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Level Two > Geometry and Measurement

# Fold and Cut

Purpose:

In this unit students explore line symmetry and the names and attributes of two-dimensional mathematical shapes. The context of this unit is students folding and cutting out shapes to make a series of mathematical shapes.

Achievement Objectives:

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:

This unit has the students using line symmetry in a practical activity.  At Level 2 the concept of symmetry is starting to be developed.  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 symetry.  In this unit, we look at shapes with reflective symmetry, i.e. objects that have one or more lines 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.

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                         - all sides are the same length, all angles 90°

Rectangle                    - 2 pairs of parallel sides, all angles 90°

Trapezium                  - 1 pair of parallel sides,

Rhombus                    - all sides are the same length, angles between 1° and 179°

Parallelogram             - 2 pairs of parallel sides, angles between 1° and 179°

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.

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.

Required Resource Materials:
Paper
Scissors
Rulers
Pencils
Activity:

#### Getting Started

1. Take a square piece of paper and fold it in half in front of the class.

Using scissors cut out the shape as shown below.  Before opening the paper ask the class,

When I open this piece of paper, what shape will the hole in the middle be?

After the students have had a chance to express their opinions, discuss how at the moment they can see half the shape.

How do you know the other half will be exactly the same?

Open the paper and open up the pice that was cut out. Talk about the attributes of the shape.

Repeat this process cutting out the following shapes.

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?

1. 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?
1. Hand out square pieces of paper and get the class to experiment and try to make some new shapes.

#### Exploring

Over the next 2 or 3 days the students need to work through the following three tasks.  At appropriate times the teacher may need to bring the class together to discuss and model, making sure the vocabulary is being developed and key ideas are emerging.

Task 1 - Straight Line Shapes

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

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

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.

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 in each shape,

Are any of the sides the same length?

or to look at the angles,

How many angles are larger than a right angle?

Task 2 – 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 stressing 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.

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

Throughout this unit the teacher needs to reinforce the understanding of what reflective symmetry is and the names and attributes of common two-dimensional mathematical shapes.  As the class works on the above tasks, small groups could be taken aside to work with the teacher.  This would enable the teacher to gauge the understanding of the students and adapt the task as necessary.

A display of the names and attributes of the shapes could available to students as a point of reference throughout the unit.

Reflecting

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

The students could be given a set of 4 shapes and asked to describe them using their own words and some of the words they have been learning about this week and decide which shapes can be made by the folding and cutting technique and which cannot.

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

## Fitness

This unit examines regular tessellations, that is, tessellations that can be made using only one type of regular polygon, and semi-regular tessellations, where more than one type of regular polygon is involved. Students are required to investigate what properties tessellating shapes must have in order to cover the plane with no gaps or overlaps.

## Inside Out

This is a level 3 geometry activity from the Figure it Out series.

## Tessellating Art

In this unit we apply our understanding of why tessellations work to form our own unique tessellating shapes. We use these shapes to create interesting pieces of art in the style of M.C. Escher.

All M. C. Escher works (C) Cordon Art, Baarn, the Netherlands. All rights reserved. Used by permission.

## Keeping in Shape

This unit examines tessellations, that is, ways of covering the plane with copies of the same shape so that there are no gaps or overlaps. Students will investigate what properties tessellating shapes must have in order to cover the plane with no gaps and with no overlapping. The tessellations investigated involve both non-regular and regular polygons.

## Fold and Cut 2

This unit involves folding and cutting paper to see the number patterns that are produced and to see what geometric shapes can be made. This unit contains similar content to Fold and Cut.  You may want to take parts from both units when planning your own teaching.