Memorizing the quadratic formula?

[MarkT]

My 8th grade DS took the placement test to get into the Algebra 2 honors class for 9th grade at the local high school.

He said that there were 14 problems to complete in 45 minutes. According to my son, three of the problems were "solve this quadratic equation using the quadratic formula" and no word problems. In his 7th grade Algebra class at his current K-8 charter school (Saxon Algebra I 4th edition) they didn't quite get to the quadratic formula because there were too many kids over-placed and the teacher was overwhelmed trying to get them to pass so she repeated some topics and also slowed it downed. I introduced the quadratic formula to my son in after-schooling but he only did a couple of problems  and did not memorize it.  I helped him create an Excel spreadsheet quadratic formula that calculated the roots based on the A,B,C coefficients input to check his factorings.

 

I feel memorizing the quadratic formula does not really show mastery of Algebra I. What do y'all think?

 

I would agree that being able to derive the quadratic formula would certainly show a honors level understanding of Algebra I.

 

 

 

 

 

[regentrude]

I feel memorizing the quadratic formula does not really show mastery of Algebra I. What do y'all think?

 

No, that alone does not show mastery of algebra 1 - but OTOH, not knowing the quadratic formula shows insufficient mastery of algebra 1. I would consider it an essential skill to have before moving to algebra 2.

 

 

I would agree that being able to derive the quadratic formula would certainly show a honors level understanding of Algebra I.

 

That most certainly. Sadly, I am not sure all algebra teachers would possess this understanding.

My 8th grade DS took the placement test to get into the Algebra 2 honors class for 9th grade at the local high school.

He said that there were 14 problems to complete in 45 minutes. According to my son, three of the problems were "solve this quadratic equation using the quadratic formula" and no word problems. In his 7th grade Algebra class at his current K-8 charter school (Saxon Algebra I 4th edition) they didn't quite get to the quadratic formula because there were too many kids over-placed and the teacher was overwhelmed trying to get them to pass so she repeated some topics and also slowed it downed. I introduced the quadratic formula to my son in after-schooling but he only did a couple of problems  and did not memorize it.  I helped him create an Excel spreadsheet quadratic formula that calculated the roots based on the A,B,C coefficients input to check his factorings.

 

I feel memorizing the quadratic formula does not really show mastery of Algebra I. What do y'all think?

 

I would agree that being able to derive the quadratic formula would certainly show a honors level understanding of Algebra I.

 

When I hear my friends talking about how their public school kids are accelerated into Algebra in seventh grade, I wonder about how much they are actually doing.  I would certainly expect memorization of and derivation of the quadratic formula in any Algebra class.  Perhaps a bigger question, though, is what other topics were skipped or glossed over in that class?  Asking the students to memorize the quadratic formula doesn't sound like a difficult teaching assignment:  if that was skipped, what else was?  Many of these topics are important for science classes down the road, so I think you should be very grateful that this placement test revealed this deficit.

[MarkT]

No, that alone does not show mastery of algebra 1 - but OTOH, not knowing the quadratic formula shows insufficient mastery of algebra 1. I would consider it an essential skill to have before moving to algebra 2.

 

Knowing when to apply it or knowing the formula?  I know my son is capable of memorizing it and sometimes when I speak to him about math he thinks that what Algebra is about (memorizing) as opposed to understanding and applying the rules of Algebra.  I would rather him learn problem solving skills. 

 

He will be taking a second half of Algebra 1 refresher course this summer.  The exam did make him realize what they did not cover in his Algebra class.   Should I emphasize learning/memorizing the quadratic formula as a part of that?

Maybe I can walk him through the derivation as a mathematical motivation.

 

[Gil]

Not knowing the Quadratic formula (either by rote or how to efficiently derive it), in my opinion, if it was covered in Algebra 1 shows that your Algebra 1 class was insufficient. WAAAAY insufficient.

 

Memorizing it though is NOT the hallmark of a good mastery of Algebra 1.

