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RIC-6098 6.9/950


Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5) Published by R.I.C. Publications® 2013 Copyright© Linda Marshall 2013 ISBN 978-1-921750-95-3 RIC–6098

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A number of pages in this book are worksheets. The publisher licenses the individual teacher who purchased this book to photocopy these pages to hand out to students in their own classes. Except as allowed under the Copyright Act 1968, any other use (including digital and online uses and the creation of overhead transparencies or posters) or any use by or for other people (including by or for other teachers, students or institutions) is prohibited. If you want a licence to do anything outside the scope of the BLM licence above, please contact the Publisher.

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All material identified by is material subject to copyright under the Copyright Act 1968 (Cth) and is owned by the Australian Curriculum, Assessment and Reporting Authority 2013. For all Australian Curriculum material except elaborations: This is an extract from the Australian Curriculum. Elaborations: This may be a modified extract from the Australian Curriculum and may include the work of other authors. Disclaimer: ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that: • The content descriptions are solely for a particular year and subject; • All the content descriptions for that year and subject have been used; and • The author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject. You can find the unaltered and most up to date version of this material at http://www.australiancurriculum.edu.au/ This material is reproduced with the permission of ACARA.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Foundation) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 1) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 2) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 3) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 4) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5) Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 6)

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AUSTRALIAN CURRICULUM MATHEMATICS RESOURCE BOOK: MEASUREMENT AND GEOMETRY (YEAR 5) Foreword Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5) is one in a series of seven teacher resource books that support teaching and learning activities in Australian Curriculum Mathematics. The books focus on the measurement and geometry content strands of the national maths curriculum. The resource books include theoretical background information, resource sheets, hands-on activities and assessment activities, along with links to other curriculum areas.

r o e t s Bo r e p ok u S Contents

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Format of this book ...................................................................... iv – v

Location and transformation .................................................... 58–115

• UUM – 1

• L&T – 1

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Using units of measurement ......................................................... 2–41 Choose appropriate units of measurement for length, area, volume, capacity and mass (ACMMG108)

Use a grid reference system to describe locations. Describe routes using landmarks and directional language (ACMMG113)

– – – – – –

– – – – – –

Teacher information ............................................................................... 2–4 Hands-on activities ................................................................................. 5–8 Links to other curriculum areas .................................................................... 9 Resource sheets .................................................................................. 10–16 Assessment ........................................................................................ 17–18 Checklist .................................................................................................... 19

• UUM – 2

Teacher information ........................................................................... 58–59 Hands-on activities ............................................................................. 60–61 Links to other curriculum areas .................................................................. 62 Resource sheets .................................................................................. 63–69 Assessment ............................................................................................... 70 Checklist .................................................................................................... 71

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• • L&T – 2

Calculate the perimeter and area of rectangles using familiar metric units (ACMMG109)

Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries (ACMMG114)

– – – – – –

– – – – – –

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• UUM– 3

• L&T – 3

Compare 12- and 24-hour time systems and convert between them (ACMMG110) – – – – – –

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Teacher information ........................................................................... 28–29 Hands-on activities .................................................................................... 30 Links to other curriculum areas .................................................................. 31 Resource sheets .................................................................................. 32–38 Assessment ............................................................................................... 39 Checklist .................................................................................................... 40

Shape ........................................................................................ 42–57 Connect three-dimensional objects with their nets and other two-dimensional representations (ACMMG111) – – – – – –

Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original (ACMMG115)

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Answers .................................................................................. 41 • Shape – 1

Teacher information ........................................................................... 72–74 Hands-on activities ............................................................................. 75–78 Links to other curriculum areas .................................................................. 79 Resource sheets .................................................................................. 80–92 Assessment ........................................................................................ 93–94 Checklist .................................................................................................... 95

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Teacher information .................................................................................. 20 Hands-on activities ............................................................................. 21–22 Links to other curriculum areas .................................................................. 23 Resource sheets .................................................................................. 24–25 Assessment ............................................................................................... 26 Checklist .................................................................................................... 27

Teacher information .................................................................................. 42 Hands-on activities ............................................................................. 43–44 Links to other curriculum areas .................................................................. 45 Resource sheets .................................................................................. 46–53 Assessment ........................................................................................ 54–55 Checklist .................................................................................................... 56

Answers .................................................................................. 57

– – – – – –

Teacher information .................................................................................. 96 Hands-on activities ............................................................................. 97–98 Links to other curriculum areas .................................................................. 99 Resource sheets .............................................................................. 100–110 Assessment .................................................................................... 111–112 Checklist .................................................................................................. 113

Answers ........................................................................ 114–115

Geometric reasoning .............................................................. 116–133 • GR – 1 Estimate, measure and compare angles using degrees. Construct angles using a protractor (ACMMG112) – – – – – –

Teacher information ....................................................................... 116–117 Hands-on activities ......................................................................... 118–119 Links to other curriculum areas ................................................................ 120 Resource sheets .............................................................................. 121–129 Assessment .................................................................................... 130–131 Checklist .................................................................................................. 132

Answers ................................................................................ 133

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

iii


FORMAT OF THIS BOOK This teacher resource book includes supporting materials for teaching and learning in all sections of the Measurement and Geometry content strand of Australian Curriculum Mathematics. It includes activities relating to all sub-strands: Using units of measurement, Shape, Location and transformation and Geometric reasoning. All content descriptions have been included, as well as teaching points based on the Curriculum’s elaborations. Links to the proficiency strands have also been included. Each section supports a specific content description and follows a consistent format, containing the following information over several pages: • teacher information with related terms, student vocabulary, what the content description means, teaching points and problems to watch for • hands-on activities • links to other curriculum areas

• resource sheets • assessment sheets.

• a checklist

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Answers relating to the resource and assessment pages are included on the final page of the section for each sub-strand (Using units of measurement, Shape, Location and transformation and Geometric reasoning).

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The length of each content description section varies.

Teacher information includes background information relating to the content description, as well as related terms, desirable student vocabulary and other useful details which may assist the teacher.

Related terms includes vocabulary associated with the content description. Many of these relate to the glossary in the back of the official Australian Curriculum Mathematics document; additional related terms may also have been added.

What this means provides a general explanation of the content description.

the teacher would use—and expect the students to learn, understand and use—during mathematics lessons.

description.

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The proficiency strand(s) (Understanding, Fluency, Problem solving Solving or Reasoning) relevant to each content description are shown listed. in bold.

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© R. I . C.Publ i cat i ons Teaching points provides a listn of the main teaching •f owhich rr evi ew pur poseso l y • Student vocabulary includes words points relating to the content

What to look watchforforsuggests suggestsany any difficulties and misconceptions the students might encounter or develop.

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Hands-on activities includes descriptions or instructions for games or activities relating to the content descriptions or elaborations. Some of the hands-on activities are supported by resource sheets. Where applicable, these will be stated for easy reference.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


FORMAT OF THIS BOOK Links to other curriculum areas includes activities in other curriculum areas which support the content description. These are English, Information and Communication Technology (ICT), Health and Physical Education, History, Geography, the Arts and Languages. This section may list many links or only a few. It may also provide links to relevant interactive websites appropriate for the age group.

r o e t s Bo r e p ok u S Resource sheets are provided to support teaching and learning activities for each content description. The resource sheets could be cards for games, charts, additional worksheets for class use or other materials which the teacher might find useful to use or display in the classroom. For each resource sheet, the content description to which it relates is given.

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Cross-curricular links reinforce the knowledge that mathematics can be found within, and relates to, many other aspects of student learning and everyday life.

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Assessment pages are included. These © R. I . C . Pu b l i cactivities at i ons support activities in the Hands-on or resource sheets. •f orr evi ew pur posesonl y•

o c . che e r o t r s super Each section has a checklist which teachers may find useful as a place to keep a record of the results of assessment activities, or their observations of hands-on activities.

Answers for resource pages (where appropriate) and assessment pages are provided on the final page of each sub-strand section.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

v


Sub-strand: Using units of measurement—UUM – 1

Choose appropriate units of measurement for length, area, volume, capacity and mass (ACMMG108)

RELATED TERMS

TEACHER INFORMATION What does it mean

Length

• The measure of a path or object in one dimension from end to end (i.e. 1-D). Area

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• The amount of surface covered (i.e. 2-D; measured in square units).

• The total area of each of the surfaces of an object added together. Volume

• The volume of an object is the total space occupied by the object (i.e. 3-D; measured in cubic units).

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• The amount of matter an object contains, commonly measured in grams, kilograms and tonnes. Weight

• Students are familiar with the common metric units of measurement for each of the attributes.

Teaching points General

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• The amount a container can hold. This is different from volume, which is how much space it takes up. An example of this difference is if you consider an esky. The amount of room it takes up in a cupboard is its volume. The amount it can hold is its capacity. Mass

• Students decide the appropriate units for each attribute, rather than the teacher telling them what to use; e.g. students decide whether to use millimetres, centimetres, metres or kilometres to measure the length of a passageway.

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• The force of gravity acting on an object, used to measure mass (actually measured in Newtons) Note 1: It is correct to use the verb ‘to weigh’. Note 2: At this year level, students may use the terms ‘weight’ and ‘mass’ interchangeably, although it is best if the teacher uses the correct terminology.

• Estimate before measuring in all measurement activities.

• The metric units for length are millimetres (mm), centimetres (cm), metres (m) and kilometres (km).

Note: the correct pronunciation of kilometre is ‘KIL-uh-mee-tuh’, not ‘ky-LOM-metre’ or ‘KY-le-metre’. When saying units, the full name is used rather than the letters of the symbols; ‘five kays’ is not really acceptable.

• The common metric units for area are square centimetres (cm2), square metres (m2), square kilometres (km2) and hectares (ha).

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Capacity

• Students are aware that within each of the attributes, there are different units that can be used, dependent of the size of the object and the level of accuracy required. For example, a bridge over a river may be measured in metres, or even kilometres, but engineers designing and building it may need to have a level of accuracy to within millimetres.

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Surface area

• Students are aware of the different attributes of an object that can be measured and that different units are used for each of them.

• The common metric units for volume are cubic millimetres (mm3), cubic centimetres (cm3) and cubic metres (m3). • The metric units for capacity are millilitres (mL), litres (L) and kilolitres (kL).

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Note: the use of the upper case ‘L’ for the abbreviation of litres; it is also used for mL and kL. This is to distinguish it from the number 1.

• The metric units for mass are grams (g), kilograms (kg) and tonnes (t). • A gap is always left between the number of units and the abbreviation of the unit—e.g. 5 cm, 8 kg, 375 mL—and no full stop is used at the end of the abbreviation (unless it is the end of a sentence). • Abbreviations of metric units never use the ‘s’ at the end; e.g. ‘5 cm’ not ‘5 cms’; ‘4 cm2’ not ‘4 cms2’; ’12 m3’ not ’12 ms3’; ‘375 mL’ not ‘375 mLs’; 8 kg’ not ‘8 kgs’. However, if the unit is written in full, the ‘s’ is needed; e.g. ‘5 centimetres’, ‘4 square centimetres’, ‘12 cubic metres’, ‘375 millilitres’ and ‘8 kilograms’. • Students need to be fluent with conversions of common metric units of measure for length, capacity and mass (see page 4). Discussion could centre on needing to know about multiplying and dividing by ten, or powers of ten. Discuss the meanings of the prefixes; e.g. ‘milli’, ‘centi’, ‘kilo’.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Sub-strand: Using units of measurement—UUM – 1

Choose appropriate units of measurement for length, area, volume, capacity and mass (ACMMG108)

TEACHER INFORMATION (CONTINUED) Teaching points (continued) Student vocabulary

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centimetres metres millilitres litres square centimetres square metres square kilometres hectares cubic centimetres cubic metres grams kilograms

• Students need to develop ‘referents’ for length, area, volume, capacity and mass. For example, students being aware that their little finger is approximately one centimetre in width; that a big stride is about a metre long; that a square centimetre is about the size of the fingernail on their little finger; that a Base Ten small cube is one cubic centimetre; and that if it was hollow, it would hold one millilitre of water. For mass, when holding an item to be estimated, most people mentally compare the item with something they know such as a tub of margarine or a bag of sugar. Capacity is often problematic because of marketing; a 2-litre cool drink bottle may look as if it holds more than a 2-litre tub of icecream. They may be familiar with the amount of liquid in a normal 375 mL can of soft drink. • Students need to know which units are appropriate for measuring different items. For example, we would not usually measure the length of a room in millimetres, the area of a book cover in square metres, the volume of a room in cubic centimetres, the capacity of a bucket in millilitres, or the mass of a balloon in kilograms.

long longer longest short shorter shortest

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• National tests usually have questions where students are required to interpret scaled instruments in length, capacity and mass. For this reason, the measuring devices used need to have a scale rather than a digital display. Students need to have had many experiences actually measuring and reading scales, not just watching others do the measuring. There is no skill in reading a digital display on a measuring device.

covers more area covers less area

Length

holds more holds less holds the most holds the least

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heavy heavier heaviest light lighter lightest

• There are 2 types of rulers: dead-end (where the ‘zero’ is level with the end of the ruler) and waste-end (where the ‘zero’ is situated a little way in from the end of the ruler). Dead end ruler

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takes up more space takes up less space

Waste end ruler

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0 1 2 3 4 cm

• Ensure that all students’ rulers are in centimetres and millimetres, not inches. • Students need to be able to express distances in terms of metres and centimetres, or just in centimetres, so that they can record distances using decimals—e.g. 1.3 metres—and know that this is the same as 1 metre and 30 centimetres. • It is impractical for students to measure long distances (i.e. kilometres), but discussion could arise, for example, when going on an excursion. Area • Areas should be measured directly, with formulas being introduced for the area of a square or rectangle arising from this experience. (See UUM–2)

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

3


Sub-strand: Using units of measurement—UUM – 1

Choose appropriate units of measurement for length, area, volume, capacity and mass (ACMMG108)

TEACHER INFORMATION (CONTINUED) Conversions

Teaching points (continued)

Length

• Finding the surface area of a three-dimensional object involves determining the outside surface area of each face of the object and adding them together. At this stage, use only simple three-dimensional objects such as cubes and rectangular prisms.

10 mm = 1 cm 100 cm = 1 m 1000 mm = 1 m 1000 m = 1 km

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Volume

• Volume is an attribute independent of shape and position. This understanding will take time to develop and will be gained through manipulation and investigation.

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1000 mL = 1 L

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• The use of cubes to investigate volume of different objects enables students to come to the understanding that the same number of cubes (e.g. 12 cubes) can be used to create different-shaped objects that have the same volume (12 units).

Capacity

• There should be no formal use for the formulas for volume of a cube or rectangular prism at this year level.

Mass

1000 g = 1 kg

Capacity

At Year 5 students would not be expected to convert between measures of area or volume.

• Students may need help with strategies for measuring capacity, particularly when not all gradations are marked on a measuring container. Many measuring containers mark only every 5 or 10 millilitres. Mass

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• When estimating and measuring mass, the teacher needs to ensure that the masses of the items to be measured are not always obvious (i.e. that students cannot determine comparisons simply by looking). To do this, you need items that have similar masses but different volumes—e.g. a golf ball and a tennis ball—and items that are the same size but have different masses; e.g. lidded tins filled with different materials such as sand and styrofoam.

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• When using metric units for mass, students may use a pan balance and weights, kitchen scales or bathroom scales. Kitchen and bathroom scales may have a dial or a digital readout. It is a useful skill to be able read a dial. There is no skill involved in reading a digital display. • Students should be given every opportunity to handle the standard units of mass. Given such experiences, students will be better able to select appropriate units of measure as well as estimate mass. Parents can be encouraged to help at home by doing regular checks of their child’s mass, and by allowing them to help when weighing ingredients for cooking.

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What to look for

• Students unsure of what attribute to measure; i.e. length, area, volume, capacity or mass. • Students using inappropriate units of measurement.

• Students using inappropriate tools to measure; e.g. using a ruler to measure the length of a basketball court, a medicine measure to calculate the capacity of a bucket or bathroom scales to weigh a pencil case. • Students having useful referents for length, area, volume, capacity and mass.

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Sub-strand: Using units of measurement—UUM – 1

HANDS-ON ACTIVITIES Length Estimate before measuring in all these activities • Students measure such personal dimensions as hand length (from wrist to fingertip), foot length and length of arm from elbow to wrist. Use these dimensions to look for similarities and differences. For many people, the length of the foot is very similar to the length of their arm from elbow to wrist. Students could investigate how many of their group had that equality, and also investigate any other relationships between the three measures; e.g. is their hand length similar to half of their arm length? Students record their results in ways of their choice.

r o e t s Bo r e p ok u S Foot length > arm length

Foot length < arm length

Greg

Alby

Gemma

Catherine

Nathan

Mark

Adam

Hand half of arm length Lana Flynn

Hand > half of arm length Weng

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Foot length = arm length

Hand < half of arm length Jake Phoebe

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• Encourage students to suggest methods of identifying a longer distance; e.g. 500 metres or one kilometre in the school grounds. Students could use a trundle wheel to measure the distance around the oval and work out how many laps of the oval would be needed to make one kilometre.

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• Are you a square or a rectangle? Students measure their height and then measure their outstretched arms from fingertip to fingertip. Compare the width and height. If the width is greater than, or less than the height, the student is a ‘rectangle’. If the height and width are close to the same, the student is a ‘square’. (Note: Some students may be selfconscious about their height. It may be that you only do the comparison for one or two students in each group so that not all students have their measurements taken.)

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• Measuring the length of curved items. Students use pieces of string to determine the lengths of objects that cannot be measured directly using a ruler, then straighten the string out against a ruler to obtain a measurement in millimetres or centimetres. • Students could be given instructions for ruling lines of different lengths. For example, Rule a line that is 5 cm long. Mark the mid-point of the line. This is a different skill from measuring a line. Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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5


Sub-strand: Using units of measurement—UUM – 1

HANDS-ON ACTIVITIES (CONTINUED) Area Estimate before measuring in all these activities • Polyominoes are puzzle pieces that can be manipulated by joining a specific number of pieces along whole sides, such as dominoes (formed by joining two squares), triominoes (three squares), pentominoes (five squares). Students investigate how many different ways they can find to join five squares (pentominoes; there are 12 different pentominoes shown below). Students explore which of the 12 can be folded along the joins to make open boxes and which cannot. This activity highlights the concept of area, as one pentomino has the same area (5 squares) as each of the others.

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Investigate how many ways six squares can be joined together. Some of these hexominoes can be folded along the lines to make a closed box (a cube); others cannot. Students investigate which ones can, and how you can determine this just by looking at the arrangement of the squares. You would not expect every student to find all 35 solutions, although they can be found on the internet. (Of the 35 hexominoes, 11 can be folded to form a cube.)

