underlying causes of students' preference of method in solving systems of linear equations

Underlying Causes of Students’ Preferenceof Method in Solving Systems of Linear

Equations

A Classroom Research Project submitted toThe Master of Arts in Teaching Program

ofBard College

byJames Cleveland

Annandale-on-Hudson, New YorkJune, 2010

Contents

1 Guiding Question and Purpose 3

2 Description of the School and Classroom Context 5

3 Background Information 7

4 Description of the Project 9

4.1 Project Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Analysis and Findings 13

6 Reflections and Implications 24

Appendices 29

A Unit and Lesson Plans 29

A.1 LP 5-6: Solving Graphically . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

A.2 LP 5-7: Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A.3 LP 5-8: Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.4 LP 5-9: No Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

A.5 LP 5-12: Solving Diagnostic . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A.6 LP 5-13: Practice I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B Interview Transcripts 62

B.1 Sarah Jane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

B.2 Amy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.3 Jack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

B.4 Rose and Donna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Contents 2

B.5 Martha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76B.6 Wilfred . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Bibliography 82

1Guiding Question and Purpose

Which of the three standard methods of solving systems of linear equations

do students prefer to use? Why?

I decided to look into this question after looking over the lesson plans for the unit on

systems. Before doing so, if someone had asked me which method I used to solve systems,

I would have most likely said, “substitution.” This seems like the “right” choice as an

advanced mathematician as it is a more versatile method and can be employed in other,

non-linear situations. This feeling was strengthened when I noticed the other math teachers

used substitution for a systems problem.

As I was looking through the problems for the applications lesson, though, my natural

instinct for solving interesting problems that I see took over. Lacking paper, I worked

in my head and, after solving one problem, I realized that the method I was using was

elimination, though slightly modified for mental math.

It was after that experience that I decided to look into student choices about their

methods. There are a number of things I wanted to look for as I researched into the

1. GUIDING QUESTION AND PURPOSE 4

topic. First, I wondered if there is some relationship between the students who are “more

mathematically inclined” with the choice of substitution, or if my perceived notion develops

later after students see other problems that require substitution (quadratics and functions).

Related to that, I also wondered if there would be a discrepancy between the expressed

choice and which method was actually used. Secondly, I wondered if there is a relationship

between the use of mental math and a preference for elimination. Thirdly, I had to consider

a relationship between the visually inclined and graphing.

Moving forward, I had three sub-questions that drove my data collection and interview-

ing.

1. Was student preference related to proficiency? Did the skill level of the stu-

dent and their skill with a particular method determine which method they used?

2. Was student preference related to efficiency? Did the layout or presentation

of the problem determine which method they used?

3. Was student preference related to execution? Did the teaching of the topic

or the form of the calculation determine which method they used?

2Description of the School and Classroom Context

Banana Kelly High School is located in the Longwood neighborhood of The Bronx. A

small school with 440 students, Banana Kelly is comprised of mostly Hispanics and Blacks,

in a two to one ratio. The younger grades have more students than the older grades, mostly

due to an imbalance in the number of male students, as the female population remains

fairly constant and slightly larger than the male.

The classroom in which I conducted my research was a 9th grade course called Thinking

Math and Science, taught by Nicola Vitale and Annie Lerew. There are two section of the

class, one a CTT class taught in the mornings with about 40% of the students having

IEPs, while the afternoon class is general ed students and tends to work at a slighter

faster pace. The classes have a block schedule so that they meet for two periods in a row

on most days. Three days out of the week, this means a class of 120 minutes, though a

shortened schedule on Fridays means class on that day is 93 minutes. Additionally, once

a week the students go on a trip to the Bronx River as part of their science curriculum,

which they have during a 180 minute block. Nic and Annie worked together as a cohesive

2. DESCRIPTION OF THE SCHOOL AND CLASSROOM CONTEXT 6

team to develop and teach the course, though Nic is officially the science teacher and

Annie is the math teacher, for the CTT class. The other class is only taught by Annie,

who fulfills both roles.

The 9th grade class acts in many ways as an introduction to the ways of high school in

general and Banana Kelly in particular. Students used notebooks to keep track of their

thinking, which helps them become more comfortable with sharing their thoughts, under-

standings, and misunderstandings. The process worked to get students used to writing,

something they are not used to doing before. This process still has a way to go with many

of the students, though, as they are often unable or unwilling more than a few sentences,

at most.

The homework turn-in rate is rather poor, which does make it difficult to gauge the

proficiency of the students in a particular method of solving and to collect data in gen-

eral. Attendance rates aren’t terrible, except for the first period in the morning, though

they could be more consistent. This does leave some holes in some students’ work, which

consequentially leaves some holes in the data when the questions are spread over multiple

days.

3Background Information

There is much written on linear equations, including students’ perceptions and thinking

about them, but there is not much written about systems of linear equations, despite the

prevalence of the topic. From a mathematical standpoint, this makes sense. Systems of

equations are merely extensions of what we already know about linear equations and how

they act. With a system, we are just examining more than one linear relationship at a

time, and as long as we truly understand how those relationships work, seeing how they

interact is no big step.

The problem then lies with the thread that is supposed to connect the earlier work with

systems of equations. This should be the same thread that connects arithmetic to algebra.

The problem lies in the fact that the thread is ill-constructed in most cases. I believe that

the root of that problem lies in students conceptions of equality.

In Kilpatrick’s Adding It Up, most students view an equals sign as an indicator to put a

solution, not as representative of a relationship. In one study, only 10% of students from

grades 1 through 6 had a conception of equality as a relationship, which is typical. The

teachers of those students were surprised by those results. But this conception can be easily

3. BACKGROUND INFORMATION 8

fixed. After a year of professional development on the topic, those teachers’ students vastly

improved. Between 66% in the first grade and 84% in the sixth grade had a conception of

equality as a relationship after that year.

Even still, those old views of equality can persist. For example, take the “equation”

below:

2.3 + 3.2 = 5.5− 1.5 = 4.0.

This is often written as a shorthand for two subsequent steps. First 2.3 and 3.2 are added

together to get 5.5. Then 1.5 is subtracted from the sum to get 4.0. But 2.3 + 3.2 6= 4.0,

which is a flaw with the equation. Older students will still use this sort of notation when

doing arithmetic and I have even caught myself doing it on occasion. I recall it coming

up during our grad-level Algebra class, which just goes to show that those grade-school

conceptions of the equals sign tend to stick around.

Once equality is understood as a relationship, though, the multiple representations of

different expressions can be better understood, and those multiple representations lead to

a better grasp of algebra and thus linear equations. For example, another problem students

often come across is discomfort with horizontal representation, which can stem from the

lack of a proper conception of equality. Students will know that 35 + 72 = 107 and that

107

−72

35

but are uncomfortable with 35 = 107 − 72. When algebra is first taught, though, the

vertical representation is usually left behind and students are expected to jump right into

horizontal representations all the time. This may explain why so many students can solve

2x+1 = 3x by subtracting 2x from both sides, underneath, but will have problems solving

1 = 3x− 2x by combining like terms.

4Description of the Project

In order to investigate the preference for the three methods of solving systems of linear

equations, I needed to first make sure a comparison was made possible during the teaching

of the methods. To facilitate this, I used the same problem (Archimedes and the Crown)

as an introduction to each method, with the same equations and same set-up. The first

lesson (LP 5-6) was on solving systems graphically, taught on April 23rd. The next lesson

(LP 5-7) was on the elimination method, on April 26th for the afternoon class, and the

third lesson (LP 5-8) was on substitution, taught on April 27th for the afternoon class.

Once this groundwork was laid out, I began to evaluate which method students prefer

and which method they are most proficient at. On April 30th (LP 5-9) the class began

with a demonstration of the elimination and substitution methods side-by-side, solving the

same problem. Though I had intended for the problem to be neutral, that is, equally simple

using both method, I accidentally biased the problem for the morning class by making

the problem simpler using elimination. (The morning class solved the system x+ 3y = 12

and 2x + 4y = 10, while for the afternoon class this was corrected by changing the first

4. DESCRIPTION OF THE PROJECT 10

equation to x = 12 − 3y.) Following this demonstration, students were asked to write

about which of the solving methods they preferred in a focused free write.

The following Friday, May 7th, was a Solving Diagnostic (LP 5-12) that I used to

answer my sub-question regarding proficiency. The front page of the diagnostic had four

systems problems. The first three problems dictated which method was to be used, so that

I could gauge which methods students were able to do. The fourth question was a neutral

question that could be solved not only with all three methods easily, but also using both

addition and subtraction with the elimination method. I could then track which method

each student used and compare to which problems the students got correct. The back page

of the diagnostic had writing prompts that asked the students to explain each method, as

another insight into their proficiency with each method. The final writing prompt asked

student to choose which method they preferred, a week after the original written prompt.

The lesson after the Solving Diagnostic (LP 5-13), on May 10th for the afternoon class,

was designed to help answer the sub-question about efficiency. This lesson began again

with a method demonstration, this time including graphing, with each teacher using one

method. (The afternoon class, which only has two teachers, had a student demonstrate

graphing.) The classwork for this lesson presented methods that were biased towards one

of the method, being more difficult for the other methods. (For example, one system was

two equations in standard form that, to solve with elimination, would have required using

the least common multiple. As this was the first problem of this type that they had seen,

it was meant to bias towards substitution.) Each system of equations was meant to be

solved using all three methods and was followed by three writing prompts, inquiring into

which method they initially chose upon seeing that problem and which method they found

to be the easiest after completing all three.

