Roll-A-Solution! (Systems of Equations Activity)

We started off our unit about systems of linear equations this week with a definition and a discussion of what a solution means. We learned how to check to see if a set of values was a solution or not in two ways – manually substituting values in (with parentheses!) or storing values in a TI-83 Plus.

I thought I would bring a little bit of a competitive aspect into their practice for this, since just checking potential solution points over and over is not super fun!

I wrote a set of 16 systems of equations that all had solutions whose x-values and y-values were both between 1 and 6 (aka, able to be rolled on a standard die). Each student received a set of cards with these 16 systems on it, a die, and this template in a dry erase pocket:

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They drew a system of equations card and placed it in its spot, then rolled their die twice for an x and a y value. They then could check their solution however they chose, and I walked around and checked in on them as they worked.

The competition part is that I had them raise their hand anytime they found a solution to a system, and if it was a true solution, that student earned a point!

A few minutes into them working, I announced that if they found a point that wasn’t a solution, but they thought they knew what the solution should be, that they could change their values to that point and check to see. Keep in mind that they haven’t learned anything about how to solve systems yet, just how to check to see if a certain point is a solution or not. This had the result I was hoping it would, that many students started to really reason with what the equations were telling them and trying to logically figure out how to adjust a false solution to find one that would work! One or two students abandoned the dice all together and were trying to just find the solution from the beginning, which was awesome!

Here’s some pictures of my students in action on this activity:

I laughed a lot at how frustrated the students got when they kept not finding solutions. We talked about how small of a chance there was of rolling the correct solution (a 1 in 6 chance for x and for y…).

Students started yelling things out like, “Miss Mastalio, 8 is NOT equal to 17 and I’m really mad about it!”

or

“This point works in the first equation and I’m going to THROW A FIT if it doesn’t work in the second one!”

Overall I had excellent engagement for this activity, and I think a lot of students really got a deep understanding of what these solutions mean.

Here is the file for both the dry erase template and the systems cards, as an editable Publisher file and as a PDF.

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Algebra 1 Unit 2 Interactive Notebooks

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.

Algebra 2 INB Posts:

Unit 1 | Unit 2 | Unit 3

Other Algebra 1 INB Posts:

Unit 1

 

Our second unit in Algebra 1 addresses A.CED.2 for linear functions only:

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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I honestly hate how I broke this unit down into skills. This unit is VERY long, because we have to cover both graphing and then writing linear equations, and there are a lot of components to both of those things. What I WISH I had done, and what I will adjust to do next year, is to split this standard into two “units” for the purposes of their INBs, and done one unit on various things they need to be able to know before working with linear functions (see skills 1, 2, and 5) and one unit on completing the actual standard. Maybe I’ll call it “Pre A.CED.2” and “A.CED.2” 🙂 I also wish that I had added a few skills or adjusted what a few of the skills say, but I’ll get to that as we get into each skill.

You can find more about my index pages in previous INB posts linked above.

Skill 1: I can identify a function and describe its domain and range

This is the first year that I have NOT EVER said ANYTHING about the vertical line test when teaching about functions, and I think it was fairly effective. I really focused on the definition of each x-value being paired with ONLY ONE y-value, and that one idea translates better to graphs, tables, mapping diagrams, and other representations. I liked the extended Frayer diagram and using a highlighter to identify what components of the non-functions made them non-functions.

We used the cards from Sarah Carter’s Function Auction as a card sort for our INBs. I had students sort them on their tables, and then we went through each relation and debated whether it was a function or not, again highlighting components that made non-functions non-functions. I had them choose three functions and three non-functions to glue into their notebooks as examples, and emphasized that they should choose three that they were potentially confused on before we discussed, because those things would probably confuse them again in the future. I also suggested that they choose one table, one graph, and one other representation for each category.

For the domain and range notes, I only wish that I had not put separate graphs for domain and range. When we got to assignments, students struggled with finding both on the same graph and kept asking if they could have another copy of things to do domain and range separately.

The function machine notes come from Sarah Carter – my students this year did a better job with not freaking out at function notation than ever before, even though I also used these notes last year!

Skill 2: I can identify when a relationship is linear

For this skill, after we took the notes on how to identify linear and discrete/continuous situations, I had them complete the card sorts on the first two pages of this Desmos activity I made. (I froze the pacing to the first two pages and then we completed the rest of the activity later as practice) Students then chose 3 examples for each case from the card sort to include in their notebooks. We went over the correct sorting as a class before they did this part, and had discussions about cases where there was disagreement.

