Designer Maths 1 - MakingMathsMarvellous -...

DESIGNERMATHS

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© Northern Territory of Australia 2005

For further information please contact

Teaching, Learning and Standards DivisionNorthern Territory Department of Employment, Education and TrainingGPO Box 4821, Darwin NT 0801Telephone (08) 8999 3707

Apart from any use permitted by the Copyright Act 1968, the Department of Employment,Education and Training grants a licence to download, print and otherwise reproduce thismaterial for educational (within the meaning of the Act) and non-commercial purposes.

This resource was originally produced by the Implementing the Common Curriculum inAboriginal Schools Program, Darwin, 1996 – 1998. It was revised in 2006.

Original Project Team:Kate Le Rossignol Education OfficerWarwick Pascoe Graphic Artist

Developed from ideas by:Sharon Namijinpa Lajamanu CECPaulina Jurkijevic Peoject Officer Profiling MathsMArk Wilson Non-contact Teacher - Mt Allan ClusterJoe Singh Gunbalanya CECRhonda Inkamala Yipirinya ScoolCheryl-anne Courtney Katherine School of the AirRobynHurley Batchelor CollegeNerissa Aguda Ramingining CEC

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Contents

Introduction Page

Do-Talk-Record................................................................................................iv

Explicit Teaching...............................................................................................v

Resources / Assessment.................................................................................vi

Possible Assessment Structure.......................................................................vii

Learning Experience 1. 2D-Shapes..................................................................1

What shapes are in your classroom?.........................................................2


Can you make a net?.................................................................................5

Learning Experience 3. Transforming Shapes..................................................7

Can you make a picture bigger?.................................................................8

Learning Experience 4. Capacity.....................................................................15

What can you fit in your classroom?.........................................................17

Learning Experience 5. Area...........................................................................19

How would you like to set up your classroom?.........................................20

Learning Experience 6. Perimeter...................................................................23

Can the perimeter of an object change

without changing the area?......................................................................24


Pop-up Cards?.........................................................................................28

Learning Experience 8. 2D-Shapes................................................................30

Which shapes will tesselate?....................................................................31

Learning Experience 9. Volume......................................................................34

Stacking Cubes?.......................................................................................35

Learning Experience 10. Volume and Surface Area.......................................39

Double Ups?.............................................................................................40

Designer Maths

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Do - Talk - Record

Do-Talk-Record is a teaching/learning model that many Northern Territory teachers haveused with success with ESL learners. The model requires learners to thoroughly discuss and doactivities until they have fully grasped the concept before recording any results.

Learners need the opportunity to flow between talking and doing as often as required, beforerecording. They may also need to repeat an activity or dicuss it further before continuingrecording.

Do an activity or experience to create a context and meaning. It should:

• be shared by the teacher and students

• provide the language for real situation and purpose

• provide lots of meaning and help learners to understand new language (it builds thecontext for the language)

• have the teacher playing an important role in planning and initiating the languageused, providing a language model, sharing cultural information and extending theuse of appropriate language

• possible for unplanned results to occur

Talk about the activity with the learners and repeat and practise the language:

• comment on what happened and what was said

• recall events

• recall the language

• practise the language

• use the language for the basis for other language development

Record the language used from the doing and the talking to show what they can do and whatthey have learned. Reflect on the activity, the learning, progress, possible changes that can bemade etc.

TALK

IDEAS

RECORD

write whathappened- negotiated- individual

- cooperative learning

- share with partnerDescribe

- sequence- process- cause- effectRetell

- observations- results

chartslabels

graphslists

gridsdiagrams

tables

interpretmeasure/calculate

observe/watchdesing the classroom

DO

mapscards tesselatescale drawing

Designer Maths

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Designer Maths

Explicit Teaching

Explicit teaching refers to active instruction. It is where new learning is to be developed deeplyand strongly. The explicit teaching stage will vary in length depending on the complexity of whatis to be learned and on the learners themselves. Active instruction involves modelling, activeteaching and learners being engaged in dialogue. Learners are to be provided with a variety oflearning tasks with the teacher guiding them to complete and learn from the tasks. Explicitteaching is used for teaching new concepts and skills.

Because a child is at school it does not mean that they know how school works or how to makethe most of posssible learning opportunities. Explicit teaching means that what is to be learnedis not implied and includes the concept of learning being ‘taught, not caught’.

Explicit teaching of numeracy means• assessing where learners are at and starting at needs level• particular needs of learners being catered for eg language needs of ESL/ESD

learners• conscious planning towards clear outcomes• all stakeholders being informed of learning outcomes

teachers knowing what, how and why they are teaching and learners knowing what,how and why they are learning ie learners are not left guesing about what ishappening and why

• negotiating the curriculum, cleary defining what learners need to know and want tolearn

• teaching the skills and processes to reach the outcomes• enabling skills, processes and concepts to be understood and applied• using a range of materials/representations/contexts for the same concept• mathematical language (vocabulary) is an integral componenet of mathematical• literacy, leading to numeracy• modelling oral language• presenting new vocabulary through shared experiences, using concrete objects etc

and providing opportunities for learners to explore and practise the new language• using appropriate mathematical equipment to undertake mathematical processes

and solve mathematical problems• modelling of numeracy strategies and/or processes eg calculation strategies• planning and programming that caters for multi-levels, cooperative learning groups

and catering for different learning styles and multiple intelligences• teaching and learning being embedded within a context that makes sense to the

learners and is linked to previous lessons and real life experiences• clear, high but realistic expectations - learners knowing what these are• learners being provided with time and opportunities to engage, practise, internalise

and reflect on learning• learners and teachers engaging in metacognitive processes including

self-evaluation and goal setting• finding a medium that is culturally appropriate• drawing out key mathematical ideas during and/or towards the end of the lesson

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Resources• ruler

• mirror

• shapes from Learning Experience 1

• pens

• pencils

• overhead projector

• overhead transparencies

• chalk

• metre rulers

• plasticine

• large construction set (optional)

• calculator

• colour pencils

• scissors

• tables

• A4 paper - white

- coloured

• coloured card

• isometric paper

• square grid paper

• 48 one centimetre cubes (MAB)

Assessment

The term ‘assessment’ in Mathematics refers to the identification and appraisal of students’knowledge, insight, understanding, skills, achievement, performance, and capability inMathematics. (Niss 1998)

Assessment is the purposeful, systematic and ongoing collection of information for usein making judgements about learners’ demonstrations of outcomes. It is an integral part of theteaching/learning process. As teachers plan learning experiences, they also need to plan howthey will collect and monitor learners’ evidence of learning.

