G r a d e 7 M a t h e m a t i c s. Statistics and Probability

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G r a d e 7 M a t h e m a t i c s Statistics ad Probability

Statistics ad Probability (Data Aalysis) (7.SP.1, 7.SP.2) Edurig Uderstadig(s): Data ca be described by a sigle value used to describe the set. Geeral Learig Outcome(s): Collect, display, ad aalyze data to solve problems. Specific Learig Outcome(s): 7.SP.1 Demostrate a uderstadig of cetral tedecy ad rage by determiig the measures of cetral tedecy (mea, media, mode) ad rage determiig the most appropriate measures of cetral tedecy to report fidigs [C, PS, R, T] 7.SP.2 Determie the effect o the mea, media, ad mode whe a outlier is icluded i a data set. [C, CN, PS, R] Achievemet Idicators: Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. Solve a problem ivolvig the measures of cetral tedecy. Aalyze a set of data to idetify ay outliers. Explai the effect of outliers o the measures of cetral tedecy for a data set. Idetify outliers i a set of data ad justify whether or ot they are to be icluded i the reportig of the measures of cetral tedecy. Provide examples of situatios i which outliers would or would ot be used i determiig the measures of cetral tedecy. Statistics ad Probability 3

Prior Kowledge Studets may have had experiece with the followig: Differetiatig betwee first-had ad secod-had data. Comparig the likelihood of two possible outcomes occurrig, usig words such as less likely equally likely more likely Creatig, labellig, ad iterpretig lie graphs to draw coclusios. Selectig, justifyig, ad usig appropriate methods of collectig data, icludig questioaires experimets databases electroic media Graphig collected data ad aalyzig the graph to solve problems. For more iformatio o prior kowledge, refer to the followig resource: Maitoba Educatio ad Advaced Learig. Glace Across the Grades: Kidergarte to Grade 9 Mathematics. Wiipeg, MB: Maitoba Educatio ad Advaced Learig, 2015. Available olie at www.edu.gov.mb.ca/k12/cur/math/glace_k-9/idex.html. Related Kowledge Studets should be itroduced to the followig: Comparig ad orderig fractios, decimals (to thousadths), ad itegers by usig bechmarks place value equivalet fractios ad/or decimals Costructig, labellig, ad iterpretig circle graphs to solve problems. Expressig probabilities as ratios, fractios, ad percets. Idetifyig the sample space (where the combied sample space has 36 or fewer elemets) for a probability experimet ivolvig two idepedet evets. Coductig a probability experimet to compare the theoretical probability (determied usig a tree diagram, table, or aother graphic orgaizer) ad experimetal probability of two idepedet evets. 4 Grade 7 Mathematics: Support Documet for Teachers

Backgroud Iformatio Studets live i a iformatio age that abouds with data. Various media cotiually offer iformatio o fashio, etertaimet, sports, fiaces, safety, health, ad world evets. Studets ecouter data regularly at school, i their marks, i sciece experimets, i social studies iformatio, ad so o. To be helpful, data eeds to be categorized ad uderstood. Statistics help reduce large quatities of data to sigle values. The sigle value makes it much simpler to coceptualize ad commuicate about the iformatio cotaied i the data. Statistics, however, are sometimes maipulated or preseted i a maer that uses facts to mislead people ad sway their opiios. By studyig statistics, studets develop their ability to uderstad ad evaluate iformatio preseted i advertisig, politics, ad ews reports, ad to commuicate their experiece with data. Measures of Cetral Tedecy I previous grades, studets collected data first had ad from electroic sources, ad leared whe to use each source. I Grade 7, studets are itroduced to three statistical measures of cetral tedecy: mea, media, ad mode. Each is a umeric value attemptig to represet a etire set of data. Each measure is a average with its ow focus, stregths, ad weakesses. The more symmetrical the set of data is, the closer the measures of cetral tedecy will be to oe aother. The more skewed the set of data is, the greater the differece betwee the values will be. The differet measures are best used i differet situatios, although sometimes all three measures provide meaigful represetatios of the data. The measures of cetral tedecy ad rage are discussed below: Mea: The arithmetic mea is commoly referred to as average, ad is commoly used to assig studet grades. The mea is the measure of cetral tedecy most affected by outliers; therefore, it is best used whe the rage of values i the set is arrow. To fid the mea, combie all the values i the set ad the evely redistribute them. The algorithm for calculatig the mea is to sum all values i the set, ad divide the combied value by the umber of values i the set. The mea ca also be foud by fidig the cetral balace poit o a umber lie. Example: Give the umbers 3, 4, 6, 3, 3, 9, 7, a) plot the umbers o a umber lie Statistics ad Probability 5

b) move the umbers toward the cetre, allowig oe move to the left for every oe move to the right c) cotiue this process util the umbers lie up o oe poit Media: The media is the middle value i a ordered set of data. The media is easy to uderstad ad easy to determie. To fid the media, place all the values i the set, icludig repeated umbers, i umerical order ad select the value i the middle. If there is o sigle middle value, add the two middle umbers together ad divide by two. Because the media is the middle value, half the values i a data set will be greater tha the media ad half the values will be less tha the media. The media represets the 50th percetile. (Note: Formal study of percetiles occurs i Grade 12 Essetial Mathematics.) The media is less affected by outliers; therefore, it is more stable. It is the most appropriate measure of cetral tedecy to represet a set of data cotaiig extreme values. Mode: The mode is the most commoly occurrig item i a set. A set of data may ot have a mode, or it may have oe mode, be bimodal, or have multiple modes. The mode may or may ot idicate the cetre of the data it represets. Geerally, outliers (extreme values at either the high or the low ed of the rage) do ot affect modes. Modes are very ustable, however, ad a small chage i the data ca drastically chage the mode. Because the mode idetifies the most typical item i a set, it is useful for predictig the case i a particular situatio. For example, if the mode for shirts sold is size 10, the buyers for a store ca use the mode to help them decide which sizes to stock i the store s ivetory. Rage: The rage describes a set of data by idetifyig the differece betwee the greatest value ad the least value i a data set. Uderstadig how ad whe to use the differet statistical values gives studets the ability to uderstad ad commuicate about data more clearly, ad to use data wisely to make iformed decisios. Whe plaig for studet learig experieces, choose learig activities that emphasize cocepts ad uderstadig. Have studets gather data for the purpose of aswerig questios. Allowig studets to ask their ow questios ad collect their ow data provides cotexts ad purposes for aalyzig data ad for explorig the differet statistics. Studets may, for example, wish to compare their classmates habits or physical skills, itegrate sciece or social studies cotet, or aswer questios about world coditios or treds. 6 Grade 7 Mathematics: Support Documet for Teachers

Mathematical Laguage average data mea measure of cetral tedecy media mode outlier rage statistics Ve diagram Learig Experieces Assessig Prior Kowledge Materials: grid paper rulers Orgaizatio: Idividual or pairs Procedure: 1. Review the cocepts of formulatig questios, collectig first- or secod-had data, ad preparig bar graphs. 2. Have studets work idividually, or i pairs, to do the followig: a) Formulate a survey questio about peers that ca be aswered with umeric values. Sample Questios: How may sibligs are i your family? How may pets (or cell phoes, televisios) does your family have? Statistics ad Probability 7

How may times a week do you eat a particular food, watch a movie, or participate i physical activity? How may hours do you sleep per ight? How tall are you? How may pairs of mittes (or shoes, pats) do you ow? How may coutries have you visited? Compare the heights (or heart rates, legths of ames) of boys ad girls i the class. b) Gather the iformatio. c) Display the data i a bar graph. d) Formulate a questio about the populatio of the survey that could be aswered usig the iformatio from the graph. Iclude a aswer key to the questio. 3. Have studets preset ad display their work. These data sets ca be used for subsequet learig experieces. Variatios: Have studets choose a questio ad create bar graphs from data you provide or from the school cesus data available olie. Rather tha havig studets coduct surveys, have them research sources to collect data to aswer specific questios (e.g., temperatures or raifall amouts over a certai period of time, sizes of farms i a regio, the price of a commodity). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Pose a survey questio that ca be aswered with umeric values. r Coduct a survey or search resources to obtai data. r Display data i a bar graph. 8 Grade 7 Mathematics: Support Documet for Teachers

Suggestios for Istructio Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Materials: bar graphs from the surveys coducted i the previous learig experiece (Assessig Prior Kowledge) presetatio board math jourals Orgaizatio: Idividual or pairs, whole class Procedure: 1. Remid studets that their surveys ad graphs reveal iterestig iformatio about their peers. 2. Ask studets to use the iformatio i their graphs to create a geeral statemet about a typical or average studet i the class, or grade, or whatever group they surveyed. Examples: How may sibligs does a typical Grade 7 studet have? How may times does the typical Grade 7 studet eat Frech fries i a week? 3. Have studets work idividually or with their parters to determie the best aswer to their questio, explai how they arrived at the aswer, ad explai why the aswer represets a typical studet. 4. Reassemble as a class ad have studets share their questios, aswers, explaatios, ad justificatios. 5. Durig the class discussio, ecourage studets to commet o ad ask questios about their classmates resposes, ad to preset alterative aswers. 6. Itroduce vocabulary related to statistics as topics preset themselves durig the sharig. 7. Record the vocabulary o a presetatio board. Iclude the three measures of cetral tedecy (mea, media, ad mode), the rage, ad outliers (if they are preset). a) If studets choose the most frequet respose to represet the average studet, itroduce mode. b) If they fid the arithmetic mea, or redistribute the items, itroduce mea ad discuss the methods they used to determie the value. c) If they use the middle value, itroduce media ad discuss the methods of fidig the media. Statistics ad Probability 9

d) If they discuss the rage of values, itroduce the rage as the differece betwee these values. e) If they metio aomalies such as a very high or very low value, itroduce outliers. 8. Iform studets that i usig oe value to represet a rage of data, they have bee explorig statistical measures of cetral tedecy. Measures of cetral tedecy will be studied i greater detail i the followig learig experieces. 9. Have studets record the ew vocabulary terms ad what they have leared about them i their math jourals. Variatios: Have studets combie the iformatio from several surveys to create a profile of the typical Grade 7 studet. Rather tha usig studet survey data, provide studets with several questios ad sets of data. For example, provide data for several styles of T-shirts, each sellig for a differet price. Ask what would be a fair price for the T-shirts if they all sold for the same price. Or supply data for the amout of moey idividual studets raised for a class trip. Ask how much moey a typical studet raised. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Demostrate a uderstadig of mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. r Demostrate a uderstadig of the rage of a set of data. 10 Grade 7 Mathematics: Support Documet for Teachers

