## Curriculum with a Message word-vomit: VA Algebra 1

In April I went to Jonathan’s NCTM session titled “Curriculum with a Message”. I had been fascinated by his ideas on Algebra 2 for a while but wasn’t fully understanding what the year looked like. I left the session really itching to do this with my next Algebra 1 or Algebra 2 course. I still don’t know what I’m teaching for the upcoming school year, but today I started to look into what this might look like for a Virginia Algebra 1 course (NOTE: we are not a CCSS state).

Before getting started, two of our standards include translating verbal expressions algebraically and representing linear and quadratic functions using multiple representations (verbal, tables, equations, graphs). I plan on doing both of those regularly throughout the course, and including concrete and other pictoral/symbolic representations as needed.

Big Idea #1: Equivalence

Our students are expected to do the following:

• evaluate expressions given replacement sets using absolute value, square roots, and cube roots
• use the laws of exponents
• add, subtract, multiply, and divide polynomials
• factor 1st/2nd degree bi/trinomials in 1 variable
• simplify square roots of whole numbers and monomial expressions
• simplify cube roots of integers
• add, subtract, and multiply 2 monomial radical expressions with numerical radicands

I would add into this part of the year solving multistep equations and inequalities (and literal equations) since we have to teach them to use properties to justify their steps. I don’t know what this big idea would look like but I’m imagining lots of open middle problems. I’m imagining this big idea taking from the beginning of school (day after Labor Day) to around Thanksgiving.

Big Idea #2: Graphically Representing and Understanding Linear and Quadratic Relationships

That’s a long big idea title but it gets the point across. I’m choosing to start with graphs first because I believe that when students have the ability to visually represent things it’s much easier to connect to the algebraic, more abstract versions.

Standards that would belong in this part are:

• slope
• graphing two-variable linear equations and inequalities
• graphing two-variable quadratic equations
• systems of two linear equations or inequalities
• characteristics of relations
• domain/range
• zeros/roots/solutions
• intercepts
• direct variation
• parallel and perpendicular lines

I think mostly with this big idea my goal would be to do a lot of word problems and get those words into a variety of visual representations; drawing, making tables (that then turn into graphs), etc. Those characteristics would be used to help us determine the reasonableness of our problem set-ups and solutions, as well as visually provide answers for word problems (like when the water will run out, when the ball is the highest, etc.). Depending on snow days, this would take until Presidents’ Day-ish.

Big Idea #3: Algebraic Representations of Linear and Quadratic Relationships

The idea here is that we would move from “nice, easy” numbers to realistic numbers, which is why graphs and other representations wouldn’t be our best option anymore.

Topics include all of the exact same things as Big Idea #2, but now we would focus on equations. Two things that would need to be included are:

• parent function transformations of y = x with m and b
• linear and quadratic regression

This would take until a week or so before the standardized test so there would be some time for test prep. The only standard that isn’t included is inverse variation. Such an annoying one to include when you don’t do rational functions. So I’d be sure to touch on that during test prep and then maybe do a mini-unit on inverse, joint, and combined variation after the test?

So basically this course would be three trimesters and each trimester would have a different theme… I’m not going to flesh it out any more until I know I’m teaching Algebra 1 next year, but if anyone has comments/questions/suggestions, I’m all ears!

## 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.

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.

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.

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.

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.

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.

## Zombie Graveyard

I haven’t posted in a couple of weeks due to some traveling but I wanted to make sure that I posted about one of my favorite (and my students favorite) review/practice problem games.

The last day of our unit on Polygons in Geometry was spent on area and perimeter. Students start learning about area and perimeter in elementary school, so most of this is review. The only new thing is the formula for area of a regular polygon. We did two problems using this formula together, and discussed the important of units before beginning the game.

Zombie Graveyard is pretty simple. The goal of the game is to be the last student alive (or the student with the most lives left if you run out of time).

