Mathematics activities Science activities

Comenius 2003/2006 Nobel projectComenius 2007/2009 SMILE project

Scuola Media Luigi StefaniniTeacher Graziano Scotto di Clemente GP Moroldo F. Tronchin

Geometrical activitiesThe geometrical image of the world

From drawing in pencil to 3D CAD A research on polyhedra (from Platon

to Euler) and the Internet resources An hypertext

The strands Give a variety of challenges and tools (physical objects,

pen, ruler and compass, DGS), to take account of the pupils’ different attitudes and learning styles( challenge, technology, support)

Stimulate interest, curiosity, motivation ( fun, motivation)

apply mathematical knowledge to real world situations ( real world)

Invite children to develop their own projects and to share each other ideas and knowledge (self guided learning, cooperation)

We have worked on: Drawings in axonometric projection many common solids:

prisms, pyramids, cones, cylinders…

Designing with a simple 3D CAD the same solids and some objects

Exploring the Internet resources to search interactive applications and informations about some particular polyhedra

Studying the geometrical properties of these polyhedra until to find the Euler’s rule

Solving traditional problems on volumes, area….

Solid geometry and “traditional aspects” in the Italian curriculum Volume, surfaces, area and other geometrical

properties (apothem…..and so on..)

The three-dimensional representation: the axonometric projection

Solid geometry and ICT: We are testing a simple 3D CAD to design polyedra and virtual objects.

The pupils have:1. To fix the shape of solid base2. To extrude the height3. To define the kind of material4. To choose the plane (x,y,z) of representation5. ….

Solid geometry and ICT

Some resources from Internet to display better the third dimension

• WRL animations• CABRI animations• Java animations• Two simple free software: Cartesio (CAD from Venice University) and Poly

Some stages in the history of geometry The Platonic solids: cube, tetrahedron,

octahedron, dodecahedron and icosahedron

The 13 Archimedean solids

The Euler’s rule (f = faces number; v= vertices number; s = edges

number)

To stimulate interest, curiosity, motivation: some links between geometry and art

Art and geometry during the Italian Renaissance

Escher Some images in the next slide

All these things are present in our hypertext, synthetically.

Leonardo da Vinci Giulia Escher

Dynamic geometry Search geometry in the real world

(for pupils 11-12 years old)

Using LOGO Using Cabri II Plus

Dynamic geometry and LOGO

Free Software: Logo for Microsoft Windows, Softronics Inc,

www.softronix.com

The Turtle’s motion and the construction of shapes:

Some cognitive aims - To be able to:

• build a sequence from a starting point to a foreseen result

• compare the concept of angle as change in direction with the Euclidean concept of angle as space between two lines starting from a common point

• compare the trace left by the turtle with the Euclidean concept of edge

• compare the idea of circle as “the limit” of regular polygons with the Euclidean definition of circle as set of all points in the plane at a given distance from a fixed point

Geometrical activitiesThe geometrical image of the world

Renew the “ancient” Calculemus by Leibniz and replace it by the new one Modelemus (J.M. Laborde cit. in Cabri2004 – Rome, September)

Using Cabri II Plus for analysing real objects

Motivation…Taking account of the pupils’ different attitudes and learning styles with a variety of challenges and tools ( the rule of technology ….)

Beginning from pupils’ curiosities and interestsOr

Beginning from real world situations

Developing their own projects and sharing each other ideas and knowledge (self guided learning, cooperation)

We are working on:•Looking around with a digital camera

•Modelling with Cabri: polygons, regular polygons, geometric transformations (symmetries, translations, rotations…)

Examples

•Searching the geometry inside real objects

What we expect The shift from a static knowledge to a

dynamic one

A better integration between perceptive-motor aspects and symbolic-representative aspects

A good training of geometric guessing

Geometrical constructions in triangles(Orthocenter, Centroid, Incenter and Circumcircle)

Teacher F Tronchin

Dynamic Geometry versus by Hand Geometry

PUPILS INVOLVED 45 pupils in their 7th school year (2^ Media) in autumn42 pupils in their 6th school year (1^ Media) in spring

