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1

While it's pretty clear from your question what method you're talking about, the title isn't so clear. When I hear factoring by grouping, I think of something like $3x^2+2x+9x+6$, and not $3x^2+11x+6$. When the student has to first come up with a way to split the middle term I've heard it called, not surprisingly, factoring by "splitting the middle term&...

3

This is a thought provoking question. Learning to think algebraically is a gradual process. It takes practice to see that $$ax+ay=a(x+y)$$ is the same as $$(x+1)x+(x+1)y=(x+1)(x+y).$$ Eventually, some students will have enough algebraic fluency to work with something like $f(x+h)$ where $f(x)=x^2-x.$ Such a successful student will have developed fluency ...

4

That is a great question that I struggled with the one year I taught Algebra 1. I came to the same conclusion that factoring by grouping problems are inauthentic because they only work when the expression is built to work. The story that got me through it is that you can use factoring by grouping to factor a quadratic expression if you manage to come up ...

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