This is part of my Unit 1 Interactive Notebook for Algebra 1. You can view the rest of it here.
1.2 – Properties of Real Numbers
Properties of real numbers. What a snooze fest, am I right? WRONG!
This set of notes turns that notion upside down and gets students involved in creating their own examples to demonstrate each property to make a lively and memorable lesson.
You might wonder, “why would I want to teach properties of real numbers?” It’s a simple way to help students start thinking algebraically and flexibly, especially after having a long summer off from math. These properties demonstrate nice rules for manipulating and moving numbers and variables around in expressions and will help them understand why certain moves are “legal” when we start solving equations. Helping students think flexibly and understand the structure of an expression is a huge step in their journey toward mathematical fluency.
This is part of my Unit 1 Interactive Notebook for Algebra 1. You can view the rest of it here.
1.1 – The Real Number System, Classifying Real Numbers & Closure
There’s nothing trickier than picking the very first topic to cover in all of Algebra 1. So many options, so what goes first?
To start the very first unit of Algebra 1 off, I begin by talking about the real number system and how we classify numbers.
Why oh why, is this where I start, you might ask?
Instilling this vocabulary is very important in helping students be able to hold fluent conversations about math. I can’t tell you how many students I’ve had in Algebra 2 (or Pre-Calc!) over the years that have asked “what’s an integer?” and they are unable to complete a problem that they otherwise would know how to do solely because they lack the basic vocabulary and don’t understand what the question is asking.
This is silly, and I want to prevent these things from happening as much as possible. Math really is its own language, and helping students learn it will allow them to be more confident and much more successful in the long run.
Starting the year off right is SO important for any class, but especially in Algebra 1, in particular. Everything that is done in the first unit lays the foundation for everything to come throughout the rest of the year, so there is is a lot riding on starting the year strong.
Here’s what to include in your first unit of Algebra 1 to start the year off right…
Students NEED to have a strong foundation, or else they’ll be fighting an uphill battle all year, which is no good. I’ve spent a lot of time thinking about what topics are most important for students to know (from vocabulary to skills), so that each following unit has a strong foundation. Here are all of the notes I used with my students during the 1st unit of Algebra 1.
If you want to look inside any of the pages included in this unit, you can take a look at these topic-specific posts for a more detailed look!
Getting started with interactive notebooks? Here’s what to do!
The logistics of starting an interactive notebook is one of the most dauting barriers for math teachers. Here’s a set of simple 3 steps to take to get started!
1. Number the pages (If nothing else, do this!)
You will want to be on the same page as your students, quite literally, when using interactive notebooks. Being on the same page allows you so many opportunities to promote study skills and refer students back to a particular topic to help them get “un-stuck.” Promoting study skills is one of the biggest benefits of interactive notebooks so don’t skip this step.
WORD OF CAUTION: Make sure to number the pages as the very first thing that you do! Interactive notebooks get very thick over the course of a semester, and it will be difficult to write in the corners by the time the notebook is almost complete.
If you want more tips on choosing a notebook, read this post about What notebook should I use for an Algebra 1 Interactive Notebook?
If you’re just getting started with using interactive notebooks with your Algebra 1 class, you might have a few questions about what type of notebook you should use, how many pages, and how to make sure all of your students have one to use. In this post, I’ll try to address them all. If I leave something out, leave me a comment so I can answer your question!
What type of notebook should I use for interactive notebooks?
As a die-hard spiral-notebook lover, it even surprised myself when I first realized that composition books are the best choice for interactive notebooks. They had never been my first, second, or even third choice for my own personal note-writing over my life, so I was a bit resistant to trying them for the first time. Here’s what I found:
Starting the year off right is SO important for any class, but especially in Algebra in particular, since everything that is done in the first unit is used throughout the entire year. Students NEED to have a strong foundation, or else they’ll be fighting an uphill battle all year, which is no good. I’ve spent a lot of time thinking about what topics are most important for students to know (from vocabulary to skills), so that each following unit has a strong foundation.
Here are all of the notes I used with my students during the 1st unit of Algebra 1.
Here are the notes I used this year for the 2nd unit of Algebra 1:
We started off the unit with a classifying variables sort. This was a good way to jog students’ memories about their prior knowledge, and it also served as a jumping point into domain and range!
From there, we went into what a relation, domain, and range is, and how it relates to independent and dependent variables.
We then made the distinction that there are two types of relations, discrete and continuous, and we must pay attention to context to determine what type of relation we have.
From there, we started to talk about all of the different ways we could represent a discrete relation, and how we find the domain and range from each representation. We used this foldable, which went over great with the students. They caught on super quickly, and they mentioned that they liked having one example to do together, and one to do on their own for each representation.
We started off with a word problem to review domain and range in a (discrete) relation.