Suggestios for Istructio Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Materials: BLM 7.SP.1.1: Fidig the Cetre of a Graph ad Comparig the Values bar graphs from the previous learig experiece or supplied data modellig clay, coloured cubes or blocks, ad/or couters grid paper (1 cm) trasparet rulers or grid strips presetatio board math jourals BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy (optioal) Orgaizatio: Idividual or pairs, whole class Procedure: 1. Distribute copies of BLM 7.SP.1.1: Fidig the Cetre of a Graph ad Comparig the Values, ad have studets work idividually or with a parter to complete it. The reassemble as a class ad discuss what studets discovered. 2. Alterately, guide the class through the steps ad have them record their learig i their math jourals. a) Have studets, workig idividually or i pairs, build a cocrete model of their ow graph, or a classmate s graph. Alteratively, supply studets with data ad have them prepare a graph ad a model for the data. Studets may use cubes or blocks, couters, ad/or 1 cm grid paper ad modellig clay. b) Whe studets have completed their graphs, ask them to do the followig: Idetify the rage for the data represeted i their graph, ad record it (subtract the least value from the greatest value). Rearrage the graph to emphasize the rage. Idetify ay outliers or extreme values i the graph. Fid the mode, or most frequet value, represeted i their graph, ad record it. Explai how the graph could be rearraged to emphasize the mode. Fid the media or middle value i their graph, ad record it. Explai how the graph could be rearraged to emphasize the media. c) Ask studets to explore, o their ow or with their parter, how to level the data ad fid its cetre, or balace poit. Emphasize that this is ot the middle value or media. Studets will be rearragig their graphs to emphasize the cetre of the data, or the mea. They record the mea. Statistics ad Probability 11

3. Have studets reassemble as a class ad share what they did to level the data, ad discuss ay questios or commets that arise. Strategies for levellig the data could iclude the followig: a) Compress the modellig clay graph, while holdig the sides ad surface firm. b) Rearrage the blocks by takig blocks from the loger bars ad placig them o the smaller bars util they are similar i height. If whole blocks caot be shared evely, it may be ecessary to share fractios of a block. c) Place a ruler perpedicular to the bars of the graph, ad adjust the positio of the ruler util there are a equal umber of blocks above the lie ad below the lie. It may be ecessary to positio the ruler withi a block if the mea is ot a whole umber. 4. Have studets compare their three values, mode, media, ad mea, ad determie whether they each represet the data well, or whether oe value represets the data better tha the others, ad why that may be. 5. Share studets reflectios ad discuss how each value is a measure of cetral tedecy or a way to represet the average value. Whe the data set has a small rage, the average values are similar, ad each represets the graph. Whe there are outliers i the data, or the rage is wide, the averages may be quite differet from each other, ad o average by itself represets the data well. Differet measures are better for differet situatios, ad sometimes more tha oe measure is eeded to represet the data. Variatios: Supply studets with graphs of data that cotrols the value of the averages (i.e., a whole-umber mea if studets are ot prepared to work with decimals or fractios), or cotrol the presece of modes or outliers. Have studets rearrage the values of the bars to create multiple bar graphs that have the same mea. Ask why the differet graphs have the same mea. Have studets explore what effect rearragig the values of the bars of the graphs have o each of the average values. See BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. r Determie the rage of a set of data. r Use reasoig ad visualizatio to determie measures of cetral tedecy. 12 Grade 7 Mathematics: Support Documet for Teachers

Suggestios for Istructio Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Materials: demostratio board magets or self-stick otes umber lies Uifix or likig cubes BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy (optioal) Orgaizatio: Pairs, whole class Procedure: 1. Review the three types of averages (mode, media, ad mea), ad how studets foud these values usig graphs. 2. Preset a set of data such as the followig: 3, 4, 6, 3, 3, 9, 7. Ask studets to use a Thik-Pair-Share strategy (thik about the questio idividually, discuss ideas with a parter, ad the share resposes with the class) to do the followig: a) Idetify the rage i the data. b) Idetify ad explai how to determie the mea without makig a graph. c) Idetify ad explai how to determie the media without makig a graph. d) Idetify ad explai how to determie the mode without makig a graph. 3. Itroduce studets to usig a umber lie to fid the cetre of the data. a) Draw a umber lie o the demostratio board from 0 to 10. b) Place a square to represet each value o the correspodig poit of the umber lie. Magets or self-stick otes work well o a chalkboard or whiteboard. (I the data set specified above, there are three umber 3s, so place three squares o 3.) c) The goal is to fid the cetre of all these values. Systematically move the selfstick otes from each ed toward the cetre util all the otes are stacked up o oe poit (e.g., a move of two jumps from the right toward the cetre must be coutered by a move of two jumps from the left toward the cetre). d) For the above data set, the blocks will all lie up o the mea 5. Statistics ad Probability 13

Variatios: Have studets practise determiig the mea ad explore the effect of differet values o averages. Alter the values i the above data set, but maitai a set of seve digits with a sum of 35. Record the measures of cetral tedecy for each set, usig BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy. Compare the values for differet sets of data. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. r Determie the rage of a set of data. Suggestios for Istructio Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. Solve a problem ivolvig the measures of cetral tedecy. Aalyze a set of data to idetify ay outliers. Explai the effect of outliers o the measures of cetral tedecy for a data set. Idetify outliers i a set of data ad justify whether or ot they are to be icluded i the reportig of the measures of cetral tedecy. Provide examples of situatios i which outliers would or would ot be used i determiig the measures of cetral tedecy. Materials: BLM 7.SP.1.3A: Simoe s Spellig Scores (Questios) BLM 7.SP.1.3B: Simoe s Spellig Performace Record Orgaizatio: Pairs or small groups, whole class (for Thik-Pair-Share) 14 Grade 7 Mathematics: Support Documet for Teachers

Procedure: 1. Distribute copies of BLM 7.SP.1.3A: Simoe s Spellig Performace (Questios) ad BLM 7.SP.1.3B: Simoe s Spellig Performace Record. 2. Preset a set of data such as Simoe s spellig quiz results, scored out of 10. Her scores for the first seve quizzes were: 8, 8, 7, 9, 6, 10, ad 8. 3. Ask studets what score best represets Simoe s spellig performace, ad why they believe it to be so. I this set of data, the mea, media, ad mode are all 8. The rage is 4. 4. Simoe writes three more quizzes, with scores of 3, 7, ad 8. Have studets idetify ad support her performace level ow (mea 7.4, media 8, mode 8, rage 7). 5. O the last three quizzes, Simoe receives scores of 9, 10, ad 0. Have studets idetify which oe umber will represet Simoe s spellig performace. Ask studets to support their choice usig measures of cetral tedecy ad rage (mea 7.2, media 8, mode 8, rage 10). 6. Discuss studets choices ad reasos. Iclude a discussio of a) the effect of outliers o the mea, media, ad mode b) the ifluece of the rage o the differet measures c) possible reasos for the outliers (e.g., did t study, called to the office, cheated, lost quiz) d) whether or ot the outliers should be icluded i the data Variatios: Use studets ow assigmet or test scores. Use a differet data set that does ot represet school scores (e.g., prices for jeas, party sizes at a pizza restaurat, sizes of shoes or clothes). Have studets research to obtai data to aswer a questio they pose. Have studets geerate radom data to explore the effects of large outliers, or rage size, o measures of cetral tedecy. Ask studets to aalyze their fidigs ad make a geeral statemet regardig circumstaces for which they recommed each measure of cetral tedecy. Statistics ad Probability 15

Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. r Determie the rage of a set of data. r Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. r Solve a problem ivolvig the measures of cetral tedecy. r Aalyze a set of data to idetify ay outliers. r Explai the effect of outliers o the measures of cetral tedecy for a data set. r Idetify outliers i a set of data, ad justify whether or ot they are to be icluded i the reportig of the measures of cetral tedecy. r Provide examples of situatios i which outliers would or would ot be used i determiig the measures of cetral tedecy. Suggestios for Istructio Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. Determie the rage of a set of data. Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. Solve a problem ivolvig the measures of cetral tedecy. Materials: BLM 7.SP.1.4: Usig Cetral Tedecy to Choose a Quarterback spiers (optioal) Orgaizatio: Idividual, small groups, whole class 16 Grade 7 Mathematics: Support Documet for Teachers

Procedure: 1. Remid studets that the mea, media, ad mode are all measures of cetral tedecy that represet a etire set of data. Studets will ow use these measures to choose a quarterback for a football game. 2. Distribute copies of BLM 7.SP.1.4: Usig Cetral Tedecy to Choose a Quarterback, ad ask studets to complete the page idividually. 3. The have studets meet i small groups to discuss their thikig. Each group will choose oe quarterback, ad a spokesperso will preset ad justify the group s choice to the class. 4. As groups preset ad defed their choices, ecourage studets to respod to presetatios with commets ad questios. 5. Discuss what to do with errors, ad whe each measure (mea, media, ad mode) is best used. Variatios: Have studets geerate additioal data by usig a spier with sectios for 0, 5, 10, 15, 20, ad 25 yards, ad the ask them to recalculate the measures ad re-evaluate their decisios. Have studets geerate data for additioal quarterbacks. Questio whether their decisios are based o the same measures each time. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the mea, media, ad mode for a set of data, ad explai why these values may be the same or differet. r Determie the rage of a set of data. r Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. r Solve a problem ivolvig the measures of cetral tedecy. Statistics ad Probability 17

Suggestios for Istructio Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. Materials: BLM 7.SP.1.2: Explorig Measures of Cetral Tedecy previously completed record sheets of data sets ad measures of cetral tedecy, icludig the iformatio from the graphs produced i the Assessig Prior Kowledge learig experiece spiers or pairs of umber cubes (regular or multi-sided) Orgaizatio: Pairs or small groups Procedure: 1. Explai to studets that they will be ivestigatig sets of data to determie geeralizatios about which circumstaces best match each measure of cetral tedecy. 2. Have studets work with a parter or i a small group. 3. Ask studets to evaluate their previous records. If they require more data, or larger data sets, they ca do the followig: a) Radomly geerate ew data sets by spiig spiers or by rollig the umber cubes ad multiplyig the displayed umbers. b) Research the legitimate aswers to actual questios (e.g., salaries eared i specific compaies, umbers of differet sadwiches sold at a fast-food restaurat, flavours of ice cream sold, quatities of differet driks sold i the school cafeteria, teams wiig champioships). 4. Have studets geerate a list of guidelies for the types of data or circumstaces they recommed for each measure of cetral tedecy. 5. Discuss the guidelies as a class (refer to Backgroud Iformatio). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Provide a cotext i which the mea, media, or mode is the most appropriate measure of cetral tedecy to use whe reportig fidigs. 18 Grade 7 Mathematics: Support Documet for Teachers

