Help needed! I have quickly learned (the hard way) this year that Algebra 2 students need mini “refresher courses” of Algebra 1 before plunging into more advanced material. I gave my students a pretest and many of them also need a review on multiplying polynomials as well. I plan to review quadratic factoring for 2-3 days in my Algebra 2 class before I jump into factoring polynomials. Since this is a quick review of a section that takes at least a couple of weeks in Algebra 1, I would love some advice.
My review plan is this –
- Day 1 – Eradicate FOIL – Multiply polynomials by using area method (boxes) and distributive property. Factor GCF’s.
- Day 2 – Factor a = 1 and special cases.
- Day 3 – Factor a > 1. Advice appreciated on methodology here.**
** What are your thoughts on factoring when a > 1 in Alg2 and above courses? The factor by grouping method is a sure thing, but is so much better than “guess and check” (the way I learned and used forever) that it worth the time? I don’t believe my students were taught this method in the past and I don’t know if teaching them now will a) take to long and b) totally confuse them since it is a more procedure heavy way of factoring. Plus, in Alg2 + how often is factoring by grouping going to be useful?
For those of you who have been there before, I would love any advice (or anything you have created).
Thanks in advance!
Julie – I’ve been there…every September for the last 4 years! Factoring goes on the list of things they should come to Algebra 2 knowing how to do but somehow don’t. Our first unit of the year is on Polynomial Operations, so while I do a 2-day review of factoring, the ideas are reinforced over and over throughout the unit. Unfortunately I have no great activity – so Row Games or Speed Dating can add some liveliness to the topic. But I did teach factoring quadratics with a>1 using the Master Product method with grouping (http://www.regentsprep.org/Regents/math/algtrig/ATV1/revFactorGrouping.htm), and it was very successful. Like you said, why DID I spend all those years with trial and error? I know whatever you do, you’ll make it engaging for you students!
Thank you so much for your adviceI It is so great to know that it didn’t take that long to teach it to Alg2 and that you feel it WAS worth it. I love that method but didn’t want to confuse them if they haven’t seen it before.
Wendy, I love the regents method, and I teach it for a=1 as well, so that kids don’t think they are learning two methods!
What about “the box method” or “X box method”?
Yes, I usually use the boxes when I teach factoring by grouping in lower grades. I wonder if Alg2 will need the boxes as well?
I really think the boxes give a solid connection to the understanding of what happens when we multiply polys. You may find that you have sudents who have glommed onto foil (I re-insert “distribute” every time I hear that term) and refuse to let go and learn something different. That hurts them as we expand binomials via Pascal!
Factoring is another can of Oligochaeta, especially those of the family Lumbricidae (I am learning new words – sorry: worms!). One of my colleagues spent weeks teaching factoring, instead of other things. Another said that teaching them to graph took care of that. I did a brief review of a=1 factoring, modeled along the lines of a>1 steps, so they only had to think about one set of steps. They didn’t get really comfortable with guess and check, but as we started finding zeros, and matching those to factors (preparing for rationals) the connections they were making helped with their understanding, I noticed.
Factoring by grouping is a valuable skill, as they encounter larger polynomials – it will help them find the non-real factors, once they have found the real, without synthetic division, but the amount of steps are about the same. There are other ways to get the info they need in Alg II. Your time limit is the thing you have to consider – what gets left out or skinnied up, if you choose to spend the time on grouping. Graphing, synthetic division, and real/non-real roots are the goal of the standards at my school, so I want my kids to understand as many routes as possible to the result! It really raises their confidence level, and that makes the whole class go better.
I don’t have worksheets so much as a loose plan and a list of good polynomials – the kind that will give clear results, at first, and then more puzzling results, where I can address the children’s misconceptions and errors. (Try a pre-calc text for some really good ones- you can then go down for lower entry points, but you can really challenge those upper level kiddos!) I also work with the graphs as we work to find factors, teaching them to read the characteristics of graphs as we go. I’ll go back through my materials and share whatever I have. I find so many awesome resources on the web, that I rarely have to make my own. I look forward to hearing what you decide and how it goes!