From there, we filled out a Frayer vocabulary model for functions, to make sure that students really understood what they are and aren’t.
Then, using the definition for function we just wrote down on the Frayer model, we made a cheat sheet to refer back to that tells us all of the different ways a relation (discrete or continuous) would NOT be a function.
We practiced classifying functions using a card sort from Amazing Mathematics. Instead of cutting and pasting, we decided to color-code instead! Love it! (In the words of one of my students, this is the page that has “fourteen thousand graphs.”)
We then filled out another cheat sheet, this time for domain and range of continuous functions. Students reasoned together through the inequalities and we talked about what a bound actually means (we used a lot of basketball references).
We practiced finding the domain and range for continuous relations (as well as determining whether or not they were a function), using the following set of notes. PS: It took me a LONG time to figure out how to make a parabola or a trigonometric wave using Microsoft’s shape tools. I feel overly proud of this set of notes! You can download them here!
We began with a recap warm-up on domain and range for continuous relations.
To make sure that students didn’t forget about discrete relations, we went back and did more practice with determining their domain and range, and also stating whether or not the relations were functions.
We started off with a reference sheet on function notation and how to read/say it.
From there, we did a lot of practice with function notation.
Inside this set of notes, we really emphasized interpreting what we were being given in a problem (input or output value) and what the problem was actually asking us to find (input or output value), before starting the problem. This helped students from making a lot of careless mistakes. After we practiced function notation in both directions (evaluating a function, and solving for an input given the function’s output), we mixed up the problems and even threw a few variables and function compositions in there!
Recap warm-up on function notation. Problems 5 and 6 both spurred amazing conversations about order of operations.
After doing this recap warm-up, we did my function notation mystery sum activity, which was a blast. It encourages students to collaborate together and it’s really high engagement each time.
From there, we continued talking about function notation, but now in terms of a graph. Interpreting what the function notation was telling us was such a huge part of the previous day’s lesson, that I wanted to see how they could do when we attached a context to the problem.
Inside, we worked on graphing functions, and using the graph to find an x-value. Some students preferred solving for x, but others were impressed by my tracing over on the graph method. To each their own–that’s the beauty of math, in my opinion.
Recap warm-up over function notation with graphs, and then we reviewed for the test.
Here’s what went into our INBs for the 1st unit of Algebra 1:
Day 1: We glued in a reference sheet for the real number system. Our textbook uses I for the set of irrational numbers. I went with the same notation this year, but I think I’m going to go with R-Q for next year, since I is used for imaginary numbers, later on.
To practice working with these definitions, we did a real number system sort, which I found from Amazing Mathematics! My students enjoyed doing it, and it spawned many great conversations about the difference (however subtle they may be), between the sets of real numbers.
For homework, students did this Always/Sometimes/Never sort, which is also from Amazing Mathematics. They were given about 20 minutes in class to begin their assignment, and then had whatever was left as their take-home assignment for the night. This one was even better than the last card sort, in terms of spurring student conversations. Students were justifying with counterexamples and providing fully flushed out reasons for where each card should get placed. It was awesome!
As a note, we also keep a binder for the class which holds extra handouts, like additional reference sheets and homework assignments that don’t go in the INB. My favorite reference sheet that didn’t go into the INB was this real numbers flowchart that I made. The day of teaching my lesson on real numbers, I noticed that using the “Venn diagram” approach wasn’t meshing well with some of my students. That afternoon, I went home and made a flowchart handout that they could refer to, in addition to their INB pages. Next year, I think I’ll just use this flowcharts in a mini-book format for notes, instead! I found that students started making more connections about the sets each number belongs to (i.e. not only is a number natural, but it’s a whole number, and an integer, and a rational number), and students were able to remember the questions they need to ask themselves when determining the best classification for a real number.
Day 2: We started off with a recap warm-up on the real number system, which we covered the day before.
From there, we did a translating expressions sort, also from Amazing Mathematics. (Can you tell I love her sorts?!).
From there, we used our key words and started defining what a variable is, and what an expression is.
For homework, students did the following problems. They had about 15 minutes of class time to get started. We color-coded “turn-around words” in pink, “parentheses-words” in green, and “equals words” in blue. Students marked the page in highlighter before beginning to translate the expressions. They mentioned that this made the process much easier for them!
We began with a recap warm-up over translating expressions.
From there, we talked about evaluating expressions and also reviewed the order of operations.
From there, we discussed the properties of real numbers and students made up their own examples for each property.
For in-class practice, students did the a properties of real numbers puzzle from Lisa Davenport. A student volunteered to glue it into my notebook. Notice the lack of glue? Notice the crooked edges? It was a very sweet offer, but I’m I don’t think it’s one I’ll be taking again any time soon.
We started with a recap warm-up over evaluating expressions and identifying properties of real numbers.
