# Solving Linear Equations in Two Variables

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1 CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Solving Linear Equations in Two Variables Mathematics Assessment Resource Service University of Nottingham & UC Berkeley Beta Version For more details, visit: MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at - all other rights reserved

2 Solving Linear Equations in Two Variables MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students are able to formulate and solve problems using algebra and, in particular, to identify and help students who have the following difficulties: Solving a problem using two linear equations with two variables. Interpreting the meaning of algebraic expressions. COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics: A-CED: Create equations that describe numbers or relationships. A-REI: Solve systems of equations. This lesson also relates to the following Standards for Mathematical Practice in the Common Core State Standards for Mathematics: 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. INTRODUCTION This lesson is structured in the following way: Before the lesson, students work individually on the assessment task Notebooks and Pens. You then review their work and create questions for students to answer in order to improve their solutions. During the lesson, students work individually on a task that requires them to interpret and solve two equations in two variables. Students then compare and discuss their solutions in small groups. In the same small groups, students evaluate some sample solutions of the same task. In a whole-class discussion, students explain and compare the alternative solution strategies they have seen and used. Finally, students use what they have learned to revise their work on Notebooks and Pens. MATERIALS REQUIRED Each individual student will need two copies of the assessment task Notebooks and Pens and a copy of the lesson task Cash Registers. Each small group of students will need a blank sheet of paper and copies of the four sheets Sample Student Work. Graph paper should be kept in reserve and used only when requested. Projector resources are provided to support the whole-class discussion. TIME NEEDED 15 minutes before the lesson, a 1-hour lesson, and 10 minutes in a follow-up lesson (or for homework). Timings given are only approximate. Exact timings will depend on the needs of the class. Teacher guide Solving Linear Equations in Two Variables T-1

7 Which approach did you like best? Why? Which approach did you find most difficult to understand? Why? Next lesson: Improve individual responses to Notebooks and Pens (10 minutes) Have students do this task at the beginning of the next lesson if you do not have time during the lesson itself. Some teachers like to set this task for homework. Return the students individual work on the assessment task Notebooks and Pens along with a second blank copy of the task sheet. Look at your original responses and think about what you have learned this lesson. Using what you have learned, try to improve your work. If you have not written questions on individual pieces of work then write your list of questions on the board. Students are to select from this list only those questions appropriate to their own work. SOLUTIONS Assessment task: Notebooks and Pens!!!! "#\$! %&&#!!! Dan is incorrect: Emma is incorrect: Dan has misinterpreted n to mean, notebooks sold rather than the number of notebooks sold. So he has read the equation 4n = p as there are four notebooks sold for every single pen sold. The equation actually means, 4 times the number of notebooks sold equals the number of pens sold, or the store sells four times more pens than notebooks. Emma has also misinterpreted n to mean notebooks rather than the number of notebooks. In the second statement, 5n does not mean, there are 5 notebooks. It means 5 times the number of notebooks. Since each notebook costs \$5, 5n gives you the amount of money taken from selling notebooks, and since each pen costs \$2, 2p gives you the amount of money taken by selling pens. So 5n + 2p = 39 means that \$39 was taken altogether by selling notebooks and pens at these prices. However, the equation does not, in isolation, tell you how many notebooks or pens were sold. Using the first equation to substitute 4n for p in the second equation gives n = 3 and p = notebooks and 12 pens were sold. Teacher guide Solving Linear Equations in Two Variables T-6

8 Cash Registers 1. The number of dollar bills is three times the number of quarters. Four times the number of quarters plus the number of dollar bills totals Possible values: (4, 12) or (8, 24). 3. Possible values: (12, 22) or (7, 42). 4. x = 10, y = 30 Comments on Sample Student Work Ava used guess and check with both equations Strengths: Her work is systematic and easy to follow. Weaknesses: Her method is inefficient and, although it is systematic, she has not reflected on each answer to determine the next set of values to check. Her lack of progress leads to her abandoning the task. Ava could add an explanation about her solution method. Ethan used an elimination method Strengths: This method can work if equations are manipulated carefully. Weaknesses: Ethan makes a mistake when rearranging the first equation. Consequently, when the two equations are added together, a variable is not eliminated, but instead Ethan has created an equation with two variables. Ethan briefly used guess and check. This gives many solutions. Ethan has simply opted to figure out two solutions. Both answers are incorrect. Ethan has not explained his working or why he was happy with the second set of values. If the first equation had been 3x + y = 0, what would still be wrong with Ethan s method? Would this method ever obtain just one solution? Teacher guide Solving Linear Equations in Two Variables T-7

