space project downloae

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Explore Math in ArchitectureMath and Hamsters?... The Culmin ating Task

Explore and Focus on A ngles and Protractors2-D Shapes... Explo ring Congr uency

Area an d Per imeter - Maximizing Space Ar ea an d Per imeter - Developing Rules3-D Figures-Exploring Shapes and Solids

3-D NetsThe Hamster House ~ Culminating Task

Culminating Task ~ Celebratin g

Including:

June 2000

Written b y:

Ontario Teachers

Math in ArchitectureGeometry, Measurement, and Patterning

An Integr ated Unit fo r Grade 5

Written using the Ontario Curriculum Unit Planner 2.0 (Sept 99) Open Printed on Jun 29, 2000 at 6:39:27 AM


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Math in ArchitectureGeometry, Measurement, and Patterning An Int egrated Uni t for Grad e 5

Ontario Teachers

Ontario Teachers

Ontario Ministry of Education, 2000

Grades 4-6 Math Implementation Resource Project

Ontario Ministry of Education, 2000

Grades 4-6 Math Implementation Resource Project

Original unit available for download at http://planner.media-x.com

Original unit available for download at http://planner.media-x.com

Based on a unit by:

This unit was written using the Curriculum Unit Planner, developed for use in the province of Ontario by the Ministry of Education and Training. The planner provides electronic access to all provincial Curriculum Expectations, an electronic

Teacher's Guide comprised of fourteen databases (including teaching/learning and assessment strategies, SpecialEducation guide, glossary, annotated bibliography) and a database of provincially licensed software for use in schools.

The Curriculum Unit P lanner offers educators a choice of three writing environments (Outliner, Lite, Open) Units writtenusing the planner can be shared and then edited electronically. This unit was printed from Version 2.0 (Sept 99), usingthe "Open" writing environment.

An Integrated Unit for Grade 5Written by:



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Math in Architecture Page 1Geometry , Measurement, and Patterning An Int egrated Un it fo r Grade 5

Task Context The students will construct the skills and knowledge necessary to propose a design for a Hamster House fora pet store chain. The chain of stores is looking to mass produce the Hamster House for sale. Hence thecompany requires a prototype, exact mathematical specifications of the house, and the materials and moneyrequired to produce a large number of the houses.

In completing the subtasks, students will work to explore the concepts of geometric shapes and solids, areaand perimeter, and mathematical patterns. Students will work collaboratively to focus on particular skills thatwill facilitate the completion of the culminating task. These skills and new understandings will be consolidatedthrough individual assignments and personal reflections on what they've come to understand. Once studentshave forged personal understandings of the required concepts, and focused on those skills applicable to thecompletion of the culminating task, the students will celebrate their achievements.

In completing the tasks, the students will be expected to create new understandings through the connectionof old knowledge structures and new information. Students will also be expected to apply these new skills

and understandings to complete the culminating task.

Task SummaryStudents will explore geometric shapes and solids, and their impact on the creation of the structures in whichwe live. Students will explore geometic shapes and solids in the everyday world. Students will then focuson key concepts related to geometic shapes and solids, such as naming, constructing and sketching. Thestudents will be encouraged to apply this new knowledge back to the world outside the classroom, in theconstruction of a Hamster House.

Developing an understanding of measurement is essential to successful completion of the culminating task.

Students will manipulate tools of measurement (protractor, ruler) and develop the skills and knowledgenecessary to use these tools in the completion of their task. They will measure angles, and discover thatcongruent shapes and angles are essential to creating stable structures. They will also discover that precisemeasurements are important to reconstructing models accurately. Students will explore the concepts of areaand perimeter, and focus on how area and perimeter will play a role in the construction and mass productionof structures.

Students will explore the mathematical patterns inherent in both geometry and mass production. Students willdetermine the patterns and relationships between length, width and area, and edges of a shape and thefaces of its corresponding prism or pyramid. Students will extend patterns, to project the total cost of massproducing the model for retail.

The students will produce a model Hamster House, and a multi-page mathematical specifications report,which will contain precise mathematical details concerning the geometric make up, cost projections, andmeasurements. The report must contain adequate enough detail for a classmate to reconstruct the modelusing only the report.

Culminating Task Assessment The students will prepare a proposal for the construction of a hamster house for the Pets 'R Us company. The proposal will include a model of the hamster house and a Mathematical Specifications Report. Eachproposal will be displayed and peer assessed. After the students have viewed the proposals they will begiven another student's Mathematical Specifications Report and challenged to recreate that student's hamster

Unit Overview



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Math in ArchitectureGeometry , Measurement, and Patterning An Int egrated Uni t for Grad e 5

Subtask List Page 1List o f Subtasks

Explore Math in Ar chitectureStudents will look at pictures of various buildings, list all the 2-D and 3-D shapes that they recognize,and organize their list into different categories. Students will discuss the concepts of area andperimeter with peers as well as search for patterns in architecture. This initial assessment can beused as a reference to gauge the students' knowledge of geometry, measurement, and patterning.

1

Math and Hamsters?... The Culmin ating TaskStudents will be introduced to the culminating task which is to design a hamster house for the Pets 'RUs company.

2

Explore and Focus on A ngles and ProtractorsStudents explore angles and protractors. Protractors are compared with rulers as standard measuringinstruments. They use a protractor to measure and construct 'pipecleaner' angles. Students drawangles of given degree measurements.

3

2-D Shapes... Explo ring Congruency

Students will construct their own knowledge of congruency and make congruent 2-D shapes usingpaper folding, Miras, and straws.

4

Area and Per imeter - Maximizing SpaceStudents will use their understanding of area and perimeter to explore whether there is a relationshipbetween area and perimeter. Through playing the Perimeter Challenge Game the students will discoverthat different shapes with the same area do not necessarily have the same perimeter. When the areais constant, changing the shape will influence the perimeter. Students will learn that as a shape getscloser to being square the space is maximized (Greatest area - smallest perimeter). This being said,measuring the area and perimeter of different shapes is essential to the design and cost of thehamster house.

5

Area an d Per imeter - Developi ng RulesStudents will explore the relationship between the length of the sides of a given shape and the area.

They will track data using a table, and identify and extend patterns in the data. Students will use theirknowledge of area and the patterns in the table to develop rules for finding the area of rectangles.

6

3-D Figures-Exploring Shapes and Solids The students will explore and sketch diagrams of different geometric solids. Through the use of at-table and their sketches, the students will explore the relationship between the number of sides in atwo dimensional shape and the number of faces in the corresponding 3-D shape.

7

3-D NetsStudents are introduced to the concept of 3-D nets. The students will use their knowledge of 3-Dfigures to explore and create the nets of familiar geometric solids. The students will apply theirknowledge of nets and geometric solids to solve related problems.

8

The Hamster House ~ Culminating Task The students will prepare a proposal for the construction of a hamster house for the Pets 'R Uscompany. The proposal will include a model of the hamster house and a Mathematical SpecificationsReport. Each proposal will be displayed and peer assessed. After the students have viewed theproposals they will be given another student's Mathematical Specifications Report and challenged torecreate that student's hamster house.

9

Culmin ating Task ~ CelebratingAs part of the culminating task, students will reflect on the process, and display their proposals forothers to see. Each student will try to reproduce another's structure by looking solely at theMathematical Specifications Report.

10



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Math in Architecture Subtask 1Explore Math in Arch itecture

Geometry , Measurement, and Patterning An Int egrated Un it fo r Grade 5 mins100

Expectations5m65 A identify, describe, compare, and classify

geometric figures;5m82 use mathematical language to describe

geometric ideas (e.g., quadrilateral, scalenetriangle);

5m83 A recognize and explain the occurrence andapplication of geometric properties and principlesin the everyday world;

5m84 discuss geometric concepts with peers anduse mathematical language to explain theirunderstanding of the concepts;

5m39 A solve problems related to the calculation of theperimeter and the area of regular and irregulartwo-dimensional shapes;

5m92 A identify, extend, and create patterns in a varietyof contexts;

5m99 A identify and extend patterns to solve problemsin meaningful contexts (e.g., leaves on trees,spirals on pineapples);

Teaching / LearningLink Math to the Real WorldStep 1Introduce the unit by explaining that 'math' is all around us.Engage the class in discussion: Often we 'see' matheveryday on our way to school. What type of math can be'seen' on the way to school? (Answers will vary: shapes,patterns, numbers etc.) Suggest that 'math', in particulargeometry, can be seen in the buildings around us.Optional Links to Texts: Read the article on page 277from the Quest 2000 Grade 5 Student Text entitled Shapes

for Buildings . Refer to pages 173 to 181 from theInteractions Grade 5 Student Text entitled InvestigatingStructures .