Suggestios for Istructio Solve a problem ivolvig the measures of cetral tedecy. Idetify outliers i a set of data ad justify whether or ot they are to be icluded i the reportig of the measures of cetral tedecy. Provide examples of situatios i which outliers would or would ot be used i determiig the measures of cetral tedecy. Materials: data sets from previous learig experieces paper Orgaizatio: Idividual, pairs or small groups Procedure: 1. Explai that i this learig activity studets will demostrate their uderstadig ad use of measures of cetral tedecy. 2. Have studets, idividually, create a realistic questio ad a accompayig data set o oe side of a sheet of paper, ad a detailed solutio to the questio o the reverse side of the paper. Questios may iclude outliers that would or would ot be used i determiig the cetral tedecy. Solutios require the rage, outliers, ad the mea, media, ad mode to be idetified. Ask studets to idetify the best measure to reflect the cetre of that data ad justify the choice. 3. Studets the share their questios with a parter or a small group ad demostrate their ability to use ad choose measures of cetral tedecy to solve problems. 4. Whe studets have solved a problem, they check their solutio ad discuss ay discrepacies with the creator of the problem. Variatios: Prepare additioal problems ad data sets for studets who require them. Provide studets with several problems ad data sets, ad ask them to provide solutios for each. Have a statistics challege i which idividual studets compete to solve the problems i frot of a classroom audiece, or i which teams of studets compete agaist oe aother to fid the best measure of cetral tedecy i each case. Statistics ad Probability 19

Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a problem ivolvig the measures of cetral tedecy. r Idetify outliers i a set of data, ad justify whether or ot they are to be icluded i the reportig of the measures of cetral tedecy. r Provide examples of situatios i which outliers would or would ot be used i determiig the measures of cetral tedecy. 20 Grade 7 Mathematics: Support Documet for Teachers

Statistics ad Probability (Data Aalysis) (7.SP.3) Edurig Uderstadig(s): Circle graphs show a compariso of each part to a whole usig ratios. Percets, fractios, decimals, ad ratios are differet represetatios of the same quatity. Geeral Learig Outcome(s): Collect, display, ad aalyze data to solve problems. Specific Learig Outcome(s): 7.SP.3 Costruct, label, ad iterpret circle graphs to solve problems. [C, CN, PS, R, T, V] Achievemet Idicators: Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% Create ad label a circle graph, with or without techology, to display a set of data. Fid ad compare circle graphs i a variety of prit ad electroic media, such as ewspapers, magazies, ad the Iteret. Traslate percetages displayed i a circle graph ito quatities to solve a problem. Iterpret a circle graph to aswer questios. Prior Kowledge Studets may have had experiece with the followig: Differetiatig betwee first-had ad secod-had data. Demostratig a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. Demostratig a uderstadig of agles by idetifyig examples of agles i the eviromet classifyig agles accordig to their measure estimatig the measure of agles usig 45, 90, ad 180 as referece agles determiig agle measures i degrees Statistics ad Probability 21

drawig ad labellig agles whe the measure is specified Selectig, justifyig, ad usig appropriate methods of collectig data, icludig questioaires experimets databases electroic media Graphig collected data ad aalyzig the graph to solve problems. Demostratig a uderstadig of probability by idetifyig all possible outcomes of a probability experimet differetiatig betwee experimetal ad theoretical probability determiig the theoretical probability of outcomes i a probability experimet determiig the experimetal probability of outcomes i a probability experimet comparig experimetal results with the theoretical probability for a experimet For more iformatio o prior kowledge, refer to the followig resource: Maitoba Educatio ad Advaced Learig. Glace Across the Grades: Kidergarte to Grade 9 Mathematics. Wiipeg, MB: Maitoba Educatio ad Advaced Learig, 2015. Available olie at www.edu.gov.mb.ca/k12/cur/math/glace_k-9/idex.html. Related Kowledge Studets should be itroduced to the followig: Solvig problems ivolvig percets from 1% to 100%. Demostratig a uderstadig of circles by describig the relatioships amog radius, diameter, ad circumferece of circles relatig circumferece to pi determiig the sum of the cetral agles costructig circles with a give radius or diameter solvig problems ivolvig the radii, diameters, ad circumfereces of circles Expressig probabilities as ratios, fractios, ad percets. Coductig a probability experimet to compare the theoretical probability (determied usig a tree diagram, table, or aother graphic orgaizer) ad the experimetal probability of two idepedet evets. 22 Grade 7 Mathematics: Support Documet for Teachers

Backgroud Iformatio The purpose of graphs is to display data. Studets come to Grade 7 with experiece i usig lie graphs to display cotiuous data, ad bar graphs, double bar graphs, ad pictographs to display discrete data. I Grade 7, studets are itroduced to circle graphs. Circle graphs are also referred to as pie charts. Circle Graphs (Pie Charts) Various media use circle graphs to display comparative data. The circle graph displays the distributio of data, ot the actual data values. The set of data is grouped ito categories, ad each category is expressed as a percet of the whole set of data. Each sector of the graph represets a part-to-whole ratio. Circle graphs emphasize the relatio betwee a category ad the whole set of data, as well as the relatio betwee differet categories withi the data set. Comparisos withi circle graphs are most clear whe the umber of categories is small ad whe there is a defiite variatio i the size of the categories. Example: Beverage Choice Data Number of Studets Percet Agle Size Juice 150 46% 166 Soda 75 23% 83 Milk 68 21% 75 Water 32 10% 36 Totals 325 100% 360 This circle graph shows that early half of the studets eatig i the school cafeteria choose juice as a luch beverage, ad that early equal umbers of studets choose milk or soda. Circle graphs may also be used to compare data sets of differet size, as circle graphs compare ratios rather tha defiite quatities. The ratios regardig studets choices of beverage i the example above ca be compared to choices made by studets i other schools or i other regios. The comparisos may be used to aswer questios or to solve problems (e.g., which school to target for a utritio educatio program). Circle graphs are also used effectively to display probability. Statistics ad Probability 23

Experiece with circles ad cetral agles (learig outcome 7.SS.1), a uderstadig of decimals, percets, ad fractios, ad the ability to perform calculatios with these values (learig outcomes 7.N.2, 3, 4, 5, ad 7) make it easier for studets to create ad iterpret circle graphs. A uderstadig of roudig is useful whe costructig circle graphs (e.g., if the majority of percets or agle sizes have bee rouded up or dow, adjustmets may be required to esure the sum of percets totals 100%, ad cetral agles represeted i the graph total 360 ). Ways to Create Circle Graphs There are may ways to create circle graphs. Several of these are described below. Each circle graph must have a descriptive title ad must be labelled with the category ames ad correspodig percets, or be accompaied by a leged. The percets represeted by the sectors must total 100%, ad the sum of the cetral agles must equal 360. Make cocrete represetatios. Divide studets ito categories, such as those who have pets, ad those who do ot have pets. Ask studets i each group to stad side by side, equidistat from each other, ad the have the two groups form a circle. Estimate the middle of the circle, ad draw a lie (perhaps usig a skippig rope) from the cetre of the circle to each poit where the two groups meet. Studets could also use tokes to represet the umbers i the two groups. The tokes could be evely spaced aroud a circle whose circumferece has bee divided ito percets, ad a lie could be draw from the cetre to the poits o the circumferece midway betwee adjacet groups. Joi bars from a bar graph. Create a bar to represet the quatity i each group. Colour the bars. The cut out each bar, ad joi the bars ed to ed with tape to create oe log strip. Brig the eds of the strip together to create a circle. Draw a lie from the cetre of the circle to each poit where a ew category begis. Use fractio circles. Choose a fractio circle that matches the umber of pieces of data. For example, if there are 10 marbles i a set, choose a circle divided ito teths. Each teth represets oe marble. If six of the marbles are blue, colour 6 of the circle 10 blue; if three of the marbles are yellow, colour 3 of the circle yellow; ad if the 10 remaiig marble is red, colour 1 of the circle red. 10 Draw lies from the cetre of the circle to the poit o the circumferece where the categories meet. Calculate percets ad use percet circles. 24 Grade 7 Mathematics: Support Documet for Teachers

Express the umber i each category as a fractio of the whole set. Covert each fractio to a decimal umber ad the to a percet. Use a circle divided ito 100ths, or ito 20ths, to represet itervals of 5% (see BLM 5 8.26: Percet Circle). Create sectors to represet the percet of each category. Calculate percets ad create cetral agles. Create a chart such as the oe below. Category Quatity Fractio of the Whole Percet of the Whole Percet Times 360 i the Circle Size of the Cetral Agle Draw a circle ad oe radius. Use the radius to measure oe of the cetral agles. Use the subsequet radii to create successive cetral agles. Mathematical Laguage agle circle graph key leged percet pie chart sectors sum sum of the cetral agles Statistics ad Probability 25

Learig Experieces Assessig Prior Kowledge Materials: grid paper markers rulers access to data sources (optioal) Orgaizatio: Whole class, idividual or pairs Procedure: 1. Use a class discussio to review the characteristics of graphs, icludig the visual display of data, descriptive titles, labellig of axes, scale, ad plots. 2. Ask studets to work idividually or i pairs to collect data o some topic, ad the display the data as a graph. Studets may obtai data through surveys or observatios, or they may research a topic (e.g., colour of clothes wor o a give day, movie, music, readig, or food prefereces, laguage(s) spoke, umber of sibligs, populatios, life spas). 3. Ask studets to write questios that ca be aswered usig the iformatio i their graphs, ad the have them write aswers to these questios. 4. Post studets graphs, alog with the accompayig questios ad aswers, aroud the room. 5. Have studets participate i a Gallery Walk to view the displayed graphs. As a class, discuss the purpose ad effectiveess of usig graphs to display iformatio about a topic. Variatios: Supply studets with data or prepared graphs, rather tha havig them collect their ow data. Supply prepared graphs ad related questios for studets to aswer. The discuss the characteristics ad purposes of graphs. 26 Grade 7 Mathematics: Support Documet for Teachers

Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Select, justify, ad use appropriate methods of collectig data, icludig questioaires, experimets, databases, ad electroic media. r Graph collected data ad aalyze the graph to solve problems. Assessig Prior Kowledge Materials: BLM 7.SP.3.1: Calculatig the Percet of the Total idex cards (optioal) calculators (optioal) Orgaizatio: Idividual or pairs, whole class Procedure: 1. Review strategies for covertig fractios to percets. 2. Distribute copies of BLM 7.SP.3.1: Calculatig the Percet of the Total, ad have studets work idividually or i pairs to fid the percets preseted i the scearios. 3. Whe studets have had sufficiet time to respod to the questios, have them compare percets with a parter ad resolve ay discrepacies i their aswers. 4. Reassemble as a class ad discuss strategies studets used to express portios of a whole as percets. Variatios: Have studets create their ow scearios ad questios regardig percets. Ask them to record the scearios ad questios o oe side of a idex card, ad the solutios o the opposite side of the card. The cards may be used for drill games, learig activities, or Exit Slips. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Demostrate a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. Statistics ad Probability 27

Assessig Prior Kowledge Materials: BLM 7.SP.3.2: Percet of a Circle protractors calculators (optioal) paper, compasses, ad multi-sided umber cubes or spiers (optioal) Orgaizatio: Idividual, whole class Procedure: 1. Review how to use a protractor to measure ad draw agles. 2. Distribute copies of BLM 7.SP.3.2: Percet of a Circle, ad have studets, workig idividually, idetify various percets of shaded circles, shade desigated percets of circles, ad draw agles to represet a percet of a circle. 3. Review ad correct studets resposes as a class, ad discuss ay questios that arise. Variatios: Provide studets with additioal practice i idetifyig various percets of shaded circles, shadig various percets of circles, ad drawig agles that correspod to a percet of a circle. Have studets use a olie computer game to idetify the percet of a circle that has bee shaded. Sample Website: Games are available o the followig website: Scweb4free.com. Circle Graphs Game. 2009. www.scweb4free.com/circle.html. I this game, studets view segmeted circles ad select a multiple-choice respose to idetify the percet of studets who prefer hamburgers. Have studets play a game i pairs, usig multi-sided umber cubes, paper, a compass, ad protractors. Each studet uses the compass or template to draw a circle, mark its cetre, ad draw a radius from the cetre to the outside of the circle. The parters take turs rollig the umber cubes. The product of the two umbers rolled equals the percet of the circle to shade. The percet 360º idicates the size of agle to draw. Studets shade each sector they draw. The first studet to shade the etire circle wis. 28 Grade 7 Mathematics: Support Documet for Teachers

Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Demostrate a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. r Demostrate a uderstadig of agles by drawig ad labellig agles whe the measure is specified. Suggestios for Istructio Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% Create ad label a circle graph, with or without techology, to display a set of data. Iterpret a circle graph to aswer questios. Materials: a large ope area with a marked cetre (e.g., the cetre of a basketball court, the pitcher s moud of a ball diamod) log cords or skippig ropes (about four) tape or pegs to hold dow oe ed of the cords grid paper markers scissors rulers demostratio board Orgaizatio: Whole class, idividual Statistics ad Probability 29

Procedure: Part A 1. As a class, review the cocept that graphs are visual ways to display data. Iform studets that for this learig activity they will use differet methods to create a graph called a circle graph. The circle graph eables them to divide a group ito differet categories, ad allows them to compare the size of each category to each other ad to the whole group. 2. Secure oe ed of the cords to the cetre of a circle i a ope area. 3. Ask studets to lie up i two categories, such as those who have pets, ad those who do ot. Record the categories ad umbers i each category. 4. Have the lies follow their leader to form a circle aroud the cetre poit. 5. Have oe studet from where the two lies meet go to the cetre of the circle ad brig the ed of oe of the cords back to the circumferece of the circle. Note how the circle has bee divided ito two sectios, those who have pets, ad those who do ot. 6. Talk about which sector is smaller, ad which is larger. Discuss whether most of the studets have pets or whether most do ot. Estimate the percet of the circle represeted by each category. Discuss a descriptive title for the circle graph. 7. Repeat the procedure with other categories (e.g., favourite colours, umber of sibligs, ethic backgrouds). The four cords accommodate four categories. 8. Stop the exercise after sufficiet examples have bee explored, ad review what studets leared about circle graphs. 9. Post the categories ad umbers i each category for each of the graphs formed i Part A. Part B 10. Demostrate creatig a circle graph for oe set of data by colourig grids to represet each category, cuttig the coloured grids ito strips, tapig the strips ed to ed, ad the joiig the eds to form a circle. Trace the circle, estimate the cetre of the circle, ad mark a poit of the circumferece where differet colours meet. Use a ruler to coect the cetre of the circle ad the poits o the circumferece. Estimate the percet of the circle represeted by each sector, record the percet, ad label the sector. Title the graph. Use the data to write comparative statemets about the categories ad the whole set of data represeted by the graph. 11. Have studets, workig idividually, select oe data set ad the create their ow circle graph ad comparative statemets for that data set. Post studets graphs. 30 Grade 7 Mathematics: Support Documet for Teachers

Variatio: As a alterative to usig the ope area, solicit questios from the class, record the data o the demostratio board, ad have pairs of studets act out scearios usig couters ad circles (as described i the Backgroud Iformatio for learig outcome 7.SP.3). Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% r Create ad label a circle graph, with or without techology, to display a set of data. r Iterpret a circle graph to aswer questios. Suggestios for Istructio Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% Create ad label a circle graph, with or without techology, to display a set of data. Iterpret a circle graph to aswer questios. Materials: BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs BLM 5 8.26: Percet Circle fractio circles, available o the followig website: Maitoba Educatio. Middle Years Activities ad Games. Mathematics. www.edu.gov.mb.ca/k12/cur/math/my_games/idex.html. coloured couters (e.g., marbles, cubes, toy cars, or toy aimals, i bags) markers or pecil crayos compasses Statistics ad Probability 31

Orgaizatio: Whole class, idividual or pairs Procedure: This learig experiece will likely take more tha oe class ad is divided ito three parts. Use the same materials to create three graphs i Parts A to C. Part A 1. As a class, review the characteristics ad purposes of circle graphs. 2. Distribute fractio circles that have bee divided ito 10ths. 3. Have studets, workig idividually or i pairs, radomly choose 10 items (or the umber of divisios o the fractio circles) from a bag. The ask studets to do the followig: a) Sort the items accordig to colour. b) Colour adjacet segmets o a fractio circle to match the umber of items of each colour. c) Draw bold lies to divide the colours ad create sectors of each colour. 4. Ask studets to label each sector with the applicable colour ad the correspodig percet, or create a leged for the categories (e.g., if 6 of the 10 cubes selected were blue, the 6 or 60% of the cubes were blue). 10 5. Have studets write a title for their graph, as well as comparative statemets relatig to the graph. 6. Have studets total the percets represeted i each sectio of their graph, ad record the totals. Ask studets to make a geeral statemet regardig the sum of the percets i each graph. Discuss why the sum is 100%. I the discussio, iclude the cocept that each category is a part of the whole set ad 100% represets the whole set. Part B 7. Distribute copies of BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs fractio circles that have bee divided ito 20ths or 100ths (see BLM 5 8.26: Percet Circle) 8. Have studets radomly select 5 to 30 coloured couters ad sort them ito colour groups. Studets the complete the followig process, usig BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs: a) Record the colours i the Category colum of the chart. b) Record the umber of couters of each colour i the Quatity colum. c) Calculate the total quatity. d) Write the quatity of each colour as a fractio of the total couters selected. e) The covert that fractio to a percet. 32 Grade 7 Mathematics: Support Documet for Teachers

f) Add the percets. If it was ecessary to roud some of the percets, it is possible that they will ot total 100%. If they do ot total 100%, determie which percets were rouded up ad which were rouded dow ad make adjustmets to the most appropriate values. (The fial two colums of the chart will be completed i Part C.) 9. Ask studets to use the percets to create a circle graph usig the percet circles. Each 100th mark represets 1% of the circle, or each 20th represets 5% of the circle. 10. Have studets label the graph, icludig a) a title for the graph b) a label for each category (if there is isufficiet room, a leged may be used istead of labels) c) the percet of the whole for each category 11. Have studets write comparative statemets related to the categories of the graph. Part C 12. As a class, review how to use a protractor ad how to draw agles of specific measures. 13. Demostrate to studets that each sector of the circle represets a agle measure there are 360º i a circle 14. Show studets how to fid the agle measure by fidig a percet of 360º. 15. Have studets calculate the agle measures ad record them o BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs. Some of the agle measures may eed to be rouded. Whe the agle measures are totalled, they may ot equal 360º. If this is the case, review the roudig, ad adjust the values up or dow as ecessary. 16. Ask studets to draw a circle graph usig the measures of the cetral agles for each category. a) Use a compass to draw a circle. b) Mark the middle of the circle. c) Draw oe radius for the circle. d) Use the radius as a startig poit to measure oe agle. e) Use subsequet radii to create successive cetral agles. 17. Have studets label the graph, icludig a) a title for the graph b) a label for each category (if there is isufficiet room, a leged may be used istead of labels) c) the percet of the whole for each category 18. Have studets write comparative statemets related to the categories of the graph. 19. As a class, discuss the applicatios for usig each method of creatig a circle graph. Statistics ad Probability 33

Variatios: Supply data for each graph, rather tha havig studets geerate their ow data. Have studets use previously collected data, or collect their ow data, to create circle graphs, usig BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs (optioal). Use techology to create circle graphs. For example, use graphig programs such as Graphical Aalysis, spreadsheet programs such as Excel or Numbers, or olie graphig programs. Sample Websites: Graphig programs are available o websites such as the followig: Math Playgroud. Circle Graphs. Math Maipulatives. 2010. www.mathplaygroud.com/piechart.html. U.S. Departmet of Educatio. Istitute of Educatio Scieces, Natioal Ceter for Educatio Statistics. Create a Graph. Kids Zoe, Learig with NCES. http://ces.ed.gov/ceskids/createagraph/. Utah State Uiversity. Data Aalysis ad Probability. Natioal Library of Virtual Maipulatives. 1999 2010. http://lvm.usu.edu/e/av/topic_t_5.html. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% r Create ad label a circle graph, with or without techology, to display a set of data. r Iterpret a circle graph to aswer questios. 34 Grade 7 Mathematics: Support Documet for Teachers