• Tangrams are another way to manipulate shapes but not alter the area of the original square from which they are formed (as long as all pieces are used and there are not overlaps). What is the same about the two shapes below? (Area) What is different? (Shape)

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• Students use nets to investigate the surface area of simple three-dimensional objects such as cubes and rectangular prisms. Once they have drawn a net for a cube, they can work out the surface area of one of the faces, then multiply this by six to determine the total surface area of the cube. Although it can be useful to have ready-made nets of objects for students to follow, it is also necessary for students to work out for themselves what the nets may look like. In the example of the hexominoes above, students will find there are a number of different nets for a cube; but often nets of cubes are only presented in one format. This activity links to the unit Shape–1 where students look at nets as twodimensional representations of three-dimensional objects. • The area of regular and irregular shapes can be measured by placing them under a transparent grid and counting the squares and part squares. The teacher could print 1 cm2 grid paper onto overhead transparencies (see page 11). Students could then use these to overlay any shapes and work out the areas. It may help to put the transparencies and the shape into a plastic sleeve. This would hold the shape steady so it’s easier for students to count the squares.

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• Groups of students could investigate how large a square metre is by joining sheets of newspaper together with sticky tape or masking tape and using metre rulers to check for accuracy. Prior to doing this, students could be asked to estimate the area of a room, then when they see the size of a square metre, be allowed to revise their estimations. If each group lay their square metres alongside each other, a physical count of the area can be made. • Once students have made their square metre of newspaper, they can investigate what happens to the area if they cut and rearrange their squares (they will all still be one square metre). This may help overcome a common misconception that a square metre must be in the shape of a square.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Using units of measurement—UUM – 1

HANDS-ON ACTIVITIES (CONTINUED) Volume Estimate before measuring in all these activities • Allow students to experiment with cubes by using them to fill boxes and other containers, then counting the number of cubes used. • Have students make various shapes using the same number of cubes. Discuss the fact that changing the layout of the cubes in the shape does not alter the number of cubes or the amount of space they take up (the volume). This may help to develop an intuitive understanding of the relationship of volume to length, width and height. At this year level, formulas for determining the volume of solids are not used.

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• Count and record the number of cubes used to fill various boxes in layers to calculate volume. Layers of cubes

1 3

Volume in cubes

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Cubes per layer

• Relate the use of Base Ten small cubes to the activities above, and discuss that each of these cubes is one cubic centimetre (1 cm3). So if they have determined that a box has a volume of 24 Base Ten small cubes, it can also be said that it has a volume of 24 cm3. • Another way to compare the volumes of their boxes would be to line the cubes for each box up alongside each other. This is a very visual way to show the different volumes.

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• Use metre sticks or sticks that are a little longer than one metre and join 12 of them with rope or tape to make a cubic metre. (There are commercial kits with joiners that can be used to a construct a cubic metre.) Prior to doing this, students could be asked to estimate the volume of a room, then when they see the size of a cubic metre be allowed to revise their estimations. It is quite difficult to move a cubic metre around the room to try to work out its actual volume, but seeing the dimensions of a cubic metre will give the students a better idea of what the volume might be.

Capacity

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• Compare various containers of fairly similar size for capacity. It would be good if there were some short, wide containers as well as tall, narrow ones. Students seriate (place in order) the containers from ‘holds the least’ to ‘holds the most’. Allow students to revise their estimates after measuring the first container. My estimate

Actual capacity

500 mL

275 mL

450 mL 250 mL

175 mL

Bottle

700 mL 500 mL

375 mL

Glass

400 mL 200 mL

200 mL

Mug Jar

We thought the glass would hold the least amount of water, but it was the jar, then the glass, then the mug. The bottle held the most.

• Investigate how many of a small container of known capacity will fill larger containers and use this to compare volumes. For example, use a 500 mL measuring jug to fill a bucket, a plastic tub and a large fruit bowl. Students convert capacities to litres. My estimate

Number of jugs

Total capacity (mL)

Total capacity (L)

Bucket Plastic tub Fruit bowl Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

7


Sub-strand: Using units of measurement—UUM – 1

HANDS-ON ACTIVITIES (CONTINUED) Mass Estimate before measuring in all these activities • Compare the masses of objects in the same containers; e.g. a yoghurt container of sand, water, gravel, styrofoam etc. This will reinforce the reality that we cannot estimate the mass of an object by just looking at it; we need to actually pick it up or handle it in some way. • Bathroom scales can be made available so students’ mass can be recorded at regular intervals. Be aware that some students may be sensitive to having their mass disclosed. For this reason, it may preferable to weigh other items such as a class pet. However, this is unlikely to offer the opportunity to measure in kilograms on bathroom scales, so other heavier objects will need to be found for weighing; e.g. a bucket of sand, a box of Base Ten Blocks, a box of books.

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• Students experiment with making their own balance scales using two plastic cups, a coat hanger, ruler and string. There are many suggestions on the internet. 1

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• Students construct lists of objects that have a similar mass. For example: What can they find that has the same mass as 500g? (2 potatoes, or 3 bananas etc.)

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

• Ask the students to choose the appropriate unit, grams or kilograms. ‘Which of these would we weigh in grams or kilograms? A piece of paper, a chair, a calculator, a bag of marbles, a person?’ • Compare the masses of various substances. One cup of ...

Estimated mass in grams

Actual mass in grams

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Rice Marbles Sand

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• Use real objects to explore the relationships between grams and kilograms and their multiples. Using balances pans to compare two 500 g bags of flour with one 1 kg bag. • Once students have experience with numbers to three decimals places, they can convert masses in grams and kilograms to just kilograms and vice versa. I kg and 500g is the same as 1.5 kg 2 kg and 225g is the same as 2.225 kg 3.2 kg is the same as 3 kg and 200 grams 1.375 kg is the same as 1 kg and 375 grams

8

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Sub-strand: Using units of measurement—UUM – 1

LINKS TO OTHER CURRICULUM AREAS English • Read Is the blue whale the biggest thing there is? by Robert E Wells. This book compares the sizes of things starting with the blue whale, which is the largest animal that ever lived. The book compares the size of a blue whale with the height of Mt Everest, then to the size of the Earth, the Sun, the star Antares, the Milky Way and the Universe.

Information and Communication Technology • A YouTube site with instructions on how to build your own balance scales can be found at <http://www.youtube.com/ watch?v=12760IJwMuU>

r o e t s Bo r e p ok u S

• Use measurement of capacity to make Oobleck. This is named after a substance described in a Dr Seuss’ book, Bartholomew and the Oobleck. The Oobleck is a strange goo that acts as both a liquid and solid, made from cornflour and water (and food colouring, if desired). A recipe and short video clip can be find at <http://www.instructables.com/ id/Oobleck/>

Teac he r

Health and Physical Education

• Students measure the lengths of sports events such as long jump, high jump and triple jump.

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• Students measure the distance a ball or beanbag has been thrown.

• If students have swimming lessons or a swimming carnival, discuss how long the pool is. For longer distances, students could work out how many laps of the pool would make one kilometre. • For Sports Day and practices beforehand, students could help with measuring out the course for a 50-metre race, a 100-metre race etc. Students could use this knowledge to look at longer distances. We had a 100-metre race. There were 10 of us in the race. Altogether we ran 1 kilometre.

Science

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

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• Students measure the length of shadows at different times of day. A metre stick could be set up and students could take measurement every hour from 9:00 to 3:00.

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• Grow plants with a fairly rapid growth pattern such as bean sprouts or broad beans. Students record the growth of the plant every day, either in a diary, logbook, or on a graph using paper tape. My plant was 4 cm tall yesterday, but today it is nearly 5 cm tall. It is about 47 mm high. • Students measure ingredients to make various recipes.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

9


Sub-strand: Using units of measurement—UUM – 1

RESOURCE SHEET

(b) 2. (a) (b)

paperclips

Join the paperclips and find out.

paperclips

Estimate how many you would need to equal one metre.

paperclips

Use your ruler to work out how many you would need.

paperclips

How many do you think you would need for 1 kilometre?

paperclips

Teac he r

3. (a)

Estimate how many joined paperclips you would need to make a chain as long as your ruler.

(c)

How did you work this out?

© R. I . C.Publ i cat i ons Compare your estimate with others in your class. Tick a box to show how your •f orr e vi ewMyp ur p osesonl y• estimate compared with others. estimate was: close to others

4. (a)

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Waist Head (b)

more than many others

fewer than many others

Use joined paperclips to fill in the chart below.

Distance Wrist

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(b)

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My estimate

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Number of paperclips

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1. (a)

Difference (+ or –)

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Were your estimates more or less than your measurements?

Remember: Separate the paperclips before packing them away.

10

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Choose appropriate units of measurement for length, area, volume, capacity and mass

Paperclip chains


Sub-strand: Using units of measurement—UUM – 1

RESOURCE SHEET

Teac he r

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r o e t s Bo r e p ok u S

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m . u

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

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CONTENT DESCRIPTION: Choose appropriate units of measurement for length, area, volume, capacity and mass

1 cm2 grid paper

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

11


Sub-strand: Using units of measurement—UUM – 1

RESOURCE SHEET Ambitious areas

You will need:

1 cm2 grid paper Ruler, pencil and scissors

1. (a)

Cut a 4 x 3 rectangle from your grid paper.

(b)

Draw in the diagonal.

(c)

Cut along the diagonal.

(d)

Place one piece over the other until they fit together exactly.

r o e t s Bo r e p ok u S

Area of rectangle equals 2 squares Area of triangle equals 1 square

2. Use diagonals across different sized rectangles in these shapes to work out the areas of the shaded parts.

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(a)

© R. I . C.Publ i cat i ons •f orr evi ew pur poses nl y• squares (b) squares (c) o squares

. t e (d) squares o c . ctwo e 3. Work out the areas of the shaded pictures. hethe r o Hint: look for the diagonals and rectangles t r s super cut in half.

(a) 12

squares

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

(b)

squares

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Choose appropriate units of measurement for length, area, volume, capacity and mass

Here is another example:

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Teac he r

You have shown: The diagonal cuts the AREA of a rectangle into two equal parts.


Sub-strand: Using units of measurement—UUM – 1

RESOURCE SHEET Boxes of eight

You will need:

8 x 1 cm3 cubes Surface area

Length

Width

Height

8 cm3

34 cm2

8 cm

1 cm

1 cm

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Teac he r

Volume

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CONTENT DESCRIPTION: Choose appropriate units of measurement for length, area, volume, capacity and mass

Sketch

. te

1. (a)

(b)

o c . Build as many different rectangular prisms as you can, each using 8 cubes, and c e h r sketch each one one the table above. Considero the length, width and height of t r s s r a cube to be 1 centimetre.u pe

Write the dimensions of each prism into the table. The first example is done for you.

2. When you have built as many prisms as you can: (a)

What do you notice about the volume of them all.

(b)

What do you notice about the surface area?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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13


Sub-strand: Using units of measurement—UUM – 1

RESOURCE SHEET Mystery tour

Start at the bus on the map on the next page. At each crossroad there is a letter. Find that letter on the list below, then work out which would be the most suitable unit to use to measure the item listed next it. Once you have decided on the unit, follow the road that has that unit until you get to the next intersection. Where will the Mystery tour take you?

The mass of an orange

C

How far it is between the shelves on a bookcase The area of your fingernail

E

The length of an ant

F

The capacity of a glass

G

The mass of a golf ball

H

The length of a running track

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D

The distance from Sydney to Newcastle

K

The height of a room

M N P

The area of grass on a football oval

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J

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I

© R. I . C.Publ i cat i ons The amount ofr water ae bucket • f or evi w holds pur posesonl y•

. te o The area of metal on a garage door c . che e r o The amount of liquid on r a teaspoon t s super The length of your hand

Q

How heavy a sack of potatoes is

R

The area of the screen on a computer

Y

How much a water tank holds

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Choose appropriate units of measurement for length, area, volume, capacity and mass

B

Teac he r

The mass of a person

L

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A


Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

A

B

C

D

E

F

G

1

2

H

D

cm

m

mm2

B

g

kg

A

m

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ml

cm

cm2

4

L

L mL

cm2 cm

R

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cm

kg

6

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m

g

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mL

P

L

m2

m2

F

LOST!

Shop

YOU ARE

g

cm

cm2

G

m

g

M

m

kg

km

cm

kg

mL

m

m . u

C A R PA R K A

kg

NOT HERE!

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m

L

Y

8

mL

g

ml

Church

kg

GO BACK!

MARKET

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cm

OOPS!

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Teac he r

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

3

CIRCUS

kg

L

m

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cm

START HERE

gr

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NOWHERE!

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MOVIE S

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km2

km

m2

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kg

N

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N E

STABLES

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15

Scale: 1 cm = 1 km 13

kg

cm

mL

m2 L

kg

r o e t s Bo r e p ok u S

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CONTENT DESCRIPTION: Choose appropriate units of measurement for length, area, volume, capacity and mass Sub-strand: Using units of measurement—UUM – 1

RESOURCE SHEET

Mystery tour map

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15


Sub-strand: Using units of measurement—UUM – 1

RESOURCE SHEET Units of length

1. Which measuring tool would you use to measure the length of each of these? trundle wheel

pedometre

0 17723

1 m ruler

(d)

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© R. I . C.Publ i cat i ons (g)r (h)n •f or(f)r evi ew pu poseso l y•

o c . che e r o t r s super

2. Write three things you would measure using each unit of measurement.

16

(a)

kilometres

(b)

metres

(c)

centimetres

(d)

millimetres

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Choose appropriate units of measurement for length, area, volume, capacity and mass

(c)

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(e)

30 cm ruler

r o e t s Bo r e p ok u S (b)

Teac he r

(a)

tape measure


Assessment 1

Sub-strand: Using units of measurement—UUM – 1

NAME:

DATE: Units of length

1. Fill in your estimates for each of the items below. Think about what equipment you will need to measure each of them and what units of measurement you will use. Item

Estimate

Actual length

Difference

Length of the classroom

r o e t s Bo r e p ok u S

Width of the classroom

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Teac he r

Height of the classroom Length of the board Height of the board Height of the door

© R. I . C.Publ i cat i ons 2. Measure only the first item, the length of the classroom. •f orr evi ew pur posesonl y• Were you close?

m . u

Write it into your table and work out the difference between your estimate and the actual measure.

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CONTENT DESCRIPTION: Choose appropriate units of measurement for length, area, volume, capacity and mass

Width of the door

3. Go back and change the estimates you made for the rest of the list if you need to. Do you need to make the distances greater or smaller?

. te o c Were these estimates more accurate? . che e r o t r s super

4. Measure the rest of the items and work out the differences.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

17


Assessment 2

Sub-strand: Using units of measurement—UUM – 1

NAME:

DATE: Boxes and more boxes about 40 x 1 cm3 cubes

You will need: 1. (a)

Build this rectangular prism. How many cubes did you use? This is its volume (24 cm3).

r o e t s Bo r e p ok u S (b)

Now look at this rectangular prism. What is its volume?

(c)

What do you notice about the two rectangular prisms? Their

is the same, but their

2. (a)

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Teac he r

cm3

are different.

Build the three prisms below. Prism 1

Prism 2

Prism 3

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(b)

Complete this table:

Prism 1 2 3 (c)

18

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Length

Width

Height

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© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

Volume

Surface area

o c . che e r o t r s super

Write about what you found.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Checklist

Sub-strand: Using units of measurement—UUM – 1

Choose appropriate units of measurement for length, area, volume, capacity and mass (ACMMG108) Choose appropriate units of measurement for ... Length

Area

Volume

Capacity

Mass

r o e t s Bo r e p ok u S

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Teac he r

STUDENT NAME

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© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

19


Sub-strand: Using units of measurement—UUM – 2

Calculate the perimeter and area of rectangles using familiar metric units (ACMMG109)

RELATED TERMS

TEACHER INFORMATION What does it mean

Perimeter

• A measure of the distance around the boundary of a two-dimensional shape. Area

• There is no direct relationship between the area and perimeter of rectangles. Two rectangles with the same area may have different perimeters and two rectangles with the same perimeter may have different areas.

r o e t s Bo r e p ok u S

• It is not necessarily the case that, as the perimeter of a rectangle is increased or decreased, the area of the rectangle is increased or decreased.

Familiar metric units

• Students should be led to ‘discover’ the formula for calculating the perimeter or area of a rectangle.

• For perimeter, use centimetres, metres and kilometres. For area, use square centimetres, square metres and square kilometres. Large areas of land may also be measured in hectares, though students would not be expected to work with this unit at this year level.

Teaching points

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Teac he r

• The amount of surface covered (i.e. 2-D; measured in square units).

• Students may measure the lengths of the four sides of a rectangle and add them together to obtain the perimeter. They may then ‘discover’ that a more efficient way to calculate the perimeter is to double the lengths of the two different sides and add them together.

• Students will have had experience in finding the areas of rectangles by counting squares. The next step is to find the number of squares in each row and then work out how many rows there are. Students should explore the relationships rather than the teacher simply telling them a formula.

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

• Teachers need to stress that the formulas for working out the perimeters and areas of rectangles only pertain to rectangles and are not necessarily appropriate for other shapes.

Student vocabulary perimeter

What to look for

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• Use the language ‘square centimetres’ not ‘centimetres squared’; they mean different things. For example, 2 cm2 (2 square centimetres) means 2 lots of 1 cm x 1 cm squares. Two centimetres squared means a square with a length of 2 cm and a width of 2 cm (which is actually 4 square centimetres).

o c . che e r o t r s super

• Students confused about the difference between perimeter and area.

• Students believing that there is a direct relationship between the perimeter and area of a rectangles.

area centimetres metres kilometres square centimetres square metres square kilometres

20

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Sub-strand: Using units of measurement—UUM – 2

HANDS-ON ACTIVITIES • Students construct rectangles on 1 cm2 grid paper (see page 11). They record the size of the length and widths and add them together to determine the perimeter. Students can be ‘led’ to realise that if they double the size of the length and the size of the width, then add those two figures together, they have found an efficient way to calculate the perimeter of a rectangle. Another method for calculating the perimeter of a rectangle is to add the size of the length and width together and double the result. 5 cm 3 cm

3 cm

The perimeter is 5 + 5 + 3 + 3, which is 16 cm. This is the same as 10 cm (2 x 5) + 6 cm (2 x 3).

r o e t s Bo r e p ok u S The length of the rectangle is 5 cm, the width is 3 cm.

Teac he r

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• Students investigate how to calculate the perimeter of a square, which is a ‘special’ type of rectangle. They could use the above method, but may realise that as all four sides of a square are equal, they only need to work out the length of one side and multiply it by four. • Find the perimeter of Geoboard shapes. These do not need to be just one rectangle or square, but can be combinations, as below.

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• The perimeter is 8 units

The perimeter is 12 units

The perimeter is 18 units

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• Use eight squares and investigate how many different shapes can be made. Some examples are shown below. Discuss which shape has the longest perimeter. Why are some of the perimeters different if all the shapes are made with eight squares? What types of shapes have the longer/shorter perimeters?

