1. Red buckets can hold 3 apples and blue buckets can hold 5 apples. If Joy has 4 red buckets of apples and 5 blue buckets of apples, how many apples does she have altogether?

2. A string is 7 yards long. Jeff needs 2 feet of string and Lori needs 15 feet of string. If they both cut their string, how much string will be left over?


3. A toy box can hold 9 toys. A toy carton can hold 6 toys. Jen brings 2 toy boxes and 5 toy cartons of toys to donate to an orphanage. How many toys did she bring?

4. A ribbon is 5 yards long. Carol uses 5 feet of ribbon for her craft. April uses 6 feet of ribbon for her craft. How much ribbon is left over?


5. A van can hold 4 adults and 2 children. A car can hold 2 adults and 2 children. If 5 vans and 2 cars go on a camping trip, how many people are able to go on the trip?

6. Wendy has 5 packages of macaroni and cheese to donate to a day care center. Each package contains 10 boxes of macaroni and cheese. If she hands out 45 boxes to the center, how are left over?


7. Kimi has an album with pokemon cards in it. Each page holds 36 cards. She has 6 pages total. If she chooses to give 10 cards to her brother, how many cards does Kimi have now?

8. Tommy has candy bags prepared for his party. There are 8 bags of candy. Each bag contains 4 pieces of candy. If little brother Harry sneaks two bags for himself. How many pieces of candy does Tommy need to buy to replace what Harry stole?


9. Keelie has 42 pieces of paper for her friends for an art project at her party. If 7 friends come over, how many pieces of paper will each person get?

10. There are 3 red baskets, 5 blue baskets, and 2 orange baskets. Each red basket has 2 gifts in it. Each blue basket has 3 gifts in it. Orange baskets can hold 10 gifts. How many gifts are there in all?

On the other hand, I also see a huge variation from one school to another.  For example, School A's Honors Geometry program is so challenging that even I can get stumped on some of their questions from time to time and I have a Ph.D. in Mathematics Education, Masters in Mathematics, etc.  The level of proofs required in School A are truly much harder than I feel is appropriate, especially considering it isn't in line with other schools and way off from the virtual school.  I tutored a student from School A who is extremely bright, knew so much about Geometry that most high school math teachers (outside of School A) who might sit down and work with this student would be very impressed with this student's knowledge of Geometry but since he attended School A, his grade for the year was a C!  If he had been in School B, he would have gotten an A, if he had taken it online, he could have slept through the course!  School B is right now the road from School A but the same math classes - and I am not just talking about Honors Geometry but all other high school math classes  - are so much easier at School B than School A.  School B requires a much more reasonable amount of homework as well.  School A requires way too much from kids and somehow thinks that if they assign 60 problems of the same type that will make the kids smarter.  My son is 11 now and smart enough to take Honors Geometry but if he has to take it at School A, I won't let him.  In fact, I am not sure I will sign him up for any honors math classes at School A because their math program is so out of line with what is reasonable - and if you happen to get a less than stellar teacher in the mix, then just forget it!

These grades students make determine many things for students in high school - they make up their GPA - this makes them competitive to get into colleges.  How does that C in Honors Geometry look to a school like Stanford?  They perceive the student as a poor student, when in fact, this student had he been down the road in School B, would have straight A's in Honors and AP math classes!  What a difference in perception and yet it is the same student, the same knowledge.  All School A did was make the student get frustrated and feel like he can't be successful in math and now this student will choose not to continue on with Honors and AP math classes that he is capable of.   I have to tell the student that it ISN'T him - I hate to put blame on outside forces with teens because it is important for teens to learn to take responsibility for their actions, however - when I work with a very bright student and watch him achieve a C (and it wasn't for not doing assignments, etc.) - there is nothing else I can do but try and help salvage the student's math self-esteem that School A has taken away from him.

Another example; a parent calls me - her son is failing - well almost, he barely has a D, in Algebra 2.  He is generally a B student in math.  She begins to relay the story.  The teacher, who gives math credit for whether a student uses the bathroom during class, is telling her that her son has only completed 47% of his homework.  Well, one would argue, if a student isn't completing their homework, that is a reason for a poor grade.  However, despite the fact that she said those exact words, the truth is that he did 100% of his homework but she graded his homework and he only got 47% of his homework correct so he has a 47 homework GRADE, not that he only did 47% of his homework.  However, isn't homework supposed to be for learning, not an assessment?  Why are we teaching a new topic, assigning homework, then grading it the NEXT day, and weighing it so heavily that it takes a student that has a B average on tests and lowers his grade to a D (almost an F) in the class?  Shouldn't you be able to come to class the next day and say, "Ms. Teacher, I didn't understand homework problems # and #, please go over these."  This is how it always worked for me.  This is how I always taught.  This teacher scores the homework and weighs it so much it fails him even though his understanding on true assessments is a B.  Now when colleges see his transcript, yet again -they think this child is a D student when his knowledge of Algebra 2 clearly indicates a B level of understanding?

