Teaching factorization of quadratic expressions in 10 minutes

[Minoosh]

I'm working with high school students, who are not currently enrolled in a math class, as they work through lessons on math-teaching software. There is a round of "benchmark testing" coming up and the principal wants to see the numbers go up. Originally this was conceived of as a relatively painless was to keep students' math skills current but now this expectation has emerged that I will give a 10-minute "mini-lesson," then students will master several methods of factoring quadratic expressions in the remaining 80 minutes.

A lot of the work so far has been in areas students are familiar with. But none of these students (freshmen) have done much with nonlinear functions.

I can show them the trick to factoring as they may see it on the test, but I am supposed to cover at least 4 techniques, without any time for students to learn why quadratic function are useful or even what they actually are. I can show them on a graphing calculator that the factored version is the same as the non-factored, and I probably can take a minute to two (literally) to give them examples. I would also like them to learn the FOIL technique for checking their answers.

I have 2 questions - can I make substantive headway in the one 90-minute session I have with my groups before the benchmark test? And, what are the merits of an overview that basically shows them the tricks, with virtually no real-world idea of what these functions are or why they are useful?

OK, three questions: What is the best way to go about this?

 

[phiwum]

I'm working with high school students, who are not currently enrolled in a math class, as they work through lessons on math-teaching software. There is a round of "benchmark testing" coming up and the principal wants to see the numbers go up. Originally this was conceived of as a relatively painless was to keep students' math skills current but now this expectation has emerged that I will give a 10-minute "mini-lesson," then students will master several methods of factoring quadratic expressions in the remaining 80 minutes.

A lot of the work so far has been in areas students are familiar with. But none of these students (freshmen) have done much with nonlinear functions.

I can show them the trick to factoring as they may see it on the test, but I am supposed to cover at least 4 techniques, without any time for students to learn why quadratic function are useful or even what they actually are. I can show them on a graphing calculator that the factored version is the same as the non-factored, and I probably can take a minute to two (literally) to give them examples. I would also like them to learn the FOIL technique for checking their answers.

I have 2 questions - can I make substantive headway in the one 90-minute session I have with my groups before the benchmark test? And, what are the merits of an overview that basically shows them the tricks, with virtually no real-world idea of what these functions are or why they are useful?

OK, three questions: What is the best way to go about this?

I have some long ago experience with teaching such basic algebra. I apologize, but I don't see how you can do it.

 

[Jules Galen]

I'm working with high school students, who are not currently enrolled in a math class, as they work through lessons on math-teaching software. There is a round of "benchmark testing" coming up and the principal wants to see the numbers go up. Originally this was conceived of as a relatively painless was to keep students' math skills current but now this expectation has emerged that I will give a 10-minute "mini-lesson," then students will master several methods of factoring quadratic expressions in the remaining 80 minutes.

A lot of the work so far has been in areas students are familiar with. But none of these students (freshmen) have done much with nonlinear functions.

I can show them the trick to factoring as they may see it on the test, but I am supposed to cover at least 4 techniques, without any time for students to learn why quadratic function are useful or even what they actually are. I can show them on a graphing calculator that the factored version is the same as the non-factored, and I probably can take a minute to two (literally) to give them examples. I would also like them to learn the FOIL technique for checking their answers.

I have 2 questions - can I make substantive headway in the one 90-minute session I have with my groups before the benchmark test? And, what are the merits of an overview that basically shows them the tricks, with virtually no real-world idea of what these functions are or why they are useful?

OK, three questions: What is the best way to go about this?

I think it's hopeless.

Just make up some phony-baloney lesson plan with all the right Buzzwords - those Buzzwords are all the school admins check for anyways - and then teach what you think best.

When your students do poorly on the Quadratic part of the test, find something to blame - even it it's stupid, and stick with the story. (Maybe you can call in sick that day)

Good luck and remember: school admins don't really want the truth, they just want some feel-good lies. So make them feel good.

 

[casebro]

It would take more than 10 minutes to teach me what the words 'factorization' and 'quadratic' means. "New" and 'Math' took years of the 5th grade.

 

[TubbaBlubba]

Um, even going over the basic algebraic rules of parenthetical multiplication that can be reversed to factorize quadratic equations will take more than that. And getting enough of an intuitive feel for them to factoroze even real quadratic equations with integer roots easily takes many hours of practice.

 

[Minoosh]

Um, even going over the basic algebraic rules of parenthetical multiplication that can be reversed to factorize quadratic equations will take more than that. And getting enough of an intuitive feel for them to factoroze even real quadratic equations with integer roots easily takes many hours of practice.

Real integer roots will be all they'll see.

Thanks to those above who think the endeavor is not realistic - you all make good points about covering my bases but the school is probably more worried about documenting that it tried than it is about actual scores. Before being handed a curriculum I took them back to a lot of basic number sense and decimal operations etc. Many needed a refresher. Their arithmetic hurts them more than their algebra.

