December 14, 2018

Apex-Math Videos

I hope to keep my videos:

  1. personal
  2. short
  3. organized
  4. study-ready

Personal:  I want to reach of you as a person, like it is a conversation between just you and me.  I am a veteran teacher and tutor.  I am going to try videos from both angles, on a large white board in front of a class and on a small white board, like we were tutoring one on one.

Short:  Each topic that I want to present, doesn’t need to be long and involved.  I want to get to the point quickly and do many videos.  Each one will allow the topics to be broken down into smaller pieces and some videos will need to be grouped as a cluster to grasp the entire “unit.”  For example, I will teach all the pieces that go along with graphing a quadratic and have one or more videos where I pull all of that together, those videos will assume mastery of the previous videos, but this will allow me to keep each video reasonably short.  No one wants to sit for an hour at a math video!  Let’s learn what to do and move on.

Organized:  Staying organized in mathematics is very important.  I encourage each of you to take notes and stay organized with me!

Study-ready:  Teachers don’t teach students how to study for math tests.  I want to provide individual videos on this topic as it is so important but also provide study notes for the student in each video.  Write down the steps, you will need them.  Very few teachers break down the information, they do an example but all those steps run together, we will always break it down and make notes about what to do.

Topics:  I have so many ideas of topics to do but eventually will start taking requests so that I can help each of you as the school year goes on.  You can always suggest a topic in the comments section of You-tube or emailing directly.  Be sure to like my videos when you leave comments and requests.  There is so much math to get to… Kindergarten through Algebra 2 is my goal but my first videos will start with middle and high school mathematics, especially some of the harder topics that don’t need to be so hard!

 

Changing Math – One Video at a time

Everyone is born with gifts.  Some people are talented at art, some are great “people” people, some are helpers who give to the needy and feed at the soup kitchen.  We all have gifts and we should use our gifts to help make the world a better place.  My gift, from the time I was a little girl, was to teach.  I can teach anything.  I can learn something one minute and teach it the next.  I have the ability to analyze something and break it down into smaller pieces that make it more understandable for people.

I like to learn, I am good at learning, and I know how to take in information, process it, then I can change it to make it easier for other people  to learn.  There are many things in this world that people learn, some are easy and some are more difficult.  One, however, that seems to come naturally for some and not for others is mathematics.  Often those who are born with the ability to understand mathematics are not also born with the ability to understand that it is not as easy for everyone else.  In fact, this very idea creates one of the biggest problems in today’s education.  Just because you understand something, very well and it makes perfect sense to you, doesn’t mean you are also able to explain that information to other people.  It is likely that you can explain the information to like minded people.  Those that grasp math easily, learn math from math teachers who are “okay” at their jobs.  They may even learn math from those that are “terrible” at their job.

However, we have many people in this world who just don’t understand math as easily as others.  It falls on a spectrum, some might be very lost when it comes to grasping math concepts while others can get there if they have strong teachers.  However, all that fall in the category of the “less mathematically minded” struggle without strong math teachers.

So why do we have math teachers that are great and amazing, some that good, some that are okay, others that are not so great, and some that are down right terrible?  Well, all the math teachers are most likely mathematically minded people.  Who would choose to teach math unless it naturally made sense to them?  However, some of those are just people who can do it easily but don’t have the gift of being able to break it down to easy to understand pieces.  Many don’t even understand that some of their student’s brains don’t work like their’s does.  Some are just in a job to get paid because it is easy for them but they have no passion or natural gift for education.  Hence, we get the large variety of math teachers in our schools.  The real problems happen when you get the not so good or worse, the terrible math teacher lined up with the student who really struggles to understand mathematics.  Math builds on itself, one bad step along the way and you can be turned off from math or left in the dust.  And remember, how many different  math teachers does a student get over their lifetime?  For those mathematically minded, what is the probability that a student will get a great/amazing or good teacher each year, year after year?

One misstep, that is all it takes to send a child off the math track.  Does our country do anything once a child shows that first sign of falling behind?  Not really and certainly not to the extent it is needed if we all had the goal to help all students succeed in mathematics.

So, I have been given that gift of being both mathematically minded and having the ability to see things from the perspective of the learning student.  I am also passionate about education, specifically math education.  I spend a lot of time watching how it all falls apart and wondering if one person can somehow scale to reach millions.  I look at what Khan Academy tried to do and it gives me hope but Khan Academy still falls in the “teaching okay,” category and if the world of mathematics education is going to get changed, it must be changed with exceptional explanations, by people who are truly extraordinary in their gift of mathematics education.  This is not to say anyone (myself included) is perfect but part of doing what is right is listening to feedback and making adjustments when your goal is not being achieved.