 

[Gil]

Knowing when to apply it or knowing the formula?  I know my son is capable of memorizing it and sometimes when I speak to him about math he thinks that what Algebra is about (memorizing) as opposed to understanding and applying the rules of Algebra.  I would rather him learn problem solving skills. 

 

He will be taking a second half of Algebra 1 refresher course this summer.  The exam did make him realize what they did not cover in his Algebra class.   Should I emphasize learning/memorizing the quadratic formula as a part of that?

Maybe I can walk him through the derivation as a mathematical motivation.

There should be a balance. Some stuff, you just need to know. You can do problem solving, and derivations, first and foremost but you should also know the more common formulas. I know that some people will live and die on the idea that math isn't about numbers and formale but concepts but really? Along with that conceptual understanding it really helps to know the basic formulas. The why and how behind them and the why and how of to use them is also important.

 

Writing out/Thinking through the steps to a problem and knowing "at this point in the process one will have a quadratic equation, but only one solution is viable in this instance since we are talking about the physical world and negative distance doesn't apply in this case" is a GOOD place to be. Actually knowing what do with the quadratic equation once you have it is a GREAT place to be.

 

WHOOPS! I didn't notice that this was in the Highschool boards.

[MarkT]

When I hear my friends talking about how their public school kids are accelerated into Algebra in seventh grade, I wonder about how much they are actually doing.  I would certainly expect memorization of and derivation of the quadratic formula in any Algebra class.  Perhaps a bigger question, though, is what other topics were skipped or glossed over in that class?  Asking the students to memorize the quadratic formula doesn't sound like a difficult teaching assignment:  if that was skipped, what else was?  Many of these topics are important for science classes down the road, so I think you should be very grateful that this placement test revealed this deficit.

 

Actually I fought the charter school when they placed him in Algebra for 7th grade because he only had whatever pre-algebra stuff they do in 6th grade.  I lost.  I did convince them to upgrade their math placement test so they did straighten that out for this past year.  A much smaller group of 7th graders qualified (along with all the 8th graders that bombed out the previous year).

 

My son ended up getting straight As in the second half of the 7th grade Algebra class with my after-schooling but I could not convince him to do any extra work. 

 

I can understand why parents want to home-school but I certainly don't have the time with my 65-70 hour work weeks.

 

My semi-retired engineer friend is going to run the refresher class so I hope that works out.

 

[regentrude]

Knowing when to apply it or knowing the formula?

 

Both. Knowing when to apply it will not do him any good if he can not actually solve the problem.

I would expect a student who has completed algebra 1 to solve problems involving quadratics, not just to identify a problem as "quadratic equation problem".

 

 

 

  Should I emphasize learning/memorizing the quadratic formula as a part of that?

Maybe I can walk him through the derivation as a mathematical motivation.

 

The equation should definitely be derived with the student, so that the student has the ability to re-derive it should he ever forget it.

I do not see any benefit in rote memorization of any formulas, and my children certainly did not learn it that way - they simply worked a large enough number of problems so that they automatically retained the formula from repeated use. In a pinch, they could derive it.

 

Math education has two components: conceptual understanding and procedural mastery. One without the other is worthless - whether it is using an equation by rote without understanding where it comes from, or understanding a concept without being able to solve problems.

[Monica_in_Switzerland]

Quadratic equations are a huge part of Algebra 1.  Not knowing the quadratic formula shows the child has not completed algebra 1 in my book. 

 

In an ideal world, the child would be capable of deriving it as well, which would show understanding of most of the complicated bit of quadratic equations. 

 

I'd be curious why there are THREE questions of plug-and-chug solving using the quadratic equatioion.  That seems overweighted to me, considering  all there is to cover in Alg1.  

[MarkT]

Of interest, when New York State (where I grew up) makes the Algebra regents Common-Core compliant they plan on providing the quadratic formula on the reference sheet see:

 

http://www.engageny.org/resource/regents-exams-mathematics-algebra-i-test-guide

 

 

[MarkT]

. I'd be curious why there are THREE questions of plug-and-chug solving using the quadratic equatioion.  That seems overweighted to me, considering  all there is to cover in Alg1.  

