Eighth Grade: Mathematics Model Lesson for Unit #1: Exploring Linear and Inverse Relationships

Transcription

1 Explring Linear and Inverse Relatinships Eighth Grade: Mathematics Mdel Lessn fr Unit #1: Explring Linear and Inverse Relatinships Overarching Questin: What des it mean when we see cnstant and predictable changes in a table f data r a graph? Previus Unit: This Unit: Explring Linear and Inverse Relatinships Next Unit: Right Triangle Relatinships Questins t Fcus Assessment and Instructin: 1. What d the slpe and y-intercept f a line represent in a realwrld situatin? 2. Hw can data in a table r scatterplt be used t predict a future utcme? Intellectual Prcesses(Standards fr Mathematical Practice) Use apprpriate tls strategically: Use graphing tls t mdel linear change in tables, graphs, and equatins. Mdel with mathematics: Identify prprtinal relatinships and make cnnectins t linear functins. Key Cncepts: slpe direct variatin cnstant f prprtinality y-intercept scatterplt line f best fit slutin prprtinal relatinship This dcument is the prperty f MAISA. Page 1 f 5

2 Explring Linear and Inverse Relatinships Lessn Abstract Students participate in tw data cllectin activities which intrduce direct and inverse variatin. In the pre-assessment, students categrize numerical, graphical and verbal representatins as linear r nnlinear and determine which linear representatins are als direct variatin relatinships. In the pst-assessment activity, students cnvey their understanding f linear, nnlinear, direct and inverse relatinships thrugh verbal and written cmmunicatin. This three-part lessn prvides pprtunities fr students t wrk individually, with partners and in small grups. During activities, mnitr student prgress in rder t make infrmal assessments f student understanding and t prvide interventin, when necessary. This lessn plan fcuses n the direct variatin prtin f the lessn. Cmmn Cre State Standards Expressins and Equatins (8.EE) Understand the cnnectins between prprtinal relatinships, lines, and linear equatins. 5. Graph prprtinal relatinships, interpreting the unit rate as the slpe f the graph. Cmpare tw different prprtinal relatinships represented in different ways. Fr example, cmpare a distance-time graph t a distance-time equatin t determine which f tw mving bjects has greater speed. Functins (8.F) Define, evaluate, and cmpare functins. 1. Understand that a functin is a rule that assigns t each input exactly ne utput. The graph f a functin is the set f rdered pairs cnsisting f an input and the crrespnding utput. 3. Interpret the equatin y = mx + b as defining a linear functin, whse graph is a straight line; give examples f functins that are nt linear. Fr example, the functin A = s 2 giving the area f a square as a functin f its side length is nt linear because its graph cntains the pints (1,1), (2,4) and (3,9), which are nt n a straight line. Use functins t mdel relatinships between quantities. 3. Cnstruct a functin t mdel a linear relatinship between tw quantities. Determine the rate f change and initial value f the functin frm a descriptin f a relatinship r frm tw (x, y) values, including reading these frm a table r frm a graph. Interpret the rate f change and initial value f a linear functin in terms f the situatin it mdels, and in terms f its graph r a table f values. 4. Describe qualitatively the functinal relatinship between tw quantities by analyzing a graph (e.g., where the functin is increasing r decreasing, linear r nnlinear). Sketch a graph that exhibits the qualitative features f a functin that has been described verbally. This dcument is the prperty f MAISA. Page 2 f 5

3 Explring Linear and Inverse Relatinships Statistics and Prbability (8.SP) Investigate patterns f assciatin in bivariate data. 2. Knw that straight lines are widely used t mdel relatinships between tw quantitative variables. Fr scatter plts that suggest a linear assciatin, infrmally fit a straight line, and infrmally assess the mdel fit by judging the clseness f the data pints t the line. 3. Use the equatin f a linear mdel t slve prblems in the cntext f bivariate measurement data, interpreting the slpe and intercept. Fr example, in a linear mdel fr a bilgy experiment, interpret a slpe f 1.5 cm/hr as meaning that an additinal hur f sunlight each day is assciated with an additinal 1.5 cm in mature plant height. Instructinal Resurces: timers r stpwatches Sequence f Lessn Activities Lessn Title: X Marks the Spt ( This link takes yu t a three part lessn. Teachers can begin reading n page 1, but the X Marks the Spt Activity actually begins n page 4. Selecting and Setting up a Mathematical Task: By the end f this lessn what d yu want yur students t understand, knw, and be able t d? Students will identify linear and nn-linear relatinships in tables, graphs, equatins, and verbal descriptins and defend their chices. Students will graph a scatterplt frm cllected data and lk fr patterns in the table and graph that cnnect the tw variables used. Students can explain the meaning f direct variatin as well as cmparing similarities and differences f direct variatin and linear relatinships. Students will lk fr the cnstant f prprtinality and cnnect it t the table f values, graph, and equatin. In what ways des the task build n student s previus knwledge? What questins will yu ask t help students access their prir knwledge? Students will draw n their previus study f linear relatinships t categrize graphs, tables, equatins, and verbal descriptins t determine whether the relatinship is linear r nn-linear. Students shuld cme t the task with an understanding f prprtinal relatinships and the cnstant f prprtinality. Make a table f values that demnstrates a linear relatinship. What key characteristic(s) did yu use t shw a linear relatinship? What des a graph f a linear relatinship lk like? Why des the graph lk this way? What numerical relatinship between variables makes linear graphs appear different frm ne anther? The same as ne anther? What specific symbls are needed fr an equatin t make a linear graph? In describing the cnnectin between tw variables, what abut their relatinship makes it linear in nature? This dcument is the prperty f MAISA. Page 3 f 5

