A few of you have asked for more explanation on using Diamond Problems and/or Bottom's Up for factoring quadratics. I figured showing you would be easier than writing about it. So here you go:
If you have any other questions, let me know and I'll be glad to see what I can do to help out.
If you have any other questions, let me know and I'll be glad to see what I can do to help out.
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9 comments:
I figured out a really sweet way to teach factoring this year. The best thing is that it's not just an algorithm (which is what I was doing before) but instead makes it clear why it works and what's happening. I start by teaching binomial multiplication in a couple ways, including the area model. Then when it's time to factor we use diamond problems to break up the middle term, and plug them into the area model, and then find the common factor of each row and column. A much better explanation is at http://mclaypool.ravenshield.com/activities/Factoring_Trinomials.pdf :)
Hi Meg
I used to teach factoring with a "non 1" coefficient that way. CPM called it "extended factoring." However, I still saw it as an algorithm. Granted, it makes the students find common factors to determine demensions, but my students still saw it as a "magical way to find an answer." I probably should show that method as well since it allows for the area model which helps some kids.
A colleague of mine teaches factoring in a similar method. However, instead of the "bottoms up" method, she still considers it a part of what she calls the "factor X." After you have your diamond filled out, she would use factoring by grouping (which is as worthwhile as any other grouping method) to finish it off. With your second example, after the diamond it would look like
2x^+6x+1x+3
2x(x+3)+1(x+3)
(2x+1)(x+3).
Pretty slick, I've never seen the "bottom's up" portion. I always teach the last step using factoring by grouping or the generic rectangle method. But, I'm all for showing kids tons of ways and letting them pick what makes the most sense to them individually.
Also, I can't recommend the diamond problems highly enough. They are a fantastic way to get kids to start thinking about the process early on. I start using diamond probs the first month of school. I start with diamonds where the sides are filled in and they have the find the product and sum. We then migrate to Diamonds where the product and sum are given. By the time we get to factoring, they think it's easy!
First time visiting your blog. I found it by doing a search for diamond problems - imagine that.
The comments here seem to be friendly and helpful. I hope my questions come across as someone seeking information/opinion. I'm trying to understand the "newer" ways of thinking.
I am wondering if there anyone left who teaches binomial multiplication using the distributive property (ie - FOIL without using the acronym)? Or who teaches factoring as finding factor pairs of the 1st and 3rd terms which produce two terms (the OI of FOIL, if you will) that combine to equal the middle term?
I've used CPM for 3 years now after teaching for 21 years with Merrill and later the Dolciani series. It seems to me that CPM's factoring method is simply a mindless algorithm. One of my goals next year is to try to get my students to fully understand not only how to use it but also why it works. Any tips?
Another thing I don't like aobut the diamond method is that it takes a long, long time. Factoring by listing/choosing pairs of the 1st and 3rd terms often gives a correct answer quickly, and reinforces that factoring is simply "undoing" multiplication (dividing).
Anyway, I'm looking forward to reading more posts. Thanks for the effort!
jmathteach
I taught CPM algebra 1 for 9 years and really appreciated the problem solving but, yeah, the factoring part seemed a bit mindless. However, I've found that this one algorithm is something I'm willing to concede. I really want to look at how I can better set the table for myself so teaching factoring a more intuitive way will be more productive.
Is there any way you can model this method with algebra tiles?
I think I may have accidentally discovered a problem that does not work with this method. The problem is shown by Brain McLogan in this video. I just thought it would be important to tell others that this method may not always work. The video I am referring to is here: https://www.youtube.com/watch?v=57kqw5DNevE. Also, this is the exact method my Algebra 1 teacher taught us and I always thought it would work until I came across Brian's video and got absolutely stumped.
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