9 Joe used a substitution method Strengths: This is an efficient method. Weaknesses: Joe failed to multiply all the terms on the left-hand side of the equation by three, so he obtained an incorrect answer. If Joe had substituted 3x for y into the second equation the solution would have been very straightforward. Mia used a graphical approach Strengths: This method can work. Weaknesses: In this case a graphical approach is not a very efficient strategy. Mia has made an error in her second table: y = 66 not 56. Mia could have used the co-ordinates (20,!10) to help plot the second line. There are no labels on either axis. The scale of Mia s graph means that the lines are not plotted accurately. Was Mia right to abandon (20,!10) as a point to be used to plot the second line? Teacher guide Solving Linear Equations in Two Variables T-8

10 Notebooks and Pens A store sells pens at \$2 and notebooks at \$5. n = number of notebooks sold. p = number of pens sold. The following equations are true: 4n = p 5n + 2p = 39 Here is what Dan and Emma think the equations mean: Dan Emma Are Dan and Emma correct? If you think Dan is wrong, explain the mistake and explain what you think the equation means. If you think Emma is wrong, explain the mistake and explain what you think the equation means. Figure out for yourself the number of pens and the number of notebooks sold in the store. Student Materials Solving Linear Equations in Two Variables S MARS, Shell Center, University of Nottingham

11 Cash Registers The drawer of a cash register contains some quarters and some dollar bills. x = the number of quarter coins in the cash register. y = the number of dollar bills in the cash register. The following two equations are true: 3x = y 4x + y = Explain in words the meaning of each equation. 2. Find two pairs of values for x and y that satisfy the first equation. 3. Find two pairs of values for x and y that satisfy the second equation. 4. Find pairs of values for x and y that satisfy both equations simultaneously. Student Materials Solving Linear Equations in Two Variables 2012 MARS, Shell Center, University of Nottingham S-2

12 Sample Student Work: Ava You are the teacher and have to assess this work. What do you like about Ava s work? What method did she use? Is it clear? Is it accurate? Is it efficient? What errors did Ava make? How might her work be improved? Student Materials Solving Linear Equations in Two Variables S MARS, Shell Center, University of Nottingham

13 Sample Student Work: Ethan You are the teacher and have to assess this work. What do you like about Ethan s work? What method did he use? Is it clear? Is it accurate? Is it efficient? What errors did Ethan make? How might his work be improved? Student Materials Solving Linear Equations in Two Variables S MARS, Shell Center, University of Nottingham

14 Sample Student Work: Joe You are the teacher and have to assess this work. What do you like about Joe s work? What method did Joe use? Is it clear? Is it accurate? Is it efficient? What errors did Joe make? How might the work be improved? Student Materials Solving Linear Equations in Two Variables S MARS, Shell Center, University of Nottingham

15 Sample Student Work: Mia You are the teacher and have to assess this work. What do you like about Mia s work? What method did she use? Is it clear? Is it accurate? Is it efficient? What errors did Mia make? How might her work be improved? Student Materials Solving Linear Equations in Two Variables S MARS, Shell Center, University of Nottingham

16 Assessing Sample Student Work You are the teacher and have to assess this work. Correct the work and write comments on the accuracy and organization of each response. What do you like about this student s work? What method did the student use? Is it clear? Is it accurate? Is it efficient? What errors did the student make? How might the work be improved? Projector Resources Solving Linear Equations in Two Variables P-1

17 Sample Student Work: Ava! Projector Resources Solving Linear Equations in Two Variables P-2

18 Sample Student Work: Ethan Projector Resources Solving Linear Equations in Two Variables P-3

19 Sample Student Work: Joe Projector Resources Solving Linear Equations in Two Variables P-4

20 Sample Student Work: Mia Projector Resources Solving Linear Equations in Two Variables P-5

21 Mathematics Assessment Project CLASSROOM CHALLENGES This lesson was designed and developed by the Shell Center Team at the University of Nottingham Malcolm Swan, Nichola Clarke, Clare Dawson, Sheila Evans with Hugh Burkhardt, Rita Crust, Andy Noyes, and Daniel Pead It was refined on the basis of reports from teams of observers led by David Foster, Mary Bouck, and Diane Schaefer based on their observation of trials in US classrooms along with comments from teachers and other users. This project was conceived and directed for MARS: Mathematics Assessment Resource Service by Alan Schoenfeld, Hugh Burkhardt, Daniel Pead, and Malcolm Swan and based at the University of California, Berkeley We are grateful to the many teachers, in the UK and the US, who trialed earlier versions of these materials in their classrooms, to their students, and to Judith Mills, Carol Hill, and Alvaro Villanueva who contributed to the design. This development would not have been possible without the support of Bill & Melinda Gates Foundation We are particularly grateful to Carina Wong, Melissa Chabran, and Jamie McKee 2012 MARS, Shell Center, University of Nottingham This material may be reproduced and distributed, without modification, for non-commercial purposes, under the Creative Commons License detailed at All other rights reserved. Please contact if this license does not meet your needs.

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