Following the completion of lesson 5-13, I decided to interview several students whose

responses and actions during the lessons intrigued me and prompted me to look further


into their thinking. Three students (Rose, Donna, Wilfred) were interviewed because they

showed weaknesses in certain methods and I wanted to look into why they chose the

methods that they did. Four students (Sarah Jane, Amy, Jack, Martha) were interviewed

because they were proficient in all the methods and I wanted to dig deeper into the writing

they did in lesson 5-13. Five students (Wilfred, Amy, Jack, Donna, Martha, Sarah Jane)

had discrepancies between what they stated as a preference and which methods they

chose to use, which is something else I asked about during the interviews. One student

(Mickey) was informally interviewed because he was the only student I witnessed using

mental math to solve systems of equations. Finally, two students (Rose, Donna) seemed

to indicate that their preference was based in the way the methods were taught, and I

wanted to get confirmation in an interview. (See Appendix B for interview transcripts.)

Student Weakness Proficient Discrepancy Execution

Sarah Jane X X

Amy X X

Jack X X

Rose X X

Donna X X X

Martha X X

Wilfred X X

Mickey X X

4.1 Project Map

April 23 (LP 5-6)

Introduced the Archimedes and the Crown problem

Taught how to solve systems graphically

April 26-27 (LP 5-7)

Continued use of the Archimedes problem


Taught how to solve systems using elimination

April 27-28 (LP 5-8)

Continued use of the Archimedes problem

Taunt how to solve systems using substitution

April 30 (LP 5-9)

Demonstrated elimination and substitution method side-by-side, though with an acciden-

tal bias towards elimination for the morning class.

First written inquiry into student preference

May 7 (LP 5-12)

Solving Diagnostic to determine student proficiency, also acted as second written inquiry

into student preference

May 10-11 (LP 5-13)

Present various situations to gauge situational preference by having students solve prob-

lems using all three methods and write about which method was easiest

Third written inquiry into student preference

May 11-13

Student interviews to question about discrepancies in their answers and dig deeper into

the meaning of their stated or inferred preference.

5Analysis and Findings

Finding: Students will ignore potential external bias when stating their pref-

erences for methods.

Based on how I have seen the students react to graphing in the past, I had hypothesized

that no students would choose the graphing method as a preference, especially considering

the amount of time that method requires. The morning class had also been biased towards

elimination (so I had thought), so I was surprised to see that, of the students who responded

to the prompt, 5 still stated a preference for substitution, 4 for elimination, and 1 for

graphing. Some of the stated reasons for their preference were interesting as well. One

student wrote that substitution was the slower method and that was why she preferred it,

because it was easier to understand. Another student wrote that elimination seemed like

it was skipping steps, so that is why she found it confusing. Many of the students stated

aloud a preference for elimination during the substitution lesson, which I believe stemmed

from the fact that the elimination lesson came earlier. They understood at that point the

elimination method and did not want to learn another way of doing it.

5. ANALYSIS AND FINDINGS 14

Finding: An unstated preference may stem from a desire to bolster weakness

in certain skills or understandings.

Sarah Jane’s reasons for choosing a method stemmed from bolstering her weak points.

She alternates which method she uses so that she has enough practice with all the methods.

JAMES: So here, we’re gonna talk about yesterday’s problems you were doing.

In this problem you had the y alone as well, but you used elimination first, you

didn’t use substitution, so why go with elimination [on 5-13] when you used

substitution [on 5-12]?

SARAH JANE: Cause I don’t go with the same things over and over, the

same order, I go with different things, especially with things I kinda need, I

struggle with, I try to work with it, so I can get myself used to using that

method also.

Though she stated that elimination was her favourite method, her decision of which

method to use when faced with the choice didn’t come from that. She consciously chose

to use methods she was less familiar with so that she would be familiar with them in the

future.

Wilfred mentioned a similar idea. On his diagnostic in 5-12, he wrote that the graphing

method was the method that he was still struggling with and disliked the most. When

given the option of method on problem #4, though, he choose graphing. Similarly, during

lesson 5-13, he used graphing first when it came time to use all three methods. When

asked about why he chose to do graphing first, he said that he wanted to “get the hard

one out of the way.”


Finding: Student preference can stem from an undiagnosed weakness in an

earlier topic.

Amy’s preference changed depending on the situation, but it wasn’t because of neces-

sarily the method that was the most “efficient.” Amy’s preference belied a weakness when

solving one equation with one unknown in situations where there is a variable on both

sides of the equation. Though Amy seemed proficient on the surface with solving, she used

different methods of solving systems to cover up her sense of unease with certain types of

single-variable problems.

JAMES: Hmm. So what I’m trying to determine is why, is how [the revised

problem] is different from [the original problem]. What’s make it so that [for]

this one elimination is the comfortable method and for this one substitution

is? This equation is equal, it’s the same as [the other] equation.

AMY: Yeah, but you see, there’s no y, so there’s not a y on the opposite side.

There’s not really a way I could substitute anything. Unless I put that in there

and, to me that would be more difficult because to me when you look at an

equation you should know which type of method to use to solve it, by knowing

how comfortable you feel with each method.

JAMES: Okay, so I think I see what it is. So if you had these two and you

used substitution, it would look like this: 3x− 7 = −3x− 1. But if you used it

over here, you would get 1 = −3x− (3x− 7).

AMY: Yeeah.

JAMES: So here, in [the first one], well, can you maybe describe, before I say

what I think, how these two are different?


AMY: Well, to me it’s different because it’s like, it’s think it’s all on, like,

what’s on each side.

JAMES: Mm-hmm.

AMY: Because, like, when you have two different equations it’s harder to just

like to put it together, you know what I’m sayin’?

JAMES: Right, because I’d have to get the x’s on one side.

AMY: Exactly.

JAMES: And on this one there’s already on the same side—

AMY: They’re on the same side, so it’s easier to put like terms together and

solve for the variable.

Amy’s discomfort with solving single-variable equations when there is a variable on both

sides led to her preference in a given systems problem.

Other students also had weaknesses in earlier topics that lead to their preference, though

the topic was solving “in terms of,” which is covered earlier in Unit 5. Donna and Rose

dislike the substitution method because they have trouble solving for one variable.

JAMES: So what is it about substitution that you don’t like?

DONNA: The way that you have to switch numbers around.

ROSE: Yeah.

DONNA: I’d rather just put them under one another and subtract or add.

Similarly, Wilfred posed an interesting case because of his stated dislike of graphing but

his apparent preference for the method, especially when, on the 5-12 diagnostic, he got the

two graphing questions correct and the elimination and substitution questions incorrect.

After being confronted with this fact, Wilfred changed his mind:


JAMES: So...based on what I’ve seen, you were mostly able to get the graph-

ing done right and so I’m trying to figure out what it was about the graphing

that made you think that it was the hardest if it isn’t actually the hardest.

WILFRED: Trying to get the y alone.

JAMES: Is that maybe substitution is the hardest now?

WILFRED: Yeah.

Wilfred shared Donna and Rose’s weakness in solving “in terms of,” which lead to his

seeming preference for elimination and showed why he was proficient in graphing when

the equations were in y = mx + b form.

Finding: Students can identify when a problem is designed to be completed

with elimination or substitution, and would use that method if they are pro-

ficient.

Those students who were proficient is all methods (Sarah Jane, Amy, Jack, Martha,

Mickey) all made statements about the set-up of a problem having an influence on which

method they would use. Each equation having identical terms (i.e. both have 2x) or addi-

tive inverses (i.e. one has 2x and the other has −2x) would indicate elimination needs to be

used. Mickey says that he uses elimination when the equations “have the same numbers.”

JAMES: You did use elimination. What made you want to use elimination?

AMY: The two 3x’s. I knew I could eliminate that.

JACK: I would say it’s because I liked graphing. But, I could have also done

elimination because it has a 3x and a 3x.


Similarly, the students said that they would use elimination on a problem where one of

the variables is solved for.

SARAH JANE: I dunno, when I see a problem like that, I just use substi-

tution.

JAMES: What do you mean a problem like that?

SARAH JANE: The y is separated, right, and then there’s the 3x and the

numbers and I find it easier than the elimination method[.]

AMY: It’s like, if I know I could eliminate one variable, then I would use

elimination. Or if I have the y alone, I would just use substitution. If I can’t

do any of those, obviously I’ll need to do the graph.

In both cases the students preferred the method that would be the least amount of

work.

JAMES: Well, I mean, [on 5-13] you did both, you used elimination, substi-

tution, and graphing. So what is it about the elimination method that was

easier in this particular case?

MARTHA: I chose this one because this one is shorter than the other ones.

Finding: Students will use methods that they are most familiar with, regard-

less of the suitability of those methods.

Jack had a preference for graphing, despite it being less accurate than algebraic methods

as well as often taking longer to complete. His preference derives from the fact that he is

more familiar with graphing:


JACK: The other ones are also easy but I think graphing is the most, how

you say it, precise?

JAMES: The most precise! Oh, I think that’s interesting. Because, take a

look at this problem you guys did the other day. You did graphing first, but

remember that your answers were things like −2.3̄ and −3.3̄. Can you really

be so precise? I mean, knowing what the answer is I can look at the graph and

go, Oh, that’s .3, that’s .3, sure. But if I didn’t know what the numbers were—

JACK: It would be a lot more difficult, yeah I know. But like, but when you

have known all about graphs since, what, the fourth grade, then yeah, it would

be a lot more easier.

JAMES: Oh, so it’s because you’re more familiar with graphing?

JACK: Yeah.

JAMES: So substitution and elimination, those are a lot newer—

JACK: Yeah, I know how to do those, too, I can do them perfectly fine, it’s

just that I’d rather recommend graphing because I’ve been doing it for a longer

time.

The length of time a student has been exposed to something can help dispel any uneasiness

they might have with a new topic. Jack could get into the new topic of solving systems

why using the old method of graphing. Similarly, while he strongly preferred graphing in

the earlier assessments, his opinions changed in the latter ones until he liked all three,

which I posit is due merely to having worked with them for an additional two weeks.

