The overall objective of this skill group is
We use our interactive student notebooks (ISNs) to record our notes and to do examples. All documents are provided at the end of this post. I have them as word documents, if you prefer PDF contact me and I can send them to you in that format.
GCF
When I taught 7th grade I used the "upside down double division" method of finding the GCF (also useful for LCM and simplifying fractions). This is such a great method! As a student I remember being totally confused by the method of writing the prime factorization of each number, etc, etc. This makes a lot more sense and is easy to remember. (this goes on our left side learning page)
Introduction to Factoring
On the same day as we do GCF, we approach the idea of factoring. I focus on how it means to "undistribute". This foldable has them cut out and sort some polynomials - some are factored, some are not factored. First they have to make two piles (factored and not factored) and then glue them in the correct column with the corresponding "other" version of that polynomial across from it. Hard to explain, but check it out here:
(I had students put this opposite the GCF, right side reflect page. But you could do just practice examples in calculating the GCF on the right page and put this on another two page spread. We were getting close to the end of our notebook and I was worried about running out of pages!).
Factoring Trinomials when leading coefficient is 1 (after factoring out GCF)
To warm students up to the thinking they need for this topic I give them this puzzle sheet:
full sheet in WORD format at end of this blog post.
I start factoring by approaching the long way. I want my students to figure out their own shortcut. I use the box method to develop overall strategy (sort of backwards from box & multiplying with a little GCF thrown in). I show my students this video. https://www.youtube.com/watch?v=X7Zedh6AQlI
He goes over three examples and then we do two more together:
- -x2 + 5x + 6 (can’t have leading coeff of -1, must factor) = (-1(x – 6)(x + 1))
- 13x – 30 + x2 needs to be in standard form! = (x + 15)(x – 2)
I tell students that there is a shortcut that they should be thinking about trying to figure out when they do practice examples & their homework.
We do this ISN foldable. I'm not sure how I feel about it. It shows how to factor this long way step by step with each step behind a door. Not sure if I'll use it next year. I was trying to be creative... (this goes on our left side learning page)
Right side reflect page:
actually some of those examples are ones we did together, so here is a better set of examples to use for "right side reflect":
- 3x2 + 3x – 36 3(x + 4)(x – 3)
- -2x2 -48 + 20x -2(x – 6)(x – 4)
- -x2 – 8x – 15 -(x + 3)(x + 5)
- 20x – 2x2 – 48 -2(x – 6)(x – 4)
Factoring Trinomials when leading coefficient is NOT 1 (after factoring out GCF)
I start the next class with a warm up of some factoring problems and while students do those I go around and check homework. I ask each student, did you notice a shortcut to factoring when you did your homework? Most students do and you can see it in their work. I ask them to think about how they might explain it. Then I have students go up to the board and work out the warm up problems using and explaining the short cut. It's pretty cool to see how they pretty much all figure it out. I never have to give them a set of rules to follow, they come up with their own way of thinking about it.
Now we go into factoring trinomials where the leading coefficient is NOT 1, even after factoring out the GCF. The technique we use is the box method used above, It's a tad bit trickier because there is a leading coefficient. It's also called the British Method and you can find a PDF here that explains the method (or search on line for other examples).
For our ISN we do a long multi-step example (that I create with 8.5x14 paper) and then "right side reflect" they have 4 examples to work out.
Special Factoring Patterns
Now we revisit those factoring patterns that we saw earlier as special products. We start with a warm up that has them factoring the "long way" some of these.
o
x2
+ 4x + 4 (x + 2)2
o
x2
– 25 (x – 5)(x + 5)
o
x2
– 8x + 16 (x – 4)2
o
4x2
– 9 (2x – 3)(2x + 3)
o
4x2
+ 4x + 1 (2x + 1)2
o
X2
– 144 (x – 12)(x + 12)
o
4x2
– 12 + 9 (2x – 3) 2
My ISN pages:
"left side learning" is another "sorting" insert where they must decide which each example qualifies as (plus finding some that don't fit either category and why they don't)
"right side reflect" is more explanation but with examples inside the booklet for them to work out.
Summarizing - all together now
Then we have a day to pull it all together. Sometimes this takes two days (I just give them the evens for HW on the first day, odds on the second day)
We start with a warm up
o
12x2
+ 5x – 2 (3x + 2)(4x – 1)
o
-2x2
– 6x + 56 -2(x – 4)(x + 7)
o
6x2
– 11x + 4 (2x – 1)(3x – 5)
o
24x2
+ 34x + 12 2(3x + 2)(4x + 3)
o
36x2
– 49 (6x – 7)(6x + 7)
o
25x2 – 30x + 9
(5x – 3)2
o
36a3b
– 24ab + 60a2b 12a2b
(3a – 2b + 5)
Which actually I think I'm going to use on the "right side reflect" next year and go right into doing the ISN insert which is a summary foldable booklet on all the different types of factoring:
This was actually a huge pain to make! But that was before I had gotten the instruction for making a booklet. I'll do that next year. But will include my old ISN document below. Each page on this is a separate page instead of folding three pieces of paper. (for a quick review of the cool booklet method see my blog post of July 9, 2015 FOLDABLE LOVE)
I use a lot of kuta materials too for HW and practice, both Algebra One and Algebra Two. And we don't be using our old falling apart textbooks this coming school year (there are not any real common core textbooks yet for high school) but I may scan pages from that book to use for HW and post them on my calendar (that's another blog post! coming soon in August 2015)
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