DescriptionStudents will look at pictures of various buildings, list all the 2-D and 3-D shapes that they recognize, andorganize their list into different categories. Students will discuss the concepts of area and perimeter with

peers as well as search for patterns in architecture. This initial assessment can be used as a referenceto gauge the students' knowledge of geometry, measurement, and patterning.

GroupingsStudents Working In Small Groups

Teaching / Learning StrategiesBrainstormingClassifying

AssessmentPart A ~ Initial Geometry

Asses sm ent

Observe students working in groups.Expect words such as: circle, square,rectangle, triangle, pyramid, cube, cylinderetc. A more advanced geometricknowledge would be indicated by wordssuch as: pentagon, octagon, hexagon,rhombus, trapezoid, triangular prism,rectangular prism, square-based pyramidetc. Although students may have thisadvanced geometric knowledgesomewhere within them, our goal is to seethe ease and precision with which theyuse mathematical language to describegeometric concepts. The students' levelof geometric knowledge can also be seenin the ways in which their ideas arecategorized. Use of words such asfaces, edges, vertices etc. may indicate amore advanced knowledge.

Have students reflect on the following intheir Learning Logs.1. What is geometry?2. Describe a time in earlier grades whenyou studied geometry.3. How are prisms and pyramids alike?How are they different?Use these particular Learning Logreflections to give you a generalunderstanding of the class' level of 'geometry knowledge'. As well, individual



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Part A ~ Initial Geometry Assessment

Explore Geometry in the Real WorldStep 2Divide the students into small groups and distribute theblackline masters (Reference BLM 1.1a to 1.1c), Geometryin Architecture (one set of three pages for each group) .Have students look at the pictures of the various buildings.Using the Brainstorming Sheet (Activity BLM 1.1) , groupswill record all the 2-D and 3-D geometric shapes theyrecognize.

Step 3Distribute the Categorizing Sheet (Activity BLM 1.2) andhave groups organize their list of geometric shapes intodifferent categories. Students should explain the

characteristics for each category.Step 4Have students debate the following question in theirgroups: In general are buildings shaped as pyramids orprisms? Use the blackline master, Pyramids or Prisms?(Activity BLM 1.3), to record the debate. Groups maypresent their debate to the class as a whole or simplydebate amongst themselves.

Step 5Have students reflect on the following questions in theirLearning Logs.

1. What is geometry?2. Describe a time in earlier grades when you studiedgeometry. What did you study? How did you learn about it?3. How are prisms and pyramids alike? How are theydifferent?

Part B ~ Initi al Measurement Assessment

Step 6 This is an explore activity. Using the Geometry in Architecture pictures (Reference BLM 1.1.a to 1.1.c) in their

small groups, have students come to a consensus on thefollowing:Which buildings do you think have the largest area?Which buildings do you think have the largest perimeter?Have groups use the blackline master, Area and Perimeter (Activity BLM 1.4), to record their discussion. In discussingthese questions, groups will have to come to a basicagreement on the definition of area and perimeter. Forexample, some groups may define 'area' as the amount of floor space in the entire building; other groups may define

students may stand out as having eitheran advanced knowlege of geometry or alack of 'geometry knowledge'.

Part B ~ Initial Measurement Asses sm entWhile each group is discussing theirdefinition of area and perimeter andarriving at a ranking, the teachercirculates and listens for the following: anunderstanding of the concepts of areaand perimeter and an ability to justify theirselection of largest to smallest.Using the group responses on theblackline master, Area and Perimeter , lookfor a general consensus.

Part C ~ Initial Patterning Asses sm entHave students make a list of all thepatterns they notice in building #3, 11, 13and 18.Use the students' Learning Log list to lookfor an understanding that patterns repeatand that patterns can be found inarchitecture. Look for a students'understanding that patterns can beshapes or numbers.

Have the students answer the followingquestion to determine their ability to extendpatterns.1. The front of building #2, has 8 pairs of shutters. If another floor was addedbetween the first and second floors, howmany pairs of shutters would be seen atthe front of the building? What if 2 floorswere added? What if 3 floors wereadded? Extend the pattern in order todetermine the number of pairs of shuttersneeded if 8 floors were added.Look for a student's ability to extendpatterns. As each floor is added, anadditional 4 pairs of shutters is needed. If 8 floors were added the total number of pairs of shutters seen would be 40.

Asses sment Str ategies



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'area' as the amount of floor space on the ground level of the building. Likewise, different definitions of perimeter mayalso arise. Arriving at a group definition will help studentsconstruct their own understanding of area and perimeter as

well as helping the teacher determine the general level of measurement knowledge of the class as a whole.

Part C ~ Initi al Patterning Assessment

Step 7Using the Geometry in Architecture pictures in their smallgroups, have students discuss the patterns that they noticein architecture.

Step 8Have students respond, individually, to the following in theirLearning Logs: Make a list of all the patterns you notice inbuildings #3, 11, 13 and 18.

Have the students answer the following question todetermine their ability to extend patterns.The front of building #2, has 8 pairs of shutters. If another floor was added between the first and secondfloors, how many pairs of shutters would be seen at thefront of the building? What if 2 floors were added? Whatif 3 floors were added? Extend the pattern in order todetermine the number of pairs of shutters needed if 8floors were added.

Extension Activi tyHave students search the internet usingwww.yahooligans.com (child's search engine) for variousbuildings. Record geometric shapes seen in the buildings.

ObservationLearning Log

Assessment Recording Devices

Anecdotal Record

Adaptations

ResourcesGeometry In Architecture 1_1.PDF

Brainstorming 1_2.PDF

Categorizin g Sheet 1_3.PDF

Pyramids or Prisms? 1_4.PDF

Area and Per imeter 1_5.PDF

Interactions Grade Five Text Ginn Publishing Canada Inc.



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Quest 2000 Grade 5 Text Addison-Wesley Publishers Limited

Yahooligans Search Engine

Corel Print Office 2000 - Clipart

Notes to TeacherUsing the results of the assessment, determine any necessary pre-teaching and/or modifications toexpectations for the class as a whole and/or particular individuals. For example: Do the students have anessential understanding of the terms prism and pyramid or will this concept require particular emphasisin later subtasks?

One copy of the five pages of Geometry in Architeture (Blackline Masters) can be given to each smallgroup during the subtask.

Additional patterning questions can be added to assess the students' knowledge.

Teacher Reflections



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Math in Architecture Subtask 2Math and Hamsters? ... The Culminatin g Task


Expectations5s86 formulate questions about and identify needs

and problems related to structures andmechanisms in the outdoor environment, andexplore possible answers and solutions (e.g.,construct a bridge that must support a given loadacross a given distance; determine whichsurface of a cantilever bridge or beam is undertension and which is under compression);

5s98 identify modifications intended to improve theperformance, aesthetic appeal, and impact on the

environment of a product they designed;5m83 recognize and explain the occurrence andapplication of geometric properties and principlesin the everyday world;

5m84 discuss geometric concepts with peers anduse mathematical language to explain theirunderstanding of the concepts;

5m70 A use mathematical language effectively todescribe geometric concepts, reasoning, andinvestigations, and coordinate systems.

Teaching / LearningMath is Truly All Around UsStep 1Remind the students of the previous discussion whichrevolved around 'seeing' math on their way to school andhow math is all around us. Explain to the students that youhave a rather open-ended question for them to answer inwhich all ideas are accepted. They are to free up theirminds and think divergently. Suggest that by now, in gradefive, they have learned so much about math that almostanything can be seen as a math problem. (Optional Activity:Read the book Math Curse to the class in which the maincharacter can't stop looking at everything in the world as if it were a mathematical puzzle.)

