Mathematics: Geometry
THE NEW ZEALAND CURRICULUM EXEMPLARS
Tessellations
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.
LEVEL 1
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.
LEVEL 2
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.
LEVEL 3
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.
LEVEL 4
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.
LEVEL 5
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.
Mathematics: Geometry
THE NEW ZEALAND CURRICULUM EXEMPLARS
Tessellations
ACCESS THE MATHEMATICS EXEMPLARS ONLINE AT
WWW.tki.org.nz/r/assessment/exemplars/maths/index_e.php
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.
Note: Each level of Figure It Out consists of a set of student books, with an accompanying Answers and Teachers
Notes for each book.
Ministry of Education (1992). Mathematics in the New Zealand Curriculum. Wellington: Learning Media.
Ministry of Education (1996). Te Whàriki: He Whàriki Màtauranga mò ngà Mokopuna o Aotearoa/Early Childhood
Curriculum. Wellington: Learning Media.