They continued struggling with the whole discrete vs. continuous idea throughout the unit – I would welcome if you have better ways to explain that! In the end, I didn’t stress too much about that because it isn’t explicitly assessed, but I do wish that they felt more comfortable with the concept.

Skill 3: I can graph a linear equation written in standard form (Ax + By = C)

This skill went pretty well, although my students struggled to retain this skill throughout the unit, so I obviously didn’t sprinkle in enough standard form practice problems later on! I don’t think I gave them quite enough structured space on the inside of this foldable to find the two intercepts, but otherwise I’m pretty happy with that. The discrete or continuous one they definitely did not have enough room to write – you can see that I barely had enough space and I am organized and know how much space to plan for and don’t have high school boy handwriting…so that one needs some adjusting.

Skill 4: I can graph a linear equation written in slope intercept form (y=mx+b)

Students did well with this one, although I wish that I had gone with my original plan to use my giant bedsheet graph and make students walk out the slope for each of the examples, because I think most of them copied my graphs from the board after we had walked through and so they didn’t physically go through the same process to get the two points as I did and then later struggled with how to create the two points themselves. Some kept wanting to back to the origin to do the slope from, some were reversing the variables in the slope, and a host of other misconceptions. I think making them walk out the graph and seeing their classmates do it would have sealed this in their brains more firmly from the beginning.

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**side note in the graphing section, I wish that I had made a separate foldable portion for horizontal and vertical lines, because even though we watched Slope Dude later and went through them within these examples, they struggled every single time they came across one of those cases.

Skill 5: I can find the slope (rate of change) of a linear function

I actually kind of screamed a little bit, on the inside but also out loud in my classroom, when I came across this slope foldable on teachers pay teachers (for free!). First of all, it gives me a space to incorporate Slope Dude, everyone’s favorite video of all time (I say frequently to groans from my students, but one of them asked if we could watch it again “if they all did well on their tests” as a reward or something, so I think it really is their favorite), and second, it almost looks like the Deathly Hallows symbol when you open it up!!!!! Yes!!!! Harry Potter Slope!

**side note: I just youtube searched “Harry Potter slope” to see if anything existed there and this is probably the funniest thing I’ve ever watched.

I wish that I hadn’t tried to include the extra examples around the outside of it though, because it just got incomprehensibly covered. I want to make a poof book foldable with extra examples, and also want to add in initial value to both those examples and the “slope in a situation” foldable.

Skill 6: I can write an equation in slope-intercept form

This is where it became obvious to me that I should have split this unit somehow. My students already felt like this unit had been going on forever, and were getting overwhelmed. It just needs to be divided into smaller pieces for them somehow.

I split this skill into three sources to write equations from: graphs, tables, and situations/scenarios. The graph notes are from Sarah Carter.

None of these were bad, persay, there isn’t really a particular thing I would point out that I really disliked about how I presented this, but I just need to rethink how I come at this next year. My students said that the pocket notes about creating all representations were “way too much writing”, but I’m honestly not sure of a shorter way to get practice with that content. Again, just some rethinking needs to happen. I welcome suggestions!

Skill 7: I can write an equation in point-slope form

Point-slope form, on the other hand, went much better than I think it ever has before. The consistency of putting the form again at the top of all three foldables helped, I think. I even had a substitute during the day they were practicing using point-slope form from x/y tables and I came back to almost all proficient practice scores! That being said, I could tell this was at the end of a very long unit for them and they were just worn out from trying to keep everything straight. If I adjust the structure of a lot of these sections for next year I think it will go much better.

If you want to use any of these files, follow the links included for those I got from other places, or download PDF’s and editable files of the ones I created here.

Algebra 2 Unit 3 Interactive Notebooks

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.

Other Algebra 2 INB Posts:

Unit 1 | Unit 2

Algebra 1 INB Posts:

Unit 1

 

Unit 3 in Algebra 2 covers F.IF.7 part e (only the exponential and logarithmic functions):

Graph exponential and logarithmic functions, showing intercepts and end behavior.

On the index page, you can see that I broke this standard down into four skills. Writing the equations from the graphs is not explicitly stated in F.IF.7e, but it’s under one of the supporting standards we are using in this unit and it is on our assessments for it.

You can find more info about my index pages in previous posts.IMG_2479

 

Skill 1: I can graph an exponential function and identify its key features

I borrowed resources from Kathryn over at Restructuring Algebra for this – I added a definition for asymptote to her description of exponentials. For the examples, I added listings of the key features for each graph, which got a little cramped, but it got the job done. All of the pink writing you see on the examples foldable I just handwrote before I made copies. I will probably type out a nicer version of this for next year. Here is Kathryn’s post about these resources.

