Systems of Equations: Take 2

We started systems of linear equations on Monday in AFDA and things could not be going better! Now don’t get me wrong, students are still struggling. But they are making sense of word problems, using different methods to solve problems, getting better at using technology, and learning how to ask for assistance. It’s great.

Monday was our first day but we didn’t actually do anything involving systems. We spent the beginning of class doing our warm-up and going over the quiz and midterm that had been taken last week. The rest of the block we worked on numberless word problems. I got this idea from Julia Finneyfrock and it was exactly what my group of students needed because they did exactly what was expected – when they didn’t think about the problem, they just added the numbers up! So we worked on reading comprehension, drew pictures, acted things out, made estimates, wrote different equations on the board… we thought. And they thought they were easy, but I know this time was well spent.

Tuesday the students started off with guess and check word problems. All I did was give each student a copy of these problems, read the problems aloud one at a time, and encouraged them while they worked. We talked strategy, used technology, students collaborated, and the pride they displayed when they finally worked out the answer was palpable.

We worked through examples using each method by writing and solving more systems using word problems on Tuesday and Wednesday as well. The class did the majority of these problems together and some students remembered some of these methods from Algebra 1 but most did (do) not. The students were then grouped into 2-4 each and each group was assigned 1-2 word problems from the matching portion of the packet linked. Each group had to identify their match, get it peer checked, and then show their work for solving and explain what the answers meant in the context of the problem.

Today, the students again worked in groups but today was more rote practice. Groups of 3 worked together to find the solution to different systems, with each member using a different method and making sure they all got the same (correct) answer. I found my worksheet on TeachersPayTeachers but you could really use any worksheet and just assign students different methods for each problem I suppose. I  don’t know whether or not the purpose of this got through to all of my students but I know it did for some; I worked through a substitution problem involving fractions with a girl and she said “All of that just for (7, 0)? Isn’t there an easier way?”

You might have noticed that missing within all of this are the different types of solutions to systems of equations. Not to worry! We’ll be addressing that on Monday with one of my favorite MAP activities: Classifying Solutions to Systems of Equations. The students will take their first quiz of the unit after that, so I guess that will be the real test. But the conversations that we and they have been having are giving me really good vibes about this unit!

Algebra 1 Unit 5

Unit 5 was our second of three units in our curriculum centered around linear functions. Students had spent all of November focusing on graphing and slope-intercept form, and now it was time to get more algebraic since the foundation had been laid. image1 (4)

 

We basically went through topics in this unit in the same order as the last unit and then added more to the end. We started off with slope, this time introducing the slope formula and how to calculate slope from coordinates, tables, and word problems.

Many of my students seemed to prefer this method for a change, because it meant they didn’t have to graph anything and they didn’t have to remember “rise over run” and they didn’t have to remember when their rise or run was positive or negative. We didn’t spend as much time on slope during this unit because we had already spent quite a few days on it during the last unit.

After slope we went right into direct variation, skipping parent functions this time. We focused more on how to find the constant of variation and how to solve word problems involving direct variation during this unit.

This was too rushed. I would have spent at least one more day working on direct variation problems. Thankfully we should have time to review these types of problems when we discuss inverse variation later this year.

I taught point-slope form for the first time. Last year we had only done slope-intercept, and we are still not doing standard form. I again don’t think that we did enough practice with this. A lot of my students are struggling to remember the formula and what is supposed to go where.

The divide in my students between those who liked point-slope and those who liked slope-intercept was very interesting. A lot of students ALWAYS wanted to solve for slope-intercept form. I tried to keep telling them that they didn’t have to do that but they didn’t seem to care. Maybe they had caught such a focus on solving for y that it was stuck in their heads. I’m not entirely sure.

After teaching point-slope form is where things started to differ from the previous unit. We did not do linear inequalities in this unit, since they are almost always represented in either standard or slope-intercept form. Instead of working with linear inequalities, we introduced a few new concepts.

We spent about a day talking about VUXHOY. Most of my students do remember this (YAY!) and are pretty good now about being able to graph and write equations that just have an x or a y.

I tried to come up with some good “real-life” situations that would have zero rate of change or an undefined rate of change but I was terrible at it and I didn’t get much back from Google. If anyone has some good scenarios for this let me know!

We worked with parallel and perpendicular lines for the next two days. During the first day, students completed an inquiry worksheet where they graphed pairs of lines and were asked to notice and wonder about the lines. Then on the second day we completed the notes and did a few more practice problems.