What are these GRADES supposed to measure?  Whether we use the bathroom?  If we could do homework the first night it was assigned?  If we can do super hard proofs when other students can get A's in the same class for basic multiple choice questions?  How is this an accurate measure of anything?  And yet, it has an impact on what college a child gets into, if they get scholarships for college?  I remember one college professor I had, he got it right.  He gave us tests, we took them and got grades (this was in math).  Our final exam was cummulative - it tested everything for the whole class.  If we knew everything on the final, then we had proven we had mastered everything we were supposed to learn in class.  So, he said to us - IF you take the final and your final exam grade is higher than your grade would be if I factor it in at 20% (or whatever the assigned weight was), I will just give you the grade you scored on the final.  So, if our grade going into the final was a D but we got an A on the final, we got an A in the class.  Why?  It made sense ... What is the purpose of a grade?  To measure your knowledge of the class content?  He didn't care WHEN you managed to "get it" - if it took you longer but you got there by the end and could demonstrate it on the final - you proved you mastered the material in the class so your grade should REFLECT your ACTUAL knowledge at the end of the course.  It was BRILLIANT!  Dr. Kenton, you are a brilliant man and teacher!

Speaking of grades - tell me if this makes sense - Wake County schools offer higher quality points towards the weighted GPA based on Honors and AP classes.  If you take a regular class and get an A, you get 4 QP, if you take an Honors Class, you get 5 QP, but if you take an AP class, you get 6 QP.  So, why do you get 6 QP for an AP class?  Well, it makes sense because AP classes are supposed to be college level classes offered in the high school.  So, college level work should be awarded more QP than an Honors level high school class, right?  That makes sense.  However, if the student actually goes TO a college and takes a college course AT a college, the county's policy is to award only 5 QP for an A.  So they equate an ACTUAL college class the same as an Honors level high school class - giving more weight to an AP class than an actual college class taken in college.  So I could take AP Calculus BC, get an A and get 6 QP but if I take Calculus III as a dual enrolled student the following semester while still in high school and get an A, the school will only give me 5 QP for it.  So it would LOWER my GPA and make me LESS competitive for colleges looking at my GPA and class rank.  Again, pointing out these grades are meaningless.

My final comparison is the grading scale used.  Most schools use a 10 point scale.  90-100 A, 80-89 B, and so on.  So if you are in states with this scale, and you get an 84, you would have a nice solid B.  However, Wake County decided that they wanted to make things more challenging for their students and now use a 7 point scale, so that same 84% would equate to C in Wake County schools.  Do colleges take this grading scale into consideration when looking at applicants?  These inconsistencies make the meaning behind grades useless.  When I taught college and graded, I preferred to think of grades this way - to me, an A meant Excellent Understanding, a B was Good Understanding, a C was Fair Understanding, a D was Poor Understanding, and an F was Little to No understanding.  After I computed a numerical grade for a student, I was looked at the student and said if I didn't have any true grades and just looked at their "understanding" and had to attach a word to their understanding - how would I define it - excellent, good, fair, poor, or little to no - I wanted to make sure their numerical score matched their TRUE understanding - luckily, it did because I was very careful with each individual assessment but this was especially helpful when students were borderline and I had to choose between two letter grades.

I chose to homeschool my son for one year of high school.  It was so liberating to not worry about grades and just have him learn for the sake of learning!  Of course, we had to "make up grades" for his transcript to send off to college.  I tried to think about what he would have gotten if had taken the class in a public school.  He always got B's in English in traditional classes, so I gave him a B in English.  Things he was passionate about and worked hard on because he just really wanted to learn and master (which he did) - those were clearly A's.  None of that really mattered to me though, he learned what he needed to and worked really hard at what was important to him.

In closing, I think back to my undergraduate years when I was minoring in Philosophy and one thing that interested me was the concept of a grade-less school.  In the book, Zen and the Art of Motorcycle Maintenance, the author wrote about a professor he had who chose not to grade his college class and instead let the students choose their grades.  It was a great read and I would encourage everyone to check it out.    I would welcome any comments on this topics.

So - what is the correct approach? Should partial credit be awarded? If so, how much should be awarded? When should it be awarded? Do teachers have the "right" to choose the 100% right / 100% wrong approach? Is it fair for some teachers to grade students this way, hence awarding a B or C to a student that might actually have a good grasp of the content when another teacher who gives partial credit would give that same student a grade 10 points higher - and hence the "unlucky" students who get the non-partial credit teachers look like they understand less (when in fact they don't) than another student who happens to have a teacher who awards partial credit?

Are all math teachers flawless? If they were not to use a calculator at all, would they never transpose a number or accidentally make a mistake? Of course not, all teachers (myself included) have been corrected by students when we occasionally make a mistake during our lessons. Yet, we are willing to subtract 8-10 points off a test grade if the student does the same?

