Geometry

Geometry Page 1

Reformed pedagogy

Geometry

The following is the next phase in the Maths in the Kimberley project and consists of some lesson experiences. It follows on from interviews with students at Kulkurriya and Yiyili schools. The focus on geometry is based on an often stated assumption that the prevalence of direction words in some Indigenous languages implies that the learning of aspects of geometry may be closer to Indigenous students’ experience than the learning of number. The following is seeking to explore this. If students do seem familiar with the concepts then it would make sense to use this aspect of the curriculum more actively to build confidence, success, and connection to mathematics and schooling. It may also be possible to incorporate visual approaches to other topics.It also seems that geometrical questions are over represented in NAPLAN assessments so some attention to this topic is likely be useful for students and the schools. An interesting aspect is that, in inspecting items in NAPLAN assessments, it seems that the majority of questions depend on knowledge of conventions as much as knowledge of space and location. One of the important conventions is the way that 3D shapes are represented on a 2D page or screen. Metropolitan based children are exposed to many such images and perhaps the remote students do not have such opportunities.The approach proposed recognises that a significant challenge for teachers is the process of connecting English language, especially specific

mathematical terms, with student experience. If concepts seem well developed in relevant language then the goal of instruction is to connect the English terms with local terms. If some concepts do not seem well developed then the goal would be to provide experiences in the use of the various concepts. There are three aspects to the following. Note that these do not address all aspects of geometry required by the curriculum. Some key aspects not addressed are transformations (symmetry etc), and angles.The three lesson sequence, each of which represents approximately one week of work, is not intended to be sequential and so can be taught in any order. They are written for upper primary classes but are probably suitable for secondary students as well.The three aspects are: Perspective Nets and properties of 3D

geometrical shapes Maps (this one is not included yet)I might add another on isometric drawing, if the first ones go well.For each aspect, the following presents: results from some data collection

that has informed the emphasis as well as a sequence of the activities.

a sequence of experiences, not written as scripted lessons, just as general ideas. It is assumed that teachers will adapt these to their own students.

The research questions at this stage are:

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Some geometrical experiences: Overview

If lesson sequences are designed to address specific geometrical concepts as identified by student interviews: Which aspects of those sequences

are engaging for students? Which aspects of those sequences

improve responses of students to geometry items on NAPLAN style assessments?

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Preliminary data collectionThere are a number of items on NAPLAN that require students to interpret diagrams and drawings from different perspectives.In an individual interview format, the students were presented with 2009 NAPLAN items on perspective and location that were relevant for their level, with the items read to the students if necessary. In summary the results were: 13/15 of the middle primary

students could choose the correct piece to go into a jigsaw

Given a prism with an L shaped cross section that required students to count faces, 4/15 of the middle primary students could choose the correct response (5/15 counted only the faces that could be seen)

Given a horizontal view drawing of an ice cream cone, 5/15 could choose a circle as the top view.

For the upper primary, given the net of a cube, 5/13 could state which face is opposite a nominated face

Shown a cube with corners cut off and asked to count the remaining faces, no upper primary student could write the answer correctly.

Given a diagram of a rectangular prism, and told that the sum of the faces, edges and vertices is 26, 3/13 could choose the total of the faces, edges and vertices of a square pyramid.

There were also some comparative items that were presented using objects that could be touched or seen and not in the test item format. The results from this were: Given a small cube, 5/28 could name

it, 4/28 could count edges, 18/28

could give the number of faces, and 17/28 could count corners.

Shown a photo of a structure made from cubes, 4/28 could state how many cubes were needed to make it, 1/28 could say how many faces would be painted (one said all of them), 25/28 could make it (development was even across the grades), 23/28 could place a yellow cube to the left of the structure (7/11 grade 4s), 14/28 could put a red cube north of the structure (development was even)

Asked to draw the bird’s eye view of the building they were in, 12/27 could do this.

Asked to draw the water tank from above 17/27 could do this

6/17 Kulkarriya were a able to draw the shape of the homemaker building seen from above, all of whom were from the upper grades

In summary, nearly all of these items were answered correctly by some students, and most students were able to do some of the tasks, including replicating a shape in a photo, and some aspects of the properties of the cube. No student was able to answer all of the items. In general, the year 6/7 students did not seem better able to do the tasks than the 4/5 students.

