My 8th graders really got factoring this year! Because of all of the time lost due to snow days, I was attempting to compact the curriculum, and decided to teach factoring and solving quadratics at the same time (instead of factoring first, and then solving afterwards). We also used Desmos to graph the equations as we went. As a result, students appreciated the point of factoring as a tool to find solutions and graph quadratics.
If you’ve read any of my previous blog posts, you will know I am a huge fan of foldables. I really believe these teach students how to organize and summarize notes, and most importantly, then refer BACK to their notes as a resource. So many 8th graders will dutifully take notes and then never look at them again, but I find they will refer to their foldables because they are organized and contain the most important topics.
For factoring we created a “waterfall” foldable using three sheets of paper. Stack the three sheets of paper and stagger them so there is about a half-inch band of color before the next paper starts. Fold the top three sheets down and crease (the middle section will have the same color twice). Staple across the top (make sure you get all three pieces of paper in the stapling). We wrote Factoring on the front, then labeled the tabs at the bottom, “GCF, 2 terms, 3 terms x^2+bx+c, 3 terms ax^2+bx+c and 4 terms.
We reviewed GCF’s last week and put these notes in the foldable on the GCF tab. I chose to skip over the 2 terms tab at this point and came back to that after factoring trinomials. We did finish the foldable (pictures at the bottom), but what made this lesson so effective this year was integrating the solving and graphing as I taught factoring.
We started with the x-puzzles that @jreulbach talks about in this post. Students love these – it usually doesn’t take long for them to get the hang of them, and soon someone says, “well these are fun, but what does it have to do with Algebra?” Yaaas.
I told students we were going to use their awesome puzzle solving skills to learn how to “undo” multiplying binomials. They multiplied (x+3)(x+4), and then we talked about how we could possibly go backwards from x^2 +7x + 12 to find (x+3)(x+4). They brainstormed a bit and then I drew the x for an x-factor and put the 7 and 12 into it. Immediately there were shouts of “positive 3 and positive 4!!” We did a few more examples, working some with negative numbers. Then we talked about what y=(x+3)(x+4) and y=x^2+7x+12 would look like if graphed. The majority of students believed they would be the same, so we graphed them in Desmos and talked about how graphing this way is a good way to check their factoring.
Next, I projected two graphs and we did a little, “what do you notice, what do you wonder?” This was followed by a great discussion about factors versus solutions, and the relationship between the two. From there we talked about the zero product property, and in one block period students had factored and solved a quadratic, and seen the relationship between factors and intercepts.
For homework they practiced their factoring skills, and the next day we expanded factoring to include quadratics with a leading coefficient greater than 1. Anyone who has taught Algebra knows this is fairly challenging for some students. Our district strongly encourages factoring by grouping as the primary method, so students took notes on how to split the middle term, and then factor out the GCF of both groups. Again, we graphed the expanded quadratic and the factored quadratic in Desmos to ensure we had factored correctly. Some students in each class noticed that the intercepts were not necessarily integers, which was a lovely tie-in to taking each of the factors, setting them equal to zero and solving for x. Here are pics of the two pages on trinomials we added to the foldable:
By the next day, I had a few students coming in just grinning that their older siblings had shown them “a much easier” way to factor. This is always so cute – I just love that A) there is a discussion about MATH going on at home (win! win!), and B) that students are excited to think they’ve outsmarted me (again, win! win!). At this point we talk about the Divide and Slide method. I personally do not care which method they use, but am just happy they have found a method they can rally behind and not get to high school pretending they’ve never even heard of factoring.
Here are the final two pages for the foldable. We went back and revisited factoring 4 terms (they already knew how to do this because of factoring by grouping), but we also talked about how a cubic function can have 3 intercepts, and how they need to look for an x^2 term to potentially factor as a difference of two perfect squares (I include an example of this on the top of that tab). Finally, we finished with the 2 term tab:
The proof will come when we come back from Spring Break on Monday when I see if they remember how to factor after a week of not doing ANY math!