G r a d e. 5 M a t h e M a t i c s. shape and space

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1 G r a d e 5 M a t h e M a t i c s shape ad space

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3 Grade 5: Shape ad Space (Measuremet) (5.SS.1) Edurig Uderstadigs: there is o direct relatioship betwee perimeter ad area. Geeral Outcome: Use direct or idirect measuremet to solve problems. SpEcific LEAriG OUtcOME(S): AchiEvEMEt idicators: 5.SS.1 Desig ad costruct differet rectagles, give either perimeter or area, or both (whole umbers), ad draw coclusios. [C, CN, PS, R, V] Costruct or draw two or more rectagles for a give perimeter i a problem-solvig cotext. Costruct or draw two or more rectagles for a give area i a problem-solvig cotext. Illustrate that for ay perimeter, the square or shape closest to a square will result i the greatest area. Illustrate that for ay perimeter, the rectagle with the smallest possible width will result i the least area. Provide a real-life cotext for whe it is importat to cosider the relatioship betwee area ad perimeter. s h a p e a d s p a c e ( M e a s u r e m e t ) 3

4 Prior Kowledge Studets may have had experiece with the followig: Estimatig, measurig, ad recordig the legth, width, ad height of objects to the earest metre ad cetimetre Estimatig, measurig, ad recordig the perimeter of regular ad irregular shapes Costructig shapes with a give perimeter Estimatig, measurig, ad recordig the area of regular ad irregular shapes with a give perimeter Estimatig, measurig, ad recordig the area of regular ad irregular shapes Costructig shapes with a give area Distiguishig betwee perimeter ad area Idetifyig ad describig patters foud i tables ad charts Idetifyig polygos Studets may also have had experiece with the followig: Perimeter is the distace aroud a shape Area is the amout of surface withi a regio related Kowledge Studets should be itroduced to the followig: Demostratig a uderstadig of measurig legth i millimetres, ad distiguish rectagles from other quadrilaterals Recogizig that all squares are rectagles BacKgroud iformatio Perimeter is the distace aroud a shape. Studets ofte cofuse this cocept with area, the amout of surface a shape covers. Ivolvig studets i actual measurig experieces ca help them distiguish betwee these two cocepts. For example, activities that have studets completely coverig shapes with square uits ca help them uderstad the meaig of area. Moreover, studets ofte have miscoceptios about the relatioship betwee perimeter ad area. Two of the most commo are the followig: 1. the loger the perimeter, the larger the area 2. perimeter ad area icrease at the same rate 4 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

5 For example, some studets have the mistake belief that if the perimeter is doubled, the area will double. Therefore, the itet of the learig experieces i this sectio is to help studets overcome their miscoceptios by havig them explore the perimeter ad area of rectagles. The learig experieces are also desiged to help studets recogize at least five geeralizatios about the relatioship betwee these two measuremets: If oly the perimeter (area) of a rectagle is give, its area (perimeter) caot be determied. Icreasig the perimeter (area) of a rectagle does ot ecessarily icrease the area (perimeter) of the rectagle. If the legth (width) of a rectagle is fixed, the icreasig its perimeter will icrease its area. The square has the largest area amog rectagles that have the same perimeter. The square has the smallest perimeter amog rectagles that have the same area. mathematical laguage Area Legth Perimeter Polygo Rectagle Square Width s h a p e a d s p a c e ( M e a s u r e m e t ) 5

6 learig experieces Assessig Prior Kowledge Materials: Cetimetre grid paper (BLM 5 8.9) Orgaizatio: Idividual a) Ask studets to use the cetimetre grid paper to draw the followig: A polygo with a perimeter of 10 cm A polygo with a perimeter greater tha 15 cm A polygo with a area of 12 cm 2 A polygo whose area is greater tha 15 cm 2 ad less tha 25 cm 2 Iside of each shape that they draw, have studets write the ame of the polygo ad its perimeter or area measuremet. b) Preset the studets with the followig problems: Kelly wats to make a woode frame for the picture his aut drew for him. Does Kelly eed to measure the perimeter or the area of the picture to fid out how much wood he eeds? What uit of measuremet do you thik he should use? Explai your aswers. Mr. Lie wats to cover the bulleti board i his classroom with a piece of paper. Does he eed to measure the perimeter or the area of the bulleti board to fid out how much paper he eeds? What uit of measuremet do you thik he should use? Explai your aswers. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: costruct a polygo with a give perimeter costruct a polygo with a give area distiguish betwee perimeter ad area uderstad that perimeter is the distace aroud a shape uderstad that area is the amout of surface iside a regio ame polygos accordig to the umber of sides that they have kow that perimeter is measured i liear uits ad that area is measured i square uits justify their selectio of a uit of measuremet 6 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

7 costruct or draw two or more rectagles for a give perimeter i a problem-solvig cotext. illustrate that for ay perimeter, the square or shape closest to a square will result i the greatest area. illustrate that for ay perimeter, the rectagle with the smallest possible width will result i the least area. Materials: Square tiles, cetimetre grid paper (BLM 5 8.9), ad recordig sheet (BLM 5.SS.1.1) Orgaizatio: Small group/whole class a) Preset studets with the followig problem: Mrs. Zah ad Mr. Stewart have gardes that are rectagular i shape. The perimeter of Mrs. Zah s garde is 16 metres ad the perimeter of Mr. Stewart s garde is 20 metres. Is Mr. Stewart s garde larger tha Mrs. Zah s? Make sure studets uderstad the problem by askig: What do you kow about Mrs. Zah s garde? What do you kow about Mr. Stewart s garde? What questio do you eed to aswer? b) Have studets write dow what they thik the aswer is, ad the share it with the other members of their group. c) Next, have studets draw rectagles o the cetimetre grid paper to show why they thik their aswer is correct, ad the share their drawigs with the other members of their group. d) Challege studets by askig: Are there other rectagles that have perimeters of 16 metres? Are there other rectagles that have perimeters of 20 metres? What are their areas? Have studets i each group use the square tiles or cetimetre grid paper to fid other rectagles that have perimeters of 16 metres ad 20 metres. e) Ecourage studets to orgaize their work by havig them record their fidigs i the table provided (see BLM 5.SS.1.1). f) Have studets i each group aalyze their tables ad record ay patters or relatioships that they fid. g) Ask each group to preset its fidigs to the other members of the class, as well as its coclusio regardig whose garde Mrs. Zah s or Mr. Stewart s is larger. s h a p e a d s p a c e ( M e a s u r e m e t ) 7

8 Observatio Checklist Check studets work to determie whether they ca do the followig: costruct or draw two or more rectagles for a give perimeter i a problem-solvig cotext recogize that they caot tell for sure whether Mr. Stewart s garde is larger tha Mrs. Zah s that is, if oly the perimeter of a rectagle is give, its area caot be determied recogize patters ad relatioships, such as the square has the largest area amog rectagles with the same perimeter the rectagle with the smallest width has the least area icreasig the perimeter does ot ecessarily icrease the area if the legth of a rectagle is fixed, icreasig its perimeter icreases its area Materials: Square tiles, cetimetre grid paper (BLM 5 8.9), ad recordig sheet (BLM 5.SS.1.1) Orgaizatio: Small group/whole class a) Preset studets with the followig problem: A farmer has 36 metres of fecig material. He is plaig to use all of the fecig material to make a rectagular pe for his sheep. What is the largest pe he ca make for his sheep? Make sure studets uderstad the problem by askig: What does the farmer wat to do? How much fecig material does he have? What do you eed to fid out? b) Have studets write dow what they thik the aswer to the problem is ad share their aswer with the other members of their group. c) Next, ask, How may differet rectagular pes ca the farmer make with 36 metres of fecig material? d) Have studets i each group use the square tiles or cetimetre grid paper to fid all the rectagles that have a perimeter of 36 uits. Have studets record their fidigs i the table provided (see BLM 5.SS.1.1). e) Have studets aalyze their fidigs ad record ay patters ad relatioships that they fid. f) Have each group share with the other members of the class its solutio to the problem ad ay other patters that it fids. 8 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

9 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: costruct or draw two or more rectagles with a give perimeter i a problem-solvig cotext recogize that the pe with the largest area is a square with sides six metres i legth that is, the square has the largest area amog rectagles with the same perimeter recogize patters ad relatioships, such as the rectagle with the smallest width has the least area the closer the rectagle is to a square, the closer the area is to the maximum area costruct or draw two or more rectagles for a give area i a problemsolvig cotext. illustrate that for ay perimeter, the square or shape closest to a square will result i the greatest area. illustrate that for ay perimeter, the rectagle with the smallest possible width will result i the least area. provide a real-life cotext for whe it is importat to cosider the relatioship betwee area ad perimeter. Materials: Spaghetti ad Meatballs for All by Marily Burs, square tiles, ad recordig table (BLM 5.SS.1.1) Orgaizatio: Whole class/small group a) Read Spaghetti ad Meatballs for All. As you read the story, have the studets use the square tiles to model what is happeig with the tables. b) Have studets discuss the problem with the table arragemets. Begi the discussio by askig, Why does Mrs. Comfort keep sayig the table arragemets wo t work? Have studets work with their parter to fid differet ways of arragig eight tables. Have them decide which arragemet is the best. c) Pose the problem: Mrs. Comfort has 24 square tables. If she pushes the tables together to form a rectagle, what is the highest umber of people she ca sit aroud the rectagle? s h a p e a d s p a c e ( M e a s u r e m e t ) 9

10 d) Make sure the studets uderstad the problem by askig them the followig questios: How may square tables does Mrs. Comfort have? What does Mrs. Comfort do with the tables? What do you eed to fid out? What is oe way that Mrs. Comfort ca push the tables together to form a rectagle? (Make sure the studets recogize that the rectagular arragemets caot have ay spaces i the middle.) How may people ca Mrs. Comfort sit aroud the table? e) Explai that the umber of tables pushed together represets the area ad the umber of people who ca sit aroud the table represets the perimeter. The, ask, Are there other rectagles that Mrs. Comfort ca make that have a area of 24 square uits? Ca she seat the same umber of people aroud each rectagle? f) Have the studets work with their parters to determie all the rectagles that ca be made with 24 tiles. Ecourage studets to record their fidigs i the table provided (see BLM 5.SS.1.1). g) Have studets aalyze their fidigs ad record ay patters ad relatioships that they fid. h) Ask studets to share their fidigs ad their coclusio as to which rectagle Mrs. Comfort should make if she wats to seat the most people with the rest of the class. 10 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

11 Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: costruct or draw two or more rectagles for a give area i a problem-solvig cotext recogize that Mrs. Comfort ca seat the most people aroud a 1 x 24 rectagle that is, amog rectagles with the same area, the oe with the smallest width has the greatest perimeter recogize patters ad relatioships, such as the followig: The closer the rectagle is to a square, the smaller its perimeter. If two rectagles have the same area, they do ot ecessarily have the same perimeter. If oly the area of a rectagle is give, its perimeter caot be determied. costruct or draw two or more rectagles for a give area i a problemsolvig cotext. illustrate that for ay perimeter, the square or shape closest to a square will result i the greatest area. Materials: Math jourals, square tiles, ad square cetimetre paper (BLM 5 8.9) Orgaizatio: Idividual/Large group a) Pose the followig problem: Mr. Satos is makig a rectagular flower garde i his backyard. If the area of the garde is 36 m 2, what is the least amout of fecig that he eeds to eclose the garde? b) Make sure the studets uderstad the problem by askig them the followig questios: What is Mr. Satos makig? How big is the garde? What do you eed to fid out? c) Tell studets that they ca use the square tiles or the cetimetre grid paper to help them solve the problem. Have the studets record their solutios i their math jourals. d) Have studets share their aswers ad the strategies they used to solve the problem. s h a p e a d s p a c e ( M e a s u r e m e t ) 11

12 Observatio Checklist Check studets work to determie whether they ca do the followig: recogize that differet rectagles ca have the same area recogize that the least amout of fecig that is eeded is 24 metres Puttig the Pieces together desig a clubhouse Purpose: The purpose of this ivestigatio is to have studets apply the cocepts of perimeter ad area to a problem-solvig situatio. I particular, it is desiged to ehace studets ability to differetiate betwee perimeter ad area costruct rectagles with a give perimeter or area maximize or miimize the area of a rectagle with a fixed perimeter maximize or miimize the perimeter of a rectagle with a fixed area I additio, the ivestigatio is desiged to ehace studets ability to commuicate mathematically coect mathematics to real-world situatios solve problems reaso mathematically Materials/Resources: Cetimetre grid paper (BLM 5 8.9), coloured cetimetre grid paper, square tiles, scissors, ad glue Orgaizatio: Whole class/small groups a) Preset studets with the followig situatio: You ad your frieds have decided to build a rectagular clubhouse. You pla to build your clubhouse i a sectio of the schoolyard with a area of 200 m 2. You have decided that your clubhouse must have the followig: The largest possible floor space Two rugs (Oe rug must have a perimeter of 24 m ad cover the largest possible area, ad the other rug must have a perimeter of 16 m that covers the least possible area.) At least two doors with a width of oe metre 12 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

13 A play area that takes up at least ¼ of the floor space. No furiture ca be placed i the play area A rectagular seatig area with a perimeter of 12 m A rectagular table with a area of 4 m 2 You also decide that the clubhouse ca have other items as log as they are ot placed i the play area. b) Explai that each group must draw up a pla for the clubhouse that icludes the dimesios of each item i the list of specificatios. Tell studets that they ca draw their pla for the clubhouse o the white cetimetre grid paper ad use the colour cetimetre paper to idicate the furiture ad the rugs. They should let each square cetimetre o the grid paper represet oe square metre. c) Help studets develop the criteria for assessig their plas for a clubhouse. d) Have studets work o their plas for a clubhouse. e) Have each group preset its desig for a clubhouse to the other members of the class. Ecourage studets to describe the dimesios of each item i their clubhouse ad how they determied its size. Observatio Checklist Use the rubric provided ad the studet-developed criteria to assess studets attaimet of outcome 5.SS.1 durig the completio of the project. s h a p e a d s p a c e ( M e a s u r e m e t ) 13

