zOne of the most important skills that a math teacher needs is the ability to look at student work and

know what questions to ask

identify what [[#|students]] understand and misunderstand, and

what actions to take in response to the work.

Experience and practice help with this but as you [[#|begin]], this part takes some time and some help to become better at it. You are encouraged to submit student work to get some feedback. Please make sure that you

include the problem being solved.

never include a student's name - black it out or put a post-it on it before you scan it.

rewrite the solution if it is hard to read.

[[#|email]] your student work to nancynpowell@gmail.com - in the subject include the words: Student Work - Green Apples. Then we'll get the discussion rolling!

To [[#|get started]], visit MathMistakes.organd see what you can learn.

Here are sample problems - please comment on 5 problems that you did NOT submit!

College Algebra - Problem 1 (PJ)

Make your observations below for College Algebra - Problem 1: (include your initials preceding your comments)
(KR) - From this math mistake, it seems that the student doesn't completely understand the concept of factoring. Instead, they memorized the "x-chart" from Algebra which was helpful when working with quadratic functions. However, the student seems to miss out on two key checking methods: 1. How many solutions should this problem have? (4) How many did I receive? (2). 2. If I multiplied the two binomials together, would the end result match the orginial equation?

(ES) I [[#|agree]] with Karianne that this student did not take the [[#|opportunity]] to check his or her work. This observation makes me wonder whether the student does not completely understand the concept of factoring or the student was rushing through the assignment. Did this student make the same mistake in every problem? Students have been factoring since Algebra I and I wonder if this student glanced at this problem, quickly concluded he or she would have to factor, and moved on in hopes of completing the assignment quickly. In this scenario, I would like to have a warm-up or exit slip with a similar problem on it the next day to see if given [[#|class]] time, the student still made the same mistake.

(GL) I feel that this tends to be a very common mistake for students to make. The student in this case seems to understand basic factoring in know that 3 and -2 add to 1 and multiply to -6. However, they failed to realize that there is now an x^4 in the front as opposed to an x^2. I look at this student as having completed the problem half-correctly. I am going to assume that in this case that students have not worked much with imaginary numbers or radicals, so it may not be the best method explaining to them that there must be four solution, because for these types of problems, that is not always the case. For this student, it may be helpful to explain to them that the x^4 can act as (x^2)^2 where we look at 'x^2' as the variable as opposed to solely 'x.' If they can understand that, it may trigger the understanding that it happens to be the exact same method. If not a new way of teaching may be in order. Perhaps a step by step process for solving these?

College Algebra - Problem 2 (PJ)

Make your observations below [[#|for College]] Algebra - Problem 2: (include your initials preceding your comments)

Geometry - Problem 1 (ES)

Course: Geometry Topic: Points of Concurrency-circumcenter, incenter, orthocenter, and centroid. Description: This is a problem from the take-home portion of the chapter test. Students were allowed to use their books, notes, and [[#|the internet]] to help them [[#|complete]] this construction.

Make your observations for Algebra II - Problem 1: (include your initials preceding your comments)
(KR) - Right away I noticed that the lines the student drew were not perpendicular to the corresponding sides of the triangle, therefore the student doesn't understand the definition of a circumcenter. The student did show some understanding in how to find the circumcenter, which makes me curious as to whether they just tried to memorize the steps rather than the properties. The student attempted to use their compass to create the "x" and they knew to draw a line between the two "x"s. They also knew that the intersection of their two drawn lines was the circumcenter. However, I believe the mistake lies in their comfort using a compass. There are two things I could see the student doing wrong, I am not completely sure which they did. (1) Either they didn't set the compass to a little over half the length of the side they were working with. or (2) they knew something about being perpendicular was involved and instead of using the two ends of the segment to make an "x" they used the point Y and then the perpendicular point to it (the midpoint of XZ) to trace the other half of the "x" and that's why the left and right "x"s are off. I do not have a copy of the test, nor a compass, but I am curious as to whether that could have been one of their mistakes since the arches seem to imply they used the midpoint of XZ as their rotation point.

