Classifying Solutions to Systems of Equations

Transcription

1 CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Classifing Solutions to Sstems of Equations Mathematics Assessment Resource Service Universit of Nottingham & UC Berkele For more details, visit: MARS, Shell Center, Universit of Nottingham Ma be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at - all other rights reserved

2 Classifing Solutions to Sstems of Equations MATHEMATICAL GOALS This lesson unit is intended to help ou assess how well students are able to: Classif solutions to a pair of linear equations b considering their graphical representations. Use substitution to complete a table of values for a linear equation. Identif a linear equation from a given table of values. Graph and solve linear equations. COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics: 8.EE: Analze and solve linear equations and pairs of simultaneous linear equations. This lesson also relates to all the Standards for Mathematical Practice, with a particular emphasis on: 7. Look for and make use of structure. INTRODUCTION Before the lesson, students attempt the assessment task individuall. You then review students solutions and formulate questions that will help them improve their work. During the lesson, students work collaborativel in pairs or threes, plotting graphs, completing tables of values and deducing equations. Then, based on the number of common solutions, students link these representations. In a follow-up lesson, students receive our comments on the assessment task and use these to attempt the similar task, approaching it with insights gained from the lesson. MATERIALS REQUIRED Each student will need a cop of the assessment tasks Working with Linear Equations and Working with Linear Equations (revisited), a mini-whiteboard, eraser, and a pen. Each group of students will need cut up Card Set A: Equations, Tables & Graphs, two cut up copies of Card Set B: Arrows, one cop of Graph Transparenc, copied onto a transparenc, a transparenc pen, a large sheet of paper for making a poster, some plain paper, and a glue stick. Provide rulers if requested. There are some projector resources to support whole-class discussion. TIME NEEDED minutes before the lesson for the assessment task, a 8-minute lesson (or split into two shorter lessons), and minutes in a follow-up lesson (or for homework). All timings are approimate. Teacher guide Classifing Solutions to Sstems of Equations T-

3 BEFORE THE LESSON Assessment task: Working with Linear Equations ( minutes) Ask the students to do this task, in class or for homework, a da or more before the formative assessment lesson. This will give ou an opportunit to assess the work and identif students who have misconceptions or need other forms of help. You will then be able to target our help more effectivel in the net lesson. Give each student a cop of Working with Linear Equations. Introduce the task briefl, and help the class to understand what the are being asked to do. You ma want to eplain to the class the term common solution. Spend fifteen minutes working individuall, answering these questions. Show all our work on the sheet. Make sure ou eplain our answers reall clearl. It is important that students answer the questions without assistance, as far as possible. Classifing Solutions to Sstems of Equations Student Materials Beta Version Februar! "!"#!%!#!"# $!& Working with Linear Equations A B C D a. Which of these tables of values satisf the equation = +? Eplain how ou checked. b. B completing the table of values, draw the lines = + and = on the grid. '$ '# " ) (! & % $ # " '" * '" '#! "!"!&!#!% $!' " # $ % &!! = + = c. Do the equations = + and = have one common solution, no common solutions, or infinitel man common solutions? Eplain how ou know.. Draw a straight line on the grid that has no common solutions with the line = +. What is the equation of our new line? Eplain our answer.! "!" $% $' " $& #! "! "!" % - 7 #$! "#! MARS Universit of Nottingham S- Students should not worr too much if the cannot understand or do everthing because ou will teach a lesson using a similar task, which should help them. Eplain to students that b the end of the net lesson, the should epect to answer questions such as these. This is their goal. Assessing students responses Collect students responses to the task. Make some notes on what their work reveals about their current levels of understanding and difficulties. The purpose of this is to forewarn ou of the issues that will arise during the lesson, so that ou can prepare carefull. We suggest that ou do not score students work. The research shows that this is counterproductive, as it encourages students to compare scores and distracts their attention from how the ma improve their mathematics. Instead, help students to make further progress b summarizing their difficulties as a list of questions. Some suggestions for these are given in the Common issues table on the net page. We suggest that ou make a list of our own questions, based on our students work, using the ideas in the table. We recommend ou write one or two questions on each student s work. If ou do not have time to do this, ou could: Select a few questions that will be of help to the majorit of students, and write these questions on the board when ou return the work to the students. Or give each student a printed version of our list of questions and highlight the questions for each individual student. Teacher guide Classifing Solutions to Sstems of Equations T-

