Hearts & Loops

Summary

Materials

• 1 wire puzzle made of 2 inter-connected pieces

Content

• spatial skills
• mathematics language
• topology / history of mathematics
• recording mathematics

Iceberg

There is no need to reveal the solution to this puzzle. Someone will find it eventually. However, if a clue is needed to calm a 'frustrated beast' then it may be helpful to suggest that some people work out how to solve it by thinking of turning a key in a lock.

The challenge for the teacher is what to do to encourage finding an iceberg in the task. The area of mathematics from which this puzzle derives is topology. Although not a mainstream topic, it is the source of many equally challenging problems. Clearly, since the pieces can be taken apart, at least one of them must be an open curve (like a classic doughnut ring with a bite out of it). Topology involves (amongst other things) identifying closed and open curves, and it has spawned other recreational pastimes as the Möbius Strip and the Königsberg Bridge Problem. In fact, the study largely grew from Euler's solution of the Bridge problem, as did the area of graph theory.

When someone does solve Hearts & Loops, their first challenge is to be able to put the pieces back together. They are not allowed to show anyone else their solution until they can master 'both directions'. However, no doubt once discovered the solution will soon travel the class 'grapevine'. But most students will still want to try it for themselves, and will frequently revisit the puzzle just to make sure they can still do it. Teachers make interesting comments along these lines:

• In our task collection, this is one of the most popular. It seems to have an inherent curiosity, perhaps because it looks so 'impossible'.
• It is one of the most common requests for a 'take home' task in our Maths Around The Kitchen Table program.
• We use it in our special education class to support students' motor-coordination skill development

Place the central dip of the heart into this loop.   Bring the large hoop(on the end of the rod) through the dip, from position A to position B.   Lift the heart out of the loop to detach the pieces.

Extensions such as this are intended to further encourage spatial and kinaesthetic 'reasoning' and the development and refinement of mathematical language.

Another challenge is to ask a successful student to tell someone else how to do it - provided the doer is the only one to hold the puzzle and the teller either sits on their own hands or keeps them behind their back. The identification and refining of spatial language used in this challenge is a lesson in itself.

• The task improved considerably in my estimation when the challenge of 'talking someone else to the solution' was added. This made the learning outcome more explicit and measurable.

Whole Class Investigation

If you have only one copy of this task it can be used as one of number at a work station which includes other spatial reasoning tasks, such as Back To Back Building. This could be part of a Menu Maths approach to a Space & Logic unit.

If you are able to have a class set (say 10 - 12) of Hearts & Loops made, the puzzle could become the focus of a whole class lesson that highlights spatial / kinaesthetic reasoning and mathematical language.

Begin by asking students to work in pairs, seated back to back. Student A is told to draw a circle, a square, a rectangle, and a triangle on a page. Student A now has to describe what is on their sheet to Student B with the aim of Student B recreating the identical design. Student B can only say 'Yes', 'No', or 'Please repeat' referring to the previous instruction (without modifications).

Teachers have observed that after the first time, the use of specific spatial language related to placement, size, and detail increases markedly. Students become very involved in the activity. They are eager to share and discuss their results and how they were obtained with the teacher or interested observer. Students are very critical of their efforts and correction is immediate. Personal and partner evaluation is inherent in this task.

The next step is to introduce Hearts & Loops.

• Let half the class learn to solve the puzzle out of sight of the others.
• Line up 10 chairs with those who know how to solve the puzzle standing behind each chair and an 'unknowing' subject sitting down.
• Students behind the chairs have to 'talk' their partner through the task to a solution, resisting the temptation to put their hands around the sides to point or prompt.

Use the lesson to develop a glossary of the most useful words and phrases that helped to solve the problem. Link this to the development of mathematical language and the need mathematicians have to communicate with others in jointly understood terms.

At this stage, Hearts & Loops does not have a matching lesson on Maths300.