Fermi book

FERMI-Questions

Directory: 1. Task: size of Abdenauer Statue (DE) 2. Solution: Adenauer Statue (DE) 3. Solution: Adenauer Statue (CY) 4. Solution: Adenauer Statue (LV) 5. Task: Art Nouveau in Riga 1 6. Solution: Art Nouveau in Riga 1 (CY) 7. Solution Art Nouveau in Riga 1 (DE) 8. Task: Art Nouveau in Riga 2 (DE) 9. Solution: Art Nouveau in Riga 2 (CY) 10.Solution: Art Nouveau in Riga 2 (DE) 11.Task: Churches of Riga (LV) 12.Task: Laima Clock in Riga (LV) 13.Task: water tap (DE) 14.Solution: water tap (DE) 15.Task: Football chain (DE) 16.Solution: Football chain (DE) 17.Task: football field (DE) 18.Solution: football field (LV) 19.Solution: football field (LV) 20.Task: ice hockey (DE) 21.Solution: ice hockey (LV) 22.Solution: ice hockey (DE) 23.Task: football shirt (DE) 24.Solution: football shirt (SP) 25.Task: basket ball (DE) 26.Solution: baket ball (DE) 27.Fermi-tasks in grade 5- lesson plan and solutions (in German)

The head of Adenauer , the first German chancellor

This is a photo from a monument of Adenauer who was the first chancellor in Germany from 1949 until 1963.

How long will the statue be if the person is represented in full size from head to toe? Explain how you got to your result!

The head of Adenauer , the first German chancellor

We found in Wikipedia that an average person, is generally 7-and-a-half heads tall (including the head). So if the statue was represented in full size it would be 7,5 times the girl’s height .

From Wikipedia :

The head of Adenauer, the first German chancellor

This is a photo from a monument of Adenauer who was the first chancellor in Germany from 1949 until 1963.

How long will the statue be if the person is respresented in full size from head to toe? Explain how you got to your result!

After my manipulations with ruler, I was able to measure the height of the girl and the statue in the picture.

Relying to the data from the internet, the maximum height of girl between 6-8 ages is 124.8 cm.

Again, relying to the data from the internet the human body and head proportion is 1:8.

148.3*8=1186.4 m whole statue’s ody height

Art Nouveau in houses of old RIGA

This is a photo from a decoration of Art Nouveau style in Strelnieku Street 9 in Riga.

What about the height if it were a statue that represents figure of the person in full length?

Art Nouveau in houses of old RIGA

An average person is 8 heads tall (including the head). And as we can see the head of the statue is approximately as high as the window is soif it were a statue that represents figure of the person in full length it would be around 8 times the window’s height.

Art nouveau in houses of old RIGA

We used Wikipedia to find out relations between head and total height (body proportions) :

http://en.wikipedia.org/wiki/Body_proportions

Art nouveau decoration at houses in the old part Riga

This is a photo from a decoration of Art Nouveau style at a house in the historical of Riga.

What about the height if it were a statue that represents figure of the person in full length?

Art nouveau decoration at houses in the old part Riga

If it were a statue that represents figure of the person in full length it would be around 8 times the balcony’s height, because the head has the same length as the balcony and an average person is 8 heads high, including the head.

Art nouveau in RIGA What about the total length of this figure?

On the picture you can see, that the balcony is quite as high as the head. We assumed that the height of the balcony ois about 1m. The heas is about a quarter higher than the balcony (ďƒ 1,25m). The usual body of a person is 8 times taller than the head. ďƒ¨ 8x1,25m=10m

The whole body of the head would be about 10m.

Private “e ondary “ hool Klasika , Riga, Latvia

Proje t More Meaningful Math

FERMI TASK Churches of Riga Rich in history and architecture Riga- the European capital of culture boasts some of

the most exquisite churches situated in the center of Riga - the Old Town.

“t. Ja o ’s Chur h The smallest church in the Old Town is St. Jacob`s Church. It was built in the beginning of the 13th century and is well known for its tall spire (86 m). It has the best preserved pyramidal shape of all the towers, characteristic to medieval times.

“t. Peter’s Chur h The highest church in the Old Town is St. Peter`s Church built in 1209. After the last reconstruction its tower is 123.25 m high and is a metal construction. There is a built - in lift which takes you up to a height of 72 m. In shape the tower is similar to St. Jacobs tower with a Gothic style pyramidal peak.

