# Haberdasher's Problem

### Task 146 ... Years 6 - 10

#### Summary

This puzzle is easy to state and easy to start, but like all good challenges, not so easy to achieve. It is connected with the history of mathematics because it derives from a puzzle popular at a time (need we say before television) when mathematics was a recreation for many. In a sense this task is the solution to the puzzle and the iceberg of the task is to take the students back to the historic challenge mentioned in the introductory paragraph.

#### Materials

• 4 pieces - 1 triangle and 3 quadrilaterals

#### Content

• properties of equilateral triangles & squares
• geometric language and construction
• measurement
• transformation geometry - rotation

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

This short Cube Tube video shows a group of Swedish teacher trainees at Högskolan Malmö attempting to solve the puzzle. These young adults do manage the square, perhaps because right angles are easier to construct, but the triangle remains elusive. The solutions are:

Encourage students to record their solutions - they will have put in a considerable effort to find them. A digital camera is one way, but an annotated sketch may be just as meaningful. Encourage them to add as much information as possible about how they 'thought it out'.

Comparing the two photos suggests how the challenge on the card - to rotate pieces to transform one shape to the other - might work. The triangle show that all the right angles of the square have been 'hidden' inside the triangle. Now look at the square and experiment with how this could be done.

If students eventually need a hint, suggest beginning with the top right triangle in the photo of the square and rotating it clockwise about its bottom corner. Because this is a midpoint, the right hand side of the top triangle will match the right hand side of the bottom quadrilateral.

Encourage further independent experimenting from there, but if necessary the remaining steps can eventually be exposed. They are:

• The top right triangle and the bottom right quadrilateral have now formed a new larger triangle. Rotate this as a whole clockwise around its bottom left corner as far as it will go.
• Rotate the top left quadrilateral anti-clockwise around its bottom corner to fit into the right angle gap. Voila! - an equilateral triangle.
Again encourage students to sketch and explain in their journals - or perhaps they could use the digital camera and create a photo slide show.

#### Extensions

1. Challenge students to pretend they are Henry Dudeney and now have to explain to someone else where to make the cuts in the triangle so it can be transformed into the square.
2. The original puzzle was to begin with an equilateral triangle and dissect it to make pieces that could be rearranged as a square. Discuss with students that Henry Dudeney must have made many experiments before he succeeded. Perhaps 5, 6, 7, ... pieces were some of his experiments. To develop a feel for the processes that he must have gone through, encourage the students to cut out the triangle from this worksheet and dissect it into a different number of pieces to see how close they can get to exactly fitting the area of the square shown on the same sheet. (The two shapes on the sheet actually do have the same area, but dissecting the triangle in any old way won't necessarily produce a square.)
3. The regular polygons in the task have the same area and the task shows that it is possible to dissect one so the pieces can be transformed into another. Could this be done with two other regular polygons of the same area? Could it be done with any two regular polygons with the same area?

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

To convert this task to a whole class investigation, each pair will need the four pieces. Trace the pieces of the task one per page, then use a photocopier to reduce each drawing. Cut and paste again to make two master pages with two pieces on each sheet. Make copies so that each person in a pair has a different page to cut out. Now the challenges begin as above with the purpose of developing geometry and measurement content in the context an historic problem and the mathematician who solved it in the most efficient way.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 86, Haberdasher's Problem, which includes fabulous Euclidean geometry-style instructions for dissecting the triangle using either a compass and pencil or ruler and pencil - a refined example of Extension 1 above.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Haberdasher's Problem task is an integral part of:

• MWA Space & Logic Years 9 & 10

The Haberdasher's Problem lesson is an integral part of:

• MWA Space & Logic Years 9 & 10