I have vivid memories in school of not remembering the difference between domain and range...that moment of fear on an exam--"Which is x and which is y?!?" For some reason there was a mental block.
As a teacher of middle school for 9 years, I never had much of an occasion to talk about domain and range. To me, it felt a bit over the heads, even to honors algebra students.
However, as more algebra has migrated to the lower grades, and student understanding in Algebra 1 has subsequently increased, I've realized that students are VERY capable of understanding the concept of a domain or range. (Even if they do continue to struggle with identifying which is which!)
Grade Level: 8-10
Course: Algebra, Algebra II
Standards: F-IF.1, F-IF.2, F-IF.3, F-IF.5
SMP: MP2, MP3, MP4, MP6, MP7, MP8
Skills: Algebra, Functions, Domain, Range, Graphs, Graphing, Relations
How to use this as a mad minute:
You have 60 seconds. Explain this math pun using accurate mathematical vocabulary and concepts.
How to use this as a warm up:
You could ask the students to consider one of the following:
1. Is this response appropriate and mathematically correct?
2. Give another example of a function (in function notation or a graph) that would have the same answer.
3. Give an example of a function (in function notation or a graph) that would NOT fulfill this answer.
4. Help, in words, pictures, notes, etc. a student understand the difference between a domain and range.
How to use this as a mini-lesson:
As an introduction to Domain and Range (Henceforth known as D&R in this post)
If I have 20 minutes, and the desire to introduce Domain and Range to students, I'm going to begin by making sure they understand what a function is, and some common variations might be. Remember, these students are just in Algebra I or beginning Algebra II. They probably know linear equations, but may not be familiar with Exponential, Quadratic, Absolute Value, Step, or many other types of functions. To make this accessible, kids need scaffolding!
I'd start by giving them only the first quadrant of a graph, and asking them to graph something like f(x)=4x-9. Hopefully this would kick start a conversation about accurate graphs, and appropriate graphing techniques, scales, etc. Of COURSE kids can graph this line when only given the first quadrant, if they think about it carefully. But, would it be more accurate to use all 4 quadrants? Would it be easier? Does it matter?
WHY do kids crave all 4 quadrants in this situation?
Because they know how to graph using the intercept and the slope! If the intercept isn't available, they really have to think about the graph in more detail. (Thinking is good, but not always easy, right?) My remedial algebra kids struggle with turning an intercept and a slope derived from a point into a full line. They like to connect two dots and stop, forgetting that this is only a line segment, and not a line that represents every solution to a linear equation. This is important because students who truly understand domain and range will understand that linear equations never end!
This is where I'd steer the conversation. Students intuitively want 4 quadrants for ease, but also because they want a "full" picture of the line, not a segment confined only to the first quadrant. If they can identify that the line will continue forever in both the x and y directions, they are ready for the D&R vocabulary. Don't make it difficult! Just point out that since the line goes on forever in the x direction, the Domain is All Real Numbers. Since it goes on forever in the y direction, the Range is All Real Numbers.
For novice students, this is a great introduction. Visually they can see it easily! Here's where I'd shift to using the visual approach for equations that they may not be familiar with, but can VISUALLY see the domain and range!
With only 20 minutes, the last 10 minutes will be a mad (and productive rush) to visually explore functions of all kinds. I did a quick Google Search for Domain and Range of graphs and grabbed several interesting graphs. Feel free to throw these into a ppt, or a worksheet, for kids to explore as a class, with a partner, in small groups, or even as a homework assignment. (I made a worksheet!)
Don't forget to share the image at the end and ask kids, now that they know what domains are, if the "pun" is accurate!
As a review of Domain and Range (D&R)
20 minutes. Ready? Let's go!
0:00-Check in--Raise your hands if you remember what domain and range are. Keep them raised if you know which one is which. Keep them raised if you feel you can define or explain them to a friend.
1:00-Those with your hands raised, spread out. Everyone else, FIND someone with their hand raised. You have 1 minute to summarize in words, pictures, etc. GO!
3:00-Ok, now MIX! Make a new group of 4 or 5 with no more than 2 people from your previous group. Discuss. Did you hear the same things? Do you "get it"? Do you agree? You have 2 minutes to discuss and compare.
