Algebra 1 Unit 7 Interactive Notebook: Solving Quadratics

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:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6

And finally, my posts from a 2nd go around I’m teaching of Algebra 1 here:

Unit 1 | Unit 2 | Unit 3 | Unit 4

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.

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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

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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

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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!

Skill 5: I can solve a quadratic using the quadratic formula

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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.

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Algebra 1 Unit 4 Interactive Notebooks [Revised]: Exponential Functions

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.

I normally teach year-long courses (47 minute classes per day). However, due to a lot of district specific things involving SPED students, a new course introduction, and graduation requirements, this year I taught a block course (94 minute classes per day) of Foundations of Algebra first semester, which covered skill gaps students would need to find success in Algebra 1. This semester, I am teaching (most of) those same students Algebra 1 on a block schedule. This means I get to start Algebra 1 from the beginning in the same year and revise my activities and INB pages!

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6

And finally, my posts from this second round of Algebra 1 here:

Unit 1 | Unit 2 | Unit 3

 

The fourth standard we cover in Algebra 1 is A.CED.2 (again, but the first time we cover it with linear functions and this time we discuss exponential functions):

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

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Skill 1: I can identify when a relationship is exponential

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As mentioned in my post from my first time teaching this unit this year, my students put everything in the wrong place with a blank inside to this foldable, so I added some typed pieces on the inside of “ex” and “look for” so they would know where to put information and examples. We used the same card sort as I did the last time, but I had them work with a partner. After they finished the sort for the second time after we discussed the definitions, they split up the cards so that each of them had 1 graph, 1 equation, and 1 table for both linear and exponential functions, which they glued in for examples. This meant I needed less copies, and also the examples fit on their page more nicely.

Skill 2: I can graph an exponential function

I edited the instructional page on this to include tips for some of the things my yearlong Algebra 1 students constantly forgot, and I think it was a nice reference for this group of students to look at. I also changed one of the original functions I had used, because I didn’t realize it had a negative multiplier when I was using my snipping tool to get examples. We just moved the parentheses when we did it last time so that it would work out, but I just chose a different problem this time. I still think the linear example on here is the most important, because my students tend to just do the thing we just learned by rote and not actually think about it, and this is a good reminder that you NEED to think about it, in case something doesn’t fit the thing you JUST learned.

Skill 3: I can write an equation to represent an exponential function

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I didn’t change anything on this page from the last edition – my students really excelled at this skill and I think there is an example for most problems they would encounter, like when the pattern is dividing or when the table is actually linear. This group of students struggles a lot more with their math facts, so we had to discuss a lot more how to figure out what the y-values were being multiplied by, since these students couldn’t just look at it and see the number.

Skill 4: I can write an equation to represent exponential growth or decay

I realized the last time I did this unit that this skill needed a lot more scaffolding. I put more examples in and put typed instructions for finding the rate/multiplier, since many of my students forgot to write the instructions last time and then were stuck. On the graph page, I included an x/y table, which was a small step that went a long way in my students remembering how to graph once they had written an equation. I like this iteration of these notes.

We also did the World Population Project again, but this time I did it before they took their test, and I think it helped them get a lot more practice before taking the test. Most of my students wrote all correct equations for the exponential growth and decay test questions!

 

You can find all of the files for these notes here, in PDF and Publisher form.

Algebra 1 Unit 6 Interactive Notebooks: Working with Polynomials

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:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6

And finally, my posts from a 2nd go around I’m teaching of Algebra 1 here:

Unit 1 | Unit 2 | Unit 3

 

I am behind, friends. I have two new posts due for the 2nd go around of Algebra 1, this one and the other unit that goes with this standard, and one for Algebra 2 that need to be written. Fourth quarter is happening, everyone.

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.

So, the working with polynomials part:

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Skill 1: I can classify polynomials by their degree and number of terms

 

This should look familiar from our Algebra 2 polynomial unit. The terms page comes from Sarah Carter, and she also uses a Frayer model. You can find her post here. Students are always annoyed by all the writing here, but then they appreciate it later when they have this reference to use when they don’t understand the words in a question or need to classify their answer. One of my class periods really enjoyed trying to guess the terms before we wrote them using their knowledge of language, which was really fun! I realized there are a lot of different prefixes that mean “two” and “one”. We also had fun brainstorming other words that start with “mono”, “bi” or “tri”.

Skill 2: I can add and subtract polynomials

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This page is also the same one I use in Algebra 2. I find it works out pretty well for both groups, working more as a review for Algebra 2 and an introduction for Algebra 1.