(GL): It appears to me that the underlying factor is that the student does not understand the definition of a circumcenter or a circumcircle. If the student understood the basic definitions, at the very least they could have drawn a free-had circle around the triangle, having it touch at each point to earn some credit. I agree with KR in the sense that the students may have tried to specifically memorize the steps as opposed to the properties. It seems to me that maybe the students chose to guesstimate the center of the triangle and use that to create the "perpendicular" lines as opposed to using the perpendicular lines to find the center. If I was teaching this student I would go right back to the basic definition as well as properties. This piece of student work just seems that the student had a general idea of what to draw on the paper, but did not particularly know why they were doing it.

Geometry - Problem 2 (KR)

Make your observations for Geoemtry - Problem 2: (include your initials preceding your comments)

(ES) In problem 27, the student does not understand that the dilation of the points should be located on the segments coming from the center of dilation. This student also does not understand how far to place the points from the center of dilation. However, the student does understand the orientation of the image and prime notation. From problem 26 above, it seems that the student has an understanding of dilations. Perhaps he or she struggled with problem 27 because it was not on a coordinate plane or because the center of dilation was not in the pre-image. I would show an interactive manipulative the next day in class to show the difference between when the center of dilation is in the figure and when it is not in the figure.

(PJ) I think that the student's attention to detail is the only thing holding them back. I suspect that they started with point A and tried to dilate it according to the instructions, and they did that fairly successfully, the only problem is that their line doesn't go through the center, which will make it harder to get the correct looking picture at the end. Which ever points they tried to dilate next were off because of the same reason. Point A' is correct that it should be on the line of dilation, but then B', C', and D' progressively get further off from where they should be. If they would have tried to draw a vertical line from the point A' in order to get D', they would have had a hard time getting a "square" looking dilation, so I think that is where they got off. The student tried to make it look like a square because the inaccuracy of their dilation lines forced them into that compromise. If you get off on the wrong foot for your very first step, it can be hard to recover later.

(GL) I agree with the above individuals. I definitely see the students general understanding of dilations. It is evident in their work on number 26. I do commend this individual on their effort though. Even though the student had difficulty solving number 27; it is good to see them actually making an attempt as opposed to leaving it blank. I do believe the student had trouble solving #27 right off the bat, and put themselves into a position where they could not recover from the error. I think as a teacher it would be good to work with the student, asking them what they already know, and use that to their advantage in reteaching them the material. As I've said in other comments, maybe this again would be another opportunity to go back to looking at definitions, as I feel maybe that would have helped them better understand what they were doing here.

Algebra II - Problem 1 (ES)

Course: Algebra II/Trigonometry

Topic: Solving Systems of Equations with Three Variables Description: Students were asked to complete a worksheet solving systems of equations with three variables. They had the option of using elimination, substitution, or Cramerâ€™s Rule.

Make your observations for Algebra II - Problem 1: (include your initials preceding your comments)
(KR) - The student clearly understood the steps in solving a system of equations with three variables. Unfortunately, a simple sign error effected the whole problem. This sign error could have been avoided as well if the student noticed that their system of equations with two variables was a perfect set up for eliminating the "c" variable. The student could have also checked their work by substituting their ordered triple back into the three original equations and observed whether the equations held true.

(PJ) I think that organization is a key aspect to solving any math problem. Right from the beginning it might be helpful for the student to put the different terms into columns, which might help them to realize that the variable c could be eliminated directly off the bat. Hopefully this would get them headed off in the right direction. Even the way that they tried to solve the problem, they could have solved for b in their second step instead of trying to multiply both equations in order to eliminate b. It is important to remember that there are always multiple ways to solve any problem, and that is why it is so important to be able to recognize what a student it thinking, but also being able to give them advice, guidance, and constructive feedback.

(GL) In my opinion, solving systems of three equations by hand is one of the most difficult things to do, since that chances of making an arithmetic mistake are so high... However let's observe what this students specifically did. Generally they have the correct idea to start. However, as I expressed it appears though they made an arithmetic error almost immediately. The student appears to think that you need to multiply a different scalar by each side of the equation (That or there was just an extra negative that was unintentionally added.) In these situations, I would just tell the student to relax, breathe, and take it one step at a time. Problems like these need to be done slowly. Even as a math professional, I still make mistakes when I do these. Also as aside note: though decimals are not necessarily incorrect answers for a system of three equations, when the answers end up in decimals (or what the students often refer to as "weird numbers") I usually tell them to go back and look at it again. If they still cannot find their error then I tell them to look at their procedure and do the best they can to earn at least some points for the problem.