4 Common issues: Student assumes that onl one of the tables satisfies the equation = + (Q) For eample: The student selects onl table A. Student makes an incorrect assumption about the multiplicative properties of zero (Q & Q) For eample: The student assumes + =. The then ma select Table B as satisfing the equation = + (Q) Student applies the rules for multipling negative numbers incorrectl (Q & Q) For eample: The student assumes + = The then ma select Table C as satisfing the equation = + (Q) Student provides little or no eplanation (Q) Student incorrectl draws the graph For eample: The student draws a non-linear graph. Student uses guess and check to complete the tables of values (Qb) Student states that the two equations, = + and = have no common solutions (Qc) For eample: The student fails to etend the line = beond the values in the table. This means the two lines do not intersect. Student provides little or no eplanation (c and ) Student either does not plot a line that has no common solutions with the line = + or plots it incorrectl (Q) Student uses guess and check to figure out the equation of the line (Q) Suggested questions and prompts: Are there more than three pairs of values that satisf the equation = +? Have ou checked the values for and in each of the tables? Is the same as? Use addition to figure out two multiplied b zero. [E.g. + =.] Is the same as? What method did ou use when checking which tables satisf the equation? Write what ou did. On our graph, is the slope alwas the same? Does this agree with the equation of the graph? How can ou check ou have plotted the graph correctl? Can ou think of a quicker method? Would changing the subject of the equation help ou figure out some of the values? What does common solution mean? Are there an other points that satisf the equation =? Plot some. Eplain wh ou think our answer is correct. Sketch two lines that have no common solutions. What propert do the share? [The lines will be parallel.] Can ou think of a quicker method? What can ou tell me about two lines with no common solution? Give me two equations that have no common solution. Teacher guide Classifing Solutions to Sstems of Equations T

5 SUGGESTED LESSON OUTLINE Whole-class introduction ( minutes) Give each student a mini-whiteboard, pen, and eraser. Maimize participation in the discussion b asking all students to show ou solutions on their mini-whiteboards. Write the equation = + on the board. Ask the following questions in turn: If = what does equal? [7] Ask students to eplain how the arrived at their answer. If a variet of values are given within the class, discuss an common mistakes and eplore different strategies. If = - what does equal? [-] If students are struggling with multipling b a negative number, ask the class to summarize the results of multipling with positives and negatives. Some students ma believe that because and are different letters, the have to take different values. Point out that here both and can both be equal to -. If = 8 what does equal? [] If = what does equal? [-⅔] Students ma either use guess and check or rearrange the equation in order to figure out the value for. You ma want to discuss these two strategies. Students often think that the have made a mistake when the get an answer that is not a whole number. Discuss the value of checking an answer b substituting it back in as, as well as emphasizing that not all solutions will be positive integers and that negative and fractional solutions can also occur. It ma also be appropriate to discuss the benefits of leaving answers in fraction form rather than converting to a decimal, especiall when a recurring decimal will result. Provide an eample of sa, ½, and discuss the difference between this fraction epressed as a decimal, and -⅔ epressed as a decimal, in terms of accurac and rounding. How can ou check our answer? [B substituting it back in as.] How can ou check that all our answers are correct? [Sketch the coordinates on a grid and see if the form a straight line.] If students work on the assessment task has highlighted issues with plotting points and making connections between solutions to a linear equation and points on a straight line graph, it ma be appropriate to ask students to check that the solutions for the equation = + form a straight line when plotted. Eplain to students that in the net activit the will be using their skills of substitution and solving equations to help them to investigate graphical representations of linear equations. Collaborative activit: Card Set A: Equations, Tables & Graphs ( minutes) Organize students into pairs. For each pair provide a cut up cop of Card Set A: Equations, Tables & Graphs and some plain paper. Teacher guide Classifing Solutions to Sstems of Equations T-