14 3 2 1 Distiguishes betwee perimeter ad area Determies the perimeter of a rectagle by fidig the distace aroud it. Determies the area of a rectagle by fidig the umber of square uits it covers. Determies the perimeter of a rectagle or the area of a rectagle. Is ot able to determie the perimeter of a rectagle. Costructs rectagles with a give perimeter Costructs a rectagle with a give perimeter. Costructs a rectagle with a give perimeter with support. Is ot able to costruct a rectagle with a give perimeter. Recogizes relatioships Recogizes that a square has the largest area amog rectagles that have the same perimeter. Recogizes that, amog rectagles that have the same perimeter, the oe with the smallest width has the least area. Recogizes that a square has the smallest perimeter amog rectagles with the same area. Recogizes that a square has the largest area amog rectagles with the same perimeter with support. Recogizes that, amog rectagles that have the same perimeter, the oe with the smallest width has the least area with support. Recogizes that a square has the smallest perimeter amog rectagles with the same area with support. Does ot recogize that a square has the largest area amog rectagles with the same perimeter. Does ot recogize that, amog rectagles that have the same perimeter, the oe with the smallest width has the least area. Does ot recogize that a square has the smallest perimeter amog rectagles with the same area. 14 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

15 Grade 5: Shape ad Space (Measuremet) (5.SS.2) Edurig Uderstadigs: all measuremets are comparisos. Legth, area, volume, capacity, ad mass are measurable properties of objects. the uit of measure must be of the same ature as the property beig measured. Geeral Outcome: Use direct or idirect measuremet to solve problems. SpEcific LEAriG OUtcOME(S): AchiEvEMEt idicators: 5.SS.2 Demostrate a uderstadig of measurig legth (mm) by selectig ad justifyig referets for the uit mm modellig ad describig the relatioship betwee mm ad cm uits, ad betwee mm ad m uits [C, CN, ME, PS, R, V] Provide a referet for oe millimetre ad explai the choice. Provide a referet for oe cetimetre ad explai the choice. Provide a referet for oe metre ad explai the choice. Show that 10 millimetres is equivalet to 1 cetimetre usig cocrete materials (e.g., ruler). Show that 1000 millimetres is equivalet to 1 metre usig cocrete materials (e.g., metre stick). Provide examples of whe millimetres are used as the uit of measure. Prior Kowledge Studets may have had experiece with the followig: Estimatig, measurig, ad recordig the legth, width, ad height of objects to the earest metre or cetimetre Describig the relatioship betwee a metre ad a cetimetre Idetifyig referets for a cm ad a m Demostratig a uderstadig of fractios less tha oe Studets may also have had experiece with the terms legth, width, height, ad perimeter. s h a p e a d s p a c e ( M e a s u r e m e t ) 15

16 related Kowledge Studets should be itroduced to the followig: Multiplyig ad dividig whole umbers by 10s, 100s, ad 1000s Readig, writig, iterpretig, ad usig decimal otatio for 10ths, 100ths, ad 1000ths Relatig fractios to decimals Describig orally ad i writig the rule for a patter BacKgroud iformatio Measuremet is the process of comparig a uit of measure with a measurable property of a object or pheomeo. The process cosists of the followig: 1. Idetifyig the property to be measured 2. Selectig a appropriate uit of measure 3. Repeatedly matchig the uit with the property or pheomea beig measured 4. Coutig the umber of uits By the ed of the 18th cetury, uits of measure varied greatly withi ad betwee coutries. The lack of stadard uits made tradig with other cultures difficult to carry out. To remedy this situatio, the Frech Natioal Assembly i 1790 asked the Academy of Sciece to develop a commo system of measuremet. The system it developed is kow as the metric system. The uits of measure developed by the academy have evolved ito the Système Iteratioal d Uités (abbreviated SI), which was established i The SI is govered by the Geeral Coferece o Weights ad Measures, which makes chages i the system to reflect the latest advaces i sciece ad techology. Eve though there are differeces betwee the two systems, SI is still referred to as the metric system. Because of its simplicity, all but a few coutries have adopted the metric system. Its simplicity arises from its use of the followig: 1. A small umber of base uits 2. The decimal system 3. A uiform set of prefixes that apply to each area of measuremet 16 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

17 These prefixes the most commo of which are show below idicate multiples or subdivisios of the base uits. prefix kilo (k) hecto (h) deka (da) deci (d) ceti (c) milli (m) Meaig 1000 uits 100 uits 10 uits 0.1 uit 0.01 uit uit I the Early ad Middle Years, studets are itroduced to legth, area, volume, capacity, ad mass. Their measuremet of these properties ivolves the uits listed i the chart below, ad ca be either direct or idirect. Direct measuremets ivolve selectig a uit ad comparig it directly with the object (e.g., usig a metre stick to measure the height of a table). Idirect measuremets are made whe a uit caot be placed directly o the object (e.g., fidig the height of a flagpole or the area of a coutry). Ofte, objects ca be measured idirectly by comparig them with thigs that ca be measured (e.g., fidig the height of a tree by measurig its shadow). Quatity Uits Symbol Legth Area Volume Capacity Mass kilometre metre cetimetre millimetre square metre square cetimetre cubic metre cubic cetimetre litre millilitre kilogram gram km m cm mm m 2 cm 2 m 3 cm 3 L ml kg g s h a p e a d s p a c e ( M e a s u r e m e t ) 17

18 Learig experieces that require studets to use measurig istrumets i realistic situatios are key igrediets i helpig them uderstad the cocepts ad skills ivolved i measuremet systems. I particular, these experieces ca help studets uderstad the followig: The measure of a uit is always 1 The uit must be of the same ature as the property that is beig measured The uit must be repeatedly matched with the property beig measured without ay gaps or overlaps (This process is kow as uit iteratio.) Oe uit may be more appropriate tha aother to measure the property of a object There is a iverse relatioship betwee the umber of uits ad the size of the uit A smaller uit gives a more exact measuremet A measuremet must iclude both a umber ad a uit Whe the same uits are used, measuremets ca be easily compared Estimatig that is, makig a reasoable judgmet about the approximate amout of a quatity also plays a importat role i the developmet of studets uderstadig of measuremet systems. A focus o estimatig eables studets to create a metal frame of referece for the size of uits ad their relatioships to each other. It also helps them judge the reasoableess of their measuremets. mathematical laguage Cetimetre Estimate Height Legth Measuremet Metre Millimetre Referet Width 18 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

19 learig experieces Assessig Prior Kowledge Materials: Assessmet activity sheet (BLM 5.SS.2.1) ad cm rulers Orgaizatio: Idividual Have studets complete the assessmet activity sheet (BLM 5.SS.2.1). provide a referet for oe millimetre ad explai the choice. Show that 10 millimetres is equivalet to 1 cetimetre usig cocrete materials (e.g., ruler). provide examples of whe millimetres are used as the uit of measure. Materials: cm rulers with mm marked o them, a cm ruler that ca be projected o the overhead or a overhead of a cm ruler, toothpicks, safety pis, idex cards, crayos, math scribblers, soda straws, ad the activity sheet (BLM 5.SS.2.2) Orgaizatio: Whole class/idividual a) Ask studets to draw a metre stick ad make sure that they iclude all the markigs. Whe studets fiish their drawigs, have them share their pictures ad explai what the markigs o their metre sticks mea. Use the discussio to determie what the studets already kow about mm so you ca clear up ay miscoceptios that they may have. b) Tell studets that they will be learig about a ew uit of liear measure called a millimetre. Place a cm ruler o the overhead ad poit out that there are 10 spaces betwee cosecutive cetimetres. Tell studets that each space represets 1 millimetre. Write the word millimetre o the board or overhead, ad show studets the symbol for the uit. c) Have studets take out their cm rulers. Ask them to fid the umber of millimetres betwee the 1 cm mark ad the 2 cm mark the 10 cm mark ad the 11 cm mark the 15 cm mark ad the 17 cm mark the 20 cm mark ad the 23 cm mark s h a p e a d s p a c e ( M e a s u r e m e t ) 19

20 d) Have studets show these poits o their rulers: 20 mm 45 mm 85 mm 120 mm e) Tell studets that millimetres are used to measure the legths of small objects. Have them idetify objects that they would measure with this uit. f) Have studets complete the activity sheet (BLM 5.SS.2.2). Remid studets of the symbol for millimetre. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: make reasoable estimates read ad use their cm rulers correctly to determie the legths of objects record measuremets correctly (e.g., recorded measuremets iclude both a umber ad the uit) recogize whe it s appropriate to use mm as a uit of measure (Note: May of the objects that studets ame could also be measured i cm or m.) 20 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

21 provide a referet for oe millimetre ad explai the choice. provide a referet for oe cetimetre ad explai the choice. provide a referet for oe metre ad explai the choice. Materials: Metre sticks ad cm rulers Orgaizatio: Whole class/parters Procedure a) Discuss the importace of estimatio i measuremet. For example, talk about how good estimatig skills ca help a idividual recogize whe a error i measuremet is made ad the cosequeces of usig a icorrect measuremet. b) Have studets idetify examples of situatios whe estimatig the legths of objects would be beeficial. For example: We eed to wrap a gift. Do we have eough ribbo to wrap the package? We wat to put a ew shelvig uit i the room. Is the room high eough for the shelvig uit? We wat to store some books. Is the box we have wide eough? c) Ask studets to estimate the legth of the room to the earest metre. Record their resposes o the board. Have studets share their strategies for estimatig the legth of the room ad discuss why their estimates varied. d) Discuss the importace of persoal referets i the estimatig process. Explai that a persoal referet is a familiar object, oe that they see or use regularly whose measure is kow. They ca thik of this object whe they are estimatig the legth of a ukow object (e.g., the legth of a baseball bat is approximately 1 metre). Whe estimatig the legth of a ukow object, they ca thik of a bat ad visualize how may bats log the object is. e) Have the studets work with a parter to fid five commo objects that are approximately 1 mm i legth, width, or height 1 cm i legth, width, or height 1 m i legth, width, or height f) Have studets share their referets for each uit with the rest of the class. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: provide reasos why estimatig is a importat skill give examples of situatios i which estimatig would be beeficial idetify appropriate referets for 1 mm, 1 cm, ad 1 m s h a p e a d s p a c e ( M e a s u r e m e t ) 21

22 Materials: Decks of 20 cards (oe side of each card should have a letter o it; the other side should have a lie segmet draw o it [BLM 5.SS.2.3]), a aswer sheet listig the legth of the lie segmet o each card Orgaizatio: Pairs a) Tell studets that they will be playig a estimatig game called Metric 210 with their parter. Explai how the game is played. 1. Shuffle the cards ad lay them face dow o the playig surface. 2. Players take turs takig a card from the top of the pile util oe of them estimates that he or she has a total legth of 210 mm ad stops the game by sayig I have the lie. This player may get rid of ay oe card that pushes the total over 210 mm. The player the states a estimate for the total legth of the remaiig cards. 3. The player uses the aswer sheet to determie his or her score. The player s score is determied by addig the differece betwee the estimate ad the actual legth to the differece betwee the actual legth ad The wier is the player with the lowest total score after five rouds of the game. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. c) Vary the game by havig studets estimate the legths of the lie segmets i cm. A roud of the game is over whe a studet thiks he or she has reached a legth of 21 cm. Observatio Checklist Observe studets to determie whether they ca do the followig: uderstad the rules for playig the game give reasoable estimates of the legths of the lie segmets calculate the total legth of the lies correctly calculate their scores correctly 22 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

23 Materials: A deck of measuremet cards (BLM 5.SS.2.4), straight edges, a cetimetre ruler, ad a piece of paper for each player Orgaizatio: Pairs a) Tell studets that they will be playig a variatio of the game Metric 210. Explai how the ew versio of the game is played. 1. Shuffle the cards ad place them face dow o the playig surface. 2. Each card i the deck represets a mm legth. 3. The first player turs over a card ad uses a straight edge to draw a lie segmet he or she estimates to be the same legth as the umber of mm o the card (e.g., if the player turs over a 30, he or she draws, without measurig, a lie segmet that he or she thiks is 30 mm i legth ad records the legth above the lie segmet). 4. The secod player turs over a card ad uses a straight edge to draw a lie segmet he or she estimates to be the same legth as the umber of mm o the card. The secod player the records the legth above the lie segmet. 5. The first player turs over a lie card, ad exteds, without measurig, his or her lie segmet the umber of mm show o the card. The first player the records the legth above the lie segmet. For example, if the first player draws a 30 ad the a 50, his or her paper would look like this: 6. Play cotiues i this fashio util oe player has a lie segmet he or she estimates is 210 mm i legth ad stops the game by sayig, I have the lie. If the player thiks the last extesio of his or her lie segmet pushes the total legth beyod 210 mm, he or she ca state a estimate greater tha 210 mm. 7. The player measures the lie segmet to determie his or her score. The player s score is determied by addig the differece betwee the actual legth ad the estimated legth to the differece betwee the actual legth ad 210 mm. 8. The wier is the player who has the lowest score after five rouds of the game. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. c) Vary the game by chagig mm to cm (BLM 5.SS.2.4). A roud of the game is over whe a studet draws a lie segmet he or she estimates is 21 cm i legth. s h a p e a d s p a c e ( M e a s u r e m e t ) 23

24 Observatio Checklist Observe studets to determie whether they ca do the followig: uderstad the rules of the game make reasoable estimates determie the legth of the lie segmets by usig ad readig their rulers correctly record measuremets correctly (e.g., recorded measuremets iclude both the umber ad the uit) calculate their scores correctly Materials: Metre sticks, cm rulers, table to record their fidigs (BLM 5.SS.2.5) Orgaizatio: Small groups Procedure a) Tell studets that they will be playig a estimatig ad measurig game with the members of their group. Explai how to play the game. 1. Players take turs amig a uit of measure (m, cm, mm) ad a object that everyoe ca see. 2. Everyoe records a estimate of the legth of the object i the stated uit. 3. The player who amed the object measures it. The player whose estimate is the closest to the actual measuremet gets oe poit. If there is a tie, all players with the best estimate get oe poit. 4. The wier of the game is the first perso to get five poits. b) Have studets record i the table provided their choice of objects, their estimates, ad the actual measuremets (5.SS.2.5). c) Demostrate how the game is played ad aswer ay questio studets might have. Have studets play the game. Observatio Checklist Observe studets to determie whether they ca do the followig: uderstad the rules of the game determie the legths of objects by usig ad readig a metre stick or cm ruler correctly record measuremets correctly (e.g., recorded measuremets iclude both a umber ad the uit of measure) make reasoable estimates select a appropriate uit of measure 24 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

25 Show that 10 millimetres is equivalet to 1 cetimetre usig cocrete materials (e.g., ruler). Show that 1000 millimetres is equivalet to 1 metre usig cocrete materials (e.g., metre stick). Materials: cm rulers ad lie segmets (BLM 5.SS.2.6) Orgaizatio: Idividual a) Give studets a copy of BLM 5.SS.2.6, ad tell them that they should measure each lie segmet twice: The first time, they should measure the lie segmet to the earest cm; the secod time, they should measure the lie segmet to the earest mm. b) Ecourage studets to record their fidigs i the table provided i BLM 5.SS.2.6. c) Ask studets to study their tables ad record ay patters they see. d) Have studets share their fidigs with the other members of the class. Ecourage studets to state a rule that describes the relatioship betwee cm ad mm. e) Check studets uderstadig of the relatioship betwee cm ad mm by askig: How may mm are i 1 cm? 2 cm? 3 cm? 4 cm? 8 cm? 15 cm? 50 cm?? If the legth of a object is give i cm, how ca you fid how log it is i mm without measurig? If 1 cm = 10 mm, what part of a cm is 1 mm? 2 mm? 4 mm? 8 mm? 10 mm? If a object is 7 mm log, how log is it i cm? If a object is 35 mm log, how log is it i cm? If a object is 83 mm log, how log is it i cm? If the legth of a object is give i mm, how ca you fid how log it is i cm without measurig? f) Show studets how to record the relatioship betwee mm ad cm. 1 cm = 10 mm 1 mm = 0.1 cm g) Help studets uderstad the relatioship betwee m ad mm by askig: How may cm are i 1 metre? How may mm are i 1 cm? If there are 10 mm i 1 cm ad 100 cm i 1 metre, how may mm are i 1 metre? Have studets use a metre stick to show why their aswers are correct. s h a p e a d s p a c e ( M e a s u r e m e t ) 25

26 h) Check studets uderstadig of the relatioship betwee mm ad m by askig: How may mm are i 1 m? 2 m? 3 m? 5 m? 8 m? 10 m? If the legth of a object is give i m, how ca you fid how log it is i mm without measurig? If there are 1000 mm i oe m, what part of a m is 1 mm? 2 mm? 10 mm? 25 mm? 100 mm? If a object is 1000 mm log, how log is it i m? If a object is 3000 mm log, how log is it i m? If a object is 6000 mm log, how log is it i m? If the legth of a object is give i mm, how ca you fid how log it is i m without measurig? i) Show studets how to record the relatioship betwee m ad cm. 1 m = 1000 mm 1 mm = m Emphasize that milli meas thousadths so 1 mm meas 1 thousadth of a metre. Observatio Checklist Observe studets to determie whether they ca do the followig: fid the legths of lie segmets by usig ad readig their cm rulers correctly record measuremets correctly (e.g., recorded measuremets iclude both a umber ad the uit) recogize the relatioship betwee mm ad cm recogize the relatioship betwee mm ad m covert cm to mm ad vice versa covert mm to m ad vice versa 26 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

27 Materials: I have, who has? cards (BLM 5.SS.7) Orgaizatio: Whole class a) Tell studets that they will be playig a metric versio of the game I have, who has?. Explai that each studet will get oe card (some studets may get two cards if there are fewer tha 30 studets i the class). Oe studet will start the game by readig his or her card, ad the perso who has the aswer to the questio posed by this studet reads his or her card. Play cotiues i this fashio util it gets back to the perso who started the game. b) After the studets have played the game several times, have them make their ow metric coversio I have, who has? game ad play it with the other members of the class. Variatio: Have studets work i groups of 2 or 3, givig them several of the cards. Play the game as a class as you would i (a) ad (b) above. This gives studets the opportuity to egage with more tha oe card. Observatio Checklist Moitor studets resposes to determie whether they multiply or divide by tes, hudreds, or thousads add, subtract, multiply, or divide umbers other tha powers of 10 kow the relatioship betwee m, cm, ad mm Materials: cm rulers, stir sticks, books, erasers, soda cas, ad pecil cases Orgaizatio: Pairs/Whole class a) Preset the followig problem to studets: Marti drew a lie that was 64 mm log. His fried Zack measured the lie segmet ad said that it was 6.4 cm log. Is Zack right? How do you kow? b) Make sure studets uderstad the problem by askig: How log is the lie that Marti drew? What else do you kow? What do you eed to fid out? c) Have studets work with their parter to solve the problem. Whe they fiish, have them share their solutios with the other members of the class, ad discuss why the two measuremets are the same. Note: Some studets will be able to solve the problem by reasoig while others will eed to use their cm rulers ad draw the lie. s h a p e a d s p a c e ( M e a s u r e m e t ) 27

28 d) Check studets uderstadig of how to express measuremets to the earest teth of a cm by askig: What is 37 mm expressed i cm? What is 93 mm expressed i cm? What is 58 mm expressed i cm? What is 116 mm expressed i cm? Have studets use their cm rulers to justify their aswers. e) Have studets make ad complete the followig table: Object Estimated Legth Legth to earest teth of a cm Legth to earest mm A stir stick Thickess of a book A eraser Distace aroud a ca of soda Width of a pecil case f) Have studets express each of the followig measuremets i mm: 25.1 cm 85.6 cm 37.9 cm 12.2 cm Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: record a measuremet to the earest teth of a cm covert mm to the earest teth of a cm ad vice versa uderstad that 10 mm is the same as 1 cm read ad iterpret measuremets expressed i decimals make reasoable estimates 28 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

29 Cautio: I some commuities, playig cards are see as a form of gamblig ad discouraged. Please be aware of local sesitivities before itroducig this activity. Materials: Deck of cards for each group Orgaizatio: Small groups of 2 to 4 studets a) Tell studets they will be playig the Metric Covert game, ad explai how it is played. 1. Shuffle the cards ad place them face dow o the playig area. 2. The umbers o the cards represet mm. Let aces = 1 mm, jacks = 11 mm, quees = 12 mm, ad kigs = 13 mm. 3. Oe player turs over a card ad places it i the cetre of the playig area so everyoe ca see it. 4. The first player to covert mm to cm correctly takes the card (e.g., if a 8 is tured over, the first player to say 0.8 [8 teths] cm wis the card). 5. If there is a tie or a error is made, the card is put back ito the deck ad the cards are reshuffled. 6. The perso who wis the cards turs over the ext card. 7. The game proceeds i this fashio util there are o cards. 8. The perso with the most cards is the wier. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. c) Vary the game by havig the studets covert from cm to mm m to cm m to mm mm to m Observatio Checklist Observe studets to determie whether they kow the relatioships betwee mm, cm, ad m calculate correctly s h a p e a d s p a c e ( M e a s u r e m e t ) 29

30 N o t e s 30 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

31 Grade 5: Shape ad Space (Measuremet) (5.SS.3) Edurig Uderstadigs: all measuremets are comparisos. Legth, area, volume, capacity, ad mass are measurable properties of objects. the uit of measure must be of the same ature as the property of the object beig measured. Geeral Outcome: Use direct or idirect measuremet to solve problems. SpEcific LEAriG OUtcOME(S): AchiEvEMEt idicators: 5.SS.3 Demostrate a uderstadig of volume by selectig ad justifyig referets for cm 3 or m 3 uits estimatig volume by usig referets for cm 3 ad m 3 measurig ad recordig volume (cm 3 or m 3 ) costructig rectagular prisms for a give volume [C, CN, ME, PS, R, V] Idetify the cube as the most efficiet uit for measurig volume ad explai why. Provide a referet for a cubic cetimetre ad explai the choice. Provide a referet for a cubic metre ad explai the choice. Determie which stadard cubic uit is represeted by a give referet. Estimate the volume of a 3-D object usig persoal referets. Determie the volume of a 3-D object usig maipulatives ad explai the strategy. Costruct a rectagular prism for a give volume. Explai that may rectagular prisms are possible for a give volume by costructig more tha oe rectagular prism for the same volume. s h a p e a d s p a c e ( M e a s u r e m e t ) 31

32 Prior Kowledge Studets may have had experiece with the followig: Usig direct compariso to compare the volume of two objects Idetifyig attributes of objects that ca be compared Demostratig a uderstadig of measuremet as a process of comparig by fillig Describig ad costructig rectagular prisms Measurig the legths of objects i m or cm BacKgroud iformatio The terms volume ad capacity are ofte used iterchageably. For the purposes of the learig experieces i this sectio ad the sectio that follows, a distictio will be made. Volume is the amout of space a object occupies or, if the object is hollow, the amout of space iside the object. Volume is measured i cubic cetimetres (cm 3 ) or cubic metres (m 3 ). Capacity is the maximum amout of liquid a cotaier ca hold. Capacity is measured i litres (L) ad millilitres (ml). mathematical laguage Cubic uit (cetimetre ad metre) Dimesio Rectagular prism Legth (width, height) Less (least) volume More (greatest) volume Same volume Volume 32 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

33 learig experieces idetify the cube as the most efficiet uit for measurig volume ad explai why. determie the volume of a 3-d object usig maipulatives ad explai the strategy. Materials: A variety of small boxes, cubes, marbles, other 3-D shapes such as a triagular prism or pyramid, ad sad Orgaizatio: Whole class/small group a) Show studets the isides of two empty boxes. Ask, Which box has more space iside? How ca we tell for sure? b) Explai that volume is the amout of space iside a cotaier or the umber of uits eeded to fill the cotaier. Ask, What uit do you thik we should use to measure volume? c) Give each group a box ad three possible uits: marbles, cubes, ad triagular prisms (or ay other shape). Tell studets that their task is to determie which uit is best for measurig volume. Explai that they will be measurig the volume of their box three times. Each time they will completely fill the box with oe of the uits. Explai that whe fillig the box they should lay the uits carefully o the bottom of the box, record the umber used, ad the fill the box layer by layer. Ask studets to record the total umber of uits used as well as their observatios o the appropriateess of the uit. d) Have studets share their observatios about the differet uits. Help them recogize that the cube is the best uit to use because it is easy to stack ad there are o gaps or overlaps whe fillig the cotaiers (e.g., have studets pour sad ito a box filled with marbles to show them that there are gaps betwee the marbles). Observatio Checklist Observe studets to determie whether they ca do the followig: measure the volume of the box correctly (e.g., completely fill the box with a uit ad cout the umber of uits used) record both the umber ad the uit of measure recogize that the cube is the most efficiet uit for measurig volume ad explai why s h a p e a d s p a c e ( M e a s u r e m e t ) 33

34 determie the volume of a 3-d object usig maipulatives ad explai the strategy. Materials: Small boxes ad cubes Orgaizatio: Pairs or small groups a) Give each group four or five boxes. Have studets label the boxes A, B, C, D. b) Have studets look at the labelled boxes, ad decide which oe they thik has the smallest volume ad which oe has the largest volume. Ask them to put the boxes i order from the smallest volume to the largest volume ad to record the order they have decided o. c) Have studets measure the volume of each box to the earest whole uit ad record their measuremets i a table like the oe show below. Have studets record the actual volume of the cotaiers ad compare it with their estimated volume. Box Estimated volume Actual volume A B C D d) Have each group share its fidigs with the rest of the class. Ecourage studets to discuss the strategies they used to determie the volume of the boxes, particularly whe the umber of cubes was ot a exact fit (e.g., whe there was some space aroud the layers). Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: compare ad order cotaiers accordig to their volume make reasoable estimates of the volume of cotaiers determie the volume of a object usig maipulatives ad explai the strategy record measuremets that iclude both a umber ad the uit use the terms more (greatest) volume, less (least) volume, ad the same volume correctly 34 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

35 idetify the cube as the most efficiet uit for measurig volume ad explai why. provide a referet for a cubic cetimetre ad explai the choice. Materials: Differet sizes of boxes, differet sizes of cubes, a white rod from a set of Cuiseaire rods, ad ceticubes Orgaizatio: Small groups a) Preset studets with the followig situatio: A box Nicky measured has a volume of 18 cubic uits. She gave the box to Cathy. Whe Cathy measured the volume of the box, she foud it had a volume of 26 cubic uits. Could both girls measuremets be correct? Why or why ot? b) Ecourage studets to devise ad carry out a pla to prove their assertios about the situatio. c) Have studets share their results ad reasoig with the other members of the class. Ecourage studets to discuss the eed for a stadard uit of measure ad the reasos why it is importat to use commo uits (e.g., to facilitate commuicatios, busiess, ad trade). d) Show studets a white rod from the set of Cuiseaire rods or a ceticube, ad explai that i the metric system a cubic cetimetre is oe of the uits used to determie the volume of a object. Show studets the word ad the symbol for the uit. e) Tell studets that they will be usig the white rods (or ceticubes) to complete the followig activity: 1. Fid a cotaier that has a volume that is greater tha 80 cm 3 less tha 40 cm 3 betwee 50 ad 60 cm 3 2. Fid as may objects as you ca that have a volume of 1 cm 3. f) Have studets share their fidigs with the rest of the class. Ecourage studets to discuss the strategies they used to fid the cotaiers ad the objects they foud that are approximately 1 cm 3. Start a class list of objects that have a volume of 1 cm 3. Ecourage studets to look outside the classroom for objects that have a volume of approximately 1 cm 3 ad add them to list. s h a p e a d s p a c e ( M e a s u r e m e t ) 35