  BY HAND

Objectives:

1. To develop manual ability by using simple school tools

2. To improve precision in geometrical drawing

3. To reinforce the knowledge of some basic geometric concepts like those of altitude of a triangle, angle bisector, midpoint of a segment and perpendicular bisector

BY USING THE COMPUTER WITH CABRI (Dynamic geometry)

Objectives1. To acquire the ability of using this software and its

specific terminology

2. To reinforce more dynamically the way to find the orthocenter, centroid, incenter and circumcircle of a triangle

3. To develop the ability of evaluating the correctness of the learned procedures

CONSIDERATIONS

The two methods develop different abilities. We don’t want to demonstrate which one is better or if we have to substitute the traditional geometry with an interactive software. By using both we want to develop complementary abilities and reinforce the knowledge of some geometric concepts.

In particular by using Cabri, because it gives the possibility to build, correct and modify figures rapidly, pupils have a more relaxed approach to geometry and their geometrical awareness probably increases as well

CHEKS

1. A systematic noting of pupils’ impressions of and reactions on the didactic activities

2. A final questionnaire to evaluate the pupils experience and highlight positive and negative aspects of the two learnng procedures

3. Possible final teachers’ remarks about the effectiveness of the learned methodology

Geometrical constructions in triangles (Orthocenter, Centroid, Incenter and Circumcircle)

By hand versus Cabri 

At the end of the activity the pupils have been asked to answer 15 questions, among them the following:

Which ones are the best Cabri features? � It is fun: 17 � It is fast: 15 �It allows me to move geometrical shapes and to understand how points do move: 16 �It makes me aware of what I am doing: 6 

And the worse? �It is automatic and does not make me think: 7 �It is boring: 1 �It is automatic and the next time I don’t remember how I have got there: 9 �There is nothing wrong with it: 5 

Which ones are the best things about the manual work? �The pleasure of building shapes a little at the time: 9 �The pleasure of working with pencil and manual tools: 6 �Because it is slow it makes me think how I am building it: 12 �None: 3

And the worse? �It is too slow: 6 �It makes me insecure because if I make a mistake I need to start all over: 13 �The drawing is never perfect: 7 �No particular defects:3

Between the best Cabri features, the pupils talked the most about how fun, fast and dynamic it is, while just few think that Cabri makes them more aware of what they are doing. This agrees with the main defect they found in the software i.e. that it is too mechanical.

Of the manual work it is appreciated most the fact that because it is slow it allows you to think about what you are doing. But in the same time it is difficult to correct the mistakes and it is imprecise. 

TEACHER’S CONSIDERATIONS ABOUT CABRI

1. Without any doubts all the pupils have fun with it and this reflects positively on the teaching and learning process

2. None of the pupils back off: i.e. the possibility this software gives to correct easily and fast the mistakes make all the students to give it a try. While in the manual work pupils who have difficulties may have hard time and get stuck right from the beginning

3. The fact that the software is “dynamic” betters the awareness especially among those that already works in a methodical way. But pupils who find it difficult to draw, or work in a mechanical way did not change much their way of working using the software

4. It was a must for me to reduce the mechanical side of the software. In order to do so, I constantly went over and over the basis of geometry, I made the pupils reflect on why to choose one way instead of another. I invited them to take measures of angles and sides to verify the validity of their supposition.

5. The dynamic look of the software does not have any equivalent in any hand made drawing and gives a big help in the learning process of the basis of geometry which are better imprinted in the memory of the pupils exactly thanks to the dynamic and visual look of it.

In CONCLUSION …

In agreement with the early statements and what the pupils say (almost all think the methodologies are both good even if some of them would have liked to work more with Cabri), it is my opinion that Cabri can not substitute the traditional geometry. Anyway especially thanks to its dynamic feature it represents an excellent tool to integrate the theoretical knowledge of geometry and its practical aspects. And because it allows the students to try several times it gives teacher the chances to have confrontations over the mistakes they make.  