 

My thoughts as well.    3 out of 14 were plug-and-chug the quadratic formula.  Why no word problems?

 

[In The Great White North]

I agree with most of the other posters; your current teacher did the class a disservice by not completing algebra I.  Any K-8 school needs to know were their kids are going to high school and make sure they are prepared.  He definitely should know the quadratic equation by heart, how to derive it, and how to use it when he finished algebra I.

 

Two of my dc had Algebra I in a Waldorf school, using a 1990's Dolciani text that had the quadratic formula as the very last chapter.  They didn't finish the book.  As far as I was concerned, the rest of the book was pre-algebra (and was covered in the 8th grade Structure and Method book I found for my youngest).  There was no way I was sending them to the local high school in Algebra II. This was one of the reasons we started homeschooling: to catch ds up with his class in math. We started homeschooling them at high school and  I made them redo algebra I with an older Foerster text that had the quadratic formula in the middle.

 

Back in the day, I remember my algebra I teacher going out in the hall and snagging a 10th grader (who had been in his algebra class the year before) and dragging her in front of our class to recite the quadratic formula.

[MarkT]

I would guess the placement test your son took is discovering who is learning outside of school, and how much.

Sorry for the confusion we live in AZ - I grew up in NYS (and have family there) so I tend to use it as a basis of comparison because they actually give a common final exam across the state.  AZ lets each school do its own thing so it is hard to understand what they actually expect after a particular course such as Algebra 1. The schools here also tend to have crappy course descriptions so that doesn't help much.

 

[MarkT]

NY does not teach the quadratic formula as part of Integrated Algebra 1. It's left for the beginning of Integrated Algebra 2/Trig.

 

The quadratic equation in one variable is in NYs CC+ standards for A1:

 

http://www.engageny.org/sites/default/files/resource/attachments/nysp12cclsmath.pdf

 

 

The reference cited is for all of high school CC math - I didn't see where it broke out the Algebra 1 course on a quick glance.

 

At least in NYS the teachers have to cover all of the topics expected on the regents final exam or they won't be employed for long.

 

[kiana]

Mark, how far did he get through the Saxon Alg 1? Completing the square is (according to the TOC I found) in lesson 104 and the quadratic formula in 110.

 

A student who actually understands completing the square should be able to derive the formula if needed. One big problem I see is that my students have dutifully memorized the quadratic formula, but have little to no understand of what it actually implies -- for example:

that if you get x = 1 and x = -3 the quadratic factors into c(x-1)(x+3), where c is a constant factor, and the quadratic in question will, if graphed, cross the x-axis at 1 and -3

on the other hand, if you get a non-rational pair of roots from a quadratic with rational coefficients, the quadratic in question does not factor over the rational numbers, however, it will still factor into c(x - root 1)(x - root 2) and thus cross the x-axis at those roots

while if you get a negative number under the radical, the quadratic in question will only factor over the complex numbers, and will not cross the x-axis at all if graphed.

[MarkT]

No, NYS teachers do not have to cover all the topics expected on the Regent's Exam.  They have to do what the local admin wants them to do. Around here, that means more time on the basics, so all will score the pass. Very few will score in the 90s, as the omitted units are necessary to score at that level. Those omitted units are only offered in the 8th grade version of the class, because that's where the top students are placed. Same scenario in science.

 

Things may have changed since when I grew up there.  My nephews school covered all the topics in their tiny NYS district where they basically have two tracks - college prep or not.  The college prep kids usually do fairly well.

 

 

[Teachin'Mine]

Sounds like the school really failed those kids in the class.  As you said, some should not have been placed in the class to begin with.  If only students who were ready for the algebra class were there, the teacher should have had no problem covering Saxon's 120 lessons within 180 days of school.  Yes, there would be additional days for tests, but it still is very easily doable. 

 

I see at lest two options, let him repeat algebra I in 8th at his new school, or complete Saxon's algebra I now and ask for a re-test when he's done, before school begins in the fall.  If your son isn't willing to do the extra work needed, then I'd do the repeat.  Algebra is important!  Doing it in 8th, he'll still have time for more advanced math in high school.

