LESSONPLAN
School : ………..
Subject : Mathematics
Gr ade/ semester : VIII/ II Number of Meeting : 5 Meetings Year of Lesson : 2011/ 2012
Standar d of Competency
Deter mine elements and par ts of cir cle and also measur ements Basic Competence
Deter mine length of tangent of tw o cir cles Indicator s
1. Sketch the gr aph of tangent pass a point and find the pr oper ty of tangent of cir cle 2. Deter mining length of tangent of cir cle
3. Under standing position of tw o cir cles
4. Under standing and sketch inter nal and exter nal common tangents of cir cles 5. Deter mining length of inter nal and exter nal common tangents of cir cle 6. Deter mining the minimum r ope r elate tw o cir cles
Lear ning Objectives :
After studying, students ar e able to:
1. Sketch the gr aph of tangent pass a point lie on cir cle and find the pr oper ty of tangent 2. Sketch the gr aph of tangent pass a point exter nal of cir cle and find the pr oper ty of
tangent
3. Deter mine length of tangent of cir cle
4. Under stand position of two cir cles and identify the cr iter ia for each position 5. Solve applied pr oblem about length of tangent and position of two cir cles 6. Under stand and sketch inter nal and exter nal common tangents of cir cles 7. Deter mine length of inter nal and exter nal common tangents of cir cle 8. Solve applied pr oblem about inter nal and exter nal common tangents 9. Deter mine the minimum r ope r elate tw o cir cles or mor e
Pr er equisite materials : Pythagor ean theor ems
Mater ials : tangent of cir cle
Headline of t opic
1. Tangent of cir cle int er sect t he cir cle at one and only one point on t he cir cle. 2. Pr oper t y of t angent of cir cle:
a. Each t angent per pendicular t o diamet er or r adius of cir cle b. Pass a point on cir cle, only a t angent can be cr eat ed c. Pass a point out side of cir cle, can cr eat e t wo t angent s.
3. Lengt h of t angent of cir cle is squar e r oot of differ ence of squar e dist ance of cent er t o a point out side t he cir cle and squar e of r adius of t hat cir cle
5. Lengt h of ext er nal common t angent is squar e r oot of differ ence of squar e of dist ance t wo cent er point of cir cle and squar e of differ ence of t wo r adii of cir cle
Time Allocations : 11 x 40 minutes
Method/ Appr oach/ Model : demonstr ation, dr ill and pr actice, constr uctivism, RME, PBI
Lear ning r esour ces :
Lear ning Activities Fir st meeting
Phase 1
Student prepares themselves to study. Teacher says learning
objectives and ask some realistic problem of circle’s tangent
Realistic problem of tangent of circle are bicycle’s chain,
pulley, bulldozer’s wheel, etc.
5 minutes
Phase 2
Exploration
Student remember definition of tangent
Tangent is line which is intersects circle exact one point
on the circle.
5 minutes
Phase 3
Student translate how to sketch tangent pass a point on a circle
5 minutes
Phase 4
Student follow teacher explanation about sketching tangent pass a
point on the circle
Sketch
tangentpass a point on a circle
10 minutes
Phase 5
Elaboration
Student does exercise page 238 latihan 1 no 1 and 2
10 minutes
Phase 6
Student translate how to sketch tangent pass a point outside the
circle
5 minutes
Phase 7
Student follow teacher explanation about sketching tangent pass a
point outside the circle
10 minutes
Phase 8
Student do exercise page 238 latihan 1 no 3 and 4
10 minutes
Phase 9
Confirmation
Conclude the property of tangent of a circle
Property of tangent of circle:
1.
Each tangent perpendicular to one diameter or one radius of
circle
2.
Pass a point on circle, only a tangent can be created
3.
Pass a point outside of circle, can create two tangents.
5 minutes
Phase 10
Conclude today lesson
5 minutes
Phase 11
Student is given homework
5 minutes
Rubric for exercise
Problem
Indicators
1 2 3 4
1
Have an idea to sketch tangent
Do the idea
2
Have an idea to sketch tangent
Do the idea
Finding property of tangent based on him/her picture
Rubric for problem sheet
Problem
Indicators
1 2 3 4
1
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
2
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
3
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
4
Finding given of the problem
Finding ask of problem
Know another topic suitable to solve problem
Can prepare solution of problem
Can solve problem
Check again each steps to find solution of problem
1= achieve the indicator less than 25%
2= achieve indicator between 25%-50%
Standard Competence
Determine elements and parts of circle and also measurement
Basic Competence
Determine length of tangent of two circles
Indicator
Determine length of tangent of a circle
LENGTH OF TANGENT OF
CIRCLE
NAME
:
…………..