The computers we usually use won't be available due to testing. I'll rustle up what I can and team them up to get peer interaction going. A few students can factor, they can work at this level while others play with online graphing calculators to see the shapes and how they transform.

 

[TubbaBlubba]

Now, I study college-level math, but my approach would be to simply teach how to multiply parentheses, and show how different coefficient signs result in different factors, finally demonstrating how to break up coefficients to find a good fit. With integer roots it's pretty straightforward, especially if the second-order coefficient is +-1.

 

[s4zando]

I think it's hopeless.

Just make up some phony-baloney lesson plan with all the right Buzzwords - those Buzzwords are all the school admins check for anyways - and then teach what you think best.

When your students do poorly on the Quadratic part of the test, find something to blame - even it it's stupid, and stick with the story. (Maybe you can call in sick that day)

Good luck and remember: school admins don't really want the truth, they just want some feel-good lies. So make them feel good.

Wow. So much becomes clear.

How about: do the best you can, with what you have to work with, don't back down from the challenge, and if it fails, for whatever reason beyond your control, leverage those factors <snicker> for a better outcome next year. And definitely don't be 'sick' that day. Be there to advocate for better education...because...you know why.

ETA:sounds like you understand this from further posts, and I am glad you are of the understanding that there is only so much you can do with that time, and I thank you in advance for your honest efforts in a political no-win. This can set the benchmark for better support next year.

 

[Trebuchet]

Quadratic formula.

There are alternative ways to do it, but this always works.

Yes, I'm an engineer, not a mathematician. Why do it the hard way?

 

[Minoosh]

Now, I study college-level math, but my approach would be to simply teach how to multiply parentheses, and show how different coefficient signs result in different factors, finally demonstrating how to break up coefficients to find a good fit. With integer roots it's pretty straightforward, especially if the second-order coefficient is +-1.

This is the approach I took with the students who got that far (I've had 4 90-minute workshops). The software first throws out perfect-square determinant problems and I had them do the "punctuation" then find appropriate constants and check through FOIL. Factoring by grouping was harder and I'm also supposed to teach "completing the square." The test actually has 2 problems on that.

In college courses I took usually the profs allowed an index card's worth of notes; these guys won't have that.

 

[TubbaBlubba]

This is the approach I took with the students who got that far (I've had 4 90-minute workshops). The software first throws out perfect-square determinant problems and I had them do the "punctuation" then find appropriate constants and check through FOIL. Factoring by grouping was harder and I'm also supposed to teach "completing the square." The test actually has 2 problems on that.

In college courses I took usually the profs allowed an index card's worth of notes; these guys won't have that.

Yeah, personally I never found factoring by grouping especially useful in single-variable algebra (it can be useful for min/max multivariable calculus, as I recall). But in single-variable I just feel that it's an extra step tacked on to explicitly demonstrate the logic behind factorization. Although maybe it has pedagogical value, I don't know, but I wonder if it will really help students develop the appropriate mental shortcuts. It feels like a slight elaboration on the basic theory to me.

There is something to be said for completing the square, though.

 

[Minoosh]

Quadratic formula.

My preference as well but I don't think they'll have a formula sheet. And, there's the completing-the-square part where they have to show an interim step. They will have calculators so the fact that they aren't great at factoring under a radical isn't critical. And they won't be seeing complex numbers.

These kids seem pretty comfortable learning the "tricks" in a completely abstract way. Practical applications, maybe they'll see those next quarter when they will have a 10-week course covering everything I'm supposed to teach them in 9 hours.

 

[TubbaBlubba]

Well, factorization is a useful skill that goes way beyond what the quadratic formula and its two higher-order equivalents can accomplish.

 

[fuelair]

I'm working with high school students, who are not currently enrolled in a math class, as they work through lessons on math-teaching software. There is a round of "benchmark testing" coming up and the principal wants to see the numbers go up. Originally this was conceived of as a relatively painless was to keep students' math skills current but now this expectation has emerged that I will give a 10-minute "mini-lesson," then students will master several methods of factoring quadratic expressions in the remaining 80 minutes.

A lot of the work so far has been in areas students are familiar with. But none of these students (freshmen) have done much with nonlinear functions.

I can show them the trick to factoring as they may see it on the test, but I am supposed to cover at least 4 techniques, without any time for students to learn why quadratic function are useful or even what they actually are. I can show them on a graphing calculator that the factored version is the same as the non-factored, and I probably can take a minute to two (literally) to give them examples. I would also like them to learn the FOIL technique for checking their answers.

I have 2 questions - can I make substantive headway in the one 90-minute session I have with my groups before the benchmark test? And, what are the merits of an overview that basically shows them the tricks, with virtually no real-world idea of what these functions are or why they are useful?

OK, three questions: What is the best way to go about this?