©Kalvis “turitis, “tudent of 8th Grade, Math Teacher Inara Vasilevska

72 m

Private “e ondary “ hool Klasika , Riga, Latvia

Proje t More Meaningful Math

FERMI TASK Suppose that the pyramidal shape of these two towers corresponds to a regular hexagonal pyramid and the side length of the base hexagon is 2 m.

Assess how high the peak of St. Peter`s Church is and calculate the number of copper plates required to cover the surfaces of both towers.

©Kalvis “turitis, “tudent of 8th Grade, Math Teacher Inara Vasilevska

Fermi question: Laima clock Have you ever had strange ideas? Looking at the Laima clock just remember tasty products of the factory Laima. Many people like to feast on sweets “Piena lasite". The more goodies, the better! What do you think, how many “Piena lasite” boxes can fit in that part of the Laima clock, where letters of the name of this factory are placed?

This part has the form of the regular prism (base - square).

SOLUTION:

ANSWER: ____________

I ter atio al tea

„water“: task 3

I think, of it as our pri ate little lake! cartoon

a) A dripping ater tap can aiste up to l of ater a day – can this be right? Make an experiment, assumptions that you explain and calculate the waste of water for a month, a year. Does your calculation fit? b) The graphic right is from the internet (link). Are these data compatible to your calculation? c) How can you visualize the amount of waisted water a year of one water tap? d) How much cost the waisted water in your school?

Questions like in task a) are called FERMI-QUESTIONS. If you are interested have a look at this webpage.

International team "water" Task 1

1. This graph describes the water level in a bath tube relating to the time on the x-axis. Tell a story that fits to the graph.

2. Draw a new graph and invent an appropriate, funny story that has to do with it.

Solution from Lara (Germany): 1.Today, Johny wanted to use his bath tube again, so he let in the water.

This took a long time and he got bored. Than he took a bucket with water and filled the bath tube with it. After that he went into it and enjoyed the warm water. But it soon got colder and he let some water out and wanted some new, warmer water. He tourned the Faucet on, but nothing came out.

He stood up to keep the bucket and some warm water. Suddently he recodnized that it was cold outside the water, so he got in the bath tube again and tried to fill the bucket. It worked, but when he wanted to distribute the bucket, it suddently fell down into the bath tube. Johny didn't care and relaxed a bit. After some time he put the bucket out again and drained the water. Than he stood up and waited for the water to drain completly.

2.

In the evening Johny's sister Margret wanted to use the bath tube too, so she told her brother to let in the water. Half an hour later, Margret came into the room and screamed: the whole floor was covered with water. She had to remove all the water with Johny's help. After that she went into the bath tube and jumped out of it again: it was very cold. Margret had to distribute and to fill the bath tube another time. Than she went into the water and relaxed. Later she heard loud noises from Johny's room. Margret got out of the bath tube and went into her brother's room. After a lot of angry words she turned around, went straight into her bed and fell asleep.

International group „sports“: Activity 1a

picture

World Football Championship takes place from 12 of June until 13 of July 2014 in Bresil.

How many kilometers will roll a ball in total a ball that is used for one match? How many kilometers will it be in total for the whole championship?

Explain how you get to your result. Results of different person may be rather different but the most important thing is to explain e.g. assumptions that you have made and data that you have used order to calculate and answer the question.

Questions like these are called FERMI-QUESTIONS. If you are interested have a look at this webpage.

Evaluation of this task: Answer the following questions in your solution.

1= I totally agree

5 =I don’t agree at all

1. It was a motivating task for me.

2. It is interesting because it is related to real life.

You additional comment to this task:

The result of the first exercise is 5,076 km. The result of the second is 324,864 km.

First I had searching in the internet the circumfence from a football (47cm). I know, that a game have 90 minutes and 90 minutes have 5400 seconds. If you assume that a ball has two turns in one second, it travels a distance of 94 cm. If you calculaIng this: 94 cm * 5400 seconds, you'll have the number of cenImeters for one game. If you convert this in kilometers, you´ll have the result. This is 5,076 km.

For the second quesIon I had look in the internet for the number of games in a championchip. That are 64. Then you have to calculate this: 5,076 km * 64= 324,846 km.