In another example, Rose and Donna both strongly preferred elimination over substi-

tution. When I asked why that was, they agreed that it was because I taught elimination

first:


JAMES: Okay. So do you think the fact that we learned elimination first is

one of the reasons you like it better?

ROSE: Yes, I think that’s why, too. Because we learned it first so when you

were showing me something else to do I was already used to elimination.

At that point, the students were just beginning to familiarize themselves with elimination

and didn’t want to be bothered learning a new method of solving. Rose was only able

to open herself up to the idea of using substitution some weeks after the original lesson.

Another student I found would also use elimination automatically, even though she said

she had no preference and was actually better with substitution, because when she saw a

system of equations, that’s what she thought she had to do.

Finding: Students will prefer the method they are best at only when they

are not proficient in multiple methods.

For all the students whose work I looked at, I found there was one method that they

were more proficient at than another. However, those students who I chose to interview

because they were able to use all three methods made their preferences based on other

factors (such as avoiding an x on both sides for Amy, or bolstering her skills for Sarah

Jane). Some, like Martha and Mickey, said that they used whatever method was best for

a particular situation.

Those students who were chosen because they had some weaknesses, though, tended to

prefer the method that they were not weak in. Both Rose and Donna preferred elimination,

which was the method they tended to do correctly. Wilfred chose graphing when the

problem was in y = form, as he was most capable at graphing in those cases, but preferred

elimination otherwise, as his second strongest.


Finding: Students with an affinity for visual elements preferred elimination,

not graphing.

Rose stated that she didn’t like graphing because she often got confused on the method

of how to graph. Though graphing makes sense from a visual standpoint, making the graph

can still be a barrier to those students who might otherwise prefer it. Instead, Rose said

that she liked the visual elements of elimination better.


JAMES: So it’s like cleaner?

DONNA AND ROSE: Yeah.

ROSE: And you could see it more visually. It’s easier for you to just, you

know...

JAMES: Oh, because of the way everything lines up nicely?


Another student, Rory, said the same thing in class, that he liked to use elimination

because “it lines up.” (It was from Rory’s statement that I got my question. Rory is in

a different class than Rose and Donna.) This matches with some things I’ve seen in the

classroom about how the students solve equations. Often when solving an equation and

a student gets to the point that they need to divide by the coefficient, they will have

written the division before they write the right-hand side of the equation. Most of the

students know something is supposed to go “beneath” what they are writing, without

always understanding why that is the case.


The preference for the “lining up” of elimination may stem from students’ discomfort

with horizontal representation, and so the vertical addition and subtraction would be

preferable to those students.[1]

Finding: Teacher support can interfere with research and student responses.

In the first set of writing about preferences, Donna stated that she liked graphing the

best, and not the others. During the diagnostic, on the other hand, Donna wrote that she

liked elimination but was struggling with graphing. I was curious about how a method

could go from favourite to struggling in a week’s time, so I asked about it.

JAMES: Okay. So, Donna, this is a question for you: originally in your note-

book when I first asked you guys what you liked best, you said you liked

graphing the best.

DONNA: Where?

JAMES: I think it was back on...5-9, yeah. I prefer graphing because it’s

easy to plot points, et cetera. But then, on your quiz, you wrote that you

were struggling with graphing and you liked elimination the best. So I want to

know what changed [two weeks ago] and [last week], where [on 5-9] graphing

was easy and [on 5-12] graphing was hard.

DONNA: I have no idea.... (exclaiming) I had help on [5-9].

JAMES: You had help on that one?

DONNA: I had help.

JAMES: So, because you had help it was easier but once you were on your

own it was harder.

ROSE: Yeah, it was easier for me then.


DONNA: It was easier when I had help but on my own I’d rather use elimi-

nation.

Because the teachers had been helping during that lesson, Donna had a false sense of

confidence in her abilities with the graphing method that fell away once she had to work

on her own during the quiz. Therefore, any attempt to collect data about what the students

are capable of or think they are capable of must take care not to bias the response by

assisting.

Finding: Mental math does not affect which method is used.

When I first decided to research this question based on my own choices for solving

systems of equations, I had hypothesized that I chose elimination when solving mentally

because it was particularly suited for mental math. The one student who solved mentally

in both my classes, Mickey, disproved that idea. Mickey would use whatever method he

thought was best for a particular problem when he was solving, which leads me to revise

my idea that mental math will lend itself to the most efficient method, the one that would

require the least amount of writing, so that less mental agility is required to solve the

problem.

6Reflections and Implications

These findings have a number of implications for my future teaching that arise from my

research. My first has to do with familiarity. The students choosing to do methods that are

more familiar to them lends credence to the “long game” attempted by math departments

of seeding topics from higher grades into the curriculum of lower grades, even if it’s just

in a cursory manner. Students “seeing” something in 9th grade that they will learn about

in 10th grade helps them accept the new topic when they see it as not being totally alien.

In this case, Jack latched onto graphing as something familiar while elimination and

substitution were new. This, combined with a question by fellow MAT student Reuben

made we think about how we can seed elimination and substitution earlier. Reuben asked

how the methods of solving were actually different. To him, the methods were the same

since they were just using properties of equality.

My response was that they use different properties of equality. The elimination method

uses the addition property: if a = b and c = d, then a + c = b + d. This is what allows us

to add or subtract the equations. But I realized that it is this same property that allows

6. REFLECTIONS AND IMPLICATIONS 25

us to solve single-variable equations (along with the multiplicative property, if a = b and

c = d, then ac = bd).

Following that train of thought, I wondered if it would be effective to teach single-

variable solving in a way similar to elimination. First, I’ll illustrate with an example.

Imagine we want to solve the equation 3x − 2 = 2x + 1. In this case, I might want to

eliminate the −2 from one side and the 2x from the other. In this case, I might write a

second equation to pair with the first, like so:

3x− 2 = 2x + 1

−2x + 2 = −2x + 2.

I can then add the two equations together, much like when using the elimination method,

to give me the solution x = 3. I think that using this method might provide a number of

benefits. Firstly, it introduces elimination earlier, as stated above. Secondly, it might help

correct a common error students make when solving. Often students will write the additive

inverse of a term twice on one side of the equals sign and not at all on the other side.

Adding equations might make it clear why that doesn’t make sense. Thirdly, introducing

this method might help clarify the meaning of equality. Finally, the vertical representation

may help ease those students who are uncomfortable with horizontal representation into

algebra.

Substitution is already seeded earlier in the form of checking answers and evaluating

expressions, but perhaps those activities could be stressed more or can include examples

where an expression is plugging in for a variable earlier on. A deeper exploration of the

transitive property might also aid in developed substitution skills.

This also has me thinking about student motivation. In fellow MAT student Sebastian’s

research, I found that those students who were motivated by passing barely passed, and

those motivated by their own futures did well. I think that finding is reflected in Sarah


Jane’s statements about why she tries the different methods. I might want to use this

topic, with its three different ways of solving a problem, to stress the idea of readiness

and preparedness for the future. Learning one method would be enough to get by, but

becoming a master of all the methods lets you be more able to take on any new challenges

you may face.

I also liked the three methods as ways into not just solving systems of equations but

different properties of equality. While all three methods were necessary to do well and were

goals of the class, it happened often that a student who didn’t understand substitution

would come to get it after reaching a fuller understanding of elimination (and vice versa).

The three methods acting as a window into student thinking was also beneficial. This

leads me to want to give students more choices in most of what they learn, because those

choices that they make can be very illuminating about their thought processes and their

knowledge and understanding of both the topic and the previous topics it builds upon.

I’ve also learned the importance of pacing when introducing new topics. Because of

the way that the TMS classes are set up, with the 2-hour blocks of math 3 days a week,

the two hours needed to introduce elimination were in one day and immediately followed

the next day by the two hours our substitution. In a different school, stretching these

lessons over the week might have helped the ideas settle in more. In any case, giving more

time between elimination and substitution, perhaps by moving an applications lesson up

in between, could give the unit some growth. In this unit, though those lessons are not

included in the Appendix, the three applications lessons can easily be paired up with a

particular method that showcases how that method is useful for solving a particular kind

of problem.

In the end, I think looking into student choice, not just in terms of methods of solving,

but in all parts of teaching, can be very instructive. People make choices and have prefer-

ences for real reasons, even if those reasons aren’t always clear to the person making the


choice. Those reasons can help us as teachers learn more about our students, as well as help

those students learn more about themselves. Armed with that knowledge, we can move

forward with ideas on how to undo misunderstandings and create new understandings.

Appendices

28

Appendix AUnit and Lesson Plans

A.1 LP 5-6: Solving Graphically

Included are only some of the classwork to show what went on. Not all of the worksheets

were necessary.

LP: 5-6 Title: Solving Graphically Objectives:

• Students will Graph using slope and y-intercept • Identify the point that makes a system of equations true • Explain the meaning of the intersection within the context of a given problem • Students will work collaboratively in small group.

Materials: • Warm Ups • Notebooks/Portfolios Presentation: a) Warm Up (20 min): -See WU 5-6 b) BEC Seller’s Recap (10 min):

-Using an image of the BEC shops equations, have students share their thoughts and discuss the meanings of the graph. Also emphasize the meaning of point of intersection.

c) Graphing Practice (15 min): -Each group member will receive a system to graph using slope and the y-intercept.