Algebra II - Problem 2 (KR)

Make your observations for Algebra II - Problem 2: (include your initials preceding your comments)

(ES) This student understands the process of factoring. He or she even shows that they checked their factored expressions by simplifying his or her answer back to the original. However, he or she did not evaluate the question. First of all, this student was given an equation. Yet, this student abandoned the equation after the second step and worked with the given information as if it was a polynomial expression. Additionally, the student did not understand the directions of "solve the following by factoring." Based on the work, this student did not understand that the directions meant he or she was supposed to find x-values that make the statement of equality true. This work makes me wonder, does this student understand how to solve these equations? Does this student understand the connection between solving for x with degree one polynomial equations and solving for x with quadratic equations? The student does have a lot of his or her work scratched out. Does this mean he or she was unsure of the process or does this mean the student was hurrying through the problems?

(PJ) I think that the student has an understanding of the factoring process, but I also agree with the observation that was made above that the student might not know what "solving" the equations meant. They were able to factor out the algebraic expression, but when it came down to setting each factor equal to zero and solving for x, they simply didn't do it. This might be something that I would make a note of next to these problems, telling them that they know how to factor, but what I am looking for are the values of x that make the original equation true. As far as the scratched out work goes, I think that this is a result of the student using the guess and check method for factoring, and then if it doesn't work, trying to write it as an expanded equation which can be factored by grouping.

(GL) I feel that in this situation one of two things happened. Either the student did not read the instructions or they do not specifically know the difference between solving and factoring. As stated above, it is evident that the student understands factoring. I think in this case the student needs to understand what it generally means to solve something (assuming they did not read the instructions). It is important to explain to the student that solving is not only factoring, but also discovering what values for 'x' would give us the answers for 0 (in this case). It ultimately is unfortunate that the student lost 6 points for this mistake, because the student could have at least made an attempt for partial credit if the followed direction/knew what it means to solve.

Algebra II - Problem 3 (GL)

This is a matrix problem that was given on a quiz

Make your observations for Problem 3: (include your initials preceding your comments)

(ES) This student did recognize that 3 + 3 =6 and 4-x +4=8-x. Therefore, I think the student does understand the process of matrix addition. I think this student knows what he or she is supposed to do when adding matrices but I think this student is struggling with adding monomials. For example, this student wrote that 2x+2=4x. Thus, evaluating the problem, I think the student's work displays an understanding of matrix addition; however, I think it is vital to further assess the student's understanding of "like terms," variables, and adding monomials.
(KR) I agree with Emily. The student clearly understands how to add matrices, but struggles with the addition of adding "unlike terms". The student doesn't fully misunderstand the properties of adding "like-terms" but still lacks some reasoning in specific areas. When presented with the problem 4-x+4, the student noticed the different terms and came up with the correct solution of 8-x. However, when the student was not aided with the already included separation of the number and variable (4-x), the idea that the terms could not be added together was forgotten. Hence they came up with x+1=2x and 2x+2=4x. This makes me wonder whether the student was just rushing through the quiz and made "silly mistakes". I am curious as to what the solution would have been if the student was given more time and was put in a less stressful situation (not a quiz). Overall, I believe the student understands the problem as a whole but overlooked a simple concept that they already had some understanding of.

(PJ) The student obviously understands the overall process of adding matrices, their mistake comes from adding unlike terms. I think that this problem is quite ironic because the student knew how to add and subtract the correct terms when they took (4-x)+4 and they got 8-x, which is correct. So I am a little confused as to why they missed the two above portions, I would have to think that it is just a silly mistake. I know that all of these things have been stated in the observations above, but I just thought that I would give my two cents.