Answer: A football covers a distance of about 325 km in one game.

International group „sports“: Activity 1e World record: A football field full of balls

A german company have got 2006 a world record. A staff of 320 people coverd the whole football field in the stadium of Borussia Dortmund with footballs.

1. How many footballs did they need for this? Explain your assumptions and calculations.

2. How many trucks were needed to bring the balls to the stadium?

3. If you look from above. What about the percentage of the field that was not covered by balls?

Evaluation of this task: Answer the following questions in your solution.

1= I totally agree

5 =I don’t agree at all

1. It was a motivating task for me.

2. It is interesting because it is related to real life.

You additional comment to this task:

Team “port Vladislav Khrushch World record: A football field full of balls.

A German company have got 2006 a world record. A staff of 320 people covered the whole football field in the stadium of Borussia Dortmund with footballs.

1. How many footballs did they need for this? Explain your assumptions and calculations.

Average size of a football field=100 m × 75 m =7500 �

Average diameter of a ball=22 cm Radius=11 cm

We do not need area of sphere(ball),we just need area of circle

Circle Area =

π • r² = ¼ • π • d²

π =3.14 s=11 x 11 x 3.14=380 �� =0.038 �

Amount of balls= Amount of balls=

.

�

�

�

=197 368 balls (but it is solution if the

ball is like a square)

b=d=75 m

diameter=0.22 m

a=c=100 m Amount of balls=balls in horizontal line × balls in vertical line Amount of the balls= balls

�

This way is more correct

x

�

=

.

x

.

=154 943

2. How many trucks were needed to bring the balls to the stadium?

Container of truck V=axbxh Average length of container a=5.2 m Average width of container b=1.85 m Average high of container

h=1.41 m

V=5.2 x 1.85 x 1.41=13.52 � if the balls is inflated:

• π • r³

We need Sphere Volume

V= x3.14x11x11x11=5 577 �� =0.005577 �

How many balls in 1 truck=

�

�

ℎ

ℎ

�

=

.

.

=2 424 balls

21,4% free space How many balls in 1 truck=2 424 -518=1906 Amount of trucks=

�

=

balls we need 82 trucks

1

9

=81.3 trucks ,but for

if the balls is not inflated: Thickness of the ball=2 cm=0.02m How many balls in 1 truck=

�

ℎ

�

ℎ

�T

�

=

21,4% free space

.

.

� .

=18 778 balls

18 778 – 4 018=14 760

Amount of trucks= balls we need 11 trucks

�

=

=10,4 trucks ,but for

Better to transport balls, if the ball is not inflated

3. If you look from above. What about the percentage of the field that was not covered by balls?

DI=DB=r =11 cm ��� =��� +��� =242 IB=15.5 cm

We have such free space near 4 balls, and so how many such squares we have =

=49 342 squares

S of square=15.15 x 15.15=240.2 ��

S of all free spaces=0.02402 x 49 342=1 185.2 � 1%

----------

X

----------

X=

.

=15.8 %

75 �

1 185.2 �

If you look from above about 15.8 % of the field that was not covered by balls!

Second way: So we have

Amount of balls=

.

=197 368 balls (but it is solution if the

ball is like a square) And

Amount of the balls= balls

�

x

�

=

.

x

.

=154 943

Free space(balls)=197 368 -154943=42 425 S of the ball=0.038 ��

S free space= 42 425 x 0.038 �� = 1 612 ��

1%

----------

X

----------

X=

75 �

1 1612 �

=21.4 % this is more correct solution

If you look from above about 21.4 % of the field that was not covered by balls!

All results are approximate!!!

Task prepared by Vladislav Khrushch

  The radius of one ball is 10cm, so the diameter is 20cm.

The footballfield is 105m x 68m big and on 1m fit 5 balls.

So 525 balls fit on 105m , because if you calculate 105m x 5 balls = 525 balls And on 68m fit 340 balls: 68m x 5 balls =

178 500

On the footballfield fit 178 500 balls.

Icehockey

How many pucks fit on a icehockey field?

Radius: 3.81cm Ice hockey: 61x30

Team “port Vladislav Khrushch Icehockey

Radius: 3.81cm Ice hockey: 61x30 = 1830 m 1.d=2r=7.62 cm Solution: Amount of pucks=pucks in horizontal line × pucks in vertical line Amount of the pucks= pucks

�

��� ��

x

��� ��

=

.

x

.