5:00--Head back to your seats. Grab a dry erase board and explain D&R with words, tables, definitions, graphs, etc. You have 1 min to get supplies and 1 min to write. Go!
7:00--Hold them up! (Here you will want to look for any that stand out. I'll hope you have a few of these to discuss and point out.) Here's one with great definitions! (Hold it up, put it on your doc cam, put it on your chalk tray, etc.) This explains.... (Review definitions.)
8:00--Here's one with great pictures! (Repeat as above.)
9:00--Here's one with equations/function notation! (Repeat)
10:00--Here's one with tables of values! (Repeat)
11:00--Ok, let me tell you a story, and you determine the domain and range. Imagine your parents help you open your first debit card account. The account will open on January 1st with $200. You can deposit money from the holidays, or from your job. You can withdraw money to spend at the movies or on clothes, etc. However, this is a very "safe" card and you can NEVER overdraft. If you try to spend more money than you have available, the purchase will be declined. Talk with a partner and then write what you believe the domain and range are on your boards.
13:00--Let's talk about it. What did you say? Why?
15:00--Great, can you think of a real life situation that would have a Domain of All Real Numbers, but a Range of y≤10? (Answers will definitely vary. It could be human height over time. Kids may debate that some day our heights COULD be over 10ft. It's just a thinking task!) Does anyone want to share ideas?
18:00--Ok, last but not least, check out this math pun I found. Remember, I'm proud to be a numbers nerd! (Show the image at the top.) What do you think?
By the way, I'd probably give them the graphs above as a homework sheet for a review assignment, or use them as a warm up the next class period.
How to use this as a full lesson?
Once again, this is tough, because it depends on how much experience your students have with D&R. Let's assume they have none. Otherwise you probably wouldn't need a full lesson on D&R and the mini lesson above as a review would be enough.
If I were to introduce D&R for the first time, I'd begin with the mini lesson above.
However, after students explored the 15 images and discussed the domain and range, I would bring them back together to both review their thinking and to offer support in HOW TO USE NOTATION CORRECTLY!
I'd review each image one at a time. I'd start by asking kids what they thought the domain and range were. I'd talk about the proper notation to show that domain and range. (Have they used inequalities recently?) I'd also ask if they've seen a graph like that before. If so, do they know what type of function it is? I would definitely throw out the appropriate vocabulary, even if the kids aren't expected to know it. It's a great prep for the future. (This is an absolute value function. This is an exponential decay function.) If the kids ask more questions, you can decide if you have time to explore, or if you want to add it to your "Let's talk about this later" list!
I would NOT plod through all 15. I'd go through the process with about 5 or 6 and then set the kids free to revise. Then I'd ask them to partner up, compare, discuss and revise. Then I'd ask them to partner with another pair, and repeat. Then I'd ask them to partner with another group of 4 and repeat. If the group WANTS to know what kinds of functions they are, they can come and ask you.
In my room, this would be a very very full 45 minute lesson, or a leisurely 60 minute lesson. If you are on the block schedule, I'd continue by exploring domain and range given actual functions. Can students identify the D&R of linear functions? Are they ready for quadratics?
If you have technology, I'd use it! Have them graph the functions and then determine the domain and range. Definitely make lists and observations so they can start to develop intuition without having to graph the functions!
How to use this as an assessment?
To me, this one is pretty obvious. It's perfectly established to slap on a quiz or test and ask the students to reflect. Of course, this will only work if you've established expectations for reflections!
You can ask a pointed question, "Is this accurate?"
You can ask an extension question, "Can you draw 2 graphs and write two functions that would also fulfill the answer All Real Numbers?"
You can spark creativity, "Add a third pane to this image. In the third pane, have the first character DISPROVE the second character by showing an equation or graph that would NOT have an answer of All Real Numbers."
I hope these ideas help you assess your students' understanding of D&R.
Please feel free to use any of these ideas and modify them to meet your needs. However, please acknowledge the original source of the items and my own lesson outlines. ©NatalieRSprigg 2013
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