A possible assessment structure for this module has been provided. The outcomes chosenby an individual teacher will depend on the emphasis taken when using the module and shouldreflect only the outcomes that will be directly monitored and for which evidence of learningwill be gathered.

Niss, M. 1998. Assessment in Geometry, In C. Mammana and V. Villani (eds). Perspectives onthe Teaching of Geometry for the 21st Century, (pp. 263 – 274), The Netherlands: KluwerAcademic Publishers)

Designer Maths

Possible Assessment Structure

2D-Shapes: What shapes are in your classroom?

Indicators – Can the student: Outcome Evidence Check

Identify and classify simple 2D shapes SS 2.1 Worksheet (Naming shapes)

Identify lines of symmetry SS 2.2 Worksheet/ Drawings

Identify congruent shapes SS 2.1 Work samples (measurements and drawings)

Think creatively and make conjectures as to the purposes shapes are used for in the real world

Cr 1 Work sample (Conclusion)

3D-Shapes: Can you make a net?


Make a 3D object based on a regular 2D shape

SS 2.1 Work sample (3D Model)

Create a viable net SS 2.1 Work sample (Physical net)

Describe a regular 3D object in terms of faces, vertices and edges.

SS 2.1 Work samples (Exploring)

Demonstrate creativity and persistence in constructing a 3D object with congruent faces

Cr 2 Observation Work sample (3D Model)

Transforming Shapes: Can you make a bigger picture?


Measure the area and perimeter of 2D shapes

MDS 3.2 Work samples (Exploring)

Determine the relationship between perimeter and area for changing scales

SM 4.3 Work samples (Exploring)

Enlarge and reduce simple pictures and patterns using a grid

SS 3.3 Worksheets (Grid pictures)

Redraw simple pictures or patterns onto a distorted grid

SS 3.3 Worksheets (Grid pictures)

List some creative uses for distorted grids Cr 1 Work sample (Conclusion)

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Capacity – What can fit in your classroom?


Estimate the number of standard objects that would fit within a given area or volume


Measure a given area in m2 and a given volume in m3


Perform calculations to solve problems involving area and volume

NS 3.3 Work samples (Working data)

Demonstrate reflective thinking in identifying the steps taken in the process of finding a solution

In 1 Work samples (Conclusions)

Area – How would you like to set up your classroom?


Measure a given area using appropriate units

MDS 3.1 Work sample (floor area/furniture)

Create a simple scale plan of a familiar location

SS 3.3 Work sample (Map)

Measure and compare lengths and distances using appropriate units

MDS 2.1 Worksheet

Explain how their floor plan is an improvement on the original

Cr 2 Presentation

Area – Can the perimeter change without changing the area?


Measure and compare the perimeter of an object as the area increases

MDS 3.2 Worksheet

Measure and compare the perimeter of an object as the dimensions change but the area remains constant

MDS 3.2 Worksheet

Determine the relationship between area and perimeter for a given shape.

NS 3.2 Worksheet

Demonstrate creativity and persistence in determining the relationship between area and perimeter

Cr 2 Observation Work sample (Conclusions)

3D Shapes – Pop-up Cards


Construct 3D shapes including a cube and a rectangular prism

SS 2.1 Work sample (Construction)

Describe the difference between a cube and a rectangular prism

SS 2.1 Work sample (Exploring)

Demonstrate creativity and persistence in finding a solution to the “Staircase” problem

Cr 2 Observation Work sample (Staircase)

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2D Shapes – Which shapes will tessellate?


Identify various 2D shapes SS 2.1 Worksheet (Naming Shapes)

Determine if a given shape will tessellate SS 3.2 Worksheet (Classifying Shapes)

Create a tessellating pattern using various numbers of shapes

SS 3.2 Work sample (Tessellating pattern)

Demonstrate reflective thinking in determining the attributes of shapes that permit tessellation.

In 1 Work sample (Conclusion)

Volume – Stacking Cubes Indicators – Can the student: Outcome Evidence Check Construct a 3D object from a 2D representation SS 3.1 Observation

Construct different rectangular prisms which have the same volume

SS 3.1 Observation

Identify congruent 3D shapes SS 3.1 Worksheet

Describe the relationship between the volume of a rectangular prism and its dimensions

MDS 3.2 Work sample (Conclusions)

Demonstrate creativity and persistence in defining the relationship between the volume of rectangular prisms and its dimensions

Cr 2 Observation Work sample (Conclusions)

Volume and Surface Area – Double Ups Indicators – Can the student: Outcome Evidence Check Construct a 3D object from a 2D representation SS 3.1 Observation

Solve problems involving the use of indices N 4.1 Worksheets and Work sample (Summary)

Describe the effect of changing scale on surface area and volume for a rectangular prism

SM 4.3 Worksheets Work sample (Summary)

Demonstrate creativity and persistence in defining the effect of changing scale on surface area and volume for a rectangular prism

Cr 2 Worksheets Work sample (Summary)

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1

2D-Shapes

What shapes are in your classroom?