They seem to understand parallel pretty well but are definitely still struggling with perpendicular. Thankfully this is technically a geometry standard here so this is just to give them some prior knowledge going into next year.

The last topic of this unit was line of best fit. Again, this was very rushed as we were heading into winter break. I would’ve liked at least one more day on this topic to do some sort of in-class activity, but the students still seemed  to grasp this concept pretty well.

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I think that next year I would hold off on ever saying the words “direct variation” until this second unit on linear functions, because there was little to no connection made by my students between the direct variation problems we did in unit 4 and this unit.

One thing that (so far) I’m really liking is that we did not go directly from this unit into systems of linear functions. Especially because we are on a block schedule (and I see these students for 85 minutes every day all year) it is very easy for them to get burnt out on topics. Linear functions for three straight months would have been terribly long I think, no matter how many activities or projects we would have tried to incorporate. This will also give them a chance to review their linear functions later in the year before we begin prepping for their state test.

Overall I really enjoyed the flow of these two units and I truly believe that it led to a better foundation and conceptual understanding of linear functions for my students. Next year though there will have to be significantly more practice on the skills required to really master these topics.

Algebra 1 Units 4

I’m super behind on this so this might be a long post but here it goes.

During our curriculum meeting at the end of last year, a teacher from another high school in the district introduced us to how he likes to teach linear functions. It was quite different than how we had done linear functions that year and how I had seen linear functions set up in various textbooks or online resources. We all agreed that this year we would adopt that flow and see how we liked it.

I loved it. I really think that intuitively it makes significantly more sense. There are some things that I would still rearrange a little, but overall these two units had a good flow. Unit 4 focused on graphs of linear functions, and Unit 5 focused on other representations of linear functions.

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The first topic was slope. I don’t think that we, as a district, focus enough on the idea that slope is a rate of change and I unfortunately didn’t realize that until after we had gotten through these two units. That’s definitely something I want to work on next year.

On the first day of slope we did a lot of discovery. I would give them two or three coordinates, and ask students to find the next coordinate in the pattern. Some students did better than others but I definitely think it helped them start thinking about “rise over run” before it was ever mentioned. The students then also worked with partners on a Gizmos activity involving slope.

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On our first day of notes, we started discussing the fact that while it was possible to find the rate of change from parabolas and other curves, we would focus on linear functions during Algebra 1. We discussed “rise over run” and how it tells us how much the x- and y-variables are changing throughout whatever problem we’re working on. Then we began discussing how to identify positive, negative, zero, and undefined slopes and drew out a slope roller coaster as well as our names.

 

We spent a day practicing how to find slope from a graph and how to identify intercepts from a graph as well.

After we had spent 3 days on slope, we began discussing linear functions a little more in depth. We started by doing multiple representations of a parent function story problem.

We had already practiced doing multiple representations during the previous unit, but now we were able to really discuss what the slope and intercept meant in the context of a situation. We also talked about what it meant to be a solution to a linear function in the context of a situation.

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Direct variation came next. This was meant to give us a good opportunity to talk about parent function transformations, since now our intercept is staying the same but we’re going to have different slopes for different situations.

The students did very well with this. We did some more practice problems on direct variation situations the next day, where I projected one of the four types of representation onto the board and they had to come up with the other three. This lead to some pretty creative verbal scenarios involving everything from puppies to aliens to stealing money… Toward the end of this lesson during the last class of the day, one of the students really understood where this was all going.

“But Miss Roberts, puppies don’t weigh zero pounds when they’re born.”

YES. I love when students do my set ups for me! So we took the exact same situation and discussed how all four of the representations would change if a puppy weighed 2 pounds when it was born. What happens to the table? The graph? The equation? We even were able to discuss how not everything is linear forever. Does a puppy continue to gain a pound every week, even once they’re an older dog? What would happen if we all continued to gain weight at the same rate that we did when we were younger? It was a really amazing conversation. I never said the words “parent function transformation” and yet the majority of the class conceptually understood and could analyze various situations that involved them.

This then obviously led really well into learning about slope-intercept form, so for the next few days we practiced how we can use slope-intercept form with equations and inequalities.

I didn’t force my students to do enough practice with this. Almost all of them still struggle to remember y=mx+b, what the m and b stand for, and/or the differences between the inequality graphs. I think that conceptually this unit flowed incredibly well, and that I was more concerned with that implementation than I was with how much they could do during this unit – something that I’ll have to change for next year.