Exterminate the principal . Clearly he is not an educator in any meaningful way.

Given your concern I suspect you will be able to do as much as can be done per his instructions, but as I recall, quadratics take some rather longer time and practice to thoroughly sink in to even young minds that are interested in math.

 

[Minoosh]

Exterminate the principal . Clearly he is not an educator in any meaningful way.

Given your concern I suspect you will be able to do as much as can be done per his instructions, but as I recall, quadratics take some rather longer time and practice to thoroughly sink in to even young minds that are interested in math.

The principal is a she, and to be fair I think my department chairwoman understands this plan is nuts, and that she (the chairwoman) would run interference for me face-to-face (though never in writing).

Even many of the good students have deficiencies in basic number sense so I think the earlier lessons could nudge scores up a bit despite the quadratic idiocy. These kids are *not* in a math class and if they were they would be getting a slow-ish buildup to exponentials, then quadratics.

 

[fuelair]

The principal is a she, and to be fair I think my department chairwoman understands this plan is nuts, and that she (the chairwoman) would run interference for me face-to-face (though never in writing).

Even many of the good students have deficiencies in basic number sense so I think the earlier lessons could nudge scores up a bit despite the quadratic idiocy. These kids are *not* in a math class and if they were they would be getting a slow-ish buildup to exponentials, then quadratics.

As should be the case!!!

 

[TubbaBlubba]

It's an ever-recurring problem that those inclined to teach math at whatever level are usually too good at it, and had too easy a time with it, to understand those who struggle.

Makes you want to call into question the basic assumptions behind elementary-mid education.

 

[Minoosh]

I was good at math when I was a teenager, but when I went back to school for math certification everything did not come flooding back. It had been decades since my last math class and I was on medication that affects cognitive function. So I have both experiences, struggling and not struggling.

The issue I'm stressing about here is, the principal has a hefty ego and I'm trying to thread a needle: Please the principal while not totally freaking out students.

Once I was tutoring in the assistant principal's office, walking students through solving a quadratic expression. He grabbed a marker, wrote the formula on a whiteboard and said, "Look, it's simple." He plugged in the values, got to the point of simplifying the radical expression and then said, "There has got to be an easier way to do this."

It was funny and somewhat gratifying as I think his mission was to show me how it was done. But he probably still thinks *I* was over-complicating the issue and I'm not about to correct him.

 

[fuelair]

I was good at math when I was a teenager, but when I went back to school for math certification everything did not come flooding back. It had been decades since my last math class and I was on medication that affects cognitive function. So I have both experiences, struggling and not struggling.

The issue I'm stressing about here is, the principal has a hefty ego and I'm trying to thread a needle: Please the principal while not totally freaking out students.

Once I was tutoring in the assistant principal's office, walking students through solving a quadratic expression. He grabbed a marker, wrote the formula on a whiteboard and said, "Look, it's simple." He plugged in the values, got to the point of simplifying the radical expression and then said, "There has got to be an easier way to do this."

It was funny and somewhat gratifying as I think his mission was to show me how it was done. But he probably still thinks *I* was over-complicating the issue and I'm not about to correct him.

I have shared your experience priorly of having to deal with incompetent admins who had no concept of how certain aspects of almost anything had to be taught to reach students. I no longer must put up with such rectums.

 

[Jules Galen]

I was good at math when I was a teenager, but when I went back to school for math certification everything did not come flooding back. It had been decades since my last math class and I was on medication that affects cognitive function. So I have both experiences, struggling and not struggling.

The issue I'm stressing about here is, the principal has a hefty ego and I'm trying to thread a needle: Please the principal while not totally freaking out students.

Once I was tutoring in the assistant principal's office, walking students through solving a quadratic expression. He grabbed a marker, wrote the formula on a whiteboard and said, "Look, it's simple." He plugged in the values, got to the point of simplifying the radical expression and then said, "There has got to be an easier way to do this."
It was funny and somewhat gratifying as I think his mission was to show me how it was done. But he probably still thinks *I* was over-complicating the issue and I'm not about to correct him.

No...it's not funny: this isn't about the math.

You know. I know. Kids know. Stray Dogs know. Everybody knows that Rule #1 for controlling children and maintaining their respect is to demonstrate Solidarity among Teachers and the Faculty. When the Principal you described petulantly blew Rule #1 with his mean-spirited display of math, he didn't do so because he was concerned that the kids weren't learning Quadratic Equations - he was attacking your credibility and potentially making it harder for you to manage a classroom. And he KNEW what he was doing. Later, he probably told other faculty that he had to step in and help you teach because you don't know Jack about Teaching.

This guy attacked you. He intentionally tried to sabotage your ability to Manage a Classroom. Do yourself a favor and don't deny it happened.

If it's any consolation, he didn't attack you because of anything you did - it's just a common technique that admins used to keep the Teachers beat-down: confused and demoralized so they can more easily control them.