Step 2Divide the students up into small groups, distribute theblackline master (Activity BLM 2.1), Hamster Math , facedown, and have them begin the task immediately withoutfurther discussion. " On your mark, get set, go! " You cancount on the suspenseful energy of the group and thechallenge inherent in the task to bring about some divergentthinking about math. (see the blackline master for an

DescriptionStudents will be introduced to the culminating task which is to design a hamster house for the Pets 'R Uscompany.

GroupingsStudents Working In Small GroupsStudents Working As A Whole ClassStudents Working Individually

Teaching / Learning StrategiesBrainstormingModel MakingOpen-ended Questions

AssessmentUse the blackline master, A New House(Activity BLM 2.2) to look for a generalunderstandiing of the culminating task(building a hamster house). Also look fora general understanding that 'doing math'will be inherent in the task.

Use the blackline master, A New House(Activity BLM 2.2) to assess the students'ability to use mathematical languageeffectively. Look for math vocabulary

particularly in the area of geometry. Lookfor the knowledge that measurement isessential in building the hamster house.Use the following rating scale, if desired,to assess the students:

Level 4 The student has used manygeometry terms correctly. The studenthas indicated at least five places in whichmeasurement would be involved.

Level 3 The student has used somegeometry terms correctly. The student

has indicated three or four places inwhich measurement would be involved.

Level 2 The student has used a fewgeometry terms correctly. The studenthas indicated two or three places inwhich measurement would be involved.

Level 1 The student has used very fewgeometry terms correctly. The student



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explanation of the open-ended question)

Step 3Have groups briefly present some of their most creative

ideas. You may want to display this group work for laterreference.

Focus in on the Culminating TaskStep 4Undoubtably the students will come up with manyinteresting ideas about 'hamster math'. Some ideas will becompletely outrageous while others will prove to be usefuland relevant. To whom would the answers to thesequestions be most useful? For example, pet owners,hamster breeders, biologists, pet store owners andemployees, pet product manufacturers etc.

Step 5Explain to the students that they have been given aparticular 'hamster math' problem to focus on. A petcompany, Pets 'R Us, is looking for new ideas for hamster houses. Often companies will solicit the help fromeveryday customers to design products. Explain thatstudents are asked to design a hamster house which will fitinto a hamster cage (If you have a small plastic hamsterhouse and a hamster cage available, they would be usefulin demonstrating the difference between a 'house' and a'cage'). See the picture of a hamster cage and house onthe cover of the unit. You may also ask students or petstore owners to bring in sample cages and houses.

Step 6Refer back to the previous subtask in which geometricshapes were found in buildings. Ask the students whatpossible geometric shapes could be used to create ahamster house.

Step 7Divide the students up into small groups again and havethem focus on the problem of designing a hamster house.Provide each group with one copy of the blackline master,

A New House (Activity BLM 2.2.), to guide group

discussion.Step 8Distribute the blackline master, A New House (Activity BLM2.2.), and have students work individually to fill in their owncopy which will become the pre-draft of their culminatingproject. Have the students keep this paper in a prominantlocation (eg. at the front of their duotang/notebook) forreference throughout the unit. During future subtasks, anidea may come to mind and can be recorded on this sheet.

has indicated one place in whichmeasurement would be involved.

Asses sment Str ategies

Observation

Assessment Recording DevicesRating Scale

Adaptations



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ResourcesHamster Math 2_1.PDF

Hamster House 2_2.PDF

Math Curse J on Scieszka and Lane Smith

Hamster House

Hamster Cage

Notes to TeacherIt is essential to make it explicit to the students that professionals design structures for a specific functionor audience. They will have to think about what their structure is being used for while planning anddesigning its construction.

A suggestion for management might be to initiate a 'professional portfolio' for the duration of the projectusing a file folder, binder, or duotang. A 'skills sheet' can be glued to the inside cover, and used for thestudent to jot things he or she learns throughout the subtasks that might help him / her in competing theproject. Rather than using a traditional mathematics notebook, the students could be encouraged to keep allof their assignments in this 'portfolio'. The student could use this as reference when planning his / herstructure.

Teacher Reflections



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Math in Architecture Subtask 3Explore and Focus on An gles and


Expectations5m78 A measure and construct angles using a

protractor;5m82 use mathematical language to describe

geometric ideas (e.g., quadrilateral, scalenetriangle);

5m83 recognize and explain the occurrence andapplication of geometric properties and principlesin the everyday world;

Teaching / LearningReal World LinkStep 1Remind students of the pictures of the various buildings andarchitectural structures that were presented earlier. Showthe pictures again and have the students point out differentangles (a point where two straight edges or lines meet) to apartner. Ask students to think, "Which buildings have themost interesting angles?" Then have students find apartner and discuss their ideas. Next have pairs sharetheir thinking with the class.

Explore AnglesStep 2Distribute a pipecleaner to each student. Demonstrate howto make an angle with the pipecleaner (by folding thepipecleaner in half; then spreading the two halves apart).Ask students to estimate the size of the following angles bymaking the angle with a pipecleaner:a) Show me the angle of the blades when scissors opentheir widest.b) Show me the angle of your arm at your elbow whenyour hand is touching your shoulder.c) Show me the angle of a hockey stick blade at its shaft.d) Show me the angle of your upper and lower leg at your knee when you're running fast.e) Show me the angle of your leg at your knee whenyou're walking.f) Show me the angle of the corner of the door at itsframe.g) Show me the angle of the hood of your car to thewindshield when it's open to change the oil.Have students suggest other angles for the class to make.

DescriptionStudents explore angles and protractors. Protractors are compared with rulers as standard measuringinstruments. They use a protractor to measure and construct 'pipecleaner' angles. Students draw

angles of given degree measurements.

GroupingsStudents Working In PairsStudents Working In Small GroupsStudents Working As A Whole Class

Teaching / Learning Strategies Think / Pair / ShareBrainstormingPeer Practice

AssessmentUse the Observation Checklist blacklinemaster during pair activities.

Students answer the following in theirLearning Log:

1. Explain the steps one must take todraw an angle of 125 o.Look for proper sequencing of stepsbeginning with drawing a baseline and

lining up the 'centre-T' of the protractorwith the end of the line.

2. Why do you think most protractorsare transparent?Look for an understanding that one needsto see through the protractor in order toline up the protractor properly and tocorrectly 'zero' the protractor with theangle. Rather than placing the actual edgeof the protractor on one side of the angle,you must measure from the zero degreeline which is marked on the protractor.

Asses sment Str ategiesObservationLearning Log

Assessment Recording DevicesChecklist

AdaptationsWritten using the Ontario Curriculum Unit Planner 2.0 (Sept 99) Open Printed on Jun 29, 2000 at 6:40:06 AM


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number on the top or the bottom) when using aprotractor? (This depends on which scale (the top or bottom) has been used to 'zero' the protractor with theside of the angle.)

Step 6Using their pipecleaners, have students measure various'pipecleaner' angles with the protractor. Have studentswork in pairs: each makes a series of 'pipecleaner' anglesfor the other to measure. Have partners check eachothers' work.

Step 7Once students are familiar with measuring angles, ask themto use their protractor to construct 'pipecleaner' angles of various degrees. e.g. Make a 45 o angle. Have studentswork in pairs: each gives angle degree measurementsfrom 0

o

to 180o

for the other to construct. Have partnerscheck each others' work and provide feedback to eachother. Mention to students that it will be difficult to obtain aprecise measurement of the pipecleaner angle since thepipecleaner is thicker than the lines on the protractor.

Step 8On the board or overhead, demonstrate how to draw anangle with a particular measurement. e.g. Draw a 60 oangle. Here is one set of instructions.a) Remind students that they must first use a ruler to drawa baseline from which to measure the angle.b) Next, place the 'centre-T' of the protractor on the end of the line so that the zero line on the protractor is lined up('zeroed') with the line.c) From the zero degree line follow the scale up until youreach 60 o . Place a dot on your paper at the 60 o line.d) Using a ruler, connect the end of the line (where the'centre-T' of the protractor lies) with the dot. You havenow drawn a 60 o angle.