1. Write down each students name on a white board that is easily accessible to the students. Give each student an X to represent each life you want them to have. The number of lives will depend on how long you want the game to last and how many student you have.
2. Students, on individual whiteboards, attempt to find the answers to a problem displayed on the board/screen. I gave my students 45 seconds to complete perimeter problems and 1-2 minutes to complete area problems. This is a faster pace than what I normally play at because this is an advanced-pace course and the nature of the topic was review.
3. Call on students randomly to show you their answers. This year I used the Class Cards app to select 5 random students. Last year I used the random number generator on my calculator and called students by their seat numbers.
4. Students who have gotten the answer correct get to go up to the white board and take away a life from anyone.
5. Once you’ve lost all of your lives, you’re dead. But that’s not the end of the game! Oh no. You have become the undead – a zombie – and zombies, should they get a problem correct when they are randomly selected, now get to take away TWO lives.
6. Keep completing problems until you only have one person left alive!

My students love this game every time we play. I think it’s by far their favorite way to do simple “plug-and-chug” practice problems, and I try to play it once a month with at least one of my classes. It does require lots of student and teacher control, but the students know that the game can end any time it gets too out of hand and we will go back to individual practice problems that are turned in for a grade.

Last year I played this for the first time with simplifying powers in Algebra I. I plan on playing it again later this month for Surface Area and Volume, and am always looking for more concepts that it can be used for!

## 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.

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).

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.

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.

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.

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.

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!

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!

## Algebra 1 Unit 2

Unit 2 goes right from expressions and operations into univariate equations and inequalities. The first day of the unit students refreshed their memories on how to solve one-step equations and the properties of equality and inequality. We tried something new this year as well – including literal equations in with the other equations and inequalities. Last year we spent a couple of days at the end of the unit trying to teach literal equations and many of the students struggled. I still have a lot of students struggling (still VERY open to teaching suggestions here) but it does seem to flow better if you slowly integrate those types of problems into the “regular” equations. During the one- and two-step days, every student was expected to check their answer to every problem by using substitution. After the first few days, we highly encouraged it but did not require it of the students any longer.

The next day of the unit we practiced solving two-step equations. Again, justifying each step using their algebraic properties and checking each solution in the notebook. Then they worked on the maze and the first so many students to correctly finish and show all of their work received a piece of candy. I enjoy activities like this maze (and those partner sheets) because it allows students to self-check their work without directly giving them the correct answer. The only problem with this one is that I suppose if a student had been smart and lazy enough they could have substituted in all of the options until one worked.

We spent two more days on solving one- and two-step problems, but these days were focused on inequalities. Friday we completed notes and practice problems in the interactive notebook and on their desks, and then Monday they did a scavenger hunt activity.

Up until this point the students were really doing great with this unit. They complained a bunch about the properties, but I only had two or three students complain about having to show all of their work. One- and two-step equations and inequalities are also taught in Foundations of Algebra, so for most of the students they had at least seen this topic before, if they didn’t remember how to do it.

The next six days were focused on multi-step problems. We did two pages on the steps to solving multi-step problems in the interactive notebook, and then one page on the different types of solutions possible.

I also did two Mathematics Assessment Project lessons during these two days. One of those lessons was a complete disaster (Building and Solving Linear Equations) while the other one went incredibly smoothly (Solving Linear Equations). My students at this point in the unit did not want to be creating their own equations step-by-step, they simply wanted to be solving them. Trying to get some of them to create their own in that first classroom challenge was one of the worst things we’ve done so far this year! The second one, on the other hand, gave students a chance to practice using their translating skills and ability to solve multi-step equations at the same time. Many students struggled at first but almost every group successfully completed the task by the end of the block.

My collaborative teacher and I needed to do more small-group instruction during this unit and of course that’s only obvious now that I can see the effects of it. We still have some students struggling to solve two-step equations, much less multi-step equations. Also there should have been more problems with fractions, variables on the same side, and multi-step literal equations. Those are things that I will change for next year, and for right now just slip them in where I can.