EsseNTial Learnings• Cr 1

Mathematics Learning Outcomes• SS 2.1• SS 2.2

This learning experience provides opportunities for learners to:-

Explore two-dimensional shapes• Carry out activities in which attributes of shapes are investigated

• sides, angles and diagonals in polygons• symmetry• dissection

• Carry out activities involving the classification of shapes• Carry out activities in which congruence is determined by measurement.• Investigate the features and functions of shape in the environment.• Use several problem solving strategies and, with some reminding, check their

solutions.

Language Focus

Exchanging information (asking questions, making statements and reacting)• Identifying• Describing

Characteristics• Shape• Physical features

Evaluation• Evaluating things seen, heard, done, eaten, etc.• Usefulness

Vocabulary

In order to be participate fully in this learning experience, students need to be familiar with theseterms and be able to use them appropriately in mathematical situations.

classify symmetrical diagonal angle side congruent

Designer Maths Learning Experience 1

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Do - Talk - Record

Initially have the students brainstorm as many 2-D shapes as possible. There couldbe races at modelling the shapes out of plasticine, to demonstrate their knowledge.Next ask them to identify and classify the shapes in their classroom. From their listthey should then be able to identify the most commonly used shape in theenvironment.

Students are to identify which shapes are symmetrical when cut in half. They needto consider if it is cut vertically, horizontally and diagonally. Using a mirror will makeit a simple task.

The final part of the activity requires the students to locate any shapes which arecongruent. So they do not have to measure every shape, they can decide whichshapes they think are congruent visually and then confirm their guesses bymeasuring the shapes.

It is important when the students are brainstorming all the shapes that they are clearwhich name goes with which shape. Have them draw and label the shapes ifnecessary and discuss how the name can show the number of sides in a shape.e.g hexagon - 6, octagon - 8, triangle - 3, quadrilateral - 4.

After doing the activity with the mirror the students may like to find other shapeswhich are not uniform, like people, to decide if they are symmetrical. They could thenmake a display of unusual symmetrical shapes.

What shapes are in your classroom?

What you need ruler

mirror

What you do Identify and classify shapes in the classroom.

Dissect the shapes to find which are symmetrical.

Measure the shapes to find which ones are congruent.

List real life uses for different shapes.


Talk

Record

Do

Do

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Getting Started

• Draw and label eight 2- dimensional shapes.

Exploring the Problem• Identify the shapes in the classroom and classify them.

Object Shape Number of Sides

• What is the most common shape?• What is the least common shape?

Working the DataSome things are symmetrical when cut in half one way, but not another, like a person’s face.Circle the triangle below that gives symmetrical shapes when cut in half.

vertically horizontally diagonally

• Draw the shapes which are symmetrical when they are cut vertically, horizonatally ordiagonally.

• Use a mirror to help you. Place a mirror on the line-of-cut. Do you see the originalshape?

• What shapes are always symmetrical no matter which way they are cut?• Can you find objects in the classroom that are congruent in shape?• Measure all the sides to be sure.• Write the shape and its measurements.

Drawing Conclusions

• What purpose is the most common shape used for?


Record

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3D-Shapes

Can you make a net?


Mathematics Learning Outcomes• SS 2.1


Explore two-dimensional shapes• Carry out activities in which attributes of shapes are investigated

• vertices, faces and edges.• symmetry• dissections and sections

Explore three-dimensional shapes• Carry out activities in which congruence of shapes is determined by measurement.• Make models and drawings of a variety of three-dimensional shapes

Including cylinder, cone, prism, pyramid, octahedron.• Contribute to discussions on different ways of using mathematics for

solving problems.

Language Focus

Exchanging information (asking questions, making statements and reacting)• Identifying

Characteristics• Shape• Physical appearance

Relationships between units of meaning• Temporal relationships (before and after)

Vocabulary

In order to do this activity, students should be familiar with these terms and be able to use themappropriately in mathematical situations.

two-dimensional (2D) faces vertices congruent

three-dimensional (3D) edges regular polyhedra


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Do - Talk - Record

The students can use a shape from learning experience one or another of theirchoice. They need to make multiple copies which have to be measured carefully soall the shapes are congruent. Even have the students check the measurement oftheir shapes after they have cut them out because they need to identify congruenceby exact measurements, not just by eye.

The 2D shapes need to be taped together into a 3D shape. The shape needs to beflattened out and traced around to make a net for a partner to make up. Once theshape is made they need to count the faces, edges and vertices. To finish they needto decide if the shape is symmetrical and if it’s faces are congruent.

Students have used the term congruent in earlier stages, but they may needreminding of its meaning.

‘Shapes are congruent if the size and shape are identical.’

‘Choose two 2D shapes that you think are congruent.’

‘A 3D shape that is made up of congruent 2D shapes has a special name - a regularpolyhedra. Can you think of a 3D shape like that?’

‘Yes, a cube, because each side is exactly the same.’

The made up nets would make an effective mobile to display students work.

Can you make a net?

What you need shapes from learning experience 1

ruler

What you do Choose a shape from learning experience 1.

Trace around it to make multiple copies.

Measure the sides carefully.

Tape the shapes together to make a 3D shape.

Make a net of your shape and give it to a friend

to make up.

Answer questions about the faces, vertices and edgesof your shape.

Is your shape symmetrical ?

Are your shapes congruent?


Talk

Record

Do

Do

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Getting Started

• Name eight 3D shapes.

Exploring the Problem

• Name a 3D shape with 6 faces.

• Does it have congruent faces?

• What was the shape/s of the faces?

• Name a 3D shape with 8 faces.

• Does it have congruent faces?

• What was the shape of the faces?

• Use the shapes from activity 1.

