I haven't taught Algebra 1 in a few years. But I was motivated to write something down in response to this thread, and in particular, Benjamin Dickman's tweet:
In particular, this.
"but I'm confident there's a way to broach factoring trinomials consonant with [e.g. youcube's description of] number sense."
I think I had a way to broach factoring trinomials that promotes number sense.
Consider this in the context of teaching kids how to multiply binomials and trinomials and such. I would emphasize the distributive property in conjunction with the area model for students who needed a bit of structure or organization. I also made a conscious effort to make connections to multiplying integers.
I also would introduce students to dividing a trinomial by a binomial. This was a natural extension to what we were doing, and the question "Can we only multiply, or can we divide, too?" seemed to always come up.
This probably took the better part of three to four weeks (so about 15 instruction days.)
Every day, in anticipation of introducing the idea of factoring after working on the idea of multiplying and dividing polynomials, I would provide my students with a short number puzzle or two as a starter problem or bell ringer or whatever you would call it. The puzzles would be along the lines of something like this:
Find two numbers whose product is 18 and whose sum is 11.
This is the same question students ask themselves when they are asked to factor $x^2+11x+18$.
I would often provide two puzzles that would complement each other. Possible second puzzles to accompany the one one above might be:
Find two numbers whose product is 18 and whose sum is -9.
or
Find two numbers whose product is -18 and whose sum is 7.
Again, these are the questions kids would ask themselves when factoring $x^2-9x+18$ and $x^2+7x-18$, respectively.
When I would introduce students to the patterns arising from multiplying squaring binomials my number puzzle would be:,
Find two numbers whose product is 25 and whose sum is 10.
When I would introduce students to the patterns arising from multiplying conjugate binomials such as $x^2-5$ and $x^2+5$, my number puzzle would be:
Find two numbers whose product is -25 and whose sum is 0.
Needless to say, the kids became incredibly proficient in solving these number puzzles and creating their own for others to try. Now is the time to throw them a challenge:
Find four numbers $a, b, c,$ and $d$ where $ac=3$ and $bd=5$ and $ad+bc=16$.
This might be the question kids would ask themselves when factoring $3x^2+16x+5=(3x+1)(x+5)$
One day, the kids would come in, and their starter puzzle would be:
Two binomials were multiplied and the resulting product was $x^2+8x+12$. What are the two binomials that were multiplied together?
So after a month of number puzzles each day to start class, factoring became just another number puzzle. As the solution to the puzzle would percolate through the room, the realization was that this is just another form of the number puzzles we had been doing. I would spend less than two weeks on factoring, because the kids had been factoring through number puzzles for a month by then.