It is time, therefore, to start one video at a time with content that teaches mathematics in easy to follow steps that a student can not only learn math from but learn how to study math.  In my next post, I will address the goals of my videos and I always welcome comments and suggestions!

Common Core Math 2 – Geometry Unit 1 Vocabulary

As I begin to help students at , a tutoring center in Apex, NC prepare for Common Core Math 2 / Geometry, I have decided to include some of the material online to make it available to all students who are beginning this course. The first thing that a student needs to know about Geometry is that one must learn all of the vocabulary and theorems in order to be successful. There is no other way to be successful in this class. Therefore, I will start with a list of the beginning vocabulary that one should learn and focus on as they begin to study for this course.

Line – Most students know what a line is but it is important to note that a line has arrows in both directions, note it is different from a line segment that terminates with end points.

Line Segment – A part of a line that terminates with a point at either end.

Ray – A mix between a line and a line segment, one end has a point, the other end an arrow.

Plane – Think of a desktop that continues infinitely in all directions.

Co-linear – Points that are co-linear would be points that fall on the same line.

Midpoint – This is the exact middle of a line segment, it divides something into two equal halves. The formula for calculating a midpoint doesn’t need to be memorized if you just remember that it is the average of the x-values and the average of the y-values.

Acute angle – A “cute” small angle that is greater than 0 degrees but less than 90 degrees.

Right angle – An angle that is exactly 90 degrees. We put a box in the corner of the angle to show it is right. In Geometry, we cannot assume an angle is right just by looking at it, if we don’t have proof or see the “box” we cannot assume it is 90 degrees even if it “looks” right.

Obtuse angle – An “obese” angle or fat angle, one that is greater than 90 but less than 180.

Straight angle – A angle that forms a line and measures 180 degrees.

Complementary – Two angles who sum to 90 degrees.

Supplementary – Two angles who sum to 180 degrees.

Adjacent angles – Angles that are next to each other.

Linear Pair – Two adjacent angles that are supplementary.

Bisect – Something that cuts into two equal pieces such as an angle bisector would cut the angle into two equal pieces.

Vertical Angles – Angles opposite each other (often form an X) – vertical angles are complementary.

Perpendicular Bisector – A line that bisects another line by hitting it at a right angle and cutting it into two equal pieces.

Distance between two points – Memorize distance formula or learn how to use the Pythagorean Theorem to find the distance between any two ordered pairs.

Perimeter – Distance around an object.

Circumference – The distance around a circle: C = PI X Diameter.

Area – The space inside a shape.

Radius – The distance from the center of the circle to the one end of the circle. Radius is half the diameter.

Diameter – The distance from one side of a circle to the other going through the center of the circle. Diameter is twice the radius.

Addition Property of Equality: When you add the same number to both sides of an equation, it doesn’t effect the equality of the equation.

Subtraction Property of Equality: When you subtract the same number from both sides of an equation, it doesn’t effect the equality of the equation.

Multiplication Property of Equality: When you multiply the same number to both sides of an equation, it doesn’t effect the equality of the equation.

Division Property of Equality: When you divide the same number to both sides of an equation, it doesn’t effect the equality of the equation.

* The above four properties are what you do when you solve an Algebraic Equation such as: 2x – 5 = 11 (add 5 to both sides: Addition property of Equality, then divide both sides by 2, Division property of equality).

Substitution Property: If two things are equal you may substitute one for the other: measure of angle 1 = measure of angle 2 and measure angle 2 + measure of angle 3 = 180, since measure of angle 1 = measure of angle 2, I can SUBSTITUTE the measure of angle 1 into my other equation making it: measure of angle 1 + measure of angle 3 = 180.

Transitive Property: (Remember as the 3 piece property) If a = b and b = c then a = c. If the measure of angle 1 = measure of angle 2 and the measure of angle 2 = measure of angle 3, then I know that the measure of angle 1 must also equal the measure of angle 1.

Reflexive Property: (Think Reflection): a=a Something always equals itself. It may seem obvious but is needed in proofs.

Angle Addition Postulate: If you have a big angle divided into two pieces the two pieces add together to equal the total angle.

