STUDENT RESPONSES
Problem #1:
Student Grade Level M/F Strategies Details
1 6th Grade F Pictures, Guess and Check, Used pictures to create grouping.  Drew two "d"'s and one "n" to create twenty-five cents.  Then grouped according to twenty-five cents until reached $1.  She then had one dime, one nickel left.  Drew chart at the end to display answer.
Chart
2 6th Grade M Guess and Check, Chart, Manipulatives, Scratch Paper Started out making a chart using guess and check.  Very quiet and diligent.  Made multiplication errors and gave up.  Then used nickels and dimes to obtain correct answer.
3 6th Grade F Guess and Check, Chart Used organized, clear chart.  Started with 1 nickel and 5 dimes, since 4 more dimes than nickels.  Much too small, so increased to 4 nickels, 8 dimes.  Incremented by one until obtained correct answer.
4 8th Grade F Algebra: Substitution, Guess and check, Calculator Set up system of equations with D + 4 = N and D + N = 1.15. Used substitution to obtain 2N - 4 = 1.15.  Since the right side had a decimal, she decided that 4 must really be 0.4, so she added the decimal.  Solving she obtained a very wrong answer.  She checked her multiplication a few times, then set up a new system:  since there are 2 nickels in a dime, she used the equation D = 2N with D + N = 115, without the decimals.  This too gave her an incorrect answer.  She finally obtained the correct solution using guess and check.
5 8th Grade F Algebra: Elimination Set up system of equations with 10D + 5N = 115 and D - 4 = N.  Using elimination, she quickly obtained the correct solution.
6 8th Grade F Algebra:  Elimination, Substitution Started out using systems with N + D = 115 and N + 4 = D.  Used substitution first with no luck.  Then, she tried elimination to get a "different" result.  She decided to come back to this problem later.  After working on the other problems, she started to guess but then realized her mistake.  She set up the correct system and used substitution to successfully solve the problem.
7 6th Grade F Chart, Guess and Check Made a clear chart of nickels, dimes, and total.  Started with 14 dimes, 10 nickels which were too large.  Guessed 9 dimes, 5 nickels next, and found the solution.  Did the multiplication by hand
8 6th Grade M Mental math, Guess and Check Did not want to write out his thoughts or count.  He did not know where to start, so I talked him through the problem.  He tried to do the problem in his head but decided to quit without a solution or verbalizing any of his guesses.
9 8th Grade F Algebra: Substitution, Elimination; Guess and Check Started with a system of equations with N + 4 = D and D + N = 1.15.  She then used substitution to obtain the equation N + 4 + N = 1.15.  Since she had a 4 on one side, she became confused.  She eventually gave up using algebra and came back to the problem at the end of the session.  This time she used guess and check.  She started with 7 * 10 and 14 * 5, then realized the restriction on number of D's and N's.  Then she made a list of multiplication until she obtained 9*10 + 5*5 = 115.
10 8th Grade M Algebra: Systems, Guess and Check, Coins Declared that he knew he had to use systems for this problem.  Wrote down D + 4 = N and D + N = 115, but could not remember how to use systems.  He eventually gave up on the algebra and did it by guess and check in his head. 
Problem #2(c):
Student Grade Level M/F Strategies Details
1 6th Grade F Guess and Check, Chart Parts (a) and (b) were done quickly using multiplication, addition, and division.  Part (c), she made a chart and started with 6 large and 3 small, which ended up being the right answer.
2 6th Grade M Guess and Check, Chart Started to draw a chart for part (c) but did not use it.  He mixed up the large and small paintings but found his mistake.  He finished the problem using guess and check.
3 6th Grade F Guess and Check, Chart Made a table starting with 1 small and 4 large, which was much too small.  She moved to 3 and 6 and found the correct solution.
4 8th Grade F Algebra: Substitution, Guess and Check Started with system: S =small, L = large; S + L = 39 and L = S + 2.  When this would not work using substitution, she used a sort of guess and check.  Starting with 39, she subtracted 3*5 to get $24.  She then took away 2 more large paintings to get $14.  Three more left $9, which left 3 small paintings.
5 8th Grade F Algebra:  Elimination Solved this problem using Systems and Elimination correctly and quickly with the system: 5L + 3S = 39 and L - S = 3.
6 8th Grade F Algebra:  Substitution, Guess and Check Started using Systems with L = S + 3 and L + 5 = 39.  When this system wouldn't work using substitution, she went back to guess and check methods, starting with 7 large and 1 small to obtain $39.  She saw this didn't work, so she tried 6 and 3, finding the correct answer.
7 6th Grade F Guess and Check, Chart Using a chart, listed number of large paintings and their total value, small and value, and the total sum of the paintings.  She started with 8 large and 5 small, then tried 5 and 2, then 7 and 4 until came to 6 and 3 which was the correct solution.
8 6th Grade M Mental math, Guess and Check Found the correct solution for part (a) but was unsuccessful for parts (b) and (c).  He had no idea and could not verbalize his thinking very well.  Eventually quit.
9 8th Grade F Guess and check Could not remember how to use algebra, so she skipped it.  Later she used guess and check to obtain the correct answer by listing 5 * 5, 2 * 3 and then 6 * 6, 3 * 3.
10 8th Grade M Mental math, Guess and Check Did this problem in his head using guess and check. 
Problem #3(c):
Student Grade Level M/F Strategies Details
1 6th Grade F Pictures, Guess and Check, Formula Sheet Started with a picture of a rectangle.  Confused herself a bit on perimeter vs. area, but figured it out.  Made a guess, then drew a rectangle to scale with one foot equal to one cm with the ruler.  Then evaluated her picture to see if it fit the requirements.
2 6th Grade M Guess and Check Confused about perimeter vs. area at first.  With some guidance, he understood the difference.  Guessed the correct answer at first, but didn't realize it and could not explain it.  Ended up giving up.
3 6th Grade F Formula Sheet, Guess and Check, Mental Math, Picture Drew a rectangle.  Went through list of values in her head and found the correct dimensions.
4 8th Grade F Picture, Guess and Check, Mental Math Drew a rectangle.  Did guess and check mentally until she obtained the correct solution.
5 8th Grade F Algebra: Elimination Set up system correctly with L-W = 4 and 2L + 2W = 24.  When using elimination to solve, she made a mathematical error and ended up with L = 7, W = 5.  She did not catch her error.
6 8th Grade F Guess and check Decided to work with totals.  First thought length totaled 20, width totaled 4, but this did not work.  Eventually found 16 and 8, which means L = 8, W = 4.
7 6th  Grade F Pictures, Guess and Check Drew several rectangles.  Started with square with 6 length and then made different combinations from this point to obtain correct answer.
8 6th Grade M Mental Math, Formula Sheet Did not have a concept of area or perimeter, so he was lost on this problem.  He did not understand the formulas and did not write anything down.  He eventually gave up.
9 8th Grade F Pictures, Guess and Check At first thought we were just adding 4x4 square to the 6x6 square.  After clarification, drew several rectangles of different dimensions with no real pattern.  Eventually found the correct solution.
10 8th Grade M Mental Math, Guess and Check Thought about the problem without writing anything down.  Used guess and check with perimeter formula to obtain correct solution on first guess.  Used perimeter formula but couldn't identify it when I asked what it was.