36 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize the eed for a stadard uit determie the volume of a object usig maipulatives ad explai the strategy idetify objects that have a volume of approximately 1 cm 3 Estimate the volume of a 3-d object usig persoal referets. determie the volume of a 3-d object usig maipulatives ad explai the strategy. costruct a rectagular prism for a give volume. Explai that may rectagular prisms are possible for a give volume by costructig more tha oe rectagular prism for the same volume. Materials: Cetimetre grid paper (BLM 5 8.9), ceticubes, scissors; tape, copies of the istructios for the activity (BLM 5.SS.3.1), ad observatio form (BLM 5 8.1) Orgaizatio: Small groups Procedures: a) Tell studets that they will be usig BLM 5.SS.3.1 to complete a ivestigatio ivolvig volume. b) Help studets determie what should be icluded i their reports ad the criteria for evaluatig them. Ecourage studets to cosider such thigs as the accuracy of their measuremets, the strategies they used to determie the volumes of the ope boxes, ad the clarity of their explaatio of what happes to the volume as the dimesios of the ope boxes chage. Observatio Checklist Use the observatio form (BLM 5 8.1) to observe how well studets work together. 36 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

37 Estimate the volume of a 3-d object usig persoal referets. determie the volume of a 3-d object usig maipulatives ad explai the strategy. Material: Ceticubes, small boxes such as a shoebox or a cereal box, a list of the volumes of the boxes, ad math jourals Orgaizatio: Small groups/whole class a) Give each group four or five small boxes ad oly eough ceticubes to cover the bottom of each box separately, plus eough to make oe stack the height of each box. b) Ask studets to estimate the volume of each box, ad to record their estimates i their math jourals. Explai that there are ot eough ceticubes to completely fill ay box; however, they ca use the ceticubes that they have bee give to help them make their estimates. Whe they fiish estimatig the volumes of the boxes, they should compare their estimates with the list of the volumes of the boxes that you have prepared. c) Have studets share the strategies they used to estimate the volumes of each box. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: explai the strategy they used to determie the volume of the boxes use a referet to make reasoable estimates of the volume of the boxes costruct a rectagular prism for a give volume. Explai that may rectagular prisms are possible for a give volume by costructig more tha oe rectagular prism for the same volume. Materials: Ceticubes, or the white rods from a set of Cuiseaire rods, or multilik cubes Orgaizatio: Pairs a) Tell studets that they will be makig rectagular prisms with their ceticubes ad determiig their volumes. Show studets a rectagular prism made with 10 cubes. Ask studets what the volume of the prism is, ad how they kow. s h a p e a d s p a c e ( M e a s u r e m e t ) 37

38 b) Have studets complete the followig activity: 1. Costruct a rectagular prism with a volume of 12 cm 3 16 cm 3 Record the dimesios of the prisms you made. 2. Build two rectagular prisms, side by side, so that oe prism has a volume of 6 cm 3 more tha aother. Record the dimesios of each prism. 3. Make two rectagular prisms with the same legth, with oe wider ad shorter tha the other, but with differet volumes. Record the dimesios of each prism. 4. Make two rectagular prisms with the same legth, with oe wider ad shorter tha the other, but with the same volume. Record the dimesios of each prism. 5. Make as may rectagular prisms as you ca that have a volume of 24 cm 3. Record the dimesios of each prism that you make. 6. Make three rectagular prisms with the followig dimesios: 6 cm x 6 cm x 6 cm 3 cm x 12 cm x 6 cm 3 cm x 9 cm x 8 cm Fid the volume of each prism. Record the dimesios of each prism ad its volume. 7. Thik about the rectagular prisms that you made. What ca you coclude about the volume of prisms? Record your observatios. Whe studets fiish each part of the activity, have them share their results with the other members of the class. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: costruct a rectagular prism with a give volume recogize that differet rectagular prisms are possible for a give volume recogize that rectagular prisms with differet dimesios ca have the same volume recogize that if oe dimesio of two rectagular prisms is the same, the volume of the prisms is ot ecessarily the same recogize that the volume of a rectagular prism is depedet o its dimesios 38 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

39 provide a referet for a cubic metre ad explai the choice. determie which stadard cubic uit is represeted for a give referet. Materials: Metre sticks, plasticie, cardboard, scissors, ad tape Orgaizatio: Small groups/whole class procedure: a) Ask studets to thik of cotaiers they see iside ad outside of school whose volume should be measured i cm 3. Keep a list of their suggestios. Fiish the discussio by askig, Are there ay cotaiers or objects that are too large to be measured with a cm 3? b) Have each group develop a list of cotaiers or objects that would require a larger uit of measure. Whe studets fiish, have them share their list with the other members of the class. c) Explai that i the metric system the volumes of very large items or cotaiers are measured i cubic metres. Ask studets to show with their hads how large they thik a cubic metre is. d) Have each group make a model of a cubic metre. Some groups ca make their cubic metre usig 12 metre sticks (or woode dowels 1 metre i legth) joied with plasticie or maskig tape, while other groups ca draw ad cut out six 1 metre squares from heavy cardboard ad joi the squares with maskig tape. e) Have studets estimate how may studets they thik will fit ito a cubic metre. Have them try it out ad the discuss how their estimates compared with the actual umber ad the reasos why they may vary. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify cotaiers or objects whose volumes should be measured i cubic cetimetres idetify cotaiers or objects whose volumes should be measured with larger uits make a cubic metre s h a p e a d s p a c e ( M e a s u r e m e t ) 39

40 provide a referet for a cubic metre ad explai the choice. determie which stadard cubic uit is represeted for a give referet. Estimate the volume of a 3-d object usig maipulatives ad explai the strategy. Materials: Models of cubic metres Orgaizatio: Whole class/small groups a) Have studets refer to their models of a cubic metre to estimate whether the followig objects have a volume greater tha, less tha, or about the same as a cubic metre: their desk a pop machie a filig cabiet a garbage ca a dump truck a stove Ecourage studets to explai their reasos for their estimates. b) Ask each group to make a list of items iside ad outside of the classroom whose volume could be measured i cubic metres. Have the groups share their lists ad explai the reasos for their choices. c) Have each group refer to its model of a cubic metre to estimate the volume of their classroom the school gym the pricipal s office Have the groups share their estimates ad the strategies they used to determie the volumes of the rooms. d) Have studets discuss the questio: Does a cubic metre have to be a cube? Note: Studets should recogize from the previous activity that a cubic metre does ot eed to be a cube sice prisms with differet dimesios ca have the same volume. e) Have studets collect ad fill oe of the cardboard cubic metres they made with a item they would like to give to charity (e.g., studets could give a cubic metre of clothes that they have outgrow). Have studets write a letter to the commuity ad other classes i the school explaiig what they are doig ad ivitig them to help collect the item they have chose. 40 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

41 Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: idetify objects or cotaiers whose volume could be measured i cubic metres give reasoable estimates of cotaiers or items whose volume could be measured i cubic cetimetres explai the strategies they used to estimate the volumes of large cotaiers ad objects s h a p e a d s p a c e ( M e a s u r e m e t ) 41

42 N o t e s 42 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

43 Grade 5: Shape ad Space (Measuremet) (5.SS.4) Edurig Uderstadigs: all measuremets are comparisos. Legth, area, volume, capacity, ad mass are measurable properties of objects. the uit of measure must be of the same ature as the property of the object beig measured. Geeral Outcome: Use direct or idirect measuremet to solve problems. SpEcific LEAriG OUtcOME(S): AchiEvEMEt idicators: 5.SS.4 Demostrate a uderstadig of capacity by describig the relatioship betwee ml ad L selectig ad justifyig referets for ml or L uits estimatig capacity by usig referets for ml or L measurig ad recordig capacity (ml or L) [C, CN, ME, PS, R, V] Demostrate that 1000 millilitres is equivalet to 1 litre by fillig a 1-litre cotaier usig a combiatio of smaller cotaiers. Provide a referet for a litre ad explai the choice. Provide a referet for a millilitre ad explai the choice. Determie which capacity uit (ml or L) is represeted by a give referet. Estimate the capacity of a cotaier usig persoal referets. Determie the capacity of a cotaier usig materials that take the shape of the iside of the cotaier (e.g., a liquid, rice, sad, beads), ad explai the strategy. s h a p e a d s p a c e ( M e a s u r e m e t ) 43

44 Prior Kowledge Studets may have had experiece with the followig: Idetifyig attributes of objects that ca be measured Usig direct compariso to compare the capacity of two objects Demostratig a uderstadig of measuremet as a process of comparig by fillig Demostratig a uderstadig of whole umbers less tha Demostratig a uderstadig of additio ad subtractio of whole umbers with aswers less tha related Kowledge Studets should be itroduced to the followig: Providig a referet for oe millimetre, oe cetimetre, ad oe metre BacKgroud iformatio The terms volume ad capacity are ofte used iterchageably. For the purposes of the learig experieces i this sectio ad the previous sectio, a distictio will be made. Volume is the amout of space a object occupies or, if the object is hollow, the amout of space iside the object. Volume is measured i cubic cetimetres (cm 3 ) or cubic metres (m 3 ). Capacity is the maximum amout of liquid a cotaier ca hold. Capacity is measured i litres (L) ad millilitres (ml). mathematical laguage Capacity More capacity Less capacity Same capacity Estimate Litre Referet Millilitre 44 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

45 learig experieces determie the capacity of a cotaier usig materials that take the shape of the iside of the cotaier (e.g., a liquid, rice, sad, beads), ad explai the strategy. Materials: A variety of cotaiers (some of which should be trasparet), fuels, water, sad (or ay other material that will take the shape of cotaiers), paper towels, spoges, ad markers Orgaizatio: Whole class/small groups a) Explai that we ofte hear expressios, such as the followig: The room was filled to capacity. They played to a capacity crowd. Ask, What does the word capacity mea? How ca we fid the capacity of a object? b) Explai that i math we use the term capacity to describe how much liquid a cotaier ca hold, ad to determie the capacity of a cotaier we eed a uit of measure. c) Show studets a trasparet cotaier. Show studets how to measure the capacity of the cotaier by usig aother smaller trasparet cotaier as the uit of measure. Repeat this activity two or three times to make sure studets uderstad how to measure the capacity of a cotaier. d) Give each group four or five cotaiers. Have studets select oe of their cotaiers to be the uit of measure ad label the other cotaiers A, B, C, D. e) Have studets look at the labelled cotaiers ad decide which oe they thik has the smallest capacity ad which oe has the largest capacity. Ask them to put the cotaiers i order from the smallest capacity to the largest capacity, ad to record the order they have decided o. f) Have studets give their uit a ame. Have them measure each cotaier ad record their measuremets i a table like the oe show below. Have studets record the actual order of the cotaiers, ad compare it with their estimated order. cotaier Estimated capacity Actual capacity A B C D s h a p e a d s p a c e ( M e a s u r e m e t ) 45

46 g) Have each group share its fidigs with the rest of the class. Ecourage them to describe how the real order of the cotaiers compared with their estimated order. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: use the terms more capacity, less capacity, ad the same capacity correctly measure correctly (studets completely fill the uit over ad over agai util the cotaier beig measured is full) record their measuremets correctly (icludes both a umber ad the uit) give reasoable estimates of capacity compare ad order cotaiers accordig to their capacities Materials: A wide variety of cotaiers, maskig tape, spoos, large plastic glasses or jars, water, markers, paper towels, spoges, ad procedure steps (BLM 5.SS.4.1) Orgaizatio: Pairs a) Have studets use the followig procedure to make their ow measurig device: 1. Put a piece of maskig tape dow the side of a glass (or jar). 2. Fill a small cotaier with spoofuls of water, keepig track of the umber of spoofuls eeded to fill it. 3. Empty the water i the small cotaier ito the glass. 4. Mark the level of the water ad the umber of spoofuls o the tape. 5. Fill the small cotaier agai. Empty the water ito the glass. Mark the level of the water ad the total umber of spoofuls. 6. Cotiue fillig ad markig the glass util the top is reached. b) Make sure the studets kow how to read ad use their measurig device. Have them use their measurig device to fid the capacity of five differet cotaiers i two differet ways. Have the studets record their fidigs i the chart provided i BLM 5.SS.4.1. c) Have studets write a paragraph describig the two differet ways they foud the capacity of their cotaiers. Have studets share ad discuss their methods with the other members of the class. d) Have studets exchage their cotaiers with aother group. Have them use their measurig device to determie the capacity of these cotaiers ad record their fidigs i a table. 46 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

47 Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: use the terms more capacity, less capacity, ad the same capacity correctly determie the capacity of a cotaier by fillig it with water usig their measurig device determie the capacity of a cotaier by fillig it, ad the pourig its cotets ito the measurig device to see how much it holds measure correctly (e.g., completely fill the cotaier) read their measurig devices correctly record their measuremets correctly (iclude both the umber ad the uit) Estimate the capacity of a cotaier usig persoal referets. determie the capacity of a cotaier usig materials that take the shape of the iside of the cotaier (e.g., a liquid, rice, sad, beads), ad explai the strategy. Materials: Cotaiers (some cotaiers should be greater tha a litre, less tha a litre, ad equal to a litre), water, sad (ad other material that takes the shape of a cotaier), studet-made measurig devices, litre measurig devices, maskig tape, ad markers Orgaizatio: Small groups a) Give each group the same two cotaiers. Have some of the groups use their measurig devices to determie the capacity of the cotaiers. Have other groups select aother cotaier to be their uit. Have these groups ame their uit ad fid the capacity of their cotaiers. b) Have studets share their measuremets. List their measuremets o the board ad ask why they differ. Ask studets what they could do so everyoe would get the same measuremet. Help studets recogize the eed for a stadard uit of measure ad the reasos why it s importat to use stadard uits (e.g., the use of stadard uits facilitates busiess ad trade). c) Tell studets that i the metric system the litre is the stadard uit of measure for capacity. Show studets a umarked litre cotaier ad tell them that a litre is the amout of the liquid it ca hold. Also, show studets how to write the word ad the symbol for the uit. s h a p e a d s p a c e ( M e a s u r e m e t ) 47