Corresponding Angles: When two parallel lines are cut by a transversal, the angles that correspond with each other are congruent.

Alternate Interior (and Exterior) Angles: When two parallel lines are cut by a transversal, the angles that are on opposite sides of the transversal line but are both inside (or both outside) the parallel lines) are congruent.

Same Side Interior (and Exterior) Angles: When two parallel lines are cut by a transversal, the angles that are on the same side of the transversal line but are both inside (or both outside) the parallel lines are supplementary.

Remembering Congruence of Angles: Vertical Angles, Corresponding Angles, Alternate Interior (Exterior) Angles
Remembering Supplementary Angles: Linear Pairs, Same Side Interior (Exterior) Angles

These are the vocabulary words one should first learn and also be able to apply to concrete problems and proof situations as they start CCM 2 / Geometry! Good Luck!

Gifted Children, our education system, and grade skipping

I have been reading a lot on the pros and cons of grade skipping. I will admit that I had some initial bias and thought grade skipping was a bad idea. I had that opinion without giving it a lot of thought, it just seemed that most people felt that way and I had been bombarded with negative thoughts on the subject. People had told me that a grade skipped child would be too socially immature with children older than them, they would reach puberty at a different time, they would likely end up a social outcast, they would “grow up too soon,” academics isn’t everything, and if they were small in size that would make things even worse! Many of these reasons seemed to have merit and since I didn’t really have the need to dwell on the subject, I didn’t. In fact, in my line of work, I always have parents asking me advice about the exact opposite – “should I retain my child?” To that I answer, “yes,” many times! The reason for retention also has so many pro’s – especially when they are children who are as far academically behind as the ones I work with at my learning center. I know that an extra year could make them so much stronger academically so that they aren’t always struggling to keep their head above water, they could feel like they are smart, and have positive self-esteem about themselves as a learner. On top of that, they would get the benefits of being the oldest in the class, more confidence and maturity, and get to do things “first,” like getting their drivers license. So with that said, why would one want to do the opposite where the child is pushed ahead? It seemed contradictory! Or so I thought.

However, every child is different. The pros for one child to be retained will give a child so many benefits but so will the pros to skip a gifted child ahead. I was so surprised to read all the positive literature on grade skipping. I found very little (other than people who just had opinions) against grade skipping for a gifted child. Most of the research done and articles written by professionals support grade skipping. The best article that shows true prospective on this topic is A Nation Deceived.  If you want to see all the myths why people think grade skipping is bad and how grade skipping actually supports the gifted child, this is a great resource and I highly recommend it.

I think the reason that so many people are against it is that they don’t walk in those shoes.  When a friend asked me about grade skipping her son, I told her I didn’t think it was a good idea.  Why?  Well, because of all the things I heard, because I didn’t live with her child, because I didn’t know her child like she knows him, and I am not qualified to be answering that question.  If you don’t have a truly gifted child and not just one that should be in an Academically Gifted program within their grade (many families have gifted children who fall in that category) you can’t understand the perspective.  Highly gifted children do have a different maturity level than their same age peers.  Does that mean they can’t play with their peers?  Not at all, but would they fit better with someone who had a higher match of maturity?  Yes.  In fact, just one grade level may not even be a match for that.  Sometimes they are a mix of maturity as their thinking far surpasses their age but in other ways their actions match their age.  Does grade skipping mean that you are putting too much emphasis on academics?  Actually, this is not that case at all either.  Many people believe this misconception.  Grade skipping is trying to find a better fit for your child as a whole – academically, socially, and emotionally.  It isn’t just about academics.  Just as people choose to retain their children – they often don’t retain them “just” because of academics – it is, as mentioned before, because of the social and emotional benefits.  Grade skipping provides social and emotional benefits to the gifted child.

So, to all of you who are considering grade skipping – know that others may not understand but as long as you know that it is the best decision for your child as a whole, their is a lot of research to support your decision.  My school district also says it isn’t a reversible decision either – if you try it and it doesn’t seem to be working out, you can always choose to go back to your child’s same age peers.

FREE WORKSHEET: Multi-Step Word Problems

After searching for some multi-step word problems on the internet, I could not find any that were hard enough to match the North Carolina homework questions that my third graders were getting and struggling with. If this is going to be a goal for 3rd graders (and it is a high goal as this is a challenge for third grade math students that struggle) then the teachers need to provide a lot more practice in class and outside of class on these types of problems. Instead, I continue to get frustrated as they just advise parents that their child can’t do these, that it is a problem, and yet push on forward. So, in an attempt to provide more practice – I hope these problems can be copied and used by others!

