Module 2: Exploring Shape and Space

By the end of this section, you will have:

• used open-ended sorting activities to explore knowledge of shapes;
• explored practical ways to introduce pupils to the language or ‘register’ of mathematical terms;
• used practical activities to develop pupils’ understanding and use of mathematical descriptions of basic geometric shapes.

Collect together as many objects of different shapes as you can. You will need at least two objects for every pupil. You could use pictures of shapes from the environment as well.

• Divide the class into groups of five or six and give each group a selection of objects (see Resource 1).
• Explain what a ‘set’ is – a collection of items with some common features, for example, the class is a ‘set’ of children, who are all taught by you. This ‘large set’ can be grouped into smaller sets – one example would be a set of boys, and a set of girls. (You may like to physically separate the pupils into these two sets to illustrate this point.)
• Explain to the groups that they have a set of different objects. You want them to sort these objects into smaller sets. Ask them the following question: How many different ways can you sort these objects into sets? This makes the task an open task, so do not specify how many sets or any criteria.
• Ask them to explain their reasons for their sorting each set.
• As they work, observe them and listen to the discussions they have in their groups, noting carefully what they say. This will help you find out who has clear ideas and who is still exploring the ideas.
• Ask each group to share the different ways they sorted their objects and note the main features on the board.

You may wish to use a double lesson for this activity.

• Introduce and practise using words for shapes through a ‘sorting’ activity (see Activity 1).
• Use a game to give pupils more practice. The game should be at the right level of difficulty for your pupils. Resource 3: Pictures for cards and Resource 4: Games give examples of the kinds of games that can be easily adjusted for different ability levels.
• Use the pictures in Resource 3 to make sets of cards.
• Using the instructions from Resource 4, demonstrate how the games are played.
• Put the pupils into pairs (or small groups if you have a large class) and give each pair/group a set of cards.
• After doing the activity with one set of cards, the sets can be passed around so that everyone has a new set.
• As the pupils play the games, go around the room and monitor what they say. Don’t interrupt unless they ask for help, but make a note of anything you want to tell the whole class afterwards.

First prepare your feely bag or box. You need a bag or box in which you put an object and the pupil can put a hand in to feel the object but not see it (see Resource 1).

You could have one feely bag for the class or, if your class is big, have more than one so that several groups can work at once. This will help more pupils participate.

Then proceed with the game.

• One pupil should feel one object in the bag/box and, without taking it out, describe it very carefully to the others. The pupil must not name the object.
• They should say things like, ‘it has all flat surfaces, it has so many corners, it has so many flat surfaces,’ etc.
• This carries on until one pupil thinks they can name the object.
• If it is the correct answer, the object is pulled out of the bag and the successful pupil is the next to do the feeling (but allowing only one chance per pupil).

Encourage your pupils to use the vocabulary they have learned in the previous activities to describe their objects. Ask them to add them to their mathematical dictionaries.

The idea of the game is to hide some interesting, different things (which are familiar to your pupils) in the feely bag/box. You could use regular shaped bowls or pots, tools, or even tins of food.

• A pupil comes to the front and feels for something in the feely bag/box. He/she doesn’t take the object out or show it to the other pupils.
• Instead, the pupil then thinks very carefully of ways to describe the thing, without mentioning its name. He/she uses the sense of touch to list and describe observations. At the same time, the pupil has to be quite scientific/mathematical. He/she has to consider the properties the object is made of. He/she also has to think carefully about the shape, size and form of the item.
• Each time the pupil makes an observation, another pupil in the class is given a chance to try to work out what the object is.
• While this is happening, the teacher can act as a scribe (or secretary) and record the observations and the inferences on the board, or on a large sheet of paper. They list the main points only.
• This carries on until someone actually works out what the item is. Then the item can be pulled out of the box and is shown to the rest of the class.
• It is important that a little time is spent discussing the accuracy of the observations – mathematical language skills, the effectiveness of the descriptions, communication skills and the quality of the inferences.