Answer:315 167 pucks fit on an ice hockey field!!

=315 167

I really like this task , because I play hockey for 10 years.

Now when I will go to next training , I can say to my teammates such information about hockey field , which does not know my coach!

Icehockey

How many pucks fit on a icehockey field?

Radius: 3.81cm Ice hockey: 61x30 The radius of one puck is 3.81cm.

R ( 3.81) x 2 = 7.62 So the diameter is 7.62cm. Now you calcute how many pucks fit on 1m.

100 : 7.62 = 13 Now you know that on 1m fit 13 balls.

The icehockey field is 61m large and 30m wide.

13 x 61 = 793 13 x 30 = 390 793 x 390 = 309.270 On the icehockey field fit on 309.270 balls.

International group „sports“: Activity 1a

Imagine that all persons who have bought a football shirt of the national German or the Spanish football team for the Football World Championship in Brasil 2014 until the end of the championship are standing side by side in a long row.

How long will be that row? How many Euros are spent for all these shirts?

Explain how you get to your result. Results of different person may be rather different but the most important thing is to explain e.g. assumptions that you have made and data that you have used order to calculate and answer the question.

Questions like these are called FERMI-QUESTIONS. If you are interested have a look at this webpage.

Evaluation of this task: Answer the following questions in your solution.

1= I totally agree

5 =I don’t agree at all

1. It was a motivating task for me.

2. It is interesting because it is related to real life.

You additional comment to this task:

Samuel Sierra Santana, Spain Imagine that all persons who have bought a football shirt of the national German or the Spanish football team for the Football World Championship in Brasil 2014 until the end of the championship are standing side by side in a long row.

How long will be that row? people that fit in a football stadium= 114.600 (approximately)

If 1 person take 2 meters (more less) 114.600 take x meters 114.600 · 2= 229.2 meters (approximately)

How many Euros are spent for all these shirts?

In a stadium fit 114.600 (approximately) People who buy a football t-shirt = x

For example In a stadium fit 114.600 (approximately) A football t-shirt cost= 65 Euros (approximately) If 10.000 people buy t-shirts

15.000 · 65= 975.000 euros spent

1It was a motivating task for me: 3 2. It is interesting because it is related to real life: 3

This activity was interesting for me because I like football, and was isteresting because searching the internet, I learned how many people fit in a football stadium.

Icehockey

How many pucks fit on a icehockey field?

Radius: 3.81cm Ice hockey: 61x30

Team “port Vladislav Khrushch Icehockey

Radius: 3.81cm Ice hockey: 61x30 = 1830 m 1.d=2r=7.62 cm Solution: Amount of pucks=pucks in horizontal line × pucks in vertical line Amount of the pucks= pucks

�

��� ��

x

��� ��

=

.

x

.

Answer:315 167 pucks fit on an ice hockey field!!

=315 167

I really like this task , because I play hockey for 10 years.

Now when I will go to next training , I can say to my teammates such information about hockey field , which does not know my coach!

Icehockey

How many pucks fit on a icehockey field?

Radius: 3.81cm Ice hockey: 61x30 The radius of one puck is 3.81cm.

R ( 3.81) x 2 = 7.62 So the diameter is 7.62cm. Now you calcute how many pucks fit on 1m.

100 : 7.62 = 13 Now you know that on 1m fit 13 balls.

The icehockey field is 61m large and 30m wide.

13 x 61 = 793 13 x 30 = 390 793 x 390 = 309.270 On the icehockey field fit on 309.270 balls.

Basketball

How many basketball balls fit on a basketball field?

Basketball The long side of the basketball field is 2800 cm long. The diameter from a basketball is 24 cm. On the long side of the field fit 116 balls.

The short side of the field is 1500 cm long. So there fit in 62 balls.

On the whole field fit 7192 balls. (116*62= 7192).

Basketball

How many basketball balls fit on a basketball field?

Radius: 12,1 cm Basketball field: 28 x 15 The radius of one ball is 12,1cm.

R (12,1) x 2 = 24,2.

So the diameter is 24,2 cm. The basketball field is 28m x 15m big and on 1m fit 4 balls.