Explain the proper method for indicating angles (a smallcurve inside the angle with the written angle degreemeasurement). Introduce the proper method for labelingangles (eg. pABC). Have students work in pairs: each

gives angle degree measurements from 0o

to 180o

for theother to draw. Have partners check each others' work andprovide feedback orally.

Step 9.... Mental Math.... Estimate AnglesProtractors should not be used during the estimation part(a) of the activity. Use the following instructions:a) Draw what you think an angle of 70 o looks like.b) Now measure your angle with a protractor.c) How close were you?



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d) Why were you higher or lower?e) How could you make your drawing more accuratewithout using a protractor?

Demonstrate an estimating strategy to the students whichwill help them draw a 45 o angle without the use of aprotractor: begin by drawing a 90 o angle and show thestudents that half of the 90 o angle would be a 45 o angle.Using this strategy, have students draw the followinganlges without a protractor: 45 o, 90 o, 135 o etc.

Have students work with their partner giving each otherangle degree measurements to draw without the use of aprotractor. Have partners measure each other's estimatedangles with a protractor. Have the students answer theabove questions orally with their partner throughout theactivity.

Step 10Have students answer the assessment questions in theirLearning Log.

Extra PracticeProvide the blackline master, Protractors: Extra Practice(Activity BLM 3.2).

ResourcesProtractor Checklist 3_3.PDF

Protractors 3_4.PDF

Protractors and Rulers 3_1.cwk

Protractors Extra Practice 3_2.cwk

pipecleaner 1

protractor 1 per person

ruler 1 per person



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Notes to TeacherStudent will have experience measuring angles with proctractors from grade four; however, constructingangles of given degrees will be a new skill.

Use the Protractors: Extra Practice blackline master to reinforce student learning.

Teacher Reflections



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Math in Architecture Subtask 42-D Shapes... Exploring Congruency


Expectations5m69 identify congruent and similar figures using

transformations;5m70 use mathematical language effectively to

describe geometric concepts, reasoning, andinvestigations, and coordinate systems.

5m80 A demonstrate congruence of figures usingpaper folding, reflections in a transparent mirror(Mira), and various computer applications;

5m81 use a computer application to explore andextend geometric concepts;

5m77 A demonstrate an understanding of congruentfigures;5m79 construct triangles given specific measures of

angles and sides, using a variety of tools;

Teaching / Learning

Real World Link and Link to Culminating TaskStep 1Refer back to the pictures of the various buildings andarchitectural structures that were presented earlier. Leadthe discussion with comments such as: What do younotice about the shapes used to make a given structure?(Students may respond that there are many differentshapes both 2-D shapes and 3-D objects; many of theshapes are exactly the same size and shape.)

Step 2As a whole class, have students focus on the 2-D shapesin the buildings. Students will generally focus on the mostobvious 2-D shapes: the windows, doors, or other surfacepatterns. Remind them that the faces of 3-D objects arealso 2-D shapes. Brainstorm reasons why structureswould be built with shapes of exactly the same size andshape. (possible answers: it looks better aesthetically,the structure is stronger or more stable). Have studentsthink back to their preliminary design of their hamster house.Will you be using any 2-D shapes of exactly the samesize and shape?

Focus on 2-D ShapesStep 3Divide students up into small groups and distribute thetwo-page blackline master, Same or Different? (Activity

DescriptionStudents will construct their own knowledge of congruency and make congruent 2-D shapes using paperfolding, Miras, and straws.

GroupingsStudents Working As A Whole ClassStudents Working In PairsStudents Working Individually

Teaching / Learning StrategiesComputer Assisted LearningMini-lessonPeer Practice

Technology

AssessmentUse the students' Learning Log responseto the question What does it mean if twofigures are congruent? to assess theirunderstanding of congruent figures. Lookfor correct use of mathematical language:sides, angles, shape, size etc. Look foran understanding that congruent figuresare exactly the same shape and size; theposition or orientation of the shape on thepage may be different.

Use the students' Mira sheet and 'straw'triangles to assess their ability todemonstrate congruence). The followingrating scale can be used:

Level 4 The student has precisely drawncongruent shapes with a Mira. Thestudent has created a 'straw' triangleprecisely congruent to that of theirpartner's.

Level 3 The student has drawn

congruent shapes with a Mira. Thestudent has created a 'straw' triangleessentially congruent to that of theirpartner's.

Level 2 With assistance, the student hasdrawn congruent shapes with a Mira andcreated a 'straw' triangle congruent to thatof their partner's.



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BLM 4.1), to each group along with a protractor and a ruler.Have students answer the questions on the blacklinemaster.

Step 4As a whole class, discuss the answers to the questions.Develop the students' understanding of 2-D shapes bysuggesting that the figures can be compared using thefollowing criteria: shape, size and position. For purposesof discussion, the 'shape' of a figure refers to the numberof sides and the degree measures of the angles. Forexample, two triangles may not be the same 'shape': eventhough they both have three sides and are, therefore,triangles; the degree measures of the angles may bedifferent. The 'size' of a figure refers to the lengths of thesides. The 'position' of a figure refers to it's orientation onthe page.

Step 5Distribute the blackline master, Congruency (Activity BLM4.2), to each student (the blackline master could also bemade into a transparency and used on an overheadprojector). Through discussion, have students come to anunderstanding of congruency and then answer thequestion, What does it mean if two figures are congruent?in their Learning Logs.

How do you make congruent figures?Step 6Explain the following to the students: Since congruentfigures are used in buildings for both aesthetic andstructural reasons, how does one go about makingcongruent figures? Architects may use rulers,protractors and computer programs to ensure theaccuracy of their blueprints. In order to build your hamster house you will need to know how to createcongruent figures. Explore the following methods of makingcongruent figures.

Step 7 The Paper Folding MethodExplain to the students that an easy way to make congruent

shapes is to simply fold a piece of paper and cut out twoshapes at once. Distribute a small piece (~ 4cm by 5cm) of coloured construction paper (all the same colour) to eachstudent. Have students fold the paper in half and cut out apair of congruent triangles. In small groups, have studentsplace their triangles on a desk and mix them up. In turnshave students select congruent pairs. (As an extensionactivity in visual arts, students can create a design usingcongruent shapes)

Level 1 With considerable assistance,the student has drawn congruent shapeswith a Mira. The student could not createa 'straw' triangle congruent to that of their

partner's. Asses sment Str ategies

Learning LogPerformance Task

Assessment Recording DevicesAnecdotal RecordRating Scale

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Step 8 The Mira MethodHave students draw a triangle with a ruler. Demonstratehow to use a Mira to draw a congruent triangle elsewhere

on the page. Have students explore with other shapes.Have students hand in this page for assessment.

Step 9Have students make a triangle using straws andpipecleaners. The straws may be cut to any length. Thepipecleaners will be used to connect the straws. Cut thepipecleaner into smaller pieces (approximately 5 cm). Tomake the 5 cm pipecleaner into a connector, fold it in half (this will allow for a tighter fit in the straw). J oin twostraws together by slipping them over either end of thepipecleaner connector. The straws are now joined andcan be moved to any desired angle. J oin the third straw tothe other two straws in the same manner. Each studentwill now have a 'straw' triangle.

Step 10Have students exchange their 'straw' triangles with apartner and challenge them to create another 'straw'triangle; one that is congruent to their partner's. Providestudents with rulers and protractors. Have students labeltheir pairs of triangles and hand them in for assessment.

Extension ActivitiesHave students use a computer application (CorelDRAW 5,AppleWorks 5.0, Geometer's Sketchpad) to createcongruent figures by drawing a 2-D shape and using thecopy/paste command to create the congruent figure. Someprograms will also allow the figure to be flipped or rotatedwhich would also demonstrate congruency.