48 d) Give each group five or six cotaiers. Have the studets label the cotaiers from A to F ad the make a list i their math joural of the cotaiers they thik are less tha a litre, the same as a litre, ad larger tha a litre. e) Have studets use the umarked litre cotaiers to measure the capacity of each cotaier. Explai that they should ot fill ay cotaier higher tha the bottom part of the eck of the cotaier. Ask studets to write the letter of each cotaier i their math joural, ad record whether its capacity is greater tha a litre, less tha a litre, or the same as a litre. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: determie the capacity of a cotaier usig materials that take the shape of the cotaier measure correctly (e.g., fill the litre-measurig cotaier ad the cotaiers they are measurig to the right levels) record measuremets properly (e.g., use the correct symbol for a litre) make reasoable estimates of capacity Materials: Litre-measurig cotaiers, a variety of cotaiers, water, sad, or ay other material that takes the shape of a cotaier, a pitcher, a water pail, ad a wastepaper basket Orgaizatio: Whole class/small groups a) Have studets provide examples of whe they would eed to estimate the capacity of a cotaier, ad discuss how they ca esure that their estimates are reasoable. b) Ask each group to fid two commo cotaiers they ca use as a referet for a litre. c) Have the groups share their referets with each other ad keep a class list of referets for a litre. d) Show studets a large pitcher, a wastepaper basket, ad a empty water pail. Ask them to thik of their referet ad the estimate the capacity of each cotaier. Have the studets check their estimates by measurig each item. e) Ask studets to estimate the capacity of a bathtub. Help them devise ad carry out a pla to check their estimates. 48 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

49 Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: provide a referet for a litre ad explai their choice make reasoable estimates of the capacities provide a referet for a millilitre ad explai the choice. Estimate the capacity of a cotaier usig persoal referets. determie the capacity of a cotaier usig materials that take the shape of the iside of the cotaier (e.g., a liquid, rice, sad, beads), ad explai the strategy. Materials: Beakers calibrated i ml, graduated cyliders calibrated i ml, a eyedropper, baby food jars, ti cas, small milk cartos, small soda cas, pickle jars, ketchup bottles, water, paper towels, fuels, ad spoges, capacity table (BLM 5.SS.4.1) Orgaizatio: Whole class/small group a) Show studets a small cotaier, such as a empty tua ca or empty baby food jar, ad ask them how they could fid the capacity of the cotaier. b) Explai that to fid the capacity of smaller cotaiers, we eed a ew uit of measure. The uit that is commoly used is the millilitre. Tell studets that the millilitre is a very small uit about the size of a drop from a eyedropper. Fill a eyedropper with water ad show studets several drops so they ca begi to coceptualize how large the uit is. c) Explai that because the uit is so small, we ofte use measurig devices that are marked off i millilitres. Show studets differet measurig devices that are calibrated i ml, ad explai how they should use them to fid the capacity of a cotaier. d) Have studets measure the capacity of each object listed below i two differet ways ad record their results i the table from 5.SS.4.1. s h a p e a d s p a c e ( M e a s u r e m e t ) 49

50 demostrate that 1000 millilitres is equivalet to 1 litre by fillig a 1-litre cotaier usig a combiatio of smaller cotaiers. Materials: A 500 ml beaker, a 250 ml beaker, a 100 ml beaker, ad a 50 ml beaker; umarked litre cotaiers, water, fuels, paper towels, ad math jourals Orgaizatio: Small groups/whole class a) Show studets the litre cotaier ad tell them that their job is to determie the umber of ml i a litre. b) Have studets estimate the umber of 50 ml beakers of water it will take to fill the litre cotaier. Have them record their estimates i a table like the oe show below. Beaker Estimated umber of Beakers Actual umber of Beakers total umber of ml 50 ml 100 ml 250 ml 500 ml c) Have studets check their estimates by fillig the litre cotaier with 50 ml beakers of water ad record their results i the table. d) Repeat the activity usig the 100 ml beaker, the 250 ml beaker, ad the 500 ml beaker. e) Have studets compare their results with aother group. Ask them what they ca coclude about the relatioship betwee a ml ad a litre. f) Give studets the followig problem ad have them record their solutio i their math jourals: Jessi has a cotaier that holds 1425 ml of liquid. Is Jessi s cotaier smaller tha or larger tha a litre? How do you kow? How much larger or smaller tha a litre is Jessi s cotaier? 50 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

51 Observatio Checklist Observe studets resposes to determie whether they ca do the followig: demostrate that there are 1000 ml i a litre usig a variety of smaller cotaiers measure correctly (fill the beakers properly) record the measuremets correctly solve a problem ivolvig the relatioship betwee millilitres ad litres Materials: Cards umbered from 0 to 9 (BLM 5 8.5), paper ad pecil Orgaizatio: Small groups/whole class a) Tell studets that they will be playig a game ivolvig the relatioship betwee a litre ad a millilitre. Explai how the game is played 1. Players should make the followig grid o their papers: 2. Shuffle the cards ad place them face dow o the playig area. 3. Tur over oe card. Players decide where they wat to write that umber o their grid. Oce a umber has bee placed o the grid, it caot be chaged. 4. The ext card is tured over ad the players ow place this umber o their grids. Play cotiues util six umbers have bee tured over ad each player has placed the umbers o his or her grid. 5. The players add the millilitre quatities ad the player or players closest to 1 litre receive oe poit. 6. Reshuffle the cards, make ew grids, ad play the game agai. 7. Cotiue playig the game. The first player to reach 10 poits is the wier. b) Demostrate how the game is played ad aswer ay questios studets might have. Have studets play the game. c) Vary the game so that the player with the sum closest to 500 ml wis a poit. s h a p e a d s p a c e ( M e a s u r e m e t ) 51

52 Observatio Checklist Observe studets to determie whether they kow the relatioship betwee ml ad litres calculate correctly Puttig the Pieces together Plaig a healthy Meal Purpose: The purpose of this ivestigatio is to have studets apply their kowledge of capacity to a real-world situatio. I particular, it is desiged to reiforce studets abilities to measure the capacity of cotaiers estimate the capacity of cotaiers record the capacity of cotaiers The ivestigatio is also desiged to ehace studets abilities to commuicate mathematically solve problems reaso mathematically coect mathematics to real-world situatios ad other subject areas (PE/HE) Materials/Resources Cetimetre measurig cubes Assorted cotaiers Food groups guide (ca be foud o the Iteret) Water, sad, or other material that takes the shape of a cotaier Cyliders or beakers calibrated i ml Paper towels Markers Orgaizatio: Whole class/small groups 52 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

53 a) Tell studets that each group will be resposible for plaig a healthy breakfast or luch. Sice the capacity of the huma stomach is approximately 1 litre, the meal they plaed should ot cotai more tha 800 ml of food. I plaig their meal, they should use the food guide to help them select foods from each food group iclude foods that are available locally idicate the quatity of each food i ml b) Have studets desig their meals. Whe they fiish plaig their meal, have them fid a cotaier with the same capacity as each item o their meu. Have studets label each cotaier by idicatig the item of food it represets ad its capacity. c) Have each group display its meu ad correspodig cotaiers. Have studets explai why their meals are utritious ad how the capacities of the differet items add up to a 800 ml meal. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: pla a meal that meets the criteria specified i part (a) make reasoable estimates of the capacities of cotaiers measure the capacity of cotaiers correctly record the capacity of cotaiers correctly s h a p e a d s p a c e ( M e a s u r e m e t ) 53

54 N o t e s 54 G r a d e 5 M a t h e m a t i c s : s u p p o r t d o c u m e t f o r t e a c h e r s

55 Grade 5: Shape ad Space (3-D Objects ad 2-D Shapes) (5.SS.5) Edurig Uderstadigs: Shapes are distiguished by their properties. Geeral Outcome: Describe the characteristics of 3-D objects ad 2-D shapes, ad aalyze the relatioship betwee them. SpEcific LEAriG OUtcOmE(S): AchiEvEmEt idicators: 5.SS.5 Describe ad provide examples of edges ad faces of 3-D objects, ad sides of 2-D shapes, that are parallel itersectig perpedicular vertical horizotal [C, CN, R, T, V] Idetify parallel, itersectig, perpedicular, vertical, ad horizotal edges ad faces o 3-D objects. Idetify parallel, itersectig, perpedicular, vertical, ad horizotal sides o 2-D shapes. Provide examples from the eviromet that show parallel, itersectig, perpedicular, vertical, ad horizotal lie segmets. Fid examples of edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, ad horizotal i prit ad electroic media, such as ewspapers, magazies, ad the Iteret. Draw 2-D shapes or 3-D objects that have edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, or horizotal. Describe the faces ad edges of a give 3-D object usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. Describe the sides of a 2-D shape usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 55

56 Prior Kowledge Studets may have had experiece with the followig: Idetifyig cubes, spheres, coes, cyliders, pyramids, triagular prisms, ad rectagular prisms Idetifyig triagles, squares, rectagles, ad circles Idetifyig the faces, edges, ad vertices of 3-D objects Sortig regular ad irregular polygos icludig triagles, quadrilaterals, petagos, hexagos, ad octagos accordig to the umber of sides related Kowledge Studets should be itroduced to the followig: Idetifyig ad sortig quadrilaterals BacKgroud iformatio Poits, lies, ad plaes are the buildig blocks of geometry. These cocepts are udefied ad, like umber, they are abstractios that caot be see or touched. Studets uderstadig of these cocepts evolves from their experieces with physical objects (e.g., the tip of a pecil, the corer of a table or block, ad the dot draw o a piece of paper suggest the idea of a poit to studets). Lies are sometimes described as a set of poits extedig edlessly i two directios. They have legth but o other dimesio. Physical models, such as a rope stretched out, a wire held taut, ad the cetre lie o a highway, ca help studets develop a uderstadig of this cocept. A lie segmet is part of a lie. It cosists of two edpoits ad all the poits betwee them. Examples of lie segmets iclude the rugs of a ladder, the edges of a box, ad the bars i a grill. A plae is two-dimesioal. Ay smooth, flat surface, such as a tabletop, a floor, or a ceilig, ca be thought of as a plae. However, each of these models is oly a part of a plae because a plae exteds ifiitely i two directios. Two lies i a plae ca itersect or be parallel to each other. Itersectig lies have oe poit i commo; parallel lies have o poits i commo. The distace betwee them is the same everywhere. Sometimes lies itersect at right agles. These lies are perpedicular. Because studets are ot itroduced to agles util Grade 6, perpedicular lies are described as two lies that form square corers. I additio, lies ca be either horizotal or vertical. A horizotal lie is a lie that is parallel to the horizo. A vertical lie is a lie that is at right agles to the horizo. Studets usually describe horizotal lies as goig across, ad vertical lies as goig up ad dow. However, their perceptio of whether a lie is horizotal or vertical might differ accordig to their perspective. 56 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

57 Whe describig the edges of prisms, some studets may thik that ay two lie segmets that do ot itersect are parallel. For example, cosider the cube show below: Some studets may thik the two dark edges are parallel sice they do ot itersect. However, these edges lie i differet plaes ad therefore are ot parallel. mathematical laguage Coe Cube Cylider Edge Face Horizotal lie Itersectig lie Lie Lie segmet Parallel lies Perpedicular lies Pyramid Rectagular prism Sphere Triagular prism Vertex (Vertices) Vertical lie S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 57

58 learig experieces Assessig Prior Kowledge Materials: A set of 3-D objects that icludes a coe, sphere, cylider, a pyramid, a cube, a rectagular prism, ad a triagular prism Orgaizatio: Whole class/idividual a) Put the 3-D objects i a place where all studets ca see them. Tell the studets that you will be askig them some questios about the shapes to fid out what they already kow about them. b) Tell studets they ca look at the shapes to help them idetify the 3-D objects or parts of objects that fit the followig clues: 1. I have six faces all the same size ad shape. (cube) 2. I am formed by the itersectio of two faces. (edge) 3. Two of my faces are circular. (cylider) 4. I am the poit where three or more edges meet. (vertex) 5. I have six rectagular faces. (rectagular prism) 6. I have o flat faces. (sphere) 7. My shape is foud o every pyramid. (triagle) 8. We are the faces foud o a triagular prism. (triagle ad rectagle) 9. I am oe face of a coe. (circle) Observatio Checklist Use studets resposes to the questios to determie whether further review o the idetificatio ad characteristics of 3-D objects ad 2-D shapes is eeded. 58 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

59 provide examples from the eviromet that show parallel, itersectig, perpedicular, vertical, ad horizotal lie segmets. Materials: Copies of the cocept descriptio sheet (BLM 5 8.2). Orgaizatio: Idividual/Whole class a) Tell studets that i the ext few lessos they will be learig about lies ad today they will be discussig what a lie is. Before begiig the discussio, you wat them to write dow what they already kow about lies. b) Have studets complete the cocept descriptio sheet. Let studets kow that it is alright if they caot thik of aythig to put i a sectio. They will have aother opportuity to complete the sheet whe they lear more about lies. c) Whe studets fiish, begi a discussio by askig, What is a lie? What are some examples of lies? As the discussio progresses, clear up ay miscoceptios studets may have about lies ad make sure they see a variety of examples ad o-examples. d) Have studets add to their cocept descriptio. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: describe the characteristics of a lie (e.g., it cotiues idefiitely i two directios) idetify examples of a lie idetify o-examples of a lie S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 59

60 Material: A log rope or a skei of yar Orgaizatio: Whole class Note: This activity could be doe i the gym or outside. a) Take studets outside to the playgroud. Stretch the rope across the playgroud. Have studets hold oto the rope ad hold it taut. Tell studets that the rope represets a lie that keeps goig forever (e.g., if they were to tie aother piece o the ed ad pull it taut, the lie would cotiue). Have studets discuss what thigs the lie would go through as it goes beyod each ed of the rope. Discuss how they thik they could show that the rope/yar would cotiue o. b) Tell studets that everyoe holdig oto the lie is a poit ad the space betwee each pair of them is a lie segmet or part of a lie. Name the lie segmets usig the studets ames (e.g., lie segmet Jack ad Josie). Call out the ames of several lie segmets. Each time you call out a lie segmet, have the amed studets hold o to the rope ad the studets betwee them let go to show how log the lie segmet is. Have studets take turs amig lie segmets. c) Ask studets to remember who is stadig o either side of them. Retur to the classroom ad draw a arrow o the chalkboard. Put a arrow o either ed to idicate that the lie goes o forever. Idicate poits o the lie ad write the ames of the studets uder them. d) Have studets discuss the differeces betwee their experieces outdoors ad the ideas represeted by the lie o the board. For example, studets should ote that the arrows o the lie idicate that the lie goes o forever, ad the poits where studets ames appear show that a lie segmet has defiite eds. e) Explai that i math we use a double arrow to idicate a lie ad capital letters istead of studets ames to idicate poits o the lie. Draw aother lie o the board. Name several lie segmets ad have studets idetify where they are o the lie. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: recogize that a lie exteds ifiitely i two directios recogize that a lie segmet is part of a lie ame lie segmets correctly 60 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