So are on 28m 112 balls: => 28x4=112 And on 15m are 60 balls: => 15x4=60 Then you calculate 112 x 60 = 6720.

On the basketball field fit 6720 balls.

Finn-Niklas & Alexandra

Fermi questions (grade 5)            

How many time do you need to write a novel of 100 pages?

How many SMS do all students of PGU write in one day/month/year?

Who many time do all students of PGU spend on tv in one month?

Ho many bikes do have all familes of students grade 9 of PGU ?

What about the amount of chalk is used to mark all football fields at one weekend of German permier league?

How long is the stripe of toothpaste of a person in one year?

How many hours does a church pastor preach in a year?

How many kilimeters do walk from school home in a year?

How often do you share hands in 10 years?

If all students of PGU woulld take hands to the student standing next how long would be this human chain?

How long will all students of PGU work on their homework in a year?

How many steps of the stairs do in school do a student climb up in one week/month?

Project: FERMI questions Our question:

Assumptions: This we have reflected, got to know by an experiment, measured, estimated …. as a base for calculation that follow …:

Our calculation:

Our answer:

Additional question and answer:

Multilateral Comenius Project

MMM_ More Meaningful Maths

MMM_Maths Lesson Plan FERMI questions Author: Monika Schwarze Level: grade 5

Maths content: - estimation, rounding Description: In this activity students will learn to follow a plan to make - dealing with grat numbers an estimation (answer)to a Fermi question and explain assumptions that - dealing with units - converting units they have made by experiment or asking experts, taking a mean, …

Aims: Students should learn - to deal with units and convert them - to make assumptions and explain it to other students - to make a poster presentation of their task - working together in a team of two Procedure: 1. Common work on a Fermi question e.g, “How many chalk will be used by all teachers in your class in one week?” or s.th. else which is interesting for the group 2. The writing on the blackboard should reflect a special structure (see here) 3. Students can choose a task out of a lot of proposals or may invent their own FERMI problem 4. Students work on the problem 5. Students exchange their solution with another group and look for advice for improving their documentation 6. students write it on a coloured paper 7. Students present their problem,the way they have made an assumption and the result of their calculation 8. Exhibition of posters in the classroom Type of activity Timing: Working in pairs for a poster presentation Topic 80-90 min. big natural numbers, rounding, units Conductor Resources/materials Grouping: in the TS (texts in German) Skills Listening pairs X Reading X Speaking, argueing X Presenting Suggestions/variations X Two groups have the same Key competences Linguistic questions (comparision of results is Knowledge and interaction with the intersting) physical world X Digital competence and information treatment Social and civic X Cultural and artistic Learning to learn X Competence on autonomy and entrepreneurship Cross-curricular

Multilateral Comenius Project

MMM_ More Meaningful Maths

Students teach students

EVALUATION REPORTS Evaluator, group, other important details, description, suggestions, variations,evaluation, …)

Report 1: from Monika Schwarze I do this lesson in each 5th grade when students lean the topic “dealing with (big) numbers and units”. The quality of the posters depends on the first example. I put a lot of attention on explaining the students that there are different methods for making assumption and that these should be as real as possible. In the first example we collect ideas for assumptions (not only for this question but for similar questions) like measuring, watching for a while (protokoll), asking different persons and then taking an avarage, experiment, ….

It always worked as described and students have fun. It is an example that mathematics has different “faces” and that results anf prediction in real life are based on assumptions.

Report 2:

Report 3:

unsere Frose:

be q&

,,b*tk§&;l

;'§*1.tut7

-t,« fl*r*A "'1,

Dos hoben wir uns vorher überlegt im Experiment herousbekommen qemessen geschötzt ... als Grundloge für die folgende Rechnung:

i:,:

rt'l* ,si,Ä ,,,

t'

,!