Links to Texts & Extra PracticeInteractions Grade 5 Student Text: Unit 3 ExaminingGeometric Relationships (especially pages 38 and 39)

ResourcesSame or Different 4_1.PDF



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Congruency 4_2.PDF

Geometry In Architecture 1_1.PDF

CorelDRAW 5 Academic

Math Trek 4, 5, 6

ClarisWorks 5.0 (Englis h)

Interactions Grade Five Text Ginn Publishing Canada Inc.

straws (small di ameter)

pipecleaners

protractors

rulers

Miras

Notes to Teacher Teachers are encouraged to use Math Trek 4, 5, 6 in order to forge a variety of connections betweenconcepts and prior knowledge. The activity TRIANGLE SEARCH lends itself to consolidating knowledgerelated to congruency and similarity of figures.

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Math in Architecture Subtask 5 Area and Perimeter - Maximizing Space


Expectations5m39 A solve problems related to the calculation of the

perimeter and the area of regular and irregulartwo-dimensional shapes;

5m55 A estimate and calculate the perimeter and areaof rectangles and squares;

5m56 explain the rules used in calculating theperimeter and area of rectangles and squares;5m58 develop methods of using grid paper to trackand measure the perimeter and area of polygonsand irregular two-dimensional shapes;

5m38 A identify relationships between and amongmeasurement concepts (linear, temporal,monetary);

Teaching / LearningReal World LinkDiscuss the implications of perimeter and area on the costof the hamster house. The cost to cover the faces of alarge house is more expensive than the cost of coveringsmaller ones. Ask the students what the best scenariowould be for them regarding cost and design. With someguidance, the students should arrive at the conclusion thatthey would want the largest area with the smallestperimeter. This would limit costs and maximize area. Youmight have to lead the discussion with guiding questionssuch as -- if you were a builder, what impact would theperimeter of a side of the house have on your costs?What would the person who is living in the house want?How can we get the best of both worlds?

Mini Lesson on the Relationship Between Perimeter and a Fixed Area1. Review the concepts of area and perimeter with thestudents -- area is the number of square units to cover asurface; perimeter is the distance around a shape.

2. Place 12 square transparent polygons in a 1x12rectangle on the overhead. Ask the students how manysquare units are being covered by the polygons -- of course there will be 12 square units of space covered.Point out to the students that this is the area. Ask students

DescriptionStudents will use their understanding of area and perimeter to explore whether there is a relationshipbetween area and perimeter. Through playing the Perimeter Challenge Game the students will discover

that different shapes with the same area do not necessarily have the same perimeter. When the area isconstant, changing the shape will influence the perimeter. Students will learn that as a shape gets closerto being square the space is maximized (Greatest area - smallest perimeter). This being said, measuringthe area and perimeter of different shapes is essential to the design and cost of the hamster house.

GroupingsStudents Working As A Whole ClassStudents Working In PairsStudents Working Individually

Teaching / Learning Strategies

InquiryMini-lesson

AssessmentObservation / Performance TaskWatch for students who aren't makingpolygons. Make sure that all shapes areclosed and edges meet. Also take noteof students who are using the conceptsexplored in the mini lesson. Studentsshould understand that they want to maketheir shapes as close to a square as they

can. This way they can take up the sameamount of space with the smallestperimeter.

Learning LogStudents answer the following questionsin their learning log using words, numbersand pictures:

1. What shapes were most useful foryou?

2. If you were building a room with anarea of 24 cm 2, what length and widthwould you want it to be? Explain.Look for students who sketched theshapes and marked their dimensions.Watch for students who indicated thatthey would want a shape close to beingsquare, but disqualified the 1x24 shape asimpractical. Also take note of studentswho accurately calculated the areaperimeter of the proposals.



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to determine the perimeter of the 1x12 rectangle. J ot thesedown on the transparency with an erasable marker.

3. Ask the students if there is a way to arrange the

squares in the form of a regular polygon, as opposed to a1x12 rectangle. The possible ways to arrange the 12squares are 2x6, 3x4, rectangles. Ask the students if theamount of space covered by the squares has changed. Ithasn't -- the area covered is still 12 square units. Ask thestudents to determine the perimeter -- what happened tothe perimeter when we changed the shape? As the shapechanged the area remained the same but the perimeterbecame smaller (1x12 rectangle - A=12, P=26; 2x6rectangle - A=12, P=16; 3x4 rectangle - A=12, P=14). Youwill notice that, given a fixed arrea, as the shape getscloser to a perfect square the perimeter gets smaller. Asquare shape maximizes area with the smallest perimeter.

4. Discuss with students how this new discovery willdetermine the layout of our houses -- rather than havinglong, thin rectangular shapes, we'll want shapes closer tobeing a square. This will allow for space to be maximized,while using the least amount of straw perimeter, hencereducing costs. Given an area, a long and thin rectangularshape has the same area as a square, only the perimeter ismuch larger. This will cost more to make (larger amount of straw needed) but will have the same amount of livingspace.

5. Place students in groups of 2. Hand out the MaximizingSpace: Area / Perimeter game sheet (Activity BLM 5.1) and2 dice to each group of 2 students.

Exploring Spatial Relationships Game -- Perimeter Challenge1. Ask each player to choose a different colour of pen withwhich to play the game.

2. Each player rolls a die -- the player with the highest rollgoes first.

3. In turn, the players will roll and add the two numbers on

the dice.4. On grid paper, outline a polygon having an area equal tothe dice sum. Remind the students that in order for theshape to be a polygon, all edges must be touching, and theshape must be closed. Polygons must not overlap on thegraph paper.

5. The game ends when both players have rolled areasthat cannot fit onto the graph paper.

3. What did you learn from this gamethat could help you with the design of you hamster house?

Look for students to indicate that a morepractical design would be one thatmaximizes the useable space. Shapesclose to being square are a more efficientuse of space than long, thin rectangularshapes.

Reinforcement -- Question and Answer Wo rksheet1. Have students complete the area andperimeter reinforcement worksheet ashomework.

Asses sment Str ategiesLearning LogPerformance TaskQuestions And Answers (oral)

Assessment Recording DevicesAnecdotal Record

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6. The player with the lowest total perimeter wins.

Group Share

7. Remind students of the class' discovery in the minilesson regarding area and perimeter. Ask guidingquestions -- What types of shapes will you want to havethe smallest perimeter? What areas are most difficult tomake shapes out of?

8. After the game have the students communicate theirknowledge by answering the learning log questions in theirmath log. Remind the students that the more ways theycan show they understand the better. Encourage the useof numbers, pictures, and words to demonstrate theirunderstanding.

ReinforcementStep 9:If desired, the students can complete pp. 212-215 in theQuest 2000 student text.

ResourcesMaximizing Space: Perim eter and Area 5_1.PDF

Math Trek 4, 5, 6

Quest 2000 Stud ent Text pp. 212-215

Graph paper / grid paper 1

Dice 3

Overhead proj ector 1

Transparent Polygon s (for overheadprojector)

1



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Notes to Teacher The circle is the shape that maximizes space, while limiting perimeter (circumference - distance around).However, for the purposes of this task, only rectangles will be explored. Students will discover that thecloser a rectangle becomes to a perfect square (four identical sides) the smaller the perimeter gets. It isimportant for students to engage the concepts of area and perimeter as they impact structures, not merelythe methodic solving of algorithms (formulae). This task will help to guide the students towards anunderstanding of spatial relationships related to area and perimeter.

It is possible to play the game on half sheets of grid paper, however, the game often lasts for a longerperiod of time, and can become somewhat tedious. Adding dice, or reducing the area can help alleviate thisproblem.

Students will often end up with an 'odd' number for the area (i.e. 5). The students can make a 1 X 5rectangle, or a 2 X 2 square, with a projecting square. The idea remains the same nevertheless--the closerto a perfect square, even if it does have one projecting piece, the better the use of space.

Students are encouraged to explore the concepts of area and perimeter using a variety of media, includingcomputer applications and other technology. The Math Trek 4,5,6 activity, Area and Per imeter Explorer is ideal for exploring area and perimeter using computer technology, and reinforcing skills learned in thesubtask.