• Measure around one of them and draw it onto cardboard.

• Make multiple copies.

• Keep the shapes congruent so you can make a 3D shape with them.

• Measure each side of the shapes after you cut them out. Are they still congruent?

• Tape your 2D shapes together to make a 3D shape.

• Does the 3D shape have a name?

• Lay your 3D shape out flat and trace around it to make a net on cardboard.

• Draw the net here and a diagram of the shape when made up.

• Give it to a partner to cut out and make up.

• How many faces does your 3D shape have?

• What are the names of the faces?

• How many edges?

• How many vertices?

• Is it symmetrical?

• Find a 3D shape which is congruent to yours.

Drawing Conclusions

• You kept all the faces of your shape congruent. Did the 3D shape have congruentfaces when it was put together?

• Find out the name of 3D shapes with congruent faces.

• Name some other 3D shapes whose faces are congruent.


Record

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Transforming Shapes

Can you make a picture bigger?


Mathematics Learning Outcomes• SM 4.3• SS 3.3• MDS 3.2


Transform shapes• Reduce and enlarge two-dimensional shapes using grids and projective light• Investigate change of shape by distortion• Make and use maps employing the ideas of scale and co-ordinates.• Extend problems by posing their own questions and identify practical applications of

mathematics around them.

Language Focus

Exchanging information (asking questions, making statements and reacting)• Comparing


Space• Location• Movement

Relationships between units of meaning• Temporal relationships (before and after)

Learning-how-to learn skills• Express their own opinions

Vocabulary

In order to do this learning experience, students should be familiar with these terms and be able to usethem appropriately in mathematical situations.

enlarge distort area grid ratio perimeter


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Do - Talk - Record

When the students are doing the first part of the learning experience, (enlarging andreducing pictures with an overhead projector), it would work best done in pairs orsmall groups. The students need to find out if there is any pattern between area andperimeter when an object is enlarged or reduced. You will need to photocopy thepictures the students choose onto overhead transparencies. Once they havecompleted the maths activities, the overheads could be used to print T-shirts.

Next the students enlarge and reduce pictures and patterns using a grid beingcareful to enlarge or reduce by length and width. The learning experience concludeswith the students redrawing objects in various distorted grids and deciding how thearea may have altered.

Discuss with the students what the object of the learning experience is.

The grid will make an effective display if they were cut out and backed on colouredcard, possibly with a heading produced on a distorted grid.

Can you make a picture bigger?

What you need pens

pencils

overhead projector

overhead transparencies

What you do Choose a picture that you would like on a T-shirt.

Copy it onto an overhead transparency.

Enlarge and reduce it on an overhead projector.

Compare the changes in the area and perimeter.

Enlarge and reduce simple pictures and patterns using agrid.

Redraw simple pictures and patterns onto a distorted grid.


Talk

Record

Do

Do

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Getting Started

• Choose a picture you would like on a T-shirt, about the size of a playing card.

• Put a border around it if there is not one.

• Measure the area.

• Measure the perimeter.

Exploring the Problem

• Using an overhead projector enlarge your picture to 2 times the size. (Make eachside twice the length of the original.)



• Enlarge the picture 10 times.



• Reduce the size of the original by 1/2.



• Reduce or enlarge your picture to a size that would suit your T-shirt.

• Is there any pattern between the size of the area and the size of the perimeter aftereach exchange?


Record

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Working the DataInstead of using an overhead projector to enlarge or reduce a picture you can use a grid

This pattern was enlarged with a ratio of 2:1.


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Redraw this picture with a ratio of 3:1 (enlarge)


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Redraw this pattern with a ratio of 1:2 (reduce)

You can change the look of a picture by redrawing it onto a distorted grid.


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Redraw this pattern onto the 2 different distorted grids below.

See what happens to the shape when it is redrawn on a distorted grid.

You can create interesting headings with letters on distorted grids.


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Drawing Conclusions• Would the area of the pictures or shapes change when drawn on a distorted grid?• How could you use distorted grids in your school work or everyday life?• In the box below make your own distorted grid which would give an interesting effect

to thepicture below:


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Capacity

What can you fit in your classroom?

EsseNTial Learnings:• In 1

Mathematics Learning Outcomes• MDS 3.1• NS 3.3


• Use concrete experience to determine areas for squares and rectangles.• Relate measurement of area to other measures.• Relate measurement of volume and capacity to other measures.• Multiply whole numbers and decimals by whole numbers.• Use several problem solving strategies and with some reminding check their

solutions.• Generate some problems of their own and identify a general solution.

Language Focus

Exchanging information (asking questions, making statements and reacting)• Enquiring about or stating facts

People, places, things, events, qualities and ideas• People• Things• Number

Vocabulary

In order to do this learning experience, students should be familiar with these terms and be ableto use them appropriately in mathematical situations.

area capacity square cubic


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Do - Talk - Record

Students find this learning experience fun when they all have to pack into a smallarea. They are surprised how many fit into one square metre. Because so manystudents are needed it is best to work as a whole class. The second part of thelearning experience (when they devise their own problem) is better to be done inpairs or small groups. To calculate the area the students should not multiply lengthby width. They should start by using repeated addition unless they discover thelength by width formula themselves. The same applies when they need to calculatethe capacity of the room. They could add on layers. The bottom layer would betreated as the floor covered with students up to one metre high. Then add on anotherlayer of one metre and then maybe a third layer, depending on the height of theroom. If your school does not have the large one metre construction rods andjoiners, you could still make a cubic metre using metre rulers and plasticine placed inthe corner of the room to give stability.

When the learning experienceis repeated using a different object the approach willremain the same - calculate how many fit in a given area first and then work out thecapacity.