61 idetify parallel, itersectig, perpedicular, vertical, ad horizotal sides o 2-D shapes. provide examples from the eviromet that show parallel, itersectig, perpedicular, vertical, ad horizotal lie segmets. Draw 2-D shapes or 3-D objects that have edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, or horizotal. Describe the faces ad edges of a 3-D object usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. Materials: Stir sticks or toothpicks, a mat or a rug Orgaizatio: Whole class/small group a) Have two or three studets come to the frot of the class. Ask them to lie dow o the floor. Explai that whe the studets are lyig o the floor they are horizotal. Now ask the studets to stad up. Explai that whe the studets are stadig up they are vertical. b) Ask differet studets to either lie dow or stad up. Have the other studets idicate whether the studets are horizotal or vertical. c) Draw a horizotal ad a vertical lie o the board. Explai that, i math, lies ca be horizotal or vertical. Lies that are lyig dow are horizotal ad those that are stadig up straight (go up ad dow o their paper) are vertical. d) Draw several lies o the board like the oes show below: Ask, Are these lies horizotal? Why or why ot? Are the lie segmets vertical? Why or why ot? S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 61

62 e) Ask studets to use the stir sticks to show the followig: A vertical lie segmet A horizotal lie segmet A lie segmet that is either horizotal or vertical A vertical lie segmet that crosses a horizotal lie segmet A horizotal lie segmet that crosses a lie segmet that is ot vertical A horizotal lie segmet that crosses three vertical lie segmets A vertical lie segmet that crosses two lie segmets that are either horizotal or vertical Four horizotal lie segmets Two vertical lie segmets ad three horizotal lie segmets A vertical lie segmet ad two horizotal lie segmets ad a lie segmet that is either vertical or horizotal f) Have studets idetify whether the edge of their desk is a horizotal or a vertical lie segmet. Have them idetify two or three other horizotal or vertical lie segmets i the room. g) Ask each group to make a list of horizotal ad vertical lie segmets that they see i the school. Have them share their lists with the other members of the class. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: use the terms horizotal ad vertical correctly make vertical ad horizotal lie segmets idetify examples of horizotal ad vertical lie segmets i the eviromet. idetify lie segmets that are either horizotal or vertical 62 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

63 idetify parallel, itersectig, perpedicular, vertical, ad horizotal edges ad faces o 3-D objects. idetify parallel, itersectig, perpedicular, vertical, ad horizotal sides o 2-D shapes. Draw 2-D shapes or 3-D objects that have edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, or horizotal. Describe faces ad edges of a 3-D object usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. Describe the sides of 2-D shapes usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. Materials: 20 stir sticks or toothpicks for each pair of studets, cubes, rectagular prisms, pyramids, ad triagular prisms Orgaizatio: Pairs/Whole class a) Show studets a square made out of stir sticks. Have them idetify the shape, as well as the horizotal ad vertical lie segmets that form it. b) Have studets make as may shapes as they ca with their stir sticks. Explai that a shape ca be made with ay umber of stir sticks, but it must have oly horizotal ad vertical lies. Ask studets to draw a picture of each shape they make. Tell studets that they should write the ame of each shape uder it, ad label the horizotal ad vertical lie segmets. c) Give a 3-D object to each pair of studets. Ask studets to discuss their shape with their parter, ad the write a descriptio of their shape i their math joural. Explai that they should write the ame of the shape, ad the use words ad pictures to explai which edges ad faces of their shape are horizotal ad which are vertical. Observatio Checklist Check studets work to determie whether they ca do the followig: costruct 2-D shapes that have vertical ad horizotal lies draw 2-D shapes that have vertical ad horizotal lies idetify the horizotal ad vertical lie segmets o a 2-D shape idetify the ames of 2-D shapes idetify the horizotal ad vertical edges ad faces of a 3-D object draw 3-D objects that have vertical ad horizotal edges ad faces S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 63

64 idetify parallel, itersectig, perpedicular, vertical, ad horizotal edges ad faces o 3-D objects. idetify parallel, itersectig, perpedicular, vertical, ad horizotal sides o 2-D shapes. provide examples from the eviromet that show parallel, itersectig, perpedicular, vertical, ad horizotal lie segmets. Draw 2-D shapes or 3-D objects that have edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, or horizotal. Materials: Stir sticks ad orage patter block squares Orgaizatio: Small groups a) Draw the followig lie segmets o the board or overhead. Explai that these lies are called itersectig lies because they cross each other. Ask studets to use their stir sticks to show three differet pairs of itersectig lie segmets two lie segmets that do ot itersect A lie segmet itersected by more tha oe lie segmet b) Explai that sometimes lies itersect i a special way. Ask studets what is special about how these two lie segmets itersect. Explai that these lies are special because they form square corers. Demostrate this by placig the orage squares at the itersectio of the lies. Tell studets that these lies are perpedicular. Have studets make three differet pairs of perpedicular lie segmets with their stir sticks. Have them use the orage squares to show that each pair of lies forms a square corer. Ask studets to make a pair of lie segmets that are ot perpedicular ad explai why they are ot perpedicular. 64 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

65 c) Ask studets what is special about this pair of lies (figure 1). Explai that lies that ever meet are parallel. Demostrate that the lies ever meet by placig orage squares betwee the two lies (figure 2) ad havig studets ote that the distace betwee the two lies is the same everywhere. Ask studets to make three differet pairs of parallel lie segmets with their stir sticks. Have them demostrate that the lie segmets are the same distace apart by usig the orage squares or their rulers. Have studets make a pair of lies that are ot parallel ad explai why they are ot parallel. d) Ask each group to idetify examples of parallel, itersectig, ad perpedicular lie segmets iside ad outside the classroom. Have them draw ad label a diagram of each lie segmet pair, ad list uder each diagram real-world examples of the lie segmet pair. e) Have each group share its examples of each type of lie segmet pair with the other members of the class. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: costruct pairs of lie segmets that are parallel, perpedicular, ad itersectig idetify lie segmets that are ot parallel idetify lie segmets that are ot itersectig idetify lie segmets that are ot perpedicular idetify real-world examples of parallel, perpedicular, ad itersectig lie segmets S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 65

66 idetify parallel, itersectig, perpedicular, vertical, ad horizotal sides o 2-D shapes. Draw 2-D shapes or 3-D objects that have edges, faces, ad sides that are parallel, itersectig, perpedicular, vertical, or horizotal. Materials: Pait or watercolours, black marker, ad samples of Piet Modria artwork (ca be foud o the Iteret) Orgaizatio: Large group/idividual a) Show studets a picture of Piet Modria s artwork. Explai that Piet Modria was a Dutch paiter who was famous for paitigs that he called compositios. b) Ask studets to describe the picture. Ecourage them to discuss the types of lies ad shapes he used to create the picture. c) Have studets use black markers ad watercolours to create a picture i the style of Piet Modria. d) Display studets artwork o walls ad coduct a gallery walk so studets ca look at each other s work. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify parallel, itersectig, perpedicular, vertical, ad horizotal lies create a piece of artwork that is comprised of parallel, itersectig, perpedicular, vertical, ad horizotal lies 66 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

67 Materials: Patter blocks Orgaizatio: Pairs a) Ask studets to use their patter blocks to complete the followig activity. Have them draw a sketch of each shape that they make. 1. Use two differet blocks to make a shape with exactly two pairs of parallel sides exactly oe pair of parallel sides o parallel sides 2. Use three differet blocks to make a shape with exactly three pairs of parallel sides exactly two pairs of parallel sides exactly oe pair of parallel sides o parallel sides 3. What is the largest umber of pairs of parallel sides of a shape you ca make with two pieces? three pieces? four pieces? 4. Ca you use six differet patter blocks to make a shape with o parallel sides? b) Have studets share their shapes with the other members of the class. Ecourage studets to idetify lies that are parallel, perpedicular, itersectig, vertical, ad horizotal. Observatio Checklist Moitor studets resposes to determie whether they ca do the followig: idetify parallel, itersectig, perpedicular, horizotal, ad vertical lie segmets o 2-D shapes draw 2-D shapes with parallel, itersectig, perpedicular, horizotal, ad vertical lie segmets S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 67

68 Idetify parallel, itersectig, perpedicular, vertical, ad horizotal edges ad faces o 3-D objects. Describe the faces ad edges of a 3-D object usig terms such as parallel, itersectig, perpedicular, vertical, or horizotal. Materials: Cubes, rectagular prisms, square-based ad triagular-based pyramids, ad triagular prisms, cards with the ames of the 3-D objects o them, oe ame per card (e.g., cube, rectagular prism, etc.), stir sticks or toothpicks, ad plasticie Orgaizatio: Small groups a) Show studets a triagular prism ad ask them to describe it. Ecourage studets to idetify the faces ad edges that are parallel, itersectig, perpedicular, vertical, ad horizotal. b) Give each group a set of the 3-D objects. Have studets take turs describig oe of the shapes to the other members of their group. Ecourage studets to poit out the faces ad edges that are parallel, itersectig, perpedicular, vertical, ad horizotal. c) Give each studet a card. Tell studets you will be describig the characteristics of 3-D objects. If the shape o their card has that characteristic, they should stad up ad show their card to the other members of the class. For example, if I say, I am a 3-D object that has three pairs of parallel faces, the studets who have cards with cube ad rectagular prism writte o them should stad up. The other members of the class have to check the cards to make sure that the right shapes have bee idetified. d) Show studets how to joi toothpicks with plasticie to build a 3-D object. Have studets use the materials to build 3-D objects that fit each set of characteristics. Each edge is perpedicular to eight other edges. The edges are ot all the same legth. (rectagular prism) There are six edges. No edges are perpedicular. (triagular pyramid) The vertical edges are perpedicular to the horizotal edges. There are three vertical edges. (triagular prism) e) Have studets discuss questios like the followig: Why are the roofs of most houses ot parallel to the groud? Why are the shelves of a bookcase parallel to the floor? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: idetify faces of 3-D objects that are parallel, itersectig, perpedicular, horizotal, ad vertical idetify the edges of 3-D objects that are parallel, itersectig, perpedicular, horizotal, ad vertical 68 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

69 Grade 5: Shape ad Space (3-D Objects ad 2-D Shapes) (5.SS.6) edurig uderstadigs: Shapes are distiguished by their properties. Geeral Outcome: Describe the characteristics of 3-D objects ad 2-D shapes, ad aalyze the relatioship betwee them. SpecIfIc LeArIG OutcOme(S): AchIevemet IDIcAtOrS: 5.SS.6 Idetify ad sort quadrilaterals, icludig rectagles squares trapezoids parallelograms rhombuses accordig to their attributes. [C, R, V] Idetify ad describe the characteristics of a pre-sorted set of quadrilaterals. Sort a set of quadrilaterals ad explai the sortig rule. Sort a set of quadrilaterals accordig to the legths of the sides. Sort a set of quadrilaterals accordig to whether or ot opposite sides are parallel. Prior Kowledge Studets may have had experiece with the followig: Idetifyig quadrilaterals Idetifyig ad explaiig mathematical relatioships usig a Ve diagram related Kowledge Studets should be itroduced to the followig: Idetifyig parallel, perpedicular, horizotal, ad itersectig lie segmets S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 69

70 BacKgroud iformatio A simple closed curve is a curve that does ot cross itself ad ca be draw by startig ad stoppig at the same poit (e.g., i the diagram below, figures (a) ad (b) are simple closed curves while (c) ad (d) are curves that are ot closed). a) b) c) d) Polygos are simple closed curves formed by the uio of lie segmets. I the diagram above, (b) is the oly polygo sice it is both a simple closed curve ad made up of lie segmets. The lie segmets that form the polygo are the sides of the polygo. A poit where two sides meet is a vertex of the polygo. Polygos ca be classified accordig to the umber of sides they have. The most commo classificatios are: triagle (three sides), quadrilateral (four sides), petago (five sides), hexago (six sides), heptago (7 sides), octago (8 sides), oago (9 sides), decago (10 sides), ad dodecago (12 sides). Other polygos are commoly referred to as -gos, where is the umber of sides. For example, a eleve-sided polygo ca be referred to as a 11-go ad a 14-sided polygo ca be referred to as a 14-go. Quadrilaterals ca be classified accordig to the umber of parallel sides that they have. The defiitio of each type of quadrilateral is give below. Trapezium A quadrilateral with o pairs of parallel sides. Trapezoid A quadrilateral with at least oe pair of parallel sides. Some texts defie a trapezoid as a quadrilateral with exactly oe pair of parallel sides. If the support material you are usig defies a quadrilateral i this way, studets should be show both defiitios. This ca help them uderstad that mathematics is ot a rigid subject ad that mathematicias do ot always agree o the defiitio of a cocept. Parallelogram A quadrilateral i which each pair of opposite sides is parallel. The opposite sides of parallelograms are also cogruet (same legth). Rectagle A parallelogram that has four right agles. Rhombus A parallelogram that has four cogruet sides. Square A parallelogram that has four cogruet sides ad four right agles. Studets ca be asked to examie how the differet defiitios affect their solutios to problems ivolvig trapezoids. Eve though the learig experieces focus o quadrilaterals that have parallel sides, some of the activities iclude quadrilaterals that have o parallel sides. This has bee doe to avoid givig studets a flawed cocept of a quadrilateral. 70 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