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

21


Sub-strand: Using units of measurement—UUM – 2

HANDS-ON ACTIVITIES (CONTINUED) • Students work out the area of a rectangle by counting the total number of squares, one at a time. Next, they count the number of squares in one row and work out how many rows there are altogether. Students then investigate any relationship between the sets of figures. To assist in this, a table could be drawn up.

r o e t s Bo r e p ok u S There are 3 rows; so the area of the rectangle is 15 cm2

Rectangle

Number of squares in each row (length)

Number of rows (width)

Total area

1

5

3

15 cm2

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Teac he r

Area of each row of the rectangle is 5 cm2

2 3

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

• Students may use a similar format to the one above, but include a column for calculating and recording the perimeters. Rectangle

Number of squares in each row (length)

Number of rows (width)

Perimeter

Total area

1

5

3

10 + 6 = 16 cm

15 cm2

2

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3 • Using a Geoboard, students investigate area and perimeter of various shapes.

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o c . che e r o t r s super

The perimeter is 8 units and the area is 3 square units.

22

The perimeter is 12 units and the area is 6 square units.

The perimeter is 18 units and the area is 9 square units.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Sub-strand: Using units of measurement—UUM – 2

LINKS TO OTHER CURRICULUM AREAS Information and Communication Technology • Students can calculate the perimeters of shapes on the website <http://www.bgfl.org/custom/resources_ftp/client_ftp/ ks2/maths/perimeter_and_area/index.html> • A shape surveyor that looks at area and perimeter of rectangles can be found at <http://www.funbrain.com/cgi-bin/ poly.cgi> • The Shape Explorer can be found at <http://www.shodor.org/interactivate/activities/ShapeExplorer/> You can choose to have only rectangular shapes for this activity. It also has the option of looking at the areas and perimeters of the shapes students have worked with in table format.

r o e t s Bo r e p ok u S

• A catchy YouTube™ song on perimeter and area can be found at <http://www.youtube.com/watch?v=D5jTP-q9TgI> It goes straight to multiplication for working out the area of a rectangle, so this could be used after students have ‘discovered’ how the formula works. • A perimeter rap song can be found on YouTube™ at <http://www.youtube.com/watch?v=wynwRcc5q_U&feature=relat ed> It shows some of the measurements in inches and others in centimetres.

Teac he r

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• An activity where students use perimeter and area to launch a ship can be found at <http://pbskids.org/cyberchase/ math-games/airlines-builder> The instructions are a little unclear, but students will soon get the hang of it. • A Design a Party planning activity where students look for particular rectangles with given areas and perimeters can be found at <http://www.mathplayground.com/PartyDesigner/PartyDesigner.html>

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© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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23


Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET Area the same

Materials:

Examples:

8 square tiles

• This square is 1 unit long and 1 unit wide. Its perimeter is 4 units.

1 cm2 grid paper

1 unit 1 unit 1 unit

• Tina the tiler has 8 tiles. She must use all the tiles so she always has an area of 8 square units.

r o e t s Bo r e p ok u S

• She has arranged her tiles like this to get a perimeter of 12 units.

1 unit

4 units

2 units

2 units 4 units

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Teac he r

1. Do the same with your tiles. Make a copy of Tina’s floor on your grid paper.

(a)

a perimeter of 12 units (different from the one above)

(b)

a perimeter of 14 units

(c)

a perimeter of 16 units

Shape

(b) (c)

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(a)

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Area

Perimeter

8 square units

12 units

8 square units

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© R. I . C.Publ i cat i ons Write the area and perimeter next to each shape and complete the table to show your • f o r r e v i e w p u r p o s e s o n l y • results.

(d) and (e) two different floors with a perimeter of 18 units.

o c . che e r 8 square units o t r s super 8 square units

(d)

8 square units

(e)

8 square units

3. Write about your results.

24

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Calculate the perimeter and area of rectangles using familiar metric units

2. Help Tina by making these 5 different floors, using all 8 tiles for each one and copy each floor on grid paper.


Sub-strand: Using units of measurement—UUM – 2

RESOURCE SHEET Perimeter the same

Materials:

Examples:

8 square tiles

• This square is 1 unit long and 1 unit wide. Its perimeter is 4 units.

2

1 cm grid paper

1 unit 1 unit

1 unit 1 unit

• Arrange 8 tiles like this.

4 units

r o e t s Bo r e p ok u S • The area is 8 square units The perimeter is 12 units.

2 units

2 units

4 units

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Teac he r

1. Do the same with your tiles. Make a copy of this floor on your grid paper. 2. Make the perimeter stay at 12 units by removing different numbers of tiles, then copy each floor on grid paper: (a)

1 tile

(b)

2 tiles

(c)

3 tiles

© R. I . C.Publ i cat i ons •Shape f orr evi ew pu r posesonl y • Area Perimeter

(a)

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12 units

m . u

8 square units

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CONTENT DESCRIPTION: Calculate the perimeter and area of rectangles using familiar metric units

Write the area and perimeter next to each shape and complete the table to show your results.

12 units

o c . (c) 12 units che e r o t r s super 3. Write about your results. (b)

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

12 units

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25


Assessment 1

Sub-strand: Using units of measurement—UUM – 2

NAME:

DATE: Twelve squares Find the area and perimeter of each shape below.

(b)

Which shapes have the same perimeter?

(c)

Which shapes have the same area?

(i)

(ii)

P=

2. (a) (b)

r o e t s Bo r e p ok u S cm

2

cm

P=

cm

P=

cm

A=

cm2

A=

cm2

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(iv)

Teac he r

A=

(iii)

(v)

P=

cm

P=

cm

A=

cm2

A=

cm2

© R. I . C.Publ i cat i ons f orr evi ew pur posesonl y• What• is its area? What is the perimeter of this rectangle?

3. Materials: 12 square tiles Make the three different rectangles that are possible with 12 tiles.

(b)

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(c)

Draw the rectangles in the first column of the table.

Complete the table.

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Sketch of rectangle

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(a)

12 square o c units . che e r o t r s super Length

Width

Perimeter

12 units

1 unit

26 units

Area

4. Write about what you found.

26

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Calculate the perimeter and area of rectangles using familiar metric units

1. (a)


Checklist

Sub-strand: Using units of measurement—UUM – 2

Compares area and perimeter of rectangles

Understands how to calculate the area of rectangles

r o e t s Bo r e p ok u S

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STUDENT NAME

Understands how to calculate the perimeter of rectangles

Calculate the perimeter and area of rectangles using familiar metric units (ACMMG109)

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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27


Sub-strand: Using units of measurement—UUM – 3

Compare 12- and 24-hour time systems and convert between them (ACMMG110)

RELATED TERMS

TEACHER INFORMATION

am (ante meridiem)

What does it mean

pm (post meridiem)

• It is expected students can tell the time to the nearest minute, but previously, have only used 12-hour clocks.

r o e t s Bo r e p ok u S

• Converting between 12-hour and 24-hour times can be difficult because of the non-metric nature of time. So 1630 is not 6:30 pm, but 4:30 pm. • Calculations of time difference can be quite difficult, again because of the non-decimal nature of time. For example, if using a timetable and calculating how long before the next bus, a calculator may actually hamper the process. If the bus arrives at 1625 and it is currently 1547, you cannot simply key 1625 into a calculator and subtract 1547; the result would be 78, which a child could incorrectly interpret as 78 minutes. In this case, the number of minutes until 1600 would be calculated first (13 minutes), and the extra 25 minutes until the desired time (1625) added to give a total waiting time of 38 minutes.

Teaching points

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(From the Latin words meaning before and after noon)

• Standard abbreviations for units for time are seconds (s), minutes (min) and hours (h). The other units do not have standard abbreviations. Note: ‘sec’ and ‘hr’ are commonly used abbreviations for second and hour, but they are not the correct ones.

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

Student vocabulary o’clock half past quarter past quarter to xx:25 (e.g. 3:25) xx:52 (e.g. 3:52) clockwise

am (ante meridiem) pm (post meridiem)

28

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• When writing 12-hour time, we use a colon between the hours and minutes; for example, two o’clock should be 2:00 not 2.00. When writing 24-hour time we generally do not use a colon, but use four digits; for example, 4:33 pm would be written as 1633. For times before 10 am, there is a zero at the start in 24-hour time in the written form; e.g.0730 for 7:30 in the morning. Whether we say ‘oh’ or ‘zero’ depends on community practice. (Note: some sources do use a colon in 24-hour time.) • Telling the time in both 12- and 24-hour formats should be practised daily and treated incidentally whenever the opportunity arises.

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• Classrooms should have both an analogue clock and a digital clock, preferably side-by-side. Regularly seeing the two different displays for the same time of day helps students realise there are two equally valid ways to read the time. There are some large clocks available commercially that clearly show the time in both formats. Clocks that display 24-hour time would also be useful. • It is generally recommended that ‘quarter to an hour’ is the only time that students deal with times to the next hour. For example, we would use 5:52 rather than 8 minutes to 6. These times ‘to’ an hour may be dealt with informally as the need arises. • The spoken time reflects the written digital time; e.g. in 12-hour time format, 11:28 would be said as eleven twenty-eight, not twenty-eight minutes after/past eleven. The time 7:31 would be said as seven thirty-one, not twenty-nine minutes to 8 or thirty-one minutes after/past seven. With times such as 11:05, whether we say oh instead of zero or whether we verbalise the zero at all, depends on community practice. However, the zero must be used in the written form. When ‘am’ and ‘pm’ are used, they are simply spoken as the letters am or pm.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Sub-strand: Using units of measurement—UUM – 3

Compare 12- and 24-hour time systems and convert between them (ACMMG110)

TEACHER INFORMATION (CONTINUED) Teaching points (continued) • When using the 24-hour time format, times on the hour generally are said as ‘hundred’; e.g. 1100 would be eleven hundred, and 0500 would be zero/oh five hundred. Other times with a zero at the end would be spoken in tens; e.g. 1120 would be eleven twenty, and 0530 would be zero/oh five thirty.

r o e t s Bo r e p ok u S What to look for

• Students unable to convert from 12- to 24-hour time

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Teac he r

• National tests usually have several questions on time, including calculating time differences.

• Students not using four digits when writing 24-hour time; e.g. writing 815 instead of 0815 for 8:15 am. • Students using the wrong base (e.g. Base 10) for time calculations. • Students using a calculator inappropriately when subtracting one time from another to find the duration of an event. • Students unable to decide which operation is appropriate when calculating time problems.

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o c . che e r o t r s super Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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29


Sub-strand: Using units of measurement—UUM – 3

HANDS-ON ACTIVITIES • Students count the times on the hour in 12-hour and then 24-hour times, saying one am, two am, three am, four am, five am, six am, seven am, eight am, nine am, ten am, eleven am, twelve am, one pm, two pm etc. Then zero one hundred, zero two hundred, zero three hundred, zero four hundred, zero five hundred, zero six hundred, zero seven hundred, zero eight hundred, zero nine hundred, ten hundred (not one thousand and of course, no longer needing the leading zero), eleven hundred, twelve hundred etc. They would see the fact that in 24-hour format, times after noon carry on from the 12:00 midday time. If viewed in isolation, students may assume, for example, 1400 in 24-hour time is 4:00 pm. • When converting from the 12-hour to the 24-hour clock for any time after 12:59 pm (that is in the afternoon), we add 12; so 5:00 pm becomes (5 + 12) which is 1700 and 11:13 pm becomes (11 + 12, and the 13 minutes) which is 2313. • Read and record start and finish times for events whenever possible, using both 12- and 24-hour time. This gives students the opportunity to see both formats for the same times. Our maths classes usually start at 9:20 am, which is 0920 in 24-hour time. The cross-country race started at 2:05 pm and finished at 2:37 pm. We could write that as 1405 for the start and 1437 for the finish.

r o e t s Bo r e p ok u S

• Students list activities, recording in both 12- and 24-hour times.

24-hour time

Got out of bed

7:45

0745

Left to come to school

8:12

0812

Lunch time

12:30

1230

Left to go home

3:10

1510

Had dinner

6:48

1848

Went to bed

9:00

2100

• Students make timelines of both 12- and 24-hour times, marked in hours. morning (am) 6 7

afternoon (pm) 4 5 6

midnight 11 12 2400

10

2300

2000

9

2200

8

2100

7

1900

1800

1700

1600

3

1500

1100

2

1400

1000

1

1300

10

1200

9

0900

0700

8

noon 11 12

0800

5

0600

4

0500

0200

0100

0000

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

3

0400

2

0300

midnight 12 1

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12-hour time

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Activity

• Discuss where 24-hour time may be used in their lives. Many communication devices such as digital recorders and music systems use 24-hour time. Bus and train timetables are often shown in 24-hour time, as are flight details. Also the military and police use this format.

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• Students make timetables and rosters for school activities such as using the computer lab, having access to the sports equipment and library times. • Students make a timetable for their perfect Saturday in 24-hour time format and work out durations of time for each activity. Activity Start time Finish time Duration

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Get up and get ready to go out

0830

0940

1 hour 10 minutes

Swim at the beach

1000

1230

2 hours 30 minutes

1230

1330

1 hour

1355

1545

1 hour 50 minutes

1600

1830

2 hours 30 minutes

Have favourite dinner

1900

2020

1 hour 20 minutes

Watch TV

2020

2130

1 hour 10 minutes

Have lunch at cafe Go to the movies Play games

• Look at guides for both the TV and cinemas. • Plan a short trip that includes travel for which they need to consult timetables. • 24-hour Time Bingo. Sets of Bingo cards on pages 36–37 are to be copied and cut out. They may be laminated for durability. Each student will need one card and nine counters. The teacher will need a set of the 12-hour time cards on page 38. The teacher shuffles the cards and either holds them up one at a time or calls them out. The students work out the 24-hour time equivalent and if they have that time on their Bingo card they cover it with a counter. The first student to cover all squares is the winner. • Memory game. Set 1 and 2 memory cards on pages 34–35 can be enlarged, photocopied and laminated, and used to play Concentration.

30

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Sub-strand: Using units of measurement—UUM – 3

LINKS TO OTHER CURRICULUM AREAS English • Use the book Tick tock by James Dunbar as a stimulus book for discussion about different time periods. • Read Just a minute! by T Slater. • Read Clocks and more clocks by Pat Hutchins. Discuss what the times in the book would be if they were shown in 24hour format.

Information and Communication Technology • A stop-the-clock format for recording 24-hour time can be found at <http://www.bgfl.org/custom/resources_ftp/ client_ftp/ks2/maths/time/index.htm>

r o e t s Bo r e p ok u S

• Stop the Clock, with a 24-hour option, can be found at <http://resources.oswego.org/games/StopTheClock/sthec5. html> Students have to match an analog clock with its digital time display.

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Languages

• Students tell the time in another language in both 12- and 24-hour formats.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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31


Sub-strand: Using units of measurement—UUM – 3

RESOURCE SHEET 12- and 24-hour comparisons

9:00

ON

OFF

GIGA-BLASTER

16:00

12-hour clock

24-hour clock

12 am

0000

1 am

0100

ON

OFF

0200

3 am

0300

4 am

0400

5 am

0500

am 6 am

0600

7 am

0700

8 am

0800

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GIGA-BLASTER

r o e t s Bo r e p ok u S 2 am

21:00

ON

OFF

GIGA-BLASTER

9 am

0900

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GIGA-BLASTER

ON

OFF

1300

2 pm

1400

3 pm

1500

4 pm 1600 . te5 pm 1700 o pm c . 6 pm 1800 e che r o 13:00 t r s1900 super 7 pm

ON

OFF

GIGA-BLASTER

08:00

8 pm

2000

9 pm

2100

10 pm

2200

11 pm

2300

ON

OFF

GIGA-BLASTER

32

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Compare 12- and 24-hour time systems and convert between them

06:00

1 pm

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1000 © R. I . C.Publ i ca t i ons 11 am 1100 •f o12 rpm r evi ew pur pos esonl y• 1200 10 am


Sub-strand: Using units of measurement—UUM – 3

RESOURCE SHEET The 24-hour clock

The am hours shown in the inner ring/ and the pm hours shown in the outer ring

24

1:00 pm

1300

2:00

1400

3:00

1500

4:00

1600

5:00

1700

6:00

1800

0600

7:00

1900

7:00

0700

8:00

2000

8:00

0800

9:00

2100

9:00

0900

10:00

2200

10:00

1000

11:00

2300

11:00

1100

12:00 midnight

2400

12:00 noon

1200

0001

1:00

0100

2:00

0200

3:00

0300

4:00

0400

5:00

0500

6:00

23 22 11 10

12

1

13 14

r o e t s Bo r e p9 21 3o 15 u k S 8 20

7 19

2

4

6 18

5 17

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12:01 am

16

Example 1

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

Example 2

1. Write the following 12-hour times as 24-hour times.

(c)

. te 12 noon

(e)

12:23 pm

(g)

9:15 pm

(a)

4:00 am

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What is 11:45 pm in 24-hour time? 11:00 pm is shown as 2300, so 11:45 pm would be shown as 2345.

