Levels 1 to 5

Geometry offers students an aspect of mathematical study that is different from but connected to the world of numbers. Geometry is strongly linked to fields such as measurement, engineering, and design and also includes concepts such as symmetry and shape, which are important in the arts.

The exemplars in this progression show the development in students' understanding of tessellations, that is, patterns of identical shapes that cover a plane without gaps or overlaps. This progression can tell teachers a lot about a student's understanding of a range of geometric concepts, including angle and symmetry, and about their spatial awareness. These geometric concepts and understandings give students a means of interpreting and describing physical environments and can also be useful tools for problem solving.

Fit shapes together to form a tessellation

At this stage in the tessellations progression, students are able to fit shapes together to form a tessellation. They are also able to see that some shapes do not tessellate, for example, circles and octagons.

Identify common shapes that tessellate

At this stage in the tessellations progression, students are able to predict and check whether a particular shape will tessellate. They have some idea of the common properties of shapes that tessellate.

Use right angles to explain the tessellation of objects

At this stage in the tessellations progression, students are able to use their knowledge of right angles to show that certain basic shapes tessellate.

Know that tessellating shapes fit together around a point

At this stage in the tessellations progression, students recognise that tessellating shapes must fit together around a point. Using their knowledge of angles, they can then argue convincingly that equilateral triangles and hexagons tessellate.

Use angles to show that shapes will or will not tessellate

At this stage in the tessellations progression, students are able to give coherent reasons to explain why a shape will or will not tessellate. By this stage, students should be using interior angles to explain tessellation.

Background to the task

The students are:

• shown a selection of shapes, including squares, rectangles, diamonds, regular hexagons, regular octagons, circles, and equilateral, right-angled and scalene triangles
• asked to identify which shapes will tessellate and say why.

The teacher tries to elicit each student's most sophisticated geometric thinking about the task.

References

Department of Education (1985-1989). Beginning School Mathematics: Cycles 1-8. Wellington: School Publications.

Ministry of Education (1992). Beginning School Mathematics: Cycles 9-11. Wellington: Learning Media, Ministry of Education.

Ministry of Education (1993). Beginning School Mathematics: Cycle 12. Wellington: Learning Media, Ministry of Education.

Ministry of Education (2001). Figure It Out, Levels 3-4. Wellington: Learning Media.

Ministry of Education (1992). Mathematics in the New Zealand Curriculum. Wellington: Learning Media.

Ministry of Education (1996). Te Whāriki: He Whāriki Mātauranga mō ngā Mokopuna o Aotearoa/Early Childhood Curriculum. Wellington: Learning Media.