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Math in Architecture Subtask 6 Area an d Perimeter - Developing Rules


Expectations5m54 A develop rules for calculating the perimeter and

area of rectangles, generalize rules, and developformulas;

5m56 A explain the rules used in calculating theperimeter and area of rectangles and squares;


5m39 solve problems related to the calculation of theperimeter and the area of regular and irregular

two-dimensional shapes;5m94 create tables to display patterns;5m93 analyse and discuss patterning rules;5m92 A identify, extend, and create patterns in a variety

of contexts;5m91 recognize and discuss the mathematical

relationships between and among patterns;5m100 use a calculator and computer applications to

explore patterns;5m101 pose and solve problems by applying a

patterning strategy (e.g., what effect willdoubling the first number have on the pattern?);

5m103 discuss and defend the choice of a patternrule;

5m104 given a rule expressed in informal mathematicallanguage, extend a pattern;

5m105 use patterns in a table of values to makepredictions;

Teaching / LearningRelating t o the Culminating TaskRemind students that they're producing models that willmass produced for sale to the general public. Ask studentswhat some of the concerns might be when producing alarge number of products -- try to guide the issues of howmuch material is needed and how much it will cost to makea large number of products. Tell the students that Pets 'R'Us will want to have all of this information if they are todecide on a prototype to use.

Draw a picture of a hamster house on the chalkboard, andask students what they'd have to know about the house if they were to make a large number of them. Refer back tothe blackline master A New House from subtask 2. Askstudents what you'd have to know in order to produce a

DescriptionStudents will explore the relationship between the length of the sides of a given shape and the area.

They will track data using a table, and identify and extend patterns in the data. Students will use their

knowledge of area and the patterns in the table to develop rules for finding the area of rectangles.

GroupingsStudents Working IndividuallyStudents Working As A Whole Class

Teaching / Learning StrategiesMini-lessonInquiryLearning Log/ J ournal

AssessmentObservation Checklis t (AssessmentBLM 6.1)1. The checklist is used both in observingthe activity in process and assessingstudent assignments submitted.

Learning LogHave students answer the followingquestion in their learning log usingpictures, words, and numbers:

1. What have you learned about therelationship of length and width to thearea of a rectangle?Look for students who comment on thefact that there was a pattern in theshapes, and that this pattern led to thedevelopment of a formula for determiningthe area of a rectangle. Also make noteof students who demonstrate theirknowledge in different formats (pictures,numbers, and words).

Reinforcement

1. Students can complete pp. 54-56 in theMy Ontario Math Workbook Grade 5

2. Students can complete pp. 110-111 inQuest 2000 Extra Practice and TeachingMasters

Asses sment Str ategiesLearning LogObservation



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large number of houses. Someone will undoubtedlydetermine that the amount of siding (material to make thefaces of the house including the bottom) is an importantissue to explore. Point out to the students that the finding

the total area of material needed is what is essential.Mini Lesson on AreaStep 1:Review the meaning of area with students -- the number of square units to cover a surface.

Step 2:Discuss with students how finding the area by counting thesquares on grid paper is fine, but we might not alwayshave grid paper to use. Finding a formula for area wouldmake it easier.

Step 3: Tell the students that they are going to explore a variety of rectangles and try to develop rules, so it will be easier todetermine the amount of material required for massproduction. Finding the perimeter of the structure will tellthem what length of straw they'll need to create the frameof the house; the area will tell them how much material willbe needed to cover the walls. These numbers can bemultiplied to determine the amount of material needed forconstructing a large number of houses.

Individual ExplorationStep 1:Distribute the Making Rules for Finding Area (Activity BLM6.1), a piece of grid paper, and a die to each student. Tellthe students that they are going to explore rectangles of different dimensions and attempt to discover a pattern.Once a pattern is discovered, they can make a rule to useeverytime they're trying to determine the area.

NOTE: Some students might already know the formula forarea. In this case simply indicate that it's alright to say whatthe formula is, but it must be substantiated or proven beforeit is used. The direction of the task changes in this case,from determining the formula to proving that the formula is

correct or incorrect.Step 2:Remind students that one way of looking at patterns is byusing a data table. Point the data table out to the studentson the blackline master.

Step 3:Students will roll the die. The first roll is the number to beplaced in the Length column of the table.

Quizzes, Tests, Examinations


Checklist

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Step 4:Students will roll the die a second time. This number isplaced in the Width column of the table.

Step: 5Once the students have the Length and Width columns,they will draw a rectangle with the correspondingdimensions on the grid paper.

Step 6:Student will estimate what the area is of the shape they'vecreated. They will print their estimation on their worksheet.

Step 7:Students will count the area of the rectangle they'veconstructed on the grid paper, and place the area in theArea column of the table.

Step 8:Students will repeat steps 3 through 6 ten times. After theexploration is complete, they'll try to identify patterns in thenumbers.

Step 9:Once students have completed ten trials, they'll use thedata they've collected to identify a pattern, and extend it bymaking a prediction as to what the area would be for a40x40 shape. Students may also use their calculators toextend their learning by completing the Calculator Areaand Perimeter (Activity BLM 6.2).

Group Share ConferenceStep 10:Ask leading questions when discussing the game-- What ishappening to the length and width to obtain the area of theshape? What dimensions give you the largest areas? Whatdimensions give you the smallest areas?

ReinforcementHave students answer the following question in theirlearning log using pictures, words, and numbers:

1. What have you learned about the relationship of lengthand width to the area of a rectangle?Look for students who comment on the fact that there wasa pattern in the shapes, and that this pattern led to thedevelopment of a formula for determining the area of arectangle. Also make note of students who demonstratetheir knowledge in different formats (pictures, numbers,and words).



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Resources Ar ea of Polygo ns 6_1.PDF

Making Rules 6_2.PDF

Rules for Area and Perim eter 6_3.PDF

Developin g Rules Checklis t 6_4.PDF

Math Trek 4, 5, 6

Quest 2000 Extra Practice and TestingMasters

pp. 110-111

Graph paper / grid paper 2

dice 1

Notes to TeacherAlthough students may have explored the formula (rule) for determining the area of a rectangle in earliergrades, it is essential to keep in mind that the understanding created by a grade 5 students is typicallydifferent from that created by the same student at a younger age. Students should be encouraged tointegrate the information collected in the exploration into their existing knowledge structures. This type of reflection and connection will facilitate the validation or discreditation of existing understandings.

Students should be encouraged to explore, identify, and extend patterns using computer technology.Students can reinforce their understandings of relationships, patterns, and rules through Math Trek 4, 5, 6 -Patterns and Sum Search acti viti es.

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Math in Architecture Subtask 73-D Figures-Exploring Shapes and Solids


Expectations5m65 A identify, describe, compare, and classify

geometric figures;5m66 draw and build three-dimensional objects and

models;5m73 sketch the faces that make up a

three-dimensional figure by looking at athree-dimensional figure;

5m75 sort polygons according to the number of sides, angles, and vertices;

5m85 A discuss ideas, make conjectures, and articulatehypotheses about geometric properties andrelationships;

5m91 A recognize and discuss the mathematicalrelationships between and among patterns;

5m92 A identify, extend, and create patterns in a varietyof contexts;

5m93 A analyse and discuss patterning rules;5m94 A create tables to display patterns;5m98 A describe patterns encountered in any context

(e.g., computer games, television show times),make models of the patterns, and create chartsto display the patterns;

5m99 A identify and extend patterns to solve problemsin meaningful contexts (e.g., leaves on trees,spirals on pineapples);

5m105 A use patterns in a table of values to makepredictions;

Teaching / LearningRelating to the World Outside the ClassroomStep 1:Review with students some of the shapes they noted in theintroductory lesson. Guide the discussion to reveal thatbuildings are made up of, or can be broken down into, acollection of geometric solids. The students will have toconsider carefully which shapes they will use to make theirhouses.

Mini Lesson on Naming 3D SolidsStep 2:Remind students of the first lesson, where they exploredthe differences between pyramids and prisms. A prism ismade up of rectangles except for the bases. The sides of a pyramid are made up of triangles. Make the students

Description The students will explore and sketch diagrams of different geometric solids. Through the use of a t-tableand their sketches, the students will explore the relationship between the number of sides in a two

dimensional shape and the number of faces in the corresponding 3-D shape.