Overall goals of this sequence of experiences on perspective and locationIt is necessary to explain to the students why they are doing this. The fundamental goal, that you can say over and over, is “imagining”.The capacity to interpret diagrams, drawing and photos is important for some topics in secondary level

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Sequence 1: Perspective and location

mathematics such as finding the volume of objects, and practical problems. It also helps in reading maps, interpreting drawing of buildings, communicating directions, describing objects and understanding instructions. These are important life skills.Hopefully some experience at these tasks will also assist students in completing future NAPLAN items as well.The following some activities intended to provide some relevant experiences.

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Explain the purpose to the students: Explain that they are going to learn about interpreting and making drawings of objects, and how those objects look different from different perspectives, and that this will help them in their lives and also in their future school learning. Explain that one of the key issues for them is to learn the words that are used to describe the ways things look.Teaching experiences:It is assumed that you will use your usual strategies for revising and introducing terms. The strategies you use in AL may be useful. The students need to hear you modelling the use of the terms, and to say the words themselves many times.Other strategies could include

- having labels on parts of the room to show north etc, left right, above, below etc.

- emphasising direction words when telling stories

- asking the AEWs or other locals what words would be used in Kriol or language and have them talk to the class about them. This could include asking specific questions such as “how do you show north? Left? near? above”

- any other strategies that you find work in literacy such as word searches

- connect the terms to what they know (“have you heard the word left before, where? How do you remember left”)

- asking what is similar and what is different about, for example, a square and a rectangle

Key terms for revision are left, right, above, below, under, on top, side, north, south, east, west …Key new words for specific and formal emphasis are face, edge, corner, vertex (plural is vertices), 2 dimensional, 3 dimensional, 2D, 3D, perspective, bird’s eye view.Note that it is suggested that you do some language activities everyday that you are doing this work.As a prompt to this discussion, you could find some images that the students might recognise (perhaps like the ones here). Ask them what the logo or graphic might represent. How do they know? How would they describe the image?

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Step 1: Revising and introducing the key terms

Step 2: Drawing and making to instructions

Explain the purpose to the students: Explain that it is often necessary to give instructions and descriptions, and it is also necessary to be able to interpret them. The following activities give experience at using and interpreting the necessary language.Teaching experiences:a. You will have to adapt this activity to

the equipment you have. The ideal is the sets of Lego blocks with a base plate so that students can hold up the shapes once they have made them. However, you can do this with any materials.Everyone has the same equipment (either individually or in pairs), including you. You make a shape but have it covered. Describe what you have made, with the students having to make what you say. For example, you might say, there is a blue block on top of the red block, there is a yellow triangle to the left of that, there is a another blue block to the north, and so on. Do this a number of times creating different

instructions each time. If they are doing well, make the instructions more complex. The review of this is important. Have the students compare what they build with the one that you built. Are they the same?You might ask one of the students to make a shape and give instructions. In fact they could give the instructions in Kriol if they prefer.

b. Give the students instructions, and ask them to draw them on a small white board. For example, “draw a square, draw a triangle on top of the square, draw a circle to the left, etc”. They can show you their whiteboards to see whether they match what you said. You can comment on their interpretation. Repeat, adapting to the level of success of the students.As before you might ask one of the students to give instructions. In fact they could give the instructions in Kriol if they prefer.

c. Have pile of cubes. Ask the students to:i. Build something using 15 cubes.

Describe what they have built.ii. Build something which is 3

cubes high and 3 cubes wide made with 15 cubes.

iii. Something like “make a tower of 3 cubes, put a yellow cube to the left, and blue cube to the north, …”

Repeat, adapting to the level of success of the students.

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Step 2: Drawing and making to instructions

Explain the purpose to the students: Explain that these activities are for students to experience the number of cubes used to construct a shape even though those cubes may not be immediately evident in the photo. One of the tasks that the students found difficult in the NAPLAN test was imagining unseen but necessary aspects of objects.Make a structure something like this photo using cubes, ideally they should be coloured. This will depend on the equipment you have. You may have linking cubes so you will have to adapt this.Ask questions like:“How many cubes did I use to make this?”“If I painted the outside black including those on the bottom, how many faces would be painted black?”Students can walk around your structure at first, but for later questions have them just look from one side.Repeat this a number of times using different structures.They could also do this in pairs, with one student making a shape and them asking the questions of the other.