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CONTENT DESCRIPTION: Compare 12- and 24-hour time systems and convert between them

What is 11:45 am in 24-hour time? Look at the 24-hour clock. 11:00 am is shown as 1100, so 11:45 am would be shown as 1145.

o c . che e r (f) t 5:25 am o r s super (h) 4:20 am (b)

2 pm

(d)

12:00 midnight

2. Write these 24-hour times as 12-hour times, using am or pm. (a)

1348

(b)

0344

(c)

2215

(d)

1805

(e)

1111

(f)

2222

(g)

0714

(h)

1621

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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33


Sub-strand: Using units of measurement—UUM – 3

RESOURCE SHEET 12- and 24-hour time memory game (Set 1)

3:58 am

0358

Teac he r

1620

r o e t s Bo r e p ok u S 0835 10:14 pm 2214

7:42 pm

1942

9:07 pm

2:20 pm

1420

11:51 am

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8:35 am

4:20 pm

2107

1:02 am

12: 56 pm

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1151

o c . c e her 12:56s r 0102 am 0056 o t super 2356

6:18 pm

1818

Note: You may wish to enlarge these before photocopying onto card and laminating. 34

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Compare 12- and 24-hour time systems and convert between them

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•


Sub-strand: Using units of measurement—UUM – 3

RESOURCE SHEET 12- and 24-hour time memory game (Set 2)

2:49 am

0249

5:23 pm

1723

1558

9:05 am

0905

8:42 pm

2042

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6:47 am

0647

10:10 pm

2210

Teac he r

r o e t s Bo r e p ok u 1:20 amS 0120 3:58 pm

. te

4:20 am

11:31 pm

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CONTENT DESCRIPTION: Compare 12- and 24-hour time systems and convert between them

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

o c . che e 0420 7:14o am r t r s super 2331

5:32 am

0714

0532

Note: You may wish to enlarge these before photocopying onto card and laminating. Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

35


Sub-strand: Using units of measurement—UUM – 3

RESOURCE SHEET Time bingo cards (Set 1)

1403 0300 1505 2345 1132 0455 1739 0006 1035 1201 0455 1302 0837 1622 1403 2001 1830 0123 0800

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Teac he r

2210 0922

r o e t s Bo r e p ok 0245 1905 1035 0744 1731 0619 u S

1622 0550 2121 1830 0922 0123 2121 0006 1905 0619 1505 1132 2345 2001 0744 1201 2210 0300

©R . I . C. Pub l i ca0240 t i on1403 s 0619 2345 0245 1302 0513 1622 1005

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0455 0300 1132 2121 0800 1302 1830 0006 2001 1731 2210 0513 0837 1905 0744 2345 0240 1201 1201 1505

. te 0006 1622 0123 1905 1035 o 0922 1505 2121 c . che e r o t r s s r u e p 2001 0123 0550 1302 2210 1403 1905 0006

0744 1830 0619 1403 0455 1739 1201 2345 0837 0300 1132 1622 1830 0922 2121 1731 0300 1830

36

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Compare 12- and 24-hour time systems and convert between them

•f orr evi ew pur posesonl y•


Sub-strand: Using units of measurement—UUM – 3

RESOURCE SHEET Time bingo cards (Set 2)

0245 1302 2210 2210 1905 1132 0619 2001 1302 1622 0513 1005 0123 0744 0455 0300 1622 1132 2345 0837

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Teac he r

0550

r o e t s Bo r e p 2121 0006 1005 o 0240 u 1505 0123 k S

1505 0619 2001 2345 0455 0006 2121 0001 2210 0744 1739 0922 1035 1622 1905 0513 1505 2001

R. I . C .Pu bl i c at i o0123 ns 0922 1830 0455 2121©1403 2210 1132 0300

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0800 1201 1731 2210 1302 0550 0744 1905 0455

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CONTENT DESCRIPTION: Compare 12- and 24-hour time systems and convert between them

•f orr evi ew pur posesonl y•

1302 0744 0245 0123 1739 1005 2001 1830 0300 1403 1403

. te 1201 1505 0800 0245 0922o0619 0619 c . che e r o t r s s r u e p 2345 1201 0922 1622 1132 1505 0123

1622 0455

0001 0550 1830 2345 1201 0619 2210 1731 1302 1905 0300 1132 0240 2121 1403 0744 2001 1005

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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37


Sub-strand: Using units of measurement—UUM – 3

RESOURCE SHEET Time bingo master cards

12:01 am 12:06 am 1:23 am 2:40 am

2:45 am

8.00 am

Teac he r

7:44 am

8:37 am 9:22 am

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3:00 am

r o e t s Bo r e p ok u Sam 5:13 am 5:50 am 6:19 am 4:55

10:05 am

. te

3:05 pm

7:05 pm

38

2:03 pm

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10:35 am 11:32 am 12:01 pm 1:02 pm

o c . c e her r 4:22 pm 5:31 pm 5:39o pm 6:30 pm t s super 8:01 pm

9:21 pm 10:10 pm 11:45 pm

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Compare 12- and 24-hour time systems and convert between them

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•


Assessment 1

Sub-strand: Using units of measurement—UUM – 3

NAME:

DATE: It’s about time Bus timetable: City to beach From city

Teac he r

0712 0733 0755 0825 0855 0925 0955 1025 1055

1105 1205 1305 1405 1505 1535 1605 1655 1722

To city 1742 1752 1822 1842 1852 1925 2025 2125

0655 0727 0747 0817 0837 0857 0917 0937 0957

1027 1047 1137 1237 1337 1437 1504 1534 1604

1634 1704 1734 1804 1834 1947 2047 2147

r o e t s Bo r e p ok u S The journey takes 32 minutes each way.

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Emma arrived at the city terminal at 3:42 pm. How long did she have to wait to catch a bus to the beach? First we have to convert 3:42 pm to 24-hour time. This is the same as 1542. Look at the timetable above. The next bus that leaves after 1542 is the 1605 one; she has already missed the 1535 bus. From 1542 to the next hour (1600) is 18 minutes; that is 60 (minutes) take 42 (minutes). Then there are another 5 minutes until the bus comes at 1605; so 18 minutes and 5 minutes are 23 minutes. Emma had to wait 23 minutes for the next bus.

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• 10:49 am. How long was it until the next bus?

Solve the problems below. Show how you worked out each of your answers.

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2. Natalie’s house was next to the beach. She needed to be home by 2:00 pm. What bus

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CONTENT DESCRIPTION: Compare 12- and 24-hour time systems and convert between them

1. Joshua was at the beach and wanted to get to the city. He checked his watch; it was

would she need to catch?

. te him at the beach at 7:30 am. Ben wanted to stay 3. Ben’s dad dropped for about an o c . chheecatch to get back to the city?r hour. Which bus would e o r st super 4. Isabella was in a surf carnival that started at 10:30. What time would she need to leave the city to get her there on time?

5. Kiara went to her friend’s beach party, which finished at 3:45 pm. What is the first bus she could get back to the city, and how long must she wait?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

39


Checklist

Sub-strand: Using units of measurement—UUM – 3

Can convert between 12and 24-hour times

Can read a 24-hour clock to the nearest minute

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Teac he r

STUDENT NAME

Can read a 12-hour clock to the nearest minute

Compare 12- and 24-hour time systems and convert between them (ACMMG110)

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Answers

Sub-strand: Using units of measurement

3. The perimeters are all the same (12 units), but the areas are different.

UUM –1 Page 10

Resource sheet – Paperclip chains

Page 26

Teacher check Page 12

1. (a) (i) P = 12 cm A = 6 cm2 (ii) P = 16 cm A = 9 cm2 (iii) P = 18 cm A = 8 cm2 (iv) P = 12 cm A = 5 cm2 (v) P = 12 cm A = 8 cm2 (b) (i) (iv) (v) (c) (iii) (v) 2. (a) 22 units (b) 10 square units 3. (a) Teacher check (b)–(c)

Resource sheet – Ambitious areas

1.–2. Teacher check 3. Bird: 19.5 squares Rabbit: 26 squares Page 13

Resource sheet – Boxes of eight

Teacher check

Assessment 1 – Twelve squares

r o e t s Bo r e p ok u S

Page 14–15 Resource sheet – Mystery tour

Sketch

You will be taken to the zoo. Page 16

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(b) (d) (f ) (h)

30 cm ruler tape measure or 1 m ruler 30 cm ruler pedometer

Area

12 units

1 unit

26 units

12 square units

6 units

2 units

16 units

12 square units

Assessment 1 – Units of length

4 units

UUM – 3

Assessment 2 – Boxes and more boxes

3 units

14 units

12 square units

4. The areas are the same for all of the rectangles, but the perimeters are different. The longer and thinner the shape, the greater the perimeter.

Teacher check

Page 33

Resource sheet – The 24-hour clock

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• Length

Width

Height

Volume

1

3 cm

3 cm

3 cm

27 cm3

54 cm2

2

2 cm

2 cm

1 cm

4 cm3

16 cm2

3

5 cm

4 cm

2 cm

40 cm3

80 cm2

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area

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a perimeter of 14 units a perimeter of 16 units

(b) (d) (f ) (h) (b) (d) (f ) (h)

1400 2400 0525 0420 3:44 am 6:05 pm 10:22 pm 4:21 pm

Assessment 1 – It’s about time

1. 10:49 am is 1049. The bus leaves at 1137. 1049 until 1100 is 11 minutes, plus the extra 37 minutes until 1137 gives 48 minutes. Answer: 48 minutes 2. 2:00 is 1400. The trip takes 32 minutes, so the latest bus she could take would be 1328 (1400–32 minutes). The bus that leaves just before 1328 is the 1305 bus. Answer: The 1305 bus 3. 7:30 am is 0730. An hour later it would be 0830. The closest bus to that time is the 0837 bus. Answer: The 0837 bus 4. 10:30 am is 1030. The trip takes 32 minutes, so the latest bus she could take would be 0958. The bus that leaves just before 0958 is the 0955 bus. Answer: The 0955 bus 5. 3:45 pm is 1545. The next bus is at 1604. It is 15 minutes from 1545 until 1600; add on the extra 4 minutes until 1604, and the wait is 19 minutes. Answer: The 1604 bus. A 19 minute wait

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Resource sheet – Area the same

1. Teacher check 2. a perimeter of 12 units

0400 1200 1223 2115 1:48 pm 10:15 pm 11:11 am 7:14 am

Page 39

(c) Teacher check

UUM – 2

1. (a) (c) (e) (g) 2. (a) (c) (e) (g)

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1. (a) Teacher check (b) 24 cm3 (c) Their volume is the same, but their shapes are different. 2. (b) Surface

P = 12; A = 8

P = 14; A = 8

P = 16; A = 8

a perimeter of 18 units

P = 18; A = 8

a perimeter of 18 units

P = 18; A = 8

3. The areas are all the same (8 square units), but the perimeters are different. Page 25

Perimeter

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Page 17

Page 24

Width

Resource sheet – Units of length

1. (a) tape measure (c) tape measure (e) trundle wheel (g) trundle wheel 2. Teacher check

Page 18

Length

Resource sheet – Perimeter the same

1. Teacher check 2. (a) 1 tile

P = 12; A = 7

(b) 2 tiles

P = 12; A = 6

(c) 3 tiles

P = 12; A = 5

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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41


Sub-strand: Shape—Shape – 1

Connect three-dimensional objects with their nets and other two-dimensional representations (ACMMG111)

RELATED TERMS

TEACHER INFORMATION What does it mean

Two-dimensional shapes; three-dimensional objects

• Whenever the curriculum mentions ‘shapes’ it is referring to twodimensions; when it mentions ‘objects’ it is referring to threedimensions.

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• A flat two-dimensional pattern that can be folded to make a model of a three-dimensional object.

• Students need experience making nets for themselves using equipment such as geoshapes and polydrons. This would come before any use of templates. • Ready-made templates for nets of three-dimensional objects usually only show one way to produce that shape. Particular nets are often chosen because they use the least amount of card or because they can fit onto a photocopier. However, there are a variety of ways of producing nets for any one shape and students need to be exposed to as many of them as possible. For example, the ‘standard’ net for a triangular prism (seen in the column on the left) is shown below (left), with a variation (right) that students should be aware of through exploration with materials.

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Net

• Cutting open a variety of boxes and cartons is a good introduction to this topic.

Prism

• A three-dimensional object with parallel and congruent end faces, with the other faces rectangles; the shape of the pair of congruent end faces names the prism; e.g. rectangular prism, triangular prism (right) etc.

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• A three-dimensional object with a polygonal base and the other faces triangles with a common vertex called the apex. It is named after its base; e.g. triangular pyramid, square-based pyramid (right) etc.

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Student vocabulary net

cube

prism

pyramid

triangular prism rectangular prism square-based pyramid tetrahedron

edges

octahedron

faces

vertices (one vertex)

• Students should become familiar with a variety of nets for cubes, prisms and pyramids.

Teaching points • Students explore the different ways to make nets for the same shape.

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• It is useful for students to produce their own nets and fold and tape them to make their shapes.

• There is more than one net for each geometric solid.

• Students look at nets with different designs on the faces and visualise what will be on each face when the solid is constructed.

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• National tests usually have a question that requires students to be able to associate three-dimensional objects with their nets. They also have questions requiring students to visualise solids and their faces from different orientations.

What to look for

• Students limited to being able to find only one or two different nets for any shape. • Students being given a net for a geometric solid and having difficulty visualising what the shape will look like when completed. • Students unable to visualise and draw models from different orientations. • Students unable to work out how many cubes are needed and then build a particular model when given different orientations. • Students having difficulty drawing three-dimensional objects on isometric dot paper.

Proficiency strand(s): Understanding

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Fluency

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

Problem solving

Reasoning

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Sub-strand: Shape—Shape – 1

HANDS-ON ACTIVITIES • Collect a variety of boxes and cartons for students to cut open and explore the resulting net. Investigate how many different ways the same box may be opened up and still be put back into the same shape. • There are two area activities in UMM–1 on page 6 that link to this description, both looking at how squares can be moved and joined together without the area changing if no pieces are added and there are no gaps or overlaps. The first is the pentomino activity where students investigate how many different ways they can join five squares (there are 12 of them) and the explore which of the 12 can be folded along the joins to make open boxes and which cannot. The following activity is similar, except that students look for the different ways six squares can be joined (hexominoes) then explore which of them could be folded to make a cube. Of the 35 hexominoes, 11 can be folded to form a cube.

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• The hexominoes activity above will lead to the realisation that there is more than one net for a cube. Another way to help with this concept is to use Polydrons or Geoshapes. Students are each given 6 squares which they connect to form a cube. The net can be sketched onto grid paper. They open out the six squares and try to find another way to join them that will still result in a cube, then again sketching the net onto grid paper. Working in groups, students try to find all 11 different nets for a cube.

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• Using triangular Polydrons or Geoshapes, students investigate the different nets that can be made for a tetrahedron (a polyhedron made from four triangles). These can be sketched onto triangular grid paper or drawn freehand.

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Sub-strand: Shape—Shape – 1

HANDS-ON ACTIVITIES (CONTINUED) • A mix of different-shaped Polydrons or Geoshapes can be used to produce other shapes. For example, a triangular prism can be made with two triangles and three rectangles or squares and a square-based pyramid can be made with one square and four triangles. In each case, students explore the different ways the nets can be joined and sketch the results. • Give students examples of nets of cubes with different images on each face, or have students design their own for their peers to construct. Students work out which images will be on adjoining faces and which ones will be on opposite faces. Students then construct the cube and check their predictions. (Note: there is often a question of this type in the national tests for Year 5.) This activity could be related to the fact that on a standard six-sided dice, the values on opposite faces add up to seven (1 + 6; 2 + 5; 3 + 4).

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When this cube is put together:

• Nets of five geometric solids are shown on pages 46–50. Note: these only offer one net for each of the five solids. Students will need to explore how many other nets can be made for each of the solids.

• Students make buildings using cubes, and sketch the resulting models from different orientations. The reverse of this could also be done, where the different views of a model are given to the students and they construct the appropriate building.

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• Students identify common objects from photos that have been taken from unusual points of view. Students could then be challenged to take their own photos from different perspectives for other students to identify.

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• Students could investigate the use of isometric dot paper to record their three-dimensional drawings. Turning the page to ‘landscape’ will assist in this process.

• Once students have had experience drawing three-dimensional objects on isometric dot paper, the students may make models from drawings done on isometric dot paper.

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Sub-strand: Shape—Shape – 1

LINKS TO OTHER CURRICULUM AREAS Information and Communication Technology • Printable nets for different solids can be found at <http://www.senteacher.org/wk/3dshape.php> Note that only one version of a net is shown for each solid, so students would still need to see other variations. • An animated display of nets and their solids can be found at <http://www.learner.org/interactives/geometry/3d_ prisms.html> Again, there is only one version of each net shown. • A similar site to the one above, but for pyramids, can be found at <http://www.learner.org/interactives/geometry/3d_ pyramids.html> • The illuminations website has an interactive page where different solids can be selected, rotated and their nets displayed. It still has the same problem as the sites above. It can be found at <http://illuminations.nctm.org/ ActivityDetail.aspx?ID=70>

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• Another website for looking at nets of solids can be found at <http://www.kidzone.ws/math/geometry/nets/index. htm>

The Arts

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• A short quiz in which students match a net with the solid can be found at <http://www.sadlier-oxford.com/math/ enrichment/gr4/EN0411b/EN0411b.htm>

• Students make models of towns, animals, robots, buildings etc. using geometric solids made from nets.

• Students decorate their geometric solids in a variety of ways; some for ornamentation, others for a purpose such as a special dice for a game. (Note: this does not need to be limited to a cube.)

Science

• Students investigate crystals.

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• A website that offers information on crystals can be found at <http://chemistry.about.com/cs/sciencefairideas/a/ aa072903a.htm>

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• Another website from the same source, which explains how to make rock candy using crystals, can be found at <http:// chemistry.about.com/od/foodcookingchemistry/a/rockcandy.htm>

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Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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This is only one of many nets that can be made for a cube. How many others can you find?

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CONTENT DESCRIPTION: Connect three-dimensional objects with their nets and other two-dimensional representations

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Net for a cube


Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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CONTENT DESCRIPTION: Connect three-dimensional objects with their nets and other two-dimensional representations

Net for a rectangular prism

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This is only one of many nets that can be made for a rectangular prism. How many others can you find?

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Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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This is only one of many nets that can be made for a square-based pyramid. How many others can you find?

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CONTENT DESCRIPTION: Connect three-dimensional objects with their nets and other two-dimensional representations

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Net for a square-based pyramid


Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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CONTENT DESCRIPTION: Connect three-dimensional objects with their nets and other two-dimensional representations

Net for a tetrahedron

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This is one of two nets that can be made for a tetrahedron. Can you find the other one?

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Sub-strand: Shape—Shape – 1

RESOURCE SHEET

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This is only one of many nets that can be made for a triangular prism. How many others can you find?

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CONTENT DESCRIPTION: Connect three-dimensional objects with their nets and other two-dimensional representations

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Net for a triangular prism


Sub-strand: Shape—Shape – 1

RESOURCE SHEET 3-D shapes and their nets

1. Complete the table to describe these prisms. Shape

Name

Front/back Other shape shape used

Number of faces

Number of edges

Number of vertices

(a) (b)

(d)

2. Complete the table to describe these pyramids.

(a)

(b)

Name

Base shape

Other shape used

Number of faces

Number of edges

Number of vertices

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CONTENT DESCRIPTION: Connect three-dimensional objects with their nets and other two-dimensional representations

Shape

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3. Draw each shape and its net. (a) cube

(c) cone

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Sub-strand: Shape—Shape – 1

RESOURCE SHEET Dicey dice

Try the questions below. If you’re not sure, cut out each net and fold it on the lines to see what the cube would look like.

1. What is opposite the ♥ on this dice?

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. temake the cube below. What shapes would be on the: o 4. This net is folded to c . che e r o (a) top of the cube? t r s super (b)

bottom of the cube?

5. Draw a net for a dice of your own, but have a different layout from the four on this page. Put your own designs on each face and then challenge a partner to work out what designs would be opposite each other. 52

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CONTENT DESCRIPTION: Connect three-dimensional objects with their nets and other two-dimensional representations

2. What symbol is on the bottom of this dice?

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Sub-strand: Shape—Shape – 1

RESOURCE SHEET Model buildings

1. Below are some designs for buildings to liven up a model train track. Use cubes to make each of the buildings, then fill in the missing details. (a)

Building 1: The hospital

top

left

right

Building 2: The station

Sketch

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CONTENT DESCRIPTION: Connect three-dimensional objects with their nets and other two-dimensional representations

(b)

back

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Front

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Sketch

using at least 8 cubes)

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back

top

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

right

Sketch

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Assessment 1

Sub-strand: Shape—Shape – 1

NAME:

DATE: Match the nets In the right hand column draw a different net for each of the solids in the centre column. Net 1 Geometric solid Net 2

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(b)

Match the nets on the left to the geometric solids in the middle.