Pre-Calculus Problem 1 (GL)

Make your observations for Pre-Calculus Problem 1 (GL): (include your initials preceding your comments)

(ES) The first thing I noticed about this problem is the fact that the student thinks that the coordinates must be subtracted as opposed to added in the midpoint formula. Therefore, does the student really understand the midpoint formula or did he or she just attempt to memorize it? Additionally, in the distance formula, the first x-coordinate is subtracted from the second x-coordinate and the first y-coordinate is subtracted from the second y-coordinate. Therefore, is this student confusing the midpoint formula with the distance formula? I also noticed that the student did not complete the second half of the problem. Perhaps this student did not complete the problem because he or she had forgotten the distance formula. If this student had been given the formulas, I wonder if they would have been able to correctly input and solve the problem. Could this student have derived the distance formula or midpoint formula using prior understanding if he or she had had the time?
(KR) The first thing that I noticed was that the student did not read the directions carefully. They could have quickly read that they were to find the midpoint and then stopped reading after that. So the first mistake was that the student does not read directions completely. I feel like this is the case rather than just not knowing the distance formula because most students would at least have attempted the problem by making up a formula or writing a number down for partial credit. I do agree with Emily when it comes to the misunderstanding of the midpoint formula. The student seems to have just tried to remember the formula rather than understanding the concept that it is the average of both the x and y distances. Students have learned how to calculate the average of numbers since grammar school, so if the student had made the connection between that and the midpoint formula I believe they would have received the correct answers. The good thing is, is that the student has a slight understanding of the midpoint formula an understands how to substitute in the values of x1,x2 and y1,y2 so this problem could easily be fixed with a recap of the formula and its meaning!

(PJ) I think that for the first problem, the student tried to find the midpoint and made a silly mistake in the process, but then they forgot to do the second part of the question, which was the distance between the two points. For the second problem, I am pretty sure they were getting the distance formula and midpoint formula mixed up, they tried to take the difference between the two x-values and the two y-values, which is what you need to do for the distance formula, but then they took the average, which correlates to the midpoint formula. Their answer is also listed as a coordinate pair, which makes me suspect that they didn't have a firm grasp on either of the concepts. One thing they could do is to visually approximate their answer, graphing it just to see if those numbers make sense because I would have to be suspicious of my answer if I found that the x-value of the midpoint is 2.5, which is not even between 3 and 8, at least not the last time I checked.

- know what questions to ask
- identify what
- what actions to take in response to the work.

Experience and practice help with this but as you[[#|students]]understand and misunderstand, and[[#|begin]], this part takes some time and some help to become better at it. You are encouraged to submit student work to get some feedback. Please make sure that you[[#|email]]your student work to nancynpowell@gmail.com - in the subject include the words: Student Work - Green Apples. Then we'll get the discussion rolling!To

[[#|get started]], visit MathMistakes.organd see what you can learn.## Here are sample problems - please comment on 5 problems that you did NOT submit!

## College Algebra - Problem 1 (PJ)

Make your observations below for College Algebra - Problem 1: (include your initials preceding your comments)

(KR) - From this math mistake, it seems that the student doesn't completely understand the concept of factoring. Instead, they memorized the "x-chart" from Algebra which was helpful when working with quadratic functions. However, the student seems to miss out on two key checking methods: 1. How many solutions should this problem have? (4) How many did I receive? (2). 2. If I multiplied the two binomials together, would the end result match the orginial equation?

(ES) I

[[#|agree]]with Karianne that this student did not take the[[#|opportunity]]to check his or her work. This observation makes me wonder whether the student does not completely understand the concept of factoring or the student was rushing through the assignment. Did this student make the same mistake in every problem? Students have been factoring since Algebra I and I wonder if this student glanced at this problem, quickly concluded he or she would have to factor, and moved on in hopes of completing the assignment quickly. In this scenario, I would like to have a warm-up or exit slip with a similar problem on it the next day to see if given[[#|class]]time, the student still made the same mistake.(GL) I feel that this tends to be a very common mistake for students to make. The student in this case seems to understand basic factoring in know that 3 and -2 add to 1 and multiply to -6. However, they failed to realize that there is now an x^4 in the front as opposed to an x^2. I look at this student as having completed the problem half-correctly. I am going to assume that in this case that students have not worked much with imaginary numbers or radicals, so it may not be the best method explaining to them that there must be four solution, because for these types of problems, that is not always the case. For this student, it may be helpful to explain to them that the x^4 can act as (x^2)^2 where we look at 'x^2' as the variable as opposed to solely 'x.' If they can understand that, it may trigger the understanding that it happens to be the exact same method. If not a new way of teaching may be in order. Perhaps a step by step process for solving these?