When appropriate, student should solve for y and put the equation in slope-intercept form. Have students check each other work.

d) Tool Kit (20 min): -Fill in Tool Kit with process of solving systems graphically.

e) Classwork (25 min): -Students will work in small groups together on contextualized and non-contextualized problems. With the contextualized problems students will give meaning of pieces of the equations and the point of intersection. f) Wrap Up (3 min): Homework: HW 5-6

Lesson Reflections:

Warm-Up 5-6 Archimedes and the Crown

Over the next few lessons we’ll be discussing the story of Archimedes and the crown and how you can use your knowledge of systems of equations and density to solve his problem. The story goes: One day the king asked for a crown of pure gold and gave gold to a blacksmith to make. When he received the crown, though, he suspected that he had been cheated and that the blacksmith had mixed in some silver to the crown, keeping the extra gold for himself. He asked Archimedes to figure out a way to determine just how much gold was used in the crown, without destroying. It was then at Archimedes figured out how to use liquid displacement to find the volume of the crown. Once he knew the mass and the volume of the crown, Archimedes could then make a system of equations to represent those values. He used x to represent the amount (volume) of gold in the crown and y to represent the volume of silver. The total volume of the crown was 140 cm3 . He also knew that the density of gold was 20 g

cm3and the density of silver was 10 g

cm3. Finally, he

measured the total mass of the crown as 2000 g. How can he set up a system of equations, one that represents volume and another that represents

mass? (Remember that D =mV

.)

Warm-Up 5-6

Archimedes and the Crown Over the next few lessons we’ll be discussing the story of Archimedes and the crown and how you can use your knowledge of systems of equations and density to solve his problem. The story goes: One day the king asked for a crown of pure gold and gave gold to a blacksmith to make. When he received the crown, though, he suspected that he had been cheated and that the blacksmith had mixed in some silver to the crown, keeping the extra gold for himself. He asked Archimedes to figure out a way to determine just how much gold was used in the crown, without destroying. It was then at Archimedes figured out how to use liquid displacement to find the volume of the crown. Once he knew the mass and the volume of the crown, Archimedes could then make a system of equations to represent those values. He used x to represent the amount (volume) of gold in the crown and y to represent the volume of silver. The total volume of the crown was 140 cm3 . He also knew that the density of gold was 20 g

cm3and the density of silver was 10 g

cm3. Finally, he

measured the total mass of the crown as 2000 g. How can he set up a system of equations, one that represents volume and another that represents mass? (Remember that D =

mV

.)

Use the slope and the y-intercept to graph each system of linear equations. Solve for y when necessary.

A)

6x + 3y = 9

y =!23x ! 3

"#$

%$


B)

y =12x ! 2

!3x ! 3y = 12

"#$

%$


C)

!3y + 9x = 3

y =34x ! 2

"#$

%$


D)

y =45x ! 2

4x + 2y = 8

"#$

%$

Graph the following system. Then write the coordinates for where both equations in the system are true. !x + y = !22x + y = 10

"#$

Describe your process for solving this system of equations:

Solving Systems Graphically TK 5-6

Names: __________________________________ Date: _______________

In your own words how would you describe what slope is? How have we seen the solutions to 1-equation and 2-unknowns represented? What are the coordinates for the point that make both equations true? Plug these x and y values back into the equation and see if it’s graphed correctly. x + y = 8x ! 2y = 2

"#$

Consider the equations we found for Archimedes earlier. x + y = 14020x +10y = 2000

What is the x-intercept representing in this problem? What is the y-intercept representing in this problem? Graph the system of equations below. How much gold (by volume) is in the crown? How much silver?

Graphing Systems - 1 CW 5-6

Group Members: ___________________________________________ ___________________________________________ Date: _______________

APPENDIX A. UNIT AND LESSON PLANS 36

A.2 LP 5-7: Elimination

Unfortunately, the pages copies from the Jacobs book are not included in the electronic

copy of the plan.

LP: 5-7 Title: Solving Systems Using Elimination Objectives:

• Students will use addition and subtraction of equations to eliminate a variable when solving a system of linear equations.

• Students will work collaboratively in small group. Materials: • Warm Ups • Homework • Classwork Presentation: a) Warm Up (10 min): -See WU 5-7 b) Writing and class discussion (15 min): - Have students write about the following question: What are things we have added in math class? Give some examples. Have a short discussion about the ideas that students come up with and catalog them on the board, overhead, or poster paper. Some answers may be:

• Positive & Negative Numbers – decimals, integers, rationals, imaginary • Variables • Expressions • Lengths

Make the connection that all of these things are found in equations and then have them write about: Can we add equations and how would you do that?

It may help to provide students with an example. 2x – 3y = 5 4x + 3y = 19 (4 , 1)

Discuss student’s responses and if possible have students highlight the mathematical errancies in their peer’s answers. Discuss what would be the advantage of doing this. Going from 2-equations with 2-variables to possibly 1-equation with 1-variable, which is something that we can solve. Reference solving poster and point out going from a tier two problem back to the first tier that we have already done so much work in.

c) Tallest vs. Shortest Man in the world problem (15 min): Copy p. 284 in Jacobs. Read through the problem together. Remind them that we are trying to find the values for x & y that make the equations true at the same time and that we can always check or solve by graphing but we are going to try adding the two equations. Either in large-group or small-group settings, help them see that by adding these equations we can eliminate 1 variable in order to obtain an equation we can solve. If there is time, have a group present their work.

d) Cats on a Scale (10 min):

Based on Jacobs example from p. 291, use physical manipulatives on the projector to represent the equation, with “big blocks” and “small blocks” representing the cats and kittens. With the two equations set up physically, ask if the students can see some way of eliminating the big blocks so that we only have little blocks left. If necessary, stress that you want the two sets of blocks to equal zero, so if there are 3 blocks in the first equation and three blocks in the second, how can I get zero? Use this elimination to find the value of the little blocks, and then “plug” in with writing to find the big blocks.

e) Solving by Elimination (10 min): Using the discussion from parts a and b, as well as referring back to the problems from c and d, explain how to combine two equations, by either adding them or subtracting them, so that we can reduce two equations with two unknowns down to one equation with one unknown, which we know how to solve (Step One). f) Classwork (20 min): See CW 5-7. g) Solving Archimedes’ Dilemma (10 min): Returning to the problem from the warm-up, it is not immediately obvious that adding or subtracting such equations would eliminate one of the variable. Discuss with students how they could alter one of the equations so that an unknown will be eliminated when they are subtracted. h) Solving by Elimination Toolkit (10 min): Fill out TK 5-7 with class, prompting students for each step. i) Wrap-Up (10 min): Allow students to finish working on CW 5-7 until end of class. Hand out HW. Homework: Skillbuilder 10 (due Friday)

Lesson Reflections:

Warm-Up 5-7 Archimedesʼ Dilemma

On Friday, Archimedes developed a system of two equations to represent the amounts of gold and silver in the crown, with one equation based on volume and the other equation based on mass. Those equations were x + y = 14020x +10y = 2000

We then solved those equations by graphing the two equations and looking for the intersection point. Archimedes has a problem, though. He lives in Ancient Greece and Descartes wonʼt invent graphing for another 1900 years. Can you think of any other ways Archimedes might be able to figure the values of x and y? Brainstorm with your group and write some ideas down in your notebooks.

Warm-Up 5-7 Archimedesʼ Dilemma

On Friday, Archimedes developed a system of two equations to represent the amounts of gold and silver in the crown, with one equation based on volume and the other equation based on mass. Those equations were x + y = 14020x +10y = 2000

We then solved those equations by graphing the two equations and looking for the intersection point. Archimedes has a problem, though. He lives in Ancient Greece and Descartes wonʼt invent graphing for another 1900 years. Can you think of any other ways Archimedes might be able to figure the values of x and y? Brainstorm with your group and write some ideas down in your notebooks.

Solve all of the following problems by using the Elimination Method, adding or subtracting the equations to get rid of one variable. Show all your work.

1) 3x + y = 223x ! y = 14

2) 5x + 2y = 714x ! 2y = 28

3) x + 4y = 506x ! 4y = 14

4) 3x + y = 17x + y = 12

5) 4x + 4y = 574x + y = 43

6)4x = 2y + 36x = 2y + 2

7) x +12y = 4!x + 8y = !64

8)5x + y = 215x + 9y = 9

Solving Systems by Elimination CW 5-7

Name: __________________________________ Date: _______________

Elimination by ____________________ x + 5y = 332x ! 5y = 6

Elimination by ___________________ 3x + y = 243x + 5y = 36

Solving Systems by Elimination TK 5-7

Names: __________________________________ Date: _______________

A) x + 4y = 25x ! y = 5

B) 2x + 3y = 8y ! x = 11


A.3 LP 5-8: Substitution

Classwork and homework are omitted from this lesson place except for the one that teaches

how to use substitution.

LP 5-8 Title: Solving Systems Day Four Substitution Objectives:

• Students will investigate how to solve simultaneous equations through addition as demonstrated by completion of problems sets and writing.

• Students will investigate how to solve simultaneous equations through subtraction as demonstrated by completion of problems sets and writing.

• Students will investigate how to solve simultaneous equations through substitution as demonstrated by completion of problems sets and writing.

• Students will work productively in small collaborative groups. Materials: • Warm-ups • Handouts • Calculators/Rulers • Notebooks Presentation: a) Warm Up (10 min):

− See WU 5-8 b) Solving Systems Recap (35 min):

− Start with CW 5-8.1 − Pass out CW 5-8.2 to students who have completed 5-8.1. Use Set IV on

Jacobs p. 310 for the bottom of CW 5-8.2. d) Solving by Substitution (45 min):

− Return to Archimedes problem from warm-up. Talk about how substitution allows Archimedes to solve for his unknowns without adding equations, as long as he can solve from one unknown in terms of the other. Pose the question: is it necessary to solve both equations for y in terms of x before we can substitute? What would happen if we substituted when we didn’t have y alone? Try as a class.

