I had three students try to solve the following question:

In the star shown here the sum of the four numbers in any "line" is the same for each of the five "lines". The five missing numbers are 9,10,11,12,13. Which number is represented by R?

Student #1

I had the student read the question and tell me what he/she thinks the question is asking for them to do. The student answered, "each line has to add up to the same number". The first line that the student looked at was 4-N-R-3 because it was easy for him/her to read from left to right. The student tried all combination of the number 9 with all the other combinations. Then the student looked at line 1-N-H-7. He/she knew that N was a common number between the two lines so they had to be the same. The student had that number as 9 and then did all the combinations. The student realized that those combinations did not work so they worked with different combinations of all the numbers The student figured out two combinations of the two lines that equal 30. The student then told me that he/she was going to find the letters that were in common in each line and have the line equal to 30. When the student added up the letters, he/she wrote them vertical each time and kept rewriting the numbers he/she already knew-1,3,4,5,7. The students told me that is was their way of thinking and adding the numbers up this way was easier for him/her to understand.

Student #2

I had the student read the question and tell me what he/she thought the question was asking for them to solve. The student though the question asked to "find the numbers that represent the letters". This students looked at the point of the star and wrote the difference between the points of the star. He/she thought the difference might help find the missing numbers in the middle of the star. then the student added the points of the star together and found those numbers thinking that might help with finding the missing numbers. The student realized that did not help him/her with the problem so he/she moved on to multiplying the point number by itself. The student found that 3 times 3 equals nine which is one of the number that could be in the middle. He/she thought that nine could be letters N and R because it lies on the line with three in it. The student then looked at the letter I and show that it was in the middle of 3 and 5, 3-I-5. Here the students stopped and i asked if they were confused. He/she said "yes", so we reread the question again. This time the student found that he/she was missing the part about the line in the question. So I asked what were the lines of the star. He/she answered 4-H-C-5 and 4-N-R-3. Then the students added the number 4,5and 3 together thinking this might help. He/she was getting frustrated again, so we reread the question again and drew a new picture of the star. We broke down the star into the different lines and then the student realized that the lines all had to equal the same sum. Here the student was confused about how four different numbers could add up to the same number. I showed him/her a simple example of 1+2+3+4=10 and 5+2+2+1=10. Each sequence is different but they add up to the same number. I then asked the student what would he/she do now that h/she has this new information. The student told me that he/she would find the missing letters in the lines that add up to the same sum.

Student #3

I asked the student to read the question and tell me what the question is asking the student to solve. As the student was reading the question he/she circled 9,10,11,12,13 and the letters because "they go together, the numbers wills will replace the letters". The student then answered my question that the problem is having him/her solve for the missing letter with the numbers given in the lines of the star. The student recognized that there was five lines in the star and each one had four numbers. The student added the two numbers that he/she knew and wrote them under the star. Then he/she put these numbers in order from smallest to largest because he/she was able to see that the numbers were only one away from each other. The student then added one random number from the list in the problem-9,10,11,12,13- to the numbers from the points being added together. He/she realized that only three of the lines add to the same sum, but then found the mistake that each line has four numbers not just three. The student then looked at what lines intersect each other at what letter. He/she wrote down the added point numbers and then put two plus sign to remind him/herself that there are two numbers/letters that have to be added together. The student wrote the letters that corresponded to the sum of the points to the left. The student then picked two random numbers to put in for I and R and found right away that those numbers would not work because both large number (point of star sum and given number) are paired together. The student then pick random number from the given list and found that these numbers wouldn't work because the intersection line needs an eight to make the lines equal and eight is not one of the numbers that was given. Then he/she picked another two numbers and found four out of the five lines equal each other, but he/she made a mistake after he/she checked the work. He/she used a given number twice. The student was so close, but he/he felt proud that she almost had the answer.

Good recount of your interactions with the students. Sounds like the problem was a reasonable one for them, and you got to see their thinking. What I'd love to see now is your description of their understanding. What can they do or do they understand, and what are they working on? (Consolidation) You could also self-evaluate: what did you do that was good for getting at understanding, where would you have wanted to ask something else?

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