6 These si cards each include a linear equation, a table of values and a graph. However, some of the information is missing. In our pairs, share the cards between ou and spend a few minutes, individuall, completing them. You ma need to do some calculations to complete the cards. Do these on the plain paper and be prepared to eplain our method to our partner. Once ou have had a go at filling in the cards on our own, take turns to eplain our work to our partner. Your partner should check our cards and challenge ou if the disagree. It is important that ou both understand and agree on the answers for each card. When completing the graphs, take care to plot points carefull and make sure that the graph fills the grid in the same wa as it does on Cards C and C. Slide P-: Card set A: Equations, Tables, Graphs on the projector resource summarizes these instructions. If students are struggling, suggest that the focus on Cards C and C first. It does not matter if students are unable to complete all si cards. It is more important that the can confidentl eplain their strategies and have a thorough understanding of the skills the are using. For students who complete all the cards successfull and need an etension, ask them to spend a few minutes comparing their completed cards: Select two cards and note on our whiteboards an common properties of the equations and/or the graphs. Repeat this for all of our completed cards. This will help ou later. While students work in small groups ou have two tasks: to make a note of students approaches to the task, and to support student reasoning. Note student approaches to the task Listen and watch students carefull and note an common mistakes. For eample, are students misinterpreting the slope and intercept on cards where the graph has alread been drawn? Do the fail to recognize an equation/graph that has a negative gradient? You ma want to use the questions in the Common issues table to help address misconceptions. Also notice the wa in which students complete the cards. Do students use the completed table of values to plot the graph or do the use their knowledge of slope and intercept to draw the graph directl from the equation? Do students first plot the line using eas values for or, and then read off values from the graph in order to complete the table? Do students rearrange the equation or do the use guess and check to solve for or? Do students use multiplication to eliminate the fraction from the equation? Do students use the slope and intercept or guess and check to figure out the equation of the graph? You will be able to use this information in the whole-class discussion. Support student reasoning Tr not to make suggestions that move students towards a particular strateg. Instead, ask questions to help students to reason together. Martha completed this card. Jordan, can ou eplain Martha s work? Show me a different method from our partner s to check their method is correct. If ou find the student is unable to answer this question, ask them to discuss the work further. Eplain that ou will return in a few minutes to ask a similar question. How can ou check the card is correct? [Read off coordinates from the line, use the slope and intercept to check the equation matches the line, etc.] Teacher guide Classifing Solutions to Sstems of Equations T-

7 For each card, encourage students to eplain their reasoning and methods carefull. How do ou know that = when = in Card C? What method did ou use? How did ou find the missing equation on Card C/C? Show me a different method. Suppose ou multipl out the equation on Card C. What information can ou then deduce about the graph? [The -intercept and slope.] Which of these equations is arranged in a wa that makes it eas to draw a graph using information about the line s -intercept and slope? [C.] What are the? [ and -½.] You ma find some students struggle when the slope of a line is negative or when dealing with negative signs, or when the slope is a fraction. Checking work ( minutes) Ask students to echange their completed cards with another pair of students. Carefull check the cards and point out an answers ou think are incorrect. You must give a reason wh ou think the card is incorrect but do not make changes to the card. Once students have checked another group s cards, the need to review their own cards taking into account comments from their peers and make an necessar changes. Etending the lesson over two das If ou are taking two das to complete the unit then ou ma want to end the first lesson here. At the start of the second da, briefl remind students of their previous work before moving on to the net collaborative activit. Collaborative activit: Using Card Set B to link Card Set A ( minutes) Give each pair two copies of Card Set B: Arrows (alread cut-up), a cop of Graph Transparenc, a transparenc pen, a large sheet of paper for making a poster, and a glue stick. Choose two of our completed cards from Card Set A and stick them on our poster paper with a gap in between. You are going to tr and link these cards with one of the arrows. The cards will either have no common solutions, one common solution or infinitel man common solutions. Select the appropriate arrow and stick it on our poster between the two cards. If the cards have one common solution, ou will need to complete the arrow with the values of and where this solution occurs. Add another completed card to our poster and compare it with the two alread stuck down. Find arrows that link this third card with the other two and stick the cards down. Continue to compare all the cards in this wa, making as man links as possible. If some of the cards are incomplete, ou will need to complete them before comparing them. Slide P-: Card set B: Arrows on the projector resource summarizes these instructions. Some students might find it helpful to use a transparenc when comparing the cards. How might ou use the Graph Transparenc on our desk to help ou to determine how man solutions the two cards have in common? If students are struggling to identif how to use the transparenc, ask them if tracing one of the graphs onto the transparenc might be helpful. Some students ma prefer to not use the transparenc. Notice how students are working and their method for completing the task. Are an students reling purel on the algebraic representation of the equation? Once students have recognized that there is one common solution, are the checking the solution algebraicall as well as using the graphs? Teacher guide Classifing Solutions to Sstems of Equations T-

8 As students work on the comparisons, support them as before. Again ou ma want to use some of the questions in the Common issues. Walk around and ask students to eplain their decisions. The finished poster produced ma look like this: Whole-class discussion ( minutes) Once groups have completed their posters, displa them at the front of the room. Based on what ou have learned about our students strategies and the review of their posters, select one or two groups to eplain how the went about addressing the task (if possible select groups who have taken ver different approaches to the task). As groups eplain their strategies, ask if anone has a question for the group or if anone used a similar strateg. When a few groups have had a chance to share their approach, consolidate what has been learned. Using mini-whiteboards to encourage all students to participate, ask the following questions in turn:. Show me two equations that have one common solution. [E.g. = + and = - ½ + ]. What are the solution values for and? [ = and =.] What happens to the graphs at this point? [The intersect each other.] On our mini-whiteboards make up two more equations that have one common solution. Don t use equations that appear on the cards. Sketch their graphs. Now show me!. Show me two equations that have no common solutions. [E.g. = + and =.] Teacher guide Classifing Solutions to Sstems of Equations T-7

9 How do ou know the have no common solution? [The are parallel lines so will never intersect.] What do ou notice about these two equations? [The have the same coefficient of /same slope.] On our mini-whiteboards make up two more equations that have no common solutions. Don t use equations that appear on the cards. Sketch their graphs. Now show me!. Show me two equations with infinitel man common solutions. [E.g. = + and = ( + ).] What do ou notice about the two graphs for these equations? [The are the same line.] Wh is this? [( + ) is + in factorized form.] The focus of this discussion is to eplore the link between the graphical representations of the equations and their common solutions, even though students ma have used both the algebraic representation and the table of values during the classification process. Help students to recognize that solutions to a sstem of two linear equations in two variables correspond to the points of intersection of their graphs, as well as what it means graphicall when there are no or infinitel man common solutions. Follow-up lesson: Working with Linear Equations (revisited) ( minutes) Give back the responses to the original assessment task to students and a cop of the task Working with Linear Equations (revisited.) Ask students to look again at their solutions to the original task together with our comments. If ou have not added questions to individual pieces of work then write our list of questions on the board. Students should select from this list onl those questions the think are appropriate to their own work. Look at our solutions to the original task Working with Linear Equations and read through the questions I have written. Use what ou have learned to answer these questions. Using what ou have learned, have a go at the second sheet: Working with Linear Equations (revisited). Some teachers give this as a homework task. Teacher guide Classifing Solutions to Sstems of Equations T-8

10 SOLUTIONS Assessment task: Working with Linear Equations a. Tables A and D satisf the equation = +. b. Table B satisfies the equation = + and table C is non-linear = + = - = + =. c. The two graphs have one common solution at =, =. This is the point of intersection of the two graphs.. Students can draw an line that has the same slope as = +. For eample = or = + etc. Lesson task: Card Sets A and B The si cards in Card Set A describe the four straight lines below: = - = + = ( + ) = ½ - = -½ + + = 8 Teacher guide Classifing Solutions to Sstems of Equations T-9

11 C = + C = ( + ) C + = 8 C = - ½ + C = C = ½... Infinitel man common solutions + = 8 and = ½ + + = 8 (C) is a rearrangement of = ½ +. = ( + ) and = + = ( + ) (C) is the factorized form of = +. No common solutions = + and = = ½ + and = ½ One common solution Equal slopes. Equal slopes. = + (or = ( + )) and = ½ + (or + = 8) have one common solution at (,). = + (or = ( + )) and = ½ have one common solution at (.,). = and = ½ + (or + = 8) have one common solution at (,). = and = ½ have one common solution at (.,). Assessment task: Working with Linear Equations (revisited) a. Tables B and D satisf the equation = +. b. Table A is non-linear and table C satisfies the equation = = + = - = + = c. The two graphs have one common solution at =, =. This is the point of intersection of the two graphs.. Students can draw an line that has the same slope as = +. For eample = or = + etc. Teacher guide Classifing Solutions to Sstems of Equations T-

12 Working with Linear Equations A B C D a. Which of these tables of values satisf the equation = +? Eplain how ou checked. b. B completing the table of values, draw the lines = + and = on the grid = + = c. Do the equations = + and = have one common solution, no common solutions, or infinitel man common solutions? Eplain how ou know. -. Draw a straight line on the grid that has no common solutions with the line = +. What is the equation of our new line? Eplain our answer. Student Materials Classifing Solutions to Sstems of Equations S- MARS, Shell Center, Universit of Nottingham