By the end of this section, you will have:

• explored practical ways to use the local environment and simple nets to help pupils understand 3D objects;
• used investigation and problem solving to extend your pupils’ thinking about the different nets to make cubes;
• used dice to encourage mental visualisation and transformation of cubic nets.

For this activity ask each pupil to bring in an empty box. You should collect some too.

• Give each group of four some glue or sticky tape and four sheets of A4 paper.

• Tell pupils that together they are going to explore how to make a box the same shape as the box (a rectangular prism – see below), using one A4 sheet and by drawing, folding and sticking.

  
• Ask them to work together and discuss how to do it before they start. Once they are happy with what they are doing, ask them to use one piece of paper to test their ideas.
• If some groups are stuck, give them a clue about how to start by suggesting they undo the box to make it flat.
• Walk around silently; only help if a group is stuck or asks for support.
• Ask each group to show their work to the class.
• In the next lesson, ask pupils to decorate their boxes and hang them from the ceiling.
• Finally, ask them to draw their plans or nets for the box they made and display these too.

Make sure pupils understand what a cube is, then ask pairs of pupils to find as many different nets for a cube as they can. They should first draw each net, then cut it out and check that it makes a cube, before trying to draw a different net.

(You may want to show one or more examples such as those below to get them started.)

You might like to set this up as a competition, with a reward for the group that can make the most nets for cubes (see Resource 2: 11 nets for a cube).

Again, do not interfere or talk too much during this lesson; make space for the pupils to talk through their ideas and to enjoy the activity. Listen carefully to them and identify how they are able to solve their own problems.

Display the finished cubes and, if there is time, allow them to decorate them to celebrate what they have achieved.

Discuss how many different nets they have found. Ask them to make a wall chart of the 11 possibilities of a net for a cube.

Before the lesson, collect up or make several dice to show your class.

• Ask pupils in pairs to look at a dice, and look carefully at the numbers – they should be able to identify that each side has a number between 1 and 6; you may have to prompt them to see that opposite sides add to 7. Allow them time to check if this rule is followed on all their dice.
• Now give each pair two sets of empty 5 x 5 square grid papers. Ask them to design different nets for a dice: a cube net with numbers written on the squares so that they obey the rules above. When they think they have solved the problem, they may cut out the nets and check that they have ‘correct’ dice.
• After the pairs have solved this problem, they could mark dice numbers on some of the other 11 cube nets that they identified.
• Ask each pair to make a poster to display the different numbering patterns for each net.
• You could extend this activity by asking your class to make a board game about shape and use their own dice to play it

Resource 5: Numbered dice net shows an example of a correct solution and a template for your more able pupils to investigate how many different ways they can place the numbers on the dice so it still works.

You may wish to use a double lesson for this activity.

By the end of this section, you will have:

• developed your own subject knowledge of regular polygons (2D shapes) and polyhedra (3D objects);
• explored practical activities to help pupils investigate the relationship between polygons and polyhedra.

Before you teach this lesson, you need to collect or make some 3D objects and keep these in a box (see Resource 1: Collecting and making shapes and objects).

Organise your class into groups of between six and eight. Ask your pupils to look carefully at the shapes and objects in the box. Ask your pupils what shapes, like squares and rectangles, they can see in the objects.

Tell them the names of the objects:

• Prism e.g. cube, cuboid, cylinder
• Pyramid e.g. cone, sphere

Ask them if they know other objects that look like these shapes around the school and near their homes.

Explain that all the solids, except the cylinder and sphere, are also called polyhedra. Ask them: ‘Why do you think cylinders and spheres are not considered polyhedra?’ (See Key Resource: Using explaining and demonstrating to assist learning  for some ideas to help you.)

Tell them that the word polyhedron is from a Greek word for ‘a seat’. Prisms and pyramids have many flat surfaces like seats but a cylinder is not a polyhedron as it has a curved ‘surface’.

Finish the activity by asking each group to count the surfaces on each object. Ask them to record their answers in their books. Share each group’s answers as a class.