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Challenge:

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CONTENT DESCRIPTION: Connect three-dimensional objects with their nets and other two-dimensional representations

1. (a)


Assessment 2

Sub-strand: Shape—Shape – 1

NAME:

DATE: Using cubes

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Front view

Back view

Top view

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Left view

Right view

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CONTENT DESCRIPTION: Connect three-dimensional objects with their nets and other two-dimensional representations

1. Liam glued five cubes together and decorated three of the faces. Draw what his arrangement will look like from the front, back, on top, left side and right side if it is standing on the three decorated faces.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

right

Sketch

Sketch

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Checklist

Sub-strand: Shape—Shape – 1

Connect three-dimensional objects with their nets and other two-dimensional representations (ACMMG111) Connects nets with geometric solids for … Cubes

Prisms

Pyramids

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STUDENT NAME

Can draw models from different orientations

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Answers

Sub-strand: Shape—Shape

Shape – 1

Page 54

Page 51

Assessment 1 – Match the nets

Net 1

Resource sheet – 3-D shapes and their nets

Geometric solid

Net 2

1. Name

Front/ back shape

Other shape used

Number of faces

Number of edges

Number of vertices

(a)

triangular prism

triangle

rectangle

5

9

6

(b)

rectangular prism

rectangle

rectangle

6

12

8

(c)

pentagonal prism

pentagon

rectangle

7

15

10

(d)

hexagonal prism

hexagon

rectangle

8

18

12

Base shape

Other shape used

Number of faces

Number of edges

Number of vertices

triangle

triangle

4

6

4

2.

(a)

triangular pyramid

(b)

rectangular pyramid

rectangle

triangle

5

8

5

(c)

pentagonal pyramid

pentagon

triangle

6

10

6

(d)

hexagonal pyramid

hexagon

triangle

7

12

7

3. (a) cube

(b)

triangular pyramid

(c) cone

(d)

cylinder

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Assessment 2 – Using cubes

1.

Resource sheet – Dicey dice

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front 2.

3. 4. 5. Teacher to check students’ designs for dice. Page 54

back

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Resource sheet – Model buildings

Building 1: The hospital

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Building 2: The station

left

right

top

left

right

3.

front

Sketch

top

back

Sketch

Building 3: The fire station

Front

back

top

left

right

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Sub-strand: Location and transformation—L&T – 1

Use a grid reference system to describe locations. Describe routes using landmarks and directional language (ACMMG113)

RELATED TERMS

TEACHER INFORMATION

Location

What does it mean

• Identifying a specific place. It is the ‘where’ of the Geometry strand.

• Being able to use simple coordinates for determining the position of an item on a map.

Coordinates (Cartesian coordinates)

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• When describing routes, it is important to be conscious of the intended meaning and to be precise and accurate. • The relative position of one feature to another needs to be correct.

Teaching points

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• A pair of numbers or symbols that represent a position on a grid. Understanding of this concept of naming coordinates is essential in later years when using a grid with negative coordinates and two or four quadrants, or later still when graphing functions in algebra and trigonometry.

• Students being able to give and receive directions to determine location

• Teacher models the use of appropriate language of location; e.g. north, south, east, west, north-west, left, right, clockwise, anticlockwise, between.

• When using coordinates for grid references, name the horizontal axis (x-axis) first, followed by the vertical axis (y-axis). So the ❂ below, is at (2,4), not (4,2). The  is at (4,2).

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• Before using maps and coordinates students need to understand: – the need for a horizontal and vertical axis – the way the two axes are labelled

– that when reading points on a map, the horizontal axis is always read before the vertical axis – that any specific location on a map can be found using both the horizontal and vertical coordinates on the grid.

• Note: In the Year 5 national tests, there is usually a question which involves coordinates. One of the possible answers is always the reverse order to the correct answer. So in the example above, if the correct answer was (2, 4), one of the choices would appear as (4, 2). • Students need to be able to follow oral direction as well as written ones. • Link to the unit on angles when discussing degrees of turn (GR–1), pages 116–131.

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Sub-strand: Location and transformation—L&T – 1

Use a grid reference system to describe locations. Describe routes using landmarks and directional language (ACMMG113)

TEACHER INFORMATION (CONTINUED) What to look for • Students who are not able to give accurate written or oral directions.

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• Students who are not able to follow written or oral directions.

• Students who confuse the terms left and right, clockwise and anticlockwise; compass points. • Students who are unable to interpret a simple scale or work out corresponding distances.

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• Students who, when using coordinates for map references, use the vertical axis (y-axis) first followed by the horizontal axis (x-axis) instead of the other way around.

• Students unable to use a legend to locate particular features on a map.

Student vocabulary coordinate horizontal axis vertical axis one axis; two axes scale legend key

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north south east west north-east north-west south-east south-west

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degrees compass

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Sub-strand: Location and transformation—L&T – 1

HANDS-ON ACTIVITIES • Students follow written or oral directions using a grid system. (See Sunny’s walk in the park on page 63.) • Students do a similar activity to the one above, but with coordinates shown on the map. Students identify the route by following the coordinates. (See Sunny’s fun at the playground on page 64.) • From coordinates, students locate cells or features on a grid or map; e.g. Start at the café at (4, 2), and find the cinema. • Students plan routes within the classroom for a partner to follow. Start at the door, turn to the left, take three steps forward then turn a quarter turn to the right. Now take five steps forward and turn 270° in a clockwise direction. Move forward four metres, then turn anticlockwise 90° to the left … • Practise giving and following directions using maps and plans. Provide students with a map of the school with none of the features identified (i.e. not showing which building is the library, canteen, office etc. and ask them to identify each of the features.

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SCHOOL FLOOR PLAN

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• Investigate the plan of a shopping mall, a large department store or a supermarket. Discuss the layout and why items are placed in certain locations. For example, Why are there usually chocolates at the checkouts, at about eye level? Why is the menswear section usually not the first thing you see when you enter a large shop? What is the purpose of the displays at the end of the food aisles? What is the arrangement for paying for purchases? Decide the route that would be the shortest way to go from one side of the shop or mall to the other without seeing any junk food.

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• Look at the maps of two small country towns of similar size and compare what features they have in common and what is different about them. Ideally one town could be in a mainly farming area and the other in a mining setting. Discuss the reasons for the similarities and differences. Students then work out routes to get from one part of a small town to another, for example, What is the best way to get from the school to the bank?

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Pi n e

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Examples of towns that are a similar size (a population of approximately 1000) for each state are: Dorrigo in NSW, Beaufort in Victoria, Millmerran in Queensland, Burra in South Australia, Roebourne in Western Australia, Zeehan in Tasmania and Wadeye in the Northern Territory.

Millmerran Bank

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Millmerran Health Services

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Sub-strand: Location and transformation—L&T – 1

HANDS-ON ACTIVITIES (CONTINUED) • Investigate the different routes that may be taken from one location to another following various criteria. For example, How would you get from the school to the park, stopping at the shops on the way? What is the shortest way to get from the park to the beach without crossing any main roads? How many different ways could you walk from the post office to the library? • Fill the square – 1 (See page 65). In this board game for 2–4 players, two different coloured 6-sided dice are thrown. One dice determines the vertical coordinate and the other the horizontal. The player with the most coloured counters on the board at the end of the game wins. • Fill the square – 2 (See page 66). This board game is played as above with a 10-sided dice and a bigger board. • Map symbol bingo (See pages 67 and 68). Students draw up a 3 x 3 bingo board, or use the blank ones on page 66 and draw in nine items chosen from the range of map symbols on page 68. The teacher then shuffles the symbol cards and holds them up one at a time. If the students have the symbol on their card they cross it out or cover it with a counter. The first student to cross out or cover all nine symbols on their board is the winner. Students discuss what each of the symbols represents.

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• Whose zoo? (See page 69). In this activity students follow compass directions to complete the outline of a zoo animal.

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Sub-strand: Location and transformation—L&T – 1

LINKS TO OTHER CURRICULUM AREAS English • Read The once upon a time map book by B Hennessy and P Joyce. This book takes a trip to six different well-known story lands with maps, coordinates, routes, hidden objects and points of interest. Each map has a grid overlay so that coordinates can be used to locate certain features. They each have a key (legend) indicating features such as rock paths, giant’s stairs and camel road.

Information and Communication Technology • Zoo maps can be found at:

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– Perth Zoo <http://www.perthzoo.wa.gov.au/visit/zoo-map/>

– Alma Park Zoo (Brisbane) <http://www.almaparkzoo.com.au/images/stories/pdfs/alma_park_zoo_guide_map.pdf> – Adelaide Zoo <http://www.zoossa.com.au/adelaide-zoo/zoo-information/zoo-map> • Interactive zoo maps can be found at:

– Melbourne Zoo <http://www.zoo.org.au/Melbourne/Zoo_Map>

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– Taronga Park Zoo (Sydney) <http://www.taronga.org.au/taronga-zoo/map-visit-planner

• Various maps of King’s Park in Perth can be found at <http://www.bgpa.wa.gov.au/kings-park/map>

Health and Physical Education • Play outdoor games that require ‘home’ territories, boundaries and bases.

History and Geography

• Use maps of places that may be visited as part of excursions etc. Students look for features of interest and plan a route through the area so they get to visit the most important features in the most efficient way.

The Arts

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• Students draw their own maps, using intuitive or more formal ideas of scale. Ideally, the relative size of features should be shown on the maps constructed by the students. They could then overlay a grid and use coordinates to identify various features and to plan routes around their maps for different purposes.

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Sub-strand: Location and transformation—L&T – 1

RESOURCE SHEET Sunny’s walk in the park

Sunny the dog went for a walk in the park. 1. Following the directions given below, how many trees did she walk past?

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CONTENT DESCRIPTION: Use a grid reference system to describe locations. Describe routes using landmarks and directional language

E–N–E–N–N–W–N–N–E–E–E–S–S–S–W–S–S–E–E–E

o c . che e r o t r s super 2. Plan a route for Sunny, from Start, so she visits each of the drink fountains once. 3. Plan a route for Sunny, from Start, so her owner can stop and have a rest at each of the seats in the park.

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Sub-strand: Location and transformation—L&T – 1

RESOURCE SHEET Sunny’s fun at the playground

Sunny the dog went to the playground. 1. Following the directions given below, mark in the route she ran when she first arrived. (1, 1) (4, 1) (4, 2) (6, 5) (7, 5) (7, 7) (4, 4) (2, 7) (5, 4) (7, 4) (7, 3) (1, 3) (5, 1) (5, 7) (1, 4)

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o c . c e 3. Sunny likes to sit under theh slides, out of the sun. What are ther coordinates for all the e o t r slides? s super

2. Sunny’s owner wants to try out all the different swings. What are their coordinates?

4. Plan the shortest route for Sunny to sniff at each tree, starting at (2, 1).

5. Think of a place on the map to bury a treasure. Write instructions on how to get there from (1, 1), and then challenge a partner to find it.

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CONTENT DESCRIPTION: Use a grid reference system to describe locations. Describe routes using landmarks and directional language

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Sub-strand: Location and transformation—L&T – 1

RESOURCE SHEET Fill the square – 1

A game for 2–4 players. You will need: A game board (below) 2–4 sets of 12 different coloured counters 2 different coloured 6-sided dice

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CONTENT DESCRIPTION: Use a grid reference system to describe locations. Describe routes using landmarks and directional language

Rules: Decide which of your two dice is the vertical coordinate and which is the horizontal. Players take it in turns to throw both dice and place one of their counters on the correct coordinate on the board. If the cell is already taken, the player misses that turn. When all the cells are filled, or when the time is up, the player with the most of their counters in the shaded square is the winner.

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1 1

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Sub-strand: Location and transformation—L&T – 1

RESOURCE SHEET Fill the square – 2

A game for 2–4 players. You will need: A game board (below) 2–4 sets of 12 different coloured counters 2 different coloured 10-sided dice

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8

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CONTENT DESCRIPTION: Use a grid reference system to describe locations. Describe routes using landmarks and directional language

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Rules: Decide which of your two dice is the vertical coordinate and which is the horizontal. Players take it in turns to throw both dice and place one of their counters on the correct coordinate on the board. If the cell is already taken, the player misses that turn. If either of the dice shows a zero (0), the player misses the turn. When all the cells are filled, or when the time is up, the player with the most of their counters in the shaded square is the winner.


Sub-strand: Location and transformation—L&T – 1

RESOURCE SHEET Map symbol bingo

Students draw up a 3 x 3 bingo board, or use the blank ones below, and then choose nine of the symbols on page 68 and draw them onto their bingo boards. The teacher photocopies the symbol cards and cuts them out. Once the cards are shuffled, the teacher holds them up, one at a time, and the students cross them out if they have the symbol on their bingo board. The winner is the student who first covers all 9 symbols on their board.

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Bingo board

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CONTENT DESCRIPTION: Use a grid reference system to describe locations. Describe routes using landmarks and directional language

Bingo board

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Sub-strand: Location and transformation—L&T – 1

RESOURCE SHEET

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Note: You may wish to enlarge these before photocopying onto card and laminating. 68

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CONTENT DESCRIPTION: Use a grid reference system to describe locations. Describe routes using landmarks and directional language

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Map symbol bingo cards


Sub-strand: Location and transformation—L&T – 1

RESOURCE SHEET Whose zoo? N

Do you know the compass directions? Check this compass on the right. Each small square on the grid paper below is 1 cm2, and for now we are going to count going diagonally across a square as 1 cm2, though it’s really a little bit more.

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Teac he r Start

Look at the example on the left:

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The instruction ‘1 E’ means that you go east for one square, and ‘1 NE’ means going northeast for 1 square. Using a ruler and a pencil, follow the instructions below; starting from the ‘start’ dot on the grid paper below.

© R. I . C.Publ i c at i ons Instructions: 1 Es •f orr evi ew pur po esonl y• 1 NE 6E

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2W

2W 10 N 1 NW 1 NE Hint: cross off each one as you do it.

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Assessment 1

Sub-strand: Location and transformation—L&T – 1

NAME:

DATE: Mystery tour revisited

Use the map on page 15 to answer the following questions. Use compass directions to help you. Also use the scale to work out how far each section is. For example, to go from intersection P to intersection F, you would head south-east for about 2 kilometres. 1. Describe the shortest route to get from the market to the circus.

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2. What are the coordinates for the zoo? (Use the end of the road as the start of the zoo.)

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3. What is at the coordinate (8, I)?

5. Draw in a symbol for each of the following at the location described: Phone box at (11, I)

Petrol station at (2, E)

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Lookout at (9, F)

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information centre at (12, H)

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6. Describe 3 different ways to get from the river to the playground. • • •

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CONTENT DESCRIPTION: Use a grid reference system to describe locations. Describe routes using landmarks and directional language

4. If you were in the bus, describe the route that would go past the shop, forest, movies, and circus and back to where you started. About how far would you travel on the trip?


Checklist

Sub-strand: Location and transformation—L&T – 1

Uses directional language to describe routes

Understands legends and keys used in maps

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STUDENT NAME

Uses coordinates to describe locations

Use a grid reference system to describe locations. Describe routes using landmarks and directional language (ACMMG113)

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Sub-strand: Location and transformation—L&T – 2

Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries (ACMMG114)

RELATED TERMS

TEACHER INFORMATION What does it mean

Transformations

• The transformations of translations (slides), reflections (flips) and rotations (turns) describe movements of a shape or object.

• There are three main types of Euclidian transformations below. In these, the length, width, angle size and area do not change. Translation (slide)

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• Line (reflectional) symmetry and rotational symmetry describe particular properties of a shape or object; whether they have one type of symmetry, both types or no symmetry (are asymmetrical). • A one-step move means that the object or shape only moves in one direction within the transformation. At this level, students only use one-step moves. • Students need to have experience transforming or moving real objects and shapes before doing so using digital technologies.

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• The movement of an object or shape that changes position in a given direction. It remains oriented the same way. The shape and size of the shape or object does not change.

• Rotations or turns can be of any size from less than 1° through to 360° (a full turn), where the object or shape ends back where it started.

Reflection (flip)

• Students need to be aware of the difference between reflectional (line) and rotational symmetry. A transparent mirrors (also known as a Mira or Georeflector) may be used when testing for reflectional symmetry, but is of no help when looking for rotational symmetry.

• The mirror image of an object or shape, so that each point of the object or shape is the same distance from the mirror line (or plane of symmetry with a three-dimensional object) as the same point on the image. The shape and size of the shape or object does not change.

Teaching points © R. I . C .Pu bl i cat i ons •f orr evi ew pur posesonl y• • The process by which a shape or object

• When an object or shape undergoes a translation, reflection or rotation transformation, its size, shape and features do not change.

Rotation (turn)

• A translation (slide) can be done in any direction, but without turning (rotating) the object or shape. • Translation (slide) transformations can be the basis of tessellations where the same shape is repeated without gaps or overlaps to create a pattern. • A reflection (flip) transformation is performed around a ‘mirror’ line, which is generally drawn in for clarity.

Symmetry (Reflectional or Bilateral symmetry):

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• The correspondence, in size, form and arrangement, of parts on opposite sides of a line, point or plane; e.g. the butterfly is symmetrical about the vertical line.

• An object or figure has rotational symmetry if it appears to retain its original orientation after turning through some fraction of a complete turn about a fixed point. The figure below has rotational symmetry of order 5 (it takes 5 turns to come back to its starting position. x

• When the object or shape has been transformed by a reflection, the mirror line indicates a line of symmetry, where any point on one side of the line is the same distance from the line as the equivalent point in the reflection. This is different from finding any line/s of symmetry within a shape, where the shape can be folded in half to show the line of symmetry. • The end result of a translation or reflection to a shape that has more than one line of symmetry can appear the same.

Mirror line

• The end result of a translation or reflection to a shape that has either one or no lines of symmetry will appear different.

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• A rotation (turn) transformation is performed about a point.

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line of symmetry

Rotational symmetry

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changes position by rotating about a fixed point through a given angle. The shape and size of the shape or object does not change.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

Mirror line

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Sub-strand: Location and transformation—L&T – 2

Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries (ACMMG114)

RELATED TERMS (CONTINUED)

Teaching points (continued) • With two-dimensional shapes, folding and cutting are common ways to determine reflectional symmetry about a line.

Transparent mirror

• A clear plastic tool used with symmetry; it has the reflective quality of a mirror, but can also be seen through, so that it reflects the front side of the shape onto the other side (also known as a Mira or Georeflector).

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• Transparent mirrors are useful tools for identifying reflectional (line) symmetry. They cannot be used for rotational symmetry. • Transparent mirrors may be used to determine whether threedimensional objects are symmetrical about a plane. • Students may need to be shown the correct way to handle a transparent mirror. Directions for this can be found on page 83. • Symmetry often occurs in the natural environment; e.g. pine cones, reflections of the landscape in a lake, and many leaves and plants.

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TEACHER INFORMATION (CONTINUED)

• Use appropriate pieces of art and craft work from different cultures as examples of the use of symmetry in design. • A pinwheel is a good example of rotational symmetry.

Clockwise

• Many logos use rotational symmetry in their designs.