 

Skill 2: I can write an exponential function based on its graph.

Again, I borrowed Kathryn’s resource talking about the y-intercept and the based, except I cut off the bottom row where she talked about the x/y table because our material only really focuses on being able to write an equation starting with a graph. Then I created a poof book from Sarah Carter’s template so we could get some practice in! I think Sarah has a more updated template for poof books somewhere possibly but this is the post I always find when I’m looking for it…

The biggest mistake my students kept making here was dividing the points in the wrong order to get the base. Definitely need to emphasize that more next year – I would probably add something about just recognizing growth or decay so that they could immediately tell if their base should be >1 or <1.

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Skill 3: I can graph a logarithmic function and identify its key features.

This is a new skill for Algebra 2 this year – previously, they had to be able to translate between logarithms and exponentials (which is the next unit I will teach), but only had to deal with graphing exponentials.

We started with a brief introduction to logarithms, since they have not seen them before. In the next unit, we get more into how they work, but a basic overview of what the calculator buttons are doing was good enough here.

The main thing that was a struggle here is that my students didn’t want to make the graphs go out far enough to get that good third point, and then their graphs weren’t good and their key features were a little off. I suppose I could have printed pre labeled coordinate grids for them to graph their assignments on, but they have to be able to label the grids themselves too!

I also did not make them find exact values when an intercept was really large (not on their graph), because we won’t learn to solve logarithmic functions until the next unit. We will revisit this thought briefly then, but for now I just had them note down that it would be a really large number somewhere off one side of their graph.

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Skill 4: I can write a logarithmic function based on its graph.

Another skill that was not previously in our Algebra 2, and that I’ve never had to teach before! I kept it to logarithms that have no scale factor, since that’s all they encounter in their assessments, and used the transformations.

This is a pretty short and sweet standard, since the next standard is about actually calculating things and solving things with exponents and logarithms. This is just focused on translating between equations and graphs, and I think my students did a pretty good job with it! They’re certainly getting better at domain and range!

You can find all the resources from this post HERE, in editable (Publisher or Word) and PDF format.

Opinion Abandonment

This time of year is always a bit stressful for teachers. It’s the time when the newness of the school year has officially worn off, most high schools have ended their first grading period of the year or are getting close to it depending on their structure, and for our district in particular, we barely have a full week of school from mid October until winter break. People think having a short week sounds great until you actually experience high schoolers on a short week of school. It’s like it short circuits their normal human behavior wires in their brains or something, having one less day in a school week. Not always bad behavior, just not their usual behavior.

Regardless, it’s a weird part of the school year. Love, Teach calls it DEVOLSON. Yesterday, one of our district teacher leaders posted this image on her facebook page and I shared it on my twitter.

The response to it tells me that I’m not alone in this feeling at the moment. I process the world best when I have lots of information about everything, and also lots of control over things that I make and do. I like to have lots of data to work from, and to be able to revise and edit and think about things thoroughly without putting them out into the world. (My staff did the Real Colors personality test at professional development last week, can you tell? I am a Green.) When the school schedule is as weird as it is, it’s easy for me to get overwhelmed by a feeling that I’m not able to process things in the way I want to process them. Letting go of things is hard for me.

I saw that yesterday, and then today the vlogbrothers (John and Hank Green, who have been making YouTube videos together since 2007 and of whom I have been a fan for almost as long) posted this video:

First, ignore the Pizza John if you don’t understand. As they say in the video, it’s not something you explain, it’s just something you accept. Second, this video made me laugh after a really weird day that I would say was a bad day except nothing really that bad happened I just felt bad about it.

Third, opinion abandonment. I think this is a thing that I desperately need to adopt. We have too many opinions! In fact, I pride myself on being very opinionated about things and being able to have a vaguely informed opinion about almost anything. But I think the thought that I don’t need to have an opinion about everything might just be super freeing. If I care so much about every little thing that happens, if I have to formulate an opinion about ALL OF IT, then no wonder I feel stressed and overwhelmed and want to detach a bit.

So I’m going to start abandoning some opinions.

I will no longer have an opinion on other teacher’s grading policies in my building.

I will no longer have an opinion about Stranger Things.

 

Phew, I feel lighter already!

(shoutout to Tedi for having an opinion on what I should abandon my opinions about!)