Suggestios for Istructio Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% Fid ad compare circle graphs i a variety of prit ad electroic media, such as ewspapers, magazies, ad the Iteret. Traslate percetages displayed i a circle graph ito quatities to solve a problem. Iterpret a circle graph to aswer questios. Materials: BLM 7.SP.3.4: Comparig Examples of Circle Graphs BLM 7.SP.3.5: Traslatig Percetages i a Circle Graph ito Quatities media sources (e.g., magazies, ewspapers, advertisemets, the Iteret) for examples of circle graphs scissors glue or tape Orgaizatio: Small groups Procedure: Part A 1. Divide the class ito small groups. Have the studets i each group search through various media sources to fid five examples of circle graphs. Have them prit or cut out the graphs, icludig ay titles, legeds, or captios that accompay the graphs. 2. Ask each group to aalyze their selected graphs to determie whether or ot each graph icludes the commo attributes of circle graphs. 3. Group members ca take turs recordig iformatio about each graph o the chart provided o BLM 7.SP.3.4: Comparig Examples of Circle Graphs. 4. Have the groups aalyze their graphs ad pose two or three questios that ca be aswered usig iformatio from their graphs. They record their questios o the first page of BLM 7.SP.3.5: Traslatig Percetages i a Circle Graph ito Quatities. Studets the prepare aswers to their questios ad record them o the secod page of the BLM. Statistics ad Probability 35

Part B 5. Preset to the class a graph prepared by oe of the groups. Demostrate how the percetages i the graph could be used to solve a problem. Example: A graph about teeage music prefereces shows the percetage of studets who prefer differet musical artists. A music store caterig to teeagers could use this iformatio to choose which products to stock i its store. If 62% of the studets prefer artist B, ad the store is spedig $9500 o ivetory this moth, how much moey should the store sped purchasig artist B s music? 6. Have studets work together to prepare problems that could be solved usig their graphs. Ask them to record the problems o the first page of BLM 7.SP.3.5: Traslatig Percetages i a Circle Graph ito Quatities. Have studets prepare solutios to their problems ad record them o the secod page of the BLM. 7. Post the completed pages for a Gallery Walk. Have studets solve their classmates problems ad verify their solutios. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify commo attributes of circle graphs, such as title, label, or leged the sum of the cetral agles is 360 the data is reported as a percet of the total ad the sum of the percets is equal to 100% r Fid ad compare circle graphs i a variety of prit ad electroic media, such as ewspapers, magazies, ad the Iteret. r Traslate percetages displayed i a circle graph ito quatities to solve a problem. r Iterpret a circle graph to aswer questios. 36 Grade 7 Mathematics: Support Documet for Teachers

Suggestios for Istructio Create ad label a circle graph, with or without techology, to display a set of data. Traslate percetages displayed i a circle graph ito quatities to solve a problem. Materials: BLM 5 8.26: Percet Circle fractio circles, available o the followig website: Maitoba Educatio. Middle Years Activities ad Games. Mathematics. www.edu.gov.mb.ca/k12/cur/math/my_games/idex.html. a variety of packages (of various sizes) cotaiig coloured items (e.g., coloured beads, cady) compasses protractors BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs (optioal) Orgaizatio: Small groups Procedure: 1. Have studets, workig i small groups, sort the cotets of the supplied packages ito various colours ad use the iformatio to create a circle graph. Studets may use BLM 7.SP.3.3: Data Chart for Creatig Circle Graphs to help orgaize their aswers. They may create their graph with a compass ad a protractor, or usig fractio circles or percet circles provided. 2. Have studets propose a questio to be solved usig the iformatio from their graph. For example, if we buy 2000 cadies, how may ca we expect to be red? 3. Ask each group to exchage their graph with that of aother group, ad solve the ew problem. 4. After solvig the problem, studets retur it to its origiators ad discuss ay discrepacies. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Create ad label a circle graph, with or without techology, to display a set of data. r Traslate percetages displayed i a circle graph ito quatities to solve a problem. Statistics ad Probability 37

Puttig the Pieces Together Geometric Flags Itroductio: Studets use iformatio from a group survey to create a flag represetative of the group. The they calculate the area of the spaces i the flag. Purpose: I this ivestigatio, studets will demostrate the ability to do the followig (coectios to learig outcomes are idetified i paretheses): Solve problems ivolvig percets. (7.N.3) Costruct circle graphs to solve problems. (7.SP.3) Perform geometric costructios, icludig parallel ad perpedicular lie segmets ad perpedicular ad agle bisectors. (7.SS.3) Perform ad describe trasformatios. (7.SS.5) Apply a formula for determiig the area of triagles, parallelograms, ad circles. (7.SS.2) Studets will also demostrate the followig mathematical processes: Commuicatio Metal Mathematics ad Estimatio Problem Solvig Reasoig Techology Materials/Resources: ruler compass protractor right triagle coordiate grid paper (for the flags) art supplies calculator (optioal) Mira (optioal) tracig paper (optioal) Orgaizatio: Small groups, idividual 38 Grade 7 Mathematics: Support Documet for Teachers

Procedure: Studet Directios 1. Survey a group of people regardig their preferred colour. You may survey groups withi your class, i differet classes, i your family, ad so o. If each perso i the group chooses a differet colour, you may wish to have the survey participats select from a list of three to five colours. You may have them rak the colours as first, secod, ad third choice. 2. Calculate the percet of the group that prefers each colour. 3. Costruct a circle graph to represet the prefereces. 4. Desig a flag for the group usig the group s preferred colours. a) Calculate the area of the flag to be covered with each preferred colour. b) Create your desig usig circles, triagles, parallelograms, ad trasformatios. Iclude parallel ad perpedicular lie segmets ad perpedicular ad agle bisectors. c) Adjust the size of each shape to match the area to be covered with each preferred colour. 5. Create a summary chart that shows the areas of the differet shapes i each preferred colour, icludig a) the total area represetig each colour b) the percet of the total area covered by each colour c) a circle graph that represets the percet of the area covered by each colour 6. Create a fial copy of your flag. 7. Prepare a presetatio about your flag. I the presetatio, prove that your flag represets the group fidigs because the area covered by each colour i the flag is the same as the percet of the group that preferred each colour. Highlight some features of your flag ad how you solved a problem i creatig the flag. Statistics ad Probability 39

Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Calculate percets. r Costruct circle graphs. r Estimate ad calculate areas of triagles, parallelograms, ad circles. r Solve a problem ivolvig the percet of a area. r Perform ad describe a trasformatio i the flag. r Costruct ad idetify parallel ad perpedicular lie segmets i the flag. r Costruct ad idetify perpedicular ad agle bisectors i the flag. r Commuicate mathematical ideas effectively. 40 Grade 7 Mathematics: Support Documet for Teachers

Statistics ad Probability (Chace ad Ucertaity) (7.SP.4, 7.SP.5, 7.SP.6) Edurig Uderstadig(s): Percets, fractios, decimals, ad ratios are differet represetatios of the same quatity. The priciples of the probability of a sigle evet apply to the probability of idepedet evets. Geeral Learig Outcome(s): Use experimetal or theoretical probabilities to represet ad solve problems ivolvig ucertaity. Specific Learig Outcome(s): 7.SP.4 Express probabilities as ratios, fractios, ad percets. [C, CN, R, T, V] 7.SP.5 Idetify the sample space (where the combied sample space has 36 or fewer elemets) for a probability experimet ivolvig two idepedet evets. [C, ME, PS] Achievemet Idicators: Determie the probability of a outcome occurrig for a probability experimet, ad express it as a ratio, fractio, or percet. Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Provide a example of two idepedet evets, such as spiig a four-sectio spier ad a eight-sided die tossig a coi ad rollig a twelve-sided die tossig two cois rollig two dice ad explai why they are idepedet. Idetify the sample space (all possible outcomes) for a experimet ivolvig two idepedet evets usig a tree diagram, a table, or aother graphic orgaizer. (cotiued) Statistics ad Probability 41

Specific Learig Outcome(s): 7.SP.6 Coduct a probability experimet to compare the theoretical probability (determied usig a tree diagram, a table, or aother graphic orgaizer) ad experimetal probability of two idepedet evets. [C, PS, R, T] Achievemet Idicators: Determie the theoretical probability of a outcome for a experimet ivolvig two idepedet evets. Coduct a probability experimet for a outcome ivolvig two idepedet evets, with or without techology, to compare the experimetal probability to the theoretical probability. Solve a probability problem ivolvig two idepedet evets. Prior Kowledge Studets may have had experiece with the followig: Demostratig a uderstadig of fractios less tha or equal to oe by usig cocrete ad pictorial represetatios to ame ad record fractios for the parts of a whole or a set compare ad order fractios model ad explai that for differet wholes, two idetical fractios may ot represet the same quatity provide examples of where fractios are used Describig ad represetig decimals (teths ad hudredths) cocretely, pictorially, ad symbolically. Demostratig a uderstadig of fractios by usig cocrete ad pictorial represetatios to create sets of equivalet fractios compare fractios with like ad ulike deomiators Describig ad represetig decimals (teths, hudredths, thousadths) cocretely, pictorially, ad symbolically. Relatig decimals to fractios (teths, hudredths, thousadths). Comparig ad orderig decimals (teths, hudredths, thousadths) by usig bechmarks place value equivalet decimals 42 Grade 7 Mathematics: Support Documet for Teachers

Describig the likelihood of a sigle outcome occurrig, usig words such as impossible possible certai Comparig the likelihood of two possible outcomes occurrig, usig words such as less likely equally likely more likely Demostratig a uderstadig of ratio, cocretely, pictorially, ad symbolically. Demostratig a uderstadig of percet (limited to whole umbers) cocretely, pictorially, ad symbolically. Demostratig a uderstadig of multiplicatio ad divisio of decimals ivolvig 1-digit whole-umber multipliers 1-digit atural umber divisors multipliers ad divisors that are multiples of 10 Demostratig a uderstadig of probability by idetifyig all possible outcomes of a probability experimet differetiatig betwee experimetal ad theoretical probability determiig the theoretical probability of outcomes i a probability experimet determiig the experimetal probability of outcomes i a probability experimet comparig experimetal results with the theoretical probability for a experimet For more iformatio o prior kowledge, refer to the followig resource: Maitoba Educatio ad Advaced Learig. Glace Across the Grades: Kidergarte to Grade 9 Mathematics. Wiipeg, MB: Maitoba Educatio ad Advaced Learig, 2015. Available olie at www.edu.gov.mb.ca/k12/cur/math/glace_k-9/idex.html. Related Kowledge Studets should be itroduced to the followig: Solvig problems ivolvig percets from 1% to 100%. Demostratig a uderstadig of the relatioship betwee repeatig decimals ad fractios, ad termiatig decimals ad fractios. Statistics ad Probability 43