4 Explring Linear and Inverse Relatinships Launch: What des it mean fr tw variables t be prprtinal? Hw will yu intrduce students t the activity s as t prvide access t all students while maintaining the cgnitive demands f the task? What will be heard that indicates that the students understd what the task is asking them t d? Students will be given a cllectin f linear and nn-linear situatins in a variety f representatins (table, graph, equatin, verbal descriptin). Individually have them srt these int tw categries, linear and nn-linear. Next have them cmpare with anther student. Have class members share their results including an explanatin f hw they determined which categry t place them in. In small grups ask students t think abut and list situatins that mdel linear and nn-linear relatinships and respnd t the fllwing questin: What abut the situatin yu prvided made the situatin linear r nnlinear? Have students lk at their linear cllectin and see if there is a way t srt them further. Prmpt them t cnsider y-intercepts. Ask what they think a direct relatinship wuld mean. Have students subdivide the linear grup int direct and nn-direct categries. Students will be able t distinguish between linear and nn-linear relatinships and describe characteristics such as straight-line graph, cnstant rate f change, always changes by the same amunt, equatin culd be a single number(cnstant) r cntains a variable such as x. In distinguishing between direct and linear relatinships, they shuld recgnize that all direct variatin prblems are linear graphs that must pass thrugh the rigin. Supprting Student s Explratin f the Task: What questins will be asked t fcus students thinking n the key mathematics ideas? What questins will be asked t assess student s understanding f key mathematics ideas? What questins will be asked t encurage all students t share their thinking with thers r t assess their understanding f their peer s ideas? What abut the relatinship between the tw variables, time and number f X s, causes the graph t appear as it des? What abut the cllectin f data may cause the graph t nt be perfectly linear? What is the difference in a table f values between linear and nn-linear relatinships? In cnducting this experiment, what are sme factrs that yu culd change s that the table results wuld mdel a nn-linear relatinship? If yu were t draw a line f best fit t mdel yur data, are there any specific pints yu wuld want t include in yur line? Why? Have students generate a list f ideas abut hw t make the utcme f the experiment nn-linear. Ask students t explain why they think each scenari listed wuld cause the desired effect. What in the descriptin f the experiment leads t a nn-linear relatinship? Hw will yu extend the task t prvide additinal Have students give an explanatin abut hw the data cllectin prcess culd be mdified fr specific cases, e.g. a negative linear relatinship, an This dcument is the prperty f MAISA. Page 4 f 5

5 Explring Linear and Inverse Relatinships challenge? expnential grwth pattern, a randm pattern, etc. Is there smething in the rate f change that was affected? What abut the manner in which the data was cllected changed the pattern? Sharing and Discussing the Task: What specific questins will be asked s that all students will: Why did this activity prduce data with a linear-like relatinship between variables? Make sense f the mathematical ideas that yu wanted them t learn? Expand n, debate, and questin the slutins being shared? Make cnnectins between the different strategies that are presented? Lk fr patterns? Begin t frm generalizatins? Des everyne s graph lk the same? What differences might yu expect t see because f different peple writing the X s? What factrs may have cntributed t the data nt being perfectly linear? Hw des a direct relatinship vary frm a linear relatinship? What abut this experiment leads t a direct relatinship? What prprtinal relatinship exists in the table f data yu cllected? Hw can this be seen in yur scatterplt? Once yu draw a line f best fit n yur scatterplt, hw culd the prprtinal relatinship f the tw variables be used t write an equatin fr the line f best fit? What will be seen r heard that indicates all students understand the mathematical ideas yu intended them t learn? Frmative Assessment: Teachers culd use questins frm the Gt Mail? assessment in the lessn. These questins shuld fcus n the characteristics f direct variatin. Anther fcus shuld be n the linear and nn-linear patterns f change and hw these are expressed verbally, in tables f values, graphs, and equatins. Sample questins frm Gt Mail? 1. Explain why the X Marks the Spt activity represents a direct variatin. Explain hw the activity culd be changed s that it n lnger results in a direct variatin relatinship. 2. Describe a situatin, ther than X Marks the Spt that represents a direct variatin relatinship. 4. Cmplete the table s that it mdels a direct variatin relatinship and write an equatin that prduces the table s values. x y Draw a graph that represents a direct variatin. 6. Draw a secnd graph that is linear, but des nt represent a direct variatin. This dcument is the prperty f MAISA. Page 5 f 5