What if the problem is testing a very difficult concept and the student gets all the concept correct, showing they clearly understood everything taught to them but they accidentally transpose a number or maybe made a silly arithmetic mistake or even lost a negative sign in all the written work required as they were focusing on the difficult concept. Are we then to reward them with no credit when in fact they clearly learned what we were trying to teach them?

Here is a quote from Brian Boley, "Avoid the "partial credit" trap when teaching middle school and high school students. Someday you may drive over a bridge which one of your students designed. Do you expect him to have calculated the loads correctly or should he get "partial credit" for getting a close answer? And all because you taught him that using the right equation was worth 90% of the problem -- and adding 2 + 5 = 8 was only 10% off."  His argument is sound, right?  Who would want to be on that bridge?  Yet, is that what partial credit is promoting?  Would we give credit for 2+5 = 8?  Of course not, that is wrong.  The entire concept that is being taught is wrong and hence no partial credit should be award in those cases.  When the student misses the concept, they do in fact lose all credit for the problem.  If they do a math problem in Algebra and have no understanding - just a few random ideas - that is not a time to offer partial credit.  We are talking about giving credit to the student who made a careless error but who clearly understood what they were doing.  Remember, we are not building a bridge, if we were, we wouldn't have a new student learning something for the first time doing the math for it - that is not how the real world works - school is a time for learning.  A "close answer" does not equal credit, what equals credit is a demonstration of the concept being tested or a partial demonstration of that concept that shows you got 1/2 the concept and you missed 1/2 the concept so we will award you credit for the 1/2 of the concept you got correct and take away credit for the 1/2 of the part of the concept that you still need to learn.  It is just a way to break up the scoring of a problem with multiple steps into multiple scoring which is a fair and reasonable thing to do.  So, when you hear arguments like Brian's, don't immediately think - yeah, I don't want to be on that bridge - don't worry, brand new Algebra students or middle school students don't build bridges.

What is our goal in teaching mathematics? Don't we want people to stop saying, "I am not any good in math." Well, we will just continue to perpetuate this problem by not rewarding students with partial credit especially when it is obvious that they grasped the concept being taught and the mistake was elsewhere! Why don't math teachers care about our students' perceptions of mathematics? How can you choose to be a math teacher when you don't care enough to make students want to feel good about math. Now, don't get me wrong, I am not a proponent of teachers who give grades for undeserved work! I met a woman who wanted her students to feel positive about math so she gave everyone a B or higher - no matter what they did. That won't help them either. They must earn their grades but if you give them positive feedback, encouragement (which includes acknowledging their efforts and what they have learned and accomplished with partial credit), they will respond with a better self-image about mathematics which in return will improve their efforts, attitude, study habits, and hence their grades.

I also don't agree that life only allows for Right and Wrong answers. If that were the case, we would all be in a lot of trouble. We are human, we make mistakes, it is a great thing for kids to learn that we acknowledge that we all make mistakes and don't expect perfection and that the world does not expect perfection. Even working a job, people will make mistakes, if you do, you figure out where your mistake is, you communicate with others, you realized the solution is not working so you rework the problem and find your own mistake, etc. Very few people do everything perfect the first time in the real world. Why would we penalize our kids psychological well-being as well as their future (see grade issue above based on the 100% wrong teachers vs. the partial credit teachers) because they made a small arithmetic mistake even though they correctly integrated this very long function?!

I also think we owe it to them to look at their work and try and find their mistakes or if teachers don't have the time, get creative. Mark it wrong and let the student come back with a test correction where they show the teacher where their mistake was and offer them partial credit back at that point based on WHY they got the problem wrong. It makes the student go back and find their own mistakes and yet still gives them partial credit.

Award partial credit appropriately. If the mistake was just arithmetic and the concept was Algebra - they lose a little. If they transpose numbers but did the whole problem right with the transposed numbers - they lose almost nothing! If they make a partial Algebra mistake - they lose much more credit, depending on how much of concept they were able to get. For example, if solving an Algebra word problem, if they got the equation right but then had no idea how to solve the equation - they would get half (or more than half as finding the equation is really the hard part) credit - if they got the equation and just made a "mistake" solving the equation but seemed to know the general process, they lose less. So partial credit is not awarded equally. If a student is solving an order of operations problem and they do the order of operations correct but state that 4^2 = 8 instead of 16, they should be awarded a large part of the credit since the problem was testing order of operations but lose some for not knowing how to evaluate an exponent. If they make that same mistake again in future problems but again solve the problem correctly, the amount lost should be minimal since obviously they will continue to make that same mistake but you already took off for it and the teacher should be looking for the main idea of the question, not marking every problem on the test wrong just because the student missed this one concept of how to evaluate an exponent even though they can solve everything else about the other problems correctly.

So, to answer the question - Should partial credit be awarded on a math test? The answer is a resounding YES. I hope this article points to the many reasons why it is important to award partial credit to students on their tests.

Author: Lynne M. Gregorio, Ph.D. in Mathematics Education
Owner: Triangle Education Center and Educator for over 23 years.