• A turn in the direction the hands on a clock move.

• National tests usually include questions about reflectional and rotational symmetry. Students need many experiences cutting, folding and turning shapes to be able to visualise the results of these actions.

© R. I . C.Publ i cat i ons • A turn that is in the opposite direction from the • way hands on r a clockr move. f o evi ew pur posesonl y• Note: In Australia, we use the term Anticlockwise

• There is an infinite number of degrees of turn that can be produced in a rotation.

Tessellations

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• Repeated patterns of shapes which completely cover a surface without gaps or overlaps.

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• A turn of 360° takes the shape or object back to where it started. • When a shape or object is rotated 180° (a half turn) it will be upside down.

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‘anticlockwise’ rather than the American equivalent ‘counterclockwise’.

• In most cases, a turn of 90° clockwise leaves an object in a very different position from a turn of 90° in an anticlockwise direction; in fact a half-turn (180°) different. • If a shape with at least two lines of symmetry is turned exactly 180°, it produces the same effect as when it is reflected (flipped). If the shape or object does not have two or more lines of symmetry, it will be upside down after a flip of 180°.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

Reflection

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Sub-strand: Location and transformation—L&T – 2

Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries (ACMMG114)

TEACHER INFORMATION (CONTINUED) What to look for • Students who alter the shape or size of the object or shape as part of their transformation.

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• Students who move their shape or object beyond the transformation to be performed; e.g. rotating a shape while sliding it.

• Students incorrectly holding and using a transparent mirror. • Students who leave gaps when tessellating multiples of the same shape, or who overlap their shapes.

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• Students who are unaware that their shape has changed in some way after a reflection, and need to reflect it back to check.

• Students who confuse half and quarter turns.

• Students who confuse the directions of clockwise and anticlockwise.

• Students who do not recognise the pattern when a shape is turned 90° or 180° several times, or who are unable to continue the pattern by drawing the next one or two elements.

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• Students are unable to identify lines of symmetry in given situations, are unable to identify more than one line of symmetry where appropriate, and are unable to identify shapes and objects with no lines of symmetry (i.e. they are asymmetrical). • Students are unable to identify planes of symmetry in threedimensional objects.

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quarter turn half turn

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anticlockwise upside down symmetry reflectional symmetry line symmetry vertical horizontal

• Students unable to identify rotational symmetry and cannot determine the Order of Rotation (i.e. how many times the shape or object has turned before coming back to its starting position including the original position).

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Sub-strand: Location and transformation—L&T – 2

HANDS-ON ACTIVITIES Transformations: translations, reflections and rotations (also tessellations) • Using manipulatives such as pattern blocks, students choose one piece and trace around it on paper. They then translate (slide) the shape in any direction and draw around it again. Discuss what has happened to the shape. What has changed? What has stayed the same? How can we record what has happened on the paper? Use the shapes again, but this time, reflect the shape. Again discuss what has happened to the shape with each transformation. What has changed? What has stayed the same? How can we record what has happened on the paper? Repeat, but this time, rotate the shape. • In pairs, using pattern blocks, one student creates a pattern or design with about 8–10 pieces. The partner now has to create the same pattern, but reflected about a line. A drinking straw could be used to designate the mirror (symmetry) line. Discuss the fact that the pattern or design has now been reflected. What has changed? What has stayed the same? How can we record action on paper? The second student makes a different pattern or design and the partner has to translate the design. A third design could be made and the partner has to rotate the design 90° around a point. Again discuss what has happened to the design with each of the two different transformations.

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• Students investigate the use of (wall and floor) tiles and look at how many ways they can be put together as a display using translations, reflections and rotations. This could also lead to discussion on tessellations; filling an area with no gaps or overlaps.

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• Students look for all possible tetrominoes (four squares joined together, with each square sharing at least one complete edge with another square). There are five variations shown below. Students investigate what happens if we use any one of the tetrominoes and make a repeating (tessellating) pattern with it. Discuss which transformations are needed to complete the pattern. Record results with information about the translation/s used.

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We only needed to translate the ‘Z’ shape to make the first pattern, but we had to rotate and translate the ‘T’ shape to make the second pattern. • Students explore tessellations using a variety of quadrilaterals and triangles. They will come to the understanding that any triangle or quadrilateral will tessellate. They may wish to experiment to see how they might arrange them so that they will tessellate. Discuss the types of transformation that are used to enable each of their shapes to tessellate. What did we need to do to make the isosceles triangle tessellate? What about the scalene triangle? The rhombus? …

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• Students investigate the angles that are formed where the pieces of a tessellating pattern meet. What will be the sum of the angles at that point? (360°) This links with the unit on measuring angles (GR–1).

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• Identify shapes that are translations of each other in a tessellation. Look at wallpaper, tiling, paving etc. • Investigate brick pattern arrangements. Students investigate how many different patterns can be made with bricks that are twice as long as they are wide.

• Transforming tiles (See page 80). In this resource sheet students are required to translate, rotate 90° or reflect decorated tiles four times. • Translation transformations (See page 81). Students describe how a shape has been translated then follow instructions to redraw a shape in a new position. • Transformations (See page 82). Students identify if shapes have been translated, reflected or rotated and then draw either the reflection, translation and rotation 90° of three different shapes. • Tessellation treatment (See page 88). Students use grid paper to show how a number of different shapes tessellate. Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Location and Transformation—L&T – 2

HANDS-ON ACTIVITIES (CONTINUED) Reflectional (line) and rotational symmetry • Students investigate reflectional (line) symmetry in common objects such as cutlery, faces, telephones, clothes etc. Check using a transparent mirror.

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• Use folding and cutting to produce symmetrical shapes. Experiment with what happens with one fold, two folds, three or four folds. Look for the lines of symmetry in each of the resulting shapes.

• Fold paper in half and half again; the use a pin to carefully prick through the paper in a design. Open out the paper to reveal the result.

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• Using 1 cm2 grid paper, students copy a half a shape and then complete the picture by reflecting this onto the other side of a mirror line.

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• The best way to test a shape or design for rotational symmetry is to cut out the shape or design and hold it steady on one point with a finger or pin, then carefully turn the shape or design until it looks the same as the original. Continue to rotate until it is back in the original position. The number of times it can be rotated and still look the same as the original gives the Order of Rotational symmetry.

Rotational symmetry of (Order 3)

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Sub-strand: Location and Transformation—L&T – 2

HANDS-ON ACTIVITIES (CONTINUED) • The simplest way to produce a pattern with rotational symmetry is to allow students to explore the kaleidoscope effect of two mirrors placed at an angle. This effect also shows reflectional symmetry. Use pattern blocks between the mirrors. Hinged mirrors are available commercially or two mirrors can be taped together. Vary the angle between the mirrors. What happens?

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• Using 6 squares and joining along a full side, students make as many different arrangements as they can that have reflectional symmetry. Record the different arrangements on square paper. Check whether they have reflectional symmetry by using a transparent mirror or by cutting and folding the arrangements. Students then look for arrangements that have rotational symmetry. To check whether they have rotational symmetry, cut out the arrangement and try turning it about a point. If it looks the same as the original after a turn of less than 360°, the arrangement has rotational symmetry. The Order of Rotational symmetry is the number of times the shape appears the same before rotating the full 360°.

Rotational symmetry (Order 2)

Reflectional symmetry

• Students identify company logos for reflectional and rotational symmetry.

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• Students investigate the types of symmetry (reflectional or rotational) in the capital letters of the alphabet. Choose a font for upper case letters that best suits this activity, such as Ariel. Students use different ways to display their findings.

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Rotational symmetry

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Reflectional symmetry only

Rotational symmetry only

Reflectional and rotational symmetry

No symmetry

A, B, C, D, E, K, M, T, U, V, W, Y

N, S, Z

H, I, O, X

F, G, J, L, P, Q, R

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• Students use the ‘Auto shapes’ formatting palette in Microsoft Word™ to draw shapes, then go to ‘Format auto shape’ and ‘Size’ to enter the number of degrees of turn to rotate their shape. Try the more common 90°, 180° and 360°. Students predict what they think their shape will look like before they make the rotation. For example, they could draw the rectangle below, then rotation it three times before it gets back to its original position.

Original shape

90° turn

180° turn

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

360° turn

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Sub-strand: Location and transformation—L&T – 2

HANDS-ON ACTIVITIES (CONTINUED) Reflectional (line) and rotational symmetry • Using a transparent mirror (See page 83). Instructions for using a transparent mirror correctly and for finding the line of symmetry are provided on this resource sheet. • Transparent mirror reflections (See page 84). This resource sheet provides practice in using a transparent mirror and finding lines of symmetry. • Symmetrical shapes (See page 85). Students are required to join three shapes made from 1 cm2 grid paper in different ways to make shapes vertical, horizontal, angled and multiple line symmetry, rotational symmetry, and with no lines of symmetry and no rotational symmetry.

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• Magic master shape (See page 86). Students place a transparent mirror on a master shape to produce new shapes and record where they placed the mirror. • Reflectional and/or rotational symmetry (See page 87). A number of different shapes are provided and students need to identify those with reflectional, rotational, both reflectional and rotational symmetry, and those with none, then draw any lines of reflectional symmetry.

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• Rotating pattern (See page 89). Students use pattern blocks to make shapes then rotate them through three quarters, recording each pattern. • Reflecting beetle (See page 90). Students use pattern blocks to make a pattern, then reflect this pattern around the mirror line and record the result. • Paper people (See page 91). Students follow instructions to create a chain of paper people.

• More paper people (See page 92). Students follow instructions to cut out a circle of paper people.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Sub-strand: Location and transformation—L&T – 2

LINKS TO OTHER CURRICULUM AREAS Information and communication technology • An animated display that explains translations, reflections and rotations can be found at <http://www.learnalberta.ca/ content/me5l/html/math5.html> • An interactive reflection activity can be found at <http://www.primaryresources.co.uk/online/reflection.swf> • A site with 10 questions to answer about reflectional symmetry can be found at <http://www.innovationslearning. co.uk/subjects/maths/activities/year3/symmetry/shape_game.asp> If a student gets a wrong answer, the explanation provided is very good. • There is an interesting tessellating program at <http://www.pbs.org/parents/education/math/games/first-secondgrade/tessellation/>

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• Another website for tessellations is <http://nlvm.usu.edu/en/nav/frames_asid_163_g_2_t_3.html?open=activities> • The Illuminations website for tessellating shapes can be found at <http://illuminations.nctm.org/ActivityDetail. aspx?ID=27>

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• A similar page from the Illuminations website can be found at <http://illuminations.nctm.org/ActivityDetail. aspx?ID=35> • An interactive kaleidoscope site can be found at <http://www.zefrank.com/dtoy_vs_byokal/>

The Arts

• Students make print designs on vegetables (potato prints, carrots etc.). They then make a pattern with their print by translating, reflecting or rotating the design a number of times. Students could investigate whether the shapes look different or the same after the transformation.

• Make rotating paper pinwheels. You will need a piece of coloured paper about 10 cm by 10 cm, a dressmakers pin, a plastic drinking straw, a small piece of blu-tac or plasticine and a ruler and pencil.

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1. Draw in the two diagonal lines using a ruler.

2. Mark 1 cm from the centre on each of the diagonals.

3. Mark in a dot in the centre, and one in one corner of the four triangles, as in the picture below.

4. Use the pin to make a small hole on each of the dots, wiggling it to make the hole a bit larger than the shaft of the pin. (A sharp pencil could be used to make the holes, but make sure the holes are not too big.)

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6. Match up the hole in each triangle to the hole in the centre.

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5. Cut along the diagonals from the corner to the mark that is 1 cm from the centre. Do not cut through to the very centre.

7. Push the dressmakers pin through the holes, and on through to the drinking straw. Place a piece of blu-tac or plasticine on to the point of the pin. You now have a rotating pinwheel.

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Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Transforming tiles

The Terrific Tiling Company has made some tiles that are to be used on the walls of different rooms in a new house. Follow the directions to complete a row of each tile. The first two are done for you.

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Rotate 90° clockwise

Reflect

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Rotate 90° clockwise

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Translate

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

Translate


Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Translation transformations

A cartoonist makes his initial sketches on grid paper, and then moves the items to where they are needed. Describe how to translate (slide) each shape to its new position. The first one is done for you. 1.

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Slide the shape …

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3.

Now follow the instructions and redraw each shape in the correct new position.

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

2.

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Slide the shape down 2 and right 4.

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Slide the shape up 1 and left 5.

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5. Slide the shape up 3 and right 7.

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Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Transformations

The shapes below have been transformed by translation (slide), reflection (flip) or rotation (turn). Decide which transformation has been used for each shape.

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

Draw the correct transformation.

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6.

Translation

7.

Rotation 90°

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

4.

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Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Using a transparent mirror

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Place the mirror on the picture of the second butterfly. Try to move the page rather than the mirror; move it until one half of the figure reflects onto the other half. Draw along the ridge with your pencil to mark in the line of symmetry. The first butterfly has been done for you.

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

Hold the mirror on your desk, as shown in the picture. There is a ‘right’ and ‘wrong’ side for looking through the mirror. The ‘right’ side has a slanted edge which is placed face down on the paper. In this position, the mirror’s slanted edge will be touching the desk and you can more it to the mirror’s line of symmetry to show you a reflection.

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Now use the mirror to check for lines of symmetry in any other shapes you can find. Some shapes and objects will have one line of symmetry, some will have more than one line of symmetry, and some may have no lines of symmetry at all (which means they are asymmetrical).

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Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Transparent mirror reflections

2. Look carefully at the shapes below. Try to guess where the line of symmetry will be. Now place your transparent mirror on the shape until one side sits exactly on top of the other. Draw a line along the mirror. This is the line of symmetry. Mark in all the possible lines of symmetry (there may be more than one).

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3. Trace around the two triangles below onto coloured paper. Cut them out and use the transparent mirror to check they are the same (they are congruent). Glue them below in such a way that you can use the transparent mirror to mark in their line of symmetry. Show the line of symmetry with a dotted line.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

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1. Place the transparent mirror along the dotted lines. Look through the mirror and use a pencil to complete the shapes below. Notice that each side of the dotted line of symmetry looks exactly the same.


Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Symmetrical shapes

1. Cut out the three shapes below from 1 cm2 grid paper.

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2. Try to find as many different ways to join these three shapes together so that they have:

• Angled line symmetry • Multiple line symmetry • Rotational symmetry

You must use all three pieces for each arrangement.

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How many different results can you get for each of the five types of symmetry above? Draw your results, and label the types of symmetry.

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

• Horizontal line symmetry

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• Vertical line symmetry

o c . che e r o 3. Can you find other ways to r put the three pieces together so that there are no lines of t s s r u e p symmetry and no rotational symmetry? Record these results.

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Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Magic master shape

Below is the Master shape.

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Using a transparent mirror on the Master shape, see if you can make each of the shapes below. Record alongside each one where you put the transparent mirror to get the effect.

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4.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

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Where the transparent mirror went

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New shape


Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Reflectional and/or rotational symmetry

1. Which of the shapes below have reflectional symmetry? 2. Which have rotational symmetry? 3. Which have both reflectional and rotational symmetry? 4. Which ones have none?

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5. Draw in any lines of reflectional symmetry that you find. Some shapes may have more than one line of symmetry. Write ‘asymmetrical’ under any shapes that have no lines of symmetry.

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

(a)

(k)

(l)

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

(m)

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Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Tessellation treatment

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

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Use the grid paper below to show how each of the following shapes tessellate (fit together with no gaps or overlaps). Fill the rest of each grid.


Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Rotating pattern

1. (a)

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Complete the pattern by rotating the shapes around the next three quarters and record each of your patterns.

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

(b)

Put pattern blocks over the shapes below.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Reflecting beetle

Complete the pattern by reflecting the shapes around the mirror line and record the result.

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(b)

Put pattern blocks over the shapes below.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

1. (a)


Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET Paper people

You will need: • A strip of coloured paper about 20 cm x 10 cm • A template

• Scissors

1. Fold your strip of coloured paper in half and half again.

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2. On the front of your paper, draw around the template.

Put the feet on the bottom of paper, and ends of the hands on the folds.

© R. I . C.Publ i cat i ons 3. Cut out the person through all the layers of paper, • f or r ev ewintact. pur posesonl y• being careful to keep thei ‘hands’

5. (a)

Which people are walking the same way?

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Which are reflections of each other?

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Write about them.

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(don’t cut fold here)

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4. Unfold your paper people. Give them names and draw in eyes and smiles.

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

10 cm

10 cm

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fol dl

fold

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line

20 cm

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Location and transformation—L&T – 2

RESOURCE SHEET More paper people

This activity will produce a circle of paper people. You will need: • A square of coloured paper about 25 cm x 25 cm

Glue

• The template below

Felt tip pen(s)

• Scissors

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Sheet paper

1. Fold your square of paper along a diagonal to make a triangle. Fold again a smaller triangle. Fold again so you have 8 layers of triangles.

fold cut

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4. Glue your circle of people onto paper. Add some faces and name your people Jane, Lara, May and Zoe.

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3. Unfold your paper people.

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Which people are reflections of each other? Write about them.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

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fold

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2. Place the template of the half person along the fold. Draw around it and cut through all the layers.


Assessment 1

Sub-strand: Location and transformation—L&T – 2

NAME:

DATE: Testing transformations

S

1. The letter S has been transformed to a new position. Describe the movement (translation, reflection, rotation, up down, left, right).

S

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2. Describe the following transformations as translations, reflections or rotations.

(c)

(d)

(e)

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(b)

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3. One of the pictures below shows the letter Q has been reflected. Circle the correct picture.

QQ

Q Q

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Q

Q

QQ

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

(a)

4. There are three cars below. Make a copy of each car according to the transformation indicated.

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Reflected

(c) Translated

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Assessment 2

Sub-strand: Location and transformation—L&T – 2

NAME:

DATE: Pattern block symmetry Make a reflection of the design and draw in the resulting shape.

(b)

© R. I . C.Publ i cat i ons Use pattern blocks to make the design below. •f orr evi ew pur posesonl y•

Make a rotation of the design through 180° and draw in the resulting shape.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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CONTENT DESCRIPTION: Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries

2. (a)

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(b)

Use pattern blocks to make the design below.

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Checklist

Sub-strand: Location and transformation—L&T – 2

Identifies rotational symmetry

Identifies line symmetry

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STUDENT NAME

Can translate, reflect and rotate 2-D shapes

Describe translations, reflections and rotations of two-dimensional shapes. Identify line and rotational symmetries (ACMMG114)

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Sub-strand: Location and transformation—L&T – 3

Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original (ACMMG115)

RELATED TERMS

TEACHER INFORMATION

Enlargement transformation

• The resizing of a two-dimensional shape such that the new shape is no longer congruent to the original, but is similar. Similar shapes

• The use of an enlargement transformation enables students to get a better idea of scale.