13 Card Set A: Equations, Tables & Graphs C = _ C + = Student Materials Classifing Solutions to Sstems of Equations S- MARS, Shell Center, Universit of Nottingham

14 Card Set A: Equations, Tables & Graphs (continued) C = C = ( + ) Student Materials Classifing Solutions to Sstems of Equations S MARS, Shell Center, Universit of Nottingham

15 Card Set A: Equations, Tables & Graphs (continued ) C = C Student Materials Classifing Solutions to Sstems of Equations S- MARS, Shell Center, Universit of Nottingham

16 Card Set B: Arrows No common solutions No common solutions Infinitel man common solutions Infinitel man common solutions One common solution when =, = One common solution when =, = One common solution when =, = One common solution when =, = One common solution when =, = One common solution when =, = One common solution when =, = One common solution when =, = Student Materials Classifing Solutions to Sstems of Equations S- MARS, Shell Center, Universit of Nottingham

17 = Graph Transparenc Student Materials Classifing Solutions to Sstems of Equations S- MARS, Shell Center, Universit of Nottingham

18 Working with Linear Equations (revisited) 8 A B C D a. Which of these tables of values satisf the equation = +? Eplain how ou checked. b. B completing the table of values, draw the lines = + and = on the grid = + = c. Do the equations = + and = have one common solution, no common solutions, or infinitel man common solutions? Eplain how ou know.. Draw a straight line on the grid that has no common solutions with the line = +. What is the equation of our new line? Eplain our answer. Student materials Classifing Solutions to Sstems of Equations S-7 MARS, Shell Center, Universit of Nottingham

19 Card Set A: Equations, Tables, Graphs. Share the cards between ou and spend a few minutes, individuall, completing the cards so that each has an equation, a completed table of values and a graph.. Record on paper an calculations ou do when completing the cards. Remember that ou will need to eplain our method to our partner.. Once ou have had a go at filling in the cards on our own: Eplain our work to our partner. Ask our partner to check each card. Make sure ou both understand and agree on the answers.. When completing the graphs: Take care to plot points carefull. Make sure that the graph fills the grid in the same wa as it does on Cards C and C. Make sure ou both understand and agree on the answers for ever card. Projector resources Classifing Solutions to Sstems of Equations P-

20 Card Set B: Arrows. You are going to link our completed cards from Card Set A with an arrow card.. Choose two of our completed cards and decide whether the have no common solutions, one common solution or infinitel man common solutions. Select the appropriate arrow and stick it on our poster between the two cards.. If the cards have one common solution, complete the arrow with the values of and where this solution occurs.. Now compare a third card and choose arrows that link it to the first two. Continue to add more cards in this wa, making as man links between the cards as possible. Projector resources Classifing Solutions to Sstems of Equations P-

21 Mathematics Assessment Project Classroom Challenges These materials were designed and developed b the Shell Center Team at the Center for Research in Mathematical Education Universit of Nottingham, England: Malcolm Swan, Nichola Clarke, Clare Dawson, Sheila Evans, Colin Foster, and Marie Joubert with Hugh Burkhardt, Rita Crust, And Noes, and Daniel Pead We are grateful to the man teachers and students, in the UK and the US, who took part in the classroom trials that plaed a critical role in developing these materials The classroom observation teams in the US were led b David Foster, Mar Bouck, and Diane Schaefer This project was conceived and directed for The Mathematics Assessment Resource Service (MARS) b Alan Schoenfeld at the Universit of California, Berkele, and Hugh Burkhardt, Daniel Pead, and Malcolm Swan at the Universit of Nottingham Thanks also to Mat Crosier, Anne Flode, Michael Galan, Judith Mills, Nick Orchard, and Alvaro Villanueva who contributed to the design and production of these materials This development would not have been possible without the support of Bill & Melinda Gates Foundation We are particularl grateful to Carina Wong, Melissa Chabran, and Jamie McKee The full collection of Mathematics Assessment Project materials is available from MARS, Shell Center, Universit of Nottingham This material ma be reproduced and distributed, without modification, for non-commercial purposes, under the Creative Commons License detailed at All other rights reserved. Please contact map.info@mathshell.org if this license does not meet our needs.