71 mathematical laguage Cogruet Polygo Parallel Parallelogram Perpedicular Quadrilateral Rectagle Rhombus (Rhombuses or Rhombi) Set Side Square Square corer Trapezoid Vertex (Vertices) learig experieces Assessig Prior Kowledge Materials: Cocept descriptio sheet (BLM 5 8.2). Orgaizatio: Idividual/Whole class a) Tell studets that they will be learig about a family of shapes called quadrilaterals, but before they begi you eed to fid out what they already kow about this shape. b) Have studets complete the cocept descriptio sheet. Let studets kow that it is all right if they caot thik of aythig to put i a sectio. They will have aother opportuity to complete the sheet after they have leared more about the shape. c) Whe studets complete the sheet, begi a discussio of their resposes by askig, What is a quadrilateral? What does it look like? As the discussio progresses, make sure studets see a variety of examples ad o-examples. I particular, studets should see examples of quadrilaterals that have o parallel sides Observatio Checklist Whe the discussio eds, have studets add to the cocept descriptio sheet to determie whether they ca do the followig: recogize that a quadrilateral is a four-sided polygo give appropriate examples ad o-examples of quadrilaterals S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 71

72 Idetify ad describe the characteristics of a pre-sorted set of quadrilaterals. materials: Five evelopes (oe labelled trapezoid, oe labelled square, oe labelled rectagle, oe labelled parallelogram, ad oe labelled rhombus), three differet cutouts of each quadrilateral, oe large sheet of paper, ad oe marker for each group Orgaizatio: Small group/large group a) Place the cut-outs ito the appropriate evelopes ad the divide the class ito five groups. Give each group a large sheet of paper, a evelope, ad a marker. b) Tell studets that each group has a differet type of quadrilateral ad that their task is to teach the other groups about their quadrilateral. To do this, they eed to look at the examples of the quadrilateral i their evelopes ad determie its characteristics. Let studets kow that they should pay particular attetio to the sides ad vertices of their quadrilateral. Tell studets that they should record the ame of their quadrilateral ad its characteristics o the large sheet of paper. c) Have each group post their sheet of paper i the frot of the room ad tell the other members of the class about their quadrilateral. Help studets idetify ay characteristics they might have missed. d) Have studets make a graphic orgaizer to help them lear the characteristics of the differet quadrilaterals ad their relatioship to each other (e.g., studets ca make a chart like the oe show below). Quadrilateral Diagram At least oe pair of parallel sides Two pairs of parallel sides All sides cogruet Opposite sides cogruet All square corers Parallelogram Square Rectagle Trapezoid Rhombus 72 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

73 Observatio Checklist Observe studets resposes to make sure that for each quadrilateral they have correctly idetified the characteristics show i the chart. Materials: Stir sticks or straws, ad scissors Orgaizatio: Whole class a) Have the studets use the stir sticks to make quadrilaterals whose opposite sides are cogruet that is ot a rhombus that has at least oe pair of parallel sides that has o parallel sides that has four square corers that is either a square or a rectagle ad has two pairs of parallel sides that has oe square corer b) Whe studets fiish makig each shape, ask them: What shape did you make? How do you kow that it is a quadrilateral? Is there aother shape that you could have made? What is it? How does it differ from the shape you made? What other characteristics does your shape have? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: costruct ad idetify a quadrilateral with the give characteristic(s) describe the characteristics of each quadrilateral that they make idetify other quadrilaterals that have the same characteristics as the oe that was give describe how squares, rectagles, parallelograms, trapezoids, ad rhombuses differ from each other recogize that there are other quadrilaterals besides squares, rectagles, trapezoids, parallelograms, ad rhombuses S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 73

74 Materials: Tagrams Orgaizatio: Idividual a) Have the studets use the tagram pieces to make two differet rectagles parallelograms trapezoids squares rhombuses Let studets kow that they ca use two or more of the tagram pieces to make each shape. b) Have studets place each shape they make o a piece of paper ad trace aroud it. Ask them to write the ame of the shape udereath their drawig ad write a setece statig which tagram pieces they used to make the shape. Observatio Checklist Check studets work to see whether they ca do the followig: make two differet rectagles, parallelograms, trapezoids, ad squares correctly idetify each quadrilateral that they made ad the tagram pieces that they used to make it spell the ames of the quadrilaterals correctly Materials: Quadrilateral cards (BLM 5.SS.6.1) Orgaizatio: Whole group a) Give each studet a card with a ame of a quadrilateral o it. b) Tell studets that they are goig to play a game called Name that Quadrilateral. Explai that you will be describig a characteristic of a quadrilateral. Studets who have a card with the ame of a quadrilateral with that characteristic o it should stad up ad show their card to the rest of the class (e.g., if I say, I am a quadrilateral whose sides are all cogruet, the studets who have square or rhombus writte o their card should stad up. The rest of the class checks to see whether studets have idetified the right quadrilaterals.). 74 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

75 c) Vary the game by selectig five studets to be pael members. Give each pael member a card with a ame of a quadrilateral o it ad tell him or her to keep it hidde from the rest of the class. Tell studets that everyoe will have a chace to ask a pael member a questio about his or her quadrilateral ad the oly questio studets ca t ask is: What is your quadrilateral? The game is over whe every studet has had a opportuity to ask a questio. The perso who correctly idetifies the quadrilateral o each pael member s card is the wier. Observatio Checklist Observe studets resposes to determie whether they idetify ad describe the characteristics of the differet quadrilaterals as show i the chart. Sort a set of quadrilaterals ad explai the sortig rule. Sort a set of quadrilaterals accordig to the legths of the sides. Sort a set of quadrilaterals accordig to whether or ot opposite sides are parallel. Materials: A set of quadrilateral cards ad a set of rule cards for each pair of studets (BLM 5.SS.6.2); two loops of strig or yar for each pair of studets Orgaizatio: Pairs a) Have the studets lay the strig loops ad the label cards at least oe square corer ad opposite sides cogruet o their workspace, as show below. b) Have studets sort their quadrilaterals ito the appropriate sets. Whe studets fiish sortig the shapes, have them describe their solutios ad explai how they kew where to place each quadrilateral. S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 75

76 c) Repeat the activity. Have the studets sort the quadrilaterals ito sets with at least oe pair of parallel sides/ all sides cogruet all square corers/two pairs of parallel sides o parallel sides/at least oe pair of parallel sides Have studets make up their ow rules for sortig the quadrilaterals. d) Vary the activity by showig studets pre-sorted sets ad askig them to describe the rules that were used to sort the quadrilaterals. For example, show studets the followig set ad ask them to idetify the rule that was used to sort the quadrilaterals. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: sort the set of quadrilaterals accordig to the stated rule explai how they kew where each quadrilateral beloged recogize the relatioships amog the quadrilaterals, such as all squares are rectagles all parallelograms are trapezoids, usig the defiitio of trapezoids as quadrilaterals with at least oe pair of parallel sides all rectagles, squares, ad rhombuses are parallelograms all squares are rhombuses 76 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

77 Sort a set of quadrilaterals ad explai the sortig rule. Materials: Quadrilateral activity sheet (BLM 5.SS.6.3) Orgaizatio: Whole class/idividual a) Ask studets to complete the activity. b) Have studets share their resposes with the other members of the class. Observatio Checklist Check studets resposes to the questios to determie whether they ca do the followig: recogize the characteristics of rectagles, squares, trapezoids, rhombuses, ad parallelograms recogize the relatioships amog quadrilaterals, such as the followig: All squares are rectagles All squares are rhombuses All rectagles are parallelograms All squares are parallelograms All parallelograms, rectagles, squares, ad rhombuses are trapezoids, usig the defiitio of trapezoids as quadrilaterals with at least oe pair of parallel sides All rhombuses are parallelograms S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 77

78 Puttig the Pieces together A Parallel World Purpose: The purpose of this activity is to have studets recogize real-world examples of lies ad quadrilaterals. I particular, the ivestigatio is desiged to eable studets to idetify real-world examples of faces ad edges of 3-D objects ad sides of 2-D shapes that are examples of parallel, itersectig, perpedicular, vertical, ad horizotal lies rectagles, squares, trapezoids, parallelograms, ad rhombuses I additio, the ivestigatio is desiged to ehace studets ability to commuicate mathematically use techology coect mathematical cocepts to each other ad the real world Materials/Resources: Digital camera or video camera* Computer Orgaizatio: Large group/small groups a) Tell studets that they will be creatig a digital scrapbook (or a video recordig). Explai that each group is resposible for takig pictures of examples of lies ad quadrilaterals that they fid either iside or outside of school. Whe they fiish takig their pictures, they will create a digital scrapbook. Each picture i their scrapbook must iclude a descriptio of the types of lies ad quadrilaterals foud i the picture. b) Help studets determie the guidelies they should follow whe takig their pictures (e.g., studets eed to cosider the amout of time eeded to fid ad take the pictures, their coduct as they move withi ad outside the school, ad the resposibilities of each group member). c) Have studets take their pictures ad create their scrapbooks. d) Have studets choose a picture to preset to the class. Ask them to explai why they chose the picture ad where i the picture they see lies ad quadrilaterals. Ecourage them to idetify the types of lies ad quadrilaterals foud i their picture. * If digital cameras or computers are ot available, have studets fid pictures of lies ad quadrilaterals i magazies ad ewspapers. 78 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

79 Observatio Checklist Use the followig rubric to assess studet mastery of learig outcomes for ad of learig (durig ad at the completio of the activity) Scrapbook icludes: a example of each type of lie a example of at least 3 differet quadrilaterals All lies are correctly idetified. All quadrilaterals are correctly idetified. Writte descriptio is clear. Mathematical terms are used correctly. Scrapbook icludes: a example of 3 or 4 types of lies a example of at least 2 differet quadrilaterals Some lies are correctly idetified. Some quadrilaterals are correctly idetified. Writte descriptio is clear. Some mathematical terms are used correctly. Scrapbook icludes: examples of 1 or 2 types of lies a example of 1 type of quadrilateral A lie is correctly idetified. A quadrilateral is correctly idetified. Writte descriptio is ot clear. Some mathematical terms are used correctly. S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 79

80 BacKgroud iformatio A simple closed curve is a curve that does ot cross itself ad ca be draw by startig ad stoppig at the same poit (e.g., i the diagram below, figures (a) ad (b) are simple closed curves while (c) ad (d) are curves that are ot closed). a) b) c) d) Polygos are simple closed curves formed by the uio of lie segmets. I the diagram above, (b) is the oly polygo sice it is both a simple closed curve ad made up of lie segmets. The lie segmets that form the polygo are the sides of the polygo. A poit where two sides meet is a vertex of the polygo. Polygos are classified accordig to the umber of sides they have. The most commo classificatios are: triagle (three sides), quadrilateral (four sides), petago (five sides), hexago (six sides), heptago (7 sides), octago (8 sides), oago (9 sides), decago (10 sides), ad dodecago (12 sides). Other polygos are commoly referred to as -gos, where is the umber of sides. For example, a eleve-sided polygo ca be referred to as a 11-go ad a 14-sided polygo ca be referred to as a 14-go. Quadrilaterals ca be classified accordig to the umber of parallel sides that they have. The defiitio of each type of quadrilateral is give below. Trapezium A quadrilateral with o pairs of parallel sides. Trapezoid A quadrilateral with at least oe pair of parallel sides. Parallelogram A quadrilateral i which each pair of opposite sides is parallel. The opposite sides of parallelograms are also cogruet (same legth). Rectagle A parallelogram that has four right agles. Rhombus A parallelogram that has four cogruet sides. Square A parallelogram that has four cogruet sides ad four right agles. Some texts defie a trapezoid as a quadrilateral with exactly oe pair of parallel sides. If the support material you are usig defies a quadrilateral i this way, studets should be show both defiitios. This ca help them uderstad that mathematics is ot a rigid subject ad that mathematicias do ot always agree o the defiitio of a cocept. Studets ca also be asked to examie how the differet defiitios affect their solutios to problems ivolvig trapezoids. Moreover, eve though the learig experieces focus o quadrilaterals that have parallel sides, some of the activities iclude quadrilaterals that have o parallel sides. This has bee doe to avoid givig studets a flawed cocept of a quadrilateral. 70 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

81 mathematical laguage Cogruet Polygo Parallel Parallelogram Perpedicular Quadrilateral Rectagle Rhombus (Rhombuses or Rhombi) Set Side Square Square corer Trapezoid Vertex (Vertices) learig experieces Assessig Prior Kowledge Materials: Cocept descriptio sheet (BLM 5 8.2). Orgaizatio: Idividual/Whole class a) Tell studets that they will be learig about a family of shapes called quadrilaterals, but before they begi you eed to fid out what they already kow about this shape. b) Have studets complete the cocept descriptio sheet. Let studets kow that it is all right if they caot thik of aythig to put i a sectio. They will have aother opportuity to complete the sheet after they have leared more about the shape. c) Whe studets complete the sheet, begi a discussio of their resposes by askig, What is a quadrilateral? What does it look like? As the discussio progresses, make sure studets see a variety of examples ad o-examples. I particular, studets should see examples of quadrilaterals that have o parallel sides Observatio Checklist Whe the discussio eds, have studets add to the cocept descriptio sheet to determie whether they ca do the followig: recogize that a quadrilateral is a four-sided polygo give appropriate examples ad o-examples of quadrilaterals S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 71

82 idetify ad describe the characteristics of a pre-sorted set of quadrilaterals. materials: Five evelopes (oe labelled trapezoid, oe labelled square, oe labelled rectagle, oe labelled parallelogram, ad oe labelled rhombus), three differet cutouts of each quadrilateral, oe large sheet of paper, ad oe marker for each group Orgaizatio: Small group/large group a) Place the cut-outs ito the appropriate evelopes ad the divide the class ito five groups. Give each group a large sheet of paper, a evelope, ad a marker. b) Tell studets that each group has a differet type of quadrilateral ad that their task is to teach the other groups about their quadrilateral. To do this, they eed to look at the examples of the quadrilateral i their evelopes ad determie its characteristics. Let studets kow that they should pay particular attetio to the sides ad corers of their quadrilateral. Tell studets that they should record the ame of their quadrilateral ad its characteristics o the large sheet of paper. c) Have each group post their sheet of paper i the frot of the room ad tell the other members of the class about their quadrilateral. Help studets idetify ay characteristics they might have missed. d) Have studets make a graphic orgaizer to help them lear the characteristics of the differet quadrilaterals ad their relatioship to each other (e.g., studets ca make a chart like the oe show below). Quadrilateral Diagram At least oe pair of parallel sides Two pairs of parallel sides All sides cogruet Opposite sides cogruet All square corers Parallelogram Square Rectagle Trapezoid Rhombus 72 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

