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.

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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.

Algebra 1 Unit 3

Unit 3 was and introduction to functions and I’m pretty sure I’ve already mentioned this but this was my students’ best unit yet. They rocked (the majority of) it.

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The first day of the unit was a review of the coordinate plane and plotting points. After completing the notes in the interactive notebook, students graphed “October” themed images and colored them. Before passing out the directions for the images, I had students put their heads down and hold up 1, 2, or 3 fingers based on how good they felt about graphing (1 meaning they still needed a teacher’s help and 3 meaning they could do it totally on their own). Students who put up 1 finger got the image that required the fewest number of points, and the students that held up 3 fingers got a really challenging Jack-O-Lantern to graph! Some of them complained about how long it was taking them but they still mostly got finished. The ones that got turned in decorated the walls throughout the month (and of course I just took these down and recycled them this morning without any pictures).

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The next week was all focused on what it meant to be a function and the multiple representations of functions. We didn’t really discuss the idea of being a relation at all and I still don’t know if that’s a good thing or a bad thing. The students simply recognize relations as either “function” or “not a function”.

We discussed what it meant to be a function and what the multiple representations looked like, then I passed out the worksheet. Instead of doing a card sort like last year, we decided to keep it all on one page and have students simply check or highlight the relations that were functions. I gave the students about 5 minutes to work on this independently, and then another 5 minutes to discuss/argue with their group members. Then I called on different students to tell me what they chose and we wrote that on the screen. After we had all of the students’ answers on the screen, I gave students a chance to argue against any of the answers up there. They were able to identify many of the relations correctly! The ones they struggled with the most were the word problems and non-linear equations, which makes total sense.

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The next day we practiced evaluating functions. Many of my students are still struggling to understand the difference between f(3) and f(x) = 3. Some of the explanations myself, my collaborator, and the students came up with were:

If you see a number where the x belongs, then you put it everywhere you see the x.

If there’s no equals sign, you substitute the value in and simplify. If there is an equals sign, you have to set up and solve an equation.

Whatever is inside of the parentheses is the input. Whatever is outside of the parentheses is the output.

Some of these worked for some students, but again there are still quite a few struggling with this idea.

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The next three days were spent creating and comparing multiple representations. There were four examples in the notebook where they were given a verbal representation, equation, table, and graph, respectively, and expected to come up with the other three representations based on that. The one they struggled the most with was verbal! Last year’s students struggled to identify patterns and create equations, but this year there was difficulty in being precise enough with their language to correctly describe the situation.

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The next week of this unit focused on x- and y-intercepts and domain and range. I just discovered this awesome lesson idea for domain and range AFTER I taught it, of course (found it on this awesome resource) but overall I think that domain and range went really well. The students understand the concept behind each, but some are still struggling with algebraically finding x- and y-intercepts and when to use set vs. algebraic notation when describing domain and range. I think the domain and range is partially due to the fact that I got called out midway through lessons two days in a row during this week of instruction… Also, interval notation is an Algebra 2 standard in Virginia, but the other Algebra 1 teachers and myself decided we would at least show it to our students.

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The last day of instruction was spent discussing inverse functions and translating word problems into equations and solving them. We had two half days during this unit otherwise I would have found some sort of fun game or two to play!

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Students then had three days in class to complete a project either with a partner or on their own. It wasn’t the most fun of projects, but it gave them a little independence and required them to use their knowledge and skills to create multiple representations and answer questions about each scenario. If anyone has a really good functions project, let me know and I’ll use it in place of this one next year!

SMART Goals

This week we started our functions unit in Algebra I, and we spent the week discussing what a function is, how we can tell if something is a function, evaluating functions, and translating between and comparing multiple representations of functions. My students overall have ROCKED it. Completely crushed it. I had to find new materials for two days because they were way ahead of last year’s students on this topic.

I made a comment during lunch about this and one of the other teachers mentioned that functions was the 8th grade math SMART goal for last year. It made so much sense! They had focused more on functions last year, so they remembered and understood more than the previous year’s students. I’m hoping that the confidence many of them had from this past week, as well as the discussions about their answers, will continue into this week’s topics of domain and range, intercepts, and inverse functions. Only time will tell.