Comparig ad orderig fractios, decimals (to thousadths), ad itegers by usig bechmarks place value equivalet fractios ad/or decimals Demostratig a uderstadig of cetral tedecy ad rage by determiig the measures of cetral tedecy (mea, media, mode) ad rage determiig the most appropriate measures of cetral tedecy to report fidigs Determiig the effect o the mea, media, ad mode whe a outlier is icluded i a data set. Costructig, labellig, ad iterpretig circle graphs to solve problems. Note: Whe workig with the probability of two idepedet evets, keep i mid that Grade 7 studets have ot yet had exposure to multiplicatio of fractios, but may have had exposure to multiplicatio of decimals. Backgroud Iformatio I our society, probability is used i makig weather forecasts to express the likelihood of precipitatio, i makig medical ifereces such as the likelihood of cotractig a ifectio or a disease, i makig correlatios betwee lifestyle habits ad health, i predictig electio results, i determiig the chaces of wiig a lottery or a draw, i uderstadig whether or ot a game is fair, ad so o. The study of probability begis i Grade 5, with studets describig the likelihood of a sigle evet occurrig ad comparig the likelihood of two possible outcomes usig the laguage of probability. Uless a evet is the oly possible outcome, or uless it is impossible for a outcome to occur, oe ca ever be certai of a outcome, or of the umber of times a outcome will occur i a give umber of trials. Thus, i Grade 6, studets determie ad compare theoretical probability ad experimetal results. 44 Grade 7 Mathematics: Support Documet for Teachers

Experimetal Probability Experimetal probability describes what actually did happe i a real situatio. Sometimes experimetal probability is called relative frequecy. Experimetal Probability of a Evet = Number of Observed Favourable Outcomes* Total Number of Trials *A favourable outcome is the outcome that the experimeter is lookig for. If, for example, a experimeter was examiig the probability of rollig a 4 whe rollig a regular umber cube, ad rolled the umber cube 50 times, ad 10 of those times he or she rolled 4s, the experimetal probability would be 10 50 or 1 or 0.2 or 20%. 5 Probability is a geeralized statemet used to predict future evets. A geeralizatio caot be trusted for makig predictios if it is based o a small umber of trials; however, studets ca icrease the umber of trials i their experimets by combiig trials coducted by differet studets, or by usig computer applets for geeratig large umbers of umber cube rolls, coi tosses, or spis. Sample Websites: Some applets are available o the followig websites: Shodor. Experimetal Probability. Iteractive. 1994 2011. www.shodor.org/iteractivate/activities/expprobability/. Select the spier, or the type of die, ad the umber of trials desired. The simulator tallies ad calculates results. Utah State Uiversity. Data Aalysis ad Probability. Natioal Library of Virtual Maipulatives. 1999 2010. http://lvm.usu.edu/e/av/topic_t_5.html. I the Grades 6 to 8 sectio, select Coi Tossig or Spiers. A larger umber of trials will permit studets to make a geeralizatio i which they ca have cofidece. The larger the sample size is, the more similar the experimetal results ad the theoretical probability will be. Theoretical Probability Theoretical probability helps studets to predict what is likely to happe i a give circumstace, but does ot foretell what will happe for sure. To calculate the theoretical probability of a evet, the evets must occur radomly, ot iflueced by ay outside force, ad the evets must be equally likely, or have the same chace of occurrig. Theoretical Probability of a Evet = Number of Outcomes Favourable* to the Evet Total Number of Possible Equally Likely Outcomes *A favourable outcome is the desired outcome. If, for example, a experimeter was examiig the probability of rollig a 4 whe rollig a regular umber cube, he or she would kow the umber of favourable outcomes is 1, ad the umber of equally likely outcomes is 6, so the probability is 1 or 016. or 166. %. 6 Statistics ad Probability 45

Theoretical probability may be used to make predictios about future evets, whe evets are equally likely. If the evets are ot equally likely, as whe tossig a object ad predictig whether it will lad right-side up, upside dow, or o its side, experimetal probability may be used to make future predictios. I Grade 7, studets express probability as ratios, fractios, or percets. Probability rages betwee impossible 0% ad certai 100% (betwee 0 1 ad or betwee 0 ad 1). 1 1 A probability lie with bechmarks may be used to illustrate the equivalet expressios. Example: Probability Ivestigatios ad Problems Grade 7 studets exted ivestigatios of theoretical ad experimetal probability to iclude two idepedet evets. They also solve probability problems ivolvig two idepedet evets. To ivestigate probability with experimets, studets will eed access to devices that geerate radom results. These may iclude spiers, items to toss (e.g., cois, bicoloured tiles, umber cubes, tacks), ad items to draw from (e.g., cards from a deck, coloured blocks, tiles, marbles from a bag). Computer-simulated applets may also be used. Sometimes probability problems ivolve situatios that may be too dagerous, too expesive, or too difficult to experimet with. I these circumstaces, a simulatio ca be chose to mimic the experimet. For example, i a situatio that requires ivestigatig the possibility of whether a birth is male or female, the two outcomes ca be represeted with the two sides of a coi. 46 Grade 7 Mathematics: Support Documet for Teachers

Orgaizig Outcomes of Probability Ivestigatios Studets may begi studyig the probability of two idepedet evets with a cocrete ivestigatio, such as predictig the outcome of tossig two cois. The obvious outcomes are two heads, two tails, ad oe of each. Some studets may coclude that there is a likelihood of 1 for each combiatio. A ivestigatio will geerate differet 3 results, ad the differece may iterest studets i orgaizig possible outcomes i a systematic way. Outcomes of probability ivestigatios ca be orgaized with the use of a tree diagram ad a frequecy table or chart: Tree Diagram: A tree diagram lists the possible outcomes for each evet i two colums ad coects them with lies to form braches. The first colum (or row) lists all possible outcomes for the first evet. The secod colum (or row) lists all outcomes for the secod evet beside each of the outcomes i the first evet. A tree diagram is used to describe theoretical probability. Example: Below is a example of a horizotal tree diagram of the possible outcomes for the idepedet evets whe tossig two cois. Number of Favourable Outcomes The Probability ofay Evet = Number ofpossible Outcomes e.g., P T, T ( ) = 1 4 Note that the probabilities for the three outcomes listed above are ot 1 3 each, but rather 1 1 4, 1 4, ad 2 respectively. Statistics ad Probability 47

Frequecy table or chart: Whe used to predict possible outcomes of a evet, a frequecy table or chart orgaizes all possible outcomes for two idepedet evets. Each cell i the table idicates oe possible outcome. Example: Number of Favourable Outcomes The Probability ofay Evet = Number ofpossible Outcomes e.g., P( assumig orderdoes 1H, 1T) = 2 = 50% * ot matter * 4 Orgaizig outcomes helps to reveal hidde outcomes, as i the examples above, where studets may erroeously believe there are oly three possible outcomes: 2 heads, 2 tails, ad oe of each. To demostrate a thorough uderstadig of probability, studets should recogize the followig: Outcomes must be equally likely. For example, a spier that is 1 2 red, 1 yellow, ad 1 4 4 blue has 2 possibilities for red. 4 The probability of a outcome is equivalet whe tossig the same umber cube six times or whe tossig six umber cubes oe time each. Or, if there are six red marbles ad six white marbles i a bag, the chace of pullig either colour is the same as if there is oe red marble ad oe white marble i the bag. For idepedet evets, the probability of a outcome does ot chage based o the previous outcome. Whe a evet such as rollig a sum of 7 o two umber cubes has repeated itself several times, there is o icreased likelihood that the ext roll will or will ot be a sum of 7. The likelihood of rollig or ot rollig a sum of 7 remais the same, regardless of the outcome i the previous roll. Kowig the vocabulary terms is importat. For example, likely meas more tha 50% of the time, ot almost all the time. 48 Grade 7 Mathematics: Support Documet for Teachers

Mathematical Laguage certai evet depedet evet evet experimetal probability favourable outcome frequecy table or chart impossible evet idepedet evet likely outcome possible outcome probability radom relative frequecy sample size sample space theoretical probability tree diagram Statistics ad Probability 49

Learig Experieces Assessig Prior Kowledge Materials: a set of two multi-sided umber cubes or spiers per group calculators (optioal) BLM 7.SP.4.1: Recordig Sheet for Fractio Decimal Percet Equivalets (optioal) paper (optioal) Orgaizatio: Whole class, small groups (of three or five) Procedure: 1. As a class, review writig equivalet fractios, decimals, ad percets. 2. Orgaize the class ito groups, ad have each group desigate a Perso A, B, ad C. 3. Demostrate a few rouds of the game usig three voluteers. The aim is to move as quickly ad accurately as possible to create 10 equivalet fractios, decimals, ad percets, usig the followig routie: a) Perso A: Roll two umber cubes. b) Perso B: Use the umbers o the umber cube to create a proper fractio. c) Perso C: Express the fractio as a equivalet decimal umber. d) Perso A: Express the umber as a equivalet percet. e) Perso B: Roll two umber cubes. The routie cotiues. f) Use BLM 7.SP.4.1: Recordig Sheet for Fractio Decimal Percet Equivalets to record results. Perso A begis the sheet after rollig the umber cube. Perso A records what Perso B says, ad the passes the sheet to Perso B. Perso B records what perso C says, ad passes the sheet to perso C. The routie cotiues, with the sheet followig behid the perso who is aswerig. 4. Groups carry o with the game, racig to see who ca be the first group to create 10 equivalet fractios, decimals, ad percets correctly. 5. Play as may rouds as seems iterestig ad useful. 50 Grade 7 Mathematics: Support Documet for Teachers