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?

Grades in school – are they meaningful?

I run a tutoring center and tutor students in Wake county, North Carolina.  I get students from many different schools, although they are all in the same county.  However, their courses, although identical in name and in “theory” content, vary greatly.  If the level of a course can range from easy to extremely difficult and yet we award a grade based on test scores to both classes, how is this fair to the student and how is this truly a measure of anything?  Here is an example.  I am currently working with a student taking Honors Geometry through Wake County Virtual Public Schools.  This is an online class given when the school is not able to provide instruction within the school.  In this case, the student is in a middle school that does not offer this course so he has to take this online version of the course.  There is only a virtual teacher who responds to questions that the students (currently 3) ask and it takes about 10-20 minutes before they get a response to each of their questions.  There are no in person lessons, just self teaching from online materials.  The students turn in assignments and their assessments are never looked at by a person, they are always multiple choice so that a computer can grade all their work.  In a typical “in house” Honors Geometry class, students are expected to do 2 column proofs on exams, however, since this is not possible in an online class (it can’t be graded by a computer) these types of problems aren’t given.  Proof type questions might be asked but in a multiple choice format, which is hardly the same as generating a proof from scratch.  The students still have to do some exercises with proofs but aren’t tested on these proofs and their exercises, I am told, count about 10%.  It seems the multiple choice questions are quite easy and a student who in a “in house” Honors Geometry class who might not be passing with the same level of knowledge, can score a B in this multiple choice testing format.

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.

Should Partial Credit Be Awarded on Math Tests?

This is a debatable subject. Math teachers seem to be on one side or the other. When asked for reasoning, I hear things such as, “No partial credit should be given because the real world doesn’t allow for things to be wrong.” Other teachers are very busy and don’t have the time to look at a student’s work in the detail needed to figure out where their mistake was or what their thinking was, so they could correctly award partial credit; it is much easier to just grade it as 100% right or 100% wrong. On the other side of the coin, teachers who award partial credit encourage students to “show their work” and want to encourage students for getting conceptual parts of the problem correct and not penalize them for making one tiny mistake in a multi-step problem that demonstrates that they have actually learned what they were being taught. We are all human after all.

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.

Progressive Math Now Available


Progressive Math Level One is now available in a hard copy. Our digital copy has been finalized and is also for sale.

This is the first book is a series of mathematical books designed to provide a fresh approach to mathematics that approaches math in small progressive steps. The goal of the course is to build a student’s new knowledge of concepts from their existing knowledge. The book provides teachers and parents with lessons on how to work with the child on these concepts and includes sample dialog. It provides many pages of practice that gradually increases in difficulty and provides constant review. The topics are carefully chosen so that they all link to topics that the student has already had exposure to.

Topics that are focused on in this book include:

  • Patterns (and applying patterns to applications such as counting money and adding without using fingers)
  • Addition Facts – we stress teaching students overall number sense and ways to learn their facts without having to count on their fingers.
  • Subtraction Facts – we use methods that allow students easier and less frustrating ways to find solutions to subtractions facts, especially harder facts such as 16-7.
  • Telling time to 5 minutes – we use the student’s previous knowledge of counting by 5’s and link this together to build the concept of telling time.
  • Counting Money – student’s use their pattern abilities and apply this with concrete visuals to learn how to easily count money.
  • Word Problems – we help students learn to look for key words to help them decide if the problem is asking them to add or subtract.
  • Getting prepared for Multiplication and Division – there are times when teaching early material lends itself to introducing concepts that prepare students for later concepts, we don’t ignore these situations, we embrace them and we introduce students to the idea that doubling a number is the same thing as multiplying times 2.
  • Place Value – In order to move forward, students need to understand place value – we have units in the book that address this issue and give students practice in locating the place value of numbers to the hundreds.

We feel our series is very different and advantageous over many of the traditional books available. We give students tools that other books do not. Other books just give practice. We teach students “tricks” and new ways to think. If they just can’t memorize that 9 + 8 = 17, what other options do they have but counting on their fingers every time? We provide them other options!

Geometry: Translations, Reflections, Rotations, and Dilations

One unit covered in Geometry deals with the concept of translations, reflections, rotations, and dilations.  This article will serve to summarize some the major points for students studying these topics.