83 Observatio Checklist Observe studets resposes to make sure that for each quadrilateral they have correctly idetified the characteristics show i the chart. Materials: Stir sticks or straws Orgaizatio: Whole class a) Have the studets use the stir sticks to make a quadrilateral whose opposite sides are cogruet that is ot a rhombus that has at least oe pair of parallel sides that has o parallel sides that has four square corers that is either a square or a rectagle ad has two pairs of parallel sides that has oe square corer b) Whe studets fiish makig each shape, ask them: What shape did you make? How do you kow that it is a quadrilateral? Is there aother shape that you could have made? What is it? How does it differ from the shape you made? What other characteristics does your shape have? Observatio Checklist Observe studets resposes to determie whether they ca do the followig: costruct ad idetify a quadrilateral with the give characteristic(s) describe the characteristics of each quadrilateral that they make idetify other quadrilaterals that have the same characteristics as the oe that was give describe how squares, rectagles, parallelograms, trapezoids, ad rhombuses differ from each other recogize that there are other quadrilaterals besides squares, rectagles, trapezoids, parallelograms, ad rhombuses S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 73

84 Materials: Tagrams Orgaizatio: Idividual a) Have the studets use the tagram pieces to make two differet rectagles parallelograms trapezoids squares rhombuses Let studets kow that they ca use two or more of the tagram pieces to make each shape. b) Have studets place each shape they make o a piece of paper ad trace aroud it. Ask them to write the ame of the shape udereath their drawig ad write a setece statig which tagram pieces they used to make the shape. Observatio Checklist Check studets work to see whether they ca do the followig: make two differet rectagles, parallelograms, trapezoids, ad squares correctly idetify each quadrilateral that they made ad the tagram pieces that they used to make it spell the ames of the quadrilaterals correctly Materials: Quadrilateral cards (BLM 5.SS.6.1) Orgaizatio: Whole group a) Give each studet a card with a ame of a quadrilateral o it. b) Tell studets that they are goig to play a game called Name that Quadrilateral. Explai that you will be describig a characteristic of a quadrilateral. Studets who have a card with the ame of a quadrilateral with that characteristic o it should stad up ad show their card to the rest of the class (e.g., if I say, I am a quadrilateral whose sides are all cogruet, the studets who have square or rhombus writte o their card should stad up. The rest of the class checks to see whether studets have idetified the right quadrilaterals.). 74 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

85 c) Vary the game by selectig five studets to be pael members. Give each pael member a card with a ame of a quadrilateral o it ad tell him or her to keep it hidde from the rest of the class. Tell studets that everyoe will have a chace to ask a pael member a questio about his or her quadrilateral ad the oly questio studets ca t ask is: What is your quadrilateral? The game is over whe every studet has had a opportuity to ask a questio. The perso who correctly idetifies the quadrilateral o each pael member s card is the wier. Observatio Checklist Observe studets resposes to determie whether they idetify ad describe the characteristics of the differet quadrilaterals as show i the chart. Sort a set of quadrilaterals ad explai the sortig rule. Sort a set of quadrilaterals accordig to the legths of the sides. Sort a set of quadrilaterals accordig to whether or ot opposite sides are parallel. Materials: A set of quadrilateral cards ad a set of rule cards for each pair of studets (BLM 5.SS.6.2); two loops of strig or yar for each pair of studets Orgaizatio: Pairs a) Have the studets lay the strig loops ad the label cards at least oe square corer ad opposite sides cogruet o their workspace, as show below. b) Have studets sort their quadrilaterals ito the appropriate sets. Whe studets fiish sortig the shapes, have them describe their solutios ad explai how they kew where to place each quadrilateral. S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 75

86 c) Repeat the activity. Have the studets sort the quadrilaterals ito sets with at least oe pair of parallel sides/ all sides cogruet all square corers/two pairs of parallel sides o parallel sides/at least oe pair of parallel sides Have studets make up their ow rules for sortig the quadrilaterals. d) Vary the activity by showig studets pre-sorted sets ad askig them to describe the rules that were used to sort the quadrilaterals. For example, show studets the followig set ad ask them to idetify the rule that was used to sort the quadrilaterals. Observatio Checklist Observe studets resposes to determie whether they ca do the followig: sort the set of quadrilaterals accordig to the stated rule explai how they kew where each quadrilateral beloged recogize the relatioships amog the quadrilaterals, such as all rectagles are squares all parallelograms are trapezoids all rectagles, squares, ad rhombuses are parallelograms all squares are rhombuses 76 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

87 Sort a set of quadrilaterals ad explai the sortig rule. Materials: Quadrilateral activity sheet (BLM 5.SS.6.3) Orgaizatio: Whole class/idividual a) Ask studets to complete the activity. b) Have studets share their resposes with the other members of the class. Observatio Checklist Check studets resposes to the questios to determie whether they ca do the followig: recogize the characteristics of rectagles, squares, trapezoids, rhombuses, ad parallelograms recogize the relatioships amog quadrilaterals, such as the followig: All squares are rectagles All squares are rhombuses All rectagles are parallelograms All squares are parallelograms All parallelograms, rectagles, squares, ad rhombuses are trapezoids All rhombuses are parallelograms S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 77

88 Puttig the Pieces together A Parallel World Purpose: The purpose of this activity is to have studets recogize real-world examples of lies ad quadrilaterals. I particular, the ivestigatio is desiged to eable studets to idetify real-world examples of faces ad edges of 3-D objects ad sides of 2-D shapes that are examples of parallel, itersectig, perpedicular, vertical, ad horizotal lies rectagles, squares, trapezoids, parallelograms, ad rhombuses I additio, the ivestigatio is desiged to ehace studets ability to commuicate mathematically use techology coect mathematical cocepts to each other ad the real world Materials/Resources: Digital camera or video camera* Computer Orgaizatio: Large group/small groups a) Tell studets that they will be creatig a digital scrapbook (or a video recordig). Explai that each group is resposible for takig pictures of examples of lies ad quadrilaterals that they fid either iside or outside of school. Whe they fiish takig their pictures, they will create a digital scrapbook. Each picture i their scrapbook must iclude a descriptio of the types of lies ad quadrilaterals foud i the picture. b) Help studets determie the guidelies they should follow whe takig their pictures (e.g., studets eed to cosider the amout of time eeded to fid ad take the pictures, their coduct as they move withi ad outside the school, ad the resposibilities of each group member). c) Have studets take their pictures ad create their scrapbooks. d) Have studets choose a picture to preset to the class. Ask them to explai why they chose the picture ad where i the picture they see lies ad quadrilaterals. Ecourage them to idetify the types of lies ad quadrilaterals foud i their picture. * If digital cameras or computers are ot available, have studets fid pictures of lies ad quadrilaterals i magazies ad ewspapers. 78 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

89 Observatio Checklist Use the followig rubric to assess studet mastery of learig outcomes for ad of learig (durig ad at the completio of the activity) Scrapbook icludes: å a example of each type of lie å a example of at least 3 differet quadrilaterals All lies are correctly idetified. All quadrilaterals are correctly idetified. Writte descriptio is clear. Mathematical terms are used correctly. Scrapbook icludes: å a example of 3 or 4 types of lies å a example of at least 2 differet quadrilaterals Some lies are correctly idetified. Some quadrilaterals are correctly idetified. Writte descriptio is clear. Some mathematical terms are used correctly. Scrapbook icludes: å examples of 1 or 2 types of lies å a example of 1 type of quadrilateral Not all lies are correctly idetified. Quadrilateral is icorrectly idetified. Writte descriptio is ot clear. Some mathematical terms are used correctly. S h a p e a d S p a c e ( 3 - D O b j e c t s a d 2 - D S h a p e s ) 79

90 N O T e S 80 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

91 Grade 5: Shape ad Space (Trasformatios) (5.SS.7) Edurig Uderstadigs: The positio of shapes ca be chaged by traslatig, rotatig, or reflectig them. Geeral Outcome: Describe ad aalyze positio ad motio of objects ad shapes. SpEcific LEAriG OUTcOmE(S): AchiEvEmET idicators: 5.SS.7 Perform a sigle trasformatio (traslatio, rotatio, or reflectio) of a 2-D shape ad draw ad describe the image. [C, CN, T, V] 5.SS.8 Idetify a sigle trasformatio (traslatio, rotatio, or reflectio) of 2-D shapes. [C, T, V] Traslate a 2-D shape horizotally, vertically, or diagoally, ad describe the positio ad orietatio of the image. Rotate a 2-D shape about a poit, ad describe the positio ad orietatio of the image. Reflect a 2-D shape i a lie of reflectio, ad describe the positio ad orietatio of the image. Perform a trasformatio of a 2-D shape by followig istructios. Draw a 2-D shape, traslate the shape, ad record the traslatio by describig the directio ad magitude of the movemet (e.g., the circle moved 3 cm to the left). Draw a 2-D shape, rotate the shape, ad describe the directio of the tur (clockwise or couter-clockwise), the fractio of the tur, ad poit of rotatio. Draw a 2-D shape, reflect the shape, ad idetify the lie of reflectio ad the distace of the image from the lie of reflectio. Predict the result of a sigle trasformatio of a 2-D shape ad verify the predictio. Provide a example of a traslatio, a rotatio, ad a reflectio. Idetify a sigle trasformatio as a traslatio, rotatio, or reflectio. Describe a rotatio by the directio of the tur (clockwise or couter-clockwise). S h a p e a d S p a c e ( T r a s f o r m a t i o s ) 81

92 Prior Kowledge Studets may have had experiece with the followig: Idetifyig triagles, quadrilaterals, petagos, hexagos, octagos, ad circles related Kowledge Studets should be itroduced to the followig: Idetifyig vertical ad horizotal lies Idetifyig rectagles, squares, trapezoids, rhombuses, ad parallelograms Measurig the legths of lies to the earest cm or mm Idetifyig equivalet fractios BacKgroud iformatio Trasformatios play a importat role i the mathematics curriculum. I the Middle Years, the study of trasformatio ca support studets work i patterig, algebra, problem solvig, geometry, ad statistics. I high school ad beyod, the study of trasformatios helps studets recogize the coectios betwee algebra ad geometry ad ehaces their uderstadig of other topics such as matrices, scalig, ad complex umbers. A trasformatio ca be thought of as a chage i the positio, size, or shape of a figure. I the learig activities that follow, studets are itroduced to three trasformatios that chage the positio of a figure. Iformally, these trasformatios are referred to as slides, flips, ad turs. Formally, they are kow as traslatios, reflectios, ad rotatios. Studets should kow both the formal ad iformal termiology. A traslatio slides a figure a fixed distace i a give directio. The figure ad its traslatio are cogruet (same size ad shape) ad face i the same directio. I the diagram show below, square ABCD has bee traslated to a ew positio represeted by square A B C D. Note that Square A B C D, which is called the image of Square ABCD, is cogruet to Square ABCD ad faces i the same directio. The arrow idicates the distace ad the directio of the traslatio. 82 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s

93 A rotatio turs a figure ay umber of degrees aroud a fixed poit called the cetre of rotatio. The cetre of rotatio may be ay poit withi or outside the figure. The figure ad its image (the result of the trasformatio) are cogruet but they may face i opposite directios (e.g., i the diagram below, the arrow ABCDE has bee rotated 90 couter-clockwise about its midpoit). The image arrow A B C D E is cogruet to Arrow ABCDE but faces i a differet directio. A reflectio flips the figure over a lie, creatig a mirror image. The figure ad its image are cogruet but have differet orietatios. The lie the figure is flipped over is called the lie of reflectio ad it is the same distace from the figure as its image (e.g., i the diagram below, petago ABCDE has bee flipped over lie k). Note that Petago A B C D E is cogruet to Petago ABCDE but faces i the opposite directio. Lie k, the lie of reflectio, is equidistat from the two petagos. mathematical laguage Clockwise Couter-clockwise Cogruet Diagoally Horizotal Image Lie of reflectio Polygo Reflectio (flip) Rotatio (tur) Trasformatio Traslatio (slide) Vertical lie S h a p e a d S p a c e ( T r a s f o r m a t i o s ) 83

94 learig experieces Traslate a 2-d shape horizotally, vertically, or diagoally, ad describe the positio ad orietatio of the image. rotate a 2-d shape about a poit, ad describe the positio ad orietatio of the image. reflect a 2-d shape i a lie of reflectio, ad describe the positio ad orietatio of the image. perform a trasformatio of a 2-d shape by followig istructios. Materials: Carpeted area or floor mats Orgaizatio: Whole class a) Have studets lie dow o a carpet or mat. Ask them to slide a short distace i oe directio. Have them repeat the movemet several times by askig them to slide up, dow, ad sideways. After each slide, ask, What chaged? What remaied the same? Emphasize that whe a slide is made, the directio i which a object is poitig does ot chage. b) Have studets demostrate flips. At the ed of a flip, studets should have chaged from stomach to back or back to stomach. Discuss the differet ways flips ca be completed. For example, studets may roll to the left or to the right, or over the feet or head. Have studets try these differet ways. After each flip, ask, What chaged? What remaied the same? Emphasize that whe a object is flipped, its orietatio chages. Ask studets how this is differet from lookig at their reflectio i the mirror. Emphasize that a true reflectio of oeself would have exactly the same image, just i a differet orietatio. c) Have studets demostrate turs. To perform a tur, studets must keep either their feet or their heads (or belly butto) at the same locatio for the duratio of the tur. If the feet are the poit (cetre) of rotatio, the the arms ad head are used to move the body. If the head is the poit (cetre) of rotatio, the feet are used to make the move. Have studets tur all the way aroud or partway aroud. Have them tur i either a clockwise or couter-clockwise directio. After each tur, ask studets, What chaged? What remaied the same? Discuss the fact that after a tur, the directio i which the head poits is differet, except whe a complete tur is made. d) Iform studets that i the ext few lessos they will be learig more about slides, flips, ad turs. 84 G r a d e 5 M a t h e m a t i c s : S u p p o r t D o c u m e t f o r T e a c h e r s