Variatio: Play the game as a baseball game. The pitcher pitches two umbers. The batter gets to first base by amig the proper fractio, to secod base by amig the decimal, ad to third base by amig the percet. By fiishig withi a time limit, the batter gets to home base ad ears a ru. Failure to respod i a give time results i a out. The game ca be played as a whole class or i pairs, usig a paper baseball diamod ad markers. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Name part-to-whole ratios as fractios ad their decimal ad percet equivalets. Suggestios for Istructio Determie the probability of a outcome occurrig for a probability experimet, ad express it as a ratio, fractio, or percet. Q Q Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Materials: BLM 7.SP.4.2: What Is the Probability? a bag of letter tiles (B, R, I, D, G, E) (optioal) Orgaizatio: Idividual, whole class Procedure: 1. Distribute copies of BLM 7.SP.4.2: What Is the Probability? 2. Have studets aswer the questios o the sheet o their ow. The questios require studets to a) idetify outcomes ad probabilities b) compare experimetal ad theoretical probability c) idetify outcomes as impossible, certai, or more likely d) create ad aswer their ow questio related to probability 3. Reassemble as a class ad have studets share their resposes to the questios, correctig ay errors. Statistics ad Probability 51

Variatios: As a class, complete BLM 7.SP.4.2: What Is the Probability? ad discuss ay questios studets may have. Have studets work i small groups. Provide each group with the bag of letter tiles (B, R, I, D, G, E), ad have them ivestigate the probability of drawig each letter i 36 trials. Combie the results of the small groups, recalculate the experimetal probability, ad compare the probability for the combied results to the results of the idividual groups. Ask studets to discuss why the experimetal results ad the theoretical probability are differet from each other. Have studets, workig i pairs, choose their ow letter tiles (or other devices) ad prepare a similar questio sheet ad aswer key. Groups exchage sheets, complete ad correct the questios, ad resolve ay discrepacies betwee the resposes. Have studets desig spiers, or situatios, that would result i give probabilities. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Describe sigle evets as impossible, possible, certai, less likely, more likely, or equally likely. r Idetify possible outcomes i a probability sceario. r Differetiate betwee experimetal ad theoretical probability. r Determie theoretical ad experimetal probability. r Compare experimetal results with theoretical probability. 52 Grade 7 Mathematics: Support Documet for Teachers

Suggestios for Istructio Determie the probability of a outcome occurrig for a probability experimet, ad express it as a ratio, fractio, or percet. Q Q Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Materials: Oe set per pair of studets: six-sided pecils otebooks markers BLM 7.SP.4.3: Experimetal Probability Tally Sheet ad Probability of Outcomes (optioal) Orgaizatio: Pairs, whole class Procedure: 1. Have studets, workig i pairs, mark each side of a six-sided pecil with oe possible outcome (e.g., umbers, studet ames, letters to form a word). 2. Studets coduct their probability experimet, proceedig as follows: a) Oe studet rolls the pecil by rubbig it betwee his or her palms, ad the lays a had o a otebook (to muffle the soud) ad allows the pecil to roll dow the had oto the otebook. b) The studet aouces the outcome that laded facig up. c) The other studet records the result o a tally sheet. d) Studets cotiue rollig ad recordig util they have completed a set umber of rolls, or util a set time has passed. 3. Studets total their tally marks, ad record the ratios of tallies for each outcome to the total umber of tallies (outcome : total). 4. Studets the record the probability for each outcome, expressig it as a fractio, a decimal, ad a percet. 5. Whe groups have completed their experimet, reassemble as a class ad ask studets what they have discovered or leared from the experiece. a) Compare the probabilities obtaied by differet groups. If studets have used the same outcomes, cosolidate the group data ad calculate the probability for the combied results. b) Studets may ote the sum of the values i the decimal colum totals about 1 (depedig o roudig), ad the sum of the percets is close to 100% (depedig o roudig). Statistics ad Probability 53

c) Ask studets for a example of a outcome that is impossible for this experimet, ad for a evet that is certai. Variatios: Have all studets mark their six-sided pecils with the same outcomes, ad combie the class results to represet a larger trial. Evets that are ot equally likely, such as a paper cup ladig upright, upside dow, or o its side, may be used for this experimet. Compare studets results ad discuss differeces observed. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the probability of a outcome occurrig for a probability experimet, ad express it as a ratio, fractio, or percet. r Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Suggestios for Istructio Q Q Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Materials: rulers (30 cm) two colours of pes or highlighters demostratio board Orgaizatio: Whole class, pairs 54 Grade 7 Mathematics: Support Documet for Teachers

Procedure: 1. Remid studets that probability is used to predict the likelihood that a evet will happe. Ask whether it is possible to predict ay evet with certaity. Aswer: The oly evets that ca be predicted with certaity are those with probabilities of 0% or 100%. 2. Ask studets for examples of evets with a probability of 0% (impossible) or 100% (certai). Example: If a draw box cotais oly odd umbers, the probability of drawig a eve umber is 0% ad the probability of drawig a odd umber is 100%. 3. Have studets create a probability lie similar to the Sample Probability Lie illustrated i the Backgroud Iformatio. Have studets iclude some or all of the followig vocabulary i their probability lie: impossible, less possible, more possible, certai less likely, equally likely, more likely less probable, more probable ever, sometimes, ofte, always 4. Have pairs of studets challege oe aother by describig situatios that could match the probability of a poit or regio o the probability lie. Example: Variatios: Studet A describes a situatio, ad asks for a evet that would match a probability descriptor: Give a bag cotaiig the letters of the alphabet, idetify a evet that would be less likely tha aother evet. Studet B replies: Give a bag cotaiig the letters of the alphabet, it is less likely you ll draw a vowel tha a cosoat. Studet B the challeges Studet A, ad may ask for a impossible evet i the same situatio. Studet A could reply: Give the letters of the alphabet, it is impossible to draw a umber. Provide a template of a probability lie with bechmarks, directioal arrows, ad blaks for studets to fill i. Q Q As a extesio, have studets idetify situatios with evets that have a 50% probability (or other value) of occurrig. Statistics ad Probability 55

Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Provide a example of a evet with a probability of 0 or 0% (impossible) ad a evet with a probability of 1 or 100% (certai). Suggestios for Istructio Provide a example of two idepedet evets, such as spiig a four-sectio spier ad a eight-sided die tossig a coi ad rollig a twelve-sided die tossig two cois rollig two dice ad explai why they are idepedet. Materials: BLM 7.SP.5.1: Which Coditios Affect Probability? cois colour couters bags ticket stubs with umbers (optioal) BLM 7.SP.5.2: Examples of Two Idepedet Evets (optioal) Orgaizatio: Small groups (of three or four studets), whole class Procedure: 1. Divide studets ito small groups. 2. Distribute copies of BLM 7.SP.5.1: Which Coditios Affect Probability? 3. Have each group determie the theoretical probability of each evet occurrig, decide whether specified coditios affect the theoretical probability of the evet, ad explai why or why ot. 4. Reassemble as a class ad discuss studets resposes. Probability is based o the umber of favourable outcomes ad the umber of possible outcomes. Oly coditios that alter the umber of outcomes (e.g., addig a extra item, ot replacig a item that has bee removed) will affect the theoretical probability of a evet. 56 Grade 7 Mathematics: Support Documet for Teachers

5. Iform studets of the followig: a) Whe oe evet does ot affect the probability of a outcome of aother evet, the evets are said to be idepedet evets. Ask studets to idetify the idepedet evets i the examples o BLM 7.SP.5.1. b) Whe oe evet does affect the probability of a outcome of aother evet, the evets are said to be depedet evets. Ask studets to idetify the depedet evets i the examples o BLM 7.SP.5.1. Have studets idetify other evets that would ot represet idepedet evets. 6. Ask studets whether rollig a umber cube before pullig a coloured block from a bag will have ay effect o the colour of the block draw. 7. Have studets retur to their small groups ad list pairs of idepedet evets, explaiig why the two evets are idepedet. Ask them to list some pairs of evets that are ot represetative of idepedet evets. Examples: Idepedet evets could iclude Variatios: choosig a card from a deck (or a marble, other couter, umber, or letter tile from a bag), returig it to the deck (or bag), ad drawig a secod item tossig two cois, umber cubes, or letter cubes radomly choosig two items (e.g., ames, umbers, meu items, clothig selectios, colours, movies, trasportatio methods, vacatio spots, games) from a selectio (The first choice must ot be removed from the set, for the evets to remai idepedet.) ay combiatio of the above items Provide cocrete opportuities for studets to experimet with determiig whether or ot probability is affected by certai coditios. For example, collect tickets or ames o slips ad coduct a draw to determie the probability of a particular wier. Coduct the experimet, first replacig a ame each time it is draw, ad the ot replacig a ame after it has bee draw. Provide a chart (such as BLM 7.SP.5.2: Examples of Two Idepedet Evets) for studets to list sets of two idepedet evets ad to explai why the evets are idepedet. Provide studets with a list of pairs of evets ad ask them to idetify whether or ot the pairs represet idepedet evets, ad to explai why the evets are or are ot idepedet. Statistics ad Probability 57

Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Provide a example of two idepedet evets, such as spiig a four-sectio spier ad a eight-sided die tossig a coi ad rollig a twelve-sided die tossig two cois rollig two dice ad explai why they are idepedet. Suggestios for Istructio Idetify the sample space (all possible outcomes) for a experimet ivolvig two idepedet evets usig a tree diagram, a table, or aother graphic orgaizer. Materials: cois (the same or differet deomiatios) demostratio board math jourals BLM 7.SP.4.3: Experimetal Probability Tally Sheet ad Probability of Outcomes (optioal) Orgaizatio: Pairs or small groups, whole class, idividual Procedure: 1. Make preparatios for havig studets ivestigate the probability of outcomes whe tossig two cois. a) Divide studets ito pairs or small groups. b) Iform studets they will be experimetig to determie the possible outcomes, ad the probability of each outcome, whe tossig two cois. Have studets create a recordig sheet for their experimet (or distribute copies of BLM 7.SP.4.3: Experimetal Probability Tally Sheet ad Probability of Outcomes). c) Tell studets they will idetify all possible outcomes ad predict the probability of each outcome. 58 Grade 7 Mathematics: Support Documet for Teachers

