Being asked to identify the domain and range of a function can often make even older and more experienced Algebra students fumble, but we’re in luck! We were so intentional in choosing the content that was included in our first three units of the year, that this unit will come together with a little smattering of everything that came before.
The functions, relations, domain & range unit will feel like the natural next step continuing off what your students have already learned, allowing for a fumble-free, stress-free unit.
Click here to jump to your list of tips on what to include in your unit of notes (and why), or keep on scrolling below.
Inequality or Interval Notation?
By the time students reach Pre-Calc, there is a clear winner for how we expect students to write out domain and range…and that is interval notation.
In Algebra 1, depending on your local standards, this decision may be made for you, but you may also have a choice.
There are pros and cons for each choice, so let’s review them.
Interval Notation:
- Pros: This is what your students will likely be asked to use in any advanced future math course, so there certainly is a “might as well learn it now” school of thought present in choosing to teach interval notation in Algebra 1.
- Cons: Domain and range are already difficult enough for many students to grasp, and now you are asking them to also learn an entirely new notation to go along with it.
Inequality Notation (or Set Builder Notation):
- Pros: This builds beautifully off of students’ pre-existing knowledge of inequalities and compound inequalities. You can build off of that to form a deep, conceptual understanding of domain and range.
- Cons: In more advanced math classes, they will not be using inequality notation and will instead use interval notation.
My experience may be different from where you teach, but we didn’t even ask students to use interval notation in our “regular” Algebra 2 class – we only introduced it with the Advanced Algebra 2 classes and our Pre-Calculus classes.
At that point, those students had such a solid foundation of understanding domain and range, that the switch to interval notation was extremely intuitive and felt like an easy and logical next step.
My 2-Cents:
If it’s up to you to decide, I encourage you to teach inequality notation in Algebra 1. It connects so wonderfully with their recent unit on inequalities, which allows students to really understand what it is that the domain and range are representing when they write it down.
Moving to interval notation is nice and concise, but students lose the connection to the true inequality meaning that it represents if it is introduced too soon. It’s always an easy add-on in a future math course, so there’s no need to rush into interval notation when students are developing their understanding of domain and range for the very first time.
Tips for Planning Your Functions, Relations, Domain & Range Unit
If you’re new to teaching Algebra 1, figuring out what is essential to include in your unit notes can be overwhelming. In the section below, I’ll walk through every topic that you should include in this unit, and I’ll also provide direction about what to emphasize while covering that topic to ensure your students are being exposed to common mistakes & misconceptions and are able to develop a deep sense of conceptual understanding.
Use this list as a prescriptive guide for your planning.
At the bottom, I have included a YouTube video that shows every page of notes I give my students during this unit to give you a better sense of how I apply all of these planning tips to create a cohesive unit of guided notes for my students.
What to Include (and WHY)
In my Algebra 1 class, solving multi-step equations is our second unit of the year. Here are the topics I include, and what I recommend emphasizing while teaching this unit and creating your guided notes.
Unit 4 – Functions, Relations, Domain & Range
Topic 4.1 – Intro to Domain & Range
Topic 4.1a – Discrete vs. Continuous Relations
We approach domain & range differently depending on whether a function/relation is discrete or continuous. The very first thing you should introduce in this unit is the difference between continuous and discrete relations so students will know how to tackle any domain & range problem accordingly.
Once that is very quickly established, move on to developing the concept of domain & range.
Topic 4.1b – Relations, Domain, and Range
In this set of notes, go over the vocabulary of relations, domain, and range. Then, dive into some examples (both discrete and continuous) where students must determine the independent and dependent variables, make a graph of the scenario (and decide whether or not it’s a discrete or continuous situation to know if they should connect the points or not), and then have them determine the domain and range being given from the examples.
This is such an amazing way to connect back to the definitions and help students engage with the relationships between domain and range.
Topic 4.2 – Representations of Discrete Relations + Domain & Range
As mentioned above, students need to deal with domain and range differently if it is discrete or continuous. Dive into all of the different representations of discrete functions & relations, but don’t worry yet about determining if they are a function or not (don’t even introduce the vocabulary word of function, yet, actually!).
Discrete representations to include: ordered pairs, tables, mappings (also known as “puddle diagrams”), and graphs.