Prerequisite:  The student needs to come into the lesson with some basic understanding of matrices.  Given a shape with points on a coordinate plane they need to be able to write those in matrix form.  It is very simple actually, you take the x-coordinates and make them the first row of your matrix and take the y-coordinates and make them the second row of your matrix.

Example: A quadrilateral with points (-4,-3), (-1,0), (1,-3), and (-3, -5) would be written as the following matrix:

$$ \left[
\begin{array}{ c c c c }
-4 & -1 & 1 & -3\\
-3 & 0 & -3 & -5
\end{array} \right]
$$

Students also need to know the identity matrix when multiplied by a matrix gives back the original matrix.  It is like multiplying a number times 1.  The identity matrix is:

$$ \left[
\begin{array}{ c c }
1 & 0 \\
0 & 1
\end{array} \right]
$$

Translations: Translations simply slide your figure around.  It is the easiest to work with since it just involves adding a value to the x-coordinates and a value to the y-coordinates.

Example: Translate the example matrix above by moving it to the RIGHT four and DOWN 1.  This would mean we just add 4 to the top numbers and subtraction 1 from the bottom numbers:

$$ \left[
\begin{array}{ c c c c }
-4 & -1 & 1 & -3\\
-3 & 0 & -3 & -5
\end{array} \right] +
\left[ \begin{array}{ c c c c }
4 & 4 & 4 & 4\\
-1 & -1 & -1 & -1
\end{array} \right]=
\left[
\begin{array}{ c c c c }
0 & 2 & 5 & 1\\
-4 & -1 & -4 & -6
\end{array} \right]
$$

Dilations: Dilations make an object bigger or smaller.  If the dilation is a number bigger than 1, the object will increase in size; if it is less than 1, it will get smaller.  Dilations require you multiple each number in the given matrix by the dilation value.

Example: Dilate the given quadrilateral by 3.

3 * $$ \left[
\begin{array}{ c c c c }
-4 & -1 & 1 & -3\\
-3 & 0 & -3 & -5
\end{array} \right] =
\left[
\begin{array}{ c c c c }
-12 & -3 & 3 & -9\\
-9 & 0 & -9 & -15
\end{array} \right]
$$

The “slightly” harder problems involve ROTATION and REFLECTION.

These simple “adjust” the coordinates according to a specific matrix.  Let’s look at some different matrices and see what they do:

Identity: Doesn’t change the value of the matrix.

$$ \left[
\begin{array}{ c c }
1 & 0 \\
0 & 1
\end{array} \right]
$$

Matrix 1: Notice this looks just like the identity matrix except it has negative 1’s rather than positive ones.  This means that it will change each sign to its opposite in the matrix.

$$ \left[
\begin{array}{ c c }
-1 & 0 \\
0 & -1
\end{array} \right]
$$

Matrix 2: This matrix looks like the identity but has a negative only in the top 1.  This means only the top row will change to their opposite signs but the bottom row will stay the same.

$$ \left[
\begin{array}{ c c }
-1 & 0 \\
0 & 1
\end{array} \right]
$$

Matrix 3: Matrix 3 is similar to Matrix 2 but the negative is on the bottom instead of the top.  This means the bottom row will change to its opposite sign and the top row stays the same.

$$ \left[
\begin{array}{ c c }
1 & 0 \\
0 & -1
\end{array} \right]
$$

Matrix 4: This matrix is a little different from the identity.  The 1’s and the 0’s have changed places.  When this happens, the whole row changes places.  Since both are positive, the numbers keep their original signs.

$$ \left[
\begin{array}{ c c }
0 & 1 \\
1 & 0
\end{array} \right]
$$

Matrix 5: Can you guess what happens in this matrix?

$$ \left[
\begin{array}{ c c }
0 & -1 \\
1 & 0
\end{array} \right]
$$

Matrix 6: How about this one?

$$ \left[
\begin{array}{ c c }
0 & 1 \\
-1 & 0
\end{array} \right]
$$

Matrix 7: And this one?

$$ \left[
\begin{array}{ c c }
0 & -1 \\
-1 & 0
\end{array} \right]
$$

Matrix 5 – the rows switch places and the top row has opposite signs.

Matrix 6 – the rows switch place and the bottom row has opposite signs.

Matrix 7 – the rows switch places and both rows also change signs.

Each of these matrices are multiplied times the matrix defined by the shape in the problem  Note that the “identity” type matrix always comes first, then the other matrix so that the dimensions match for multiplying.

