Fold and cut


In this unit students explore line or reflective symmetry and the names and attributes of two-dimensional mathematical shapes. They fold and cut out shapes to make shapes that have line symmetry.

Achievement Objectives
GM2-3: Sort objects by their spatial features, with justification.
GM2-7: Predict and communicate the results of translations, reflections, and rotations on plane shapes.
Specific Learning Outcomes
  • Explain in their own language what line symmetry is.
  • Describe the process of making shapes with line symmetry.
  • Name common two-dimensional mathematical shapes.
  • Describe the differences between common two-dimensional mathematical shapes in relation to number of sides.
Description of Mathematics

A shape that can be folded down a line to produce two matching halves is said to have line symmetry or reflective symmetry. The fold-line is called a line of symmetry. A line of symmetry can also be described as a mirror line or line of reflection because the part of the object that is on one side of the line is reflected onto the other side of the line. 

line of symmetry

The goal at this level is to have the students being able to describe reflective symmetry in their own language and understand this concept. This creates a foundation on which to build a more complex understanding of symmetry at higher levels of the curriculum, e.g. the order of reflective symmetry and rotational symmetry. 

Learning the names and attributes of common two-dimensional mathematical shapes is important and necessary as students develop a geometry vocabulary. The following are common two-dimensional mathematical shapes and their attributes that could be introduced in this unit.  Not all these shapes need to be presented to all students in the class.  Teachers need to select the ones appropriate, based on the readiness of the students.  

  • Polygon - a shape with straight line sides
  • Triangle - a shape with 3 straight sides
  • Equilateral triangle - all sides the same length and all angles 60°
  • Right angle triangle - one inside angle is a right angle, 90°
  • Isosceles triangle - two sides are the same length and two angles are the same
  • Scalene triangle - all sides are different lengths, all angles are different
  • Quadrilateral - a shape with 4 straight sides
  • Square - a shape with 4 sides all the same length, all angles 90°
  • Rectangle - a shape with 2 pairs of parallel sides, all angles 90°
  • Trapezium - a shape with 4 sides, including 1 pair of parallel sides
  • Rhombus - a shape with 4 sides all the same length, angles may or may not be right angles
  • Parallelogram - a shape with 2 pairs of parallel sides, angles may or may not be right angles
  • Pentagon - a shape with 5 straight sides
  • Hexagon - a shape with 6 straight sides
  • Octagon - a shape with 8 straight sides

Note that pentagons, hexagons and octagons are any shapes with 5, 6 or 8 straight sides.  The length of sides do not need to be the same nor do the angles need to be the same.

Pentagon, hexagon and octagon

Pentagons, hexagons and octagons with sides the same length and angles the same are called regular pentagons, regular hexagons and regular octagons.  A square is a regular quadrilateral and an equilateral triangle is a regular triangle.

Opportunities for Adaptation and Differentiation

This unit can be differentiated by varying the scaffolding of the tasks or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:

  • Provide templates that students can use to create symmetrical shapes.
  • Have students use mirrors to help them think through how particular shapes can be made by folding and cutting.
  • Provide students with a reference list of the names and attributes of a range of shapes they could work with throughout the unit.

The context in this unit can be adapted to recognise diversity and student interests to encourage engagement. Students can identify familiar objects with line symmetry, such as skateboards, swimming goggles, and running shoes. They may also like to investigate line symmetry that can be found in kōwhaiwhai patterns or other cultural motifs.

Required Resource Materials
  • Paper
  • Scissors
  • Rulers
  • Pencils

As students work through these activities the teacher may need to bring the class or small groups of students together from time to time to discuss and model. Make sure an understanding of what line symmetry is and the names and attributes of common two-dimensional mathematical shapes is developing, alongside appropriate vocabulary.

Teachers may also like to generate a class display of  the names and attributes of the shapes over the course of the unit.

Session 1

  1. Take a square piece of paper and fold it in half in front of the class. 
  2. Using scissors cut out the shape as shown below.  Before opening the paper ask the class:
    How do you know the other half will be exactly the same?
    When I open this piece of paper, what shape will the hole in the middle be?
    cutting paper
  3. Open the paper and open up the piece that was cut out. Talk about the attributes of the shape.
  4. Repeat this process cutting out the following shapes.
    paper with cut out shapespaper with cut out shapepaper with cut out shape
  5. Discuss the shapes when the paper was folded in half and when it was unfolded.  The aim of this discussion is to find out what the students know and notice. Questions like the following could be used:
    Why did it work like that?
    How many sides and how many angles?
    What do you notice about the length of the sides?
    Are any angles the same?
    Does anyone know the name of this shape?

  6. Challenge the students:  What other shapes could be made by folding a square piece of paper in half and cutting? and  What shapes do you think are impossible to make?
  7. Hand out square pieces of paper and get the class to experiment and try to make some new shapes.

Session 2 - Straight Line Shapes

How many different straight line shapes can be made by folding a square piece of paper in half and cutting?

For most of this unit the focus is on straight line shapes. Using a ruler to draw the straight lines onto the folded paper before cutting is suggested. Working in small groups, the students are to make as many of the following as they can.

Make . . .

  • 4 different looking shapes with 3 straight sides
  • 4 different looking shapes with 4 straight sides
  • 4 different looking shapes with more than 4 straight sides

Place these shapes into three piles.

  1. Shapes with 3 straight sides
  2. Shapes with 4 straight sides
  3. Shapes with more than 4 straight sides

Once as many different shapes as possible have been made assign a category of shapes to pairs of students, e.g. shapes with 3 straight sides.  The pairs sort their shapes according to the way they look.  The students then share with the rest of the class why they sorted their shapes as they did.

Pairs who need help, could be encouraged to look at the length of sides or the angles in each shape:
Are any of the sides the same length? 
How many angles are larger than a right angle?

Session 3 – Make the Shapes

How many of the following shapes can you make by folding and cutting?

Ask students to fold a square piece of paper in half  and cut out a shape so that when they unfold it the hole will be one of the shapes below.

Model doing one in front of everyone emphasising that you are looking for a line in the shape that you could fold on so both halves would be the same. 

Get the students to predict, before they start on this task, which shapes will be the easiest to make, the hardest to make and whether any will be impossible.  Ask why they think they will be easy, hard or impossible.




Make some more challenges like the ones above for others in your class.

Session 4 - Alphabet Shapes

Make as many letters of the alphabet as you can by folding and cutting.

Session 5 - Reflecting

Ask the students to think about the things they have learnt this week, the names of shapes and about their reflective or line symmetry.

Give pairs of students 4 of the shapes listed below.  Ask them to describe them using their own words and the words they have been learning this week.  Also ask them to identify which shapes have line or reflective symmetry. 

Equilateral triangle, Right angle triangle, Isosceles triangle, Scalene triangle, Square, Rectangle, Trapezium, Rhombus, Parallelogram, regular and non regular Pentagon, Hexagon and Octagon.

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Level Two