For homework, ask them if they can see any of these shapes on their way home – or at home – and report back the next day.

You will need copies of Resource 3, paper, scissors and sticky tape or glue. If you only have enough materials for one group to work at a time you can spread this activity over a week.

• Explain to your pupils that pyramids can have bases of any number of sides – the simplest have equilateral triangles on all four surfaces, but pyramids can be made with any regular polygon as a base: the groundnut pyramids are made of triangular sides, but have square bases.
• Give out the nets of triangular- and square-based pyramids, and ask pupils to cut, fold and glue these to make paper pyramids. Mount a display of them.
• Next, place some straws or matches on each group’s desk and ask if they can, using string or sticky tape, make a pyramid out of these materials. Go around and support the groups while they work. Let them share what they did to make their pyramids.

First, consolidate pupils’ learning from earlier activities. For this, you will need your box of shapes and objects and charts to record results (see Resource 5: Recording results) or ask your pupils to draw the two charts in their books.

Organise your pupils into pairs or small groups. Give them one of the items from your shapes box, and ask them to carefully look at the shape and complete the chart as best they can.

Suggest they complete a row at a time. Ask them to return their shape to the box and take another one until they have looked at every shape.

After an appropriate time, ask one pair or group to give their answers to the class for one shape. Go round the class until all the shapes’ features have been shared and each pair has been able to check their answers against everyone else’s.

Ask them if they noticed any patterns in their observations. Which shapes and objects are related?

Display their charts.

You might want to use the game ‘Shape Challenge’ in Resource 5 to finish this topic and assess their understanding as they play. You could divide the class into teams to play.

You may need to use a double lesson for this activity.

By the end of this section, you will have:

• used group work to help develop pupils’ understanding of symmetry, including multiple lines of symmetry and orders of rotational symmetry;
• developed a range of strategies including using open-ended questions to develop thinking skills around symmetry;
• worked across curriculum areas to extend ideas about symmetry.

Before the lesson, collect some natural objects that have approximate symmetry: these could include leaves, flowers or vegetables. You could even use local animals (but you must ensure they are well treated) or you could use photos of them (you might ask your pupils to help you). Resource 1: Examples of symmetry found in nature has some useful photos and you may want to collect more from magazines and newspapers, or some samples of local fabrics.

• Divide the class into small groups of five or six and ask each group to consider the objects or images and try to identify all the lines of symmetry. Share their answers as a class (see Key Resource: Using group work in the classroom  to plan how to do this).
• Ask your groups to think of other objects from everyday life that are symmetrical. Suggest that on the way home they try to find other examples and either note these down or bring a sample in if possible.
• In the next lesson, ask each group to make a poster of six different objects that they have found that have lines of symmetry and draw the line(s) of symmetry on them. They could draw or perhaps stick on some objects.
• Display the posters for the whole class to see and discuss their ideas after a day or so to remind them.

You will need enough paper and pencils or paints for each pupil to make a colourful mask, string or elastic to tie the masks on, and pieces of cardboard big enough to make the masks with. You may have to spend some time collecting these resources before you can do the activity but your pupils may be able to help you gather materials together (see Key Resource: Being a resourceful teacher in challenging circumstances).

Explain to the pupils that they are going to make masks, but that both the shape of the mask and any drawing or painting on it should be symmetrical. Suggest that they do a rough design before they start working. You could show them some local masks. Perhaps they could gather resources and do a rough design in one lesson, and make the mask in the next lesson or two.

Suggest they make masks of people, leaves, animals, wings, imaginary creatures, or tribal masks. This could be a decision you leave to each pupil, or one you decide for the whole class.

Think about what resources might help the pupils design their masks (such as photos or objects – see Resource 2: Examples of symmetry in Kenyan masks). What other creative activities could pupils do to consolidate their understanding of symmetry?

You will need a page of polygon shapes (see Resource 5: Polygons) for each small group of pupils.

