The circle

Maths course exercises

Vittoria International School – Torino

The circle

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Politiche Linguistiche della Commissione Europea 

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a cura di Serenella Iacino

The circle

Indice modulo clilStrategies – Before•Prerequisites•Linking to Previous Knowledge and Predicting•Glossary Italian/English

Strategies – During• Video• Keywords riferite al video attraverso esercitazioni mirate • Conceptual Map

Strategies – After• Exercises: - Multiple Choice - Matching - True / False - Cloze or Completion - Flow Chart - Think and Discuss

• Summary and Summary Questions

• Web References di approfondimento come input interattivi per test orali e scritti e per esercitazioni basate sul Problem Solving

Answer sheets

The circle

The circle

Maths

1

the prerequisites are

Equation of first degree Equation of second degree

System of first degree System of second degree System of forth degree Equation of a straight line

Equation of a set of straight line

Strategies BeforePrerequisities

The circle

Strategies beforeLinking to Previous Knowledge and Predicting

1. Do you know the definition of Geometric Locus?

2. How many conditions do you need to solve a system of equations?

3. Do you know the equation of a set of straight lines passing through a point ?

4. Are you able to solve a system of first, second, or fourth degree?

5. When does a straight line pass through a point

P ( x ; y )P ( x ; y )

P ( x ; y )P ( x ; y ) ?? P PP P

P PP P

The circle

Strategies Before

Glossary Italian/English

asse - axis asse radicale - radical axis centro - centre circonferenza - circle circonferenza concentrica - concentric circle circonferenza degenere - degenerate circle coefficiente angolare - angle coefficient combinazione lineare - linear combination conica - conic section curva - curve determinante - determinant discriminante - discriminant distanze - distance equazione - equation

The circle

Strategies Before

Glossary Italian/English

esterna - external fascio di circonferenze - set of circles luogo geometrico - geometric locus punti base - base points Radici coincidenti - coincident roots radici distinte - distinct roots radici immaginarie - imaginary roots raggio - radius retta - straight line retta normale – normal line secante - secant tangente - tangent

The circle

• angle coefficient • axis • base points • centre • circle • set of circles• coincident roots • concentric• conic section • curve • degenerate circle • determinant • discriminant • distance

• distinct roots • equation • external • geometric locus• imaginary roots • linear combination • normal line• radical axis• radius• secant• straight line • tangent

Strategies DuringKeywords

The circle

Key Words

1. Circle the odd one out Geometric locus, centre, radius, secant, tangent, external,

straight line, triangle, equation, discriminant, radica axis, function, base points.

2. Circle the odd one out Curve, conic section, determinant, distance, axis, coincident

roots, imaginary roots, distinct, sin function, normal line, set of circles, angle coefficient, linear combination, degenerate circle, concentric.

Strategies DuringKeywords

The circle

The circle

Set of circles

radius

another circle

constantsa,b,c

defined by

Depends on

straight line

base points

In whichinside

secant

Depends on

external

tangent

Strategies DuringConceptual Map

center point Pradical axisexternal tangent secant

Complete the conceptual map using the following words:

The circle

Strategies After

What is the radius of the circle having this equation?

x² + y² – 4 x – 2 y – 3 = 0x² + y² – 4 x – 2 y – 3 = 0

Multiple Choice

22 22 22 4422

The circle

Which of the following equations doesn’t represent a circle?

x² + y² – 2 x – 4 y – 7 = 0x² + y² – 2 x – 4 y – 7 = 0

Strategies AfterMultiple Choice

x² + y² – 2 x – 4 y – 2 = 0x² + y² – 2 x – 4 y – 2 = 0

x² + y² – 2 x – 4 y + 7 = 0x² + y² – 2 x – 4 y + 7 = 0

x² + y² – 2 x – 4 y + 3 = 0x² + y² – 2 x – 4 y + 3 = 0

The circle

What is the radius of the circle having centre C (3;2) and

passing through the point P(2;3)?

22 55 33None of None of the abovethe above

Strategies AfterMultiple Choice

The circle

a – b – c = 2a – b – c = 2

Which of the following conditions expresses the passage of the

circle x² + y² + a x + b y + c = 0 x² + y² + a x + b y + c = 0 through the point P (1; - 1)?