## College Algebra - Problem 2 (PJ)

Make your observations below

[[#|for College]]Algebra - Problem 2: (include your initials preceding your comments)## Geometry - Problem 1 (ES)

Course: GeometryTopic: Points of Concurrency-circumcenter, incenter, orthocenter, and centroid.Description: This is a problem from the take-home portion of the chapter test. Students were allowed to use their books, notes, and[[#|the internet]]to help them[[#|complete]]this construction.Make your observations for Algebra II - Problem 1: (include your initials preceding your comments)

(KR) - Right away I noticed that the lines the student drew were not perpendicular to the corresponding sides of the triangle, therefore the student doesn't understand the definition of a circumcenter. The student did show some understanding in how to find the circumcenter, which makes me curious as to whether they just tried to memorize the steps rather than the properties. The student attempted to use their compass to create the "x" and they knew to draw a line between the two "x"s. They also knew that the intersection of their two drawn lines was the circumcenter. However, I believe the mistake lies in their comfort using a compass. There are two things I could see the student doing wrong, I am not completely sure which they did. (1) Either they didn't set the compass to a little over half the length of the side they were working with. or (2) they knew something about being perpendicular was involved and instead of using the two ends of the segment to make an "x" they used the point Y and then the perpendicular point to it (the midpoint of XZ) to trace the other half of the "x" and that's why the left and right "x"s are off. I do not have a copy of the test, nor a compass, but I am curious as to whether that could have been one of their mistakes since the arches seem to imply they used the midpoint of XZ as their rotation point.

(GL): It appears to me that the underlying factor is that the student does not understand the definition of a circumcenter or a circumcircle. If the student understood the basic definitions, at the very least they could have drawn a free-had circle around the triangle, having it touch at each point to earn some credit. I agree with KR in the sense that the students may have tried to specifically memorize the steps as opposed to the properties. It seems to me that maybe the students chose to guesstimate the center of the triangle and use that to create the "perpendicular" lines as opposed to using the perpendicular lines to find the center. If I was teaching this student I would go right back to the basic definition as well as properties. This piece of student work just seems that the student had a general idea of what to draw on the paper, but did not particularly know why they were doing it.

Geometry - Problem 2 (KR)

Make your observations for Geoemtry - Problem 2: (include your initials preceding your comments)

(ES) In problem 27, the student does not understand that the dilation of the points should be located on the segments coming from the center of dilation. This student also does not understand how far to place the points from the center of dilation. However, the student does understand the orientation of the image and prime notation. From problem 26 above, it seems that the student has an understanding of dilations. Perhaps he or she struggled with problem 27 because it was not on a coordinate plane or because the center of dilation was not in the pre-image. I would show an interactive manipulative the next day in class to show the difference between when the center of dilation is in the figure and when it is not in the figure.

(PJ) I think that the student's attention to detail is the only thing holding them back. I suspect that they started with point A and tried to dilate it according to the instructions, and they did that fairly successfully, the only problem is that their line doesn't go through the center, which will make it harder to get the correct looking picture at the end. Which ever points they tried to dilate next were off because of the same reason. Point A' is correct that it should be on the line of dilation, but then B', C', and D' progressively get further off from where they should be. If they would have tried to draw a vertical line from the point A' in order to get D', they would have had a hard time getting a "square" looking dilation, so I think that is where they got off. The student tried to make it look like a square because the inaccuracy of their dilation lines forced them into that compromise. If you get off on the wrong foot for your very first step, it can be hard to recover later.

(GL) I agree with the above individuals. I definitely see the students general understanding of dilations. It is evident in their work on number 26. I do commend this individual on their effort though. Even though the student had difficulty solving number 27; it is good to see them actually making an attempt as opposed to leaving it blank. I do believe the student had trouble solving #27 right off the bat, and put themselves into a position where they could not recover from the error. I think as a teacher it would be good to work with the student, asking them what they already know, and use that to their advantage in reteaching them the material. As I've said in other comments, maybe this again would be another opportunity to go back to looking at definitions, as I feel maybe that would have helped them better understand what they were doing here.