CLASS
:
………….
Direction!
1.
Read summary topic first!
2.
Answer exercise based on structure problem solving steps that your teacher explains!
Summary of Topic
Tangent of circle is line which is intersects circle on one point at the circle.
Length of tangent of circle is square root of difference of square distance of center to a point
outside the circle and square of radius of that circle
Exercise
1.
Length of tangent of a circle is 24 cm, distance between center to a point outside of circle is
26 cm. Determine radius of circle!
Solve using structure problem solving steps
Given
Ask
Solve
2.
Look at the picture!
Given radius of a circle, r = 6 cm and length AB = 10 cm.
Determine length of tangent of circle and find another
tangent of circle pass point B!
Solve using structure problem solving steps
Given
Ask
Solve
3.
Determine the area of triangle POQ based on the following picture!
Solve using structure problem solving steps
Given
Ask
Solve
B
A O
Q
P
O 25 cm
4.
Two woods which are look like a circle is bundled with 144 cm rope. If radii of two roots are same, determine the radius!Given
Ask
Solve
Second Meeting
Phase 1
Student discuss their homework at Erlangga book page 61 no 1-5
15 minutes
Phase 2
Student prepares themselves for studying. Student knows learning
objectives of lesson today
Understanding position of two circles
5 minutes
Phase 3
Exploration
Student make figure of some position of two circle based on
Erlangga book page 62
No Pict ure
1.
2
.
3.
20 minutes
s r1
r2
r1
r2
s
s r1
4.
5.
6.
Phase 4
Student pays attention to teacher’s explanation. Teacher show
some realistic condition of position of two circles and guide
student to conclude position and characteristic of each example
that is given.
1.
Position of coin Rp 100 and Rp 500 if it is patched
over.
2.
Position of minute hand and hours hand of watch
5 minutes
Phase 5
Elaboration
Student give the name of each position and also its characteristic
No Pict ure Posit ion Caract erist ic
1. Int ersect ion r1 + r2 > s
2. Cont iguous
out side
r1 + r2 = s
3. Cont iguous
inside
r1 + r2 > s
30 minutes
s r1
r2
r1
r2
s
s r1
r2
s r1
r2
M r2
r1
N
s r1
4. Disjoint out side
r1 + r2 < s
5. Concent ric r1+r2=0
6. Disjoint
inside
r1 + r2 > s
Phase 6
Student do the problem sheet
10 minutes
Phase 7
Confirmation
Discuss the problem sheet
5 minutes
Phase 8
Student know their homework to translate how to sketch internal
and external common tangents
5 minutes
Assessment :
problem sheet
Third Meeting
Phase 1 Student review their knowledge about position of two circle
Show in power point
5’
Phase 2 Student prepares themselves for studying. Student knows learning objectives of lesson today
The students are able to:
1. Understanding internal and external common tangents 2. Sk etching internal and external common tangents
5’
Phase 3 Student translate how to sketch internal common tangents 10’
Phase 4 E xploration
Student sketching internal common tangents
No Steps Picture
1. Sketch a circle where center is P and the radius is r1, and then circle
with central point is Q and radius is r2.
25’
s r1
r2
M r2
r1
N
s r1
r2
P Q
r1 r
2. Join center of both circle
3. Sketch arc of circle with center point P and point Q with same radius. Both of arc intersect on point A and B.
4. Join point A and point B, so that its intersect line PQ on point C.
5. Sketch a circle with center point is point C and diameter CP.
6. Make an arc with radius r1+ r2 and
central point is point P at the top and at the bootom.
Intersection of these arc are H and I. Joint point P and H and also point P and I. Intersection of line PH and circle P is point D. Intersection of line PI and circle P is point E .
7. Make arc to circle Q with radius DQ and the central points are D
P Q
P Q
A
B
P Q
A
B C
P Q
A
B C
P Q
A
B C D
E
F
G
P Q
A
B C D
E
F
8. Joint point D and G and also point E and point F.
Line DG and E F are internal common tangents.
Phase 5 Student mention some realistic example of external common tangents
Phase 6 Student sketching external common tangent
No Steps Picture
1. Sketch circle with center point is point P and radius r1, then circle
where center is Q and radius r2.
2. Join both center of that circle.
3. Sketch arc of circle with center point on point P and point Q and have same radius. Both arc intersect on point A and point B.
