Hex

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.

Weekly Problem 7 - 2006

It takes four gardeners four hours to dig four circular flower beds, each of diameter 4 metres. How long will it take six gardeners to dig six circular flower beds, each of diameter six metres?

Plex

Plex lets you specify a mapping between points and their images. Then you can draw and see the transformed image.

Who Is the Fairest of Them All?

Stage: 3 Challenge Level:

Why do this problem?

This problem is the third of three related problems. The first problem is Mirror, Mirror... , and the second is ...on the Wall . All three problems ask students to consider the effect of combining two transformations, and then challenge them to describe the single transformations that produce the same results.

This problem follows on from the other two but works equally well on its own. It could be used as extension work for students learning to draw enlargements from a centre of enlargement, using fractional scale factors.

Alternatively, this problem/series of problems could provide suitable task(s) for a unit of work on combined transformations. Some knowledge of vector algebra might be useful for students attempting a full proof.

Possible approach

As an extension task, all that is needed is to provide the problem as a worksheet to pairs of students who could then make sense of it together. When they have established the combined transformation for one specific example, a teacher intervention may be appropriate, to move the focus to the general case, asking the key questions below.

With a full class, encourage different students to start with different flag positions. The teacher intervention above could become a full class discussion. In theory, all students will have the same combined transformation, which should be a perfect moment for a comment on evidence versus proof.

Key questions

What if the flag was in a different place?

What if you used the other point first?

What if the points were moved?

What, precisely, does the final position of the flag depend on?

Possible extension

Ask students to summarise their findings in exactly 20 words (!) then ask if there is anything further that might be varied.

Ask (suitably experienced) students to create a dynamic geometry file that demonstrates their findings.

Possible support

Spend time drawing accurate enlargements. In theclass/groupevery enlargement could be drawn on paper with full details written on and signed; then every student could alternate between doing an enlargement and checking one off the pile.

smartphone. Generalising. Creating expressions/formulae. Interactivities. Compound transformations. Enlargements. Visualising. Translations. Mathematical reasoning & proof. Scale factors.

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Hex