Geometry and shapes for CTET: content, van Hiele levels and teaching

Show a class of eight-year-olds a square standing on one corner and many will call it a "diamond", not a square. Show them a long, thin triangle and some will insist it is not a triangle at all, because "triangles look like this", and draw one sitting on a flat base. These are not silly answers. They tell you the child is judging shapes by how they look, not by their properties, and that single idea sits behind many of the geometry pedagogy questions in CTET.

Paper I covers "shapes and spatial understanding" and "solids around us" at the NCERT Class 1–5 level. Paper II adds lines and angles, triangles, polygons, symmetry and constructions. This guide covers the content, then the van Hiele levels and the teaching moves CTET expects.

Solids: faces, edges and vertices

A face is a flat surface, an edge is where two faces meet, and a vertex (corner) is where edges meet. Curved solids need care: a cylinder has curved surfaces and curved edges but no vertex.

Euler's formula holds for polyhedra, solids with only flat faces: F + V − E = 2. Cube: 6 + 8 − 12 = 2. Triangular prism: 5 + 6 − 9 = 2. Do not apply it to cylinders, cones or spheres.

Worked example. A hexagonal prism has 8 faces (2 hexagons and 6 rectangles) and 12 vertices. How many edges? 8 + 12 − E = 2, so E = 18. Check by counting: 6 edges round each hexagon and 6 joining them, 6 + 6 + 6 = 18.

Paper I also tests spatial understanding. The top view of an upright cylinder is a circle and its side view a rectangle; a cone's side view is a triangle. A net is a flat shape that folds into a solid: a cube has 11 different nets, and a row of six squares is not one of them.

Lines and angles

• Complementary angles add to 90°; supplementary angles add to 180°.
• Vertically opposite angles are equal.
• When a transversal cuts two parallel lines, corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side add to 180°.

Worked example (clock). What is the angle between the hands at 3:30? The minute hand points at 6, which is 180° from 12. The hour hand moves 30° every hour, so at 3:30 it is at 3.5 × 30 = 105°. The angle is 180 − 105 = 75°, not 90°, because the hour hand has moved halfway towards 4.

With only a ruler and compasses you can construct 60° and 90°, and by bisecting and combining them, 30°, 45°, 75°, 105°, 120° and 135°. Angles such as 70° or 50° cannot be made this way.

Triangles and polygons

• The angles of a triangle add to 180°. An exterior angle equals the sum of the two opposite interior angles.
• Triangle inequality: the sum of any two sides is greater than the third. Sides 3, 4 and 8 cannot form a triangle, because 3 + 4 < 8.
• By sides: equilateral, isosceles, scalene. By angles: acute-angled, right-angled, obtuse-angled.
• Pythagoras (right triangles only): the square on the hypotenuse equals the sum of the squares on the other two sides. For 6, 8, 10: 36 + 64 = 100.
• Angle sum of a polygon with n sides = (n − 2) × 180°. The exterior angles of a convex polygon add to 360°.
• Number of diagonals = n(n − 3)/2. A hexagon has 6 × 3 / 2 = 9.

Worked example. The angles of a triangle are in the ratio 2 : 3 : 4. That is 9 parts = 180°, so one part is 20°, and the angles are 40°, 60° and 80°.

Worked example. Each interior angle of a regular polygon is 144°. Each exterior angle is 180 − 144 = 36°, so the polygon has 360 ÷ 36 = 10 sides.

Quadrilaterals as a family

Every square is a rectangle (four right angles) and a rhombus (four equal sides), and every rectangle and rhombus is a parallelogram. Not every rectangle is a square. Children who learnt shapes as separate pictures find this hard, which is why it appears in questions on levels of geometric thinking.

Symmetry

A regular polygon with n sides has n lines of symmetry. Capital letters are a favourite: A, M, T and U have one vertical line; B, C, D and E have one horizontal line; H has two; N, S and Z have no line of symmetry but look the same after a half turn.

The van Hiele levels of geometric thinking

The Dutch mathematics educators Pierre van Hiele and Dina van Hiele-Geldof described, in the late 1950s, how geometric thinking develops through a series of levels. Most CTET material numbers them 0 to 4; some books number them 1 to 5, so read the options carefully.

Three features are asked about:

• The levels are sequential. A learner cannot skip one.
• Progress depends on experience and teaching, not on age alone. This is the main contrast with Piaget's stages.
• Each level has its own language. If the teacher talks at level 2 to a child at level 0, the child may memorise the words without understanding them.

Most primary children work at levels 0 and 1, and upper-primary teaching aims to move them towards level 2. The van Hieles also described five phases of learning that carry a learner from one level to the next: information, guided orientation, explicitation, free orientation and integration.

Children's misconceptions about shapes

Teaching geometry well

• Handle real solids before drawing them: boxes, balls, cans, dice, cones.
• Paper folding for symmetry, right angles and angle bisectors.
• Tangrams: seven pieces (two large, one medium and two small triangles, a square and a parallelogram). They build spatial sense and show that different-looking shapes can have the same area.
• Geoboards and dot paper: make a shape, change it, notice which properties stay.
• Sorting shapes by the children's own rules, then explaining the rule.
• Nets: open out real boxes, then fold nets into cubes and cuboids.
• Language: move from "corner" and "side" to "vertex" and "edge" as understanding grows.

See maths teaching aids for these materials and perimeter, area and volume for measuring shapes.

Worked pedagogy question

A Class 6 student says, "A square has four equal sides and four right angles, but it is not a rectangle." In van Hiele's terms the student is most likely at:

(a) Level 0 (b) Level 1 (c) Level 2 (d) Level 3

Answer: (b). The student reasons with properties (level 1) but does not yet relate one class of shapes to another (level 2).

Practice set

1. How many edges does a square pyramid have?
2. A polyhedron has 6 faces and 8 vertices. How many edges does it have?
3. Which can be the sides of a triangle: (3, 4, 8), (2, 5, 7), (5, 6, 10), (1, 2, 4)?
4. What is the angle between the hands of a clock at 4:00?
5. The supplement of an angle is three times the angle. Find the angle.
6. How many lines of symmetry does a regular pentagon have?
7. Which angle cannot be constructed with ruler and compasses alone: 75°, 105°, 135° or 70°?
8. A child sorts cards and says, "These are rectangles because they look like doors." Which van Hiele level is this, and what should the teacher do next?

Answers

1. 8.
2. 6 + 8 − E = 2, so 12 (a cube or cuboid).
3. (5, 6, 10), since 5 + 6 = 11 > 10. (2, 5, 7) fails because 2 + 5 = 7 is not greater than 7.
4. 4 × 30 = 120°.
5. 180 − x = 3x, so 4x = 180 and x = 45°.
6. 5.
7. 70°.
8. Level 0, recognition by appearance. Next: activities that focus on properties, such as checking corners with a right-angle tester and including rectangles that look nothing like doors.

What to do next

• Fill in the solids table from memory and check each row with Euler's formula.
• Write the five van Hiele levels with one "what you hear" line for each.
• Make a paper tangram and form a square, a triangle and a parallelogram from all seven pieces.
• Continue with perimeter, area and volume, then patterns.