Here is a summary of when to use each matrix:

Reflection over y = x: use matrix 

$$ \left[
\begin{array}{ c c }
0 & 1 \\
1 & 0
\end{array} \right]
$$

Reflection over x-axis: use matrix   

$$ \left[
\begin{array}{ c c }
1 & 0 \\
0 & -1
\end{array} \right]
$$

Reflection over y = -x: use matrix 

$$ \left[
\begin{array}{ c c }
0 & -1 \\
-1 & 0
\end{array} \right]
$$

Reflection over y-axis: use matrix 

$$ \left[
\begin{array}{ c c }
-1 & 0 \\
0 & 1
\end{array} \right]
$$

Rotation of 90 degrees: use matrix   

$$ \left[
\begin{array}{ c c }
0 & -1 \\
1 & 0
\end{array} \right]
$$

Rotation of 180 degrees (same as rotation over Ho): use matrix

$$ \left[
\begin{array}{ c c }
-1 & 0 \\
0 & -1
\end{array} \right]
$$

Rotation of 270 degrees: use matrix

$$ \left[
\begin{array}{ c c }
0 & 1 \\
-1 & 0
\end{array} \right]
$$

Example: Given A(2,5) and B(1, -2) and C(-2,3).

Find a rotation of 270 degrees:

$$ \left[
\begin{array}{ c c }
0 & 1 \\
-1 & 0
\end{array} \right] *
\left[
\begin{array}{ c c c }
-2 & 1 & -2 \\
5 & -2 & 3
\end{array} \right] =
\left[
\begin{array}{ c c c }
5 & -2 & 3\\
-2 & -1 & 2
\end{array} \right]
$$

Note:  The two rows switched, then the bottom row switches signs.

Find a reflection over the y-axis:

$$ \left[
\begin{array}{ c c }
-1 & 0 \\
0 & 1
\end{array} \right] *
\left[
\begin{array}{ c c c }
2 & 1 & -2 \\
5 & -2 & 3
\end{array} \right] =
\left[
\begin{array}{ c c c }
-2 & -1 & 2\\
5 & -2 & 3
\end{array} \right]
$$

Note: The rows did not switch, but the signs on the top row changed to their opposites.

To summarize:

Dilations: Multiply matrix through by the amount of the dilation.

Translation:  Adjust each x by the change in the x-axis of the translation and adjust each y by the change in the y-axis  of the translation.

Reflections and Rotations:  Find the corresponding matrix for each reflection or rotation and multiply the matrix by the correct “identity-type” matrix listed above.

Happy Transformations!

If you found this article helpful, please let us know!

This article was brought to you by:

www.apex-math.com

Part 2 and 3 of the video can be found under Videos on the homepage.

Teaching Multiplication Facts

Teaching Multiplication Facts

Multiplication facts are part of the North Carolina Standard Course of Study curriculum for the third grade.   Eventually we hope that students will just “know” their facts – in other words, they are memorized but when first learning facts, it is best to teach students strategies for finding facts.  This allows students to always have a fall back plan in case they “forget” the fact and it makes them quicker to learn. The order in which facts should be taught is given below.  The reason for this is that the easier ones get learned first and then they can rely on their “partner fact” (3X4=4X3) for some of the harder facts.

Here are some methods for each fact:


X0 Fact: Anything X0 is 0. This is a very easy fact since students just need to learn that anything times 0 is 0.  Remind them that 0 sets of something is 0.

____________________________________________________________________

X1 Fact: Anything X1 is itself. Again, another very easy fact.  For example,  1 X 7, this means 1 set of 7, which is just 7 items.

_____________________________________________________________________

X2 Fact: Circle the number that is not 2, double that number. Since the student is very good at doubling, this is an easy fact.

__________________________________________________________________

X4 Fact: Circle the number that is not 4, double the number and double again.   The student should have doubling down well before starting, so X4 facts will come very easy to them.  The only one that might be difficult is X9 – tell them that they can wait and use the “9’s trick” on that one instead of the 4’s trick if they don’t know 18 + 18 since we didn’t really drill that double.

___________________________________________________________________

X10 Fact: Add 0 to the number that is multiplied by 10.

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X5 Fact: There are two strategies that work here.  Lots of kids like to just count by 5’s because they are good at it.  That takes longer when they are counting by 5 eight times for numbers like 5 X 8.  For these bigger numbers (especially the even ones) they can try this other strategy:   multiply the number times 10 (see 10 fact) and then take half of that number:  so for 5 X 8 we would do 10 X 8 = 80 then take half = 40.

