Season 1 • Episode AP06 • Using Geogebra to solve a problem

Season 1 • Episode AP06 • Using Geogebra to solve a problem

Using Geogebra to solve a problem

Season 1

Episode AP06 Time frame 1 period

Prerequisites :

Objectives :

•

Materials :

•

•

1 – Part A 20 mins

Using geogebra and working by pairs, Students have to draw a gure and to onjeture

the maximalvalue of anarea.

2 – Part B 20 mins

Studentshavetondout the funtioninvolved inthe problemandthentodrawit(using

geogebra) in order tond the maximal value of the area.

3 – Part C 20 mins

In this part, Students have to onrm their previous answers, this time using omputa-

tions.

Using Geogebra to solve a problem ^{Episode AP06}

Document ICT

In the gure below,

ABC

A

BC = 8

M N P Q

△ ABC

BM

x

in m.

The aim of this session isto nd the position of the point

M

the retangle

M N P Q

B N

A

P

C

M Q

x

Part A – Using Geogebra

1. Use Geogebra to draw the gure. Don't forget to display the area of the retangle

on the gure.

2. Movethepoint

M

BC

of

M N P Q

Part B – Using the graph of a function

1. What is the minimal value for

x

2. What is the maximal value for

x

3. What are the measures of the angles of

△ ABC

△ BM N

of the triangle

BM N

M N

x

Season 1 • Episode AP06 • Using Geogebra to solve a problem

4. Let

f

x

M N P Q

f (x) = 8x − 2x ²

5. UseGeogebratodrawthe graphof

f

f

of

x

6. Dedue the maximalarea of

M N P Q

M

reahed.

Part C – Using computation

In this part, let's admit that the area of

M N P Q

8x − 2x ²

where

x

BM

1. What is the sign ofthe expression

− 2(x − 2) ²

2. Expand the expression

−2( x − 2) ²

3. Usingthetwopreviousquestions,dedue themaximalvalue ofthe areaof

M N P Q

In the gure below,

ABC

A

BC = 8

M N P Q

△ ABC

BM

x

in m.

The aim of this session isto nd the position of the point

M

the retangle

M N P Q

B N

A

P

C

M Q

x

Part A – Using Geogebra

1. Use Geogebra to draw the gure. Don't forget to display the area of the retangle

on the gure.

2. Movethepoint

M

BC

of

M N P Q

The maximum area for

M N P Q

8

²

Part B – Using the graph of a function

1. What is the minimal value for

x

The minimalvalue for

x

0

x

2. What is the maximal value for

x

The maximal value for

x

4

M

midpoints.

3. What are the measures of the angles of

△ ABC

△ BM N

of the triangle

BM N

M N

x

△ ABC

A

∠BAC = 90 ^o

∠ABC =

∠BCA = 45 ^o

In

△ BM N

∠BM N = 90 ^o

M N P Q

∠M BN =

∠CBA = 45 ^o

△ BM N

B

We dedue that

M N = BC = x

Season 1 • Episode AP06 • Using Geogebra to solve a problem

4. Let

f

x

M N P Q

f (x) = 8x − 2x ² f ( x ) = M N × M Q = x × (8 − 2 x ) = 8 x − 2 x ²

5. UseGeogebratodrawthe graphof

f

f

of

x

Aording to the graph, the maximal value of

f

8

x = 2

6. Dedue the maximalarea of

M N P Q

M

reahed.

Thefuntion represents thearea sothe maximalvalue ofthe area seemstobe

8

²

BM = 2

Part C – Using computation

In this part, let's admit that the area of

M N P Q

8x − 2x ²

where

x

BM

1. What is the sign ofthe expression

−2( x − 2) ²

( x − 2) ²

−2

we dedue that

− 2(x − 2) ²

2. Expand the expression

− 2(x − 2) ²

− 2(x − 2) ² = − 2(x ² − 4x + 4) = − 2x ² + 8x − 8

3. Usingthetwopreviousquestions,dedue themaximalvalue ofthe areaof

M N P Q

Aording tothe rst question

− 2(x − 2) ²

− 2(x − 2) ² 6 0

Aordingtotheseondquestion,weget

− 2x ² +8x − 8 6 0

− 2x ² +8x 6 8

The lastinequationmeans that

f ( x )

8

In the gure below,

ABC

A

BC = 8

M N P Q

△ ABC

BM

x

in m.

The aim of this session isto nd the position of the point

M

the retangle

M N P Q

B N

A

P

C

M Q

x

Part A – Using Geogebra

1. Use Geogebra to draw the gure. Don't forget to display the area of the retangle

on the gure.

2. Movethepoint

M

BC

of

M N P Q

Part B – Using the graph of a function

1. What is the minimal value for

x

2. What is the maximal value for

x

3. What are the measures of the angles of

△ ABC

△ BM N

of the triangle

BM N

M N

x

4. Let

f

x

M N P Q

f ( x ) = 8 x − 2 x ²

5. UseGeogebratodrawthe graphof

f

f

of

x

6. Dedue the maximalarea of

M N P Q

M

reahed.

Part C – Using computation

In this part, let's admit that the area of

M N P Q

8 x − 2 x ²

where

x

BM

1. What is the sign ofthe expression

− 2(x − 2) ²

2. Expand the expression

−2( x − 2) ²

3. Usingthe twopreviousquestions, dedue the maximalvalue of the area of

M N P Q

and the position ofthe point

M