
one die is rolled
2. Three coins are tossed simultaneously 100 times. The following outcomes are recorded. Outcome
three tails
Frequency
two tails
one tail
no tail
28
23
26
23
Find the probability of coming up more than one tail. 3. An experiment is conducted. Probabilities of an event are calculated by some students. Which of the following could be a correct answer? (a)
5
1
1
Frequency 42
(c)
−
2
(d) 1.3 4 3 3 4. A die is rolled 300 times and the following outcomes are recorded. Outcome
(b)
2
3
4
60
55
53
(i)
Find the probability of getting an even number
(ii)
Find the probability of getting a number more than 4.
5. 500 cars passed the building on a certain day. 142 cars were from Hyundai, 125 from Tata and rest from other brands (i)
Find the probability of seeing a car from Hyundai.
(ii)
Find the probability of seeing a car from other brand.
Follow up: Assignment 1. Define a sample space. 2. What is a random experiment ? Explain. 3. Give some example of statements involving probabilistic situations. 4. The probability of occurence of an event = ________________________
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5. Two coins are tossed 200 times and the results are given below: Both Head
One Head and One Tail
Both Tails
56
96
48
(a)
What is the probability of the event "Both Head"?
(b)
What is probability of the event "Both Tails"?
6. What are the limits of probability of occurence of an event. 0 ≤ P(A)≤ 1 7. At a crossing, 1000 vehicles passed on a particular day: Two wheelers
Three whellers
310
270
Four wheelers 420
(a)
What is the probability of a two wheeler passing from that crossing on a day?
(b)
P (A three wheeler) = ________________________
(c)
P (A four wheeler) = _________________________
8. In hospital number of patients that were admitted was as follows Age Group (yrs.)
0-10
10-20
20-30
30-40
Above
No. of Patients
53
39
28
32
68
(i)
What is the probability that the patient admitted is not above 40 yrs of age.
(ii)
What is the probability that the patient admitted is below 20 yrs of age.
9. The percentage of marks obtained by a students in four tests are given below. Test
I
II
III
IV
% of Marks
67
90
78
52
Find the probability that a student gets less than 52% 10. The following table shows the marks scored by 50 students in mathematic test of 50 marks Marks
0-10
10-20
20-30
30-40
40-50
No. of Patients
13
8
12
9
8
Find the probability that the students obtained 30 or more marks.
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Task - 2 : MCQ Worksheet Topic
Probability
Nature of task
Content Oriented
Content Coverage
Empirical Probability
Learning Outcomes
The students will be able to: l
define and to calculate empirical/ experimental probability
l
differentiate between empirical and mathematical probability
l
appreciate that probability of an event lies between 0 and 1 (0 and 1 inclusive)
Execution of task
Teacher can conduct a 20 minutes MCQ.
Duration
1 Period
Assessment Criteria
F or correct response mark can be given.
Follow up
Discussion on MCQ can be followed by An Activity where students can actually conduct an experiment to show outcomes and to calculate the probability. The students may perform different experiment (e.g. with dice) to verify and to understand the formula of empirical probability thoroughly.
Multiple Choice Questions Choose the correct answer: 1. A coin is tossed 1000 times and 560 times a "head" occurs. The empirical probability occurrence of a Head in this case is (a)
0.5
(b)
0.56
(c)
0.44
(d)
0.056
2. Two coins are tossed 200 times and the following out comes are recorded HH
HT/TH
TT
56
110
34
What is the empirical probability of occurrence of at least one Head in the above case (a)
0.33
(b)
0.34
(c)
0.66
(d)
0.83
3. A die is thrown 200 times and the outcomes are noted, with their frequencies:
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Outcome
1
2
3
4
5
6
Frequency
56
22
30
42
32
18
Worksheet (i)
What is the empirical probability of getting a 1 in the above case? (a)
(ii)
0.50
0.28
(c)
0.15
(d)
0.21
(d)
0.52
What is the empirical probability of getting a number less than 4? (a)
(iii)
(b)
0.50
(b)
0.54
(c)
0.46
What is the empirical probability of getting a number greater than 4? (a)
0.32
(b)
0.25
(c)
0.18
(d)
0.30
4. On a particular day, the number of vehicles passing a crossing is given below: Vehicle
Two wheeler
Frequency
Three wheeler
52
Four wheeler
71
77
What is the probability of a two wheeler passing the crossing the on that day? (a)
0.26
(b)
0.71
(c)
0.385
(d)
0.615
5. The following table shows the blood-group of 100 students Blood group
A
B
O
AB
B+
Number of Students
12
23
35
20
10
One student is taken at random. What is probability that his blood group is B? (a)
0.12
(b)
0.35
(c)
0.20
(d)
0.10
