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**a help sheet explaining how to solve equations by balancing them**, here's the result:

Though this student made a mistake on the first line, her logic was perfect. She started out with one nickel, which meant she needed 5 dimes (4 more). By multiplying the number of nickels by 5 and adding this value to the product of the number of dimes and 10, she was able to obtain her total. Using guess and check, she continued to increment her values for nickels until she arrived at a total of $1.15.

Four of the 6^{th} graders obtained the correct
solution to this problem. Each was able
to explain his or her work well, and their strategies remained fairly
consistent throughout the problem-solving sessions.

__8 ^{th} Grade Responses:__

__ __

To my surprise, the 8^{th} graders showed the most
difficulty with this problem. Each
student started out using systems of equations to model the problem. Four of the five algebra students developed
a version of the following system:

*N = nickels, D =
dimes*

*D = N + 4*

*N + D = 115*

Problem #1 was consistently the most difficult and time
consuming for the 8^{th} graders, partly because of their stubborn
efforts to force the problem to work.
When I asked one stumped student what her equations meant, she replied,
I dont know! I just put them
down! These students could not explain
their work well. To provide insight
into the struggles on this problem, consider this 8^{th} grade (who
maintains an A in the class) students work:

From the beginning, she incorrectly set up her systems by defining the equation:

*D + N = 1.15*.

Her problems only became worse. Using the method of substitution, she replaced *D* with *(N-4)*,
only to realize that she had a decimal value on one side of the equation and a
whole number on the other side. She
realized that this was a problem, but simply added a decimal in front of the
4. She had no concept of what these
values represented or even what her equations meant. Though she repeated her calculations several times, she could not
get a satisfactory result. So she tried
a different approach:

She knew that the value of a dime is equal to two
nickels. Thus, she replaced her *N =
D+4* equation with *D=2N*. It
seems clear that she also has little understanding of her variable
meanings. In the same system, *D*
stands for a monetary value $0.10 as well as the number of dimes. Once again, she could not explain her work
well, and in the end obtained the correct answer using guess and check methods.

The other students displayed similar difficulties, although one of the four eventually remedied her mistake. Three of the five total students reverted to guess and check to obtain their final solution.

** Problem #2**:

*1.
**Elinor loves to paint.
She paints pictures to sell in her mothers store. She earns $3 for her small paintings and
$5 for her large paintings. *

*a.
**How much money would Elinor earn if she sold 6 large
paintings and 11 small paintings?*

*b.
**One Monday, Elinor sold only small paintings. She earned $36. How many small paintings did she sell?*

*c.
**On Tuesday, Elinor sold 3 more large paintings than
small paintings. In total, she earned
$39. How many of each size painting did
she sell?*

* *

This example produced results very similar to the first
problem. Each student was able to solve
parts (*a)* and *(b)* successfully or with only insignificant
arithmetic errors. Part (c) proved the
most challenging.

__6 ^{th} Grade Responses__:

Again, the 6^{th} grade students utilized guess and
check methods to solve this problem.
Several used charts to organize their work, one would write an answer
and erase it until he found the correct solution, and one student was unable to
solve the problem. The techniques were
very similar to the first problem.

__8 ^{th} Grade Responses__:

Again, only one algebra student successfully used algebra to solve part (c). The other students set up the system with the same error:

*S = Small, L =
Large*

*S + L = 39*

*L = S + 3*

These students eventually solved the problem using guess and check methods.

__Problem #3__:

Arwen wants to plant a tulip garden. To keep out the rabbits, she must set up a fence around the flowerbed. She has 24 feet of fencing.

*a.
**If Arwen makes a square flowerbed with her fencing, how
long will each side be? What is the
area of this flowerbed?*

*b.
**In the square flowerbed, Arwen wants to plant 2 tulips
for every square foot inside the flowerbed.
How many tulips will she plant in the flowerbed from part(a)?*

*c.
**Arwen decided to make the flowerbed a rectangle. She made one side 4 feet longer than the
other side. How long is each side of
her flowerbed?*

This problem is geometry-based but can be solved algebraically. Surprisingly, only half the students drew a diagram while working on this problem. Most students had little trouble with parts (a) and (b) outside of an occasional lack of conceptual understanding of area and perimeter. Again, the most interesting part of Problem #3 was part (c).

Each student that drew a picture also employed the guess and
check method. Most of these students
used charts; one student drew the rectangles to scale using a ruler and 1 inch
= 1 foot. One 6^{th} grade
student was quick to solve this style of problem in the previous two algebraic
examples, but had difficulty seeing any relationship between those problems and
this geometry problem. In the end, she
was successful, but could not see the same algebraic relationships that she had
used in the previous problems.

Only two of the students missed this problem. The first was the 6^{th} grade
student who was unable to understand any of the previous similar
situations. Ironically, the second was
the only student who set up the problem correctly using algebraic methods. Each of the other students used guess and
check methods to obtain their solutions.
The algebra students again set up the system incorrectly, and one did
not even attempt to use algebra by this point.

This is an example of the correct system of equations used
by an 8^{th} grade student: