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

Remark Alternative Information history Information about the class to be selected Which class do you want to teach?

Remark Alternative Information history How important are the following goals for your coming lesson? very rather a littlenot 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

Remark Alternative Information history 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) You train with your pupils the detection and construction of arithmetical patterns by investigating possible continuation of number sequences. 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.

Remark Alternative Information history 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

Remark Alternative Information history 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 thefolding 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

second branch Remark Alternative Information history Assignment for the pupils Make a table and put your results in it as far as possible 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

Remark Alternative Information history collaboration disturbance

Remark Alternative Information history 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 5 th 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. disturbance collaboration motivation

Remark Alternative Information history 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 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 very rather Comparison selected goals and pupils’ learning and teachers’ instruction arrangements outcomes very rather a little very rather consistency flexibility convergence Richness of problem investigations disturbance back rather Remark Alternative Information history

Remark Alternative Information history How the pupils should work together? group-work partner-work single-work

Remark Alternative Information history The pupils work out the next folding disturbance collaboration motivation

Remark Alternative Information history disturbance collaboration motivation The pupils put the number of holes they figured out into the table. There are a lot of difficulties when tackling the 6 th folding. Cutting the folded paper can only.be performed 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 The present work is interrupted

Remark Alternative Information history disturbance collaboration motivation End of the problem solving session Many pupils are disappointed because they don’t succeed in doing the 6 th folding. Especially some boys compete in doing the 7 th folding. It is very difficult for the pupils to develop reasonable conjectures, there is no more motivation to continue the work on the problem.

back Remark Alternative Information history very rather Comparison selected goals and pupils’ learning and teachers’ instruction arrangements outcomes very rather a little very a little consistency flexibility convergence Richness of problem investigations disturbance 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