Try and errorAn introduction to “Equations”

For 11-12 years old pupils

Try and error in a closed setAdditive pairs

x + y = N

N {0, 1, 2, 3….., N}

Which are the correct pairs ?

Try and error in a closed setMultiplicative pairs

x * y = N

N {0, 1, 2, 3….., N}

Which are the correct pairs ?

Try and error An introduction to “Equations”

For 13-14 years old pupils

Try and error: algebraic equationsStudy an equation with a single variable

as:

ax+ p = bx +q

where (a,b,p,q) are real numbers

Try and error: algebraic equations

Search the solution of the equation with

Cabri II plus

Science: hand–on activities We live in a “technological sea”: searching an initial

situation (from a problem in the daily life)

Questions… Prior ideas

Selection of questions

Science: hand–on activitiesInvestigations

Direct experimentations conceived and realised by children

Material realisation (search for a technical solution)

Direct or instrument-aided observation, with or without measurement

Science: hand–on activitiesDocumentary research and divulgation

Pupils like teachers

Construction of know-how, knowledge and cultural benchmarks

Science, hands–on activities

Investigations

Direct experimentations conceived and realised by children

Material realisation (search for a technical solution)

Direct or instrument-aided observation, with or without measurement

How a pile generates electricity

Science: hands–on activities

1. Which couples of metals ….

Some questions during the hands-on activity:

2. Acid or basic or neutral environment…..3. How to connect the circuit….

….trying and checking different hypothesis….

Science: hands–on activitiesAt the end of activities……the best solution.

Science: hands–on activitiesAims

1. to stimulate and raise pupils' curiosity about the technological world

2. to develop process skills, such as observing, questioning, hypothesizing, predicting, investigating, interpreting, and communicating, that play an essential role in helping children develop a scientific way of thinking

3. to promote a direct, active participation of the pupils4. to improve the children’s ability to work independently 5. to take account of the pupils’ different attitudes and learning

styles 6. to create a positive effect on the attitudes the pupils have for

science

Science: hand–on activitiesFrom the mobile – battery

To the lemon/tomato – batteries

An example of activity

Science: hand–on activities and mathematics

Slowbicycle race

Too slow!

This is an activity in which students (11-13 years old) compute the speed at which they rode a bicycle.

Objectives:In this activity students will:

work in teams ride a bicycle as slowly as possible use a stopwatch to measure time record data use multiplication and addition to determine team’s total distance and time use division to calculate team’s average speed compare average speed with other teams to determine which team was the slowest create a bar graph to depict each team’s average speed

Questions to Ask:

1. What was the most difficult part about riding the bike?2. What happens to the speed as the time increases?3. What happens to the speed as the time decreases?

Travel Book Activities: Writing About a Trip

Slowbicycle race

Science activitiesSome experiences on: Electrostatic Elementary electric

circuits Elementary

electromagnetism

With the use of everyday materials

Science activities

Practical executions with:1. A representation, either

schematic or axonometric 2. A design with the proper

dimensions, writing a list of the necessary materials…..

At last the construction and the testing

Science activitiesUsing of ICT to communicate the results to

other pupils:

Short videos hypertext

Dynamic Geometry versus by Hand Geometry

Teacher F. Tronchin

Science: Hands-on Activities in Physics

Teacher GP Moroldo

What factors (mass, length, angle

of oscillation, shape…) affect the

period T of oscillation of a

pendulum?

THE PENDULUM

To answer this question it was important to...

1) …conduct tests always on

small oscillations;2) …use heavy and dense objects

in order to make friction and resistance negligible;

3) …vary one factor at a time.

Pendula with different masses

but the same shape and the

same length have allowed us to

exclude that the mass is a significant variable in

determining T.

The sample weights for scales were o.k.

... but lead fishing weights were our favorites!

We easily verified that length is the most important variable: the change of T is evident!