If a student chooses a small object such as a marble, it will be a little difficult to covera square metre let alone fill a cubic metre. In that case have the students use arealistic measure - e.g. 10 square cm and 10 cubic cm.

The students need lots of discussion about the best method to calculate the areaand capacity of the classroom. Do all the activities for the area before mentioningcapacity.

‘We haven’t got enough students to cover all the floor so how can we work out howmany are needed?’

‘If we find out that 15 fit in 1 square metre then how can we find out how manysquare metres in the classroom?’

‘Yes, we could add the square metres from one side of the room to the other untilthere is a long line of them. Then we could add the lines until we covered the wholefloor in square metres.’

Once the students have done their calculations they could draw a picture of theirclassroom full of elephants or basketballs or whatever they chose.


Talk

Talk

Record

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What can you fit in your classroom?

What you need chalk

metre rulers

plasticine

large construction set (optional)

calculator

What you do Estimate, then find how many students are needed to cover1 square metre.

Measure how many square metres are needed to cover theclassroom floor.

Calculate how many students are needed to cover the wholefloor.

Estimate then find how many students are needed to fill 1cubic metre.

Measure how many cubic metres are needed to fill theclassroom.

Calculate how many students are needed to fill the whole

room.

Follow the same procedure to find out how many of anotherobject are needed to fill the classroom, eg peas, elephants,

footballs.


Do

18

Getting Started

• Estimate how many students would fit in your classroom.

Exploring the Problem• Mark a square metre with chalk on the floor.• Estimate how many students can fit on your square metre.• Get as many students as you can to stand in your square metre.• How many were there?• How many square metres will it take to cover the floor of your classroom?• Now multiply the number of students who fit on a square metre by the number of

metres that cover the floor.• How many students would it take to cover your classroom floor?

Working the Data• If you wanted to fill the whole classroom with students floor to ceiling with students,

estimate how many would fit.

• Calculate how many students will fit in a cubic metre.• You could use construction rods or metre rulers held together with plasticene.• How many cubic metres in your classroom?• Multiply the number of students who fit in a cubic metre by the number of cubic

metres in your classroom to find how many students are needed to fill the wholeroom.

• What else could you use to fill your classroom? eg smarties, marbles, basketballs,peas etc

• Choose something and do the calculations.• Estimate how many will fit on a square metre.• Check by covering the square metre with your object (if it is very small you may need

to just cover 1/10 of it and multiply by 10).• Now estimate how many will fit in a cubic metre.• Is it possible to check? If so, now calculate how many will fit into the whole room.

Drawing Conclusions• Did knowing how many things fitted into a square metre help your estimate for a

cubic metre?• How did you work out how many things fill the whole room?• Were your estimates more accurate for the second group of things you planned to fill

the classroom with?• Why?


Record

19

Area

How would you like to set up your classroom?


Mathematics Learning Outcomes• MDS 2.1• MDS 3.1• SS 3.3


• Investigate position and layout in the environment.• Make scale models and drawings of familiar structures and areas.• Carry out activities involving measurement to the nearest metre, centimetre and

millimetre.• Use concrete experience to determine areas for squares and rectangles.• Use several problem solving strategies and, with some reminding, check their

solutions.• Generate some problems of their own and identify a general solution.

Language Focus

Exchanging information (asking questions, making statements and reacting)• Enquiring about or expressing likes, dislikes, preferences.


Space• Location

Vocabulary


area grid scale


20

Do - Talk - Record

As there will only be a few possible ways to layout your classroom, and the finaldesigns will have had thought and discussion put into them, it is recommended thatthis learning experience is done in groups. You may decide to have an award for thebest design. If the students did learning experience 4 they will not have to remeasurethe area of the classroom and can outline it straight onto the grid paper. They willneed to carefully measure and mark in items such as windows and doors.

Every object in the classroom which takes up floor space needs to be carefullymeasured and drawn onto grid paper using the same scale as the classroom. Makesure each object is clearly labelled. Have students produce their own classroomdesign. Have them think about different table /desk arrangements because it canaffect the number of available seats. If students are familiar with a logo programsuch as Microworlds, this program could be used to scale down and move objects.

Before the students begin working in their groups to design their new layout, havethem list all the essential aspects of their classroom, eg floor seating, a quiet area,display tables.

‘Imagine everything is out of the classroom. What furniture do we need so we canwork?’

‘We can list the things that we need to keep’

‘Do we have to leave some sort of spaces?’

‘Is there anything new that you would like added to our classroom?’

‘Maybe you have a special idea you want to share with your group that will go in yourdesign.’

The students design will make a great display, but over the space of a term, theclass could be rearranged to trial each of the designs. The students could then voteon the most successful design with an award going to the designers.

How would you like to set up your classroom?

What you need ruler

colour pencils

scissors

What you do Measure the area of your classroom.

Draw it on grid paper.

Measure and mark permanent fixtures in the room.

Measure the objects in your classroom.

Draw them on grid paper and cut them out.

Arrange them on the classroom grid to find how muchspace they need.

Arrange them on the classroom grid to find the bestdesign.


Record

Do

Talk

Do

21

Getting Started

• Estimate how much floor area would be covered by everything in your classroom.

Exploring the Problem• Measure the length and width of the classroom floor.• Calculate the area of the classroom floor.• Measure up some grid paper to represent your classroom.• You could use a scale of 1 sq cm =1 sq mm.• Mark in the door/s, windows, whiteboard - anything else that is permanent around the

outside of the room.• You will need to measure the length of all these things.• To put them in the right place on the wall, measure the distance from the corner.