Teaching experiences:

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Step 3: How many cubes?

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Explain the purpose to the students: Explain that sometimes drawings and photos show only one perspective of a shape, yet it is necessary to imagine what the rest of the shape might be. There is a particular view that is called “bird’s eye”.Teaching experiences: a. Ask them to draw what the school

bus would look like to a bird flying directly above it. What would be school bus look like to a dog sitting on the road as the bus approached.

b. Show a picture from google earth of the community (on the next page – actually you might be able to get a better one). You might be able to put this on the smartboard. Ask them to name the various buildings or objects they can see. Ask questions like “how do you know that is the river?” Explain that this image is old. What has been build since this photo was taken?

c. Draw a circle on the board. Sat “This is an object seen from above, What might be the object?“ (Note that there is a range of possible anwers to this question so consider all reasonable suggestions. Repeat this for different shapes such as rectangle, etc?

d. Have a photo of something around the school. Ask students to say where the photo was taken from. Ask them to draw what the building would look like if the photo was taken from the opposite side? … if the photo was take from above.

e. Have drawing of a building from above. Ask them to suggest what building might your drawing be?

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Step 4: Taking a bird’s eye view

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Explain the purpose to the students: Explain that this activity will give them the opportunity to revise the work from the previous lessons, and to match up some different views of the same object made from cubes. Emphasis that they will have to explain their reasoning.Teaching experiences: There are two sets of cards: an easy one to start with, and a harder set if they can do this well.The set of cards has 5 different representations of objects made from what they mean. You could ask them to read the individual cards together. Ask them to interpret the information on the cards. These are provided. Have some large versions to read through, and discuss The idea is that students, in pairs, match up the different representations of the objects, and then describe what they have done. Try to give them as little help as possible.You will need to decide whether this is too difficult for the students depending on how they have gone on the earlier steps. If you feel it is too difficult, you might simplify the card set by, for example, removing some of the cards.After they have done that, there is a need to review their answers. One possibility would be to put the cards onto the smartboard. Another might be

to have a large version of the cards. Another might be to have a set with Velcro on the back.

Explain the purpose to the students: This worksheet gives some revision and also extends their knowledge of shapes.Teaching experiences:The students can work on the sheet individually or in pairs. It is quite difficult, so you may need to adapt it.

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Step 5: Sorting the cards

Step 6: Worksheet

I have 5 faces

I have 4 faces and 6 edges

I am a triangular

prism

My net is

I have 6 vertices and 9 edges

I have 12 edges

I am square pyramid

My net is

I have 4 vertices

I have 6 faces and 8 vertices

I am an octahedron

My net is

I have 12 edges

I have 8 edges and 5 vertices

I am a tetrahedron

My net is

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I have 8 faces and 6 vertices

I have 5 faces

I am a rectangular

prism

My net is

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Revision sheet for perspective and location Name ________________________

1. Draw the bird’s eye view of the building you are in.

2. Draw what your house looks like if you standing on the road.

3. Draw a square. Draw a triangle on top of the square. Draw a circle so that it is inside the square. Draw another triangle on the right hand side of the square.

4. Look at this drawing.

a. How many cubes would be needed to make this building?

b. Draw the bird’s eye view of the building

c. If you painted the outside of this building black, (the bottom is painted as well), how many faces would be painted black?

d. Which of these might be the view from the front?

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Preliminary data collectionThere were 3 questions requiring analysis of nets or related tasks in the Year 5 2009 NAPLAN. The questions were asked individually to each student, and read to them if necessary, to minimise the chance that any difficulties were with the language rather than the geometry. The questions were: Given the net of a cube, when asked

to find the face opposite a given face, 5/13 of the upper primary students could answer (this was a write-in answer).

Given the net of a pyramid, and asked to match a drawing with a net, 4/13 of the upper primary students could do this (choosing from 4 options)

Asked to find the sum of number of faces, edges, and vertices of a square pyramid, 3/13 of the upper primary students could do this (choosing from 4 options).