## Algebra II - Problem 1 (ES)

Course: Algebra II/TrigonometryTopic: Solving Systems of Equations with Three VariablesDescription: Students were asked to complete a worksheet solving systems of equations with three variables. They had the option of using elimination, substitution, or Cramerâ€™s Rule.Make your observations for Algebra II - Problem 1: (include your initials preceding your comments)

(KR) - The student clearly understood the steps in solving a system of equations with three variables. Unfortunately, a simple sign error effected the whole problem. This sign error could have been avoided as well if the student noticed that their system of equations with two variables was a perfect set up for eliminating the "c" variable. The student could have also checked their work by substituting their ordered triple back into the three original equations and observed whether the equations held true.

(PJ) I think that organization is a key aspect to solving any math problem. Right from the beginning it might be helpful for the student to put the different terms into columns, which might help them to realize that the variable c could be eliminated directly off the bat. Hopefully this would get them headed off in the right direction. Even the way that they tried to solve the problem, they could have solved for b in their second step instead of trying to multiply both equations in order to eliminate b. It is important to remember that there are always multiple ways to solve any problem, and that is why it is so important to be able to recognize what a student it thinking, but also being able to give them advice, guidance, and constructive feedback.

(GL) In my opinion, solving systems of three equations by hand is one of the most difficult things to do, since that chances of making an arithmetic mistake are so high... However let's observe what this students specifically did. Generally they have the correct idea to start. However, as I expressed it appears though they made an arithmetic error almost immediately. The student appears to think that you need to multiply a different scalar by each side of the equation (That or there was just an extra negative that was unintentionally added.) In these situations, I would just tell the student to relax, breathe, and take it one step at a time. Problems like these need to be done slowly. Even as a math professional, I still make mistakes when I do these. Also as aside note: though decimals are not necessarily incorrect answers for a system of three equations, when the answers end up in decimals (or what the students often refer to as "weird numbers") I usually tell them to go back and look at it again. If they still cannot find their error then I tell them to look at their procedure and do the best they can to earn at least some points for the problem.

## Algebra II - Problem 2 (KR)

Make your observations for Algebra II - Problem 2: (include your initials preceding your comments)

(ES) This student understands the process of factoring. He or she even shows that they checked their factored expressions by simplifying his or her answer back to the original. However, he or she did not evaluate the question. First of all, this student was given an equation. Yet, this student abandoned the equation after the second step and worked with the given information as if it was a polynomial expression. Additionally, the student did not understand the directions of "solve the following by factoring." Based on the work, this student did not understand that the directions meant he or she was supposed to find x-values that make the statement of equality true. This work makes me wonder, does this student understand how to solve these equations? Does this student understand the connection between solving for x with degree one polynomial equations and solving for x with quadratic equations? The student does have a lot of his or her work scratched out. Does this mean he or she was unsure of the process or does this mean the student was hurrying through the problems?

(PJ) I think that the student has an understanding of the factoring process, but I also agree with the observation that was made above that the student might not know what "solving" the equations meant. They were able to factor out the algebraic expression, but when it came down to setting each factor equal to zero and solving for x, they simply didn't do it. This might be something that I would make a note of next to these problems, telling them that they know how to factor, but what I am looking for are the values of x that make the original equation true. As far as the scratched out work goes, I think that this is a result of the student using the guess and check method for factoring, and then if it doesn't work, trying to write it as an expanded equation which can be factored by grouping.

