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