− Group practice – see CW 5-8.3 (Jacobs p. 323-325) e) Wrap Up (10 min):

− Allow students who have finished classwork to begin working on homework. Homework: See HW 5-8. Have the back page of HW 5-8 be Set IV from Jacobs page 327.

Warm Up 5-8

Archimedes and the King

Using the information he discovered about the crown, Archimedes went back to the king of Syracuse and told him that he had, in fact, been cheated. When Archimedes tried to explain his elimination method for solving for the equations, though, the King didn’t understand. The King did not believe that you could add equations together. Because of this, Archimedes needed another way to explain his logic. He said, “The amount of gold in the crown is the same as the volume of the crown less the amount of silver. The mass of the gold is the same as the mass of the crown less the mass of the silver. So I can say that the amount of gold is the same as the mass of gold divided by 20, its density. Since I know have two expressions that represent the amount of gold, I can let them be equivalent and then deduce the amount of silver.” Can you translate what Archimedes said into equations, so that we can help the King solve them? Look back at WU5-6 and WU5-7 for the values named above.

Warm Up 5-8

Archimedes and the King

Using the information he discovered about the crown, Archimedes went back to the king of Syracuse and told him that he had, in fact, been cheated. When Archimedes tried to explain his elimination method for solving for the equations, though, the King didn’t understand. The King did not believe that you could add equations together. Because of this, Archimedes needed another way to explain his logic. He said, “The amount of gold in the crown is the same as the volume of the crown less the amount of silver. The mass of the gold is the same as the mass of the crown less the mass of the silver. So I can say that the amount of gold is the same as the mass of gold divided by 20, its density. Since I know have two expressions that represent the amount of gold, I can let them be equivalent and then deduce the amount of silver.” Can you translate what Archimedes said into equations, so that we can help the King solve them? Look back at WU5-6 and WU5-7 for the values named above.

Solve each of the following systems of equations by writing the equations and then making the indicated substitution. Show all your work.

Solving Systems by Substitution CW 5-8.3


A.4 LP 5-9: No Solution

Only includes plan and warm up.

LP 5-9 Title: Solving Systems Day Five Substitution II and No Solution Objectives:

• Students will be able to (SWBAT) solve two equations with two unknowns as demonstrated through small group discussion, problem solving and writing.

• SWBAT identify when a system of equations has no solutions or multiple solutions as demonstrated through graphical interpretation.

• Students will work productively in small collaborative groups. Materials: • Warm ups/Templates/Notebooks • Calculators/Rulers • Homework/Classwork Slips Presentation: a) Warm Up (15 min):

− Students will consider when a solution for a system of equations does not make sense in the real world context of the problem.

b) Substitution review (15 min):

− Solving systems practice. Students review 2 problems that are already solved, explain the steps that are taken and critique for correctness.

− In your own words: Have students write (to share) what substitution is in their own words.

c) Summary of solving methods (15 min):

− Last week and this week we've seen strategies for solving systems graphically, by adding or subtracting equations and by substitution. Annie, James, and one volunteer demonstrate solving one system on the board, each using a different method. Emphasize that sometimes it's more efficient to use one strategy over another, but you could also use the strategy that you're most comfortable with.

− Focused free write – which solving strategy do you prefer and why? Which confuses you the most?

d) Group Investigation (15 min):

− Groups will try to solve a system through any method they choose. − On the board the teacher will graph the system (which will not intersect). − Discussion: What if the functions don’t intersect? What do you notice about

functions that don’t intersect when they are in slope-intercept form? − Repeat with two equivalent equations. − In your own words: How many solutions are there?

e) Group work (30 min):

− Jacobs 313-318 #4, 9, 10 (there will be slips for these three) − problems that demonstrate one solution, no solution or equivalent

equations. f) Wrap Up (5 min) Homework: See HW5-9

Warm-Up 5-9 The Blacksmithʼs Defense

Given the evidence before him, provided by Archimedes, the King of Syracuse decided to bring in the blacksmith for questioning about the false crown. The blacksmith made the following argument: “I agree that the mass of the crown is 2000 g, and I admit that I did use some silver, but I only used a very small amount to add structural support, and all of the gold is still there. I did not use nearly the amount that Archimedes claims. In fact, I find his liquid displacement methods to be suspicious; by my calculations, the volume of the crown is 90 cm3 .” At that point, Archimedes shouted, “Hold it! Youʼre lying! Those numbers are impossible, and I can prove it!” Can you prove Archimedesʼ claim, that the numbers are impossible? Look back at WU5-6, WU5-7, and WU5-8 for clues.

Warm-Up 5-9 The Blacksmithʼs Defense

Given the evidence before him, provided by Archimedes, the King of Syracuse decided to bring in the blacksmith for questioning about the false crown. The blacksmith made the following argument: “I agree that the mass of the crown is 2000 g, and I admit that I did use some silver, but I only used a very small amount to add structural support, and all of the gold is still there. I did not use nearly the amount that Archimedes claims. In fact, I find his liquid displacement methods to be suspicious; by my calculations, the volume of the crown is 90 cm3 .” At that point, Archimedes shouted, “Hold it! Youʼre lying! Those numbers are impossible, and I can prove it!” Can you prove Archimedesʼ claim, that the numbers are impossible? Look back at WU5-6, WU5-7, and WU5-8 for clues.


A.5 LP 5-12: Solving Diagnostic

This is the diagnostic often referenced to in the interviews.

LP: 5-12 Title: Solving Systems Applications III Objectives:

• Students will be able to solve two equations with two unknowns as demonstrated through problem solving and writing.

• Students will solve systems of equations using three methods, graphing, substitution, and elimination.

Materials: • Warm Up • Diagnostic • Calculators Presentation: a) Warm Up (10 min): See WU 5-12 b) Review for Diagnostic (25 min): Students write solutions to the past two homework assignments on the board. They have the opportunity to ask questions about specific solving methods. b) Diagnostic (50 min): Graph paper will be provided. c) Wrap Up/Clean Up (8 min)

Lesson Reflections:

Differentiation within lesson: -Students who request extra time to complete the diagnostic will be allowed to do so outside of class. Student Goals (9-2): - Groups 1 and 5 will be encouraged to complete all problems with the designated method, but will be allowed to use the method of their choice if they are struggling.

Warm Up 5-12

Write in response to the following prompt in your notebook. How do you think you did this marking period? Did you earn the grades that you would have liked to earn? How are you understanding Unit 5 so far? Do you think you will do well on today’s diagnostic? Is there anything about your study habits or behavior that you think you will change for Marking Period 6?

Warm Up 5-12


Warm Up 5-12


Warm Up 5-12


Warm Up 5-12


Solving Systems of Equations Diagnostic Name: _______________________ 1. (5 pts.) Use elimination to solve the following system of equations: 2x + 4y = 242x ! 4y = 0

x = ______y = ______

2. (5 pts.) Use substitution to solve the following system of equations: y = 20 ! 3x2y +10x = 60

x = _______y = _______

3. (5 pts.) Graph the following system of equations and find the solution: y = 2x ! 2y = !x ! 5

x = _______y = _______

4. (5 pts) Solve the following system of equations using any method: y = 3x ! 7y = !3x !1

x = _______y = _______

Writing Portion 5. (5 pts.) What is this unit “solving systems” all about? How would you sum it up to someone who isn’t in your class? 6. (5 pts.) How would you explain the elimination method? Are there certain times when it is easier to use this method, or certain times when it is harder? 7. (5 pts.) Explain how you solve a system of equations graphically. Is there always a solution? 8. (5 pts.) What is the substitution method all about? How would you explain it to someone who isn’t in your class? 9. (5 pts.) Which of the methods we explored in this unit did you like the most? Why? Which, if any, are you still struggling with?


A.6 LP 5-13: Practice I

The problems to be investigated were modified for the second class. Only two problems

were presented, with the second system of equations changing to

2x + 3y = 12

3x + 2y = 13.

LP 5-13 Title: Solving Systems Practice I Objectives:

• Students will reflect on their semester two progress and determine what their marking period six average will need to be in order for them to achieve their desired semester two average.

• Students will be able to solve two equations with two unknowns as demonstrated through small group discussion, problem solving and writing.

• Students will be able to identify when a system of equations has no solutions or multiple solutions as demonstrated through graphical interpretation.

• Students will work productively in small collaborative groups. Materials: • Warm ups • Classwork • Notebooks Presentation: a) Warm Up (20 min): -See WU 5-13 -Semester two goal setting. Students use MP4 and MP5 grades to calculate what they’ll need to get in MP6 to receive their goal grade for semester two. b) Introduction (5 min): - Today we will be work on our solving systems skills and exploring different kinds of systems. I want to find out which method of solving each of you prefers and if they can change based on what we see today, as well as help you develop your skills in the methods you are least confident in. b) Method Demonstrations (15 min): -The following system will be used in the example: y+1=2x + 5 and y= -2x+2 -Each teacher demonstrates a method (James elimination, Annie substitution, Nic graphing). c) Classwork (35 min): -CW 5-13: Students solve a set of four problems using each method (graphing, elimination, substitution), and then write about the ease of each method in a given situation and their preferences for each system. Annie (and Nic for 9-2) will circle among the groups, providing support, while James observes the groups work and takes notes for feedback. d) Discussion (15 min)

- Discuss whether or not students’ preferences for different methods change based on the situation, and if they discovered certain things about certain set-ups for systems. Have students share their writing. If students do think that the situation changes which method they should use, talk about the value of efficiency in math as they move forward into 10th grade and move on to the Wipe Off Game. If the students do not think the situation affects which method they use, talk about the value of knowing their strengths when it comes to problem-solving and move on to the Relay Race Game.

e) Relay Race Game (20 min): - Students will work as a team with their small groups to solve systems of equations,

sending one student at a time to the board to take the next step in the solution. Groups can talk/discuss which steps to take next.

f) Wrap-Up (10 min): -Hand out skill builder 11, due Friday. -Fill in the appropriate pages of the student planner. Homework: Skill builder 11 (due Friday)

Differentiation within lesson: - Students are encouraged to use the method of solving that they prefer during multiple points in the lesson. -Students reflect upon the solving methods and how well they’re progressing with each as individuals. -During the warm up students use a self-chosen number for the sum of the three marking periods. Student Goals: -Some students will be focusing on the graphing method, particularly in choosing the best scale for the problem and graphing points correctly on the coordinate plane. -Some students will analyze efficiency within the solving methods.