We also asked some more direct questions to the middle and upper primary students. Shown the net of a cube, 1/13 could

draw the net of a square pyramid.In summary, while there are issues with names of shapes, it seems that some students are able to answer most of these questions, suggesting that such concepts are accessible at this level. That the majority of students could not, even in the one on one interview situation,

indicates that it is worth seeking to improve their experiences in this aspect of geometry.

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Sequence 2: Nets

Overall goals of this sequence of experiences on netsIt is necessary to explain to the students why they are doing this.The fundamental purpose is for students to experience the conventions of connecting 2D representations of 3D shapes. It seems a useful step in doing this is through some experiences with nets, in that these show directly how a 2D sheet is converted to a 3D shape.There is also a pleasurable aspect to this, somewhat like an Art lesson, in that the shapes when made are nice to look at. There are also some problem solving aspects of what is proposed.There are 9 steps in this sequence. I have no real idea how long each step will take (and will want you to tell me that) but I think that all of the steps will be needed.Note that there is some great software that goes with these types of activities which would be ideal on the smartboard. You probably have some of these already, but there is also plenty available free on the web.

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Explain the purpose to the students: Explain that they are going to learn about shapes and the words we use to describe them. Once they do this, it will be easier to understand and describe what they are doing in maths.Teaching experiences:It is assumed that you will use your usual strategies for revising and introducing terms. The strategies you use in AL may be useful. The students need to hear you modelling the use of the terms, and to say the words themselves many times.Other strategies could include having posters with the shapes drawn

and labelled having labels on everyday objects

(such as this is a vertex, this is an edge, ...)

matching labels with the objects emphasising shape words when telling

stories connect the terms to what they know

(“have you heard the word cube before, where?”)

asking students to “tell me everything you can about this shape”

asking the AEWs or other locals what words would be used in Kriol or Language and have them talk to the class about them. This could include asking specific questions such as “what is the difference between a square and a cube?” “what would you call this shape?”

sorting sets of shapes into groups and describing the groupings

asking what is similar and what is different about, for example, a square and a rectangle

any other strategies that you find work in literacy such as word searches

Key terms for revision are square, rectangle, triangle, …Key new words for specific and formal emphasis are cube, net, face, edge, vertex (this is a better word than corner), 2 dimensional, 3 dimensionalFurther words that need to be developed include rectangular prism, square pyramid, tetrahedron (triangular prism).Note that it is suggested that you do some language activities everyday that you are doing this work.

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Step 1: Revising and introducing the key terms

Explain the purpose to the students: Explain that whenever they see pictures or drawings of 3D objects they have to imagine what the object is like, not just what they can see directly in the picture.Teaching experiences:Put a big die on the table. Ask students to tell you everything they can about the die. Extend them with questions like “what is the mathematical name for this shape?” “can you tell which number is on the bottom without picking up the die?” “what is the number of the north side of the die?”, what is the number to the left of the die?’ “what number would a bird flying overhead see?” etc.

Explain the purpose to the students:

Explain that one of the key ideas in mathematics is seeing ways that 3D objects (like a box) are sometimes made out of 2 D shapes (like a square or a rectangle). Teaching experiences:Find some boxes, ideally one in the shape of a cube and another that is a rectangular prism (a box shape that is not a cube). Focus on the cube first. Ask them to work out how many faces there are, how many edges, how many corners. Cut it up along the edges. You might need to prepare this beforehand. Ask the students to try to reassemble the box. If the class is big, you might need to have a few boxes.Repeat this for the rectangular prism box.

