This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.
My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.
You can find my Algebra 1 (year long class) INB posts here:
And my Algebra 2 INB posts here:
And finally, my posts from a 2nd go around I’m teaching of Algebra 1 here:
The 6th standard we teach in Algebra 1 I split into two units. It is A.REI.4:
Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions.
I split it into a unit where I teach the background skills needed for solving quadratics – basically working with polynomials and being able to use vocabulary, add, subtract, and multiply them. Then the second unit contains all the actual methods for solving quadratic equations. This is that second unit.
Skill 1: I can solve a quadratic by factoring
We began with an exploration of factoring using algebra tiles. We discussed the first problem pictured as a class, and then they worked for an entire 47 minute period on the rest. I think allowing students to productively struggle with a just a few problems is really helpful when we discuss the algebraic method of factoring, because they have something to tie it to. I gave them a chance to share and discuss their answers at the end of class.
When we started using the algebraic method of factoring, I decided to give them a chart with factors of every number from 1 to 100. With many of my students lacking a firm grasp on their multiplication facts, factoring can become a miserable process quickly if you’re missing that one factor pair that will give you the correct factoring. I was always taught factoring as a sort of guess-and-check process in high school, and learning about factoring by grouping, and then the Box Method for factoring in college revolutionized my world. The Box Method directly ties into what the students did the day before with the algebra tiles, so it becomes easy for them to understand. The only change I would make is the small print where it says “factors” along the edges of the box, I want to put in instructions for how to get these factors (“what number goes into both numbers in the row/column”? I need to find a way to say this with less words) because that is the thing my students forgot most often.
Also in hindsight, I wish I had included a prime quadratic in their notes, because my students continuously stumbled when they encountered these and thought they had done something wrong instead of remembering that not all quadratics can be factored. I think I will put two examples on the last part of the foldable to add this next time.
When using this method, I created a dry erase template for my students to use, so that they didn’t have to draw boxes every time. It’s included in the files at the end of this post. I just print them and slip them into page protector sleeves, and leave a stack of them on the bookshelf at the back of my classroom!
Skill 2: I can solve a quadratic using inverse operations
I used to teach this method first, but decided to move it after factoring since factoring is more obviously the reverse process of multiplying, which they had just learned. I like this sequence also because using inverse operations leads right into completing the square, since completing the square transforms a quadratic that cannot be solved with inverse operations into one that can. Looking back on these notes, the “is squared somehow” blank is not really necessary, because if nothing is squared, it can still be solved with inverse operations, it just isn’t a quadratic.
The poof book of examples is made using a template from Sarah Carter. My students this year actually hate poof books – the folding is just incomprehensible to them no matter how many times I walk them through it, so I have mostly avoided them since the first half of the year, but it just fit so well for this situation that I made them do it anyways. They were mad. Oh well. I included an example that cannot be solved by inverse operations, but the new thing I did this time that I haven’t done previously is that I had them solve that one by factoring. I tried to incorporate throughout our practice in this unit that students would still need to use previously learned methods, so that they weren’t just practicing each method in isolation. I think this made them more confident when taking the final assessment when they could choose their own method, because they were used to doing a mixture of methods.
Skill 3: I can solve a quadratic by completing the square
Completing the square is not my strong suit, teaching wise. I worked really hard this year to make my teaching more conceptual. I think the exploration I created was a pretty decent introduction to this concept – we had a class discussion about what a square is, then talked through the first page of the exploration and discussed WHY it was useful to create a square. They then spent the rest of the period working through the 4 other examples with their tiles. I had them fill out the table to find their pattern, which was very difficult for a lot of students. I prompted them to think through what they physically did with the x tiles in their exploration, which helped them to describe the pattern. The final challenge was really included for students who breezed through the exploration, and I told my students not to worry about it unless they had extra time, but it was a good challenge for those who reached it.
Once we began doing it algebraically, I tried to tie it back to the tiles as much as possible. I think the format I used to visualize the idea was pretty effective, because my students this year have been the most proficient at completing the square out of all my students! I also created a dry erase template for this, which you can find in the files for this post. I ended up putting these in the same sheet protector sleeves as the factoring one, so that they were back to back and students could just grab a sleeve and flip it to the side for the method they wanted to use!
Skill 4: I can use the discriminant to determine the number of real solutions to a quadratic
This lesson started with me putting the following question up on the board: “What does the word ‘discriminate’ mean?” My students were so confused. “I thought this was math class!” they complained, but we had a good discussion about the word. They came up with definitions that included the words “sort”, “separate” “treat someone differently”. Then I added the word “discriminant”, and told them that we were going to use this number to “sort” quadratics based on what their solutions looked like. I honestly think this little discussion to start class helped them to understand the point of the discriminant really well! We also watched this video to introduce imaginary numbers, and my students responded with great curiosity and interest instead of frustration. My tip is to play this video at 0.75 speed because the girl talks very quickly, and to warn students before you start that she will sound very weird because of this and her accent.
As you can see, I put the options positive, negative, and zero in a stupid order, and I will change those next time to be positive, zero, negative. Otherwise this foldable is very straightforward and does its job in helping students interpret the discriminant. It also makes the Quadratic Formula seem like less work when it gets introduced, since it includes a step they already know.
We closed out class with me deriving the Quadratic Formula through completing the square, which is something my district wants students to have seen. It was a great way to blow the students’ minds before they left for the weekend, but they really tried to follow along as much as they could and were even able to get a few of the steps on their own!
Introduced imaginary numbers to my Algebra 1 students today as we calculated discriminants. They were so inquisitive and I wore the perfect shirt today for them to ask more questions about how they work. #teach180 pic.twitter.com/AQQsu0t3NC
— Liz Mastalio (@MissMastalio) April 13, 2018
Skill 5: I can solve a quadratic using the quadratic formula
I tried to split the process of finding solutions using the quadratic formula up into very clear steps. Yes, to help my students understand, but also because I was gone for two days and knew that my substitute teacher would have to be able to follow and help the students fill in these pages. I like the steps I ended up with, although I would emphasize that -B should be considered as “the opposite of B” since that seems less confusing when B is already negative.
Skill 6: I can select and apply an appropriate method to solve a quadratic
I stressed that “most effective” is not the same as “easiest” when we filled out this flowchart. We discussed what the word effective means, and it turns out many of my students did not actually know, so that was a good conversation to have. We then solved three quadratics, comparing the “most effective” method to another method. We had a lot of good discussions about what methods different students preferred, and how you needed to know how to use either the completing the square method or the Quadratic Formula because sometimes the other two methods don’t work for the quadratic you’re looking at. The thing that was hardest for me was to resist guiding them towards certain methods when they chose “another method”. We solved the first example by completing the square in two classes, which just ends up being ridiculous honestly, but I was emphasizing that THEY COULD CHOOSE the method, so I let them choose. One class chose factoring for a prime quadratic and we had to abandon ship and use a THIRD method to solve it. I think those were good lessons though in paying attention to the structure of the equation before just starting to solve it without thinking.
Files for these pages can be found here, in PDF and Publisher form. Dry Erase templates are also included there.