Skill 3: I can multiply polynomials

 

This is the first time my students have worked with algebra tiles, and this exploration becomes pretty important because we then do explorations with algebra tiles for both factoring and completing the square. It’s good to let the students struggle with this for a bit and to go around and individually discuss with them whether their answers make sense or not. Even though many of my students profess to hate the tiles, they also admit that it makes the algebraic method seem much easier and that it makes more sense when they’ve done the tiles first.

I also really liked the side by side comparison I did with the tiles vs the box method in the second page, because it helped the students who DID really like using the tiles see why the tiles would not always be helpful and that there were faster, more efficient ways to do it. I think that this year I have really gotten better in relating our work with algebra tiles to the actual algebraic method, and it’s actually improved my own understanding of the algebra, so I think that’s a win!

 

The only thing I want to change about this unit in the future is in the practice, not the INB pages. When we took a quiz over this, my students pretty much across the board either added/subtracted every problem, or multiplied every problem, without paying attention to what they were actually being asked to do. I think adding more practice where addition, subtraction, and multiplication were mixed would clear this up pretty quickly.

 

You can find the files for these pages here, in PDF and Publisher forms.

Focus (Lent 2018)

Lent and the Easter season are my favorite times of the year. I love the preparation time of Lent and participating in the prayer, fasting and almsgiving to prepare ourselves to rejoice at the resurrection of Jesus on Easter. It’s like New Year’s, only more meaningful for me. A chance to reform yourself, to start something new.

However, every year when Lent rolls around, I struggle with what to “give up”. I really dislike the superficiality of the “sacrifices” many people around me make for Lent, and I want my Lenten sacrifice to be really meaningful for me. When I was younger, I gave up things like chewing gum, chocolate, or drinking pop. I gave up pop for several years, which actually kickstarted my vow to not keep any pop in my home anymore, which has transformed into me currently being over a year pop-free entirely! So these more superficial things definitely can have an effect on your life and making it richer and healthier and better. But this year, I couldn’t think of anything to give up that would really affect me that much.

My first thought was to give up superfluous spending. Then I realized how stressed out about money I’ve already been this year and realized that putting more monetary restrictions on myself would not be beneficial to my mental health. I thought of fast food, which I’ve also done in the past, then realized that the only times I eat fast food anymore are when I’m traveling or occasionally when I have a really awful day. I just wasn’t coming up with anything.

Then I thought of the years where I’ve ADDED something, and immediately knew I was on the right track. My facebook “on this day” had started giving me memories of the year I posted a Bible verse as my status every day of Lent, and I remembered the time I did Liturgy of the Hours every day as well. My church going habits were quite strong when I was in college, but dropped off a lot when I started teaching for many reasons. I want to get back into being involved with my church, which is something I realized during the meditation and mindfulness course my staff did for 8 weeks recently. When I am more involved in Church, I feel calmer, more peaceful, and get all those health benefits you get from meditation or mindfulness practices.

I have had a habit for a few years now of listening to Christian music every Sunday on my drive to and from church, to set the tone. I decided that for Lent this year, I would extend that. Any time I’m in my car, only Christian music. Nothing else. Can’t change the station.

I was not expecting it to be as impactful as it was. Forcing myself to listen to the talking parts and not station flip constantly, listening to the positive messaging of the songs, I found myself with so much more focus as I was driving throughout Lent. It started to spread beyond the car, too, as I found myself humming the songs when I was preparing lessons, listening to them while I worked on grad school assignments, singing them to myself as I got ready for bed. I could focus on other things because this one decision was always made for me, and I didn’t have to form opinions on every song that came on or decide if I wanted to hear it. This definitely became a meditation practice for me, and changed my temperament drastically through the season of Lent. In general, I’ve felt calmer, and even when I’ve gotten stressed, I’ve been able to come  back to my normal mood more quickly.

Easter came and went last weekend, and I got in my car on Monday…and I didn’t want to change the radio station! It felt weird. I’ve changed it a few times since, but I keep gravitating back towards the Christian station.

I was surprised that such a small change had such a big impact on my life. I’ve been able to focus on thinking about other things in my life while I’m in the car, and like I said, the meditative/mindful health benefits have extended into the rest of my days. I think considering our actions more mindfully and choosing to make one small change could really reinvigorate us for the rest of the school year!

Is there a location or a part of your day that you could make a small change in? Maybe you’ll decide to end or start your day with a different routine, or pick the car as your place for change also…leave me your ideas in a comment!

I’ll leave you with my favorite song that I’ve kept coming back to again and again…several times during Lent I found myself sitting in my car in my garage or in a parking lot, listening to the end of this one and taking the time to be thankful and peaceful – at least for the duration of the song.