4. Join point A and point B, so line AB intersect line PQ on point C.
5. Sketch a circle where center is C and radius CP.
20’
P Q
A
B C D
E
F
G
P Q
r1 r2
P
Q
P Q
A
B
P Q
A
B C
C A
B
6. Sketch a circle with center point is point P and radius r1-r2. This circle
intersect circle with central point is point C on point D and point E .
7. Join point P and point D so that intersect circle with cetral point is point P and radius r1 on point M.
After that join point P and point E , so that intersect circle with center point is point P with radius r1 on point N.
Name point K and point L for intersection of circle with the center is point C and point Q.
8. Join point M and point K, and
then join point N and point L. Line KM and line LN are external common tangent of circle with center points are point P and point Q.
Phase 7 E laboration
Students try to sketch internal and external common tangent for another circles
10’
Phase 8 Confirmation
Students conclude the lesson
5’
Performance Assessment
1.
Internal common tangents
No Steps Picture 0 1 2
1. Sketch a circle where center is P and the radius is r1, and then circle
with central point is Q and radius is r2.
P Q
A
B C D
E
K
L
P Q
A
B C D
E M
N
K
L
P Q
A
B C D
E M
N
P Q
r1 r
2. Join center of both circle
3. Sketch arc of circle with center point P and point Q with same radius. Both of arc intersect on point A and B.
4. Join point A and point B, so that its intersect line PQ on point C.
5. Sketch a circle with center point is point C and diameter CP.
6. Make an arc with radius r1+ r2 and
central point is point P at the top and at the bootom.
Intersection of these arc are H and I. Joint point P and H and also point P and I. Intersection of line PH and circle P is point D. Intersection of line PI and circle P is point E .
``````````````````````````````````````````````````````` ```````````````````````````````````````````````
7. Make arc to circle Q with radius DQ and the central points are D
P Q
P Q
A
B
P Q
A
B C
P Q
A
B C
P Q
A
B C D
E
F
G
P Q
A
B C D
E
F
8. Joint point D and G and also point E and point F.
Line DG and E F are internal common tangents.
2.
External common tangents
No Steps Picture 0 1 2
1. Sketch circle with center point is point P and radius r1, then circle
where center is Q and radius r2.
2. Join both center of that circle.
3. Sketch arc of circle with center point on point P and point Q and have same radius. Both arc intersect on point A and point B.
4. Join point A and point B, so line AB intersect line PQ on point C.
5. Sketch a circle where center is C and radius CP.
6. Sketch a circle with center point is point P and radius r1-r2. This circle
P Q
A
B C D
E
F
G
P
Q
r1 r2
P
Q
P
Q A
B
P Q
A
B C
P Q
A
B C D
E
C A
B
intersect circle with central point is point C on point D and point E .
7. Join point P and point D so that intersect circle with cetral point is point P and radius r1 on point M.
After that join point P and point E , so that intersect circle with center point is point P with radius r1 on point N.
Name point K and point L for intersection of circle with the center is point C and point Q.
8. Join point M and point K, and
then join point N and point L. Line KM and line LN are external common tangent of circle with center points are point P and point Q.
Note:
0= student don’t do the step
1= student do the step but not suitable 2= student do the step correct
For th meeting
Phase Activities Time
1. Students pr epar e themselves to study 5’
2. Explor ation
Students mention some object w hich r elated to inter nal and exter nal common tangents
5’
3. Students know the lear ning objectives of the lesson and the r ule of lesson for today
5’
4. Students r emind how to sketch inter nal and exter nal common tangent One of the students sketch the tangents on w hite/ blackboar d
5’
5. Elabor ation
Teacher br ing students to find r ight tr iangle on inter nal common tangent and dir ect students to use the concept of Pythagor ean theor em to deter mine the length of inner common tangents
K
L
P Q
A
B C D
E M
N
K
L
P Q
A
B C D
E M
6. By using same w ay, students find the length of exter nal common tangents
7. Students solve some pr oblem about inter nal and exter nal common tangents
Teacher help students to solve their pr oblem 8. Confir mation
Students discuss the most difficulties pr oblem on class discussion
9. Student conclude lesson today and know their homework
There are 2 outer common tangents if two circles are disjoint outside
L ength of outer common tangent is square root of difference of square of distance two center point of circle and square of difference of two radii of circle
Assessment of exercise
1. Radii of two circles are 20 cm and 4 cm. determine the distance between two central points
if known length of outer common tangents is 26 cm. (1 point)
Given Radii of two circles are 20 cm and 4 cm.
