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# Fold and Cut 2

Keywords:
Purpose:

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.

Achievement Objectives:

Achievement Objective: GM2-7: Predict and communicate the results of translations, reflections, and rotations on plane shapes.
AO elaboration and other teaching resources

Specific Learning Outcomes:
• fold paper systematically
• cut shapes from folded paper
• find number patterns derived from folding and cutting using a table
Description of mathematics:

The aim of this activity is to explore shape and number using a creative activity. In the process of the unit the students should get aesthetic pleasure from producing pleasant shapes and they should reinforce their knowledge of basic shapes and have their understanding of geometry enhanced.

One of the central mathematical ideas is that of the relation between fold and symmetry and the spatial effect that different folds and cutting have on the shapes as a whole.

Required Resource Materials:
Scissors
Several pages of newspaper per student
Activity:

Session 1

In this session the students get an opportunity to experiment with folding and cutting.

1. Bring the class together and talk about folding and cutting paper. Show them some ways of folding and cutting paper to make interesting shapes and patterns. One interesting method is to take a rectangular piece of paper and fold it in half one way, then fold it in half the other way. Now fold the paper along a diagonal through the point of the paper that is as close as possible to the centre of the original paper. Fold the paper in half again through a similar diagonal. Now cut a person shape so that the hands go through the opposite side of the fold to the body. When you open up the folds you should find that you have 8 people in a circle holding hands.  This will be much easier with a large piece of paper.

2. The cutting and folding for the 8 people is shown in the diagram. But there are lots of other ways that you can fold paper. We have suggested some in the sessions below. And you can always take any of these and fold in half perpendicular to any other folds.

3. Talk with the students about how the paper might be folded. Then talk about the cutting that they might do. Let them talk about the possibilities. Experiment with a few ideas together.

4. Tell them that you want them to go away and experiment with all sorts of folds and all sorts of cuts to see what they can come up with. Give them all scissors and lots of paper. Let them know that they can draw the shape they plan to cut before they do the cutting. Before they go off to work, remind them that you will want to get together in a little while to see what they have done. As part of this it is important that they can tell the other members of the class how to produce what they have done. So it might be useful to make some notes.

5. While they are working, move around and see how they are going. Give them encouragement and help and praise them for interesting shapes and risk taking. Did that shape you made surprise you?

6. Bring the class together and let students talk about what they have achieved. Display the work in some suitable location.

Session 2

In this session we use cutting to find a pattern in the number of shapes produced by ‘concertina’ folding.

1. Demonstrate to the class how they can ‘concertina’ a piece of paper.  Start by folding a piece of paper in half, then fold the loose ends down to meet the first fold line.  Show them how to do more and more folds using this method (this is a bit fiddly). Talk about cutting pieces (i) from one side but not totally through; (ii) from the other side but not quite totally through; and (iii) totally through from both sides.
What do you get?
How many shapes do you get?

2. Let the students experiment by themselves with different shapes and different numbers of folds.

3. Bring the class back together to discuss what they have so far. Ask them:
How do the number of shapes depend on the number of folds?

4. Talk to them about the value of a table for collecting evidence.

5. Send them back to get evidence for a number pattern.

6. Discuss their results. Let them design a poster to display these results.

Session 3

In this session we use cutting to find a pattern in the number of shapes produced by repeatedly folding in half along the same axis.

1. Demonstrate to the class how they can fold a piece of paper in half several times, while keeping the folds parallel.  Discuss how this method of folding is different to the one in the previous session. Talk about cutting pieces (i) from one side but not totally through; (ii) from the other side but not quite totally through; and (iii) totally through from both sides.
What do you get?
How many shapes do you get?

2. Let the students experiment by themselves with different shapes and different numbers of folds.

3. Bring the class back together to discuss what they have so far. Ask them:
How do the number of shapes depend on the number of folds?

4. Talk to them about the value of a table for collecting evidence.

5. Send them back to get evidence for a number pattern.

6. Discuss their results. Let them design a poster to display these results.

Session 4

In this session we use cutting to find a pattern in the number of shapes produced by ‘perpendicular’ folding.

1. Demonstrate to the class folding how they can fold paper in half one direction and then fold it in half the other direction (perpendicular to the first fold). Show them how to do more and more folds using this method.  Discuss how this method of folding is different to the ones in the previous sessions.  Talk about cutting pieces (i) from one side but not totally through; (ii) from the other side but not quite totally through; (iii) totally through from both sides; and (iv) from the corners.
What do you get?
How many shapes do you get?

2. Let the students experiment by themselves with different shapes and different numbers of folds.

3. Bring the class back together to discuss what they have so far. Ask them:
How does the number of shapes depend on the number of folds?

4. Talk to them about the value of a table for collecting evidence.

5. Send them back to get evidence for a number pattern.

6. Discuss their results. Let them design a poster to display these results.

Session 5

In this session they use their folding and cutting to make prescribed numbers of specific shapes.

1. Recall the activities of the previous sessions. Tell them that this time they are to make various shapes but they have to decide how to fold and cut the paper to produce the shapes that we list below.

• one diamond
• one kite
• two diamonds
• four diamonds on the corners of a square
• a square
• a square using only two cuts
• four circles in a row
• regular hexagon (six sided figure with all sides and angles equal)
• a big disk with a smaller disk removed (an annulus)
2. In pairs the students go off to complete their tasks. (If some students finish well ahead of the others, check that they can describe how they produced the required figures. Then get them to invent some tasks of their own.)

3. Have a reporting back session.