First, ask pupils to write in their books three column headings: ‘polygon sides’ ‘lines of symmetry’ ‘rotational symmetry’. Then ask them to look at the shapes and, for each polygon, count and record:

• How many sides it has.
• How many lines of symmetry they can find.
• How many orders of rotational symmetry they can find.

After the first few shapes, some pupils may begin to spot a pattern and be able to complete their table without counting; others may not see the pattern. If this happens, ask the pupils who have seen a pattern to explain how it works to those who have not.

Use questions like: ‘How many lines of symmetry would a polygon of [n] sides have? And how many orders of rotational symmetry?’ ([n] could be any whole number.)

Ask each group to complete the chart you have drawn out on a sheet of newsprint and display their charts in the classroom (see Resource 6: Recording symmetry).

By the end of this section, you will have:

• introduced pupils to transformation, congruence, translation and reflection;
• used cut-out shapes as a means to develop the mental transformation of geometric shapes;
• considered the challenges of differentiating this work for older and younger pupils, and tried some different approaches.

To complete this activity, you will need a piece of cardboard and a pencil and ruler for each pair or small group of pupils, and several pairs of scissors.

• Ask your pupils to draw three different straight-sided shapes on their card and then cut out their shapes. They should number each of their cardboard shapes 1, 2 or 3.
• Next, on a separate piece of paper, ask your pupils to draw around each shape; then move the shapes any way they like without overlapping what they have already drawn, and draw round them again. Repeat this until the page is full of shapes, then label inside each outline with a letter (e.g. a, b, c…). (The finished work should be similar to Resource 2: Examples of congruent shapes.)
• Ask pupils to swap their work with another group. Can they find the outlines that were made with the same shape? (Younger children might need to use the cardboard shapes to help them.) Ask them to write down what they think is the answer – e.g. shape 1, outlines a, b, d, g.
• Using the cut-out shape, can they show you what has to happen in moving from one outline to another? Can they describe this in their own words?
• Early finishers can colour in their work, using one colour for outlines from the same shape. You could display these on the classroom walls, headed ‘Congruent shapes’.

Make sure pupils understand how to give x-y coordinates, through whole-class teaching. To differentiate the task for older or younger pupils, see the notes on differentiation in Resource 4.

• Ask pupils to draw and cut out a triangle, square and rectangle from a piece of squared paper: emphasise that each corner (or vertex) of their shapes should be at one of the ‘crosses’ on their grid paper by drawing an example on the board. No side should be more than 10 squares long.
• On a second piece of grid paper, ask pupils to draw and label x-y axes at least 20 squares long (see Resource 4).
• Putting one of their cut-out shapes on the paper so that its corners are on the ‘crosses’ of the grid, they should mark the vertices (a, b, c & d as appropriate), then draw the shape and write down the coordinates of each vertex.
• Ask them to move their shape to a new position (keeping it the same way up) and repeat this process.
• Ask your pupils: ‘What happens to the x coordinates between the two positions? Does the same thing happen to each coordinate? What happens to the y coordinates?’

What parts of this activity caused difficulty for your pupils? How will you support them next time?

Your pupils could reuse the shapes they cut out of grid paper for Activity 2, or make some more if necessary.

• On a second piece of grid paper, ask pupils to draw and label x-y axes at least 20 squares long (see Resource 4).
• Putting one of their cut-out shapes on the paper so that its corners are on the ‘crosses’ of the grid, they should mark the vertices (a, b, c & d as appropriate) then draw the shape and write down the coordinates of each vertex (corner).
• Ask pupils to draw a vertical or horizontal mirror line on their grid. They should then draw the reflection of the shape on the other side of the mirror line (remind pupils that they may use the cut-out shape if it helps them) and write down the coordinates of the reflection.
• Challenge your pupils to work out the reflection coordinates without using the cut-out shape. Ask them to explain how they did it. Practise using lots of shapes so that pupils become confident.

How well did you introduce and explain this work? How do you know this?