Problem 2: Interpreting Data and Identifying Linear Equations

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1 Problem 2: Iterpretig Data ad Idetifyig Liear Equatios This algebra problem assesses studets abilities to iterpret data from a table ad use that iformatio to idetify the equatio of a lie, the slope, ad the y-itercept. Studets i grade 8 are expected to model the relatioships betwee ad amog data, testig to determie whether the relatioship is liear or oliear. They should be egaged i describig situatios that are liear or oliear ad use fiite differeces to determie whether data i a table are liear. The data i the table show the cost of retig a bicycle by the hour, icludig a deposit. Retig a Bicycle Hours (h) Cost i dollars (c) If hours, h, were graphed o the horizotal axis ad cost, c, were graphed o the vertical axis: Are the data liear or oliear? How do you kow? What is the equatio of the lie that best fits the data? If the data are liear, what is the slope of this lie? What is the y-itercept? Source: Adapted from STAR (Stadardized Testig ad Reportig), Califoria Departmet of Educatio, Sample Test Questios, Grade 8, 2009 Curricular focus This problem aligs with the followig Focal Poits, Priciples ad Stadards, ad Commo Core State Stadards. Focal Poits Algebra: Aalyzig ad represetig liear fuctios ad solvig liear equatios ad systems of liear equatios. Studets use liear fuctios, liear equatios, ad systems of liear equatios to represet, aalyze, ad solve a variety of problems. Copyright 2011 by the Natioal Coucil of Teachers of Mathematics, Ic. All rights reserved. 1

2 Priciples ad Stadards Represet ad aalyze mathematical situatios ad structures by usig algebraic symbols. Use symbolic algebra to represet situatios ad to solve problems, especially those that ivolve liear relatioships. Commo Core State Stadards Uderstad the coectios betwee proportioal relatioships, lies, ad liear equatios. Aalyze ad solve liear equatios. Assessig prior kowledge Sample rage questios: a. Give data i a table, how might you determie whether the relatioship is liear? b. Give data i a table that are liear, how might you determie the slope? c. The total cost, c, i dollars of retig a motor scooter for d days is give by the equatio c = d. List everythig you ca determie from the equatio. d. List everythig you ca for the expressio a Assessig studets at work Ways to assess studets iclude hige questios, gallery walks, accoutable talk, ad observatio protocols. Liste for accoutable talk. Ecourage studets to iclude appropriate mathematical laguage that uses the terms slope, y-itercept, liear, variable, table, idepedet variable, ad depedet variable. Gallery walk: Hag a series of easel-sized paper sheets with various tables aroud the room. (See the followig sample tables.) For each table, istruct the studets to do the followig: Idetify whether the data represeted are liear or oliear. Write a equatio of the lie represeted by the data if possible. Sketch a graph that best represets the data i the table. Create two questios that someoe might ask about each table or create a situatio that ay table represetig a liear relatioship ca describe. Sample tables follow (be sure to use variables other tha x ad y so that studets realize that other variables act i the same maer as x ad y ad may better defie the situatio beig represeted): Copyright 2011 by the Natioal Coucil of Teachers of Mathematics, Ic. All rights reserved. 2

3 s t c d a b m Use your observatio protocol to determie which studets show cofidece with the mathematics ad which studets eed further experieces with the differet represetatios. For studets who demostrate proficiecy, challege them to write problems that the data i the tables could represet. Studet thikig For each of the followig studets we suggest appropriate questios or strategies that may help them make better sese of the problem ad/or how to represet the data give. Studet A Studet A made oe of the most commo errors. Give a table, may studets simply examie its output sectio to determie what the relatioship is, perhaps because most of their experieces iclude a icrease of 1 i the iput. Whe the iput icreases by 1, studets do ot eed to thik of the ratio betwee the iput ad the output; rather, they thik additively istead of multiplicatively. Directios requestig that studets explai their thikig or justify their work would also have bee helpful. Copyright 2011 by the Natioal Coucil of Teachers of Mathematics, Ic. All rights reserved. 3

4 Studet B Studet B chose to use a graph to determie whether the relatioship is liear or oliear. The studet appears to have draw all her coclusios from the shape of the graph. The studet also drew i the rise ad the ru, showed the ratio betwee them, ad simplified the ratio ad used that as the basis for the equatio. It appears as if the studet foud the y-itercept by ispectio. Agai, sice the problem did ot require justificatio, we ca oly speculate o what the studet was thikig. Likig assessmet to istructio You ca use exit cards, gallery walks, iterviews, or itervetios based o studet miscoceptios to lik assessmet to istructio. Copyright 2011 by the Natioal Coucil of Teachers of Mathematics, Ic. All rights reserved. 4

5 Sample Exit Cards Explai at least two ways i which you ca determie the slope of a lie give a table with a costat rate of chage. Explai what it meas if you get a costat rate of chage after fidig a secod fiite differece i a data table. What are two differet ways of thikig about the slope of a lie? Does this table represet a liear or oliear relatioship? Explai your reasoig. r s What are some cosideratios you must make whe determiig the equatio of a lie from a table? (Assume that the relatioship is liear.) Copyright 2011 by the Natioal Coucil of Teachers of Mathematics, Ic. All rights reserved. 5