a – b – c = - 2a – b – c = - 2

a + b + c = - 2a + b + c = - 2

a – b + c = - 2a – b + c = - 2

Strategies AfterMultiple Choice

The circle

Strategies AfterMultiple Choice

Given two circles having the equations

and the radical axis is the straight line:

x² + y² + 2x - 2y - 4 = 0x² + y² + 2x - 2y - 4 = 0

x² + y² - 2x - 4 = 0,x² + y² - 2x - 4 = 0,

X + 3Y = 0X + 3Y = 0

2X – Y= 02X – Y= 0

2X - Y + 1= 02X - Y + 1= 0

It doesn’t existIt doesn’t exist

The circle

Match the constants a, b, c, and the pictures

Strategies AfterMatching

a b = 0b = 0CCCC

CC

CC

CCCC

1 2 3

4 5 6

b c = 0c = 0

c a = 0a = 0

d a = b = 0a = b = 0

e a = c = 0a = c = 0

f b = c = 0b = c = 0

The circle

Match the equations and the pictures

1

CC

CC

CCA B C

x² + y² - 2x + 2y - 8 = 0x² + y² - 2x + 2y - 8 = 0

2 2x² + 2y² - 4x + y - 6 = 02x² + 2y² - 4x + y - 6 = 0

3 2x² + 2y² - 2x - 7y - 4 = 02x² + 2y² - 2x - 7y - 4 = 0

44

-1-1 22

22

-2-2 44

-4-4-2-2

-1-1 33

Strategies AfterMatching

The circle

if the distance between the centres is greater than the sum of the radii

CC C’C’

if the distance between the centres is equal to the sum of the radii

CC C’C’

if the distance between the centres is less than the sum of the radii

CC C’C’

Link the sentence to the right picture

A

B

C

1

3

2

Strategies After

Matching

The circle

• If a circle passes through two points A and B, its centre belongs to the perpendicular straight line to AB and passes through the middle-point of AB

Strategies AfterTrue/False

• The equation represents for each a = 0 a circle having the centre on the Y axis.

x² + y² + a x = 0x² + y² + a x = 0

T F

FT

The circle

• If the distance of a straight line from the centre of the circle having equation is 3, then the straight line is a tangent to the circle.

• If a straight line r is a tangent to a circle in the point P, then the centre of the circle belongs to the straight line passing through the point P and perpendicular to r.

x² + y² - 4x - 5 = 0x² + y² - 4x - 5 = 0

T

T

F

F

Strategies AfterTrue/False

The circle

• The set of circles, generated by circles and has two base points

• If two circles are tangent within then the sum of their radii equals the distance between the centres

x² + y² = 4x² + y² = 4

x² + y² - 4x = 0x² + y² - 4x = 0

• The radical axis of two esternal circles is not perpendicular to the line joining their centres

T F

T

T F

F

Strategies AfterTrue/False

The circle

Complete these sentences

• A straight-line is secant to the circle if……………………………………………………………………………………………………………………………………….………………………………………………………………………………….

• In a set of non concentric circles, the radical axis is considered …………………………………………………………………………………………………………………………………………..………………….………………….………………………………………….

• The base points of the set of circles are points for which ………………………………………………………………………………………………………………………………………………………………………………………………………………………………

Strategies AfterCompletion Exercise

The circle

Complete these sentences

• If the straight line having equation is tangent to the circle having equation then the ………………………. of the equation solving the system between the circle and the line is equal to ……………………………………………………………………………………………………………………………………………………….………………….…………………………………………………

• If the distance between the centres of two circles is greater than the sum of the radii, then the circles ………………….………………….……………………………………………………………………………………………………………………………………………………………………………………..

y = mx + qy = mx + q

x² + y² + ax + by + c = 0x² + y² + ax + by + c = 0

Strategies After

Completion Exercise

The circle

Strategies AfterFlow Chart

Complete the flow chart in order to determine the equation of a straight line tangent to a circle in its point

start

end

P ( x ; y ) P ( x ; y ) P PP P

Set of straight lines

Discriminant is equal to 0

Equation of the circle

Defining a system

Equation of the 2nd degree

The circle

Strategies AfterThink and Discuss

The following activity can be performed in a written or oral form. The teacher will choose the modality, depending on the ability (writing or speaking ) that needs to be developed.

The contexts in which the task will be presented to the students are:

A) The student is writing an article about the use of the circle in the modern Architecture;B) The student is preparing for an interview on a local TV about the use of the circle in the

Architecture of the Renaissance.

The student should:

1) Write an article about the use of the circle in the roman buildings: the Pantheon;

2) Prepare the article or the debate, outlining the main points of the argument, on the basis of what has been studied;

3) If the written activity is the modality chosen by the teacher, the student should provide a written article, indicating the target of readers to whom the article is addressed and the type of magazine / newspaper / school magazine where the article would be published;

4) If the oral activity is the modality chosen by the teacher, the student should present his point of view on the topics to the whole class and a debate could start at the end of his presentation.