Length of outer common tangents is 26 cm
0.5 0.5
Ask Distance between two central points 1
Solve Suppose
Radius of fist circle = r1= 20 cm
Radius of second circle = r2 = 4 cm
Length of outer common tangents = t = 26 cm Distance between two center point = s
Length of outer common tangent is square root of difference of square of distance two center point of circle and square of difference of two radii of circle
Its mean
=
−
(−
)26 =
−
( 20−
4) 26 =−
( 16) 262= 2−
1622 = 262+ 162
2 = 676 + 256 2 = 932
=
√
932 s = 30.51
1
1
1
2. Given radii of two circles are 16 cm and 17 cm. Distance of two central points is 24 cm.
Calculate the length of outer common tangents! (1point)
Given Radii of two circles are 16 cm and 17 cm.
Distance of two central point is 24 cm
0.5 0.5
Ask Distance between two central points 1
Solve Suppose
Radius of fist circle = r1= 16 cm
Radius of second circle = r2 = 17 cm
Distance between two center point = s = 24 cm Length of outer common tangents = t
Length of outer common tangent is square root of difference of square of distance two center point of circle and square of difference of two radii of circle
Its mean
=
−
(−
)= 24
−
( 16−
17) = 24−
( 1)=
√
576−
1 =√
575 t = 23.91
1
1
1 1
1 1
3. Suppose radii of two circles are 12 cm and 5 cm. Distance of two central points is 20 cm.
Calculate the length of outer common tangents! (1 point)
Given Radii of two circles are 12 cm and 5 cm.
Distance of two central point is 20 cm
0.5 0.5
Ask Distance between two central points 1
Solve Suppose
Radius of fist circle = r1= 12 cm
Radius of second circle = r2 = 5 cm
Distance between two center point = s = 20 cm Length of outer common tangents = t
Length of outer common tangent is square root of difference of square of distance two center point of circle and square of difference of two radii of circle
Its mean
=
−
(−
)= 20
−
( 12−
5) = 20−
( 7)1
1
1
=
√
400−
49 =√
351 t = 18.71 1 Fifth meeting
Phase 1 E xploration
Student remember all concept of tangents of circle
Property of tangents of circle
1. E ach tangent perpendicular to one diameter or one radius of circle 2. Pass a point on circle, only a tangent can be created
3. Pass a point outside of circle, can create twotangents.
L ength of tangent of circle is square root of difference of square distance of center to a point outside the circle and square of radius of that circle
Position of two circle
No Picture Position Caracteristic
1. Intersection r1 + r2 > s
2. Contiguous
outside
r1 + r2 = s
3. Contiguous
inside
r1 + r2 > s
4. Disjoint outside r1 + r2 < s
5. Concentric r1+ r2= 0
15’
s r1
r2
r1
r2
s
s r1
r2
s r1
r2
M r2
r1
6. Disjoint inside r1 + r2 > s
There are 2 inner common tangents if two circles are disjoint outside
L ength of inner common tangents is square root of difference of square of distance two center point of circle and square of addition of two radii of circle
There are 2 outer common tangents if two circles are disjoint outside
L ength of outer common tangents is square root of difference of square of distance two center point of circle and square of difference of two radii of circle
Phase 2 Student is given double homework because no body collect it on time.
Phase 3 Student discuss their homework no 7
Dua buah lingk aran bersinggungan di luar. L ingk aran besar memilik i jari-jari x cm dan lingk aran yang lainnya berjari-jari 7 cm. Jik a panjang garis singgung persek utuan luarnya 30 cm, tentuk an panjang x dan luk is garis singgung persek utuan luar tersebut!
15’
Phase 4 Student prepares themselves for studying. Student knows learning objectives of lesson today
The students are able to:
1. Using concept of tangent of circle on solving the problem
2. Understanding the minimum rope relate two circles
5’
Phase 5 Student mention some realistic example of minimum rope relate two circles
Bundle of PV C, bundle of wood
5’
Phase 6 E laboration
Student discuss how to determine minimum rope related based on problem given
10’
Phase 7 Student concludes how to determine minimum rope related.
Student determines the minimum rope related by joint all central points of the object.
5’
Phase 8 Student solve the problem on Erlangga book page 20’
Phase 9 Confirmation
Student discuss the problem 76-77
10’
Phase 10 Student conclude lesson today and know their homework
Solving the minimum rope related by sketching the problem, and then joint each central points and make parallel line to the outer side.
5’
Assessment : assessment of homework
For each problem:
1. Student gets 3 points if they find given and ask from the problem 2. Student gets 7 points if their answer correct
s r1