Object Length Distance from Corner

door 1

door 2

windows 1

windows 2

blackboard

pin board


Record

22

Measure the surface area of the desks, chairs, tables, bookcases, display shelves, tray trolleysand any other furniture in your classroom.

length width length width

student desks/tables teacher’s desk

chairs teacher’s chair

bookcase 1 table 1

bookcase 2 table 2

bookcase 3 display shelf

tray trolley 1

tray trolley 2

Using the same scale as for the room, calculate the area for the objects and mark them on• a separate piece of grid paper.• Label and colour each item.• Cut out each item from the grid paper.

Working the Data• Arrange the objects on the grid paper scaled to the size of the classroom.• Put everything in one corner - what area is covered?• How close to your estimate?• What percentage of area is covered?• How much area is left in the room to move around?

• What are some things which have to be included in your classroom design?eg an area for everyone to sit on the floor.

• Now arrange the objects in a way that would be easy to work.• Draw your final copy on 1 cm grid paper.

Drawing Conclusions• Did your design include everything it needed?• Did you add anything new or take away anything?• Is it easier or harder to move around the room with the changes?• Convince the others in your class that yours is the best design.


23

Perimeter

Can the perimeter of an object change without changing the area?

EsseNTial Learnings:• Cr 2 Translates innovative thinking into action and is willing to

take risks when challenged by setbacks.

Mathematics Learning Outcomes• MDS 3.2 Explain and use the relationship between units of measurement

and between length, area and volume when solving problems

• NS 3.2 Use order of operations and number relationships to solveequations; identify and explain the rule used to generate anumber pattern

This learning experience provides opportunities for learners to:

• Measure and compare the perimeter of an object as the area increases.• Measure and compare the perimeter of an object as the dimensions change but the

area remains constant.• Determine the relationship between area and perimeter for a given shape.• Demonstrate creativity and persistence in problem solving

Language Focus

Exchanging information (asking questions, making statements and reacting)• Comparing


Relationship between units of meaning• Logical relationships (cause and effect)• Comparison

Vocabulary

In order to do this activity, students should be familiar with these terms and be able to use themappropriately in mathematical situations.

arrangement area perimeter


24

Do - Talk - Record

In this learning experience the students will learn that the relationship between areaand perimeter does not remain constant.

The learning experience is planned around class tables with a width that is half thelength. First establish that a table is one (1) unit of area and it is possible to fit 6chairs around it. Then progress experimenting with different table arrangements andthe affect they have on the number of chairs which can go around. Where the activityrequires the use of 9 tables have the students draw the possible arrangements ifthere are not enough tables. The students then need to explain which tablearrangement is best and why.

Have the students work in groups and before they make up each arrangement, havethem estimate the numbers of chairs which will fit. Have them talk about what ishappening with the chairs.

‘One table can fit 6 chairs around it. How many will fit around 2 tables?’

‘If we put the 2 tables together, how many chairs will fit around?’

‘Why are there less chairs now the tables are together?’

‘What happens when there are 3 tables together?’

The learning experience can be finished by relating back to Learning Experience 5 todemonstrate where the information can be used.

Can the perimeter of an object change without changing the area?

What you need tables

rulers

What you do Find how many chairs equal the perimeter of a table.

Find the change in number of chairs when a pair of tablesare arranged different ways.

Find the change in number of chairs when three tables arearranged different ways.

Use the information to help you work out how many chairsare needed for 9 tables.

Find which design gives the most seats.


Talk

Record

Do

Do

25

Getting Started

• Place 6 chairs around a classroom table.

• Have you done it this way?• Sit at the table, everyone must have enough space.• The perimeter equals 6 chair spaces• The area equals 1 desk.

Exploring the Problem• Place 2 desks together like:

• How many chairs can you place around this arrangement?• You probably found 10 chairs fitted like the diagram below.

Area = 2 (2 tables)Perimeter = 10 (10 chairs)

• Put the 2 tables together another way.• How many chairs can you use?• Draw it.• Can more, less or the same number of children sit around this arrangement?


Record

26

Place 3 desks end-to-end.

Area = 3Perimeter =

• How many chairs can be placed around 3 tables?• Find the 2 other ways of arranging 3 tables.• Draw them and find the number of chairs.

Area = 3 Area = 3Perimeter = Perimeter =

Working the Data• Place 9 tables in the 3 different arrangements and place chairs around.• Draw them below.

1 2 3

• Explain which arrangement would be best.

Drawing Conclusions• What have you discovered about the changing of area and perimeter?• Does the arrangement of the tables affect how many students can sit?• How many students can be seated with your room design from the last learning

experience?• Is that enough for your class?• What arrangement with the tables in your room would allow the most students to sit?

• Draw it.


27

3D-Shapes

Pop-up Cards

Essential Learnings• Cr2 Translates innovative thinking into action and is willing to take risks when

challenged by setbacks.

Mathematics Learning Area• SS 2.1 Recognise, describe, draw and make a range of 2D shapes and 3D

objects and use some geometric language to describe theirfeatures and functions


• Construct 3D shapes including a cube and a rectrangular prism• Describe the difference between a cube and a rectangular prism• Demonstrate creativity and persistence in problem solving

Language Focus

Exchanging information (asking questions, making statements and reacting)• Asking for and giving information• Describing

Organising and maintaining communication• Seeking of confirmation• Asking someone to explain what they just said


Vocabulary


fold cube rectangular prism


28

Do - Talk - Record

The first part of this learning experience needs to be done as a demonstration withthe students working along with you.

It would be a good idea to have samples of what you are wanting the students toachieve. Take the students through step-by-step as explained in the student’ssheets. Have them place another A4 piece of paper of a different colour at the backto help create the other sides of the cube. Next have the students progress ontomaking a rectangular prism themselves, but make sure they are aware the sidelengths will change.

Once they have managed to make a rectangular prism allow them to progress ontosmall prisms coming from large ones or staircases. If they are made in 2 colours ofcard, they can be quite effective. With the designs that work, glue the outside edge ofthe 2 pieces of card and when the students close the card make sure the pop-uppart is folded the right way.