The circle

Strategies AfterSummary

The circle is a conic section and is defined as the geometric locus of the points P of the Cartesian plane which are equidistant from a fixed point called “the centre”. In its equation there are three constants a, b, c and if we want to find the equation of a circle we need three independent conditions, that is one condition for each constant: for example the coordinates of the centre, the radius or the coordinates of a point P for which the circle passes through.According to the values of the constants a, b, c we can have particular circles with the centre at the origin, or on the x axis, or on the y axis, or even passing through the origin.The position of a straight line in relation to a circle may be secant, tangent or external, and from an algebraic point of view we must solve a system of equations, one for the circle, one for the straight line; from this system we obtain an equation of second degree, from which, if we have the straight line secant, tangent or external.

Δ > 0, Δ = 0, Δ < 0Δ > 0, Δ = 0, Δ < 0

The circle

The tangents to a circle from a given point P can be real, coincident or imaginary if the point P is outside the circle, on the circumference or within the circle.The relative position of two circles is external, tangent from the outside, secant, tangent within, inside, if the distance between their centres is greater than the sum of the radii, equal to the sum of the radii, less than sum of the radii, equal to the difference between the radii and less than the difference between the radii.From an algebraic point of view we must solve the system of equations of the two circles; from this system we obtain the equation of a straight line called Radical Axis of two circles that is perpendicular to the line joining their centres.From a linear combination between the equations of two circles we obtain the equation of a set of circles that pass through the common points to the circles, called Base Points.The radical axis is considered as a particular circle of this set, having an infinite radius, and is called Degenerate Circle.

The circle

Strategies AfterSummary Questions

1. How do you define the circle?2. How many constants do you find in its equation?3. How many conditions do you need to determine its equation?4. Which is the position of a straight line in relation to a circle?5. What do you solve to determine the equation of a tangent to a circle?6. Which is the relative position of two circles?7. How do you obtain the equation of a set of circles?8. What are the Base Points?9. What is the Radical Axis?

Write a short abstract of the summary ( max 150 words )highlighting the main points of the video.

The circle

Solve the following problems on the circle:1) Study the set of circles of equation and find k so that:• the circle passes through the point • the circle has centre equal to • the circle is tangent to the line of equation x = 1

2) Given the equations of two circles :

• write the equation of the set of circles determined by them;• find the base points, and the radical axis;• find the circles of the set having radius equal to 5.

x² + y² + 4 x + k y – 5 – 3 k= 0x² + y² + 4 x + k y – 5 – 3 k= 0

P (1;1)

x² + y² – 3 x + y + 2 = 0 ,x² + y² – 3 x + y + 2 = 0 ,

x² + y² = 1x² + y² = 1

C (-2;3)

Activities Based on Problem Solving

13

The circle

http://www.visualmathlearning.com/Website designed to provide parents and classroom teachers with the means to better employ visual imagery.

http://www.videomathtutor.com/This site is intended to help students from secondary school through college. Teachers, other tutors, and parents will also find this site to be very useful.

http://www.aaaknow.com/AAA Math features a comprehensive set of interactive arithmetic lessons. Unlimited practice is available on each topic which allows thorough mastery of the concepts.

http://www.icoachmath.com/iCoachMath offers students the opportunity to engage in meaningful Math learning that leads students to expand their knowledge on their own and explore in detail different areas of mathematics.

Web References

The circle

The circle

Set of circles

radius

Another circle

Constants a,b,c

Defined by

center

Point P

Depends on

Straight line

Base points

Radical axis

In which

inside

secant

external

tangent

Depends on

secantexternal

tangent

Answer sheetsConceptual Map

The circle

Multiple choice:2 2 ;

Matching:1c; 2e; 3d; 4f; 5b; 6aA3; B1; C2A3; B1; C2

True or false:T; F; T; T; T; F; F

Completion exercise:“It meets the circle in two real different points”;“A degenerate circle”;“All circles pass through”;“Discriminant is zero”;“Are external”

Problem solving:1) The circles of the set are tangent; k=1/2; k=-6; k=0 e k=12

x² + y² – 2 x – 4 y + 7 = 0 ; x² + y² – 2 x – 4 y + 7 = 0 ; 2 ; a – b + c = - 2 ;a – b + c = - 2 ; 2x – y= 02x – y= 0

2) (1+k) x² + (1+k) y² – 3 x + y + 2 - k = 02) (1+k) x² + (1+k) y² – 3 x + y + 2 - k = 0; ; A(1;0) B(4/5;-3/5);A(1;0) B(4/5;-3/5); 3x-y-3=0; k=7/6 e k=7/83x-y-3=0; k=7/6 e k=7/8

Answer Sheets

The circle

Complete the flow chart in order to determine the equation of a straight line tangent to a circle in its point

start

end

P ( x ; y ) P ( x ; y ) P PP P

Making System

Equation of the 2nd degree

Discriminant is equal to 0

Set of straight lines

Equation of the circle

Answer SheetsFlow Chart Solution

The circle

end