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Step 2: Exploring the cube

Step 3: Taking apart some boxes to show what they

look like flat

Explain the purpose to the students: These experiences are the reverse of the previous one. It might seem like a waste of time, but actually it is the most important in this sequence because the students need to have the experience. The goal is that they see how the 3D geometrical shapes can be made. It also gives them practice as being careful, and doing things artistically. Teaching experiences:Give them the net of a cube (nets are provided). I have provided two sets: one set has the flaps and the other does not. The nets with the flaps are made by gluing the flaps to the relevant face. The ones without flaps are made by using sticky tape. Choose which ever one you think will work better. Maybe you could try both to see which one works. Emphasise to them that they need to be careful, in doing this, and fold carefully along the lines. Ask them to: decorate the faces in some way, while

it is flat, including with their name fold and make into the cube, - you will

need to explain the need to fold carefully, the ways to tape the edges, etc

once they have done this, ask them to describe their cube to the class;

choose a few of the cubes and ask formal questions like: What is the shape of one face, how many faces, point to an edge, how many edges, point to a vertex, how many vertices (note the plural form – they might see this in the next NAPLAN),

ask problem solving questions like “What is on the face opposite the (you choose something)”

Explain the purpose to the students: This worksheet gives the students practice at imagining which nets can be made into the cube. There is some interactive software that does this. It is also a problem solving type experience. Teaching experiences:The students can work on the worksheet (attached) to imagine which nets can be made into cubes and which cannot. They need to explain their reasoning. You perhaps turn this into a poster and do it as a whole class.

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Step 4: making a cube from a net

Step 5: Imagining with nets

Just imagining

1. Which ones of these can be folded to make a cube?

a) b)

c) d)

e) f)

g)

2. This is the net of a cube.

When it is folded which number will be opposite the 1?

When it is folded which number will be opposite the 2?

When it is folded which number will be opposite the 3?

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4

561 2

3

When it is folded which number will be opposite the 4?

When it is folded which number will be opposite the 5?

When it is folded which number will be opposite the 6?

Explain the purpose to the students: This is the same as the making cubes from nets but it makes a different object. Teaching experiences:Given the net of a rectangular prism (two sets of nets provided). The same comments about nets and flaps etc above apply.

o decorate, including writing in the name

o describe what you think this might look like when it is folded

o fold and make into the rectangular prism taping the edges,

o ask them to tell you everything they can about the rectangular prism,

What is the shape of one face, How many faces, how many edges, how many corners,

What is on the face opposite the (yellow flower?)

Explain the purpose to the students: This is taking the same idea one step further. If you need to skip any activities, then this is the one to skip. However if the students are doing well to this stage, I suggest you do it. It is more important for seeing the beauty of these shapes, but less important in terms of their future NAPLAN tests. Teaching experiences:Give different nets to different students, such tetrahedron, square pyramid, octahedron (again two sets are provided)

decorate, including writing in the name

describe what you think this might look like when it is folded. Maybe have some pictures or even models of these.

fold and make ask them to tell you everything

you can about the shape, What is the shape of

one face, How many faces, how many edges, how many corners,

What is on the face opposite the (yellow flower?)

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Step 6: Making a rectangular prism from a net

Step 7: Making other shapes from nets

Explain the purpose to the students: This is a problem solving activity that requires them to connect the different words, descriptions and nets, and pictures together. Explain that this gives them a chance to use the knowledge they have learned in the previous experiences.Teaching experiences:There is a set of cards, with 5 different representations, of 3D objects. I have included some sets of cards and also a set of large version cards.Read through the large versions with them, and discuss what they mean. You could ask them to read the individual cards together. Ask them to interpret the information on the cards.The idea is that students, in pairs, match up the different representations of the objects, and then describe what they have done. Try to give them as little help as possible.You will need to decide whether this is too difficult for the students depending on how they have gone on the earlier steps. If you feel it is too difficult, you might simplify the card set by, for example, removing some of the objects represented, or removing some of the cards.After they have done that, there is a need to review their answers. One

possibility would be to put the cards onto the smartboard. Another might be to have a large version of the cards. Another might be to have a set with Velcro on the back.

Explain the purpose to the students: This worksheet gives some revision and also extends their knowledge of shapes..Teaching experiences:The students can work on the sheet individually or in pairs. It is quite difficult, so you may need to adapt it.

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Step 8: Card sort

Step 9: Worksheet

REVISION OF NETS WORKSHEET Name ______________________________

1. Write down everything you can about this object

2. Complete the missing parts of this tableName No. of faces No. of edges No. of

vertices

3. On the back of this page, draw the net of this object

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4. I am imagining an object that has at least one face that is a triangle. Draw what this object might be, and draw its net.

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