GroupingsStudents Working In PairsStudents Working IndividuallyStudents Working As A Whole Class

Teaching / Learning StrategiesCollaborative/cooperative LearningWorking With Manipulatives

AssessmentShape to Soli d Works heetLook for students who discuss therelationship between edges of the 2Dshape and faces of the corresponding 3Dfigure. Notes should also be made forstudents who notice how the relationshipdiffers between prisms and pyramids.

Learning LogWhat did you notice about 2D shapeswhen they are used as bases in 3D

solids?Look for students who noticed that thereis a relationship between the number of sides of the 2D base and the faces of itscorresponding pyramid (faces = sides +1)or prism (faces = sides +2).

How are you going to use yourknowledge of 3D figures when planningyour structure?Answers will vary, but take note of students who identify the need to keepthe bases of the roof and bottom of the

house congruent.Identify ing Solids Pencil / Paper

Assignmen tLook for students who correctly matchedthe solids with the appropriate name.

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aware of how important the base is to naming the prism orpyramid (A prism with a triangular base is a triangularprism).

Exploring 3D SolidsStep 3:Organize the students into groups of 4 or 5. Distribute theblackline masters of the nets of geometric solids(Reference BLM 7.1). Be sure to tell the students that eachgroup member should have two blackline masters.

Step 4:Have the students begin to construct the geometric solidsusing the nets.

Step 5:As the students are working to construct their geometricsolids, distribute the Shapes to Solids (Activity BLM 7.2). The students will work collaboratively to complete the charton the blackline master. Each student should have his / herown record of the group's findings in their learning log.Guide the students with questions such as, What patternsdo you see in your chart? What is the same about eachbase and the corresponding 3D solids?

Group Share ConferenceStep 6:Initiate a discussion with the students about what patternsarose in their data tables. Also discuss overallcharacteristics of the 3D figures. Attempt to makeconnections between the activity and their final project --they'll need to have both a prism and a pyramid in their finaldesign, and the bases must be congruent for the model towork. Students will have their choice of what kind of prismand pyramid they will use to construct their house. Theycould refer to their exploration in subtask 2 to help themdecide what solids they'll use. They might choose to usetwo prisms, although typically houses are made up of aprism (base) and a pyramid (roof).

Step 7:Have students complete the learning log questions: What

did you notice about 2D shapes when they are turned into3D solids? How are you going to use your knowledge of 3D figures when planning your structure?

ReinforcementHave the students complete the Identifyng Solids (ActivityBLM 7.3) as a reinforcement activity.

ExtensionHave the students explore with Polydrons, Frameworks or

Learning Log


Adaptations



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other geometry manipulatives.

ResourcesMets of Geometric Solid s 7_1.PDF

3-D Solids 7_2.PDF

Identifying Solids 7_3.PDF

Scissors 1

Scotch Tape 1

Polydron Sets

Frameworks Sets

Notes to TeacherStudents can be encouraged to create a rules for finding the number of faces on a pyramid or prism, giventhe shape of the base. Push the students to relate their understanding of congruent shapes to this newinformation, and to the construction of their Hamster House.

Anecdotal Records should reflect the expectations and student response to product and process.

Teacher Reflections



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Math in Architecture Subtask 83-D Nets


Expectations5m65 identify, describe, compare, and classify

geometric figures;5m66 A draw and build three-dimensional objects and

models;5m71 identify nets for a variety of polyhedra from

drawings while holding three-dimensional figuresin their hands;

5m72 A construct nets of cubes and pyramids using avariety of materials;

5m73 A sketch the faces that make up athree-dimensional figure by looking at athree-dimensional figure;

5m75 sort polygons according to the number of sides, angles, and vertices;

5m77 demonstrate an understanding of congruentfigures;

5m82 use mathematical language to describegeometric ideas (e.g., quadrilateral, scalenetriangle);

Teaching / LearningDemonstr ation / Teacher Led Investigation1. Review the characteristics of 3D figures from theprevious subtask, paying particular attention to the numberof faces on each solid.

2. Ask students if it is possible to display all faces of a 3Dfigure/object on a 2D piece of paper ( a 3D picture on paperwill hide at least one face).

3. Explain to students that one way to represent a 3Dfigure/object on paper is to draw its NET (A 2D shape thatcan be folded into a 3D figure is a NET of that figure) .

4. Demonstrate for students how any 3D figure can betransformed into a 2D net: Use a straw model of a cube(Reference BLM 8.1). Tape it to the blackboard. Open upeach side and tape them to the blackboard. The resultantshape after all sides are opened up will be the net of thesolid (however, because straws are being used, there willbe two missing sides per face. J ust add these with chalk).

5. Tell students that each square on the board representsa face of the solid. The net can be folded up to create a

DescriptionStudents are introduced to the concept of 3-D nets. The students will use their knowledge of 3-D figuresto explore and create the nets of familiar geometric solids. The students will apply their knowledge of

nets and geometric solids to solve related problems.

GroupingsStudents Working As A Whole ClassStudents Working In Small GroupsStudents Working Individually

Teaching / Learning StrategiesDemonstrationCollaborative/cooperative Learning

AssessmentSELECT RESPONSE1. Students will submit their notebooks,which will contain their hexonimosketches (arrangements of six squares tocreate the net of a cube). Watch to besure that the student has identified thosehexonimo arrangements that will create acube, as well as those that do not form acube. Also take note of students whohave the nets from other 3D solids, whichthey shared with their group members.

2. Students will also submit the 'Net or Notthe Net' blackline master. Look to be surethat the student has suggested one netthat makes up the solid/object, and onethat does not.

LEARNING LOG:3. What did you notice about hexominoesand cubes?Look for students who noticed that thereare different ways to 'unfold' a solid anddraw its net. However, simply drawing

the faces of the solid does not alwayswork as its net.

TEST ON NETSHave students complete theaccompanying test.

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cube.

Collaborative Exploration:6. In small groups, have students explore all of the possible

hexonimoes (arrangements of six squares to create the netof a cube) using straws and pipe cleaners.

7. Allow students to sketch different hexonimoarrangements.

8. Have students create the 3D figure from each hexominoarrangement and determine which will fold to create acube.

8b. Ask the students to remember the shape theyconstructed last subtask. Ask them to use straws andpipecleaners to construct the solid (if they created a squarebased pyramid last subtask, they'll create a square basedpyramid using straws during this subtask). Students whoare constructing a solid with many faces (i.e. pentagonalprism or pyramid) might have to cut their straws. J ust makeit explicit to the students that the straws must all be cut tothe same length. After the students have constructed thegeometric solid, have them 'unfold' the straws (as in theteacher demonstration) and determine one net for the solid.

The students will sketch this net in their notebook or projectportfolio.

8c. Have the students share the nets they've found withthe rest of the group. Have each group member sketch thenets in their notebook, for use later in the unit - namelywhen they have to design the levels of their house. Makesure that students are sketching the nets, and look forstudents who determine that there are several ways to'unfold' the solid.

Group Share ConferenceHave the students share the nets that fold to form thedifferent solids. Students could sketch them on theblackboard. Encourage students to copy sketchesdifferent from their findings into their notebooks. Studentswill discover that there are several nets to each solid. They

will also discover that any arrangement of the faces won'tnecessarily fold to create the solid.

ReinforcementStep 9:Have the students complete the The Net or Not the Net(Activity BLM 8.1). Given a 3D figure, the students willhave to construct a net for the shape, and one net thatcan't be folded to make the shape.

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Students may also complete p. 139 of Quest 2000 ExtraPractuce and Teaching Masters , or pp. 44-45 of Interactions.

ExtensionHave the students explore with Polydrons, Frameworks orother geometry manipulatives.