I do something similar to the grouping method, but after finding the factors we write binomials using the same signs, divide by “a”, simplify, and “bottoms up” (the new denominator in the constant term of binomial becomes coefficient for variable). This is the only way I teach them now because it works for all the cases (special, a=1, a>1). I love it so much more than the guess and check I learned many moons ago.
I teach guess and check but there is a method to my madness. 1) 99% of the time, either a or c is prime. Lock those factors in (I.e. Don’t waste your time switching those around). 2) factors that are closer together are “luckier.” (I heard this from another teacher and I don’t know why, maybe textbook companies are lazy?). 3) if your original does not have a GCF, neither of your binomials can either (let that sink in for a moment). That means don’t bother trying (2x +6)(5x+1) because of the 2 and 6. I think this rule is the most helpful and mind blowing. I find it helps students to write the outer and inner products below to check that the sum matches the middle term, and then I try to wean them away from having to write it.
Doing the first day on mini whiteboards also takes out some of the frustration of erasing. They will try a bajillion guesses as long as they are writing with dry erase markers. 🙂
I actually taught both “bottoms up” and guess and check one year and I found that my weaker kids chose bottoms up, but then when they multiplied AC the number was pretty big to quickly find factors. The other issue with bottoms up is next year in Precal, 99% of them forget the last step and since they don’t “check” they think the answer is reasonable.
I taught grouping my first couple of years, and I can’t really remember why I stopped. Maybe it *was* too procedurey?
Also you are not allowed to work on school this much during the break!
Hey Julie, my Factor Draft game might be good for reviewing factoring! I never did get around to making a version for a > 1, though…
I don’t subscribe to any canned method for finding the right combination of factors during a trinomial factoring problem. But I do share one nugget which helps minimize a search, and also demonstrates understanding of factors:
Say we had 6x^2 +25x + 24. This problem would tend to be nasty for students, as 24 has so many factor pairs. Let’s say we try 2x and 3x as the leading terms first, and now need to test factors of 24. Here’s the idea: There is now way that (2x+6), (2x+4), (2x+12), (2x+8) could be correct factors in this problem. Why? All of those binomials have a GCF, so if any of those were factors of the originial trinomial, the entire trinomial would have a GCF…which it does not. Thinking about factor pairs which would cause GCF’s and never checking them in the first place can minimize a search greatly.
On a side note, it’s easy to get wound up in factoring. But I’ve always held that if we are giving problems which are just plaing cruel (like having 144 for c), then maybe we need to rethink what we really want to accomplish. Just my 2 cents…
Hope this helps.
Actually just realized that Meg offered the same idea I propose here. Great minds…. 🙂
Dear Meg and Bob,
You two are amazing. You saved me with transformations, and now this. I love it. Thank you!
I am with those suggesting x-box (though I had never heard it called that before). It ties factoring pretty clearly to multiplying polynomials, which is a big plus. It also has the added benefit of setting them up for polynomial long division using the box, which is MUCH easier than any othe method I have seen.
Lastly, X box method works well for leading coefficient of 1 AND leading coefficient of >1. Kids who do enough and are keen to notice patterns will start to recognize that they can skip the box part with a leading coefficient of 1, while the rest won’t be confused by needing to learn two different techniques. For that reason, I usually teach factoring of trinomials without differntiating between leading coefficients situations. Many of my kids end up thinking leading coefficient of 1 is actually a bit harder, because they have to remember that the coefficient is even there.