Recap warm-up over distributing and combining like terms.
What is a solution? What does it mean to be a solution? What does it look like?
Up next, we focused on solving and verifying solutions to 1-step and 2-step equations. I’ve found that verifying a solution is a skill that students struggle with more than solving (at least in Algebra 1), so I wanted to make sure it got emphasized.
We filled out a foldable for solving 2-step equations. Those pesky fractions are going to be our friends by the end of today!
Making things for interactive notebooks can be tedious, at times. If you’re like me, you use a composition notebook so students will (hopefully) resist the urge to tear out pages for scratch paper. The issue with composition notebooks, however, is their sizing. A full sheet of paper is much too large to fit, but a half sheet makes the page feel a bit empty.
Also, unless you want to make everything from scratch to perfectly fit in your interactive notebook, you’re a bit stuck on what to do to get full-sized materials you may have used in the past to fit.
My hack: print any normal sized paper at 80-85% the size and, after cutting out the paper, it will fit PERFECTLY into a composition interactive notebook. Use this hack to make the world your oyster.
Here’s how to do it:
1. Make sure your document has been saved as a PDF.
2. When you go to print, select the following setting:
Rule of Thumb: If the margins on the original paper are 1″, print at 85%.
If the original margins on the paper are .5″, print at 80%.
If the original margins on the paper are at .25″, print at 75% (not common).
Here’s the difference it makes:
This has saved me a TON of time making interactive notebook pages, and also allows the writing space to be much larger for students. Sometimes a half-sheet can be cramped. Hopefully this teaching hack can help save you a ton of time, like it does for me!
Recently, I reached out to the MTBoS looking for fun ideas for practicing solving literal equations. I had searched pretty thoroughly to find any pre-existing activities on the internet, but there wasn’t a lot available. On top of that, what was there, required way more pre-existing skills (SO MUCH FACTORING!) than my Algebra 1 students currently had a month and a half into the school year. Unfortunately, the MTBoS and I were pretty stuck.
Farther down in this Twitter conversation, however, it was mentioned that someone recently used BetterLesson’s lesson for teaching literal equations. At that point I had already taught the lesson and most of my students caught onto solving them quite quickly, but I still was looking for a fun way to get a bit more practice in. While exploring what BetterLesson had, I found this worksheet that gave me inspiration for a game I could play with my students. After a little bit of brain-storming, I created what I’m calling a Connect 4 Activity. Essentially, it’s BINGO, but 4×4 instead of 5×5.
How to play:
Before game: print enough game cards so each student has one, and cut apart the 16 problems. I fold the problems in half (the problem number to the inside) and put them into a plastic bin. (When printing from your computer, make sure it says “print double sided, flip on long-edge.”)
To start off the game, each student gets a game board, on which they randomly place the numbers 1-16. Students then pull out a piece of scratch paper, where they will be doing their work.
The teacher brings the plastic bin containing the 16 equations around the classroom, letting a student volunteer pick a problem at random. (They LOVE getting to pick!)
The teacher then places the problem under the document camera (or writes it on the chalk/white-board if you’re at a low-tech school) for students to solve.
After all students have solved the problem, discuss the solution as a class.
Once all students are silent, the problem number is revealed for students to cross off on their game card. (The excitement levels usually explode at this point, hence the moments of silence in between.)
Repeat for as much class time as you have available, or until all 16 problems have been solved.
Each time a student gets 4 in a row, they bring up their card and their work for inspection (they showed their work and corrected any mistakes for each problem), and are allowed to choose a small piece of candy (Jolly Rancher, a Starburst, etc.).
Reasons why I LOVE this game:
It is super easy to set up and is so adaptable for other topics. This has probably been the lowest prep activity I have made for my students, yet it has been one of the most successful.
Students felt much more confident about their skills and were able to get nearly-instant feedback about how they’re doing.
Students LOVED it. The class begged me to continue letting them play the game through passing time.
Download the game here:More Literal Equations Activities: (Updated September 2017) This year I wanted to find more ways to practice literal equations with my Algebra 1 students. We teach literal equations the week before Halloween, so I wanted to make something really fun and “Halloween-y.” I made a Carving Pumpkins activity that’s self-checking and SUPER fun! I couldn’t wait to try it out, so I gave it to my Algebra 2 students mid-September (patience never was my virtue) since they review literal equations in their first unit. Students though it was fun, and they also found it really comforting that it’s self-checking. To quote a group of boys, “this is super dope, we should do this for all of the holidays!”
Students are given 12 literal equations to solve for a specific variable. Depending on what their answer was, they “carve” color the corresponding pumpkin in a particular way. In the end, each of the pictures should end up looking the same, as far as the color and carvings go.
I’ll be making more activities, and will update the post!