Students need to be able to discuss what is happening.

‘What sort of shape appears to be here?’

‘Even though there is nothing on all the faces?’

‘What makes us think something is on the faces?’

‘What is going to be different with the rectangular prism from the cube?’

‘How are you going to work out the length of the lines?’

‘Try to picture it, the faces of the rectangular prism coming out of the card. Whichpart has the short edge?’

The final cards will be a wonderful display done in two colours of card.

Pop-up Cards

What you need scissors

A4 paper - white

- coloured

coloured card

pencil

ruler

What you do Follow directions to make a pop-up cube.

Make your own pop-up rectangular prism (with help).

Make a pop-up with 2 rectangular prisms.

Make a staircase pop-up card.


Talk

Record

Do

Do

29

Getting Started

• How many faces does a cube have?• How many faces does a rectangular prism have?

Exploring the Problem• Fold an A4 piece of paper in half.• Open it up with half flat on your desk and the other half straight up.

• Treat the fold as one of the edges of a cube.• Measure 10 cm in the middle of the fold.• Measure 10 cm at right angles from both ends of the 10 cm line.

• Cut along the two 20 cm long lines. (only)• Push the centre piece out and fold the centre fold back the other way.

(Try putting a piece of paper of a different colour behind.)

• Join the lines

• Is there paper for every face?• Count the faces.• What sort of shape do you have?

Working the Data• What is different from a rectangular prism to a cube?• Using the method shown to make the cube, make a rectangular prism.

either or

• Make another rectangular prism, this time with a smaller one coming off the biggerone.

• If you have time you could make a staircase (which is lots of rectangular prisms) withcoloured paper to be a pop-up card.

Drawing Conclusions• Although there is no paper on some of the sides, how do we see a cube or a

rectangular prism?• What did you find was different from when you measured the sides of the cube to the

rectangular prism?• What did you have to do to make the small prism fit the bigger one ?


Record

30

2D-Shapes

Which shapes will tessellate?

EsseNTial Learnings• In 1 Uses own learning preferences and meta-cognitive processes to

optimise learning.

Mathematics Learning Area• SS 2.1 Recognise, describe, draw and make a range of 2D shapes and 3D

objects and use some geometric language to describe theire featuresand functions.

• SS 3.2 Create and describe patterns and designs based on symmetrical ortessellating figures.


• Identify various 2D shapes• Determine if a given shape will tessellate• Create tessellating patterns using various numbers of shapes• Demonstrate reflective thinking in assessing their own learning

Language Focus

Exchanging information (asking questions, making statements and reacting)• Identifying• Describing

Characteristics• Shape• Physical appearance• Colour

Space• Location• Movement

Vocabulary


parallelagram rombus trapezium hexagonquadrilateral pentagon tesselate angle


31

Do - Talk - Record

This allows the students to become familiar with more shapes whilst increasing theirunderstanding of why various shapes tessellate. Have the students cut the shapesout and experiment until they find the groups of shapes which will tessellate.

Have the students discuss the reasons why certain shapes will tessellate.‘The square and the right angle triangle will tesselate, why do they fit together?’

‘What is different about the equilateral triangle so that it will tesselate with thehexagon?’

‘Which group would you put a rhombus in?’

‘Where does the pentagon belong?’

Allowing the students time to colour the tessellations will also highlight the patterns.Try not to use excessive number of colours or the effects will be lost.

Which shapes will tessellate?

What you need scissorspencilcolour pencils or textasisometric papersquare grid paper

What you do Name the shapes.Cut out the shapes.Sort the shapes into groups which will tessellate.Make a tile pattern with 2 shapes on isometric paper.Make a tile pattern with 3 shapes on square grid paper.

Make a tile pattern with 4 shapes on isometric paper.


Talk

Record

Do

32

Getting Started

• What does tessellate mean?

Exploring the Problem• Name the shapes below.

• Photocopy this page and cut out the shapes.


Record

33

• Not all shapes will tessellate.• Divide the shapes into groups which will tessellate and write the names.

(You may use a shape in more than one group.)

Group A Group B Group C Group D

Working the Data• Using the isometric paper make a pattern with two shapes.• Using the square grid paper make a pattern with three shapes.• Using the isometric paper make a pattern with four shapes.

Drawing Conclusions• What helped you to work out which shapes would tessellate?


34

Volume

Stacking Cubes

EsseNTial Learnings• Cr 2 Translates innovative thinking into action and is willing to take risks when


Mathematics Learning Area• SS 3.1 Recognise different 2D representations of 3D objects, including nets and

cross-sections, and use geometrical language to describe, compare andclassify features and functions of 2D shapes and 3D objects whencomparing, classifying, drawing and constructing these.

• MDS 3.2 Explain and use the relationship between units of measurement andbetween length, area and volume when solving simple problems.


• Construct a 3D object from a 2D representation• Construct different rectangular prisms which have the same volume• Identify congruent 3D shapes• Describe the relationship between the volume of a rectangular prism and its

dimensions• Demonstrate creativity and persistence in problem solving

Language Focus

Exchanging information (asking questions, making statements and reacting)• Responding and reacting to requests for information, statements and comments.

People, places, things, events, qualities and ideas• Number


Cognitive Processing Skills• Draw conclusions, using given information.

Vocabulary


length width height volume prism


35

Do - Talk - Record

Using the concrete items (the cubes) rather than numbers helps students tounderstand how to calculate the volume of an object. The tables used to present theanswers will also make it clearer for students. The diagrams with the cubes inperspective may prove difficult for some students to draw. If so give them someisometric paper to draw their diagrams on and then they can be cut out.

Spend time talking about the difference between length, width and height and how torecognise them.