I just thought of this idea:
cubic ax^3 + bx^2 +cx + d
multiply by a^2 to get a^3x^3 + a^2bx^2 + a^2cx + a^2d
which is (ax)^3 + b(ax)^2 + ac(ax) + a^2d
and change the variable to z = 3x
giving z^3 + bz^2 +acz + da^2
Now we have a cubic with leading coefficient equal to 1
Solve using the a=1 methods and for each z root the x root is z/3
Now with numbers:
I took 3x^3 + 7x^2 -2x – 8, which factorizes to (x+2)(x-1)(3x+4)
but we don’t know that !
Multiply by 3 squared
and get 27x^3 + 63x^2 -18x -72, better written as (3x)^3 + 7(3x)^2 – 6(3x) – 72
Now change the variable to z = 3x
and get z^3 + 7z^2 – 6z -72
Solve by any method for z, and the x is z/3
This deals with the difficult case easily, and also is an introduction to changing the variable(s),
which is a very very useful technique (spoiled in calculus by calling it “u – substitution”).
ps I studied maths for 7 tears, taught maths, statistics, and then control systems engineering for the last 17, and NEVER had to factorize a cubic polynomial ! ! ! ! ! ! ! !
yup, didn’t know it, but I use the Regents method, I always thought it was called the British Method. In Algebra 2 I wouldn’t separate a = 1 from a > 1 . They are big kids and just need one method, eventually they will see which ones are one steppers and which are by grouping. Now to get my Precalc kids from thinking (4x+9)^2 is 16x^2+81. I see what you mean by making too many assumptions…
They should be “forced” into putting numbers in as a check. x = -1 is the best, but they might have trouble with the x^2 !
I teach Alg 1 and Alg 2. In Alg 1, I teach the Xbox method. By algebra 2, I review it. At this point, they have kind of figured out that guess and test is faster but some of them are able to do it and need the more visual procedure of XBox. I teach factor by grouping too for 4 terms.
I have used the diamond method for the last few years with my Alg I classes. I think this is probably the same as the X or box method. It has been very effective.
I have a way I love, but it will be hard to describe. It’s something that needs to be seen. It’s essentially guess and check, but organized. One the kids get used to it, they can factor quickly.
I start with the polynomial written out.
I factor (guess) the coefficient a and write them one under the other under the a.
I factor the c (guess) and write them one under the other under the c.
Here’s the check:
I cross multiply the top factor of a with the bottom factor of c. Then the bottom factor of a with the top factor of c. When I demo, I draw arrows showing what’s being multiplied. The products go in a column to the right of the c factors.
I add those products to check against the middle term.
If it doesn’t check, I switch the c factors and try again. If that fails, I 5 other factors for c or a.
What’s nice is that this provides an organized visual method for the students. Once they get the hang of it, they’re getting correct factors on first or second guesses. They’re basically guessing and checking before they write anything down.
Julie, you have already received great ideas. Like others, my algebra 2 students needed to be taught factoring. I started with the X puzzle and box method because that’s what they are familiar with it. I taught grouping (split the middle using the X puzzle) to move them forward. I taught both methods in 90 minutes. On the next day, students created a digital poster comparing the box method to grouping to emphasize that they really aren’t that different. Some students did an excellent job with their analysis.
I don’t teach patterns until I get to the polynomials … I teach difference of squares, difference, sums of cubes … with an emphasis on patterns.
Other than the digital poster, the ‘activity’ was whiteboarding.
I like doing factoring by grouping and a>1 in one step, because it then makes factoring when a = 1 a special case that students look for. The guessing process is then reduced to finding the factors, which is much easier to do and check algorithmically than the process of multiplying out the binomials.
Factoring in Alg 2 is always a headache! I’ve relied on my Algebra 1 teachers for this. I work in a school where all the math teachers work together-from Algebra 1 to Calculus. We try to teach things in a cohesive manner. We have all agreed that factoring by grouping when a>1 is the best method–it’s very useful in PreCal and calculus. I also noticed that after we used grouping for quadratic trinomials, my students were able to factor polynomials a much easier. And it helps them with looking for the GCF (my number one step when we factor is look for a GCF…I have my kids trained to respond “GCF!” when I ask “what’s the first thing I do when factoring?”). Now, here’s hoping they remember how to factor after Christmas break because we’ve got to simplfy rational expressions when we return in January…
Do the students get asked to factorise a cubic which does not have a nice linear factor?