*'i' ( §

rJr

Y

Unsere Antwort: n -{--lt

i,

''), n

-p?t' \/t*, l; {,:g17 .t;.. t'- I ; 1t' ,: t s.

t'tlc

),*;n,7

!;',

-(' !* ' !v'

f-*r""i*&

Eine weitere Froge mit Antwort:

lt

'i r.*",.dqi

ff{§

i*,

{l

;r,1

* ': {,'}

"

,i u4, -lah{€.n

i-:

Unsere Froge:

afe,Pulu

Vf/o-./", I ä"r[l :

n c^.jr,,

,1;;^,ru,> .r,

rna

i'ri

^ i* Joh. .ff"ß|.

l,/ld

e '',

'

Qsrt

Dos hoben wir uns vorher überlegt im Experiment herousbekommen gemessen geschätzt . . . ols Grundloge f ür die folgende Rechnung: Lr)','

,

i,.,,,tro.,*, *:i).;ri i"i;ir)t:,:-:r.*, :3,*,ä#il.;1,*i

ntf/ d#t Ffnqu,r, J*o Ü

J*nrr. jro(ruo dfr

Lä* qutl€tYru.§§'/1' ; .-'

Unsere Rechnung:

bLiss..z: fr6|n tlrq norl' l-lousu 4 , +OO

; i.t ,//^q.

J2

pLS: tu,4sulp.t

4€2.tttoc

,16

492,4 oo

oä lt 4

25r,a ü s

%eE*€

TS

f,8 oo

Unsere Antworf:

,T$At ?§\,§**r"* 3-_?# u,v\§Ä" (ren,\ pfrLt ,.*d, tt** Eine weitere Froge mit Antwort:

**,Ä*

f{#

-f* Sr$"'3# k44 ^&^", ^{s,,, '*d{t {r" t-$""t

hro.r\ ,-Bü4e .ä§\l8o*ku -'*l \§*L?sKry1

p,

e* daL'

dü,r

kJ"u!A+

r qnd Pi

#r+rrrt"Unsere Froge: \,,i i

;

n v;e i e

_,1

,-1,,,äen

i v;,-{,',1'l €in Li

i r.f

f,*^#r

1 \l

',,

q,''.., {'

,A

Das hoben wir uns vorher überlegt. im Experiment herausbekommen oemessen geschätzt ... ols Grundloge für die folgende Rechnung:

tr.li

r Lo'U*n

L\ \

,l

q {'', 'i!1

rr

r r*,

r,

lL n fo:tor

Unsere Rechnunq: r '),/-, ij( ./ " r-1 a ,l^_-;

.:

',\

\: ..

iI

ii"

/i,i )0

'

-1.

,.<'Lt

Erfl(i'r

,1 '1,i'lrrr,{;-i* ,i ,, / lil,-, r'\'1 i 3 '/^r1, \.r'Yr{ i -l 5t unc{e ;r I li' \ ') " rl -,

.

-: r't,",..-,i. i \J

i_-j

1

\

\

1- ' '' r ," u \a

\

*'t

.*1

.*.j

i,,)

r.,t

'"i

rl

l-rr:-:i 'i..

i,

Unsere Antwort:

[n

siner,r,r

]o-

€rfi

f*-t,s

Antwort: ;

ir

"

r ; .-f,'.' Cr r1 \rrrh 'N'*lU !it"\" i:" ' '{'{^' \1; \ 'r, \'tlnr(t'i. \,,it'rr:-i *,i i.Ai lCr.t xl ;g '' n'' (-r:'* ii --\' i]-:-'1'*\1'*:'\YIl, r\utlr'rlr#nsg rcq'\'es Lt'r[\ut!'c\i spni'hi i."n J"^, "Ji:::; i;; ,: Eine weitere Froge mit

'

I

i

\

Unsere

Froge:

t*/fr'," .,+,'., -

.}apr

,

'{

i,4 ''"'

'-)

''"' 'Ä/i'--"'-'l

/V?gr..aü',O,"tg-€tie,Ü/$gt' J-l**he*{€ *'r no''h ;t 7"t5 i": ':'

Clis,-.i

t$"t'

l:,.j,l...i

{

/al

,

Y,§

i-{l:

r,nr',

Dos hoben wir uns vorher gemessen geschätzt .. . ols Grundloge für die folgende Rechnuns

Unsere Rechnung: '

J: p i #u$ü* n "tt ': f-gl.r"6r C '"' 4 -'' 'rt, .

*4,2

4d r,> *}.T5b

;a'

'

1:

'r

-i. '; :) tt

}t))

'i

:<.t{

'

}r**,60 I l;'"-t

'.-., ),--ft.'..\"':

,l j

i)

,r

,i

-,'t'

t

'+