d) Discuss whether the procedure for tossig the cois (tossig both cois at oce, or tossig them oe after the other) will affect the results of the experimet. Neither will affect the probability; however, good experimetal techique esures that the same procedure is followed as closely as possible throughout a experimet. 2. Have studets coduct the ivestigatio i pairs or i small groups. a) Distribute two cois to each group. b) Have studets coduct their experimet for a give umber of coi tosses, or for a set period of time. c) Have studets total their results. d) Cosolidate studets data o the demostratio board. 3. Compare studets results to their predictios, ad discuss differeces or similarities. If studets predicted a probability of 1 for each outcome, the results will likely ot 3 support that predictio. This raises a opportuity for discussig how to determie the sample space (umber of possible outcomes) for two idepedet evets. 4. Calculate probability as the umber of favourable outcomes out of the umber of possible outcomes (the sample space). Number of Favourable Outcomes P Number of Possible Outcomes a) Agree that the favourable outcomes are two heads, two tails, or oe of each. b) Ask studets to idetify all possible outcomes. 5. Ask studets to demostrate, or demostrate for them, systematic ways to orgaize the sample space for two idepedet evets i a way that esures all possible outcomes are idetified. If studets do ot see the beefits of orgaizatioal charts, have them idetify the sample space for evets with multiple outcomes, such as rollig two umber cubes. 6. Orgaizers iclude the followig (see Backgroud Iformatio for examples): a) tree diagrams b) frequecy charts or tables c) orgaized lists 7. Have studets make a math joural etry illustratig methods to idetify the sample space (all possible outcomes) for two idepedet evets. Variatios: Use computer applets to simulate coi tosses. Replace the itroductory ivestigatio with a guided example of tossig two cois as two idepedet evets. Demostrate calculatig the sample space with a tree Statistics ad Probability 59

diagram, ad the with a frequecy chart or table, as illustrated i the Backgroud Iformatio. Demostrate usig orgaizers for idetifyig the sample space for a experimet ivolvig two idepedet evets, ad the have studets idetify such a experimet ad use two methods to idetify the correspodig sample space. Studets may use evets from the list they created i the previous learig experiece. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Idetify the sample space (all possible outcomes) for a experimet ivolvig two idepedet evets usig a tree diagram, a table, or aother graphic orgaizer. Suggestios for Istructio Determie the theoretical probability of a outcome for a experimet ivolvig two idepedet evets. Coduct a probability experimet for a outcome ivolvig two idepedet evets, with or without techology, to compare the experimetal probability to the theoretical probability. Materials: maipulatives (e.g., cois, umber cubes, colour couters, marbles, letter tiles, bags) software programs or olie applets (such as those listed i the Backgroud Iformatio) BLM 7.SP.6.1: Frequecy Chart for Orgaizig Outcomes for Two Idepedet Evets (optioal) Orgaizatio: Idividual or pairs Procedure: 1. Iform studets they will desig ad coduct a probability experimet ivolvig two idepedet evets, as outlied i the followig steps. a) Choose two idepedet evets. Previous lists provide examples of evets to choose from. b) Determie the sample space (all possible outcomes) by drawig a tree diagram, a frequecy chart, or aother orgaizer. BLM 7.SP.6.1: Frequecy Chart for Orgaizig Outcomes for Two Idepedet Evets may be provided as a template. 60 Grade 7 Mathematics: Support Documet for Teachers

c) Determie the theoretical probability of a outcome for the experimet. d) Coduct the experimet usig maipulatives or computer applets or software programs. e) Determie the experimetal probability for the outcome. f) Compare the experimetal probability to the theoretical probability. Iclude descriptive statemets ad umerical statemets i the comparisos. Propose a explaatio for variatios i the results. 2. Have studets preset their ivestigatios to each other, or post the ivestigatios for studets to view. Variatios: Assig studets two idepedet evets to use for the experimet. Have all studets work o the same ivestigatio, ad compile their results for comparig the theoretical ad experimetal probability. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie the theoretical probability of a outcome for a experimet ivolvig two idepedet evets. r Coduct a probability experimet for a outcome ivolvig two idepedet evets, with or without techology, to compare the experimetal probability to the theoretical probability. Statistics ad Probability 61

Suggestios for Istructio Solve a probability problem ivolvig two idepedet evets. Materials: demostratio board display board Orgaizatio: Whole class, pairs Procedure: 1. Idetify games of chace that ivolve two idepedet evets, such as the followig: Rock, Paper, Scissors (played with two people) games that ivolve rollig umber cubes (e.g., Lucky Seve) games that ivolve drawig marbles from a bag (e.g., Player A draws a marble from a bag cotaiig a give assortmet of marbles, ad returs the marble to the bag. Player B the draws a marble. If the colours match, Player B scores a poit; if they do t match, Player A scores the poit.) 2. Ask whether a game is fair, ad how oe judges whether or ot it is fair. 3. As a class, ivestigate a game of chace, such as Rock, Paper, Scissors, to determie whether or ot the game is fair. 4. Have idividuals or pairs of studets determie whether or ot other games of chace are fair. Have them provide evidece for their coclusios. 5. Provide opportuities for studets to play the games. 6. Use studets results to create a display etitled Would You Play This Game? Variatios: Have pairs of studets play a game based o the probability of two idepedet evets occurrig. After a certai time has passed, ask studets who thiks the game is fair. Challege studets to support their opiio usig what they kow about probability. Have studets desig a fair game based o the probability of two idepedet evets occurrig. Have them provide evidece that the game is fair. Host a Game Fair i which studets itroduce ad play their games with others. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a probability problem ivolvig two idepedet evets. 62 Grade 7 Mathematics: Support Documet for Teachers

Suggestios for Istructio Solve a probability problem ivolvig two idepedet evets. Materials: BLM.7.SP.6.2: Probability Problems Ivolvig Two Idepedet Evets file cards (optioal) Orgaizatio: Idividual, pairs Procedure: 1. Preset studets with probability problems ivolvig two idepedet evets, such as those o BLM.7.SP.6.2: Probability Problems Ivolvig Two Idepedet Evets. 2. Ask studets to complete the problems idepedetly ad the compare their resposes with those of a classmate, resolvig ay discrepacies. Variatios: Ask studets to desig their ow probability problems ad solutios. Have studets cosolidate problems as a assigmet sheet ad a aswer key, ad share questio sheets with others. Studets compare their resposes to the aswer key, ad discuss ay discrepacies with the authors. Place studet-created probability problems ad solutios o large file cards. Studets ca pick a card or two, ad complete the problems as part of a learig activity cetre or as a Exit Slip. Play olie probability games requirig players to solve probability problems ivolvig both simple ad idepedet evets. Sample Website: For a sample probability game, refer to the followig website: Math-Play.com. Probability Game. 2007. www.math-play.com/probability-game.html. Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Solve a probability problem ivolvig two idepedet evets. Statistics ad Probability 63

Puttig the Pieces Together Defiig the Average Potato Itroductio: Studets collect data, determie measures of cetral tedecy, cosider the rage ad the effect of outliers, ad aalyze circle graphs i order to defie the average potato. Purpose: I this ivestigatio, studets will demostrate the ability to do the followig (coectios to learig outcomes are idetified i paretheses): Determie measures of cetral tedecy (mea, media, ad mode) ad rage. (7.SP.1) Determie the most appropriate measures of cetral tedecy to report fidigs. (7.SP.1) Determie the effect o the mea, media, ad mode whe a outlier is icluded i the data. (7.SP.2) Costruct circle graphs to solve problems. (7.SP.3) Express probabilities as ratios, fractios, ad percets. (7.SP.4) Studets will also demostrate the followig mathematical processes: Commuicatio Coectios Metal Mathematics ad Estimatio Problem Solvig Reasoig Techology Materials/Resources: Each studet will require a potato a data sheet a way to label each potato (e.g., adhesive tape) Each group will require a tape measure or strig a ruler (cm) a mass scale 64 Grade 7 Mathematics: Support Documet for Teachers

a setup for measurig volume (e.g., water, a basi, a cotaier with a wide mouth filled to the brim that sits iside aother cotaier with a pour spout, a measurig cup to measure the overflow, towels for cleaig up spills) a calculator a compass a protractor Orgaizatio: Small groups (of four or five) Procedure: 1. Esure each member of your group has a potato, a assiged letter of the alphabet, ad a copy of the Data Sheet for Determiig the Average Potato. 2. Label your potato with the letter of the alphabet you were assiged. Give your potato a ame that begis with that letter, ad record the ame o your data sheet. Cout the umber of eyes o your potato, ad record the umber o your data sheet. 3. Measure ad record the legth aroud (cm), the breadth (cm), the mass (gr), ad the volume (ml) of your potato. 4. Whe the measurig is complete, have each perso i your group clearly read ad repeat his or her data aloud. Liste carefully to your classmates data ad record it accurately. 5. After all the data has bee collected, work as a group to defie the average potato. a) Have each group member take resposibility to calculate the measures of cetral tedecy for oe of the potato measures. b) Idetify the rage ad ay outliers, ad discuss the effect of the outliers o the measures of cetral tedecy. c) Prepare circle graphs for each potato measure to help defie the average potato. d) As a group, agree o a defiitio for a average potato. e) Determie the umber of potatoes that would fit your defiitio of a average potato. What is the experimetal probability that a potato i the group would be a average potato, accordig to your defiitio? Explai. f) Prepare to preset your group s defiitio ad a defece of that defiitio to the class. 6. Preset defiitios to the class, ad take part i a discussio. Does the class agree o a defiitio for a average potato? Explai. Statistics ad Probability 65

Observatio Checklist Liste to ad observe studets resposes to determie whether studets ca do the followig: r Determie measures of cetral tedecy. r Determie the affect of outliers icluded i the data. r Determie the most appropriate measures of cetral tedecy to report fidigs. r Costruct circle graphs to solve problems. Extesio (Optioal): Studets desig a average-potato lottery ad compare the experimetal results to the theoretical probability of wiig. Purpose: I this extesio, studets will demostrate the ability to do the followig (coectios to learig outcomes are idetified i paretheses): Express probabilities as ratios, fractios, ad percets. (7.SP.4) Defie the sample space for the probability experimet. (7.SP.5) Compare the theoretical ad experimetal probability of two idepedet evets. (7.SP.6) Procedure: Use the collected data ad the agreed-upo defiitio to hold a potato lottery. Defie the criteria for the lottery. The wier could receive the potatoes. Radomly select five or six potatoes from your group. Calculate the theoretical probability of selectig a average potato from the group twice i a row, give the first potato is retured to the buch after it has bee selected. Determie the experimetal probability of the same evet by holdig a class lottery. The potatoes could serve as the lottery prize. 66 Grade 7 Mathematics: Support Documet for Teachers

Data Sheet for Determiig the Average Potato Name Legth aroud (cm) Breadth (cm) Mass (gr) Volume (ml) Number of Eyes A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Total Mea Media Mode Statistics ad Probability 67

N o t e s 68 Grade 7 Mathematics: Support Documet for Teachers

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