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Information. history. Alternative. Remark. The Paper- Folding - Problem.

The Paper- Folding - Problem

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The Paper-Folding-Problem

A sheet of usual rectangular typing paper (A4) is halved by folding it parallel to the shorter edge. The resulting double sheet can be halved again by folding parallel to the shorter edge and so on.After nfoldings the corners of the resulting stack of paper sheets are cut off. By unfolding the paper, it will be detectable that (for n > 1) a mat with holes has resulted.

Find out and explain a connection between the number n of foldings and the number A(n) of holes in the paper.

In most states of Germany fifth graders are gathered in new classes and new schools. The lesson should be taught in the middle of the first term in grade 5. At this time most pupils have adjusted to the new environment, to new teachers and new classmates. Experiences from the primary school might still have some impact.

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Information abouttheclasstobeselected

Which class do you want to teach?

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How important are the following goals for your coming lesson?

very

not

a little

rather

The pupils should be able to

detect the number of holes for the next cut-folding

detect the number of holes for the folding-cuts of higher orders

give reasons for a rule to determine the number of holes in the paper

find conjectures

give reasons for conjectures and relations

Identify and use structures

to follow the explanations of the teacher precisely

to investigate the problem independently

to enjoy working mathematically

to stand also frustrations when working mathematically

If there should be another important goal, please click here

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Before teaching the folding problem, do you want to prepare the class by teaching one of the following problems ?

“Paper-strip-problem” (Formulation not for pupils)

Take a sufficient long paper-strip. Halve the length of the strip by folding it parallel to the smaller side and repeat this type of folding for n times. Finally, make a triangular cut at the edge without the two “free ends” of the paper-stack and unfold it. You get a strip with holes. The pupils should figure out a relation between the number A(n) of holes and the number of foldings.

You train with your pupils the detection and construction of arithmetical patterns by investigating possible continuation of number sequences.

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How would you like to present the problem to the pupils ?

The pupils investigate the first folding guided by the teacher

The teacher presents some foldings to the pupils. Then they get some task to work on

The teacher presents the problem-text without any comments to the pupils

The teacher presents the problem to the pupils by demonstrating the folding procedure

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history

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How would you like to present the problem to the pupils ?

The pupils investigate the first folding guided by the teacher

The teacher presents some foldings to the pupils. Then they get some task to work on

The teacher presents the problem-text without any comments to the pupils

The teacher presents the problem to the pupils by demonstrating the folding procedure

The teacher makes demonstrations until the

folding

To fix the results

a table is used (at the black board or on the work sheet

papers which were already worked on, are fixed at the black board

a figure is made at the black board

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Assignment for the pupils

Make a table and put your results in it as far as possible

secondbranch

Figure out the number of holes after the next folding without cutting off the corners.

Make a figure how the paper would look like after the next folding-cut

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collaboration

disturbance

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End of the problem solving session

Many pupils rise their hands very vivid and loud:

Sandra: * I am sure there will be 18 holes in the paper*

Andrea: * There will be 6 holes additionally, but now 3 left and right, respectively.*

Fabian: *I have the same opinion as Andrea*

The pupils don’t have more conjectures.

The 5th folding is carried out together and the result is compared with the conjectures of the pupils. But the pupils do not have an approach for an argument for the number of holes, they gained by action.

There is no more motivation to continue this work.

motivation

collaboration

disturbance

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Comparison selected goals and pupils’ learning and teachers’ instruction arrangements outcomes

Comparison selected goals and estimation of pupils’ learning results and teacher’s arrangement of instruction

The pupils should be able to

detect the number of holes for the next cut-folding

very

rather

detect the number of holes for the folding-cuts of higher orders

rather

give reasons for a rule to determine the number of holes in the paper

find conjectures

very

give reasons for conjectures and relations

rather

Identify and use structures

rather

to follow the explanations of the teacher precisely

a little

to investigate the problem independently

very

very

to enjoy working mathematically

to stand also frustrations when working mathematically

rather

consistency

Richness of problem investigations

flexibility

disturbance

convergence

back

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Howthepupilsshouldworktogether?

group-work

partner-work

single-work

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The pupils work out the next folding

motivation

collaboration

disturbance

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The pupils put the number of holes they figured out into the table.

There are a lot of difficulties when tackling the 6th folding. Cutting the folded paper can only.beperformed by considerable effort . In many cases the paper-stack falls into pieces because the pupils cut off corners which are too large.

Many pupils don’t know how to continue their work. Some start to become nervous , start to do something else or repeat the same folding they had done already.

Possible reactions of the teacher

The pupils continue their present work. The teacher is roaming from one group to another and might give little hints

motivation

collaboration

The present work is interrupted

disturbance

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End of the problem solving session

Many pupils are disappointed because they don’t succeed in doing the 6th folding.

Especially some boys compete in doing the 7th folding.

It is very difficult for the pupils to develop reasonable conjectures, there is no more motivation to continue the work on the problem.

motivation

collaboration

disturbance

Information

history

Alternative

Remark

Comparison selected goals and pupils’ learning and teachers’ instruction arrangements outcomes

The pupils should be able to

detect the number of holes for the next cut-folding

very

rather

detect the number of holes for the folding-cuts of higher orders

rather

give reasons for a rule to determine the number of holes in the paper

find conjectures

very

give reasons for conjectures and relations

rather

Identify and use structures

rather

to follow the explanations of the teacher precisely

a little

to investigate the problem independently

very

very

to enjoy working mathematically

to stand also frustrations when working mathematically

a little

consistency

Richness of problem investigations

flexibility

disturbance

convergence

back