 

With Saxon, there should be no skipping of problems, much less lessons!  Regarding the 3rd edition at least, the student who completes all problems should easily at least know the formula and how to apply it to problems.

[JanetC]

I agree that memorizing the quadratic formula is a necessary part of algebra I.

 

A good sequence for quadratics in algebra 1 is several weeks on strategies for factoring quadratics.  The student is learning when you can solve a quadratic quickly without the quadratic formula.  Those "shortcuts" will come in handy later, but it's hard to learn those if you know the brute force way already.  Then comes the quadratic formula.  Including both understanding and memorizing the formula, and understanding the discriminant.

 

Memorizing formulas in high school is at least as important as memorizing the basics like times tables was in elementary math.  In my college level math classes, the tests usually consisted of some "basic applications" and then the "tricky ones" that required more analysis and problem solving skills.  Memorizing the formulas and knowing the basics by rote means you have the time to really attack the "tricky ones." 

 

Getting into the habit of memorizing and drilling in high school algebra lays the foundation for that.

[flyingiguana]

The formula can be sung to the tune of Row Row Row Your Boat. Memorizing it is a cinch with that.

 

However, I don't know that having it memorized is really all that essential to Alg1. Algebra 2, yes, one should definitely memorize it. There aren't that many things one has to memorize. That should probably be one of them.

 

Is he sure he wasn't just supposed to factor them and get the answer that way? That would seem to be way more appropriate for an Alg 1 test.

[MarkT]

1) Mark, how far did he get through the Saxon Alg 1? Completing the square is (according to the TOC I found) in lesson 104 and the quadratic formula in 110.

 

2) A student who actually understands completing the square should be able to derive the formula if needed. One big problem I see is that my students have dutifully memorized the quadratic formula, but have little to no understand of what it actually implies -- for example:

that if you get x = 1 and x = -3 the quadratic factors into c(x-1)(x+3), where c is a constant factor, and the quadratic in question will, if graphed, cross the x-axis at 1 and -3

on the other hand, if you get a non-rational pair of roots from a quadratic with rational coefficients, the quadratic in question does not factor over the rational numbers, however, it will still factor into c(x - root 1)(x - root 2) and thus cross the x-axis at those roots

while if you get a negative number under the radical, the quadratic in question will only factor over the complex numbers, and will not cross the x-axis at all if graphed.

 

1)  I looked back at his notebook from 7th grade. They covered up to lesson 87 which is pretty pathetic. A considerable amount of time (estimate 15-20%) was spent doing worksheets for the AZ 7th grade standardized math test. I certainly realized at the time that they should have covered completing the square and the quadratic formula so I did cover those after the school year but he did not want a lot of HW because it was summer vacation. Also at his school they post the "formulae" at the front of the classroom so it discourages learning them.  I encouraged my son to "internalize" these to go faster. I also volunteered tutored for the Algebra class late afternoon every other Friday and saw that many of the students had no business taking Algebra in 7th grade. I don't blame the teacher much since the administration places the students.  She agreed with my assessment. 

 

2) Most of the key points you make here I believe are covered in most Algebra 2 courses (at least that I am aware of).

 

[MarkT]

Sounds like the school really failed those kids in the class.  As you said, some should not have been placed in the class to begin with.  If only students who were ready for the algebra class were there, the teacher should have had no problem covering Saxon's 120 lessons within 180 days of school.  Yes, there would be additional days for tests, but it still is very easily doable. 

 

I see at lest two options, let him repeat algebra I in 8th at his new school, or complete Saxon's algebra I now and ask for a re-test when he's done, before school begins in the fall.  If your son isn't willing to do the extra work needed, then I'd do the repeat.  Algebra is important!  Doing it in 8th, he'll still have time for more advanced math in high school.

 

With Saxon, there should be no skipping of problems, much less lessons!  Regarding the 3rd edition at least, the student who completes all problems should easily at least know the formula and how to apply it to problems.