Lesson Reflections:

Name________________ Semester Two Average

In order to receive credit for the second semester your marking period four (MP4), marking period five (MP5) and marking period six (MP6) averages must add up to at least 195 points because 195/300=.65 or 65% and you need at least a 65% to receive credit. In the space provided below write down your MP4 and MP5 averages and then try to figure out what your minimum MP6 average is.

MP4 = MP5 = Try to figure out what number MP3 must be to make the following equation true.

MP4 + MP5 + MP6 = 195

MP3 =

The number you just found for MP3 is the LOWEST grade you can get and still receive credit for semester 2. Repeat the process above, but replace the 195 with the number of your choosing. Take your goal semester two average and multiple it by three. Replace the 195 with the product you just found.

Name________________ Semester Two Average

In order to receive credit for the second semester your marking period four (MP4), marking period five (MP5) and marking period six (MP6) averages must add up to at least 195 points because 195/300=.65 or 65% and you need at least a 65% to receive credit. In the space provided below write down your MP4 and MP5 averages and then try to figure out what your minimum MP6 average is.

MP4 = MP5 = Try to figure out what number MP3 must be to make the following equation true.

MP4 + MP5 + MP6 = 195

MP3 =

The number you just found for MP3 is the LOWEST grade you can get and still receive credit for semester 2. Repeat the process above, but replace the 195 with the number of your choosing. Take your goal semester two average and multiple it by three. Replace the 195 with the product you just found.

Classwork 5-13 Solve the following systems of equations in your notebooks using elimination, substitution, and graphing methods. For each system, answer the three writing prompts. 1) Which method of solving did you decide to use upon first seeing the problem? What made you decide to use that method? Is it the method you always choose first? 2) What the method that you chose the easiest or simplest method? If not, which one was? 3) Why do you think your answer for #2 was the “easiest” for this problem? Do you think it will be the easiest for every problem, or is there something about this problem that makes it easier?

y = 2x + 8y = !x +1

2x + 3y = 865x + 4y = 187

x = 2y + 5y = 3x ! 5

3x2

!10 = 7y + 3

x2+ 2y = 3

Classwork 5-13 Solve the following systems of equations in your notebooks using elimination, substitution, and graphing methods. For each system, answer the three writing prompts. 1) Which method of solving did you decide to use upon first seeing the problem? What made you decide to use that method? Is it the method you always choose first? 2) What the method that you chose the easiest or simplest method? If not, which one was? 3) Why do you think your answer for #2 was the “easiest” for this problem? Do you think it will be the easiest for every problem, or is there something about this problem that makes it easier?

y = 2x + 8y = !x +1

2x + 3y = 865x + 4y = 187

x = 2y + 5y = 3x ! 5

3x2

!10 = 7y + 3

x2+ 2y = 3

Appendix BInterview Transcripts

B.1 Sarah Jane

JAMES: So, Sarah Jane, you recognize what this is, right? Do you remember what

method you used to solve [#4 on 5-12]?

SARAH JANE: Substitution.

JAMES: So why did you use substitution for this problem?

SARAH JANE: Because it was easier, you could just, um, substitute 3x − 7 for the

y.

JAMES: Okay, um, so why was it easier than, say, elimination.

SARAH JANE: I dunno, when I see a problem like that, I just use substitution.

JAMES: What do you mean a problem like that?

SARAH JANE: The y is separated, right, and then there’s the 3x and the numbers

and I find it easier than the elimination method, and I just want to try other methods

than the elimination method.

APPENDIX B. INTERVIEW TRANSCRIPTS 63

JAMES: Okay, because on the back of the page you wrote that elimination was your

favourite method, so I would have thought that, since elimination was your favourite, you

would have used elimination for this problem. So that’s what I was wondering why you

didn’t use elimination.

SARAH JANE: I wanted to try something else.

JAMES: Because you wanted to try something else, okay. So you thought substitution

worked well because you had sorta the y alone, right?

SARAH JANE: Mm-hmm.

JAMES: So here, we’re gonna talk about yesterday’s problems you were doing. In

this problem you had the y alone as well, but you used elimination first, you didn’t use

substitution, so why go with elimination [on 5-13] when you used substitution [on 5-12]?

SARAH JANE: Cause I don’t go with the same things over and over, the same order,

I go with different things, especially with things I kinda need, I struggle with, I try to

work with it, so I can get myself used to using that method also.

JAMES: I think that’s a good thing to do.

Okay, so then, do you remember the second problem you did yesterday?

SARAH JANE: Yes.

JAMES: I don’t know if you finished it, but the first thing you did was solving it with

elimination, and you had to do all that multiplying, was elimination still an easy method—

SARAH JANE: Yes.

JAMES: –even with all of those extra steps—

SARAH JANE: Yeah.

JAMES: Okay. Did you actually solve it—

SARAH JANE: With substitution? I was getting to it, getting the y alone, but I

found it kinda hard.

JAMES: You found it hard. What was so hard about it?


SARAH JANE: I found it hard was to get y alone, especially when there’s a number

where the y is—

JAMES: A coefficient?

SARAH JANE: Yeah, a coefficient, and I was kinda starting with the division so it

kinda took me some time and I just like, once I got the x alone, because I tried with x,

and I got kinda confused so I wasn’t really sure where to go.

JAMES: Right.

SARAH JANE: So even though I did the elimination method, I tried to do the

substitution so I could get used to it because not always it elimination going to work.

JAMES: Right.

SARAH JANE: I try to do things I can’t do just in case someone asks me, can I show

them. So I have to get myself...[trails off]

JAMES: Yeah, that’s good thinking. So, looking at this particular problem, so elimi-

nation is easier because you only have to multiply but with substitution you have to do

dividing and dividing is a lot harder than multiplying?

SARAH JANE: It depends on the number.

JAMES: It depends on the number, right. I guess, yeah, the dividing by 2 could be

easy but the dividing by 3 could be harder.

SARAH JANE: Yeah.

[cut]

JAMES: So what do you think of the graphing method? We’ve been talking about

elimination and substitution, but I about graphing?

SARAH JANE: The thing is...it’s the same as the solution with y = 3x − 7, it’s is

easy to graph, we have the slope and y-intercept, so it would be easy to find it. But again

it depends on...sometimes it’s hard, if you need to get the y alone or the x alone, or if it’s


between numbers, it’s kinda hard to figure that out, but if it’s by itself it’s easy to figure

out, but I don’t find it that hard neither. It’s okay.

JAMES: It’s okay. Would you ever choose to use instead of one of the other two, if you

could only use one?

SARAH JANE: Yeah, why not?

JAMES: [laughter] Why not? Well, realistically, if you had a whole quiz like this where

you could use any method for every problem, would you ever actually choose graphing?

SARAH JANE: Depends on the number. Depends on the solution.

JAMES: So if you’ve done three that were all elimination you’d want to try graphing

to change it up?

SARAH JANE: Yeah. It’s kinda, for me, I try to do as many things as I can. Even if

I don’t like it, I have to get used to it because in the future this might come out.

JAMES: That’s true.

SARAH JANE: Or someone might ask me and I wouldn’t know what it is, and they’d

be like “What’d you go to school for?” So I have to get my mind set on that one.

JAMES: I like the way you think, Sarah Jane.

SARAH JANE: [laughter]

B.2 Amy

JAMES: So part of what we’re talking about is what we’re doing on the quiz and what

we did the other day, was it, Monday? So first, I know I’ve asked this a lot of times but

I’ll ask it again, what is your favourite method for solving systems of equations?

AMY: Um, I would have to say elimination.

JAMES: Elimination? Okay. So that’s interesting because here, on the quiz, you said

substitution—


AMY: I like both of them.

JAMES: You like both of them? Is that something that happened since the quiz to

change your mind or you’ve always like them both equally?

AMY: It’s just, I like them equally. I don’t like graphing.

JAMES: Right, graphing. Why don’t you like graphing?

AMY: Cause it’s mad stuff to do and, like, it’s difficult to figure out how to put the,

how to figure out how to make it y = mx + b.

JAMES: Well, when you’re doing substitution, don’t you usually have to get y by itself

anyway?

AMY: Yeah, but it’s like different, because, depending on the kind of question I have

I know when to use each method.

JAMES: So, I wanna talk more about that. What is it about the question that lets

you know which one to use?

AMY: It’s like, if I know I could eliminate one variable, then I would use elimination.

Or if I have the y alone, I would just use substitution. If I can’t do any of those, obviously

I’ll need to do the graph.

JAMES: So then for this one, this is the one you can use any method, this is from the

quiz, so, just looking at that question, what would you have to do to solve it?

AMY: I believe I used elimination.

JAMES: You did use elimination. What made you want to use elimination?

AMY: The two 3x’s. I knew I could eliminate that.

JAMES: Okay, so, you also said that when you have the y alone or the x alone you’d

use substitution, so it would be easy to use substitution in that case.

AMY: But then, you see the difference is that, there’s no y on the other side [the right

side], both of the y’s are on the other side [left side].