‘If you had to match these three words with these three, which would you do?long, wide, high - length, width, height’

‘When do we most often hear height talked about?’

‘How do we tell the difference between length and width?’

‘So which is the width?’

An extension learning experience could be for the students to continue with the 48cubes finding all the possible arrangements.

Stacking Cubes

What you need 48 one centimetre cubes (MAB)pencil

What you do Arrange the MAB cubes to match the diagram.Count the number of cubes used for length, width and height.Make the 3 other possible prisms.Record your results.With a partner make all possible rectangular prisms with abase of 12.Record your results.


Talk

Do

Do

36

Getting Started

• We can build rectangular prisms using MAB blocks.• What is a rectangualr prism?

Exploring the Problem• Using 12 cubes build the prism below.• You will find the length is 4 cubes, the width is 3 cubes and the height is 1 cube.

• Using 12 cubes make more prisms.• You will only be able to make 4 different prisms.• All others will be congruent to one of the four, but on a different base.

e.g. 4 x 3 x 1 prism is the same as 3 x 1 x 4 prism

In the table over the page, draw a diagram of the prisms you made.• Count the length, width and height of each prism.• Multiply the three numbers together for each prism.


Record

37

Diagram Length Width Height L x W x H

• How many were congruent to the first prism?

• How many times did you change the base for the second prism with its area stillstaying the same?

4 3 1 4x3x1=12


38

Working the Data• Do this learning experience with a partner.• You will need 48 cubes. Build prisms with a volume of 48, but all must have

a base of 12. (Always 12 cubes resting on the table).• Record as you did the last ones.

Drawing Conclusions• What did you learn about the finding the volume of solids?

Diagram Length Width Height L x W x H

6 2 4 6x2x4=48


39

Volume and Surface Area

Double Ups

EsseNTial Learnings• Cr 2 Translates innovative thinking into action and is willing to take risks when


Mathematics Learning Outcomes• SS 3.1 Recognise different 2D representations of 3D objects, including nets and

cross-sections, and use geometrical language to describe, compare andclassify features and functions of 2D shapes and 3D objects whencomparing classifying, drawing and constructing these.

• SM 4.3 Use relationships between area and vbolume, ltime and distance, anglesand lines, and Pythagoras’ theorem: use and interpret scale, detailedplans, maps and other representations of objects.

• N 4.1 Represent, compare, order and manipulate numbers including fractions,decimals, percentages, directed, ratios, surds, pi and indices


• Construct a 3D object from a 2D representation• Solve numerical problems involving the use of indices• Describe the effect of changing scale on surface area and bolume fo a rectangular

prism• Demonstrate creativity and persistence in problem solving

Language Focus

Exchanging information (asking questions, making statements and reacting)• Asking for and giving information• Describing

Organising and maintaining communication• Seeking of confirmation


Vocabulary


model face surface area prism volume square cube cubic


40

Do - Talk - Record

This learning experience should follow on quite well from the previous one. Thestudents will be reinforcing terms from there and earlier work. Being able to build themodels and draw them will allow the students to see the changes from a singlemodel to a double model.

The students should have already done some work with indices - square numbers.This activity takes it a step further and introduces cubic numbers, using the visualcues to aid understanding.

As with Learning Experience 10 the students may have difficulties drawing therectangular prisms in perspective. So supply them with isometric paper to do theirdiagrams on, and then cut and paste them onto the table.

There is a lot of text in the work sheet. Work through the first two sections of it as aclass, spending time making sure the students all understand the terms.

‘What is volume?’

‘How do we know it is a cube?’

‘What is the surface of something?’

‘How many cubes do you think we need to make a double cube?’

‘ Yes, it wasn’t 2 or 4, but 8.’

The completed table with all the diagrams will be a good record of the work. You mayneed to enlarge the table as the diagrams may become awkward to fit on once thestudents are dealing with the larger numbers.

Double Ups

What you need A supply of MAB unit cubes

What you do Count the faces of a MAB unit.

Double the size of the cube.

Count the number of cubes and write it as a sum.

Count the faces and write it as a sum.

Make models using 3, 4 and 5 cubes and then double modelsof each.


Talk

Record

Do

Do

41

Getting Started

• Count the faces of one of the cubes.• Check your answer with the one in the table over the page.

Exploring the Problem• Make another cube, but double the size.• Double the length, double the width and double the height.• Count the number of cubes and the number of outside faces (surface area).• Record in the table.

• How many unit cubes in the double size cube?

1 x = 8

The double cube has double (x 2) the length, double (x 2) the widthand double (x 2) the height.

So the double cube has 2 x 2 x 2 = 8 times as many cubes.

2 x 2 x 2 = 23because 2 is multiplied by itself 3 times.

We say 2 is the base and 3 is the index.

Complete the table below

13 = 1 x 1 x 1 =

23 = 2 x 2 x 2 = These are cubic numbers

33 = 3 x x 3 =

43 = 4 x x =

53 = x x =


Record

42

• Look at the shaded faces in the table.• One face becomes 4 when it is doubled.• The surface area increases 2 x 2 = 22 = 4 times

when the solid is increased to double.

• Complete the table below

12 = 1 x 1 =

22 = 2 x 2 =

32 = 3 x = _____ These are square numbers

42 = x =

52 = x =

Working the Data• Make the rest of the shapes in the table.• Use 2, 3, 4 and 5 cubes as shown.• Make the model first and then the double model.• Count the number of cubes and the number of surface areas.

Drawing Conclusions• Look at the table.

• If you double the model, how much does the surface area increase by?

times

• If you double the model, how much does the volume increase by?

times

• Try building some models of your own using 5 or 6 cubes.

• Add them to the table (be sure to make the double models too).


43

Model

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24

2

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23

=

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44

Model

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Desi

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