6. In a bag, there are 100 bulbs out of which 30 are bad ones. A bulb is taken out of the bag at random. The probability of the selected bulb to be good is (a)
0.50
(b)
0.70
(c)
0.30
(d)
None of these
7. On a page of telephone directory having 250 telephone numbers, the Frequency of the unit digits of those number are given below: 0
1
2
3
4
5
6
7
8
9
18
22
32
28
40
30
30
22
18
10
(a)
A telephone number is selected from the page at random. What is the probability that its unit digit is (i)
2
(a)
0.16
(ii)
More than 6
(a)
0.20
(b)
0.128
(c)
0.064
(d)
0.04
(b)
0.25
(c)
0.32
(d)
0.16
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(iii)
Less than 2
(a)
0.16
(b)
0.18
(c)
0.22
(d)
0.32
Suggested Probability Activity Activity 1 : Toss two coins simultaneously 20 times and record your observations in the following table No. of times the two coins are tossed
20
No. of times no head comes up No. of times one head comes up No. of times two heads come up The experimental or empirical probability P(E) of an event E is given by =
No. of trials in which the event happened Total no. of trials
= _____________
Calculate A=
B=
C=
No. of times no head comes up Total no. of times two coins are tossed No. of times one head come up Total no. of times two coins are tossed No. of times two head come up Total no. of times two coins are tossed
= = =
Activity 2: Use table 1 given below to complete table 2 and answer the questions that follow Table 1 COIN TOSSES
206
H
T
T
T
NUMBER OF HEADS
T H H H H T H T H H T H H H
T
H
T
12
H T
T T
H H H T H H H H T H H H
H
T
H
13
H
H H
H T
T H T
T
T
T H T T
T
H H
T H T H T
H
T
T
H
T T
H T
H
T
T
9
H H T
H H
T
H
H
9
T H H H T
H H
H
T
T
10
H H T H H T T
T T
T
T
T
H
T
H
8
H T
H H T H H H T
H T
H H T
H H
H
H
H
15
T
T
T
T H H H T
H H H T
H H T
H T
H
T
T
10
T
T
H
T H H H H T T
T H H T T
H T
T
H
T
9
H
H T
H H
H
H
H
11
H
T T
T T
T H H T
H T T
T
T T
T T
T
H H T H T
H T
H T
Table 2
Row 1
No. of Trails
No. of heads
No. of tails
20
12
08
Row 2 Row 3 Row 4 Row 5 Row 6 Row 5 Row 6 Row 7 Row 8 Row 9 Row 10 Total
200
1. What is the probability of getting a head if a coin is tossed 200 times. 2. What is the probability of getting a tail if a coin is tossed 200 times. 3. Consider just the first 100 outcomes : What is the probability of getting a head. 4. Consider just the last 100 outcomes : What is the probability of getting a tail. 5. Observe the difference in the results of Q1 and Q3. Probability of getting a head when a coin is tossed just once is ½. Why is the difference in the results obtain.
Task - 3 : Remedial Worksheet Topic
Probability
Nature of task
Post–content
Content Coverage
Complete chapter
Learning Outcomes
The students will be able to revise the contents especially problem areas
Task
Step by step explanation of method.
Execution of Task
Sheets can be distributed to the students with blanks for steps. They can be given 15 minutes to complete the task.
Duration
15 minutes
Assessment Criteria
Teacher may use the strategy of peer checking in pairs.
Follow up
All questions shall be discussed in the class after the assessment.
207
Worksheet 1. A coin is tossed 1000 times ; with the following out comes : Head : 455, tails : 545 Compute the probability for each event. P(getting Heads) =
=
P(getting Tails) =
=
What is P(getting Heads) + P(getting Tails) =
+
= __________________
Note : Head and Tail are only two possible outcomes of each trial. Important Points to Note :
2.
(i)
The probability of an event lies between 0 and 1.
(ii)
The sum of all the probability of all possible outcomes of an experiment is 1.
Consider the following data. Bag
1
2
3
4
5
Balls
40
48
42
39
41
There are 5 bags in all containing number of balls mentioned above respectively. What is the probability of having more than 40 balls in a bag? Observe : In how many bags there, are more than 40 balls? {Hint : Observe bag 2, 3, 5,} So, P(having more than 40 balls in a bag) =
{There are total 5 bags)
= _______________
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Task - 4 : Oral Assessment Topic
Probability
Nature of task
Post - content
Content Coverage
All Concepts Learn in Topic
Learning Outcomes
The students will be able to: l
tell the probability of a given event using given frequency distribution table
l
understand that sum of the probabilities of all possible outcomes of an experiment is unity
l
access the students understanding and proficiency to apply all the concepts learnt in the 'probability'
Execution of task
Teacher can conduct an oral assessment in the class. Every student can be called individually and can be asked 4 to 5 questions and graded according to rubric discussed for oral assessment. At the time of oral assessment other students can be given a simple worksheet and a reference concept sheet so that they get time to revise their concepts thoroughly. These sheets may not be graded.