## Algebra 1 Unit 1

Today is a teacher workday (the first election day since I turned 18 that I’m not voting on… I’m a little disappointed in myself). I’ve gotten the next unit planned out in my interactive notebook, but before I teach and post that I’m going to start posting all of our Algebra I units. I’ve taken most of these ideas from other websites (mostly Math=Love) and will try to keep track and give credit where credit is due in the future.

Unit 1 in our curriculum is basically a review of some of the more important things from the Foundations of Algebra (middle school math) courses they have taken over the past three years. We title ours Expressions and Operations, and it discusses topics from VDOE SOLs A.1, 4, and 5.

Our first day of instruction is the second day of school, and we spent the majority of the time discussing how to set up and use the interactive notebooks. Next year when I do this, however, I’m definitely going to use Sarah Hagan’s new table of contents idea combined with the concept-based grading scale that I already have in place to create the unit dividers. After that we also talked about a few different vocabulary terms. I asked students to define them before we wrote down any definitions and examples to see what they already knew.

We finished the first day by creating a foldable of some key terms for addition, subtraction, multiplication, and division in preparation for the next day’s lesson.

During our next lesson, we practiced translating expressions. We did a few examples together as a class using this Gizmos activity, and then students worked either independently or in their small groups (tables of 3) on the practice problems in their notebook. When we went over the practice problems as a class, we wrote down all possibilities that students were willing to share and discussed which ones worked and which didn’t. Some students seemed confused when there was more than one answer but a lot of students liked arguing for or against certain translations.

The next day we began talking about evaluating expressions. I asked students “Who remembers what the order of operations is?” and about half of the hands went up. I responded with “Good, because for the next 5-10 minutes you don’t need it.” and was met with some puzzled looks. I wrote on the board an expression that involved all parts of the order of operations and told the students to pretend that there was no order of operations. If it didn’t exist, what answers could we get for this problem? Then they worked to come up with as many answers as they could. Some students only came up with one or two, others had six or seven. As a class we looked at about five options, and then discussed why it was important to use the order of operations. Those who had originally said they forgot what it was definitely seemed to remember and understand by the end of the activity.

So then I introduced the “new” order of operations. Many of them had been taught PEMDAS – but we like to call it GEMDAS. The “G” stands for grouping, and it allows us to include more than just parentheses into that first step. We did an example as a class, and then students worked on a practice worksheet for the remainder of the block.

We finished talking specifically about evaluating the next day, where we introduced substitution. We completed a couple of examples in the notebook before students worked on a partner activity. It was one of those partner activities where they each have different problems but if they do them correctly they get the same answers (I love these types of worksheets). After the students completed the partner activity, we started discussing properties.

If anyone has a good way to teach properties, PLEASE let me know. My students groaned about them last year and never really understood how or why to use them. They (so far) have been doing the same thing this year. They are a boring concept that I cannot figure out how to get across to my students!

We spent the next day working on some more evaluating problems, doing pull-out small group instruction for students who were still struggling with the order of operations, and took notes on a couple more properties.

The last day of explicitly teaching properties was also spent reviewing absolute value. This is, again, something that they had seen in middle school so we did a couple of examples together as a class and then did another worksheet. Next year at this point I’d like to play Zombie Graveyard (which I will post about as soon as I DO play it this year!) but we were still working on classroom management and student self-discipline and didn’t get around to it.

The last two days of our unit were spent practicing identifying and combining like terms and justifying simplifying expressions using properties. I’ve tried highlighting and circling and boxing and underlining for combining like terms but I still have some students that do not get it. They don’t comprehend the idea that “a” is not the same thing as “b”, because to them they are both unknowns so they should be the same. I’m hoping that this will change when we get to linear functions soon, but if anyone has a really good way of teaching combining like terms I will use it in a heartbeat!

At this point I was not doing concept-based grading with my Algebra 1 classes, so we took a quiz that included translating and evaluating expressions, and then we had a test that also included properties at the end of the unit.