…the fa

stest

→

…the slowest →

The shortest…

The longest…

MAIN DIFFICULTIES:

1. It’s hard to catch the initial and final time of an oscillation

2. When the length of the pendulum is short, the period is too brief to observe!

SOLUTIONS: time several consecutive oscillations (usually 10) five or six people perform the same test in order to

make an average of the times (possibly discarding “wild” data).

Andrea: 2.61Alex: 2.60Sara: 2.63Luca: 2.59

Francesk: 2.64Gucci: 2.66

G.Paolo: 2.81(discarded)

The arithmetic mean of acceptable values is

2.62 s and it will be used in subsuquent data

processing…

For example, when L = 1,70 m we timed (s)

RESULTS The results of our experiments are shown in the table on the right.

It’s clear that when L increases, T also increases… but which mathematical function can represent this growth? For example, can we predict the period of a pendulum 3.5 m long? (our ceilings are not high enough!)

L (m) T (s)0.00 0.00

0.10 0.71

0.16 0.81

0.18 0.83

0.21 0.93

0.23 0.97

0.32 1.12

0.53 1.45

0.78 1.78

0.86 1.84

1.09 2.08

1.14 2.16

1.39 2.36

1.70 2.62

1.72 2.63

1.79 2.71

1.91 2.77

1.94 2.78

2.08 2.89

2.20 2.98

2.42 3.14

2.76 3.30

3.00 3.46

The pair (0, 0) was actually added by us becouse on

decreasing of the length the period T becomes

smaller and smaller…

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

L (m)

T (s)

The couples (L ;T) have been analyzed by an appropriate

software(*) which recognises the mathematical function that

represents growth. (3.5 m ; 3.7 s)

(*) Graph 4.3 http://www.padowan.dk/graph/

We can forecast, for example, that a pendulum 3.5 m long will have a period T of about 3.7 s

L (m) T2 (s2)0.00 0.000.10 0.500.16 0.660.18 0.690.21 0.860.23 0.940.32 1.250.53 2.100.78 3.170.86 3.391.09 4.331.14 4.671.39 5.571.70 6.861.72 6.921.79 7.341.91 7.671.94 7.732.08 8.352.20 8.882.42 9.862.76 10.893.00 11.97

We obtained a line that describes how T changes when L changes.

We are used to study mainly straigth lines, some examples of hyperbola and of parabola… but this is so different!

… we had a nice surprise when

we tried to discover how the length determins

the squared period!

0 1 2 30

1

2

3

4

5

6

7

8

9

10

11

12

L (m)

T2

We finally found something familiar: we recognised that T2

is directly proportional to L and the constant of proportionality

is 4 T2 = 4.L

(0 ; 0)

(1 ; 4)

(2 ; 8)

(3 ; 12)

(0.5 ; 2)

(1.5 ; 6)

Our work concerned only a small part of what pendulum

studies can suggest…

SOME IDEA ABOUT WORKING WITH PENDULUM…

Science: CO2 – an environtmental problemWhat can we study about CO2 in a

school lab, and how?

CO2 and living beings CO2 and rocks

CO2 and human activities

How these different points are linked between them?

Science: CO2 – and living beings

Experiences on:

respirations fermentations

photosyntesis

Verifying the production or the consume of CO2 in different living beings:•Human beings•Yeasts•Green plants

Beer yeasts, used for home pizza

Science: CO2 – and rocksExperiences on the

natural deposits of CO2:

From the gas to the rock

From the rock to the gas

Science: CO2 – and human activities The chemistry of combustion

Global warming – greenhouse effects – CO2……a ” carbon footprint calculator “

A website to analyse our CO2 emissions

http://www.safeclimate.net/calculator/index.php

The first input

The second input about our home transportation

The answer by the CO2 calculator for an input

1. Is it possible to use the CO2 calculator in different classes?

2. What is the perception of the climate changing in the families (interviewing old people, collecting old photos…)

3. What are the data on the climate changing in the different countries?

Science: CO2 – and human activities

An hypothesis of cooperation between classes:

using a Skype conference to describe their researches to another class.

Science: CO2 – and human activities