Topic 4.3 – Identifying Functions (Determining if a Relation is a Function or Not)
Introduce the definition of a function, and then have students take a look separately at discrete and continuous relations. Allow them to determine if the examples given are functions or not. Also, make sure to include the Vertical Line Test (or VLT, for short) and allow students to explain how that connects back to the definition of function.
Topic 4.4 – Domain & Range for Discrete Relations
Now that students have been exposed to how to find the domain and range of discrete relations as well as how to determine if a relation is a function or not, it’s time to give them more practice by combining the two.
Make sure to have students justify why or why not a function is a relation for every example. The more they have to write about it, the better! Their explanations should also go back to the definition of what it means to be a function.
Common justifications for why a relation IS a function:
- “Each x goes to exactly one y”
- “It passes the VLT”
Common justifications for why a relation ISN’T a function:
- “An x goes to more than one y”
- “It fails the VLT”
Topic 4.5 – Domain & Range for Continuous Relations
Finally, give students a chance to work on finding the domain and range of continuous relations. Remind them of their knowledge of writing inequalities from their last unit, and keep having them justify whether or not the graph they are being given is a function.
Providing your students with a reference flowchart can be very helpful for those students who need that extra support.
Topic 4.6 – Interpreting & Sketching Graphs of Real-World Scenarios
This is an optional add-on to your unit, but it’s worth including if you have time because it helps to deepen your students’ understanding of functions and the relationship between independent and dependent variables.
Give students a graph and have them come up with a short story to describe what is happening in the graph. Also make sure to go the other direction where you give students a very short story, and then ask them to create the graph that was described init.
Topic 4.7 – Function Notation
This is such an essential topic to Algebra 1, that it is worth spending multiple days on this topic to really ensure your students build a strong understanding of and competency in using function notation.
Topic 4.7a – Intro to Function Notation
Start by reviewing the definition of a function, and then introduce function notation.
Be sure to point out that f(x) does NOT mean f times x. This is a very common (and understandable) misconception for novice algebra students.
Also, go over the difference between being asked to find f(5) and being asked to find x if f(x)=5.
Topic 4.7b – Function Notation
Dive into the meat and potatoes of function notation. Give students tons of examples that will force them to encounter common mistakes and stuck-points when using function notation.
Test how well your students understand what function notation means, by having them get all of their answers through reading from a table as well as by having students translate function notation into coordinate points.
Include a few examples where students must work backward to solve for x, given what f(x) equals.
Wrap up by doing a lot of mixed practice and even apply their learning by having them find something like 2*f(5) or by finding f(7)-2.
Topic 4.7c – Function Notation with Graphs
There are so many layers to deeply understanding function notation, that I’ve found it best to give function notation & graphs their own day.
Start by having students evaluate a function in function notation by using a graph.
To level up, have students do that again with a graph that has a scenario written about it. Have students interpret what each of their answers means by writing a sentence connecting it to the scenario. For example, d(2)=5 could mean “after 2 hours he was 5 miles from home.”
Finally, allow students to work with an equation in function form, like f(x)=3x-4, and create a table of values and then plot them to make the graph.
Topic 4.8 – Analyzing Functions
This is another optional add-on topic that is perfect if you have a more advanced group. You can go over increasing and decreasing intervals, end behavior, maximum(s) and minimum(s), and so much more to help students think more deeply about functions!
See Every Page of Notes
If you want to take a look at every page of notes I include in Unit 4 of my Algebra 1 interactive notebook, check out this video. You can speed it up or pause at any point to view a specific page.
Want it D-O-N-E for you?
Let me lighten your planning load!
If you’d like to grab the daily guided notes and recap warm-ups I have created for this unit, you can find the interactive notebook version here and the binder note version here.
All of the amazing planning tips from above have been incorporated to make an incredibly cohesive and thorough unit that will help your students reach grade-level standards.
This unit includes all of the notes you would need to teach your daily lessons, along with color-coded answer keys for every set of notes. There are also daily recap warm-ups included along with a pacing guide so you know just what to cover each day!
You can grab my entire year of interactive notes & warm-ups for Algebra 1 here. You can also get the exact same set of notes in a print-and-go binder format here.
Math by the Mountain 