 

Algebra was in 7th grade - he is in Geometry this year in 8th with the students that did well in Algebra so that teacher can move at a faster pace and is covering just enough Saxon Geometry (separate text).  He is going into 9th grade next year (his charter is K-8).  My son tested as gifted at the end of 7th grade.

 

My current plan is to have him take a 2nd half Algebra I topics review over the summer. The instructor is a friend of mine. I trust him. DS is a little more mature this year so I believe he can catch up this summer.  I will continue my after-school support efforts with him for 9th grade. Mostly I try to get him to think not memorize.

 

The one thing I need to avoid is to have him with the riff-raff students that do not want to be in class and spend the time disrupting the teacher, etc.  That is why at the large public high school he needs to be in with the honors students.  You may get the occasional smart aleck but for the most part they want to be in class and belong there.

  

 

[kiana]

1)  I looked back at his notebook from 7th grade. They covered up to lesson 87 which is pretty pathetic. A considerable amount of time (estimate 15-20%) was spent doing worksheets for the AZ 7th grade standardized math test. I certainly realized at the time that they should have covered completing the square and the quadratic formula so I did cover those after the school year but he did not want a lot of HW because it was summer vacation. Also at his school they post the "formulae" at the front of the classroom so it discourages learning them.  I encouraged my son to "internalize" these to go faster. I also volunteered tutored for the Algebra class late afternoon every other Friday and saw that many of the students had no business taking Algebra in 7th grade. I don't blame the teacher much since the administration places the students.  She agreed with my assessment. 

 

2) Most of the key points you make here I believe are covered in most Algebra 2 courses (at least that I am aware of).

 

 

Because they are covered does not mean emphasis is placed on them or on understanding them.

 

Please note, I am not blaming the teachers nor even the students. I am blaming a system that is designed to bring everyone up to mediocrity and no higher, which encourages teaching monkey-see monkey-do memorization in order to get them a just high enough score on the exam in order to move to the next grade and so on.

 

I think that your ideas -- 1) covering the second half of algebra 1 and 2) putting emphasis on understanding are both good ideas and will help set your son up for success both in the honors algebra 2 and beyond.

 

Please note, I am not blaming the teachers nor even the students. I am blaming a system that is designed to bring everyone up to mediocrity and no higher

 

 

I think the reality is much more complicated and nuanced that this.  If this school just wanted to get everyone to the same mediocre level, they wouldn't have offered this accelerated "Algebra" to 7th graders.  I'm curious if this 7th grade Algebra covered the same material as the 8th grade class (or 9th grade), or if was a special "Algebra for 7th graders"?

 

I think that parents see that college admissions are getting more and more difficult, and that public school education is getting less and less rigorous, so they cry out for more rigor in the classroom.  Unfortunately, "rigor" is a pretty fuzzy concept, and it is difficult for parents to verify that the Algebra class their ninth grader is taking is a really solid, rigorous math class, which will teach them concepts and techniques that they will use for years to come.  So, the school answers these cries with the simplest, most tangible, visible response they can:  They offer "acceleration", and set up classes for younger students to take subjects formerly only offered to older ones.

 

(Aside:  I think it is perfectly age appropriate for many 8th and 7th graders to study Algebra, but only if they have had preparatory classes the previous few years, which I don't think happens in many public schools.  Indeed, if this school is so concerned about the 7th grade standardized math tests for these advanced students that they spent 20% of class time preparing for the "normal" 7th grade test, it doesn't sound like the school is altogether confident about their baseline skills)

 

So, the school gets to brag that they aren't like all those other "bad" schools, that they offer accelerated classes, and the parents are assuaged.  But, what really happens is that many of these kids didn't have the preparation to succeed in the class (and in this case, 15-20% of the time was wasted preparing for a standardized test), so the teacher has to slow down and dumb down the class to meet the students.   Instead of Algebra 1, the class ends up being "Algebra 0.75".  

 

What is much worse, though, is the students and their parents don't know that this happened.  They both think the student got a good grade in Algebra 1, and that all such classes are pretty much interchangeable.  Then they run into trouble later in the sciences when they need to apply these tools.