Duration
1-2 Period
Assessment Criteria
For correct response marks can be given
Suggested Oral Questions 1. Illustrate some events which relate to probability. 2. What is a random experiment? 3. What is an event? 4. What is an out come? 5. What is the difference between experimental probability and mathematical probability? 6. Is it correct to say that Probability of occurrence of an event
=
No. of cases favourable to an event Total number of cases
7. If a coin is tossed 100 times and the event "head" occurs 60 times, what is the experimental probability of occurence of a Head?
209
Task - 5 : Hands on Activity
210
Topic
Probability
Nature of task
Post - content
Learning Outcomes
The students will be able to: l
find the probability of a given event by repeated experimenting
l
understanding that sum of the probability of all possible outcomes of an experiment is unity
l
differentiate between empirical and experimental probability
Execution of task
l
Each student is asked to make a eubotahedron using OHP sheets and cello tape with the help of the net given to them. They are then asked to observe and state the probability of it landing on one of its square faces and then the probability of it landing on one its triangular faces. They are then made to experiment actually by throwing the cuboctahedron 20-30 times and find out the probability of two cases again.
Duration
1-2 Period
Assessment Criteria
No assessment in terms of grading required here. Task can be used to create interest and for diagnostic purpose only.
The net of a cuboctahedron is given. It consists of 6 squares and 8 triangles. Make this 3dimensional object using either an HOP sheet or a card paper.
Answer the following questions: 1.
If this objects is thrown, what do you think will be the probability of it landing on (i)
one of its square faces?
(ii)
one of its triangular faces?
2.
Throw the object (at least 20 times) and estimate these probabilities.
3.
How close are they to your original estimates?
211
Project Work Creative Mathematics Project Ideas (SAMPLE) General Guidelines: l
Each student is required to make a handwritten project report according to the topic allotted. Please note down your project number according to your Roll Number. Roll Number
Project Number
1-5
1
6-10
2
11-15
3
16-20
4
21-25
5
26-30
1
31-35
2
36-40
3
41-45
4
46-50
5
Note: l
A project has a specific starting date and an end date.
l
It has specific objectives.
l
List the sources of the information collected. General lay-out of the project report has the following format
212
Page Number
Content
Cover Page
Your Name, Class, Roll No., Title of project
1
Table of contents - Page titles
2
Brief description of project. How would you proceed?
3-10 (may change)
Procedure (With pictures)
11
Mathematics used/involved
12
Conclusion/Result
13
List of resources (List of encyclopedia, websites, reference books, journals (etc.)
14
Acknowledgements
Project 1
Objectives
Useful Link:
Exploring Fibonacci 1. Fibonacci numbers are a sequence of numbers i.e 1, 1, 2, numbers. 3, 5, 8, 13, 21, 34.... The first number of the sequence is 1, the second number is 1, and text term is equal to the sum of the previous two numbers of the sequence itself.
http:/britton.disted.camosum. bc.ca/fibslide/jbfibslide.htm
http://ewaysmathematics.blog spot.com/search/label/Fibonac ci
Description
2. Write the next 20 terms of the sequence generated by it. 3. History of the mathematician who gave this concept. 4. Explore in nature the things that correspond to Fibonacci numbers with pictures.
213
For example : When counting the number of petals of a flower, it is most probable that they will correspond to one of the Fibonacci Numbers. It is seen that: (a) White calla lily has one petal (b) Euphorbia have two petals (c) Trillium have three petals (d)Columbine have 5 petals Explore more such examples with pictures from internet. (give atleast 8 examples) Project 2 Useful Link : http://ptril.tripod.com/
Exploring Pascal Triangle
Description (take upto 10 row) 1. Definition 2. History 3. How to construct it 4. Mention about the properties (a) The sum of the numbers in any row is 2n, when n is the number of the row. (b) Property related to prime number. (c) Hockey stick pattern (d) Fibonacci sequence located through Pascal triangle. 5. Make a model on Pascal triangle.
214
Project 3
Description
Useful Link :
Making 3D Snowflakes Make a model with project report having contents http://www.wikihow.com/Ma ke-a-3D-Paper-Snowflake (a) What is 3-D snowflake? (b) Its applications in daily life. (c) Mathematics involved in it. (d) Procedure of the model. Project 4 Useful Link :
Description Making platonic solids
http://mykhmsmathclass.blog spot.com/search/label/Platoni c%20Solids
(1) Introduction (2) Mention about 5 platonic solids and its properties. (3) History (4) Procedure of making Platonic solids. (5) Verify Euler's Formula for each of the solid.
Project 5
Objective
Useful Link :
Exploring Mathematics 1. Look around yourself. around us. In the house....
http://mykhmsmathclass.blog spot.com/search/label/Mathe matics%20Around%20us
Description
In the garden... In the market... In a bank... In the nature... 2. Click photographs using a digital camera/mobile and explore the hidden mathematics. 3. Click minimum 20 photographs.
215
Assessment Criteria: The project may be assessed on the following parameters.
216
l
Identification and statement of the project
l
Procedure/processess adopted
l
Write-up of the project
l
Interpretation of result
l
Viva
l
Presentation
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