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• There is a link to the concepts of area and perimeter when doubling or trebling two-dimensional shapes using a grid. (See UUM–1) • There is a link to reading maps using coordinates, and understanding the scale of maps. (See L&T–1) • There is also a link to transformations in that similar shapes are not necessarily shown in the same orientation and may have been translated, reflected or rotated. (See L&T–2)

Teaching points • Start by enlarging very simple shapes.

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• Shapes that are of the same proportion, but with different lengths to the sides. The corresponding angles of the two shapes will be the same (congruent); and the corresponding sides will be in the same ratio. In the example below, the two triangles are similar. The lengths of each of the sides have been doubled, but the sizes of the corresponding angles remain the same.

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What does it mean

• Discuss the relationship between the area of a shape where the lengths and widths have been doubled and the area of the original shape. It will be four times the area (two by two times larger).

© R. I . C.Publ i cat i ons •andf o rr evi ew pur posesonl y• Note: all regular polygons all circles

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Congruent shapes

• Shapes that have exactly the same shape and size. In the example below, the two triangles are congruent because they have corresponding sides the same length and corresponding angles equal. The different orientations (one has been rotated) make no difference to their congruence.

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• When doubling or trebling the dimensions of a shape, the perimeter of the new shape, or the lengths of any of the sides, will be double or treble the original. • The relationships between the area and perimeter of the new shapes compared with the original shapes will vary if only one of the dimensions is enlarged; e.g. if only the height of a shape is doubled.

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are similar, regardless of their dimensions; i.e. all squares are similar, all regular hexagons are similar etc.

• If the dimensions of a two-dimensional shape have been trebled, the area of the new shape is three by three times larger; i.e. nine times larger.

• If all the dimensions of a shape are doubled or trebled, the resulting shape will be similar to the original (see note on the left). • If the corresponding lengths of a two-dimensional shape are the same and the corresponding angles are the same, the shapes are congruent (equal in every way). The orientation of the shape, whether they have been translated, reflected or rotated, makes no difference to their congruence.

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• There is usually at least one enlargement or scale question in the national tests at Year 5.

What to look for Student vocabulary enlargement proportional congruent congruence similar

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• Students not retaining the same proportions in their enlargements. • Students not recognising similar and congruent shapes.

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Location and transformation—L&T – 3

HANDS-ON ACTIVITIES • Big and bigger 1 and 2 (Pages 100 and 101). On 1 cm2 paper, students draw outlines of familiar two-dimensional shapes a particular number of times larger that the original (e.g. double or triple the dimensions). Start with simple shapes that follow the grid lines, such as squares and rectangles, then introduce shapes that cross diagonals, such as triangles. Compare the area and perimeter of the before and after shapes. What is the same about both shapes? What is different? Why?

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P = 8 cm A = 4 cm2

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P = 16 cm A = 16 cm2

P = just over 12 cm A = 8 cm2

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The perimeters of the new shapes are twice as long as the original ones. The areas of the new shapes are 4 times as big as the original ones. The before and after shapes are similar.

• Big and bigger 3 and 4 (See pages 102 and 103). Students use a larger grid paper to copy shapes and to report on relative size. • Big and bigger 5 (See page 104). Students construct double scale models of different shapes on grid paper and complete a table comparing perimeters and areas. • Growing triangles (See pages 105 and 106). Triangle paper is used to construct triangle with different length sides and perimeters and areas recorded and compared. • Rover (See page 107). Students use a grid to double the size of a dog shape.

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• Discuss why, when the lengths and widths of a shape have been doubled, the area is four times larger and when they have been tripled, the area is nine times larger. • Students make drawings of a simple shape and then double only one dimension; for example, the height only. What happens to the area in this instance? What about if only the widths are doubled and not the heights?

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• Triple scale with curves (See page 109). Later students can enlarge shapes that involve curves. This would lead to students taking a design they like, such as a favourite cartoon or image and being able to enlarge it by using an overlaid grid on the original, then a second, larger grid for the enlargement and copying across the design square by square.

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• Students look at photographs of familiar objects on an interactive whiteboard or computer, then copy and enlarge the image alongside and discuss what has changed and what has stayed the same. • Use model cars and planes to discuss the scale factor and compare the model to the original. The length, for example, of a model racing car could be compared to the actual length of the car by laying out a length of rope that is the true length and placing the model alongside it. The same could be done with the width of the car at its widest point. A rectangle could then be marked out that roughly defined the area of the car when viewed from above and the model placed in one corner to show how much larger the area of the car is.

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• Change the scale of maps using a grid system to enlarge them. For example, students could enlarge the map of Tasmania (see below) by making a grid that is twice as long and twice as high as the one shown, then copying each section of the map, square by square, onto the enlarged grid. The southernmost section has been started below. The overall area of the new map will now be four times as large.

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Sub-strand: Location and Transformation—L&T – 3

HANDS-ON ACTIVITIES (CONTINUED) • Many students would be familiar with the use of GPS systems for location. Discuss the scale of the images. Students may also be familiar with Google™ maps and again be aware of the facilities for enlarging (and reducing) the images. • Students construct shapes that are congruent and shapes that are similar. Discuss the difference. Link the shapes that are similar to the enlargement of shapes, as long as both dimensions (length and width) are enlarged by the same factor; e.g. both doubled or trebled. If only one dimension is doubled or trebled, the resulting shape will not be similar.

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• Point projection (see page 108) is another enlargement technique. Students draw a shape and then make a point outside the shape. They then rule a line from the point to each of the vertices of the shapes and beyond. To double the scale of the original shape, the same distance from the point to a vertex is measured and marked on the other side of the vertex. This is done for each vertex. Finally the new vertices that have been marked are joined. The lengths of each side of the new shape will be twice the lengths of the original shape and it will have four times the area.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Location and transformation—L&T – 3

LINKS TO OTHER CURRICULUM AREAS English • Read The once upon a time map book by B Hennessy and P Joyce. This book takes a trip to six different well-known story lands with maps, coordinates, routes, hidden objects and points of interest. Each map has a grid overlay so coordinates can be used to locate certain features. They each have a key (legend) indicating features such as rock paths, giant’s stairs and camel roads. Discussion could focus on the scale of the maps and what that means in ‘real life’ measures. They could look at a particular feature on a map—for example, the distance of a path—work out its actual length, pace it on the oval and lay it out with rope. The measures on the maps include: Pirate miles, Munchkin miles, Rabbit hops and Genie steps. • Read Knee high Nigel by L Anolt. This is a story about five giants, one of whom though still a giant, is considerably smaller than the others. They argue over the building of castles and go their separate ways with unsuccessful results. Again this book could be used as above to compare the sizes of each of the giants and the buildings they construct.

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• Read Jim and the beanstalk by Raymond Briggs. This book is a humorous take on the Jack and the Beanstalk tale. In the story, the giant demands that Jim gets him glasses, false teeth and a wig. This could lead to a discussion on the relative sizes of these items and the idea of scale. The illustrations show Jim standing on the giant’s hand and carrying the glasses, teeth and wig one at a time. From this students could consider how tall the giant would be and how big each of the items he gets for him would be. Using cardboard, students could construct a pair of glasses that would fit the giant. They could also make items not included in the story, such as what size they would need to make shoes or a shirt to fit him. • The story Six feet long and three feet wide by J Billington and N Smee uses the idea that a bed to be made for a princess is measured out by the king’s foot size then built by an apprentice with smaller feet. It could be used to discuss the difference in scale between the two people’s feet. • Another book with exactly the same premise is How big is a foot? by R Myller.

• The book Actual size by S Jenkins has fantastic drawings of animals (actual size, large and small). Be sure to get the English edition in centimetres.

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

Information and Communication Technology

• There is a good tutorial on similar shapes, with quiz questions interspersed, which can be found at <http://www.carmel. org.uk/elearning/similar_shapes/index.htm> • A short practice quiz on congruent and similar shapes can be found at <http://au.ixl.com/math/year-5/similar-andcongruent>

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• A game in which students decide whether shapes are similar or congruent can be found at <http://au.ixl.com/math/ year-5/similar-and-congruent> • A website where students check for congruence of shapes can be found at <http://www.learner.org/courses/ teachingmath/grades3_5/session_02/section_02_b.html>

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History and Geography

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• Discuss early Egyptian wall paintings, where designs were first made on papyrus on which small squares had been drawn before large squares were chalked onto the wall to be decorated. The artists then copied the design within each square on to the equivalent enlarged square of the wall to achieve the desired decoration. The outlines of the designs were called ‘cartoons’.

The Arts

• Students design their own ‘cartoons’ for enlargement using grids. They could swap designs with a partner and each replicate a partner’s enlarged design on a larger grid. • Groups of students could make a design on grid paper that they copy onto paving squares using washable chalk. They could end up with a walkway where the pavers are decorated according to a theme. Other classes could be invited to view their creations.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Location and transformation—L&T – 3

RESOURCE SHEET Big and bigger – 1

The shapes below have been drawn on 1 cm2 grid paper.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

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Make a new version of each shape, but twice as long and twice as high as the originals.


Sub-strand: Location and transformation—L&T – 3

RESOURCE SHEET Big and bigger – 2

The shapes below have been drawn on 1 cm2 grid paper.

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CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

Make a new version of each shape, but twice as long and twice as high as the originals.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Location and transformation—L&T – 3

RESOURCE SHEET Big and bigger – 3

The shapes below have been drawn on 1 cm2 grid paper.

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(b)

How much longer is each of the lines of your new shapes?

(c)

How much bigger is the area?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

Make a new version of each of the shapes, but on the larger grid paper.

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1. (a)


Sub-strand: Location and transformation—L&T – 3

RESOURCE SHEET Big and bigger – 4

The shapes below have been drawn on 1 cm2 grid paper.

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Make a new version of each of each of the shapes, but on the larger grid paper.

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CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

1. (a)

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(b)

How much longer is each of the lines of your new shapes?

(c)

How much bigger is the area?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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103


Sub-strand: Location and transformation—L&T – 3

RESOURCE SHEET Big and bigger – 5

1. (a)

On 1 cm2 grid paper, construct a double scale model of each shape below.

1.

4.

© R. I . CPerimeter .Publ i cat i onsArea double Area original double scale •f orr evi e w pur posesonl yscale •

Complete the table below.

Shape 1

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Perimeter original

3

(c)

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How do the perimeters of the original shapes compare with those of the double

o c . che e Explain why. r o t r swith those of the double scale? supcompare er How do the areas of the original shapes scale?

(d)

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Explain why. (e)

What do you think the perimeter of Shape 1 would be if we made a new shape that was triple the scale?

(f)

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What do you think its area would be?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

(b)

Note: the length of a line that cuts a square in half diagonally is really a bit more than 1 cm long, but for this activity we’ll count it as 1 cm.

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3.

2.


Sub-strand: Location and transformation—L&T – 3

RESOURCE SHEET Growing triangles

1. (a)

On the triangular paper (see page 106), draw around one triangle and cut it out. One side of this triangle is 1 unit long. How long is the perimeter? The area is 1 triangle.

2. (a)

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Cut out another triangle that has sides 2 units long like this. How long is its perimeter?

(c)

How many small triangles is its area?

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(b)

3. (a)

Cut out triangles with sides 3, 4 and 5 units long. Glue all the triangles onto a chart. Colour them in.

(b)

Copy the table below onto your chart and complete the information for each triangle. Length of sides

Perimeter

Area (number of small triangles)

©1 R. I . C.Pub l i cat i ons1 3 •f orr vi ew pu6r poseson y• 2e 4l 4 5

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3

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CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

(b)

o c . che e r o t r s uper Now look for the numbers patterns.

When you see the pattern, complete the table for triangles with sides of 6, 7 and 8 units. You do not need to cut out these triangles. Title your chart Growing triangles.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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105


Sub-strand: Location and transformation—L&T – 3

RESOURCE SHEET

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

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Triangle paper


Sub-strand: Location and transformation—L&T – 3

RESOURCE SHEET Rover

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Draw a bigger picture of Rover, below, making the lengths and widths of each line twice as big. For example, this Rover’s nose is one square wide and one square long, so the new Rover will need his nose to be two squares wide and two squares long.

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© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

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CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

1. (a)

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How many times bigger is your Rover than the original Rover?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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107


Sub-strand: Location and transformation—L&T – 3

RESOURCE SHEET Point projection

A1

Step 1: Draw the shape you wish to enlarge. (This is triangle ABC) Step 2: Place a dot outside the shape (D).

A

B

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Step 4: Do the same for all the other vertices.

C

D

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Step 5: To double the scale of the shape, measure the distance from D to A. Mark a new point A1 that is the same distance, but from A to A1 on the line you’ve already drawn.

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Step 6: Do the same for all the other vertices and mark the new points B1 and C1. Step 7: Join the vertices for the new shape. (This is triangle A1 B1 C1) 1. (a)

What can you say about the lengths of the sides of the new shape?

(b)

What can you say about the area of the new shape?

(c)

Are the two triangles congruent, similar or neither?

© R. I . C.Publ i cat i ons f oonto rr evi ew pu r pos sothe nl y•as 2. Copy these• shapes paper and use point projection toe enlarge shapes directed.

Double the lengths of the sides of these two shapes.

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These two shapes are:

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congruent

. neither te o Triple the length of the sides of these two shapes. c These. two shapes are: che e r o t r s congruent super similar

similar

neither

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(c)

Write the lengths of all the sides on each of your shapes, both the original and new shapes.

(d)

Can you work out the areas of each of the shapes?

(e)

Tick the boxes to show if each pair of shapes are congruent, similar or neither.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

C1

B1

Step 3: Rule a line from the dot (D) to the first vertex and beyond.


Sub-strand: Location and transformation—L&T – 3

RESOURCE SHEET Triple scale ... with curves

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Fit your curved lines into squares and rectangles that are 3 times as wide and 3 times as high as the squares and rectangles the small drawing is based on.

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CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

Make a triple size (3 times) scale drawing of this penny-farthing bicycle. Begin at the bottom of your square paper.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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109


Assessment 1

Sub-strand: Location and transformation—L&T – 3

NAME:

DATE: Making bigger

© R. I . C.Publ i cat i ons •f orhas r e vi e wonp1u p seMake so l y •of it, The shape below been drawn cmr grido paper. an new version 2

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but on the larger grid paper.

o c . che e r o t r s super How much longer is each of the lines of your new shapes?

(c)

110

How much bigger is the area?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

2. (a)

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1. The shapes below have been drawn on 1 cm2 grid paper. Make a new version of each of the shapes, but twice as long and twice as high as each of the originals.


Assessment 2

Sub-strand: Location and transformation—L&T – 3

NAME:

DATE: Jumbo

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Draw a bigger picture of Jumbo below, making the lengths and widths of each line twice as big. For example, our Jumbo’s ear is half a square wide and one and a half squares long; so the new Jumbo will need his ear to be one square wide and three squares long.

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CONTENT DESCRIPTION: Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original

1. (a)

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How many times bigger is your Jumbo than the original Jumbo?

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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111


Checklist

Sub-strand: Location and transformation—L&T – 3

Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original (ACMMG115) Students enlarge two-dimensional shapes using ...

larger grids

point projection

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STUDENT NAME

grids of the same size

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Answers

Sub-strand: Location and transformation—L&T

L&T – 2

L&T – 1 Page 63

Resource sheet – Sunny’s walk in the park

Page 80

Resource sheet – Transforming tiles

1. 6 2.–3. Teacher check. Answers will vary. Page 64 1.

Translate

Resource sheet – Sunny’s fun at the playground

Rotate 90°

7

Reflect

6

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5

4

Rotate 90°

2

Reflect

1

1

2

3

4

5

6

7

2. 5, 1 2, 6 5, 4 3. 4, 1 2, 5 6, 4 7, 7 4.–5. Teacher check. Answers will vary.

Page 69

Resource sheet – Whose zoo?

Translate

Page 81

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3

Resource sheet – Translation transformations

1. Slide the shape down 2 and right 4. 2. Slide the shape up 1 and right 4. 3. Slide the shape down 2 and right 8. 4.

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

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Assessment 1 – Mystery tour revisited

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1. Go west 3 km, turn left and continue on road for 7.5 km. Turn right and go 3 km. Turn left and go 5.5 km to the circus. 2. 11, C 3. church 4. Go east for 5.5 km, then turn right and go southwest for 4 km to the forest. Return north-east to the intersection and go east for 9.5 km. Take a sharp right to pass the movies. Head south for 9.5 km, turn right and continue 17 km west to the circus. Finally, head north 10 km to where the bus started. Total: 59.5 km. 5.–6. Teacher check.

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Resource sheet – Transformations

1. Rotation 3. Translation 5.

2. Reflection 4. Rotation 6.

7. Page 84

Resource sheet – Transparent mirror reflections

1.–3. Teacher check

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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113


Answers Page 85

Sub-strand: Location and transformation—L&T

Resource sheet – Symmetrical shapes

(c)

(d) Reflectional and rotational symmetry

Note: There may be solutions other than the ones given below. 1. Teacher check 2.

Reflectional and rotational symmetry

vertical (e) horizontal

(f )

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angled

(i)

Reflectional symmetry only

(j)

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multiple

Reflectional symmetry only

No reflectional or rotational symmetry

Reflectional symmetry only rotational

(k)

BOB

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Reflectional and rotational symmetry

Reflectional symmetry only

(g)

3.

H

Reflectional symmetry only

(l)

No rotation

Reflectional or rotational symmetry

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

Page 86

(m)

Resource sheet – Magic master shape

1.

No reflectional or rotational symmetry

Page 89

Resource sheet – Rotating pattern

1. (b)

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1. (b)

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Resource sheet – Reflectional and/or rotational symmetry

(b), (c), (d), (e), (f ), (g), (h), (i), (k), (l) (a), (b), (c), (d), (f ), (l) (b), (c), (d), (f ), (l) (j), (m)

Z Rotational symmetry only

(b) Reflectional and rotational symmetry

Resource sheet – Reflecting beetle

Page 93

Assessment 1 – Testing transformations

1. Translation 3 down and 7 right. 2. (a) Reflection (b) Translation (c) Translation (d) Rotation (e) Reflection 3. Q Q Q QQ

Q

1. 2. 3. 4. 5. (a)

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4. (a) (b)

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Q Q

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

(c)

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Answers

Sub-strand: Location and transformation—L&T

Page 94

Assessment 2 – Pattern block symmetry

1. (b)

Page 105

Resource sheet – Growing triangles

1. (a) Teacher check 2. (a) Teacher check 3. (a) Teacher check (b)

Perimeter

Area (number of small triangles)

1

3

1

2

6

4

3

9

9

4

12

16

5

15

25

6

18

36

7

21

49

8

24

64

(c) The perimeters are all triangular numbers and the areas are all square numbers.

Page 107

Resource sheet – Big and bigger – 1

Resource sheet – Rover

1. (a)

Teacher check Page 101

Length of sides

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L&T – 3

Page 100

Resource sheet – Big and bigger –2

Teacher check

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• Resource sheet – Big and bigger – 3

(b) My Rover is two times bigger than the original Rover.