JAMES: So you would use substitution for that case?


AMY: Yeah.

JAMES: So if it was like y = 3x− 7 and then 1 = −3x− y, you’d use substitution in

this case?

AMY: Yeah.

JAMES: So why wouldn’t you still use elimination?

AMY: No, knowing me I would still use elim—I would use substitution because it’s

easier for me to work it out with substitution.

JAMES: So why is it going—so what is it about substitution that makes it easier to

work out?

AMY: I would just have to plug in the stuff.

JAMES: So why wouldn’t elimination work here?

AMY: It would, but I feel more comfortable using substitution.

JAMES: Hmm. So what I’m trying to determine is why, is how [the revised problem]

is different from [the original problem]. What’s make it so that this one elimination is the

comfortable method and for this one substitution is? This equation is equal, it’s the same

as [the other] equation.

AMY: Yeah, but you see, there’s no y, so there’s not a y on the opposite side. There’s

not really a way I could substitute anything. Unless I put that in there and, to me that

would be more difficult because to me when you look at an equation you should know

which type of method to use to solve it, by knowing how comfortable you feel with each

method.

JAMES: Okay, so I think I see what it is. So if you had these two and you used

substitution, it would look like this: 3x − 7 = −3x − 1. But if you used it over here, you

would get 1 = −3x− (3x− 7).

AMY: Yeeah.


JAMES: So here, in [the first one], well, can you maybe describe, before I say what I

think, how these two are different?

AMY: Well, to me it’s different because it’s like, it’s think it’s all on, like, what’s on

each side.

JAMES: Mm-hmm.

AMY: Because, like, when you have two different equations it’s harder to just like to

put it together, you know what I’m sayin’?

JAMES: Right, because I’d have to get the x’s on one side.

AMY: Exactly.

JAMES: And on this one there’s already on the same side—

AMY: They’re on the same side, so it’s easier to put like terms together and solve for

the variable.

JAMES: That’s interesting.

[cut]

Well, I think that’s it. You said some really interesting things.

AMY: Because I’m a genius.

B.3 Jack

JAMES: So my first question to you is, um, what’s your favourite method for solving

systems of equations, elimination, substitution, or graphing? I have some answers you’ve

given before, but I want to see what you think right now.

JACK: Um, I would say graphing.

JAMES: [laughter] You would say graphing? Why are you smiling when you say that?

JACK: Because it’s mad easy.

JAMES: Okay, so, what is it about graphing that makes it easy?


JACK: Finding the point.

JAMES: Finding the point?

JACK: Yeah, the intersecting point.

JAMES: The intersecting point. So, uh, can you tell me why graphing is easy but the

other ones aren’t easy?

JACK: The other ones are also easy but I think graphing is the most, how you say it,

precise?

JAMES: The most precise! Oh, I think that’s interesting. Because, take a look at this

problem you guys did the other day. You did graphing first, but remember that your

answers were things like −2.3̄ and −3.3̄. Can you really be so precise? I mean, knowing

what the answer is I can look at the graph and go, Oh, that’s .3, that’s .3, sure. But if I

didn’t know what the numbers were—

JACK: It would be a lot more difficult, yeah I know. But like, but when you have

known all about graphs since, what, the fourth grade, then yeah, it would be a lot more

easier.

JAMES: Oh, so it’s because you’re more familiar with graphing?

JACK: Yeah.

JAMES: So substitution and elimination, those are a lot newer—

JACK: Yeah, I know how to do those, too, I can do them perfectly fine, it’s just that

I’d rather recommend graphing because I’ve been doing it for a longer time.

JAMES: So you’re more comfortable with it.

JACK: Yeah.

JAMES: Okay, that’s interesting.

[cut]

JAMES: Okay, so when you have a question like this, this is #4 on the quiz where you

could choose any method, you chose to do graphing.


JACK: Mm-hmm.

JAMES: Is there anything about the way the problem was presented that made you

want to do graphing, or just because you liked it, liked graphing the most?

JACK: I would say it’s because I liked graphing. But, I could have also done elimination

because it has a 3x and a 3x.

JAMES: Right, so could you also have done—

JACK: Substitution as well? Yeah.

JAMES: Yeah. I picked that question for a reason, I designed this question so that all

three of them would be good choices. So you decided to do graphing because it was the

one you liked best.

JACK: Yes.

JAMES: So when you have a chance to graph, you’ll do graphing?

JACK: Yeah.

JAMES: Now, what if I gave you a question like...[writing] a question like this one?

Would you want to graph it? [The question is 2x + 3y = 4 and 3x− 2y = 6.]

JACK: Um....probably, if I actually knew how.

JAMES: [laughter] If you knew how—

JACK: Well, actually, what I would do is get the y by itself first, I just remembered

that. I would get the y by itself, start from there, and I would probably try the-the-the—

JAMES: So why don’t you try it right now, try graphing—not necessarily graphing,

try solving it right now. Or, at least get started.

JACK: [working silently]

JAMES: So, so far you took the first equation and got y by itself.

JACK: Yeah, so like—

JAMES: Now what would you do?

JACK: So obviously you would divide these two, it would be 1 point, is it 1.1?


JAMES: 1.333333. I like leaving at a fraction so I don’t have to have a repeating

decimal, so I just leave it as four thirds.

JACK: So 43 minus 2

3 . So, like, on the graph I would go, like, 1.3 on the y-axis, yeah

on the y-axis, I would go up—

JAMES: So do you want to graph it, or would another method be more useful here?

JACK: It would be a lot more useful because you could see, it just took a pretty long

time.

JAMES: So if you didn’t want to graph it, what would you do now?

JACK: Let’s see...probably solve.

JAMES: What do you mean “solve”?

JACK: Like, see what I would get if y would be by itself, or x.

JAMES: Well, you have x by itself for the first equation. I want to talk about the

whole system.

JACK: Hmm. [thinking] Well, I’m not really into it right now.

JAMES: [laughter] You’re not really into it right now? Okay.

So, here on the other questions you wrote that graphing isn’t the one you always pick

first, you pick substitution, because substitution is the easiest. You didn’t really answer

my third question about that, but, so let’s talk a little bit about substitution. Why would

substitution be easier than elimination?

JACK: Well, because if I have this, then I can use substitution because I can replace

it with one of the y’s up there, and then I could just solve it from there, just get the x or

the y by itself.

JAMES: Okay, so then why do you prefer doing that instead of elimination?

JACK: Because with elimination I would have to, get like, one of the two variables the

same.

JAMES: Is that harder than getting y by itself?


JACK: Yeah, because I would have to multiply it by a specific number, getting it to

the other one, and I would basically have to do the entire process again. This is 2x, and

I’d probably get it to 3 or something...and then I would have to start this entire process,

just to figure out the specific answer.

JAMES: And that’s difficult because [referring to substitution] this is something you

know how to do and figuring out which number to multiply isn’t so easy?

JACK: Yeah, not so much. Not all the time.

JAMES: Can you think of times when it would be easier to use elimination than

substitution?

JACK: Elimination would be easier in a problem like this one that you gave, because

it did have 3x and −3x. And this one too. [#1 on 5-12.]

JAMES: If I had switched these [#1 and #2 on 5-12], I made [#1] a substitution

problem and [#2] and elimination problem, it would probably be a much harder quiz.

JACK: Yeah, a little bit more. But, um, you know.

JAMES: So say on the final exam you see a problem like this one—

JACK: But it says substitution.

JAMES: Yeah—no, it says any method, would you use elimination?

JACK: If it was a problem like that, then yeah.

JAMES: And why is that?

JACK: Because both x’s have the same number. −2x, 2x, −4y, 4y.

JAMES: So basically for a question you would take the least number of steps.

JACK: Yeah, it depends on how you see the problem, the variables and stuff.

JAMES: Okay, okay. Well, thank you Jack.


B.4 Rose and Donna

JAMES: So my first question for you two is: what is it about the elimination - we’re

talking about solving systems of equations – what is it about the elimination method that

makes it easier?

DONNA: It’s not complicated.

JAMES: What do you mean by complicated?

DONNA: The other ones are harder to do.

ROSE: Yeah. Graphing, to me, with problems like that, it’s harder, like I get confused

when it comes to graphing.

JAMES: Okay.

ROSE: And I’m not good with, like, negative numbers and positive numbers—

JAMES: You don’t know which way is which?

ROSE: Yeah. So I get little confused with that, so elimination is better. And—

DONNA: Substitution, I don’t like it.

ROSE: I don’t like it either.

JAMES: So what is it about substitution that you don’t like?

DONNA: The way that you have to switch numbers around.

ROSE: Yeah.


JAMES: So it’s like cleaner?


ROSE: And you could see it more visually. It’s easier for you to just, you know...

JAMES: Oh, because of the way everything lines up nicely?



JAMES: Okay. So do you think the fact that we learned elimination first is one of the

reasons you like it better?

ROSE: Yes, I think that’s why, too. Because we learned it first so when you were

showing me something else to do I was already used to elimination.

JAMES: Okay. So, Donna, this is a question for you: originally in your notebook when

I first asked you guys what you liked best, you said you liked graphing the best.

DONNA: Where?

JAMES: I think it was back on...5-9, yeah. I prefer graphing because it’s easy to plot

points, et cetera. But then, on your quiz, you wrote that you were struggling with graphing

and you liked elimination the best. So I want to know what changed [two weeks ago] and

[last week], where [on 5-9] graphing was easy and [on 5-12] graphing was hard.

DONNA: I have no idea.... (exclaiming) I had help on [5-9].

JAMES: You had help on that one?

DONNA: I had help.

JAMES: So, because you had help it was easier but once you were on your own it was

harder.

ROSE: Yeah, it was easier for me then.