For example x^3 + x^2 +1 = 0
Do they know what to do when the methods fail?
Can they spot cases of 3 linear factors?
(..from (x+a)(x+b)(x+c) = x^3 + (a+b+c)x^2 +(ab+bc+ca)x +abc..)
Do they ever use numerical methods for finding roots ?
Do they realise that the majority of cubics with a=1 and integer coefficients do not have nice roots?
Julie, the above comment was not specifically about your problem but a general inquiry about the whole factorizing business. I wish you a happy new year, and my next comment is about a method I just devised to do the job !
A new method for integer roots of a cubic.
The cubic: x^3 + fx^2 + gx + h
A factor (x + a), corresponds to a root x = -a
The other (quadratic) factor: x^2 + bx + c
The cubic is the product of these two, which is:
x^3 + (b + a)x^2 + (c + ab)x + ac
So f = b + a, g = c + ab, and h = ac
Now it is “obvious” that a is a factor of h, and then c is the other factor.
So to apply a test we must find all the factor pairs of h, not forgetting that if a and c is a factor pair then -a and -c is also a factor pair.
From the equations above we see that b = f – a, and this can be calculated.
So we construct a table for the values a, b, ab, c, c+ab
One row for each factor value of a.
Then if the value of c + ab is equal to the g in the cubic (the x coefficient) then x + a is a factor.
We may find one factor, or all three factors, or none at all
Example x^3 – 2x^2 – 5x + 6 (for which the grouping method doesn’t work)
f = -2, g = -5, h = 6
Table: (dots to preserve spacing)
a, …b, . ab, . c, c+ab
6 .. -8 -48 .. 1 . -47
3 .. -5 -15 .. 2 . -13
2 .. -4 .. -8 .. 3 … -5 —> factor x+2
1 .. -3 .. -3 .. 6 …. 3
-1.. -1 … 1 .. 1 … -5 —> factor x-1
-2 .. 0 … 0 . -2 … -3
-3 .. 1 .. -3 . -3 … -5 —> factor x-3
-6 .. 4. -24 .-6 . -25
If there is only one “integer” factor then the b and c in that line are the coefficients of the remaining quadratic factor.
The arithmetic is so much quicker than when evaluating the cubic directly,
even if using the efficient method ((x + f)*x + g)*x + h
Thanks so much for all of your help!
If you are going to use the area model for multiplying polynomials which I also use – then why not continue the use when your class factors. This idea of area is something they really do understand and they can learn so many things from it. The use of the area model is really factoring by grouping when you write down the steps of what is being done mathematically through the model. I also use the grouping method with all factoring until my students see that when a=1, then they can use this method without writing everything down if they read it backwards… for example x^2 + 3x – 10…factors of 10 that subtract to be 3. Which is the same as the thinking that takes place when factoring by grouping. I would suggest that you do not separate the two days of factoring. If you teach grouping with the additional support of the area model, then there is no need to separate days. Students that can factor with guess and check or patterns will continue to do this but the students that you are trying to reach probably never really “got” factoring and will need the grouping and/or the area model to be successful. When they are ready, they will move to the other methods naturally because they will “see” it and they don’t want to write down all of the steps that are required with other methods. I have found from my experience in teaching both Algebra 2 and College developmental classes that the more methods I introduce that don’t seem to connect the more confusion that I brought into my classroom. Hope this helps.
I have a simple question.
Are we finding factors in order to get practice with algebra or to find the zeros/roots of the polynomial equation?
Completing the square for quadratics is a very mathematically appealing and clever way to go, but what is preventing the development of the quadratic roots formula, it is only one step further?
Apologies, that is two questions !