(GL) I feel that in this situation one of two things happened. Either the student did not read the instructions or they do not specifically know the difference between solving and factoring. As stated above, it is evident that the student understands factoring. I think in this case the student needs to understand what it generally means to

solvesomething (assuming they did not read the instructions). It is important to explain to the student that solving is not only factoring, but also discovering what values for 'x' would give us the answers for 0 (in this case). It ultimately is unfortunate that the student lost 6 points for this mistake, because the student could have at least made an attempt for partial credit if the followed direction/knew what it means to solve.## Algebra II - Problem 3 (GL)

This is a matrix problem that was given on a quiz

Make your observations for Problem 3: (include your initials preceding your comments)

(ES) This student did recognize that 3 + 3 =6 and 4-x +4=8-x. Therefore, I think the student does understand the process of matrix addition. I think this student knows what he or she is supposed to do when adding matrices but I think this student is struggling with adding monomials. For example, this student wrote that 2x+2=4x. Thus, evaluating the problem, I think the student's work displays an understanding of matrix addition; however, I think it is vital to further assess the student's understanding of "like terms," variables, and adding monomials.

(KR) I agree with Emily. The student clearly understands how to add matrices, but struggles with the addition of adding "unlike terms". The student doesn't fully misunderstand the properties of adding "like-terms" but still lacks some reasoning in specific areas. When presented with the problem 4-x+4, the student noticed the different terms and came up with the correct solution of 8-x. However, when the student was not aided with the already included separation of the number and variable (4-x), the idea that the terms could not be added together was forgotten. Hence they came up with x+1=2x and 2x+2=4x. This makes me wonder whether the student was just rushing through the quiz and made "silly mistakes". I am curious as to what the solution would have been if the student was given more time and was put in a less stressful situation (not a quiz). Overall, I believe the student understands the problem as a whole but overlooked a simple concept that they already had some understanding of.

(PJ) The student obviously understands the overall process of adding matrices, their mistake comes from adding unlike terms. I think that this problem is quite ironic because the student knew how to add and subtract the correct terms when they took (4-x)+4 and they got 8-x, which is correct. So I am a little confused as to why they missed the two above portions, I would have to think that it is just a silly mistake. I know that all of these things have been stated in the observations above, but I just thought that I would give my two cents.

## Pre-Calculus Problem 1 (GL)

Make your observations for Pre-Calculus Problem 1 (GL): (include your initials preceding your comments)(ES) The first thing I noticed about this problem is the fact that the student thinks that the coordinates must be subtracted as opposed to added in the midpoint formula. Therefore, does the student really understand the midpoint formula or did he or she just attempt to memorize it? Additionally, in the distance formula, the first x-coordinate is subtracted from the second x-coordinate and the first y-coordinate is subtracted from the second y-coordinate. Therefore, is this student confusing the midpoint formula with the distance formula? I also noticed that the student did not complete the second half of the problem. Perhaps this student did not complete the problem because he or she had forgotten the distance formula. If this student had been given the formulas, I wonder if they would have been able to correctly input and solve the problem. Could this student have derived the distance formula or midpoint formula using prior understanding if he or she had had the time?

(KR) The first thing that I noticed was that the student did not read the directions carefully. They could have quickly read that they were to find the midpoint and then stopped reading after that. So the first mistake was that the student does not read directions completely. I feel like this is the case rather than just not knowing the distance formula because most students would at least have attempted the problem by making up a formula or writing a number down for partial credit. I do agree with Emily when it comes to the misunderstanding of the midpoint formula. The student seems to have just tried to remember the formula rather than understanding the concept that it is the average of both the x and y distances. Students have learned how to calculate the average of numbers since grammar school, so if the student had made the connection between that and the midpoint formula I believe they would have received the correct answers. The good thing is, is that the student has a slight understanding of the midpoint formula an understands how to substitute in the values of x1,x2 and y1,y2 so this problem could easily be fixed with a recap of the formula and its meaning!

(PJ) I think that for the first problem, the student tried to find the midpoint and made a silly mistake in the process, but then they forgot to do the second part of the question, which was the distance between the two points. For the second problem, I am pretty sure they were getting the distance formula and midpoint formula mixed up, they tried to take the difference between the two x-values and the two y-values, which is what you need to do for the distance formula, but then they took the average, which correlates to the midpoint formula. Their answer is also listed as a coordinate pair, which makes me suspect that they didn't have a firm grasp on either of the concepts. One thing they could do is to visually approximate their answer, graphing it just to see if those numbers make sense because I would have to be suspicious of my answer if I found that the x-value of the midpoint is 2.5, which is not even between 3 and 8, at least not the last time I checked.