ResourcesNets of Geometric Solids 7_1.PDF

Sample Model 8_1.PDF

The Net o r Not The Net 8_2.PDF

Quest 2000 Extra Practice and TestingMasters

p. 110

Interactions Student Text pp. 44-45

Straws 20

Pipe Cleaners 10

Scissors 1Scotch Tape 1

Polydron Sets

Frameworks Sets

Notes to Teacher

Teacher Reflections



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Math in Architecture Subtask 9The Hamster House ~ Culmin ating Task


Expectations5m39 A solve problems related to the calculation of the

perimeter and the area of regular and irregulartwo-dimensional shapes;


5m57 A estimate the area of irregular polygons andmeasure the area by dividing the polygons intoparts, using grid paper;5m65 identify, describe, compare, and classifygeometric figures;

5m66 A draw and build three-dimensional objects andmodels;

5m70 use mathematical language effectively todescribe geometric concepts, reasoning, andinvestigations, and coordinate systems.

5m72 A construct nets of cubes and pyramids using avariety of materials;

5m73 A sketch the faces that make up athree-dimensional figure by looking at athree-dimensional figure;

5m77 A demonstrate an understanding of congruentfigures;

5m78 A measure and construct angles using aprotractor;

5m82 use mathematical language to describegeometric ideas (e.g., quadrilateral, scalenetriangle);

5m84 A discuss geometric concepts with peers anduse mathematical language to explain theirunderstanding of the concepts;

5m85 discuss ideas, make conjectures, and articulatehypotheses about geometric properties and

relationships;5m92 A identify, extend, and create patterns in a varietyof contexts;

5m93 A analyse and discuss patterning rules;5m94 A create tables to display patterns;5m95 A apply patterning strategies to problem-solving

situations.5m100 A use a calculator and computer applications to

explore patterns;5s93 cut, join, and rearrange pliable and rigid

Description The students will prepare a proposal for the construction of a hamster house for the Pets 'R Uscompany. The proposal will include a model of the hamster house and a Mathematical Specifications

Report. Each proposal will be displayed and peer assessed. After the students have viewed theproposals they will be given another student's Mathematical Specifications Report and challenged torecreate that student's hamster house.

GroupingsStudents Working In Small GroupsStudents Working IndividuallyStudents Working In Pairs

Teaching / Learning StrategiesConcept Clarification

BrainstormingMini-lesson

AssessmentSection One ~ PlanningStudents reflect on the following in theirLearning Log:1. Describe your hamster house.Look for a proper understanding of therequirements for the house ie. that twodifferent 3-D shapes must be used.2. What aspect of the project do youforesee as being the most difficult for you? Explain your thinking?Look for a general understanding of theoverall project.

Section Two ~ BuildingIn the Mathematical Specifications Report,look for precision: an accurate drawingof the two nets, correct anglemeasurements, correct labeling of sideswith the intended measurement of theside, and an accurate highlighting of which lines in the net represent straws.

In the model, look for good constructiontechnique: a solid structure, properangles, and strong connections betweenstraws.

Learning LogHave students reflect on the followingquestion in their Learning Log:



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materials to make an object (e.g., cut wood at a45 angle to make a mitre joint; make a mould fora face mask);

Teaching / LearningSectio n One ~ Planning

Re-Introduction of the Culminating Project ~ TheHamster HouseStep 1Set the context for the students:

The Pets 'R Us company is looking for new ideas to promoteits new line of small rodent toys. You have beenapproached by the Pets 'R Us company to design a new

style of hamster house. Your task is to prepare a proposalfor a hamster house that would fit into most cages. Yourproposal will consist of a life-sized model of the hamsterhouse along with a report on the mathematicalspecifications of your house in order that the house can bemass produced. Discuss the sizes of typical cages(approximately the size of a filing cabinet drawer) and thesize of most hamster houses (the floor area no smaller thanthe size of a CD case).

Have students look at the blackline master, A New House(Activity BLM 2.2) which they completed in Subtask 2.Suggest that they have learned quite a bit since completingthe sheet and that perhaps some of their thinking haschanged. Have students discuss with a partner their initialthoughts about the Hamster House (which were recordedon the blackline master, A New House). Furthermore, havestudents add to their sheet and revise their plan toincorporate new ideas that they may have.

Review Activity BLM 9.1 ( Pets 'R Us ~ Hamster HouseProject ) with the students. This sheet will explain in detailwhat is required in each area of the proposal. Discuss theproject and clarify any misunderstandings with the wholeclass.

Think About ItStep 2In small groups, have students brainstorm possible 3-Dfigures that could be used for their hamster house. Remindthe students that their final model will have to beconstructed out of straws and pipecleaners. Havestudents make a final decision on which two 3-D figuresthey will use (one for the base, one for the roof). Possiblebase/roof combinations are: rectangular prism/triangular

What steps have you taken in your Mathematical Specifications Report toensure that your model can bereproduced accurately?

Section Three ~ Mass ProducingUse the rubrics to assess the students inthe strands of Geometry, Measurementand Patterning & Algebra.

Use the first page of Section Three in theMathematical Specifications Report todetermine if the student can sketch thefaces that make up a three-dimensionalobject by looking at a three-dimensionalobject (Geometry strand 5m73).

Asses sment Str ategiesLearning LogObservationPerformance Task

Assessment Recording DevicesAnecdotal RecordRubric

Adaptations



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accurately, distribute blackline masters of nets (found withSubtask 7 - Reference BLM 7.1) from which they can makea tracing. Some students may prefer to draw the netsaccurately on their own; in which case, a protractor should

be used to ensure correct angle measurements. The netsneed not be drawn life-size; however, an attempt should bemade to draw the nets proportionately (ie. mathematicallysimilar to the actual net made from straws).Labeling the Mathematical Attrib utes of the NetsUsing a protractor, students will measure all the angles inthe net and record these measurements properly. Labeleach side with its actual measurement in the completedmodel.Highlight th e 'Straws'With a highlighter or another colour, students are to indicatewhich lines on their nets represent actual straws in thefinished model.

Exchange Feedback with a Partner Step 3On their own, students will check for accuracy betweentheir report and their model using the Self EvaluationChecklist (Assessment BLM 9.1). In partners, students willcompare their partner's model against the MathematicalSpecifications Report and provide feedback orally byanswering the following questions: e.g. Was I carefulenough in my measuring? Where could I improve? Doyou think you could construct my model by looking at myreport alone?

ReflectionStep 4Have students reflect on the following question in theirLearning Log:What steps have you taken in your MathematicalSpecifications Report to ensure that your model can bereproduced accurately?

Section Three ~ Mass Produc ing

Step 1Explain to the students that the third section of theMathematical Specifications Report (Activity BLM 9.2e and9.2f) will be used by the Pet's 'R Us company to determinethe feasibility of mass producing the hamster house forretail sales. Once the company approves a model, thehouse will be constructed out of plastic. (As far as thestudents are concerned, the actual construction of thehouse out of plastic is theoretical).



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Mathematical Specifications Report ~ SectionThreeHow Much Plastic is Needed?Step 2In section three of the Mathematical Specifications Report(Activity BLM 9.2e and 9.2f), students must draw eachunique face of their structure and determine how many likefaces (congruent faces) their structure has. Demonstrateto the students how the surface area of their structure canbe calculated by finding the area of each face. Todetermine the area of each face, students will use the rulesdeveloped earlier for calculating the area of rectangles andthe grid paper method for calculating the area of otherpolygons. By adding up these areas, the total amount of plastic needed by the company for one house will be found.

Determine Quantiti es for Mass Producti onStep 3Explain to the students how to use patterning strategies inorder to determine the quantities for mass production of their house. Students answer the questions at the end of section three of their report. Allow students to use acalculator. See examples in the blackline master,Exemplar: Specification Report .

Discuss w ith a Partner Step 4Have students discuss the pattern rules with a partner andrecord them on the back of section three of the

Mathematical Specification Report.The EntranceStep 5Explain to the students that the hamster will need some sortof hole in one of the walls of the house. This hole will becut out from the plastic during the manufacturing process.

The hole will not actually affect the amount of plasticneeded to make the house because this plastic will still beused (make an analogy with donuts and donut holes... thebatter is used and then cut away... however, the 'hole' isstill usefu

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