 

I think this same problem happens at many levels.  Over and over again, I hear of "AP" classes in the high school where no student bothers to take the AP test because so few ever score well on the test from that school.

[kiana]

GG: I agree with everything that you've said. I will furthermore add, that imo 7th grade students should only be taking algebra if they are prepared to get, at the worst, a solid B. 7th graders who score lower than that (and I do not include homework in this calculation -- they should get roughly an 85% on a final, including partial credit) should take a transitional class -- not necessarily repeating algebra 1 -- but reviewing and extending it -- rather than continuing straight into geometry. I see no reason to hurry through a mediocre education.

 

The complaining about the system was not specifically aimed at this case, but rather the large numbers of students I see who have been taught something fundamental like the quadratic formula without really understanding it, its derivation, and what else it implies. It sounds like Mark's son had a rather average (non-honors) alg 1 class that was just taught early, and in my opinion that's a crying shame. It's a good thing that his son has a concerned dad to help him out.

[In The Great White North]

 

 

. I will furthermore add, that imo 7th grade students should only be taking algebra if they are prepared to get, at the worst, a solid B. 

 

I would go further than that.  If they get a B (80%), that means they miss/don't understand 20% of the material.  If they can't do all (>95%) of the material, they should not be taking it early.

 

Missing 20% of the material is what happens to kids who are unable to keep up with the lowest available class - not what you intentionally do to the best and brightest.

[kiana]

I would go further than that.  If they get a B (80%), that means they miss/don't understand 20% of the material.  If they can't do all (>95%) of the material, they should not be taking it early.

 

Missing 20% of the material is what happens to kids who are unable to keep up with the lowest available class - not what you intentionally do to the best and brightest.

 

I will have to disagree with this. It is pretty common for a student who *does* understand the material to occasionally make computational errors which cost them points. It is especially common among students who have been bright but underchallenged in the past.

 

I think that if a student gets an 85% (the number I used in my post) on a comprehensive final, they know enough of the course material that any remaining holes can be filled in during the next class.

[MarkT]

I think the reality is much more complicated and nuanced that this.  If this school just wanted to get everyone to the same mediocre level, they wouldn't have offered this accelerated "Algebra" to 7th graders.  I'm curious if this 7th grade Algebra covered the same material as the 8th grade class (or 9th grade), or if was a special "Algebra for 7th graders"?

 

I think that parents see that college admissions are getting more and more difficult, and that public school education is getting less and less rigorous, so they cry out for more rigor in the classroom.  Unfortunately, "rigor" is a pretty fuzzy concept, and it is difficult for parents to verify that the Algebra class their ninth grader is taking is a really solid, rigorous math class, which will teach them concepts and techniques that they will use for years to come.  So, the school answers these cries with the simplest, most tangible, visible response they can:  They offer "acceleration", and set up classes for younger students to take subjects formerly only offered to older ones.

 

...

 

So, the school gets to brag that they aren't like all those other "bad" schools, that they offer accelerated classes, and the parents are assuaged.  But, what really happens is that many of these kids didn't have the preparation to succeed in the class (and in this case, 15-20% of the time was wasted preparing for a standardized test), so the teacher has to slow down and dumb down the class to meet the students.   Instead of Algebra 1, the class ends up being "Algebra 0.75".  

 

What is much worse, though, is the students and their parents don't know that this happened.  They both think the student got a good grade in Algebra 1, and that all such classes are pretty much interchangeable.  Then they run into trouble later in the sciences when they need to apply these tools.

 

I think this same problem happens at many levels.  Over and over again, I hear of "AP" classes in the high school where no student bothers to take the AP test because so few ever score well on the test from that school.

 

You pretty much nailed it.

The charters here have to "compete" with the BASIS charter school  (where even the 5th graders do several hours of HW each night) so they accelerate. My son was only at his charter for 7th and now 8th grade.  He went to the neighborhood elementary school.  I chose the charter because I liked the English and Literature program (and still do).

 

We ended up with Algebra "lite" less filling and does not taste great.