1. (a) Teacher check (b) double (twice as long) (c) four times the area Page 103

Page 108

1. (a) They are double the original lengths. (b) It is double the original area. (c) similar 2. (a) similar (b) similar (c)–(e) Teacher check

Resource sheet – Big and bigger – 4

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Resource sheet – Big and bigger – 5

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(Note: the perimeters and areas given for Shape 4 may vary slightly, but the double scale perimeter should be twice as long as the original, and the double scale area should be four times as big.) 1. (b) Perimeter Area Shape

10 cm

2

Page 109

Resource sheet – Triple scale ... with curves

Teacher check

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Perimeter double original scale

1

Resource sheet – Point projection

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(c) 4

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2. (b)

(b) 3 units (b) 6 units

Page 110

Assessment 1 – Making bigger

1.

Area double original scale

20 m

4 cm2

16 cm2

11 cm

22 cm

51/2cm2

22 cm2

3

8 cm

16 cm

4 cm2

16 cm2

4

10 cm

20 cm

81/2cm2

34 cm2

(c) The perimeters are twice as long, as each line has been doubled. (d) The areas are four times as big, as it is 2 x 2 the original size. (e) The perimeter would be 30 cm. (f ) The area would be 36 cm2.

2. (a) Teacher check (b) Two times longer

Page 111

(c) Twice as big

Assessment 2 – Jumbo

1. (a)

(b) Two times bigger

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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115


Sub-strand: Geometric reasoning—GR – 1

Estimate, measure and compare angles using degrees. Construct angles using a protractor (ACMMG112)

RELATED TERMS

TEACHER INFORMATION What does it mean

Angle

• Two rays with a common endpoint called a vertex and the extent of rotation about a point. Protractor

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• Right angle: 90°.

• Students will need to be shown how to use a protractor correctly. It is usually better to use a 180° protractor as it is the one most commonly used. Students may be introduced to a 360° protractor (preferably with a moveable line) if desired.

Teaching points

• Angles are classified by their size in their relationship to the right angle (90°)—see definitions left.

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• A device for measuring the size or an angle in degrees. It is usually made of clear plastic and is in the shape of a semi-circle, full circle or square.

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• Students are introduced to the protractor in Year 5. This should come after they have a sound understanding of the relative size of angles compared to the right angle.

• Show all angles in different orientations, especially right angles. This is to help avoid a common misconception that we can have right angles and ‘left angles’.

• Acute angle: less than 90°.

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• • Obtuse angle: greater than 90°, but

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• Straight angle: exactly 180°.

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• Reflex angle: greater than 180°, but less than 360°.

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• It is important to make angles with different arm lengths so that students realise that the length of the arms does not affect the size of an angle. • National tests often include a question on angles in which students identify the largest or smallest angles from a set of angles that have different lengths on the arms. There is also often a question where students are asked to identify types of angles; e.g. acute, obtuse, right angle.

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• Students who don’t know the different types of angles and their properties.

• One rotation: a full turn to end up at the start; 360°.

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• Students confused by the length of the arms of an angle; thinking that an angle with short arms is less than an angle of lesser degrees but with longer arms. Students with this misconception would judge that the first angle below is larger than the second, because of the length of the arms. In fact, the second angle is larger.

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Sub-strand: Geometric reasoning—GR – 1

Estimate, measure and compare angles using degrees. Construct angles using a protractor (ACMMG112)

TEACHER INFORMATION (CONTINUED) What to look for (continued) • Students who don’t know which scale to use on a protractor (the inner or outer scale).

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• Students who don’t know that the centre mark must be on the vertex.

• Students unsure how to construct particular angles using a protractor.

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• Students who can’t identify if the angle they are measuring is greater than or less than 90° (i.e. whether it is acute or obtuse), or whether it is greater than 180° (i.e. a reflex angle).

• Students constructing an incorrect angle due to misreading the scale on a protractor; e.g. constructing an angle of 135° instead of 45°.

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Student vocabulary

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right angle (90°) acute angle

obtuse angle reflex angle

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straight angle (180°) full rotation (360°)

degrees protractor base line centre mark inner scale outer scale

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Geometric reasoning—GR – 1

HANDS-ON ACTIVITIES • Students practise measuring angles using a protractor (see pages 122 and 123), and then practise constructing angles (see page 124). Begin measuring acute and right angles, move on to obtuse angles and finally, if students are ready, measure reflex angles. (See page 125 and practice sheet on page 126.)

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• Comparative size of angles in a set of pattern blocks. The angles on a set of pattern blocks are all proportional. This is particularly useful as we can use the square as a starting point (knowing that each angle is 90°) to work out the angles of the other pieces. Another starting point is for the students to know the angles on the green triangle are each 60° (as it is an equilateral triangle). If not they could lay three triangles on top of two squares. This shows 180° (a straight angle), and the adjoining angles of three triangles together make this angle, so each must be 180° ÷ 3, which is 60°. There are two different-sized angles in the blue rhombus: the acute angle is the same size as the angles on the triangle (60°), while the obtuse angle is the same as the angle on two triangles, therefore being 120°. The two different-sized angles on the trapezium are the same as the two different-sized angles on the blue rhombus, so they are 60° and 120°. The angles on the hexagon are the same as the obtuse angle on the rhombus, so they are 120°. Finally, the two acute angles on the white (tan) rhombus are the same as one of the acute angles on the blue rhombus, so each is 60° ÷ 2, which is 30°. The obtuse angle on this last white rhombus is the same as laying a square and triangle together; so it is 90° + 60°, which is 150°.

120°

60°

60°

60°

60°

60°

120°

120°

120°

90°

90°

90°

90°

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30° 120°

120°

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120°

120°

120°

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60°

60°

30°

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• Using the above information, students could investigate the sum of the internal angles of a quadrilateral. They have the angle measurements of the square (4 x 90° = 360°), the blue rhombus (120° + 120° + 60° + 60° = 360°), the white rhombus (150° + 150° + 30° + 30° = 360°) and the trapezium (120° + 120° + 60° + 60° = 360°). From this, they could investigate the internal angles of other quadrilaterals such as a kite, parallelogram and rectangle as well as other quadrilaterals that are none of the above. Students will need to be able to use a protractor to be able to complete this (see page 122). An activity that looks at the internal angles of a square is on page 129. • The internal angles of a triangle will always add up to 180. In the pattern block activity above, the only triangle available is an equilateral triangle where all the angles are 60°, so the three angles add to 180°. (Also see activity on page 128.) • Link to L&T–3 where students apply enlargement transformations to various shapes. Students could measure the angles of any polygons before and after enlargement and compare them. Corresponding angles should be the same size, which reinforces the idea that the size of an angle is not influenced by the length of the arms (or in this case, the length of the sides of the polygons).

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Geometric reasoning—GR – 1

HANDS–ON ACTIVITIES (CONTINUED) • Link to activity on page 6 (UUM–1) on tangrams. Students could investigate the size of the angles of each of the seven pieces. This could be done by comparing them to a right angle (90°), without the use of a protractor.

• The square has four angles, all 90°.

r o e t s Bo r e p ok u S

• The parallelogram has two acute angles that are the same size as the acute angles on the triangles which is 45°. The other two angles are obtuse. If you lay the square on top of this angle and then lay the acute angle of any of the triangles next to it, they will add up to the same as the obtuse angles on the parallelogram; i.e. 90° + 45°, which is 135°.

Tangram pieces

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• There are 5 triangles, all similar right-angled isosceles triangles: two large, one medium and two small. So one angle of each triangle is 90°, the other two angles on each of the triangles are equal, therefore they are half of 90°, which is 45°.

• We could check by adding the four angles of the parallelogram: 45° + 45° + 135° + 135° = 360°.

• We know that the internal angles of all quadrilaterals add up to 360°, so these angles must be correct.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Geometric reasoning—GR – 1

LINKS TO OTHER CURRICULUM AREAS English • Read What’s your angle Pythagoras? by Julie Ellis. This book is a fictional look at how Pythagoras may have found out about right-angled triangles. Although it looks at Pythagoras’ theorem, which is beyond Year 5, it would provoke an interesting discussion on the usefulness of knowing about angles.

Information and Communication Technology • There is an animated demonstration on how to draw a 50° angle using a 180° protractor. A virtual protractor can then be used for students to have a go for themselves. It can be found at <http://www.mathsisfun.com/geometry/ protractor-using.html>

r o e t s Bo r e p ok u S

• Another website that allows students to measure angles using a 180 protractor can be found at <http://www. mathplayground.com/measuringangles.html> It has a tolerance of only 1°. • Yet another website that gives a short tutorial, then some practice measuring angles with a 180° protractor can be found at <http://lrrpublic.cli.det.nsw.edu.au/lrrSecure/Sites/LRRView/13231/applets/protractor.htm>

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• There is an interesting ‘Alien angles’ game at <http://www.mathplayground.com/alienangles.html>

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• Mission 2110 Roboidz game: This game uses angles, but it is somewhat corny. It includes the fact that the three angles of a triangle equal 180°. It can be found at <http://www.bbc.co.uk/bitesize/ks2/maths/shape_space/angles/play/> • Wally the Penguin demonstrates turning 90°, 180°, 270° and 360° at <http://www.kerpoof.com/#/view?s=iai1000001> • The National Library of Virtual Manipulatives has a site where students track a ladybug through a maze by clicking on directional keys with icons such as turn right or one step forward. The site can be found at <http://nlvm.usu.edu/en/ nav/topic_t_3.html> Scroll down to Grades 3–5 and find ‘Ladybug mazes’.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Mix and match angles

Cut out the 24 cards below. Mix them up and try to put them back in the correct groups.

90°

r o e t s Bo r e p ok u S 180°

Acute angle

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Obtuse angle

Reflex angle

45°

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360° ©R . I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

Full turn

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CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

Straight angle

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Right angle

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200°

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Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Using a 180° protractor

A standard 180° protractor is used to measure angles between 0° and 180°. Its features are the: • base line

• outer scale marked from left to right

• inner scale marked from right to left

• centre mark.

r o e t s Bo r e p ok u S

base line

© R. I . C. Pu bl i cat i ons centre mark To measure a particular •f oangle: rr evi ew pur posesonl y•

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1. Place the protractor with the base line along one arm of the angle to be measured and the centre mark on the vertex (the corner itself).

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2. Decide whether the angle is acute (less than 90°) or obtuse (between 90° and 180°). 3. Use the scale that will give the correct reading for the type of angle.

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4. Read the number on the scale where the second arm of the angle lies.

This angle is an acute angle. It is 30°. 122

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

inner scale

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outer scale


Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Measuring acute and obtuse angles

1. Use your knowledge about measuring angles using a protractor to measure the following six angles. Remember to work out beforehand whether you would expect each angle to be an acute angle, a right angle or an obtuse angle. Circle your answer.

r o e t s Bo r e p ok u S

°

(b)

right angle

right angle

obtuse angle

obtuse angle

Teac he r

acute angle

° © R. I . C.Publ i cat i o n s (d) acute angle angle • f o rr evi ew pur posesacute on l y• °

right angle

right angle

obtuse angle

obtuse angle

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(e)

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(c)

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acute angle

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CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

(a)

°

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acute angle

acute angle

right angle

right angle

obtuse angle

obtuse angle

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Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Constructing angles

1. Using a ruler and pencil, draw a straight line about 15 cm long.

80 70

100

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30 10

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4. Remove the protractor. Use a ruler to join the dot to the end of the line where the centre mark had been.

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0

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10

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10

170

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180

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0

30

5. Use the protractor to check that the angle is correct.

0

0

90

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0 13

40

110 0 12

50

6. In the space below, make these angles using the method above. Remember to look at whether the angle is acute (less than 90°) or obtuse (more than 90°). (a) 65°

(d) 12°

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(c) 110°

(e) 165°

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

30

10

170

0

0

110

0 12

50

180

180

10

170

160

20

80

70

60

3. Read around the scales of the protractor until you reach the size of angle you want to construct. Mark the paper with a dot at the point.

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14 0

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0 13

40

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110

60

50

0 14

2. Place the base line of the protractor on the line with the centre mark in the middle. Mark it with a dash.


Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Measuring reflex angles – 1

Reflex angles are angles that are greater than 180° but less than a full turn (360°). This is a reflex angle (ABC). It is equivalent to 360° minus the acute angle. A

This is a reflex angle (JKL). It is equivalent to 360° minus the obtuse angle. J

r o e t s Bo r e p ok u S B

K

To measure a reflex angle:

1. Extend one arm of the angle beyond the vertex. J D B

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K

A

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D B

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M © R. I . C.Publ i ca t i ons 2. Use a protractor to measure the angle from the extended arm to the other arm of the •f orr evi ew pur posesonl y• angle. C

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CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

A

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C

L

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K

JKM = 45° o c . c eneed to add the straight 3. Finally, add 180° to the measured angle. This is because we h r e o angle that forms part of ther refl ex angle. r s upest C

M

ABD = 135°

J

A

L

D B

K M

C 135° + 180° = 315°

45° + 180° = 225°

So angle ABC is 315°

and angle JKL is 225°

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Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Measuring reflex angles – 2

Use your knowledge about measuring angles using a protractor to measure the following six angles. Remember to work out beforehand exactly which angle is to be measured. °

°

(a)

°

(c)

° (d)

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

r o e t s Bo r e p ok u S

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(b)


Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET Which angle?

1. Look at the sketches below. In each of them, a protractor is shown placed over an angle. Decide which of the two options is correct and explain why.

70 60

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14 0

40 14 0

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Explain.

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(d)

Is the measure of the angle 89° or

Explain.

Explain.

91°?

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Is the measure of the angle 5° or

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

150

180

40

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0

10

50

Explain.

0

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0

175°?

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13 50

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Is the measure of the angle 35° or 145°?

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1

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or 170°?

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Is the measure of the angle 10°

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0 0 14

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10

(b)

14

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13 50

40

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30

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13

Explain.

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0

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• 70

180

110 70

0

Is the measure of the angle 60°

60

(e)

100 80

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or 120°?

(c)

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30

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20

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30

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0 50

1

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(a)

CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

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0 14

170

100 80

1 30

0

90

100

20

50

180

80

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Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET What’s your angle? – 1

Remember – a straight line is an angle of 180°

180°

– the angle around a point is 360°. 360°

A

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C

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1. Make 3 copies of the triangle (right) onto coloured paper and cut them out. Label the angles A, B and C on each triangle.

Bu © R. I . C. l i cat i ons AP C b angle Ai +e angle + angle C =o 180° •f orr ev wBp ur p sesonl y•

3. Draw your own different shaped triangle below and make 3 copies of it.

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Label the angles.

Cut the triangles out and place one of each together to see if the three angles equal 180°.

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Glue the triangles on a sheet of paper, and don’t forget to label the angles and write your results. L

M

N

° angle L + angle M + angle N = 128

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

2. Place the 3 triangles together so you have one of each angle joining the next. They form a straight angle, which is 180°. Glue the 3 triangles together on a sheet of paper and write:


Sub-strand: Geometric reasoning—GR – 1

RESOURCE SHEET What’s your angle? – 2

1. (a)

(b)

B

A

Place the 4 squares together so you have one of each angle joining the next. They form a full turn angle, which is 360°.

r o e t s Bo r e p ok u S

Glue the 4 squares together on a sheet of paper as shown below and write:

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(c)

Make 4 copies of the square on coloured paper. Label the angles.

2. (a) (b)

D A B D C

C

© R. I . C.Publ i cat i ons Make 4 copies ofv it,i on coloured paper cut them out label all• the angles. • f o rr e e w pu r p o se sand on l y Draw your own different quadrilateral (a shape with four straight sides).

Fit one of each of the corners together.

(d)

What angle do they make? Glue them onto a sheet of paper and write:

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(c)

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CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

angle A + angle B + angle C + angle D = 360°

L

K

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N

M

° angle K + angle L + angle M + angle N = Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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Assessment 1

Sub-strand: Geometric reasoning—GR – 1

NAME:

DATE: Labels for angles Label each of the angles below. Choose from:

acute angle

right angle

obtuse angle

straight angle (b)

reflect angle

full rotation

Explain why each one is this type of angle.

r o e t s Bo r e p ok u S Type of angle

Explanation

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Angle

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© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

o c . che e r o t r s super 2. In each of the shapes below, put a  in any angles that are right angles. Put a

in any angles that are acute; and a in any angles that are obtuse. One has been done for you.

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

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CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

1. (a)


Assessment 2

Sub-strand: Geometric reasoning—GR – 1

NAME:

DATE: Measuring and constructing angles with a protractor

1. Using a protractor, measure each of the angles below. Write whether the angles are acute or obtuse. (a)

(b)

(c)

r o e t s Bo r e p ok u S °

°

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°

B

© R. I . C.Publ i cat i o ns ° C •f orr evi ew pur poses° o nl y• D °

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° E

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°

A

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CONTENT DESCRIPTION: Estimate, measure and compare angles using degrees. Construct angles using a protractor

2. Using a protractor, measure each of the angles in the shape below and write it in the shape. Write whether the angles are acute or obtuse.

3. In the space below, construct the following angles using a protractor.

o c . che e r o t r s super 35°

22°

Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

120°

102°

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Checklist

Sub-strand: Geometric reasoning—GR – 1

Constructs angles using a protractor

Measures angles using a protractor

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STUDENT NAME

Compares angles to a right angle

Estimate, measure and compare angles using degrees. Construct angles using a protractor (ACMMG112)

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Australian Curriculum Mathematics resource book: Measurement and Geometry (Year 5)

R.I.C. Publications® www.ricpublications.com.au


Answers

Sub-strand: Geometric reasoning—GR

2.

GR – 1 Page 123

Resource sheet – Measuring acute and obtuse angles

(a) 45° (acute) (d) 90° (right) Page 124

(b) 75° (acute) (e) 135° (obtuse)

  

Resource sheet – Constructing angles

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Resource sheet – Measuring reflex angles – 2

(a) 350° (d) 319°

(b) 280° (e) 270°

Page 127

(c) 195° (f ) 254°

Resource sheet – Which angle?

1. (a) 60° (d) 89°

Teac he r

(b) 170° (e) 5°

(c) 145°

Resource sheet – What’s your angle? – 1

Teacher check Page 129

Resource sheet – What’s your angle? – 2

Teacher check Page 130

Assessment 2 – Measuring and constructing angles with a protractor

1. (a) 23°, acute (b) 105°, obtuse (c) 154° obtuse 2. A: 71°, acute B: 119°, obtuse C: 100°, obtuse D: 127°, obtuse E: 124°, obtuse 3. Teacher check

Assessment 1 – Labels for angles

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

Angle

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Type of angle

Explanation

Full turn

360°, full circle

Reflex angle

Greater than 180°, less than 360°

Right angle

Exactly 90°

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Page 131

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Teacher check Page 126



(c) 15° (acute) (f ) 127° (obtuse)

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Obtuse angle

Greater than 90° but less than 180°

Acute angle

Less than 90°

Straight angle

Exactly 180°

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Australian Curriculum Mathematics - Measurement and Geometry: Year 5 - Ages 10-11