DONNA: It was easier when I had help but on my own I’d rather use elimination.

JAMES: Okay, so, my last question is: this problem here, how would you solve it?

ROSE: Okay, these two problems?

JAMES: Yeah. If you want a separate piece of paper, here you go.

ROSE: You want us to like solve it right now, or just how would we solve it?

JAMES: Well, just tell me how you’d solve it...

ROSE: Personally, me, I’d switch it up a bit. 2y here, +x= 5.

JAMES: But are those two equations equivalent? Wouldn’t that have to be −2y?

ROSE: It’s confused me. But elimination is easier for me, I dunno why.


JAMES: Well, then why don’t you try elimination on this one and see what you would

get.

ROSE: It’s harder when it’s like that.

JAMES: It is harder like that.

DONNA: Yeah.

JAMES: That’s true.

ROSE: Like, elimination like this [pointing to previous problem], when they’re equiv-

alent to each other, is easier for me. (trails off)

JAMES: So elimination is hard for this kind of problem, when it’s like this. Do you

think substitution might be an easier way for this type of problem?

ROSE: Yeah. Yep.

DONNA: Elimination is good for some certain problems, not for all.

JAMES: Not for all. Okay, so, but would you choose to do substitution for this one,

when it looks like it’s easier, or would you do elimination because you like it better?

DONNA: I think I’d do substitution.

JAMES: You would do substitution.

ROSE: I don’t really understand the different between substitution and elimination.

JAMES: Well, the difference is that elimination , we’re adding or subtracting our

equations to get rid of a variable. With substitution we know what one is equal to so we

can plug it in and replace it in the other equation. So we only deal with one equation at

a time. Elimination you’re dealing with both at once.

ROSE: That makes it easier.

JAMES: It’s easier to deal with both at once instead of one at a time?

ROSE: Yeah.

JAMES: Okay.


B.5 Martha

JAMES: So, remember I had these questions about which methods you like the best,

which ones were the easiest. The first time I asked you guys that, you said that you liked

substitution the best. But here [on 5-13] you wrote that you liked elimination the best,

that it was the easiest. So what I wanted to know is, what is it, what made you change

your mind and made you like elimination better for this problem?

MARTHA: Um, I dunno.

JAMES: You dunno?

MARTHA: I just tried to use a method that is easier.

JAMES: Well, I mean, [on 5-13] you did both, you used elimination, substitution, and

graphing. So what is it about the elimination method that was easier in this particular

case?

MARTHA: I chose this one because this one is shorter than the other ones.

JAMES: So it’s shorter. So just whatever one takes less time would be the easier one?

MARTHA: Yes.

JAMES: Okay. So my next question is, um, let me get a pen. So, if you had this

question, y = 3x− 7 and y = −3x− 1, how would you solve it?

MARTHA: At first I would...mmm...I would, um, take this from, the −3x from the

3x.

JAMES: So you would—so what method are you doing?

MARTHA: Elimination.

JAMES: Elimination. So what do you get when you do that? You can write it down.

MARTHA: Zero.

JAMES: Write down what you would do.


[cut. Martha works out the problem down to one equation with one unknown, with lots

of scratching out, but cannot seem to solve it from there.]

JAMES: It seems like you’re having a problem because of how messy the equations

are. Let me rewrite it. Here you have 0 = 6x− 6.

MARTHA: So, you can add 6 to both side.

JAMES: Mm-hmm. See, see how messiness causes problems?

MARTHA: So 6x = 6, and you can divide by 6. So x = 1.

JAMES: So I’m not going to ask you to find y, it’s not particularly important. So this

particular problem, when I wrote it it was designed to be a problem that works pretty

easily for any of the methods: substitution, elimination, or graphing. So why in this case

did you choose elimination?

MARTHA: I dunno. I just try...

JAMES: So you just tried that one first? If it didn’t work you would have switched?

MARTHA: Mm-hmm.

JAMES: Okay. Was there something about the problem that made you think, Oh,

elimination, that’s what I should do.

MARTHA: I dunno.

JAMES: So, here you wrote that it’s easiest for this problem, and for this problem too,

but you don’t always think it’ll be the easiest. So what would a problem have to look like

for elimination to not be the easiest?

MARTHA: I dunno.

JAMES: You dunno?

MARTHA: No.

JAMES: Let me think of a different way to put it. This problem, you could use elimi-

nation because you said Oh, I have 3x and −3x. So they might cancel, or the two y’s, they


might cancel. So I’m wondering, if I didn’t have those two things, would you still choose

elimination?

MARTHA: No.

JAMES: So like, say, this problem here [the second problem on 5-13], we didn’t get to

it in this lesson but you would—

MARTHA: I’d use substitution.

JAMES: Because it’s not so obvious that elimination would work. Okay. Thank you,

Martha.

B.6 Wilfred

JAMES: So I want you to tell me what you think about the graphing method for solving

systems.

WILFRED: The graphing method for me is the hardest one.

JAMES: Why is it the hardest one?

WILFRED: It’s hard to get the y alone. And I really don’t know how to do the y

intercept and how to move on the graph.

JAMES: So then, remember the lesson we did where we did all three methods?

WILFRED: Yes.

JAMES: You said that you did graphing first because you wanted to get the hardest

one over with. My question is, here, this is your quiz, on the problem where you could

do it any method you want, you chose to do it with graphing, even though you said that

graphing was the hardest.

WILFRED: I chose graphing because it had y on both sides.

JAMES: Because it had y on both sides...even though graphing was the hardest, you

still thought it was better than the other ones?

WILFRED: Did I get the answer right?


JAMES: You did get it right.

WILFRED: I told you cause didn’t know how to do it with these two, so I did it with

graphing. [inaudible] just to try to get the answer right.

JAMES: Right, and the interesting thing is (and you probably noticed this when you

got your quiz back) is that you got [#1 and #2] wrong, and the two ones you did with

graphing—

WILFRED: Was right.

JAMES: —were right. Don’t you think that’s interesting?

WILFRED: Yeah.

JAMES: (laughing) So can you really say that you’re struggling the most with graphing

when the graphing ones are the ones you got correct?

WILFRED: What?

JAMES: Would you still say that graphing is the hardest knowing that you got the

graphing ones right and the non-graphing ones you didn’t get right?

WILFRED: Nah, I would say the substitution one is harder.

JAMES: So you’re saying substitution is the hardest. You’re changing what you think.

WILFRED: Yeah.

JAMES: So...based on what I’ve seen, you were mostly able to get the graphing done

right and so I’m trying to figure out what it was about the graphing that made you think

that it was the hardest if it isn’t actually the hardest.

WILFRED: Trying to get the y alone.

JAMES: Is that maybe substitution is the hardest now?

WILFRED: Yeah.

JAMES: Because you have to get y alone in substitution?

WILFRED: Yeah.


JAMES: So the problem is the Step Two part. Because Step Three you can handle,

and Step One you can handle, but solving is terms of is hard. So what do you think of

elimination?

WILFRED: That’s an easy one for me. Like, right here? All you have to do is eliminate

one letter to get the other ones alone.

JAMES: And that’s easy because there’s no solving in terms of—

WILFRED: Yeah, and once you get the y you just substitute the letter and that’s it.

JAMES: So could you use elimination for this one [#4 on 5-12]?

WILFRED: No, well, you could to get y’s, and then this.

JAMES: Mm-hmm. Yeah, you could. How would you do it?

WILFRED: Um. Is that a negative sign or a take-away sign?

JAMES: Well, are you adding or subtracting?

WILFRED: I’m adding. So that’s 8, take away 8, and that’s it. I’m thinking you

could—

JAMES: Well, wouldn’t that be 2y?

WILFRED: Yeah. So then you gotta divide by 2, and y = −4. So then you gotta

substitute in.

JAMES: Yeah, good, don’t worry about that. So it seems like you have a good grasp of

elimination, but you still chose to graph it because it was in [y = mx + b] form. I wonder

what you’d do if it looked like something else. Let me come up with a new problem.

(writing) Something like this.

WILFRED: Try to solve it?

JAMES: Yeah. How would you solve that?

WILFRED: Eliminate the x.

JAMES: How would you eliminate the x?

WILFRED: Add it.


[cut. Wilfred works on the problem.]

WILFRED: I don’t know how to do it.

JAMES: You don’t know how to do it.

WILFRED: That’s my problem, I can’t get the y alone.

JAMES: You don’t have to get the y alone. You can still eliminate the y.

[cut. Wilfred finishes the problem.]

JAMES: So the reason I gave you this one was because it would be sorta messy. After

you eliminated you would have x’s on both sides. You would still prefer to use this method

instead of, say, substitution?

WILFRED: Um, I think substitution would be, like, easier, because all you have to

do is take away the one and substitute in for x.

JAMES: So substitution would be easier, then?

WILFRED: For this problem.

JAMES: For this problem. What is it about this problem that makes substitution

easier but not for other problems.

WILFRED: Not for other problems because it wouldn’t have y alone. If you have y

alone it would be easier to substitute.

JAMES: Well, y isn’t alone here, do you mean, I think you mean, you’re saying y

doesn’t have a coefficient?

WILFRED: Yeah.

JAMES: It’s less steps to get y by itself?

WILFRED: Yeah. For other problems it was be hard to get y alone and this problem

not that much because all you gotta do is take away the one.

JAMES: Okay. Thank you, Wilfred.

Bibliography

[1] Jeremy Kilpatrick, Jane Swafford, and Bradford Findell (eds.), Adding it up: helpingchildren learn mathematics, National Academy Press, Washington DC, 2001.

[2] Harold R. Jacobs, Elementary Algebra, W. H. Freeman, 1979.

underlying causes of students' preference of method in solving systems of linear equations

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