_____________________________________________________________________

X11 Fact: Circle the number that is not 11, write the number twice – 11 X 3 = 33.

______________________________________________________________________

X9 Fact: Use the finger trick – hold up all ten fingers, count from the left and bend down the finger of the number you are multiplying by 9. For example, for 9X3, bend down your third finger. Now count the number of fingers on the left side of the bent finger, write that number down – then count the number of fingers on the right side of the bent finger, that is your second digit. With the 9X3 example, you have 2 fingers to the left of the bent finger and 7 fingers on the right side of the bent finger, the answer is therefore 27.  Also show your child all the different patterns in the 9’s.  The sum of the digits of all the 9 facts add to be 9.  The tens place of the 9’s facts is always 1 less than the number you are multiplying by:  if you multiply 9 X 6 then you know that your answer is going to be Fifty – something since 5 is one less than 6.

_____________________________________________________________________

X3 Fact: This is where we get the harder ones – students should know the commutative property and therefore only need to learn: 3X3, 3X6, 3X7 and 3X8. There is no good trick for these, tell the child to double the number and add one more. 3X3 = (double 3) + 3 = 6 + 3; 3X7 = (double 7) + 7 = 14 + 7 = 21.  They can also learn to count by 3′s.

______________________________________________________________________

X6 Fact: With the commutative property, you will only need to learn 6X6, 6X7, and 6X8. 6X6=36 and 6X8 can be taught using rhythm (tap them out on the table as you say them).

_____________________________________________________________________

X7 Fact: For this group, you will need to learn 7X7 and 7X8. To memorize 7 X 8, I use the visual:  5 6 7 8 these numbers go in order of counting and you can just put = and x in the middle and get your fact 56 = 7 x 8. Help your child see this visual and use it as a cue to help them to remember the fact.  7 X 7 = 49 – sorry, just got to memorize that one.

_____________________________________________________________________

X8 Fact: You only need to learn 8X8=64. You can use “bend down touch the floor, eight times eight is 64” or “Skate X Skate = Sticky Floor.”  You can also do “double, double, double if the student can double 3X.  This triple double works really well for 8 X 3 as it is a harder 8 fact but is easy to double 3X.  Double 3 gets to 6, double 6 gets to 12, double 12 gets to 24.

You will want to work on 1 fact at a time, do lots of practice with that fact before moving to the next one. Once you move to the second fact, you should work on that alone and then provide a mix of that with all previously learned facts, and so on.

TABLE of FACTS – shows strategies for each of the different facts:

1 2 3 4 5 6 7 8 9 10
2 Double Double Double Double Double Double Double Double Add Zero
3 Double Count by 3’s Double Double Again Count by 5’s Double 6 + 6 Double 7 + 7 Double 3 times Nines Trick Add Zero
4 Double Double Double Again Double Double Again Double Double Again Double Double Again Double Double Again Double Double Again Double Double Again Add Zero
5 Double Count by 5’s Double Double Again Count by 5’s Count by 5’s Count by 5’s Count by 5’s Nines Trick Add Zero
6 Double Double 6 + 6 Double Double Again Count by 5’s Rhythm Memorize Rhythm Nines Trick Add Zero
7 Double Double 7 + 7 Double Double Again Count by 5’s Repeat Memorize 5678 Nines Trick Add Zero
8 Double Double 3 times Double Double Again Count by 5’s Rhythm 5678 Rhyme Nines Trick Add Zero
9 Double Nines Trick Double Double Again Nines Trick Nines Trick Nines Trick Nines Trick Nines Trick Add Zero
10 Double Add Zero Add Zero Add Zero Add Zero Add Zero Add Zero Add Zero Add Zero
    ANY CHARACTER HERE

You will want to work on 1 fact at a time, do lots of practice with that fact before moving to the next one. Once you move to the second fact, you should work on that alone and then provide a mix of that with all previously learned facts, and so on.  Note that the facts above are written in the order in which they should be taught.  Don’t teach them in numerical order but instead from easiest to hardest so that students can use the commutative property to their advantage and have a feeling of accomplishment.

We are getting ready to launch our Online Multiplication Curriculum. This online program will be interactive and affordable. It will teach your child each strategy mentioned and give them practice with instant feedback using the strategies. We hope to post it on our site soon but for those who can’t wait, feel free to contact us about our pre-release version.